Three Empirical Essays on Asymmetries in the Monetary Policy Transmission Mechanism

Cregoire T k a a

Department of Economics McGill University Montreal, Canada.

A Thesis submitted to the

Faculty of Graduate Studies and Research

in partial fulfilment of the requirements of the degree of Doctor of Philosophy

O Gregoire Tkacz 1999

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Abstract

The objective of this dissertation is to ascertain empkically whether there are non-

linearities between interest rates and selected macroeconomic variables in the monetary

po lic y transmission mechanism. This is accomplished b y using recentl y-developed

econometric techniques to uncover, and model, the non-linearities. We consider

relationships between five variables along the path of the transmission mechanism

(commercial interest rates, consumption, investment, output growth and inflation

changes) and interest rates over which the central monetary authority has significant

control (such as a long-short yield spread or a short-term money market rate). Our

findings reveal that there is evidence in favor of non-linearities, with expansionary policy

having a srnaller impact on the key variables than a contractionary policy in the United

States. In some instances we are successfully able to capture these non-linearities using

either threshold or neural network models.

Sommaire

Le but de cette thèse est d'analyser empiriquement si il y aurait des relations non

linéaires entre les taux d'intérêts et quelques variables clés dans la transmission de la

politique monétaire. Nous accomplissons cette tâche à l'aide d'outils économétriques

récents pour découvrir, et de capter, ces non linéarités. Nous considérons les relations

entre des variables tels que les taux d'intérêts commerciaux, la consommation,

l'investissement, la croissance du produit intérieur brut et les changements au taux

d'inflation, et les taux d'intérêts que les banques centrales peuvent controler avec certain

succès (tels que les écarts entrent les taux a long et a court-terme, ainsi que du niveau

d'un taux à court-terme). Nos résultats démontrent qu'il y a de fortes preuves que

certaines relations soient non linéaires. Une politique monétaire expansioniste semble

avoir un effet mitigé sur le taux d'inflation comparé à une politique restrainte aux États-

Unis. Dans certains cas nous pouvons capter ces non linéarités en incluant un effet de

seuil dans nos modèles, ainsi qu'avec des modèles de réseaux neurales.

Table of Contents

. . Sommaire 11

. -. Table of Contents 111

List of Tables vi

-.. List of Figures vu1

List of Symbols ix

2. Motivation 1

3. The Monetary Policy Transmission Mechanism 2

4. Structure of Dissertation 3

ESSAY 1: ESTIMATION AND INFERENCE FOR ASY~I&IETRIC EFFECTS BETIVEEN INFLATION CHANGES AND INTEREST RATE DIFFERENTIALS 6

1. Introduction 6

2. Literature Review 8

3 - The Linear Mode1 3.1 Theory 3.2 Data 3 -3 Estimates

4. Non-linear Analysis 4.1 Theory 4.2 Graphical Analysis: Kemel Densities 4.3 Residual-based Non-linearity Tests

4.3.1 RESET and Keenan Tests 4.3.2 Tsay Test

4.4 Specific Non-linearity Tests

4.4.1 Threshold Alternative 4.4.2 Non-parametric Alternative 4.4.3 Neural Network Alternative

4.5 On the Power of Non-linearity Tests

5 . Modeling Non-linearities 5.1 Non-parametric Models 5.2 Threshold Modeis 5.3 Neural Network Models 5.4 Cornparing the Models

6. Conclusion

Appendix 1 : Data

Appendix 2: Tables and Figures

ESSAY 2: TESTING FOR ASYMMETRY IN THE LINK BETWEEN THE YIELD SPREAD AND OUTPUT IN THE G-7 COUNTRIES

1. Introduction

2. Literature Review

3. Data and Linear Models

4. Non-linear Analysis 4.1 Kemel Densities 4.2 General Non-linearity Tests 4.3 Threshold Tests 4.3 Neural Network Models 4.5 Non-linearity Tests on Non-linear Models

5. Conclusion

Appendix 1 : Data

Appendix 2: Tables and Figures

ESSAY 3: THE FEDERAL FUNDS RATE, COMMERCIAL RATES, AND THRESHOLD E F F E ~ S ALONG THE INTEREST RATE CHANNEL

1. Introduction

3. Li terature Review 2.1 Prime Rate Papers 2.2 Other Asymmetries

3 . Interest Rates 3.1 Data 3.2 Granger Causality Tests 3 -3 Cointegration Anaiysis 3.4 Linear Models 3 -5 General Non-linearity Tests 3.6 Threshold Tests

4. Consumption 4.1 Data 4.2 Linear Models 4.3 Threshold Models

5. Investment 5.1 Data 5.2 Linear Models 5.3 Threshold Models

6. Conclusion

Appendix 1: Data

Appendix 2: Tables and Figures

List of Tables

ESSAY 1: ESTIMATION AND INFERENCE FOR ASYMMETRIC EFFECTS BETWEEN INFLATION CHANCES AND INTEREST RATE DIFFERENTIALS

Table 1 : Unit Root Tests Table 2: Long Memory Tests and Fractional Integration Parameters Table 3: Linear Models TabIe 4: Bootstrap MuItimodality Tests Table 5 : General Non-Iinearity Tests Table 6: Non-Linearity Tests with Specific Alternatives Table 7: Threshold Models Table 8: Non-linearity Tests on the Non-linear Models Table 9: Neural Network Models Table 10: Root Mean Squared Errors

ESSAY 2: TESTINC FOR ASYMMETRY IN THE LINK B E ~ V E E N THE YIELD SPREAD AND OUTPUT IN THE G-7 COUNTRIES

Table 1 : Unit Root Tests Table 2: Linear Models Table 3 : Bootstrap Multimodality Tests Table 4: General Non-linearity Tests Table 5: Hansen Threshold Test Table 6: Neural Network Models Table 7: Non-linearity Tests on the Non-linear Models

ESSAY 3: THE FEDE~WL FUNDS TE, COMMERCIAL TES, AND THRESHOLD EFFECTS ALONG THE INTEREST I TE CHANNEL

Table 1 : Unit Root Tests Table 2:Granger Causality Tests Table 3 : Johansen Cointegration Tests Table 4: Interest Rates - Linear Models Table 5: Interest Rates - General Non-linearity Tests Table 6: Interest Rates - Threshold Models Table 7: Interest Rates - Non-linearity Tests on Threshold Models Table 8: Consumption - Linear Models Table 9: Consumption - Johansen Cointegration Tests Table 10: Consumption - Error-Correction Models Table 1 1 : Consumption - General Non-linearity Tests Table 12: Total Consurnption - Threshold Models Table 13 : Durable Consumption - Threshold Models

Table 1 4: Consumption - Non-lineanty Tests on Threshold Models 158 Table 15: Investment - Linear Models 159 Table 1 6: investment - Johansen Cointegration Tests 160 Table 17: Investment - Error-Correction Models 161 Table 18: investment - General Non-linearity Tests 162 Table 19: Fixed Private hvestrnent - Threshold Models 163 Table 20: Gross Private Domestic Investment - Threshold Models 164 Table 2 1 : investment - Non-linearity Tests on Investment Threshold Models 165

List of Figures

ESSAY 1: ESTIMATION AND INFERENCE FOR ASYMMETRIC EFFECTS BETWEEN INFLATION CHANCES AND INTEREST RATE DIFFERENTIALS

Figure 1 A: Kernel Densities for Linear Mode1 Residuals (Policy Horizons) 63 Figure 1 B: Kernel Densities for Linear Model Residuals (Non-Policy Horizons) 64 Figure SA: Sequence of LM Statistics for Threshold Test (Policy Horizons) 65 Figure 2B: Sequence of LM Statistics for Threshold Test (Non-PoIicy Horizons) 66 Figure 3: Fitted Cumes, Policy Horizons (Short Rate = 3 Months) 67 Figure 4: Fitted Curves, Policy Horizons (Short Rate = 6 Months) 68 Figure 5: Fitted Cuves, Policy Horizons (Short Rate = 12 Months) 69 Figure 6: Fitted Curves, Non-Policy Horizons (Short-End of Yield C w e ) 70 Figure 7: Fitted Curves, Non-Policy Horizons (Long-End of Yield Curve) 71

ESSAY 2: TESTINC FOR ASYMMETRY IN THE LINK BETWEEN THE YIELD SPREAD AND OUTPUT IN THE G-7 COUNTRIES

Figure 1 : Kernel Densities for Linear Model Residuals Figure 2: Sequence of LM Statistics for Threshold Test

ESSAY 3: THE FEDERAL FUNDS RATE, COMMERCIAL RATES, AND THRESHOLD EFFECTS ALONG THE INTEREST RATE CHANNEL

Figure 1 : Interest Rates, 197 1 - 1998 166 Figure 2: Interest Rates - Enor-Correction Terms 167 Figure 3: Interest Rates - Sequence of LM Statistics for Threshold Test 168 Figure 4: Consumption - Error-Correction Terms 169 Figure 5: Consumption Levels - Sequence of LM Statistics for Threshold Test 170 Figure 6: Consumption Growth - Sequence of LM Statistics for Threshold Test,

Interest Rate Threshold 171 Figure 7: Consumption Growth - Sequence of LM Statistics for Threshold Test,

Error-Correction Threshold 172 Figure 8: Investrnent - Error-Correction Terms 173 Figure 9: Investment Levels - Sequence of LM Statistics for Threshold Test 174 Figure 10: investment Growth - Sequence of LM Statistics for Threshold Test.

Interest Rate Threshold 175 Figure 1 1 : Investment Growth - Sequence of LM Statistics for Threshold Test,

Error-Correction Threshold 176

viii

List of Symbols

The following symbols are used throughout this dissertation:

: When placed over a variable is meant to denote a fitted, or estimated, value.

- : Used to identi& a threshold variable.

Rm - Rn : A nominal yield spread between an m-period bond and an n-period bond, where both m and n represent months and m > n.

ii" - if : The difference between an m-period annualized infiation rate and an n-period annuaiized inflation rate, where m > n.

6, 7 : Reserved symbols for mode1 errors.

r : Symbol used to denote the location of a threshold.

y : S ymbol used for the parameter of a threshold variable.

6 : Symbol used for the parameter of an enor-correction term.

Ac knowledgements

Financial assistance for my graduate education fiom the Quebec Fonds pour la Formation de chercheurs et 1 'aide à la recherche ( 1 994-98) and the McGill Institute for the Study of Canada (1 994-97) is gratefully acknowledged.

Several individuals have played pivotal roles in the development of this thesis, and in my graduate education in general. My principal supervisor, John W. Galbraith, deserves much credit for the elaboration of my thesis topic, for reading nurnerous drafts and for painlessly coaching me through my graduate studies. He also CO-authored the chapter entitled Tesringfor Asymmetry in the Link Berween the Yield Spread and Output in the G- 7 Counlries.

Victoria Zinde-Walsh, the second member of my thesis cornmittee, also merits credit for reading through my cira&. She was instrumental in bringing me to McGill in 1994, and played a major role in my graduate econometrics training.

Through the years 1 have also spent several stints at the Bank of Canada - my home away from McGill. The Bank allowed me to use its library and data resources, and provided me with vast cornputing power. More irnportantly, several Bank researchers also helped develop my burgeoning ideas about economics. First among these is the late Barry Cozier, who led me towards interest rate spreads and their impact on the economy. Credit should also be given to Joseph Atta-Mensah and Ben Fung, who have been CO-authors for several Bank projects. Working on different areas of the literature, 1 am hopefül that within a few years al1 our work can be synthesized so as to provide a clearer picture of the empirical links of the interest rate channel of the monetary policy transmission mechanism.

Preliminary versions of the essays that comprise this thesis have been presented at conferences or seminars:

Essay 1 was presented at the annuai meetings of the Canadian Economics Association at Mernoriai University in St. John's, Newfoundland, in June 1997, and at McGill in November 1997. Thanks to Mingwei Yuan and other participants for valuable comments. Essay 2 was presented (by John Galbraith) at the CREFE (UQAM)/Bank of Canada con ference New Approaches in the Estimation of Monetav Policy Effecrs in April 1996, and by myself at the Bank of Canada in August 1996. Thanks to Jean-Pierre Aubry, Benoît Carmiehael, Tiff Macklem, Simon van Norden and other participants for valuable comments. Essay 3 was presented at the annual meetings of the Société canadienne des sciences économiques at École des Hautes Études Commerciales in Montreal in May 1997. Thanks to Jacques Raynauld and other participants for vaiuable comments.

The procedures for the threshold test and the feasible GLS estimator were written by J.W. Galbraith, and that of the kemel density estimator by G. Biirdsen. Al1 other programs and routines were written by the author in RATS 4.24.3 and Gauss 3.

Finally, my family merits much credit for the endless encouragement directed towards yours tnily. Although this dissertation, and indeed much of my work in economics and central banking, remains an enigma to them, they have al1 contributed to the final product in some manner. More precisely: Gregoire Sr. inadvertently directed me towards the field of economics sometime in 1985 when 1 was in Grade 8 (so he S the one to blame); Roman and Danny taught me the vaiue of hard work; Angie always wanted me to attend McGill, so there you go (an excellent decision); Robert's disdain for statistics gave me an added incentive to master the field (fiom this work 1 hope you will find that there îs a use for the 2 distribution); John's cornmon-sense arguments on economic matters aiways pushed my theoretical training to its limits; Grandmutî's gift of $1 on each of her visits was an early lesson in econornics (which, given the prices in effect at the time, was equivalent to two comic books); and the late Grazina Montvydaite instilled in me a love for books and learning that will undoubtedly last a lifetime,

GENERAL INTRODUCTION

Are the relationships between instruments of monetary policy (short-term interest

rate levels and spreads) and some transmission variables in the monetary policy process

(commercial interest rates, consumption, investment, output and inflation) asymmetric?

That is! do these relationships differ in any systematic way in different regimes? If so, can

we modei such relationships? This dissertation will empirically demonstrate that there are

indeed asyrnmetries between such variables, and that such non-linearities can sometimes

be captured using recently-developed modeling techniques.

2. Motivation

Interest rates and interest rate differentiais play a key role in the macroeconomy.

For esample, interest rates affect aggregate consumption because they capture the rate of

tirne preference in consumption decisions of economic agents. The higher is the current

rate of interest. the higher is the opportunity cost of current consumption. Investment is

also affected by interest rates, since investment projects almost always require agents to

borrow. The higher the rate of interest, the higher the cost of borrowing, and the lower the

espected net return on investment projects. Thus, there should be a negative relationship

between investment spending and interest rates.

Consumption and investment are key components of national income. When they

are affected, then so too is GDP. The level of actual GDP, in relation to its theoretical

trend level, dictates the level of economic activity, from which we can determine whether

aggregate demand is weak or strong, If there is a Iink between aggregate economic

activity and pnces through a Phillips Cwe-type relationship, then we can expect that

differences between actud and trend output will affect inflation. Because of these

important macroeconomic relationships, it is important to have a solid understanding of

how interest rates affect the macroeconomy.

This dissertation proposes to explore empirically whether the effects of monetary

policy on the economy are asymmetric, emphasizing the role of interest rates as the

primary instrument of policy. in theory, asymmetries can arise if nominal wages are

sticky downwards. in such instances the labor market does not always clear, implying that

kinked supply curves can emerge in the short-m. which can give greater weight to

negative demand shocks. If these demand shocks onginate fiom a monetary policy action,

then monetary policy can have asyrnmetric effects on macroeconomic variables.

3. The Monetary Policy Transmission Mechanism

We begin from the premise that the primary instrument of monetary policy is the

ability to influence short-term interest rates.' These rates, in turn, influence the exchange

rate. managed interest rates, consurnption, investrnent, output and, ultimately, the

inflation rate. This is a view closely shared by Blinder (1996, 1997, 1998). The diagram

on the next page describes this transmission mechanism in more detail.

' Blinder (1997:7) notes that the controversy between advocates of monetary aggregates and interest rates as the primary instrument of rnonetary policy "is now history. Virtually al1 major central banks nowadays use the overnight interbank rate (in the United States, the Federal funds rate) as their central policy instrument." Blinder (1998:30) adds: "In the contemporary United States, virtually al1 acadernic and market observers agree that the Federal fùnds rate - the overnight rate in the interbank market for reserves - is the Federal Reserve's central policy instrument. And so does the Fed."

Xnterest Rate Channel of the Monetary Policy Transmission Mechanism

Policy Action

i Short-Term Rate

Other Rates (Commercial, Long-Term) & Exchange Rate I +

Consumption & Investment

Current Output

Output Gap I

If there are asymrnetries between interest rates and infiation, then they can potentially

originate from any link within this transmission mechanism.

4. Structure of Dissertation

This dissertation is comprised of three essays, each analyzing a different

relationship bettveen interest rates and transmission mechanism variables. To test whether

non-1 inearities are present, we follow the philosophy of Granger (1 99 1, 1 993) who argues

that one should not rely on a single method or test to detect non-linearities. This will

mavimize our chances of detecting asymrnetries should they exist, as it is difficult for any

test to retain high power against al1 fonns of non-linearities under the alternative

hypothesis. As such, we rely on graphical techniques, general diagnostic non-linearity

tests as well as tests that stipulate a specific f o m of non-linearity under the alternative

hypothesis, most particularly threshold models.

Essay 1 expands the mode1 of Mishkin (1990) that relates changes in inflation to

interest rate spreads, and shows that asymmetries c m arise in theory if expected real

interest rates at different horizons differ. Focusing solely on the United States, we find

that thresholds appear for values of the long-short interest rate spread around -1.5, with

the relationship being three times more pronounced below the spread than over. Non-

pararnetric and n e d network models also reveal changes in the relationship around that

value. This indicates that when monetary policy is dready tight and that a m e r

tightening ensues, the marginal benefits in terms of a reduction in inflation will be greater

than if such a tightening were conducted when monetary policy would be considered

espansionary. These findings should be of use to policy-makers, as it details how changes

in the instrument variable affect changes in the ultimate target variable.

Essay 2 is devoted to testing for thresholds between interest rate spreads and

output growth in the G-7 countnes. Several authors have found that the terrn structure is a

useful predictor of output growth in industrial countries, with Laurent (1988), Rudebusch

( 1 995) and others noting that a monetary policy interpretation can be attached to the yield

spread. This chapter expands on that literature by testing whether the output-spread

relationship differs above and below some unknown threshold. Our findings are that

threshold effects appear to be present only for the United States. More general non-

linearity tests reveal that significant non-linearities exist in the reduced-form mode1 for

France, although such asymmetries do not appear to originate fiom the spread-output link.

The final essay explores some relatively neglected links in the transmission

mechanism, namely how changes in short-tenn market rates affect commercial interest

rates, consumption and investment. The purpose here is to anempt to uncover the sources

of the non-linearities that we observe in the first two essays. We find that commercial

interest rates, such as the prime rate and mortgage rates, respond less quickly to

movements in money market rates when money market rates are low. We also detect

asymmetries between interest rates and consumption and investment, but most of the non-

linearities seem to emerge in the growth rates of these variables rather than the levels.

In the conclusion of this dissertation we summarize our main findings and suggest

avenues for future research. Our general conclusion is that the effects of monetary policy

on the economy are indeed asymmetric, with expansionary policy having a weaker impact

on commercial interest rates, consumption, investment, output and inflation than a

contractionary policy. The main findings of al1 three essays are consistent with this

conclusion. However, there are still many links in the transmission mechanism worth

exploring in order to uncover the precise source of these non-linearities.

ESSAY 1:

ESTIMATION AND INFERENCE FOR ASYMMETRIC EFFECTS BETWEEN INFLATION CHANGES AND INTEREST RATE DIFFERENTLALS

1. Introduction

This essay investigates whether non-linearities exist in the relationship between

inflation changes and interest rate differentials in the United States. Given that price

stability is a long-term objective of the Federal Reserve, and since interest rates represent

a target variable in the impiementation of monetary policy, this line of research will assist

policy-makers in determining the amount of liquidity that should be made available to

money markets under different regimes.

This work extends and complements that of Mishkin (1 990a, 1990b) and others.

These studies have found that the terni structure of interest rates at the middle or upper

end of the yield curve contains usefùl predictive information about inflation changes. This

paper attempts to improve on these findings by accounting for possible non-linearities in

the relationship. which cm &se due to different expected fùture real interest rates. We

begin by presenting a simple theoretical model for the inflation-spread relationship that is

built upon the Fisher equation. Using monthly data begiming as far back as 1953, we

then estimate this (linear) model, which we subject to numerous non-linearity tests. Our

findings reveal that the linear model is inappropriate in modeling the relationship, as it is

incapable of capturing salient rnovernents in inflation changes. We then undertake the

task of constructing an appropriate non-linear model. Threshold models offer an

interesting alternative, since they allow the inflation-spread relationship to differ

according to the magnitude of the spread. htuitively, when credit markets are tight and

bond yields are bid up, changes in the spreadinflation relationship may exist, since

movements in the spread in this case may not reflect changes in overall economic

conditions.

In addition, threshold models lend themselves nicely to the estimation of rnodels

that rnay be subject to changes in regime. The sample period in our andysis covers the

weli-documented changes in the Fed's operating procedures between 1979 and 1982;

Mishkin (1990b) notes that the dope parameter on the interest rate differential changes

during that period, which implies that a regime-switching model, such as a threshold

model, would be more appropriate than a standard Iinear model.

We also consider non-parametric and neural network models, which both allow

for richer non-linearities. These complement the threshold models and re-affirm our

finding that the impact of the yield spread on inflation changes can be even more

sensitive to regime changes. The models reveal that the relationship is quite strong when

the spread is negative, but aimost non-existent when positive, indicating that the

esplanatory power of the spread is due almost exclusively to its ability to esplain

inflation changes in tight policy regimes. The pattern of non-linearities that we find is

similar across al1 three non-linear models.

This chapter is structured as follows. The next section examines some of the

previous work on the infiation'spread relationship. Section 3 develops a formal (linear)

model. emphasizing the underlying theory, and presents the empirical estimates of the

linear model parameters. Section 4 presents the various non-linearity tests that we

p e h r m , and implernents these on our linear model. Building on our non-linear test

results, Section 5 sets about the task of building an adequate non-linear model. The last

section concludes.

2. Literature Review

In general, authors have found interest rate differentials to be significant

predictors of inflation changes, for the United States and other countries. However,

results seem to be dependent upon the area of the yield curve under consideration, as

interest rate differentials on securities at the short end of the yield curve appear to contain

less information than those differentials that utilize long-tenn rates.

MisNUn (1990a) examines differences between yields on government securities

dong the short-end of the yield curve (one, three, six, nine and twelve month Treasury

bills). while Mishkin (1990b) focuses on long-short yield spreads on securities with

maturities of one to five years. In these papers the author derives a forecasting equation

for inflation changes from the original Fisher equation. Previous work has found that

interest rates predict inflation, and that the term structure predicts interest rates, thus it

follows that the tenn structure should predict inflation. He finds that for maturities of six

months or Iess, the term structure contains almost no information about the path of future

inflation, while the longer end of the term structure is a much stronger predictor of future

inflation. Similar results for several OECD countries are presented in Mishkin (1 99 1) and

Jonon and Mishkin (1991).

Some researchers have tried to refine Mishkin's methodology by relaxing the

assumption of a constant slope for the reai term structure. Fran!!el and Lown (1994) use

information dong the entire length of the yield c w e to forecast inflation. Their

frarnework requires only 2 assumptions: (1) that the expectations hypothesis holds; and

(2) that the steady-state inflation rate becornes incorporated into nominal rates over time

(i.e. money is not neutral in the short-ru@. They state that long-term interest rates refiect

espected inflation more fülly than short-term rates and the slope of the yield curve can be

used to extract indicators of real interest rates, state of monetary policy and expected

inflation. By simply using two extreme points on the yield curve, they improve the fit by

over 50% when compared to Mishkin's model. When using their alternative measure,

which uses information dong the entire curve, the results are marginally improved.

Fama (1990) uses the spread between five-year and one-year bond rates to predict

changes in the one-year spot rate, one-year inflation rate and the expected real return on

one-year bonds. The results indicate that the term structure is an adequate predictor of

espected inflation and real retums. However, both variables rnove in opposite directions.

The result is that the spread is a poor forecaster of the one-year spot rate.

The inflation indicator properties of t e m spreads have also been recognized

through the estimation of simple astructural models. For example, Emery and Koenig

( 1 992) find that the fit of traditional indicator models of inflation and inflation changes

generally improve when interest rates are included. Bernanke (1990) runs a "horse race"

to see which interest rates and spreads are best at predicting a number of econornic

variables, including inflation. He fin& the risky spread to be the best based on three

established criteria (R-bar-squared, the Akaike Information Criterion and the Schwarz

Criterion).

Some work on this topic has also k e n performed for countries other than the

United States. Lowe (1992) and Day and Lange (1997) employ the Mishkin methodology

for Australia and Canada respectively, and both find that the term spread is a useful

predictor of inflation fiom two to five years in the future. Robertson (1992) finds that the

term structure may be usefiil for explainhg changes in the U.K. inflation rate for up to

four years. The term structure model produces better forecasts than a simple AR model,

but other alternatives are not examined. This paper differs fiom the Mishkin model in that

the inflation rate is assumed to follow a random walk.

Kozicki (1997) uses a long-short yield spread to forecast inflation up to four years

in the future for ten OECD countries. She finds that the spread is only significznt for

about half the countries in her sample (Australia, Canada, Itaiy, Sweden and United

Kingdom). with ma~imum forecasting performance being achieved at the two to four-

year horizons. The spread is insignificant or only marginally significant for al1 countries

at the one-year horizon.

3. The Linear Mode)

3.1 Theory

In this section we develop the basic theoretical model which links inflation

changes to interest rate differentials. The approach used follows closely that of Mishkin

(1 990a), who derives a term structure-based forecasting equaticn fiom the foundations of

the Fisher equation. This basic linear model will be estimated for various forecasting

horizons, and then its residuals will be subjected to various tests designed to detect

possible non-linearities in the data.

We begin by defining the annualized m-period inflation rate,a:' as

x,"' = log(4+, / P , )x 1200/m, where m can take the values 3 (3 months), 6 (6 months), 12

( 1 year), 36 (3 years), 60 (5 years) or 120 (10 years). We then note that the rn-period real

interest rate, r,"' , must satisfy the condition (1 + r,"') = (1 + Rrm)/(1 t a:), where Rr is the

rn-period nominal interest rate. Solving for the nominal interest rate we get

Since the last term in this expression is negligible for small values of r*"' and K I ,

it is usually dropped for sirnplicity.' Solving for the idation rate we therefore get

' However, when the objective o f a study is to test for possible non-linear relationships behveen variables. such terms may play a signifiant role. We expand on this point in Section 4.

11

Equation (2) states that the exposr m-period inflation rate is equal to the m-period

nominal interest rate less the ex posr m-period real interest rate. Taking an expectations

operator through (2), we derive an expression for the ex ante inflation rate:

The above relationship is the familiar Fisher Equation, proposed by Fisher (1907, Chapter

V). Thus, the ex anse rn-period inflation rate is equal to the m-period nominal interest rate,

less the ex ante nt-period real interest rate. We know also that the expost inflation rate is

equal to the ex ante inflation rate and an inflation forecast error term:

Substituting (3) into (4) we get

rr; = RI" - E,rrrn + &Y

~ v h i c h states that the ex posr m-period inflation rate is equal to the rn-period nominal

interest rate less the ex anse nr-period real interest rate and an inflation forecast error. A

similar equation can be derived for the n-period expost inflation rate, where ncm:

Subtracting (6) fiom (5) we get

~ r p -z: = ( R f -R:)-(E,r ," - E l r l n ) t ( ~ ~ - & y ) .

Let us now decompose the ex anre real rate into the mean of the ex anre real rate

over the sampIe period and deviation fiom that mean:

E,r," = Y" + (E,rln - F n ) = 7' + u:

Inserting (8) and (9) into (7) we get

By assuming that the difference between the means of the es ante real interest

rates does not change, and by combining a11 the error terms, we get the following inflation

c hançe forecasting equation

where a,, equals zero and Pm, equals one, under the assumptions that the means of the

ex ante real interest rates at horizons rn and n are equai and that expectations are rational.

In the former case this implies that the differences in forecast erron (u< - tC) in (10)

must be zero, implying that qrfl = (E: - E : ) in (1 1). Mishkin (1990a) notes that the

implications of rational expectations are that the forecast mor s of inflation at horizons m

and n, E< and&:, must be unforecastable conditional on al1 information at time r , which

includes the nomina1 interest rates Rrmand Rtn . As such, the error term in (1 1) must be

uncorrelated with the regressor on the right-hand side. Hence, inflation changes, and the

slope of the real term structure, should move one-for-one with changes in the slope cf the

nominal term structure.

The joint hypothesis of constancy of real rate differentials and rational expectations

(Le. a,,, = O and p,, = 1 ) can be readily tested. Should it be rejected, and if P , , is still

significantly different from zero, then the nominal tenn structure can still contain

information about changes in intlation. Thus, (1 1 ) is the basic (linear) relationship that we

choose to estimate. The residuals fiom (1 1) wilt then be used in forma1 tests to detect

possible departures from linearity. Insignificance of the spread in esplaining inflation

changes at some forecast horizons for the linear models will not deter us fiom estimating

non-linear modeis in those cases, as the insignificance may simply be due to the

inadequacy of the linear relationship.

3.2 Data

Ali the data for this study were obtained fiom the Federal Reserve Bank of St.

Louis FEED database, with samples consisting of monthly observations beginning as far

back as 1953 and extending through 1996. lnterest rates consist of 3 and 6-month

Treasury bills, as well as 1, 3, 5 and 10-year government bond rates. The dependent

variable consists of changes in CPI inflation. More precise variable definitions and

sarnpies are provided in Appendk 1.

It is customary when analyzing time series data to begin by performing tests for

the stationarity of the variables used. In the present case this practice is imperative. as

some of the non-linearity tests used in Section 4 are designed for use with stationary

regressors. To this end, we perform three different unit root tests the inflation changes

and yieid spreads: the Augmented Dickey-Fuller (ADF), Phillips-Perron (PP) and

Kwiatkowski et al. (1992) (KPSS) tests. The first two of these specifL a unit root process

under the nul1 and a stationary process under the alternative, while the KPSS test reverses

these hypotheses.

The KPSS test is straightforward to implement. We first run a regression of the

variable of interest on an intercept and a time trend., ailowing us to obtain the residuaIs

2, for 1 = 1 , . . . , T. The partial sum process of the residuals is

I

S, = C e , , fort =1,2 ,..., T

and the test statistic is constructed as

The likelihood of rejecting the nul1 of stationarity increases as the sum of squared

residuals increases, indicating significant deviations fiom the behavior of a trend-

stationary process.

The results of the unit root tests are given in Table 1, with the lags in the ADF test

chosen by minimizing the Schwarz Criterion. Both a constant and a trend are included in

the regressions. The lags used in the Phillips-Perron tests reflect the fkequency of the data.

Twelve lags are used to estimate the variance of the error term. &,' , in (13). as KPSS

show rhat this test has good power with a large number of lags for sarnples comparable to

ours.

Beginning with the interest rate spreads, we can reject the nul1 hypothesis of a unit

root at the 5% level for al1 spreads, with the exception of the three-year less six-month

spread. for which we can only reject the null at the 10% level. According ro the Phillips-

Perron test, the unit root hypothesis is rejected for al1 the spreads. The null of stationarity

is rejected for three spreads at extreme ends of the yield curve, narnely one year less three

months. one year less six months, and ten years less five years.

For the inflation changes, we can reject the unit root hypothesis for ten of the

fifieen changes exarnined using the ADF test, with roughly similar conclusions being

reached by the KPSS test. Intuitively, if the price level is 1(1), then inflation is I(O),

implying changes in inflation m u t also be I(0). Some researchers (e.g. Johansen (1995))

would instead argue that the price level is I(2), implying that inflation must be I(1). If this

is the case, then differences in inflation rates at different horizons can either be [(O) or

I ( l ) , which is what we appear to be witnessing. Meanwhile, the Phillips-Perron test

reveals that only two of the inflation change variables are non-stationary.

Since our unit root tests reveal that some of our variables contain unit root, it

would be instructive to perform some tests for long memory to determine whether the

variables may in fact be stationary over the long-run. This feature cm make it difficult for

unit root tests to distinguish a long-memory stationary process from a non-stationary one

(see e.g. Diebold and Rudebusch (1991)). The first test we consider is the Modified

Rescaled Range (or ) test of Lo(199 1). Defining the range as

and the variance as

where c, is the jth-order sample auiocovariance of y, and w,(q) are the Bartlett window

weights of

The test statistic is then constnicted as

for which Lo (1991) tabulates the critical values. For a test with the nul1 being short-

memory and the alternative being long-memory, the 95% confidence interval is

[0.809,1.862] and the 99% interval is [0.721,2.098]. In Table 2 we present the results of

the Lo long-memory test. For the interest rate spreads we find that the test statistics al1 lie

within the confidence region, and can therefore be considered short-memory processes.

For the inflation change variables, only the differences between the ten-year and three-

year and ten-year less five-year inflation rates appear to be long-memory processes.

Interestingly, these are the only variables for which the Phillips-Perron unit root test fails

to reject the nul1 of a unit root.

Finally, for completeness it is instructive to obtain estimates of the formal

fractional integration parameters. Geweke and Porter-Hudak (1983) propose a frequency-

domain method fiom which we can estimate the fractional differencing parameter, d,

from an ARïiMA(O,d,O) process. To find this garameter we use TI" spectral ordinates,

CO,, for j = 1, ..., TI", fiom the periodogram of each variable y, that is IJo,), and run the

following regression

The fractional-differencing parameter d is then found to be -6. In Table 2 we

present the fractionaI integration parameters for al1 our variables. In almost al1 cases the

estimated FI parameter is less than one, consistent with short-rnemory processes. The

exceptions are once again the differences betweeen ten-year and three and tive-year

inflation rates, where the FI parameter is estimated to be greater than one.

From this analysis we c m conclude that al1 but two of our variables are short-

memory processes, and that the unit root tests may be affected by the presence of long-

memory in the data, Considering that the unit root nul1 is only rejected for the long-

memory variables, we can proceed with ow analysis that al1 Our variables are trend-

station^.. albeit only over the very long-run for our long-memory variables, x,"' -n16

and n:?' - z,60 .

The estimation results of (1 1) for different values of m and n are given in Table 3.

The ordering of the (m,n) combinations follows the reasoning that we divide them into

policy and non-policy horizons. The Federal Reserve essentially implements monetary

policy through its control of short-term interest rates (see, for exarnple, Blinder (1 996)),

whereas long-term rates can only be infiuenced by the Fed through its effect on inflation

espectations. As such, the difference between long and short-term rates may act as a

prosy for the direction of monetary policy, a proposition fonvarded by, for exarnple,

Laurent (1988, 1990) and is discussed as a possible (but not exclusive) reason why the

yield spread is such a good predictor of economic activity by Bonser-Neal and Morley

( 1997) and Kozicki (1 997).

In our study we are examining six different interest rates, and as such we can

define the three shortest maturities (3, 6 and 12 months) to be "shon" interest rates, and

the three longest maturities (36, 60 and 120 months) to be '-long" rates. Thus, a yield

sprezd constmcted as the difference between a "long" and a "short" rate can be thought of

as providing a reasonable proxy for the direction of monetary policy, since the Fed can

influence the magnitude of the spread through its control of the "shon" rate. Contrast this

~vi th the difference between two "short" rates, such as the 3 and 6-month rates, and two

"long" rates, such as the 60 and 120-month rates. in the former case a monetary action

will have a significant impact on both 3 and 6-month rates, therefore the spread between

these rates will not change much, failing to capture the effect of the policy action. In the

case of the spread constmcted as the difference between two long rates, monetary policy

has little effect on either; therefore once again it is difficult to attach any policy

interpretation to such a spread as it fails to capture the policy action. As such, we have

nine policy horizons since we have nine long-short spreads. and six non-policy horizons

eiven the three short-short and three longlong spreads. C

The residual diagnostics reveal that these models suffer from serial correlation,

non-normality and heteroskedasticty for the most part. This therefore implies that

inference on the spread parameter cannot be conducted using the usual t-statistic. For this

reason the reported "t-statistics" have been corrected for serial correlation. Since the

observations are monthly, there is a moving average term of order (m-1) buiIt into our

data. If estimating (1 1 ) using OLS, we would expect our residuals to follow a MA(m-1)

prosess as a result. For esample. the one-year inflation rate is constructed by comparing

changes in the price level between periods r and c+ 12. 14 and t+l3. and so on. For this

reason we choose to use the well-documented Ne\vey and West (1987) covariance matris

in order to obtain consistent estimates of the standard errors.

The spread parameter P,., provides the change in inflation that will occur given an

increase in the longer-term rate relative to the shorter-term rate. For example, in the case

of t ~ 3 6 and n=j7 we find that fibv3 = 1.091. This implies that if the difference between

the three-year and three-month interest rates increases by 100 basis points (that is, the

yield curve steepens) then the inflation rate over the nest three years will be 1.091%

higher than the inflation rate over the next three months, implying an acceleration in

inflation.

For the nine policy horizons, which are the first nine rows in the Table, we find

that the yield spread is a significant predictor of inflation changes. The corrected t-

statistics range from 6.8 to 15.8, and the estimated slope parameters are in al1 but one

case greater than 1, indicating that there is at least a one-to-one relationship between yield

spreads and inflation changes at these horizons.

In terms of explanatory power, the best fits occur when using infoxmation in the

middle of the yield curve, in particular yield spreads between 5-year and 1-year, 3-year

and 1 -year and 10-year and 1 -year rates, respectiveiy. This corroborates the finding of

Mishkin (1990b), who notes that the middle portion of the yield curve provides the most

esplanatory power. One possible reason for this may be the lags with which monetary

policy is thought to affect inflation, which are generally thought to be between 18 and 24

months (Blinder (1996)). As such, the middle portion of the yield curve may be providing

more explanatory power because it coincides more closely with the time fiame that a

policy action affects inflation.

Rows 10 through 12 in Table 3 provide the regression results for horizons which

use only information along the short-end of the yield curve. As previously mentioned, the

yield spreads for these horizons would not be adequate proxies of the direction of

monetary policy, since policy actions tend to affect short rates proportionally. The result

is that for al1 three cases the spread parameter is insignificant, and the fit of the models is

very poor. These findings are similar to those of Mishkin (1 990a).

Finally, in rows 13 through 15 we use information dong the longer-end of the

yield curve to explain changes in inflation. Again, since the Fed's policy variable is the

Fed Funds rate, the impact of changes in this short rate on longer rates is much less

pronounced, with the result that longer rates can only be viewed as intermediate variables

in the monetary policy transmission rnechanism that respond to changes in inflation

expectations, and can therefore not be controlIed with any degree of precision by policy-

makers. We frnd that the estimated dope paramater is positive and significant for

(m=60,n=36), negative and insignificant for (m=120,n=36), and negative and significant

for (m=120,n=60). That is, when the difference between 10-year and 5-year rates widens

by 100 points, we expect inflation to fa11 between 5 and 10 years into the future. Although

the succession of events linking the yield spread to inflation changes at this horizon is far

less cIear, one may infer that the negative impact on economic activity brought about by a

higher 10-year rate would ovenvhelrn the beneficial impact of a lower 5-year rate, thereby

causing the economy to slow and for inflation to decelerate. Although not directly linked

to monetary policy, suc h a relationship may nevertheless be exploited by policy-makers to

provide valuable forecasts of inflation in the (very) long-run.

Having presented the estimates of the linear models, we will now test these for

possible asymmetries. Ideally, if the residuals show thizt asymrnetries exist, we would li ke

to build a model that captures these effects, thereby improving the fit of the model and

cleansing the residuals.

4. Non-linear Analysis

In this section we seek to determine whether a non-linear relationship c m feasibly

arise between inflation changes and interest rate spreads. We begin by explaining how a

non-linear relationship can arise in theory, and then propose a stnictured three-step

approach to ascertain empiricaily whether the data lends credence to a non-linear model.

The first step is to analyze graphically the residuals fiom our linear models in

Section 3. Such graphs may be of use in detecting any non-linearities and, if present, the

f o m that the non-linearities may take. We examine kernet densities of the residuals fiom

the linear models. We are especially interested in ascertaining whether the densities are

multimodal, which would indicate a clustering of residuals, implying that the linear

models are consistently missing salient relationships in the data.

The second step consists of testing the nuIl hypothesis of linearity against

alternatives consisting of various forms of non-linearities. Such tests, like the well-known

RESET model specification test, are of use in detecting general forms of non-linearities.

If we rejsct the nuli hypotheses of linearity in favor of the alternatives, then we can

estimate general non-linear models using non-parametric regression techniques.

Finally, the third step involves testing the nul1 of linearity against alternative

hypotheses that have a specific non-linear form, such as threshold models. If the nul1 is

rejected, then the non-linear models stipulated under the alternative hypotheses will be

used as viable non-linear models in Section 5. The rationale for this third step is that we

have the opportunity to specie models that can ailow for regime changes.

A major strength of the above strategy in the detection of non-linearities is our

reliance on a multitude of tests and techniques. Granger (1993) notes that the srnaIl-

sample properties of most tests for non-linearities are still not well understood. In

addition, some tests have good power against some forms of non-linearities under the

alternative hypotheses, but low power against others. Thus, by relying on several tests, we

maximize our chances of detecting non-linearities should they exist as some tests will

have greater power against the (unknown form of the) non-linearity in the data. Of course.

when the form of the non-linearity is unknown, the power of some tests will be reduced.

Hence, failure to detect non-linearities using any of the tests does not imply that the true

underlying relationship is necessarily linear.

4.1 Theoty

In building non-linear modefs, one is theoretically confionted with an infinite

number of possibilities. Fortwiately, economic theory can help us narrow the choices by

directing us towards models that can feasibly arise in economics. Threshold models are a

suitable candidate, with relationships between variables changing according to different

regimes.

In the spread-output relationship non-linearities can feasibly arise in through (I),

where we drop a term in order to obtain a simple linear relationship between inflation

changes and the spread. If we do not drop the last term in (11, we find that (10) becomes

instead

I f the expected real rates of interest at horizons rn and n are zero, then this again

simplifies to a linear structure (without a constant). In fact, even if the expected real

interest rates at different horizons are non-zero but equal, such that E,r, = E,rrm = E,rln,

then we still get a linear relationship:

If the expected real rate is positive, then we expect the coefficient on the spread to

be iess than 1; if negative, it will be greater than 1. However, in (20) we see that the slope

\vil1 only be constant if the ex anre real interest rate is constant. This of course need not be

so. If the ex onte real rate is below the mean real rate, then the slope parameter will be

larger; if it is above the mean real rate, the slope parameter will be smaller. Thus, it would

be desirable to estimate a model that allows for changes in the slope parameter. One such

model is a threshold model, where the slope parameter depends on the level of the yield

spread. Fama (1990) notes that nominal yield spreads can contain information about real

espected bond retums, and therefore it would seem reasonable to suspect that the ex ante

real interest rate, and therefore the slope parameter, would depend upon the level of the

yield spread.

We can be more generai still if we relax the assumption that the expected real

rates at different horizons mut be equal in (19), which may be excessively stringent if the

difference between the horizons m and n is sizeable. Therefore estimating more general

non-linear models, such as non-parametric or neural network models, may alIow us to

uncover some more information about the true underlying relationship. To summarize,

non-linearities can arise fiom differences in expected real rates, and presumabty are more

likely to occur the larger the difference between the long and short horizons.

4.2 Grap hical Analysis: Kernel Densities

As a first step towards detecting non-linearities between inflation spreads and

interest rate differentials, a graphical analysis of the residuals from the Iinear models

helps us determine whether there are some neglected non-linearities in the residuals, and

the possible fonn of the non-linearities.

In Figures 1A and 1B we plot the kernel density fùnctions of the residuals fiom

the Iinear models. Non-linearities may exist in the residuals if the kemel densities are

bimodal (or multimodal), implying that residuals tend to be clustered around two

(multiple) points. This rnay be an indication that our linear models are unable to account

for some systematic effects between inflation changes and interest rate differentiais,

which implies that a model allowing the estimated parameters to differ in different

regirnes might be better able to account for such effects.

The fifteen kernel densities in Figures 1A and 1B correspond to plots of the

residuals of our fifteen linear models, for al1 possible combinations of rn and n, divided

according to whether they reflect policy or non-policy horizons, as defined earlier. The

most syrnmetric graph would appear to be the one for m=6, n=3, which is nicely centered

around zero. The graph for rn=12, n=3 also shares some of these features. This indicates

that the residuals of these linear models do not have a tendency to cluster around multiple

points, and nor are they apparently skewed to one side or the other. However, we should

recall that the parameter on the term spread for these models was insignificant in Table 1,

consistent with the finding of Mishkin (1990a) and others that the short end of the term

structure has little explanatory power for inflation changes. The symmetric plots of the

densities of the residuals represent preliminary evidence that there may not be any non-

linear relationships that we can exploit either.

For the other matuities we note that most densities appear to display multiple

peaks. The case of m=120, n=60 is the most noteworthy with the distinct peaks near 1 .O0

and -2.50 being very far apart. A multiple-regime model, allowing for differential effects

above and below a certain value of the term spread, would try to capture these large

residuals by allowing for a sizable difference in the effect of the term spread above and

below the threshold. In cases where the peaks are near each other, such as the case where

m=60, n=3, we would expect the difference in the slope parameter above and below the

threshold to be less dramatic.

Of course, such graphs can be misleading since they are drawn for a particular

value of the window width, h. By increasing h, the curves necessarily becomes smoother,

implying that a unimodal kemel density can be achieved using a high value for h. For our

graphs h was chosen according to the equation h = (4/3)-+ x a x T + , where a is the

standard deviation and T the nurnber of observations. This is the optimal window width

for a normal distribution with known variance.

Fortunately, to circumvent the issue of the window width, we can resort to a

bootstrap test presented by Efion and Tibshirani (1993:227-232) to conduct credible

inference. The idea behind the test is as follows: In a given data set we can achieve

unimodality for h = h^; we would like to determine the probability that the true h lies

above our estimated 6 , in which case h ̂ would be undersrnoothing, and so unimodality is

a reliable finding. To calculate such a probability, we: (i) draw B bootstrap samples from

Our estimated density; (ii) for each sample we calculate the smallest h for which

unimodality is achieved; and (iii) calculate the p-value, which is simply the number of

estimated bootstrap h's which exceed h , divided by the nurnber of bootstrap samples B.

For exarnple, our kernel densities were constructed using a default grid of 128

observations for the residuals. These data points c m be denoted by x,, for i = 1 to 128.

From these 128 observations we then take B samples with replacement of 128

observations; the observations in the samples c m be denoted as y*,, for i = 1 to 128. We

then can obtain the actual bootstrap observations, constructed as

where h^, is our initial estimate, &* is the estimate of the variance fiom our initial data

(the 128 residuals), and thes, are standard normal random variables. Notice that the

above transformation implies we are not simply sampling with replacement fiom the data,

but rather sampling from the smooth estimate of the population. Hence, this procedure is

known as a srnooth bootstrap.

The results of implementing this test on the residuals fiom our linear models are

given in Table 4. We test two different hypotheses: (ij Ho: # of Modes = 1; Hl: f: of

Modes > 1 ; (ii) Ho: f: of Modes = 2; Hi : # of Modes > 2. For the latter hypothesis instead

of selecting the smallest h for which unimodality is achieved, we instead select the

srnallest h such that bimodality is achieved. Thus, this test can easily be extended to test

any number of modes under the null hypothesis.2

' The rationale for conducting the second test is that should we reject the null of unimodality, we woüld not know the number of modes of the true density. If there are two modes, then a No-regime model (constructed with one threshold) would be adequate in capturing the term spread-inflation change relationship. Should there be three modes, then a three-regirne (two-threshold) mode1 would be adequate. Thus, testing for two versus more than two modes is relevant for modeling purposes.

In our search for the smallest h for which the number of modes in the density

under the null hypothesis is achieved, we allow the grid to increase in increments of 0.0 1.

The reported p-values in the table are based on 5,000 bootstrap replications. For most

cases we find that the probability of not rejecting the null of unimodality is very high,

with the exception of the two models at the Iongest horizons (m=120,n=36 and

m=120,n=60). For these models the probability of unimodality is only 0.23 and 0.24

respectively. These low probabilities are consistent with o w observation that the distinct

peaks in the kernel densities are very far apart for these models when compared to those

of the other models.

When repeating the test using bimoddity as the null, we find that almost al1 the p-

values increase, becoming very close to l. Those for the final two models converge to

0.99 and 0.77 respectively, implying that more than two modes are unlikely. in the

parlance of a modeler? this implies that a single-threshold, two-regime mode1 would

likely suffice for any of the m,n combinations.

There would appear to be some contradiction between the visual inspection of the

densities and the actual test results, with the former denoting multiple peaks and the latter

pointing to unimodality- However, we know little of the power properties of the test,

especially when the peaks are relatively close to each other as is the case in some of our

models. Therefore the foregoing analysis should be viewed only as a preliminary, yet

informative, step in detecting non-linearities. More forma1 tests on the residuals are

presented in the following sections.

There are essentidly two broad classes of non-linearity tests in the literature. With

the null hypothesis being linearity, the alternative can either be formulated as a general

form or a specific form of non-linearity. By "general" we mean that the model under the

alternative hypothesis simply tries to capture some of the neglected non-linearities by, for

example, modeling the residuals of the linear model against higher powers of the

residuals. Typically, it has been found that the power of tests with specific forms of non-

linearity as alternative hypotheses is greater than tests which propose a generai form of

non-linearity, provided that the true underlying non-linearity in the data is the one

postulated by the specific test. in Section 4.4 we examine tests that postulate specific non-

linear rnodels under the alternative hypothesis.

43.1 RESET and Keenan Tests

The first non-linearity test we consider is the Ramsey (1969) Regression Error

Specification Test (RESET). The test is used to determine whether the functional form of

Our mode1 fits the data wel!. It accomplishes this by regressing the residuals of the model

under the null hypothesis (usually the linear rnodel) against the original explanatory

variables and powers of the fitted values fiom that model. The motivation here is to

dcterrnine whether the additional powers are significant at explaining the residuals. If

they are, then a non-linear model would be superior to a linear one. Formally, the test is

implemented in the following steps:

We estimate the linear model, which in our case is (1 1 j, and obtain the residuals

4"" , fined values (5: - 2:) and sum of squared residuals. The next step is to run the

following regression:

We wish to test the significance of the @ - 1) parameters w; thus, we can

construct an F-statistic for such a purpose. The number of powers to consider @) is open

to question, although p=4 is ofien used in the Iiterature- In our application we use p=2?

p=3 and p=4; in the case of p=2 the F-statistic reduces to a X2 statistic, and is commonly

known as the Keenan Test (e.g. see Keenan (1985)). The results for the tests applied to

the residuals of our linear models from Table 3 are presented in Table 5 .

We find that the nul1 of linearity is clearly rejected for al1 nine policy horizons

(rows 1 through 9). Contrast this to the non-policy horizons that use information solely

along the short-end of the yield curve (rows 10 through 12) where we cannot reject the

nul1 of linearity. In these cases we found in Table 3 that the models fit the data poorly,

and that the interest rate spreads were not significant in explaining inflation changes. This

is also consistent with our observation of the kernel densities of the residuals in Figure 1,

where the densities for small values of rn and n appeared to be symmetric and unirnodal.

For the non-policy horizons at the longer-end of the yield curve (rows 13 through

15) evidence of non-linearities again emerges. The case of m 4 2 0 and ~ 3 6 is

particularly interesting since there is evidence of non-linearities, yet the spread parameter

for this horizon in Table 3 was insignificant. This indicates that the results of these tests

are not dependent on whether the spread parameter in the linear model is significant, and

may be one instance in whiich only a non-linear model would be capable of capturing any

relationships between the spread and inflation changes.

43.2 Tsay Test

Tsay (1986) proposes a test that specifies a linear mode1 under the null, and a

Logistic Smooth Transition Autoregressive (LSTAR) model under the alternative, where

reçime changes are specified as smooth shifts rather than discrete jumps. To implernent

the test one need only regress the residuals fiom the linear model (1 1) on the original

regressors and al1 cross-products of the regressors. Since we have but one regressor in the

linear model, namely the spread, it follows that we need only to regress the residuals on

the spread and the spread squared. An LM statistic can then be constructed to test the

significance of the additional regressor. Fomally, to perform the test we estimate the

following regression

in'" = a,, + Pnl., (Km - Rrn + Y m , n (R< - R: 1' + ut

and test the significance of y,,,. The results for this test are given in Table 5. We find that

the results corroborate those of the RESET and Keenan tests, namely that non-linearities

appear to be quite prevalent at aimost al1 horizons. The notable exceptions are once again

the non-policy horizons at the short end of the yield c w e .

4.4 Specific Non-linearity Tests

So far the tests discussed have been residual-based and have focused on testing

the nul1 of linearity against the alternative of a general fonn of non-linearity. Such tests

are best used as residuai diagnostics to determine whether the residuals are fiee of any

non-linearities. In this section we perform tests that speciQ a specific modei under the

alternative hypothesis which one can then estimate should the null of linearity be rejected.

4 - 4 1 Threshold Alternative

The first such test we consider is the test of Hansen (1996), which ailows us to

test the null of linearity against the alternative of a threshold model, with the precise

location of the threshoid being unknown a priori. Such a test was used by Galbraith

(1 996); we apply it here on (1 1).

When testing for a threshold we seek to determine whether the relationship

between inflation changes and the spread differs when the spread exceeds a certain

threshold. r. Unfortunately, the value of r is unknown. Thus, we are forced to perform a

grid search in order to locate the most likely value of the threshold. To determine whether b

the estimated threshold is significant, we must compute the p-values using simulation

methods, since the distribution of the test statistic is unknown when the threshold is not

provided a priori. We therefore estimate the following regression for 200 different values

of the threshold

- (x< -z:) = a,,, +fln, (Rtm - R3+6,,(Rtm - Rtn) + qmn

where

- (R," - R:) = (RI - R:)I[(R;" - R,") S r]

and

For each threshold value we compute an LM statistic in order to test the null of

linearity. Figures 2A and 2B plot the sequence of LM statistics that were obtained when

conducting this test on the fifieen models, and Table 6 provides the maximal statistics

with the associated thresholds. We notice that the null is clearly rejected for al1 nine

policy horizons (rows 1 through 9) with p-values of 0.000, and that the thresholds

themselves are located for values of the spread between -0.62 and -2.26. When short-

term rates exceed long rates in such a mariner monetary policy is usually considered to be

tight, implying that the relationship between yield spreads and inflation changes is

different when monetary policy is tight than when it is expansionary. In Section 5 we

estimate threshold models to determine the magnitude of this asymmetry.

For the non-policy short rates (rows 10 through 12) we do not find evidence of a

threshold at the 5% significance level, consistent with the previous non-linearity tests.

Thus, even the addition of a threshold does little to unearth any relationships between

yield spreads and inflation changes at these horizons.

Finally, for the non-policy long rates (rows 13 through 15) we again find

significant threshold effects. The noteworthy differences here are the locations of the

optimal thresholds, which are positive for the 10-year less 3-year and l 0-year less 5-year

spreads. Again, this is evidence of an asymmetric relationship between yield spreads and

inflation changes, although it is dificult to attach policy interpretations to the different

regimes since the Fed cannot immediately affect the three, five and ten-year rates, and

therefore carmot affect the spreads between these rates.

4.3.2 Non-parametric Alternative

Wooldndge (1992) shows that a Davidson and MacKimon (1 98 1) J-type test used

for testing non-nested hypotheses is asymptoticaily valid when used to test the nul1 of a

linear mode1 venus a non-parametric alternative- With the nul1 hypothesis given by (1 1),

the alternative is

Here mh is some non-parametric fùnction with bandwidth h. To implement the test we

need to run the following artificial regression

where 61, denotes the fitted values of the non-pararnetric regression for bandwidth h .

The test is simply a t-test on the parameter cr, if it is significant, then we reject the linear

nul1 in favour of the non-parametric alternative-

in Table 6 we present the results of the Wooldridge test for both an optimal

bandwidth chosen through cross-validation, h =h * (see, for example, Hadle (1 990)) and

for a fixed bandwidth h=0.30, to veri@ whether the results are robust to the selected

bandwidth. For the policy horizons, we reject the linearity null for al1 cases when h=0.30,

but are unable to do so at the 5% level for policy horizons where m=120. For the short

non-policy horizons (rows 10 through 12) we are unable to reject t&e null regardless of

the selected bandwidth. Finally, for the long non-policy horizons (rows 13 through 15)

both tests favor linearity for (m=60,n=36), the non-pararnetric alternative for

(m=120,n=36) but are are divided for the case of (rn=120,n=60). Based on this exercise,

we c m conclude that there is substantial evidence that non-parametnc models are of use

in modeling the relationship between inflation changes and yield spread for a11 but the

shortest horizons.

4.4.3 Neural Network Alternative

Another test designed to detect non-linearities is the Neural Network test,

proposed by White (1989). The regression used in the test is

where y ( y ; (R,"' - )) = [1 + exp(-y ; (R: - R," ))]-' is the logistic function. The test is

inspired by the way information is thought to be processed by the brain, and is based on

*'hidden units" (unknown variables) that link the yield spread to inflation changes

w , = ( y ( y ; (R: - R: )), . . . , yl(y (R: - R: ))) where the y, need to be chosen

independently of the data (Lee, White and Granger (1993)). In practice, they are chosen

from a uniform distribution in the interval [-2,2], and we follow that tradition here. To

impiement the test we first need to rescale both the interest rate differentials and the

inflation change data so that they lie in the [0,1] interval.

Furthermore, since q is usually quite large, a multicollinearity problem arises.

Therefore the test needs to be canied out using a few principal components. Following

Lee. White and Granger (1993), we set q=IO, and use q*=2 principal components (the

second and third largest)). To implement the test, we regress the residuals from the linear

mode1 on the spread and iy*, which is similar ;O iy except that it consists of only two

principal components. Thus, we have three regressors. The test is then an ordinary LM

test, with the test statistic computed at T X R ~ from the auxiliary regression, and is

distributed asyrnptotically as X2(q*).

The resuits are presented in Table 6 . Since the test statistic is based on random

draws from a unifonn distribution, we perform it three times in order to veri@ the

-

5 We also experimented with q*=l and 3, and q=20. The results were qualitatively similar.

39

robustness of our results, but only report the first test since the test statistics do not Vary

much. For the policy horizons (rows 1 through 9) we frnd that the linearity nul1 is rejected

for al1 but one case. More specifically, we notice that for policy horizons with m420 the

p-values tend to rise noticeably, the same phenomenon witnessed for the Wooldndge test

with h=h*. For the non-policy horizons the same pattern emerges as with the previous

tests, with the exception of the case (m=12,n=6) where we find evidence in favor of the

non-linsar alternative.

4.5 On the Power of Non-linearity Tests

Having presented a number of tests, the question now arises: Which test is the

most credible? The answer depends crucially on the true form of the data-generating

process. If the DGP is a threshold model, then a test which specifies a threshold model

under the alternative hypothesis will have greater power than one which specifies a

general alternative. On the other hand, if the DGP is in fact not a threshold model, then a

test with a general alternative will have greater power than one which specifies an

incorrect non-linear model under the alternative hypothesis.

Through simulation studies, Lee et al. (1 993), Teravirta et al. (1 993), Terasvirta

(1996) and de Lima (1997) examine the power of a number of non-linearity tests. In

general, the consensus is that no single test dominates al1 the others under al1

specifications. However, some tests, in particular the Neural Network test, perform as

well as or better than the others under most situations. Since the nul1 of linearity has been

rejected for almost al1 cases using at least one test statistic, we proceed under the

assurnption that it may be possible to improve upon the modeling of the inilation-spread

relationship for al1 forecasting horizons.

5. Modeling Non-linearities

In this secti~n we propose and estimate a nurnber of non-linear models. We begin

by specifjing the most generai non-linear model possible which is then estimated using

non-pararnetric regression techniques. We then draw upon the results of our threshold

tests to estimate threshold models. The residuais fiom the non-linear rnodels are then

subject to the same non-linearity tests applied to the linear models, with the aim of

determining whether the residuals have been cleansed of their non-linear properties.

5.7 Non-parametric Models

Apart from specieing the variables which enter a model, the non-pararnetric

models we consider impose no functional form on the data. The models are specified as

There are nurnerous methods to estimate (30). We choose to use the popular kernel

method, using a Gaussian kernel. From Hardie (1990:25), each fitted value is found from

where the function Kh is the kemel with window width h. Naturally, the goodness-of-fit

of the regression curve will depend crucially on h, with lower values of h fitting the data

more closely. In our work we set h=0.30 for al1 the cases, which allows for easier

cornparison across models. We also select h by minimizing a cross-validation function

(Htirdle (1990: 152-154)), but only plot the cuves for which h 4 . 3 0 . Higher values for h

will produce smoother curves, whereas lower values will be produce more jagged curves.

The estimated non-parametric regression curves, dong with the curves fitted

through the other methods, are s h o w in Figures 3 thomgh 7. For the policy horizons

(Figures 3 through 5) we find a consistent pattern in the curves, narnely a steep portion

for negative values of the yield spread and a flat portion for positive values of the spread-

This can be interpreted as indicating that the relationship between yield spreads and

inflation changes is far greater when the yield curve inverts, and that there is a very weak

link between these variables for upward-sloping yield curves. If the yield curve inversions

can be attributed to a tightening of rnonetary policy, then such asyrnrnetries c m be said to

esist in monetary policy's effect on inflation changes.

We note that for the three short-term non-policy cases (m=6, n=3; m=12, n=3; and

nz=12. n=6 in Figure 6) the curves are very flat. implying that inflation changes do not

greatly depend on interest rate differentials at those horizons. This agrees with two other

findings: (i) when estimating the linear models at these horizons, we found the spread

pararneter to be insignificant; and (ii) several non-linearity tests showed that no non-

linear relationships appeared to exist at these horizons. We can therefore state that there

does not appear to be any noticeable relationship, either linear or non-linear, between the

term spread and inflation changes at the shorter-end of the term structure. For policy

purposes, the Fed should therefore use the term spread to monitor inflation changes at the

medium and longer horizons.

Fed actions are believed to take about 18 months before being felt on the economy

(consistent with the interest rate channel of the monetary policy transmission mechanism,

see Blinder (1996)), so the lack of any relationship at the shorter end of the term structure

may sirnply be due to the ineffectiveness of Fed monetary actions in the short-run. Thus,

this is entirely consistent with the view that only interest rate spreads between longer and

short rates represent good indicators of monetary policy. This is a view shared by, for

esample, Laurent (1 988,1990) and Bernanke and Blinder (1 992).

Finally, it is worth noting the c u r e s depicted for the cases m=60, ~ 3 6 ; m=120.

t7=36; and m=120, n=6O (Figure 7). Here we find the curves to be either flat or sloping

dot~mtvards. The Fed has little influence over these rates, insofar as it can affect inflation

eq~ectations.~ Instead, market forces more likely determine such rates. The results show

that when, Say, the spread between the five and ten-ycar rates widen, inflation changes

between these two horizons actually falls. This corroborates the negative coefficients we

found on the spread in Table 3. However, due to the Fed's lack of influence over these

rates, it is dificult to attach any policy interpretation to the results.

'' See Mehra ( 1 996) for some empirical evidence and Akhtar ( 1 995) for a survey.

43

5.2 Threshold Models

in th is section we formally estimate the threshold model (24) using the estimated

thresholds from Table 6. The results are shown in Table 7. We see that the threshold

parameter is significant in the majority of cases considered, resulting in superior fits over

the linear models. Regarding the parameter estimates, beyond the one-year horizon we

see that they are positive. Recall in (26) we defined the indicator fùnction to take a value

of one when the spread is below the threshold, and zero otherwise- As swh, since the

parameter on the threshold variable is positive, this can be interpreted as a srrengthening

of the effect that the spread has on inflation changes when the spread lies below the

threshold. For example, consider the case where m=60 and n=3. From Table 6 the

estirnated threshold for this model is -1 -29. Thus, when the yield curve inverts and the

three-month rate exceeds the five-year rate by more than 129 ba i s points, changes in the

spread will have greater impacts on the five-year less three-month inflation change than it

othenvise would.

More specificaliy, if the current spread is less than -129 basis points, a 100 basis

point decrease in the spread will cause the five-year less three-month inflation difference

to fa11 by 2.45 1 per cent. Alternativeiy, if the current spread is greater than -129 bais

points. a 100 ba is point drop in the spread will cause the inflation difference to fa11 by

only 1.128 per cent. Since the Fed can control the three-month rate with relative ease, but

has little control over longer rates such as the five-year rate, the 100 basis point drop in

the spread can be readily achieved by a tightening of monetary policy. What we have

uncovered through the threshold model is that the benefits of a tighter policy, as measured

by a reduction in the gap between long and short-term inflation rates, will be felt more

strongly in a regime when policy is already considered tight (which is what we would

have if the spread were less than - 1.29).

The threshold curves are plotted in Figures 3 through 5 for the policy horizons.

Note that the slope is steeper in al1 cases when the spread is located below the threshold.

For the non-policy horizons (Figures 6 and 7) the slope ofien changes sign in order to

better capture the relationships in the data, but the "kinks" are only significant for the last

two curves in Figure 7, leading one to believe that a negative relationship exists between

yield spreads and inflation changes for extreme positive values of the yield spread.

To help determine whether threshold models may be useful at capturing some of

the non-linearities in the data, we can perfom our general Keenan, RESET and Tsay tests

on the residuals of these models. If we fail to find evidence of non-linearities, then this

may provide evidence that the threshold rnodels are adequate at capturing the inherent

non-lineax-ities between yield spreads and inflation changes.* in Table 8 we find that, for

the policy Iiorizons (rows 1 through 9), the threshold models are incapable of capturing

al1 the non-linearities in the data, since the nul1 of linearity in the residuals f?om these

models is rejected in each case. However, some progress has been made to model the

non-linearities, as demonstrated by the fact that several of the test statistics are lower than

their counterparts obtained for the linear model residuals in Table 5 .

For the short non-policy horizons (rows 10 through 12) there is mixed evidence

for non-linearities, although note that for many cases the test statistics are in fact higher

than in Table 5. This is not surprising if we recall that the thresholds are insignificant in

the models at these horizons.

For the long non-policy horizons (rows 13 through 15) we find that residuals have

been cleansed of their non-linearities by the threshold model. Compare, for instance, the

Keenan statistic for the m= l2O,n=6O horizon. For the threshold model it is 0.994, whereas

for the Iinear model in Table 5 it was 28.41. This therefore Iends credence for a model

that allows for both a positive and negative relationship to exist between the data in

di fferent regimes.

5.3 Neural Network Models

With recent advances in computing technology, computationally-intensive

techniques such as neural network models have begun to gain prominence as a method to

capture non-linearities in complex data sets. It allows one to rnodel relationships between

one or several input (independent) variables and one or several output (dependent)

variables.

The relationship between inputs and outputs, however, need not be direct. In the

relationship between interest rates and inflation, for example, one c m argue that there are

rnost likely several interrnediate variables in the transmission from interest rate changes

Note that such tests cannot be performed on the residuals of the non-parametric models, since these have been derived fiom a re-ordering of the data which was necessary to derive the non-parametric curves.

to inflation changes. lnterest rate changes can first affect durable consumption and

investrnent, output, the output gap, and ultimately inflation. in neural network models

these intermediate stages can be captured by a "black box" in which weights between

input variables and (unknown or unobserved) intemediate variables are computed, as are

weights between the intemediate variables and the output variable. The unknown

intermediate variables are commonly referred to as hidden units.

The nurnber of intermediate variables and, indeed, the nurnbers of stages or layers,

are not necessarily known, and therefore some experimentation is required on the part of

the modeler in order to adequately capture the salient feautures of the data. The neural

network model that we estimate has the following f o m

whereg(rr) = l/(l +e- ' ) is the logistic function which serves as a smooth squashing

function (analogous to a smooth? as opposed to discrete, threshold model that allows for

regime changes), Pkl is the connection strength (weight) between the yield spread and the

hidden unit k, K is the the total number of hidden units (= 2 in our work) and a k is the

n-eight of unit k on the output, (n," - ~r: ) .

To estimate the parameters Pkl and ak, one minimizes the sum of squared

deviations between the output and the network:

To estimate (33) we use backpropagation to re-calculate the parameters until we achieve

the pre-specified convergence Ievel. In our applications we speci5 a mean squared error

that we wish to achieve. The number of epochs (replications) required to achieve the

desired levels of convergence are provided in Table 9. Through our experimentation we

found that convergence after some tens of thousands of epochs yielded sensible curves

that balanced our desire to capture the underlying non-linearities in the data without over-

fitting, and therefore chose the convergence levels with this goal in mind.6

The neural network curves for the policy horizons are plotted in Figures 3 through

5. Two features are immediately apparent for al1 nine of these curves: (a) they dl closely

resemble the non-parametric curves (although the neural network cuves seem less

influenced by the end-points, where data is scarcer); and (b) they al1 have similar shapes.

being quite steep for values of the spread between -1.5 and 0.0, and fiat for positive

values of the spread. Again, if we assume that the Fed can influence these spreads

through its actions, then when the spread is positive and the Fed tightens (causing the

long-short yield spread to fall), the eflect on inflation changes rvill be minimal. However,

should it tighten when the spread is already negative (usually associated with a policy

Neural nehvork rnodeling in general is still in its relative infancy, and therefore requires a great deal of judgernent and experimentation on the part o f the modeler in order to be properly perforned. From this

regime that is aiready considered tight), then rhe effecr on i n r i o n changes will be

pronortnced as we are moving down the steep portion of the fitted curve. Such

information can be vital to policy-makers who view the yield curve as a good indicator of

hture inflation, as these fmdings show that most of the positive correlation between the

yield spread and inflation changes are due to a strong relationship that exists in a

relatively narrow range of the yield spread.

in Figure 6 neural network curves are fitted for short-mn non-policy horizons.

Coinciding with the other models we find the neural net curves to be very flat, indicating

that there does not appear to be any discernible relationchip between yield spreads and

inflation changes at these horizons. By contrast, in the long-run non-policy horizons

(Figure 7) we find some relationships in the data- Confirming what we haC found earlier

using kernel densities and threshold and non-paramateric models, the reiationship

between yield spreads at the long-end of the yield curve and inflation changes appears to

be highly non-linear. For m=120 and n=36 we find the fitted curve to be bell-shaped,

indicating that some quadratic function might be appropriate in rnodeling the relationship.

When applying some non-linearity tests7 to the neural net models (Table 8), we

find that these models are unable to cleanse the residuals of al1 their non-linear properties

for al1 the policy horizons. For the non-policy horizons, however, some progress appears

perspective it can be considered an art as much as a science. See Masters (1993) and Mehrotra, Mohan and Ranka ( 1 997) for effective expositions.

Note that the Tsay test i s not applied since it can only be used on parametric models.

to have been made for the last two cases considered as the test statistics are noticeably

lower, as rejection of linearity in the residuals is very marginal for m=120, ~ 3 6 .

5.4 Comparing the Models

In this chapter we have estimated numerous models, both parametric and non-

pararnetric, in order to gain a better understanding of the empirical relationship between

yield spreads and inflation changes. In Table 10 we present the root mean squared errors

of al1 the models in order to determine which fit the data best. The noteworthy findings:

The threshold models fit the data at least as well as the linear model for dl horizons

considered;

The non-parametric models often have the best fit, but this is mostly a fünction of the

bandwidth. A lower bandwidth always provides a better fit;

The neural network model fits the data noticeably better than the linear model at

several horizons, especially for the policy horizons (rows 1 through 9). For example,

compare the neural net root mean squared error of 2.47 versus that of 2.86 for the

linear model at the m=120, n=6 horizon.

6. Conclusion

This paper first constnicts a theoretical model, relying on the foundations of the

Fisher equation, to link changes in inflation to interest rate differentials. A linear model is

estimated, and through a three-step procedure we are able to demonstrate that the

inflation-spread relationship displays rnarked non-linearities for a number of horizons.

The main exceptions are for horizons of one year and under, in which case there appears

to be no discemible relationship between inflation changes and interest rate differentiais.

We constmct three types of non-linear models: non-pararnetric, threshold and

neural network. In al1 cases we are able to capture some of the non-linearities in the data,

and demonstrate that the relationship is more pronounced when interest rate differentials

are negative. This is s h o w through marked steepness in the non-parametric and neural

network curves and a number of thresholds in the vicinity of values of -1 for the spread.

However, we also note that in spite of these findings, there is still scope for improvement

in the models as the residuals fiom the non-linear models still have not been completely

cleansed of their non-linear properties, at l e s t for the policy-relevant horizons. At the

short-run non-policy horizons we find no relationship between yield spreads and inflation

changes, whereas in the long-run marked non-linearities are apparent, and our models are

capable of capturing much of these non-linearities. in future work we hope to capture

more of the non-linearities for the policy-relevant horizons.

Appendix 1: Data

The data used in this study was obtained fiom the Federal Reserve Bank of St- Louis FRED database. The fiequency is monthiy, and al1 observations end in October 1996. The variable definitions, dong with their starting dates, are as follows:

CPIAUCSL: Consumer Price index, Al1 Urban Consurners, SeasonaIly Adjusted, January 1947.

GS 1 : One-year Treasury, Constant Maturity Rate, April 1953.

GS3: Three-year Treasury, Constant Matunty Rate, April 1953.

GS 5 : Five-year Treasiuy, Constant Maturity Rate, April 1953.

GS 1 O : Ten-year Treasury, Constant Manirity Rate, April 1 953.

TB3MS: Three-month Treasury Bill Rate, Secondary Market, January 1934.

TB6MS: Six-month Treasury Bill Rate, Secondary Market, December 1958.

In Our empirical work we use April 1953 as the staring date for al1 models, with the exception of models for which the six-month rate is used, in which case we are constrained to begin in December 1958.

Appendix 2: Tables and Figures

Table 1: Unit Root Tests

ni. n Augrnented Dickey-Fuller Phillips-Perron KPSS

120.60 -2.186 24 -4.340 2 - 1 -742 -4.2 15 0.426 0.22 1 The number of lags (k) for the ADF test were chosen by minimizing the Schwarz Criterion, with a maximum allowable lag of 24. The lags used for the PP and KPSS tests are 12, correspondhg to the fiequency of the data. The critical values for rejection of the unit root hypothesis for the GDF and PP tests are -2.57 at the 10% level; -2.88 at the 5% level and - 3.46 at the 1% level- The critical values for rejection of the nul1 of stationarity for the KPSS test are 0.1 19 at the 10% level; 0.146 at the 5% levei; and 0.216 at the 1% level.

Table 2: Long Memory Tests and Fractional Integration Parameters

m.n Modified Rescaied Range Geweke-Porter-Hudak (months) FI Parameter

I20,60 27.95 1 .O06 1.372 0.4 10 The 95% confidence interval of the Modified Rescaled Range Statistic of Lo (1 99 1 ) is [0.809,1.862]; the 99% confidence interval is [0.72 1,2.098]. The GPH fractional inteption pararneter is the pararneter (d) in an ARFIMA(O,d,O) model. The nurnber of spectral ordinates used in the GPH regressions = 7''.

Table 3: Linear Models

n: -a: = a,, +&(R,? - R:) + rf"

(4.8 19) (-5.87 1) Newey and West (1987) corrected t-statistics are in parentheses. Runs is a non-pararnetric test for the randomness of the residuals, and is asymptotically normally distrubuted. zs is the Bowman- Shenton test for non-normality, distributed as X2(2); z~ is an LM test for 12th-order autocorrelation and is distributed as X'(12), while z~ is a modified Breusch-Pagan LM test for heteroskedasticiry, distributed as $(î). (*) denotes signifkance at the 5% level.

Table 4: Bootstrap Multimodality Tests

nt. n Ho: # of Modes = i &: # of Modes = 2 (months) HI: # of Modes > 1 HI: R of Modes > 2

6 p-value h p-value L

120,60 0.65 0,242 0.32 0.77 1 k is the smallest window width for which the number of modes under the nui1 hypothesis is achieved. The p-value is the probability of not rejecting the nul1 hypothes is. 5,000 bootstrap replications were used for each test.

Table 5: General Non-Linearity Tests

m,n Obs. Keenan RESET RESET T,=y (months) ( T ) F( 1, T-3) F(2, T-4) F(3. T-5) x'( 1 1

120,60 403 28-41 ** (*) denotes rejection of the nul1 of linearity at the 5% Ievel; (**) at the 1% level.

Table 6: Non-Linearity Tests with Specific Alternatives

Threshold Non-pararnetric Neural Network

m,n Estimated Hansen Estimated Wooldridge Wooldridge Lee, White (months) Threshold Sup-LM Bandwidth t-stat t-stat & Granger

(h*) (h=h4) (h=O. 3 0 ) F(29T-4)

54.02 (0.000) 79.60

(0.000) 44.42

(0.000)

43 -90 (0.000) 62.05

(0.000) 30.58

(0.000)

144.0 (0.000) 171 -8

(0.000) 80.07

(0.000)

5.147 (O. 128) 7 -3 62

(0.052) 4.28 1

(O. 152)

27.0 1 (0.000)

13.55 (0-004) 61.38

-4.008 (0.000) -4.049 (0.000) - 1 -456 (O. 146)

-2.555 (0.01 1) -2.848 (0.005) - 1.953 (0.052)

-3.559 (0.000) -4.147 (0.000) -0.748 (0.455)

-0.167 (0.868) 0.029

(0.977) -0.1 16 (0.907)

- 1.670 (0.096) 3 -759

(0.000) 7.067

7.547 (0.000) 14.36

(0.000) 1.535

(0.2 17)

7.1 15 (O .O0 1 ) 19.85

(0.000) 3 -248

(0.030)

15-48 (0.000) 15.70

(0.000) 4.1 12

(0.0 17)

0-07 1 (0.93 1) 2.536

(0.080) 5.64 1

(0.004)

1.750 (O. 175) 7.127

(0.00 1) 13.73

(0.000) (0.000) (0.2 1 O) (0.000) The nul1 hypothesis for each test is linearity, with p-values in parentheses. The estirnated thresholds were located by searching over a grid of 200 equispaced gridpoints in order to maximize the test statistic. 3,000 replications were used to simulate the asymptotic distribution of the threshold test statistic using the method suggested by Hansen (1996). We present two different sets of Wooldridge (1992) non-pararnetric tests: The first for a bandwidth h* which was chosen through cross-validation, and a second for a constant bandwidth h=O.jO.

Table 7: Threshold Model

m,n Obs. am.n Pm. n Threshold R~ (rnonths)

(7.293) (-7.646) (6.114) Newey and West ( 1 987) corrected t-statistics are in parentheses.

Table 8: Non-linearity Tests on the Non-linear Models

ïhreshold Mode1 Neural Nctwork Mode1 m.n Obs. Keenan RESET RESET Tsay Ksenan RESET RESET

(months) (T ) F(I,T-3) F(2,T-4) F(3,T-5) 1%) F(l,T-3) F(2,T-4) F(3, T-5)

120.60 403 0.994 1.223 0.835 1.378 10.20** 6.657** 7.249** (*) denotes rejection of the nui1 of linearity at the 5% level; (O*) at the 1% level.

Table 9: Neural Network Models

-- -

m,n Obs. Epochs Mean Squared (months) (T ) Error (x 1 O")

One hidden layer and two hidden neurons are used for each rnodel.

Table 10: Root Mean Squared Errors

nt, n h* Linear Threshold Non-Parametric Non-Parameuic Neural (months) ( k h * j (k0.30) Nenvork

I20,60 0.04 1.193 1.150 1.131 1.203 1.154 il is the bandwidth used for the non-parametric model, and h* is the optimal bandwidth chosen bv cross- validation. The srnallest RMSE for &ch forecasting horizon is depictedin bold digits.

Figure 1A:Kernel Densities for Linear Model Residuals (Policy Horizons)

Figure 1B: Kernel Densities for Linear Mode1 Residuals (Non-Policy Horizons)

Figure 2A: Sequence of LM Statistics for Threshold Test (Policy Horizons)

Figure 2B: Sequence of LM Statistics for Thresbold Test (Non-Policy Horizons)

Figure 3: Fitted Curves, Policy Horizons (Short Rate = 3 Months)

Figure 4: Fitted Cumes, Policy Horizons (Short Rate = 6 Months)

Figure 5: Fitted Cumes, Policy Horizons (Short Rate = 12 Months)

Figure 6: Fitted Curves, Non-Policy Horizons (Short-End of Yield Cuwe)

- - - N o n - P a r i m e t r i c

L i n e a r .-... Tnresnoia - - - N o n - P a r a m efr ic H e u r a t - -- u e l i o r k

Figure 7: Fitted Curves, Non-Policy Horizons (Long-End of Yield Curve)

o r J

0 4 - o z -

- 3 8 O i

---. . T h r e l h o l d

1 ., - - - N o n - P a r a m a c r i c

-- - - - - y- -Lertear - . - . ~ n r e s n o i d -. - - - N o n - P a r a m e t r i c

H e u r a t N e i r o r k

1 -

5.

5 , O 4 1 C O - - 1 . - - -0 5 - C -

- L ~ n e a r . t 5 - - - . . - T h r e s h o l d - - - Non-Param e t r i c -2 - N e u r a l U e t w o r k - - --

-2 5 - 11.11 Spread

ESSAY 2:

TESTING FOR ASYMMETRY IN THE b N K BETWEEN THE YIELD SPREAD AND OUTPUT IN THE G-7 COUNTRIES

1. Introduction

There has been a good deal of recent interest in the link between yield spreads and

agçregate economic activity, for several related reasons. First, the yield spread,

specifically the diflerence in yieids between long-term and short-term interest-bearing

securities, has been found to be one of the most usefid business cycle leading indicators;

see, e.g. Estrella and Hardouvelis (1991), Bernanke and Blinder (1 992) and Cozier and

Tkacz (1994). Second, there is a popular argument (espoused by Laurent (1988), for

example) to the efTect that the interest rate spread acts as an indicator of the direction of

rnonetary policy. To the extent that this is true, the value of the spread in senring as a

leading indicator of aggregate activity could be the result of its value in surnmarizing the

current impact of monetary policy, which affects aggregate output in the future.

A third important point is that the yield spread (or dope of the yield cuve) is a

variable that can be observed imrnediately, and with virtually no measurement error or

approximation error arising from the use of an index, which distinguishes it from many

other indicators and is one possible explanation for its empirical usefulness as a business

cycle indicator,

The present essay examines this yield spread-utput link, and in particular the

possible existence of asymmetries in the relationship. The examination of asymmetries is

suggested by, and in part derives its importance fiom, the fiequently-reported finding of

asymmetries in derived measures of current monetary policy or money supply changes,

such as those of Cover (1992), Morgan (1993) and Karras (1996). At the sarne time, this

paper uses data from the entire G-7 group of countries, rather than the U.S. alone.'

Asymmetry, if present, implies that the information content of the spread cannot be fdly

exploited in a linear model. Less formally, the existence of asymmetry here, as in other

contexts where policy can affect events, would imply that we should anticipate greater

proportionate impacts for some values than for others; policy actions and forecasts should

be adapted accordingly.

In several countries monetary policy has the dual objective of achieving low

inflation while attaining high levels of growth. In the previous chapter we investigated

the relationship between monetary poIicy and inflation- This chapter focuses on the

second objective of monetary policy, namely high levels of growth. Thus, this chapter

can be viewed as being complimentary to the first.

This chapter also serves the objective of verifjing whether asymmetties in the

monetary policy transmission mechanism arises due to an asymmetric response of output

growth to interest rate changes. Failure to find evidence of asymrnetry could lead one to

conclude that asymmetries between inflation changes and interest rate differentials may

be caused by an asyrnmetric relation between output growth and inflation (Le. a non-

linear Phillips curve), which is the last link of the transmission mechanism.

To test whether the yield spread has an asymmetric impact on output, we proceed

as in the previous chapter, namely we apply several general and specitic non-linearity

' Karras ( 1 996) also pursues this strategy in examining asymmetries in derived measures of money supply shocks, but aggregates the data across the G-7 rather than examining individual country effects.

73

tests. Central among these tests is a test for a threshold effect in the relation, as it allows

for a direct test of asymmetry between the yield spread and output growth. To implement

such a test we treat as unknown the threshold beyond which the effect of the yield spread

becornes greater (or smaller); evidence of a threshold effect is evidence in favour of

asymmetry (or of some other non-linearity which can be approximated in this way). We

use the test proposed by Hansen (1996), which allows testing for a threshold effect

without a priori knowledge of the threshold value. This procedure is similar to that used

by Galbraith (1996) in testing credit rationing models which predict threshold effects

between measures of money and output growth. Treating the threshold as unknown has

the advantage that it ailows us to consider the likelihood of asyrnmetry contingent upon a

number of different threshold points, and also requires us to use a test which explicitly

accounts for the fact that the choice of threshold is based on the likelihood. Earlier test

procedures that implicitly or explicitly use threshold values that maximize the likelihood

of finding asymmetry invalidate the nominal distributions used for inference; we will

return to this point below. Moreover, by leading us to consider a set of possible

threshoIds, this method gives us a more general overview of the usefuhess of the

asymmetry hypothesis in descnbing the relation between yield spreads and output.

In examining the relationship in this way, we find a distinction between U.S. and

non-U.S. data. This suggests that caution is required in importing information about the

U.S. transmission mechanism even to other developed countries, and aIso suggests that it

could be fruithl to attempt to understand the reasons for apparent differences in the form

(as well as the strength) of this link across counules. We aiso offer a few conjectures

about these differences.

The next section discusses in more detail the use of the yield spread, or slope of

the yield curve, as a leading indicator, and as an indicator of monetary policy. Section 3

presents the data and Iinear models. General non-linearity tests are reported in Section 4,

while Section 5 presents sorne specific non-linearity tests. Section 6 concludes.

2. Literature Review

Economists have been interested in business cycle indicators since at least the

seminal work of Mitchell and Burns (1938). A good indicator is valuable to poiicy-

makers primarily in helping to reduce the policy recognition lag, and is particularly

important where aggregate output is concerned, given both the time required for initial

GDP estimates to be released, and the fact that revisions to the initial estimates are ofien

substantid.

A good indicator should be, among other things, timely and precisely measured.

The yield spread meets both of these requirements very well, being available

immediately, daily, and being virtually free of the measurement errors that plague many

macroeconomic data series. Moreover, although the yield spread was not examined in

the early NBER work on indicators, a nurnber of recent studies have concluded that it is

one of the better business cycle indicators available, and it has been successfully used to

predict output growth in the US. and other countries. This is the finding of Stock and

Watson (1989) and Estrella and Hardouvelis (1991) for the U.S., Cozier and Tkacz

(1994) and Harvey (1997) for Canada and Harvey (1991) for the G-7 countries. in most

cases these authors fmd that the yield spread leads output growth by about eighteen

months, and that it outperforms a number of competing indicators such as monetary

aggregates and stock indexes. In a parallel strand of the literatwe, Estrel!a and Mishkin

(1 998) and Atta-Mensah and Tkacz (1 998) find that the yield spread is the best predictor

of recessions in the U S . and Canada, respectively. Thus, this is yet further evidence that

the yield spread is a usefiil indicator.

One hypothesis ofien suggested to explain the value of the yield spread as a

leading indicator for aggregate output is that it may be serving as a current indicator of

the direction of the effect of monetary policy on the aggregate economy. This is the view

of, for example, Laurent (1988). The argument in favolur of the hypothesis holds that the

central bank exerts a considerable amount of control over short-term rates, but is only

capable of influencing long-term rates insofar as it c m influence inflation e~~ectations.'

As a result, a tightening of monetary policy results in higher short-term rates, causing the

gap between short and long rates to narrow. As mentioned above, authors such as Estrella

and Hardouvelis (1991) find that the yield spread leads output growth by about eighteen

months, which is consistent with the effectiveness lag of monetary policy implied by the

interest rate channel of the monetary policy transmission mechanism. Blinder (1 996): for

esarnple, notes that the time lapse occurring between a monetary policy action and

changes to actual output is roughly in the range of one and a half to two years.

' For recent empirical evidence on the Fed's weak control of long rares, see Mehra ( 1 996); Akhtar (1995) offers a comprehensive survey of the literature.

The yield spread has advantages over the use of changes in a short rate alone,

since the effects of extemal shocks can be dampened, leading to a better indicator of

deliberate policy actions. For example, the oil shocks of 1973 and 1979 affected the

yields of US. securities at al1 maturities, represented by an upward shift of the entire

yield curve. Although the spreads between long and short rates widened, the effects of the

shocks on the spread were mild relative to changes in the federal funds rate alone. If

therefore the monetary authority has significant influence in the determination of short-

tenn rates through the control of bank reserves, and little control over long-tenn rates,

then the spread should be a good indicator of policy; the collection of papers by the

centra1 bankers of several industrial countries in BIS (1995) suggests that such a pattern

of control over interest rates does indeed hold for rnost of the G-7 countries. A notable

exception is Japan where, as Tatewaki (1 99 1) notes, financial markets were heavily

regulated until the early 1980s. In particular, rates on long-tenn government bonds were

set by the issuer until 1978, when a public tender formula was finally adopted. Thus, we

might expect the yietd spread to be a less-than-ideal indicator of monetary policy for

Japan, at least pior to 1978. With this possible exception, another desirable feature of the

spread is the ease with which comparable indicators of monetary policy cari be

constructed across countries.

For al1 of these reasons, the spread is a useful and interesting indicator. We now

consider whether, in parallel with the literatures involving other measured aspects of the

monetary transmission process, we can detect asyrnrnetry in the effects of this variable.

To the extent that there is such evidence, we look for the diminished effectiveness at

strongly expansionary values which has been detected in measures of money supply

shocks, particularly in the U.S.

3. Data and linear models

For each country we assemble a data set containhg t h e senes of real output

(gross domestic product), real govemment expenditure, short- and long-term interest

rates. Precise variable definitions and data sources are described in Appendix 1. Al1

senes except interest rates are transformed to logarithms. For countries other than the

U.S., we also consider U.S. GDP as a potential explanatory factor for local GDP. In order

to constnict a consistent Germa. data series, we use data applying to the pre-unification

borders of West Gennany through 1994, the last year for which we could obtain data

reported on this basis.

The yield spread is simply the difference between the long- and short-term

interest rates at any point in time. However, because GDP data are only available at

quarterly frequency, we must construct a quarterly spread variable from higher-frequency

interest rate data. As well, GDP measurements reflect developments over the entire

quarter rather than pwely at a point in time; we therefore use a quarterly average for the

spread rather than a point-in-time value. The current quarterly-average spread is defined

to be the mean of the present month's spread and two monttily lags; the previous quater's

spread is the mean of the monthly spreads in months three to five preceding the last

month of the current quater, and so on.

The tests used in Section 4 require stationary regressors. Unit root tests to

confirrn this were performed, with the resuIts presented in Table 1. To achieve

stationarity we use the fint difference of both real GDP and governrnent expenditures,

while the spread variable is defined as a three-quarter moving average of the difference

between long- and short-term interest rates. For each of the seven countries, therefore, we

begin by speciQing a mode1 of the changes in the logarithm of real output. The models fit

the forrn

P k

Ay, = an, + x a, AY,-< [+ 2 pl Ay:] + ~.&.-. + 6% + &, I=I 150 <=O

where y, is the logarithrn of real GDP, gr is the logarithm of real govenunent

expenditure, and w, is a function of the spread tu be defined below.

Table 2 summarizes the results of fitting mode1 (1) for each of the seven

countries. These models reflect separate specification searches to select lag lengths for

each country, and a base-case definition of ta,., as a three-quarter moving average of the

quarterly spread defined above. The dating of this variable at 1-1 reflects the fact that the

current-quarter spread is not incorporated into any definition of the spread variable used

below, consistent with its possible use as a leading indicator.

The results indicate that 'the spread variable is statistically significant' for al1

countries, ~6th the exception of Japan. This may not be surprising considering that the

earIier discussion of Japanese interest rates, as long-term governrnent bond rates were set

by the issuer until 1978. For the remaining countries we note that the spread is most

significant in the United States and Canada.

4. Non-linear Analysis

The aim of this section is to carry out tests for asymmetry, and to estirnate models

that may be able to capture the non-linear relationships in the data. We begin with a

graphical analysis of the residuais of the linear models as we had done in the previous

chapter. If the kernel densities of these residuals prove to be multi-modal, then this may

be evidence of residual clustering and that non-linear models may prove to be useful

alternatives. We then present general tests for non-linearities in the relationship. Such

tests serve as usefiil diagnostic tools. We follow this with tests of statistical significance

of the threshold variable 6,-, , without specifying a priori the value of the threshold.

Doing so offers two potential advantages. First, the process of searching across a set of

threshold values offers the best chance of finding any threshold effect which does exist:

we can compute the point around which any asymmetry is maximized. Most existing tests

take zero as the threshold; it is possible, however, that a distinction will be more readily

apparent between values above and beIow some other point. Of course, such a procedue

i nvalidates standard inference.

Finaily, we estirnate neural nenÿùrk models with the aim of captunng any non-

Iinearities in the relationship between output growth, the spread and the growth of

govenunent spending. If there is no significant threshold effect between the yield spread

and output growth, and yet there is still evidence of some non-linearities, then the

(unhown) non-linear relationship may be captured by the neural network models.

4.1 Kernel Densities

As stressed in the previous chapter, examining the distributions of residuals from

the linear models can prove to be enlightening in detecting potential residual-clustering, a

clue that Tong (1990) notes rnay be evidence of non-linearities. The densities plotted for

each country use the residuals obtained after estimating (1). The window width (h) for

each country is the smallest window width for which we have bimodality. The larger the

window width, the smoother the density- Visudly it appears that very few of these are

normaliy-distributed, with the exception of France. However, note the range of residuds

for this country: They begin around -0.09 and end at -0.075. Compare this to Canada's

residuals which are much sinaller, ranging from 4.024 to 0.024.

To ascertain statistically whether the bimodality of these densities is significant,

we resort to the bootstrap multimodality test presented in Efron and Tibshirani (1993),

which was also irnplemented in the previous chapter. In Table 3 we present the p-values

for testing the hypotheses of one mode versus more than one mode, and two modes

versus more than two modes. With the exception of France, the nul1 of one mode is

clearly not rejected, as p-values are al1 greater than 0.9. France has a p-value of only

0.297; aithough tbis e s not correspond to usual confidence levels foi hypothesis tests, it

is interesting to note that the p-value for France is clearly an outlier when compared to

those of the other countries. When testing the nul1 of two modes, the results are more

consistent across ail countries, as p-values are now al1 over 0.88. Thus, we conclude from

this exercise that bimodality in the linear model residuals, and thus non-linearities, are

most likely to be uncovered for France.

4.2 Geneal Non-linean'fy Tests

The general non-linearity tests considered are the KeenanRESET tests, the Tsay

(1 986) test, and a Logistic Smooth-Transition Autoregressive (LSTAR) test, which is

presented in Granger and Teravirta (1993). These tests can be considered general

diagnostic tools for detecting non-linearities in the basic relationship. 'The

KeenadRESET test was discussed in the previous chapter, and in the present context it is

implemented by estimating the model

where the dependent variable is the residuals fiom the linear model (l) , X is a vector of

explanatory variables specific to each country, and the mode1 is augmented by powers of

the fined values from the linear model. The test is implemented by testing the

significance of the parameters ry,; if they are, then we reject the nul1 of linearity.

n i e ~ s a g t e s t also regresseç the linear mode1 residuals a d the original regressors,

but augments the mode1 with al1 cross-products and squares of the regressors in order to

capture potential negiected non-linearities. The significance of these additional regressors

is then jointly tested, We therefore have

For exarnple, in the case of Canada we have five original regressors, making X a five-

variable vector, The cross-products and squares of these regressors therefore yield 15

addi t ional regressors-

The LSTAR test specifies a smooth transition model under the alternative

hypothesis, with the logistic function being used to specify the smooth regime changes.

The regime changes themseIves are triggered by lagged values of the dependent variable.

Since the "trigger lag", or delay pararneter, is not known, the test can be attempted with

different values of the deiay parameter d. The estimated model is

k 3

Ay, = xp+ C ~ w , x . ~ ~ L d +ut

Again. X is a vector of dimension k, with k being the number of original explanatory

variables specific to each country; d is the lag of the dependent variable that triggers the

regime changes (we present results for d = 1 and 2). The additional regressors here are

the origi&l variables rnultiplied by powers of the laggod dependent variable, with the

lags specified by the delay parameter. For instance, for Canada we have k = 5 explanatory

variables. Since we use three different powers of the lagged dependent variable, the test

therefore involves testing the significance of 5 x 3 = 15 additional regressors. Denvations

of the Tsay and LSTAR statistic can be found in Granger and Terkvirta (1 993) or Tkacz

( 1 997).

The results for implementing these general diagnostics on the linear models of al1

G-7 countries are presented in Table 4. We can immediately note that the nul1 of linearity

is not rejected in any of the tests for Canada and Germany. At the other extreme* the nul1

of Iinearity is rejected using al1 the tests for France. For the four remaining countries the

evidence is somewhat mixed, as some tests detect non-linearities while others do not.

This is in fact the purpose of implementing several non-linearity tests: Some tests have

stronger power against specific forms of non-linearity, and therefore we should not

espect that al1 tests would be able to detect non-linearities when they exist.

4.3 Threshold Tests

In this section we implement a test that can detect asymmetries directly between

the yield spread and output growth. The previous section implemented general diagnostic

tests for non-linearities on the whole model; the non-linearities rnay or may not be caused

by the relationship between the yield spread and output growth.

The threshold test again begins with the linear model (1) specified under the nul1

hypothesis. It requises that we defne an additional threshold effect, that is,

- o,-, = w,-,l[tq-, 2 r ] , where s is the threshold parameter and where the indicator

O, for O+, > r 1, for a,-, S r

Models based on (1) will then be estimated which include as regressors not only the

original spread variable ut-, but also the threshold variable t%,-, , with a coefficient that we

ivill cal1 q. If the yield spread variable has a linear effect on output, then v = 0.

In the current context we do not specim the threshold variable a priori. Doing so

offers two potential advantages. First, the process of searching across a set of threshold

values offers the best chance of finding any threshold effect which does exist: we can

cornpute the point around which any asymmetry is maximized. Most existing tests take

zero as the threshold; it is possible, however, that a distinction will be more readily

apparent between values above and below some other point. Of course, such a procedure

invalidates standard inference.

Fortunately, there are tests whose distributions can be computed given a threshold

value chosen, in effect, to maximize the likelihood fiuiction of the model. Standard tests,

by contrat, treat the value of the threshold parameter as known. It is of course true that a

standard test (e.g., a t-test) of the nul1 of symmetry, treating the value of the threshold

parameter as known, is valid when that value is selected independently of the data.

However, one might doubt that such independent selections could be made in data sets as

widely explored as those involving output and monetary variables in major industrial

countries. A second advantage of the type of test used here is that, by allowing explicitly

for maximization of the likelihood over the threshold parameter, we deal with the implied

potential for pre-test b i s . This probtem can imply a large departure fiom the nominal

distribution; see Galbraith (1 996) for a simulation example.

The test procedure that we use, which explicitly accounts for the fact that the

threshold parameter is not identified under the nul1 hypothesis, is that of Hansen (1996).

Hansen's procedure allows us to simulate the limiting distribution of suprernum statistics

which ernerge from rnaximization over the values of the threshold parameter.3 From the

simulation results, the asymptotic distribution can be estimated, and p-values obtained.

Simulations in Hansen (1996) suggest good size and power performance of the tests

based on either the supremum or the average of LM and Wald statistics. Here we report

the maximum of the sequence of LM statistics. The threshold parameter is aIIowed to

take, in turn, each value in the sample range of the spread variable; folIowing Andrews

(1 999, this set of values is then trimrned by 15% of the sample at each end before

computing the sup(LM) statistic. The p-values from the asymptotic distributions are then

obtained from 1000 replications of the simulation procedure.

' Andrews and PIoberger (1995) prove asymptotic admissibility of tests based on the sup(LR). and asymptotic equivalents such as the sup(LM), for models of this type.

Table 5 gives these asymptotic p-values, corresponding to the nul1 of no threshold

effect, for each of the seven countries and for four possible specifications of the threshold

variable. The frrst three cases correspond to the specification

where q = 2,3,4 respectively. The last case does not use an average of this type, but

instead takes a single quarterly lag of the spread, defrning w,, = s,+ The particular lag is

chosen to maximize the likelihood function. The estirnated models described in Table 2

correspond to the first case, and apart from the varying definition of wi, the Table 5

results al1 use the models given in Table 2.

Sensitivity to other changes in specification was found to be small relative to

sensitivity to specification of the spread variable; hence the varying choices of the latter

reported in Table 5. In performing this sensitivity analysis with respect to functions of the

spread, Our aim is to conserve test power by concentrating the predictive content in a

single variable, and also to avoid dependence on a particular weightinç of lagged spreads

in computing test statistics. The results are representative of those obtained with similar

weightings on lagged values of the spread.

There are several noteworthy points. First, at the conventional level of 0.05 or

5%, there is no statistically signifiant evidence of threshold effects outside the U.S. In

the Japanese case, however, we must bear in mind (frorn Table 2 and similar results) that

the yield spread was insignificant in each case. Our inability tc detect a departure from

the linear ef3ect may be attributable to the irrelevance of the yield spread as an indicator

there, even in a linear form.' For the U.S. the threshold is significant at a level of less

than 5% on al1 but one specification, and at 7% on that one.

Second, the results are not greatly sensitive to the choice of specification. While

some variation in the p-values does occur across models, no values for countries other

than the U.S. are significant at a Ievel of 5%.

In order to explore these results M e r , and in particular to examine the points at

which maximal LM statistics occur, it is necessary to consideï the full sets of LM

statistics for each country. Figure 2 therefore record these sequences for each of the seven

countries, and for the base specification (three-quarter moving average of the spread)

recorded in Table 2. Note fiom the U.S. sequence of statistics the significant value occurs

near the upper extreme of the range of spreads; this is true of the U.S. statistics for al1

four specifications, although we do not record the sequences for all four. No other

threshold points corne close to generating the values produced by the large positive

spreads. To the extent that any deviation from a linear effect is indicated, therefore, it

implies a threshold relevant oniy beyond very large positive values of the spread.

Examining the other figures, it is clear that the maximal statistics do tend to occur

at fairly large positive values of the threshold, but, again, none of these is significant.

Recall from Section 2 that possible reasons for this rnay be found in Tatewaki (1991).

88

Overall, evidence of asymmetry is very weak for al1 countries other than the U.S., and

quite strong for the U.S.

Since the threshold tenn Gr-, is significant in the US., it is interesting to compare

estimates of the base model, augmented by that term, with the estimates from Table 2.

The augmented model of the U S data is estimated as

Only small changes in other coefficients appear, but the coefficient on the spread

falls from 0.002 to 0.001, with the threshold variable taking a coefficient of 0.002. Given

the definition of the threshold in (S), this implies a total effect of the spread variable of

0.003 for spreads below the threshold, and 0.001 above. This result can be interpreted as

follows. Viewed as a leading indicator, a higher spread always suggests higher output

growth, but the proportionate impact that the spread should have on a forecast is reduced

for Iarge spread values; additional declines in short-term rates relative to long-term rates

have diminished marginal impacts once the threshold level is crossed. To the extent that

the link is viewed as causal because the spread represents monetary policy, the effect of

strongly expansionary yield spreads is proporrionately lower than that of more moderate

values, but the marginal impact of more aggressively expansionary policy is always

positive. The estimated threshold parameter (the value of the yield spread at which the

maximal statistic occurs for the U.S.) is 2.214.

4.4 Neural Network Models

In the previous section we have found that threshold effects between the yield

spread and output growth are only significant for the U.S. However. we detected potential

non-linearities for four other countries, namely France, Italy, Japan and the U.K. If the

non-linearities do not originate in the spread-output relationship, then non-linearities

must be present between other variable combinations. In this section we propose

estirnating neural network models in order to attempt capturing as much of the non-

linearities as possible. These models can then be used by policy-makers for forecasting

purposes, and as tools to better explain the movements of output gowth.'

For any given country we have K explanatory variables. In the terminology of

neural nets these represent K "inputs". These are linked to output growth (the "output") in

a non-linear marner, but first pass through a hidden layer of J (unobserved) variables,

giving us J "hidden units". For example, the link between the spread variable and output

growth is most certaidy not direct, as changes in the yield spread must first affect

intermediate variables, such as consumption and investment, pnor to influencing output

growth. The mode1 is specified as

- -

' The exposition that follows is inspired by Campbell, Lo and MacKinlay (1997).

90

The parameters are interpreted as follows: Pb is the "co~ec t ion strength", or

weight, behveen input k and hidden unit j. The larger (in absolute terms) the value of this

parameter, the greater is the impact of a change in input k on the hidden unit j. g(-) is the

'.activation fùnction", and in our case this is represented by the logistic function

g(u) = 1/(1+ e '" ) . We have previously encountered this fùnction in our discussion of the

Tsay and LSTAR tests, as it ailows for smooth (as opposed to discrete) threshold effects

(or regime changes). This function serves the exact same purpose within the context of

the neural net model, as it dictates whether node k is activated.

The pararneters a, represent the weights on each hidden unit frorn hidden unit j to

the output. Again, these give us the relative strength of each (unobserved) hidden unit -

the higher the absolute value of a,, the larger the relative impact of a change in hidden

unit j on the output.

To estimate (8) we choose the a, and Pt, so as to minimize the surn of squared

deviations between the output and the network:

This is achieved using backpropagation, as different values of a, and & are tried until we

obtain a pre-specified level of convergence (i.e. we "train" the network). Several

thousand iterations (or "epochs") are required to achieve a reasonable convergence level.

Details on the convergence levels and epochs for each country are given in Table 6.

In models of this type some experimentation is necessarily required, since we do

not know a priori how many hidden units are required. In our final andysis we used two

hidden units for al1 countries, except for the U S . for which we used three. The choices

ivere motivated by the speeds of convergence to the desired significance levels. The

significance levels themselves were chosen again through some experimentation. We

typically find that in the training process the mean square errors fa11 rapidly afier the first

few thousand epochs, but then tend to level out at some significance Ievel, at which point

even after several hundreds of thousands of epochs the mean square error changes only

marginalIy, indicating that the marginal benefits of M e r training is minimal. Thus, the

pre-specified convergence criteria were chosen such that they are the levels at which

further training is only marginally beneficial.

The parameters for al1 countries are presented in Table 6 . The ordering of the

parameters reflect the ordering of the variables in each country's respective linear model,

given in Table 2. As mentioned, the alphas reflect the weight fiom the input variable to C

the hidden units; the betas are the weights from the hidden units to the output. The a,~ are

the bias weights on hidden node j; Po is the bias weight on the output node. Although

difficult to interpret on their own, it is worth comparing some of the parameters across

countries. Of special interest are the parameters on the spread variable, ut-,. From Table 2

we recall that the spread parameter was the 1s t variable to enter each model, thus its

parameters are given by the last ajk for each country. The parameters on the spread are

almost universally negative, and they tend to be quite small for France, Gennany, Italy

and Japan, when compared to the larger yield spread parameters obtained for Canada, the

United Kingdom and the United States. This indicates that changes in the spread variable

should have a greater impact on economic growth for these countries.

Also of note are the simiiarities in the signs for the parameters Pl and pz. These

give us the weights fiom the first and second hidden units to the ouput growth variable.

For al1 countries for which we use two hidden nodes we find that B i is negative, while pz

is positive. This indicates that increases in the value of the first (unknown) factor will

have a negative impact on output growth, while increases in the value of the second

(unknown) factor will have a positive impact on output growth. It is interesting that the

impacts of these factors should impact output growth in the same direction for so many of

these countries. This implies that changes in our explanatory variables (the yield spread,

govenunent spending, U.S. output growth and lagged output growth) should follow a

similar transmission mechanism when they eventually affect current output growth.

4.5 Non-linearity Tests on the Non-linear Models

Having tested for non-linearities and constructed non-linear models, it is of

interest to determine whether these models have adequately captured the non-Iinearities

in the data. In Table 7 we present the results of Keenan and RESET tests on the neural

net model for each G-7 country and the threshold model for the U.S. Contrary to the tests

on the linear models in Table 4, we see that the nul1 of linearity almost universally cannot

be rejected when we use the fitted values from the non-linear models in our tests. This

includes the threshold mode1 for the U-S ., implying that the inclusion of a threshold effect

captures much of the non-linearity for that country.

The clear exception is France. where we consistently favor the non-linear

alternative. This implies that the neural net mode1 for France does not adequately capture

the non-linearities in the data. Suspecting that the non-linearities may be due to a non-

linear relationship between government spendinç and output growth, we repeat the

Hansen threshold test of Section 4.3. Afier 5,000 replications we obtained a p-value of

0.258. not sufficiently low to reject the linearity nul1 at the usual significance levels. In

lighr of this evidence. we can simply state that more extensive rvork is required to fully

esplain output growth in France.

5. Conclusion

The purpose of this chapter is to detect possible non-linearities between the yield

spread and output grouth. We find that the use of some transformation of the spread

between long-term and short-tsrm interest rates is successful in predicting changes in

output in al1 G-7 countries except Japan. Other evidence has also suggested that such

information generally provides a good measure of the direction of monetary policy.

Ho\{-ever. using this indicator on G-7 data produces little evidence of asymmetric effects

of the yield spread. escept in the US.

The result suggesting asymmetry in the U.S. is consistent with a good deal of

previous literature conceming the impacts of money supply shocks in the U.S., aithough

the procedure here offers the advantage of providing a test valid with data-based selection

of the point around which asymmetry is observed. In the U S , the evidence points to

asymmetry around a large positive value of the spread, but not around values near zero.

That is, the linearity of the effect of the yield spread on output, or on appropriate

forecasts of output, appears to break down for large (strongly expansionary) values of the

spread. This is also consistent with the results on U.S. data fiom similar inferential

procedures in Galbraith (1996), where money supply measures were used in examining

credit rationing models.

Since there is little evidence of asymmetry in the form of a threshold effect

between the yield spread and output growth in the other G-7 countries, it is interesting to

ask why the U S . appears to differ. Possible explanations might lie in the world-currency

role of the US. d o l ~ a r , ~ and in the relatively low degree of dependence on foreign trade of

the US. economy. In many countries, monetary policy operates through both interest

rate and exchange rate channels; expansionary policy that lowers interest rates also tends

to cause a depreciation of the domestic currency, thereby expanding domestic output. In

the U.S., because of both the relatively Iow importance of trade, and the fact that many

comrnodities are priced in US. dollars, the exchange rate channel might assume less

importance. It is possible that outside the U.S., exchange-rate effects can mitigate a

diminished effectiveness of monetary policy through the interest-rate channel when

spreads become very high (policy attempts to be very expansionary); we may conjecture

that this exchange-rate mechanism is of little importance in the U.S., so that the

diminished effectiveness at high spreads does appear through a threshold efTect.

Of course, it is dways possible that the negative results outside the U.S. rnerely

reflect our inability to detect the effect (a lack of test power), but our use of a test which

selects the optimal threshold point for the asymmetry conjecture, together with the fact

that the same procedure produces strong results in the US., suggests that genuine

differences may be present.

With several tests revealing that non-linearities are present in the data, but that

they do not seem to onginate fiom the relationship between the yield spread and output

growth, we then constmcted neural network models in an attempt to capture these non-

Iinearities. With the exception of France we were successfûl in achieving this, and noting

that the general link between the explanatory variables and output growth appears to be

quite similar across countries. In the case of F m c e we suspect that our models omit

variables unique to France's macroeconomic experience, and this may explain why we

are unable to capture the apparent non-linearities in the data.

We conclude that the evidence for asymmetry of yield spreads on predicted output

is strong in the US., and weak outside, and that firther research is necessary in order to

understand the reasons for such differences to exist.

We thank J.-P. Aubry for suggesting this explanation.

Appendix 1: Data

The definitions of the variables used in the paper are given below. Al1 variables, with the exception of interest rates, are trmsformed to natural logarithms. National accounts and price data are seasonally adjusted. AI1 data used are quarterly; interest rates and consumer prices were converted to quarterly observations by averaging the monthly observations within each quater. Data were obtained fiom the International Monetary Fund's International Financial Statistics, the OECD's Main Economic Indicators, the FRED database at the Federal Reserve Bank of St. Louis, Statistics Canada's CANSIM database and the Bank of Canada.

1. Canada (1954:3 to 1995:4)

Short-terrn interest rate: Ovemight rate (Bank of Canada: Amour et al., 1996). Long-tenn interest rate: Average yield of IO-year and over Govemment of Canada marketable bonds (CANSIM 8 140 13)

Nominal GDP (CANSIM D20000) Price level: GDP deflator, 1986=100 (CANSIM D20556) Government spending: Nominal govemment consurnption and invesment (CANSIM D20181 + D20192)

2. France (1 965: 1 to 1995:4)

Short-term interest rate: Monthly average of rates for day-to-day loans against private bills (IMF I32F6OBZ) Long-tem interest rate: Average yield to redemption of public sector bonds with an original maturity of more than five years (IMF 132F6 12) Output: Nominal GDP (IMF 132F99CBZ) Price level: Consumer prices, 1990-100 (IMF l32F64Z) Government spending: Nominal govemment consumption (IMF 132F9 1 CFZ)

3. (West) Germany (1960: 1 to 1994:4)

Short-term interest rate: Monthly average of ten daily average quotations for day-to- day money (IMF 134F60BZ) Long-term interest rate: Weighted average of al1 public sector bonds with an average remaining life to maturity of more than three years, or four years before January 1977 (IMF 1 MF6 12) Output: Nominal GNP (Bank of Canada) Pnce level: Consumer prices, 199 1=100 (Bank of Canada) Government spending: Nominal government consumption (Bank of Canada)

4. Italy (1971 :1 to 19953)

Short-term interest rate: Three-month interbank rate ( M F 136F60BZ) Long-tenn interest rate: Average govenunent bond yield, 9-10 years (1 5-20 years pnor to Apnl 1991) (MF 136F6lZ) Output: Nominal GDP ( M F 136F99CBZ)

e Pnce level: Consumer pnces, 1 !WO=lOO (IMF 136F642) Government spending: Nominal government consurnption (IMF 136F9 1 CFZ)

5. Japan (1 966:3 to 19953)

0 Short-term interest rate: Lending rate for collateral and overnight loans in the Tokyo Cal1 Money Market; pnor to November 1990, lending rate for collateral and unconditional loans ( M F 1 S8F6OBZ) Long-tenn interest rate: 10-year central govemment bond rate (OECD S0050562OOOAH)

a Output: Nominal GDP (OECD S005000100002) a Price level: GDP deflator, 1 WO=lOO (OECD S 1 OSOOO43OO9.J) a Government spending: Nominal government expenditures (OECD S 10500 12000 12)

6. United Kingdom (1 96 1 : 1 to 1 9954)

Short-terni interest rate: 3-month Treasury Bill rate (IMF 1 12F60BZ) Long-term interest rate: Theoretical g r o s redemption bond yields. Issue at par with 20 years to maturity (IMF 1 12F6 12) Output: Nominal GDP (OECD S l28OOO~OOOOS) Price level: GDP deflator, 1 99O=lOO (OECD S l28OOO43009.J) Government spending: Nominal government consumption (OECD S 12800 12 10072)

7. United States (1 954:3 to 1995:4)

Short-terrn interest rate: Federal Funds Rate (Federal Reserve Bank of St. Louis) Long-term interest rate: 10-year government bond rate (Federal Reserve Bank of St. Louis) Output: Nominal GDP (Bank of Canada) Price level: GDP deflator, 1987=100 (Bank of Canada) Govement spending: Nominal govement consumption and gross investrnent (Bank of Canada)

Appendix 2: Tables and Figures

Table 1: Unit Root Tests

Country ADF (k) Phillips-Perron (4 Lags) AY~ A8r or AY~ Agr @f

Canada -7.142 (1) -1 1.68 (1) -3.502 (9) -9.91 7 -17.51 -3 -369

France -8.795 (1) -7.237 (1) -2.868 (8) -1 5.43 -1 1.30 -3.598

Italy -5.138 (1) -3.634 (1) -1.141 (12) -7.161 -4.3 19 -2.868

U.K. -7.927 (1) -3.849 (2) -2.259 (1 O) -1 2.24 -10.35 -2.460

U.S. -6.904 (1) -9.298 (1) -2.936 (12) -9.087 -1 1.41 -3 -260 The number of lags (k) used for the ADF tests are given in parentheses, and were selected by minimizing the Schwarz Criterion; the lags for the Phillips-Perron correspond to the fiequency of the data. The critical values for rejection of the unit root hypothesis are -2.57 at the 10% level; -2.88 at the 5% level and -3.46 at the 1% level.

Table 2: Linear Models

Coefficient Canada France Gennany Italy Japan UK US

Constant

AYI- 1

AYI-2

AY,-3

A-v,"'

43

QI- l

Y- 1

Obs. d. f. R'

- - .v - -A

-,, . 3 1.3 9.76 16.3 34.4 12.4 r-statistics are in parentheses. 2.v is the Bowman-Shenton test for non-nonnality, distributed as zA is an LM test for fourth-urder autocorrelation and is disuibuted as &4), while z~ is an LM test for heteroskedasticity, distributed as x2(0bs - d.f.). A (*) denotes significance at the 5% level.

Table 3: Bootstrap Multimodality Tests

Country Ho: # of Modes = 1 Ho: # of Modes = 2 HI: # of Modes > 1 HI: # of Modes > 2 * h p-vaiue h p-value

Canada 0.0020 1 .O000 0.00 18 0.9993 France 0.02 13 0.2970 0.0086 0.9687

Germany 0.0083 0.9 170 0.0046 0.9930 ItaIy 0.0037 9.9967 0.003 5 0.8837

Japan 0.0032 1 .O000 0.0029 0.9930 UK 0.0050 0.9387 0.0028 0.9967 US 0.0026 1 .O000 0.0022 0.9960 -

h is the srnailest window width for which the number of modes under the nul1 hypothesis is achieved. The p-value is the probability of not rejecting the nul1 hypothesis. 3,000 bootstrap replications were used for each test.

Table 4: General Non-linearity Tests

country Obs. (7) Keenan RESET RESET ';"Y LSTAR 1 LSTAR2 F( 1.7'-3) F(2, T-4) F(3 ,T-5) x (r) F(dfl ,do) F(df1 ,dQ)

Canada 162 0.274 0.877 0.586 15.55 (15)

France 120 33.8 1 ** 16.78** 13.45** 47.18** (6)

Germany 136 0.003 0.092 O. 134 6.337 (6)

Italy 95 0.089 0.8 1 1 5.170** 4 1.99** (6)

Japan 113 1.134 0,577 3.035* 26.98* (5 )

UK 136 7275** 3.684* 2.703* 19-05 (10)

US 162 1 .O09 0.566 1.389 14.76* (6) (9,149) (9,149)

A (*) denotes significance at the 5% Ievel; (**) at the 1% IeveI. Degrers of freedom are given in parentheses.

Table 5: Hansen Threshold Test

Country Obs. (7) Estiamted p-val ues Threshold 3-Q MA 2-Q MA 4-4 M A Lag

Canada 162 0.869 0.09 O. 12 0.24 0.26

France 120 0.773 0.62 0.48 0.7 1 0.65

Germany 136 3.167 0.32 0.54 0.30 0.30

Italy 95 1.333 0.67 0.66 0.34 0.08

The p-values for different specifications of the spread variable are reported. The base case is a three-quaner moving average of the spread Lag is the case in which a single lag of the level of the spread is used. The Esrirnated Threshold is the value of the spread in the base case for which the maximal LM statistic is achieved. The higher the p-value. the smaller the probability of not rejecting the nuIl of linearity.

Table 6: Neural Network Models

Parameter Canada France Germany Italy Japan UK US

MSE x 10" 8.800 For each country we estimate neural network mode1 (8) with two hidden units, with the exception of the U.S. for which we use three hidden units. We present the estimated parameters, the number of epochs (passes rhrough the data) required to achieve the desired convergence criterion, given by the Mean Square Error (MSE). The ordering of the parameters reflect the ordering of the explanatory variables in Table 2.

Table 7: Non-linearity Tests on the Non-linear Models

Country Obs. (T ) Keenan RESET RESET ( 1 3 ) F(2, T-4) F(3, T-5)

Canada 162 1227 1.508 1 .O8 1

France 120 28.42** 14.69** 9.769**

Gennany 136 0.005 0.254 O. 169

1 taly 95 0.087 0.665 0.682

US (Thre.) 162 1.907 1.260 1.195 For each country we analyze the residuals of the neural network models, with the exception of the U.S., where in addition we analne the residuals of a threshold model. A (**) denotes significance at the 1% level. Degrees of fieedorn are given in parentheses.

Figure 1: Kernel Densities for Liaear Mode1 Residuals

Figure 2: Sequence of LM Statistics for Tbreshold Test

ESSAY 3:

THE FEDERAL W S RATE, COMMERCIAL RATES, AND THRESHOLD E ~ C T S ALONG THE INTEREST RATE CHANNEL

1. Introduction

In the Federal Reserve Act, the United States Congress has directed the Federal

Reserve to promote "maximum employment, stable prices, and moderate long-term

interest rates", with the third goal k i n g somewhat redundant since it is usually achieved

when price stability is achieved (Blinder (19965)). In the implementation of monetary

policy, the Fed uses its abiiity to control money market interest rates to achieve these

goals. Former Vice Chairman of the Fed, Alan Blinder. espouses the view that changes in

inflation are largely determined by the differences between actual output (over which the

Fed bas control) and potential output (over which the Fed has no control). The

transmission mechanism of monetary policy can thus be descnbed as follows: The Fed

affects interest rates; interest rates affect spending; spending affects actual output; actual

output determines the output gap; and the size of the output gap (positive or negative)

thrn determines changes in prices. Blinder (1 996:6) believes that, given this transmission

mechanism. the effects of monetary policy actions on inflation take about two years.

The rate over which the Fed has most control is the discount rate, which is the rate

charged to commercial banks on short-term loans by the Fed. Other money market rates,

such as Treasury bill rates and the Federal Funds rate, change in response to the Fed's

actions in short order. The reason for the rapid changes to money market rates rests in the

fact that thousands of money market dealers around the world immediately respond to

changes in the scarcity (or abundance) of securities at various maturities. However, for

Fed actions to have any impact on actual spending in the economy, the rates charged on

loans for consurnption and investment purposes must also change in response to the Fed's

actions. Most such loans are tied to the Frime Rate, which is defined as the base rare on

corporate Zoans posred by ar Ieasr 75% of the nation's 30 Iargesl banks (Wall Street

Journal).

A number of recent studies (cg. Cover (1992), Morgan (19931, Rhee and Rich

(1995)) arrive at the conclusion that the effects of monetary policy on output are

asymmeuic; that is, they differ depending on whether monetary policy is contractionaty

or expansionary. A marner in which such asyrnmetries c m arise- stems fiom the link

between market rates and "managed rates", that is, rates that are set by financial

institutions. Intuitively, a contractionary policy would cause money market rates to rise,

prornpting banks to increase the rates charged on commercial loans. Failure to raise such

rates would imply that the difference between what banks are paying to borrow funds and

what is being charged on loans to customers is narrowing, thereby reducing profit

margins. On the other hand, an expansionary policy would lower market rates; should

managed rates not quickly respond, then the profit margins of banks would presurnably be

increasing, and consumption and investment would not rise until such rates fall. Thus, in

this situation an expansionq policy might not have the stimulative impact desired.

The objective of this chapter is to uncover the possible sources of asyrnmetries

dong the interest rate channel of the monetary policy transmission mechanism. In

previous chapters we have uncovered marked asyrnmetries between interest rate variables

over which the Fed had some influence, and ultimate target variables such as inflation

changes and output growth. Here we focus our attention on the responsiveness of

intermediate variables to changes in short-term money market rates, namely cornmerciai

lending rates, consumption and investment. A clearer understanding of the behavior of

these variables to interest rate changes can be beneficial to policy-makers. For non-linear

modeling purposes we rely exclusively on threshold models, since these are especially

convenient at detecting potential floors for rnanaged interest rates.

This chapter is organized in the following manner: in the next section we review

some of the literature on modeling managed interest rates, in particular the prime rate. We

also examine some studies that have attempted to uncover asymrnetries between money

n~arket rates and managed rates, consumption and investment. in Section 3 we analyze

the relationship between money market and managed interest rates using weekIy data

frorn 1971 to 1998. Sections 4 and 5 are respectiveiy devoted to uncovering potential

asprnmetries between interest rates and consumption and investment. Section 6

concludes.

2. Literature Review

In this section we review selected articles that have attempted to mode1 the prime

rate and other managed rates. Papers searching for potential asymmetries between interest

rates and intermediate variables along the transmission mechanism are also reviewed.

2.1 Prime Rate Papers

Goldberg (1982) provides a detailed historicd account of the setting of the prime.

He notes that from the 1930s the prime became a posted rate which did not react to

imbalances in supply and demand. instead, it was altered by means of announcements,

typicaily fiom New York banks. When one bank announced its intentions, other banks

would usually follow. In 1971, a formula for the setting of the prime was adopted by the

banks, making the prime in effect a "floating prime". In his empirical work, the author

models the prime with the current and two monthly iags of the three-month certificate of

deposit (CD) rate, which acts as a proxy for the general condition of the economy and the

marginal costs of the banks' managed liabiIities.

The objective of Goldberg (1984) is to examine if the relationship between the

prime and money market rates of interest has changed over time with the development of

the commercial paper and other direct credit markets as alternative sources of short-terrn

credit for bank customers. His sarnple extends fiom 1972 to 198 1, and he also analyzes

the four 3%-year sub-periods within the sample. He distinguishes between penods where

money market rates are increasing and decreasing through the introduction of dummy

variables. He finds that asymmetries exist in the relationship, which may partly be

esplained by below-prime rate Ioans that some banks offer.

Levine and Loeb (1989) use simple statistical tests to detennine whether

asymmetries around zero exist between the prime and CD rates. They estimate two

regressions: One using data for which rnoney market rates are nsing and one for when

they are falling; tests for equality of parameters in the two regressions are rejected.

Using the Federal funds rate as the proxy market rate, Arak, Englander and Tang

( 1983) find that the prime has moved asymmetrically over an interest rates cycle since the

mid- 1970s. Financial innovations and new forms of borrowing by large corporations are

offered as possible explanations.

Forbes and Wilhite (1991) use back-of-envelope calculations to conciude that the

relationship between the 90-day CD rate and the prime rate between 1983 and 1989 was

symmetric. Indicators used to reach this conclusion include average times from any

change in the prime to a subsequent increase or decrease (increase: 12.07 weeks;

decrease: 9-34 weeks), as well as speeds of adjustment of the prime to increases and

decreases in market rate (the authors find no statistically significant difference).

For Canada, Hendry (1 992) finds that the most important determinant of the prime

rate between 1975 and 1989 is the Bank rate. Other variables of note include short-term

money market rates such as the 90-day Commercial Paper rate, and spreads between

prime and short-term rates. Such spreads are especially usehl at detecting threshold

effects. should they exist.

2.2 Other Asymmetries

In an innovative study, Ausubel (1990) finds that there is an asymmetric response

in several bank interest rates to changes in money market rates. Deposit rates tend to be

more upwardly sticky than downwardly sticky, whereas the converse is true for lending

rates. He finds that the degree of stickiness is related to particular characteristics of each

type of account. In particular, the relative sophistication of the clientele appears to be a

significant factor in explaining rate stickiness, as rates on accounts aimed at fmancially

sophisticated clients appear to be more responsive to changes in money market rates.

Johnson, Montplaisir and Verdier (1 997) examine the responsiveness of durable

consumption and investment to money market rate changes in Canada. They find little

evidence of asymmetries, although they confine their search to an a priori zero threshold

for interest rate changes.

3. Interest Rates

in this section we examine the relationship behveen a representative money

market rate over which the Federai Reserve has a certain amount of influence, the Federal

funds rate. and two commercial interest rates. The first is the prime rate which. as

previously discussed, is a rate that has historically been used as a floor for other

commercial rates. In recent years, however, banks have increasingly been offet-ing loans

at below-prime, thereby making the prime rate a somewhat less usehl gauge of

commercial lending rates. For this reason we also consider another commercial interest

rate. namely a conventional mortgage rate. We would therefore expect changes in the

Fed Funds rate to lead to changes in the prime, which itself would cause the mortgage

rate to change. Hence, this section examines whether such relationships are linear or

asymmetric, using several of the testing and modeling techniques of the previous

chapters.

3.1 Data

Data on the Federal funds rate, prime rate and mortgage rate were obtained fiom

the Federal Reserve Board. The mortgage rate is the current period fixed rate on 30-year

conventional mortgages. We use weekly data in our analysis, beginning the week of

March 3 1, 1971, and ending with the week of March 25, 1998, yielding 1409

observations. The sample length is constrained by the availability of the mortgage rate.

The data are plotted in Figure 1. We notice that the prime rate is almost always

above the Fed funds rate, and that the mortgage rate is almost always above the prime.

Since 1991 we also notice that the spread between the prime and Fed funds rate widened

as the funds rate fell below three per cent, and that the prime rate never fell below six per

cent. As the funds rate began increasing again in 1994, the prime rate quickly followed,

keeping the historically wide spread intact. Meanwhile, the mortgage rate fell below

prime in 1995, and has remained below it almost every week since.

The three unit root tests in Table 1 reveal that there is strong evidence that these

interest rates follow unit root processes. For the Augmented Dickey-Fuller test we see

that the null of a unit root cannot be rejected for the data in levels, but is rejected in first

differences; this finding is the sarne regardless of whether we select the lag length by

minimizing the Schwarz criterion, or if we fix it at 53. The Kwiatkowski er al. (1992)

(WSS) test reverses the null and alternative hypotheses, with the null being stationarity

and the alternative being a unit root. Here we find that the null of stationarity is strongly

rejected for the data in levels, but is not rejected for the data in first-differences,

confirming once again that the data follow unit root processes.

However, the Phillips-Perron tests reach different conclusions, as we find that the

unit root nul1 for the data in levels is rejected- Intuitively, one would not expect interest

rates to contain a unit root, since such a supposition implies that they would have an

infinite variance with the consequence that they could easily have negative values.

Economically, of course, negative nominal interest rates can never occw, thereby ruling

out the unit root story on intuitive grounds. Nevertheless, the unit root story cannot be

ignored, at least over our sarnple, given the ADF and KPSS test results. For this reason,

vie will proceed with our analysis under the view that our interest rates follow unit root

processes, but \vil1 remind the reader that this contentious issue remains unresolved.

3.2 Granger Causality Tests

Up to this point we have proceeded under the assumption that managed

commercial rates follow the lead of money market rates. We now empirically test this

supposition using Granger causality tests. The test involves running the following

regression

and testing the joint significance of the P,. If they are significant, then X is said to cause Y.

However, to state that "X Granger-causes Y', we must aiso test whether Y causes X; if we

reject this hypothesis, then X Granger-causes Y.

In Table 2 we present the results of the Granger-causality tests on both the levels

and first-differences of the data. The lag rn was each set to one, and was found by

minimizing the Schwarz criterion with maximum allowable Iags of m = 24. The results

for the variables in levels indicate that the f h d s rate Granger-causes the prime rate; the

prime rate Granger-causes the mortgage rate; and that the fimds rate Granger-causes the

mortgage rate. This is the direction o f causality we would expect. For the data in first-

differences, however, the results are arnbiguous as the tests are unable to detect the

direction of causality. This could be due to the frequency of the data we are using; when

the funds rate changes in any significant manner, we would expect both the prime and

mortgage rates to change within the same week. As such, these tests are unable to identiQ

the lead rate at this frequency.

3.3 Coin fegration A nalysis

In modeling the relationship between money market and commercial interest

rates. it may be possible to exploit some long-run relationships between these variables in

our models. We have already seen that there is evidence that these rates may be random

walk, or I(l), processes; as such, if they are cointegrated then we can find a linear

combination between the rates that is stationary, or I(0). With the estimated cointegration

vectors we can then constnict error-correction modeis that can be subject to various non-

Iinearity tests.

To verifL whetfier the variabIes are cointegrated, we use the full-information

1 ikelihood methodology of Johansen (1 988, 199 1) to estimate the cointegration vectors.

We begin by specifying a two-variable VAR (vector autoregression) with Gaussian errors,

\vhere X, = [xi, xi],', and xi and xz are either the Fed funds rate, prime rate or mortgage

rate: p is a constant vector; and the errors are i.i.d. Np(O,X), where p is the nurnber of

variables in the system and k is the order of the VAR. In o w applications we have p = 2,

and k = 3, 4 or 5.Writing (2) in error-correction form yields the vector error-correction

mode1

AX, = r ,s ,- , +. . .+I -~-~AX~-~+, + n ~ , - ~ + P + E , .

The hypothesis of cointegration is then

Hi ( r ) : ï ï = ap'

where a and p are p x r matrices, and a test for cointegration is then simply a test on the

rank of p. This hypothesis States that under certain conditions, when AX, is stationary and

X, is non-stationary, there exists a p such that P X r is stationary. In other words, p i s a

cointegration vector. When multiplied by X, we have a linear combination of the

variables which is stationary. Hence, (4) is the hypothesis of cointegration.

To test (4), one can use the trace test devised by Johansen. The trace test for

cointegration against the null of no cointegration is given by

where 2, represents the estimated eigenvalues of P, obtained by solving an eigenvalue

problem required to estimate P from a likelihood function concentrated with respect to

the parameters in (3). The test then involves detemining whether we have r cointegration

\*ectors (i.e. r independent rows or columns in P) versus the null of no cointegration

vectors.

The estirnated cointegration vectors are presented in Table 3.' The orders of the

V.4Rs are found by minimizing a multivariate version of the Schwarz criterion, details of

~vhich can be found in Judge et al. (1985:687). The results indicate that each pair of

variables is significantly cointegrated, and the cointegration vectors themselves have

plausible ~ i ~ n s . ~ To interpret the vectoa, consider the system [Prime, S'cid, Conîtunt].

The est imated vector is

Prime, - 1 -03 8 FedFunds, - 1 -649 = O, or

Prime, = 1 -03 8 FedFunds, + 1.649 .

The vector indicates that there is almost a perfect one-to-one relationship between the Fed

funds rate and the prime rate in the long-run, as given by the Fed fünds coefficient of

1.038, and that, in equilibrium, the prime rate is 1.649% higher than the funds rate- That

the prime is higher than the fimds rate in equilibrium is to be expected, since the

difference between the rates at which the banks lend and borrow should be positive. For

the [Mortgage, FedFunds, Consfunf] system, note that in equilibrim the mortgage rate is

more than 2% higher than the funds rate. Interestingly, for the [Mortgage, Prime,

Conslant1 system, the equilibrium spread between the mortgage rate and the prime rate is

-0.36%. indicating that the mortgage rate is lower than the prime in equilibrium. This

may be due to the fact that the prime rate is sticky downwards as noted when examining

Figure 1 , or that the term-to-maturity of loans made at prime is not directly comparable to

the 30-year mortgage rate that we are using.

In Figure 2 we plot the error-correction terms obtained from the cointegration

vectors. These can be interpreted as deviations from equilibrium, with positive values

' The vectors were estimated using CATS in RATS; see Hansen and Juselius (1995).

depicting periods in which the dependent rate is above its equilibrium level. For instance,

in the first panel we plot the error-correction term for the Prime / Fed fünds relationship.

What is most striking is that the deviation fiom equilibrium has been virtually steady

since 1991, with no signs of the gap closing. Recall fiom Figure 1 that this is the year the

spread between the Fed f h d s rate and the prime widened due to an increasingly low

funds rate and a sticky prime. M e n the fünds rate began to rise again, the prime quickly

followed, thereby maintainhg the wide gap. The gap will only be closed if the h d s rate

rises more quickly than the prime, or if the prime falls more quickly than the h d s rate.

In the last two panels, which depict the deviations fiom equilibrium between the

prime and mortgage rates and the Fed fùnds and mortgage rates respectively, we notice

that both graphs are very similar. We therefore would not expect substantial differences

when we use either the funds rate or the prime rate as an explanatory variable in models

of the mortgage rate.

3.4 Linear Models

In Table 4 we estimate six different linear models: three models have the variables

specified in levels, and three in first-differences. The models in first-differences are in

fact error-correction models, as the error-correction terrns of Figure 2 are included as

explanatory variables. In these models al1 variables are I(O), therefore we have balanced

regressions. For the models in levels we are dealing with two I(1) variables, which c m

potentially lead to spurious regression results. However, since we have found that these

' Note as well that we have inciuded a constant in the cointegration space, since economically we expect an

120

variables are cointegrated, a mode1 incorporating these variables in levels can still be

viewed as an acceptable practice; see Banerjee et al. (1993, Chapter 6) for a more

thorough discussion.

The three regressions using data in levels can be given a partial adjustment

interpretation, as was done by Ausubel (1990). The parameter represents in each case a

speed of adjustment coefficient; the closer it is to one, the quicker is the dependent rate's

adjustment to its target d u e . The farther this value is fiom one, the more sluggish is the

adjustrnent of the dependent rate to its target value.

To estimate these linear models we use a feasible generalized least squares

(henceforth FGLS) estimator that captures some of the senal correlation in the models,

thereby yielding more efficient estimates. To obtain an estimate of the covariance matrix

? we first need to estimate the order of the process by which the OLS residuais are

generated. Once this is achieved, the FGLS estimator can then be constructed as V

p* = (x'?x)-' x'?Y, where X is a T x k matrix of explanatory variables, and Y is a T

x 1 vector for the dependent variable.

To identi@ the ARMA@,q) process generating the least squares residuals we

estimate several different processes for different values of p and q, and choose the one

that minimizes the Schwarz criterion. With the ARMA@,q) process of a series w, being

equilibriurn spread to exist between any pair of interest rates.

represented as a(L)w, = B(L)E, , the AR parameters a(L) and MA parameters B(L) are

obtained using the simple estimator proposed by Galbraith and Zinde-Walsh (1 997). This

estirnator relies on identi%ing the true ARMA parameters from an AR(k) approximation

bj. considering the distance between these two processes in the Hilbert metric. The order

of the finite AR process usrd for identification purposes, k, must at least be greater than

the sum of the number of AR and MA components, p + q, and should increase with the

sample size T. In this section k is set to 21 with T k i n g over 1400; in Sections 4 and 5, k

is set to 6 as we have just over 100 observations. Our initial searches for the most

appropriate ARMA@,q) process considered cases where p +- q was as high as 12: but we

found that the Schwarz criterion was typically minimized for much lower values. For this

réason most of the empirical work that follows is restricted to searches where p i- q = {O.

. . . 5) inclusively.

We see for the three regressions in levels that pl equals, respectively, 0.898,0.967

and 0.962. This implies that the prime rate is more sluggish to funds rate changes than is

the mortgage rate (0.898 versus 0.962).' Funhermore. the mortgage rate adjustrnents to

prime and funds rate changes are almost identical (0.967 versus 0.962).

For the error-correction models the P parameters can be interpreted as expiaining

the short-nin dynamic adjustment of the dependent variable, while the enor-correction

pararneter 8 indicates how the dependent variable adjusts in response to a deviation from

' Non-nested hypothesis tests on the equality of for the three models confirm that the prime rate is statistically more sluggish than the mongage rate.

122

the long-nin equilibrium. For example, in the case of the model in which we model

changes in the prime rate by changes in the funds rate, a positive value of the error-

correction tenn (EC) implies that the prime rate is currently above its long-run

equilibriurn relative to the funds rate, and therefore it must fa11 in order to return to its

equilibriurn; hence the negative value for 6. It is interesting to note that the magnitude of

this parameter is almost four times larger for the model in which the prime rate is the

dependent variable than for the mongage rate models. This indicates that the prime rate

re-adjusts more quickly in response to a disequilibrium than does the mortgage rate. In

fact. we can notice this in Figure 2 as the error-correction tenn in the top panel fluctuates

more closely around 0.0 than do the mortgage rate error-correction terms.

3.5 General Non-linearity Tests

Having estimated the linear models, we now impiement our usual general non-

linear diagnostic tests on these models. Rejection of the nul1 of linearity implies that the

movements of the interest rates may be more adequately captured by a model aliowing for

regime changes.

The first tests considered are, as with the previous chapters, the Keenan and

RESET tests. They are implemented by estimating the model

where Ê, are the residuals fiom the linear modeis (either in levels or in error-correction

form), X is the vector of independent variables specific to each model, and j, represents

the fitted values for each linear model. The test involves testing the joint significance of

the powers of the fined values; if they are significant, then non-linearities may be present.

In Table 5 these tests reveal some evidence of non-linearities. The Keenan test, for

which p = 2, indicates non-linearities at the 5% level for the error-correction models. The

RESET test presents somewhat mixed evidence, although the evidence in favor of non-

linearities seems somewhat greater for the error-correction models. AI1 three tests favor

non-linearities in the error-correction model in which changes in the mortgage rate are

modeled against changes in the funds rate.

The next test considered is the Tsay (1986) test. As before:

Analogous to the RESET test, we are testing the joint significance of the additional

regressors, which are simply rhe cross-products of the original list of regressors. In Table

5 these additional regressors are quite significant as s h o w by the large values of the test

statistics. Thus, this is again evidence of non-linearities in the relationship.

The final diagnostic is the LSTAR test. Regime changes, and hence non-

linearities, are triggered by lagged values of the dependent variable. The mode1 required

to implement this test is:

where y is the dependent variable? and d is the delay parameter. In our tests we use d = 1.

The test again involves testing the significance of the additional regressors. In Table 5

both LSTAR tests agree with the conclusions of the Tsay test, narnely that non-linearities

are present in al1 models.

To surnmarize the results of this section, we have found evidence of non-

linearities in the six linear models under consideration. This indicates that, either in levels

or in first-differences, the movements of managed commercial interest rates rnay be

regime-dependent. In the next section we try to forrnally identie whether multiple

regimes exist, and the precise location of the breakpoints.

3.6 Threshold Models

Researchers have long been interested in interest rate floors. We know that one

must at least exist for money market rates at zero per cent. For commercial lending rates

these floors must of course be higher, since the rates must at Ieast cover administrative

costs and default risk. Retuming to Figure 1 again, we notice that over the 1971 to 1998

penod the mortgage seldomly fell below 7.5 per cent, the prime rate below 5 per cent, and

the funds rate below 2.5 per cent. Thus, we would expect managed commercial rates to

become stickier once chey approach their (unknown) floors.

The present section is not directly concemed with locating interest rate floors, but

rather with locating thresholds that can be of use in determining when managed rates

become more sticky. We use once again the test of Hansen (1996). that allows us to

locate the most significant threshold location. As in the previous chapters, we augment

Our six linear models by a variable capturing the threshold effect, G,-, = of-, l[w,-, 5 r] . where r i s the threshold parameter, and the indicator fünction is

O, for a,-, > 7

1, fora,-, I z

The threshold models to be estimated are of the form

where y is sorne managed commercial rate (prime or mortgage) either in levels or first-

differences, x is an explanatory varïabIe (Fed fùnds rate or prime) in levels or first-

differences and EC is an error-correction term (used only in the error-correction models).

in o w study we consider two possible threshold variables- The first is of course

the interest rate x; should a threshold effect exist, then we c m iocate the level of x at

which the movements in the rate y become more or less sluggish. The second type of

threshold variable is the error-correction term, EC. Evidence of a threshold effect in this

case allows us to determine whether the adjustment of y to a disequilibrium depends on

the magnitude of the disequilibrium itself. Intuitively, one might be inclined to believe

that significantiy large departures fiom equilibrium, brought about by some exogenous

shock, might cause variable y to adjust more rapidly in order to re-establish equilibrium

than it would under ordinary circumstances. This model can therefore be thought of as a

threshold error-correction model, which is a specific class of more general non-linear

error-correction models discussed by Granger and Teriisvirta (1 993 :59-60).

In Figure 3 we plot the sequence of LM statistics for the Hansen threshold test.

The first row presents the statistics for the three models in levels, the second row for the

error-correction models with an interest rate threshold variable, and the third row for

enor-correction models with the error-correction terni acting as the threshold variable. It

is instructive to examine the plots of these statistics as it gives us a sense of the

significance of our estimated thresholds as compared to other potential alternatives. For

esarnple, for the first error-correction model in which x = MedFunds and y = APrime, we

find a distinct peak at a value of -1.23 for x, and the sequence of test statistics faIls

markedly as x increases, indicating that the likelihood of a threshold effect falls as well.

Compare this to the corresponding model for which the error-correction tenn is used as a

threshold variable instead. We see two distinct peaks, around 1.37 and -2.60, indicating

that the choice of threshold here is more controversial.

in Table 6 we use the thresholds found fiom the sup-LM statistics in each graph of

Figure 3 and estimate the formal threshold models. Beginning with the first model in

ievels, we see that a threshold is located at the point where the Fed funds is equal to 1 1.16

per cent. Given how we defmed the indicator fùnction in (9), the threshold parameter y

must be added to the fbnds rate parameter in order to determine how the prime rate

reacts to fùnds rate when the funds rate is below the threshold. Since yis negative, the

funds rate has a smaller effect on the prime rate when the b d s rate is below 1 1.16. More

specifically, when the fiuids rate is below 1 1.16 and it falls by 100 basis points (in a given

week), then the prime rate would fa11 by 9.2 basis points. Conversely, if the prime is

above 1 1.16 and rises by 100 bais points, then the prime rate will rise by 10.3 basis

points. In short, the prime rate is less responsive to fùnds rate movements when the funds

rate is low.

Similar stones can be told regarding the other two models in levels. The mortgage

rate is less responsive to the prime when the prime is below 10.05, and less responsive to

the hnds rate when it is below 15.13. In this case the slope parameter is 0.009 below the

threshold, and 0.01 5 above it. Compare this to the estimated slope of 0.013 found for the

linear model in Table 4, and one can then note a definite kink in the relationship.

Although such difierences do not appear very large, in percentage terms the mortgage rate

is 67% more responsive to funds rate changes when the hinds rate is above 15.13 per

cent, while the prime rate is 12% more responsive to the funds rate when it is above 1 1.16

per cent.

For the error-correction modeIs using interest rate changes as the threshold

variable, we find none of the threshold variables to be statisticaily significant. However,

the enor-correction thresholds appear to be significant. In the case of the prime / Fed

funds error-correction model, we h d that the threshold exists when the prime rate is 1.37

per cent above its equilibrium level. For every 100 basis points above this departure fiom

equilibrium, we expect the convergence back to equilibrium to be of the magnitude of a

10.8 basis point drop per week. Below this disequilibrium threshold, the convergence to

equilibrium only occurs with a 2.4 bais point drop in prime changes. Hence, as

suspected' the greater the magnitude of the disequilibrium, the greater the speed of

adjustment towards equilibrium.

Note for the mortgage / prime enor-correction model that the threshold is

estimated to lie at a point where the mortgage rate is 4.54 per cent beloiv its equilibrium

level. To return to equilibrium the mortgage rate must therefore increase. Below 4.54 per

cent the magnitude of the prime changes for every 100 basis point disequilibrium is 2.1

basis points per week; above the threshold, when the mortgage rate is closer to its

equilibrium value, the magnitude of the changes would be 1.4 basis points.

In Table 7 we perform our usual non-linear diagnostic tests to determine whether

the introduction of thresholds contributes towards eliminating non-linearities in the

models. We perfonn the tests on the three models in levels, three error-correction models

using the interest rate as the threshold, and three error-correction modeIs with enor-

correction thresholds. It is clear that the nul1 of linearity is rejected in almost each case,

indicating that the thresholds do li ttle to eliminate the non-linearities. This indicates that

the non-Iinearities in these models are more complex than a two-regime threshold model,

and that more work is required to firlly capture these non-linearities.

We conclude this section by noting that there are significant non-linearities

between the levels of money market and managed commercial interest rates. As noted by

the threshold models, the responsiveness of managed interest rates to changes in money

market rates is lower when money market rates are low. This is consistent with potential

interest rate floors, as managed rates become reluctant to cross them. The fact that

managed rates are less responsive at lower rates has obvious policy implications if these

rates are thought to be important determinants of consurnption and investment. The

impact of a stimulative monetary policy on the prime rate, as interpreted as a 100 basis

point drop in the Fed funds rate, will be 12 per cent lower when the h d s rate is below

1 1.16 per cent than when it is above this level.

Our next task is to examine the effects of these interest rate changes on

consumption and investment. If we find no evidence of non-linearities, then this would

imply that much of the non-linearities on output and inflation, detected in earlier chapters,

may be caused by non-linearities between money market and commercial interest rates.

However, if we do find further non-linearities between the interest rates and consumption

and investment, then this would indicate that these must be taken into account whenever

policy-makers wish to determine the impact of a policy action on the ultimate target

variable, which is inflation. The implication is that there would seldomly be any reliable

rules of thumb on which one could rely, as the current levels of interest rates,

consumption, investment and output would have to be considered when contemplating

the ultimate policy impact.

4. Consumption

Along the transmission path fiom interest rates to inflation, we have consumption

as an intermediate variable. In this section we speciQ simple models of consumption that

we then test for non-linearities.

4.1 Data

We use two diffèrent definitions of consumption, namely total consumption and

consumption of durable goods, which is more interest-sensitive. Both are deflated by the

consumer price index. As explanatory variables we use lagged real GDP and our three

interest rates, narnely the Fed fùnds, prime and mortgage rates. Although more likely

relevant for investment purposes, the mortgage rate is used here as a proxy for typical

commercial loan rates. The sample begins in 1971 and ends in the first quarter of 1998.

Interest rates were converted to the quarterly frequency by averaging the monthly

observations over a given quarter. Data on consumption and GDP are well-known to have

a unit root, therefore no unit root tests are performed on these series. Both consumption

variables and GDP are specified in logs. Precise variable definitions are given in

Appendix 1.

4.2 Linear Models

The theory of consumption and its determinants is well-establishedS4 The most

important explanatory variables are typically income and interest rates. As such, our

general consumption mode1 is specified as

~vhere CONS is either total or durable consumption, and RATE is one of our three interest

rates. The lag specification follows a minimization of the Schwarz criterion over a

maximum of eight potential lags for each variable. The chosen specification is also

almost perfectly consistent with a general-to-specific modeling methodolo~, where we

began with eight lags for each variable and proceeded to eliminate the statistically

insignificant lags. We deviate from the general-to-specific specification only when we

choose to include a statistically insignificant variable if we feel it may have something to

contribute within the non-linear framework we wish to develop.

' See, for example, Chapter 12 of Branson (1989) for formal theories o f consumption and its other potentially usefül determinants, such as wealth.

Estimates of (1 1) are presented in Table 8. For total consumption we rernark that

the interest rate parameters have the requisite negative sign and are significant, with the

exception of the mortgage rate. For durable consumption al1 three interest rate parameters

are significant, but more important are the magnitudes of the parameters, as they are

noticeably larger than for the total consumption models.

Estimating an equation such as (1 1) is only acceptable if the variables are

cointegrated. In Table 9 we present the estimated cointegration vectors found using the

Johansen methodology. To interpret these long-mn relationships. consider the first system

consisting of [Tord Consurnprion, GDP, Fed frinds]. The lonr cointegration vector

estimated is [1.000, -0.985, 0.0 161, implying

so that the long-run marginal propensity to consume is 0.985. and the semi-elasticity of

interest rates on consumption is -0.ûi6. Both signs are consistent with theory.

For the systerns consisting of durabIe consumption and either the Fed funds or

prime rates, we find two statistically significant cointegration vectors. It is only the

second vector in each system that actually has the relevant signs and magnitudes for the

long-run parameters. Thus, when estimating error-correction models we use two error-

correction terms. since the true equilibriurn relationship may be a linear combination of

these two cointegration vectors.

In Figure 4 we plot al1 the error-correction terms derived from these cointegration

vectors. Interestingly, the only enor-correction terms that appear to be fluctuating around

a zero equilibriurn are the second error-correction terms of the durable consumption

systems. The other terms appear to be either noticeably above or below equilibrium.

In Table 10 these error-correction terms are used in the estimation of error-

correction models. The error-correction parameters are almost aiways significant, and of

these they have negative signs. Positive deviations from equilibriurn require a fail in

consumption to re-establish equilibrium, hence the negative pararneters. in terms of fit, it

is clear that the models for which the Fed fùnds rate is used as the representative rate have

a bener fit.

When applying our general non-linearity tests to the consumption models, we

notice from the test statistics in Table 11 that the results are somewhat mixed. For the

modeis in levels the Tsay test tends to reveal the presence of non-linearities, whereas the

other tests are less iikely to detect them.

For the error-correction models the results are again mixed, as in no instance do

any of the models pass al1 the diagnostic tests. We can therefore proceed with our analysis

under the assurnption that there is scope for irnprovement over Our linear specifications.

4.3 Threshold Models

Having found in the previous section that non-linearities may be present in the

data, we now implement threshold tests to discover whether these asymmetries originate

from the interest rate variables. in Figures 5, 6 and 7 we plot the sequence of LM

statistics used to isolate the most likely threshold values for d l our consumption models.

In Figure 5 the plots for the models in levels are presented. It is immediately apparent that

the values of the sup-LM statistics for the dürable consumption models are higher,

indicating that the likelihood of asyrnmetries between interest rates and durable

consumption is higher than it is for total consumption. It is also interesting to note that the

pattern of test statistics is very similar for both durable and total consumption, indicating

that the sources of asymmetries in the consumption-interest rate relationship is most

likely operating through durable consumption.

In Figure 6 the sequence of statistics using interest rate changes in the error-

correction models are plotted. Interestingly, we now find that the values of the sup-LM

statistics are higher for models of changes in total consumption. In Figure 7 we find more

likely evidence of threshold effects between the error-correction terms and durable

consumption rather than total consumption. For the first two durable consumption models

in these diagrams note that we are using the second error-correction term as the threshold

variable, as this appears to give the most sensible equilibrium characterization. In fact,

these are the only error-correction ternis in these diagrams that test for thresholds for both

positive and negative deviations from equilibriurn. In the case of the total consumption

error-correction terms, we are constrained to testing for thresholds only among negative

deviations fiom equilibriurn since, as we observed in Figure 4, these error-correction

terms are never positive over our sample.

Estimates of threshold mode1 parameters are presented in Tables 12 and 13. in

Table i 2 we present the estimates for the total consumption models, and in 1 3 the durable

consumption models. In Table 12 we observe that for the models in levels, none of the

thresholds are significant. By contrast, for the error-correction models with interest rate

thresholds, a significant threshold is found for changes in the mortgage rate.

For durable consurnption in Table 13, some stronger asymmetries are detected for

the models in levels. When the h d s rate is below 9.83% the interest serni-elasticity is -

0.002; above it is -0.005. Thus, not only is durable consumption more sensitive to

interest rate movements than total consumption, but it is also more susceptible to

asymrnetric responses to rate changes than the latter.

For the error-correction models, there do not appear to be any asymmetries

between changes in the b d s rate and changes in durable consumption. However, we find

interesting threshold effects between the error-correction terms and changes in durable

consumption. For the rnodel that includes the fbnds rate as the interest rate variable, when

the error-correction terni is below -0.035, the parameter on the error-correction term is

-0.05 1; when above this value it is -0.306. Since this threshold is close to 0.0, we can

state that adjustments to durable consumption changes are more pronounced when

durable consumption is above its equilibrium level than below. That is, if the level of

durable consurnption is hi& relative to its equilibrium level, the growth of durable

consurnption will FALL more quickly to restore durable consumption equilibrium

RELATIVE to the growth of durable consumption that must FUSE when durable

consumption is below its equilibrium level. In other words, it is easier for durable

consurnption to fdl during a boom (when durable consumption is hi&) than it is to rise

during a recession (when durable consumption is low).

in Table 14 we perform our non-linear diagnostics on al1 these threshold models.

In general we find that the Keenan and RESET statistics fa11 somewhat, although for a

number of models, especidly those in Levels, the null of linearity is rejected using the

Tsay test. For the error-correction models the best result in tenns of eliminating non-

linearities is the durable consumption error-correction model with an error-correction

threshold. In Table 1 1 the Keenan test found evidence of non-Iinearities, whereas in Table

14 the null of linearity cannot be rejected for any of the tests. Hence, the asymmetric

adjustment of durable consurnption to different disequilibrium regimes appears to be the

source of non-linearities in this relationship, and is adequately captured by a threshold

model.

To summarize, the main results of this section are as follows: (1) There are

noticeabIe threshold effects between the LEVEL of durable consumption and the Fed

funds rate; consumption is roughly three times less responsive when the latter is below

9.75%. (2) We fmd that adjustments to disequilibriurn of durable consumption is more

rapid when durable consumption is above its equilibrium level than when it is below.

5. lnvestment

The second major macroeconomic variable to be influenced by interest rates dong

the monetary policy transmission mechanism is investment. This section proposes to

examine whether there are any asymrnetries between interest rates and investment, and if

so, whether they can be captured using threshold models. As with consumption, it is

realistic to believe that monetary policy regimes, as defined by interest rate thresholds,

may have asymmeîric effects on investment. Issues such as policy credibility and inflation

uncertainty are more likely to be felt in high interest rate periods (or tight monetary policy

regimes), thereby causing investrnent to fa11 more rapidly in response to interest rate

changes than during low-rate regimes. Johnson et al. (1997) find little evidence of

asymmetry between investment and interest rate changes in Canada, but they only test for

asyrnmetry around a zero threshold for interest rate changes, leaving open the possibility

that asymmetries may manifest themselves for larger (either positive or negative) interest

rate changes.

5.1 Data

We again focus on quarterly US. data, fiom 197142 to 199744. We use the

sarne three interest rates as before, namely the Fed fùnds, prime and mortgage rates. Our

real income definition is d s o unchanged. To ver@ the robustness of our findings, two

different investment rneasures are used: Fixed Private Investment (henceforth FPI) and

Gross Private Domestic Investment (GPDI). Both are measures of total investment,

although GPDI is slightly more volatile. Precise definitions are given in Appendix 1.

5.2 Linear Models

The linear models of investment that we consider are very similar to the

consumption models. They are of the form

where INV is either FPI or GPDI, and RATE is one of our three interest rates. The

combination of two investrnent variables and three interest rates implies that there are six

different models to estimate in levels. The present lag specification differs fiom that of

the consurnption models as we now add a second (significant) lag of real income.

In Table 15 we present the estimates of (12) for the six possible combinations of

investment and interest rate variables. Al1 interest rate variables are significant, and we

also find the magnitudes of the incorne parameters to be higher for the GPDI models,

implying this investrnent masure is more sensitive to income changes than is FPI.

However, the fit of the estimated GPDI models is slightly lower than the FPI models, due

to the fact that GPDI is a more volatile series to model. Furthermore, the fit of the models

are as good or better when we use the commercial rates as explanatory variables instead

of the hnds rate.

The validity of the regressions in IeveIs in Table 15 is confïrmed by the

cointegration vectors in Table 16. Here the existence of a single cointegration vector for

each system is quite clear, given the size of the first eigenvalue relative to the second, and

the high trace statistics for the existence of one cointegration vector. The vectors

themselves have theoretically plausible signs and magnitudes. For example, the long-run

income elasticity is 0.706 and the interest semi-elasticity is 4 .014 for the system

consisting of [FPI, GDP, Fed Fun&]. It is also worth noting that the interest semi-

elasticities are highest for systems in which the mortgage rate is included. Thus, changes

in the mortgage rate will have greater long-run impacts on either FPI or GPDI than

similar increases in the funds rate or the prime rate. This is not surprising considering that

mortgage rate-sensitive residential housing investment is included within our dependent

variables.

In Figure 8 we plot the error-correction terms derived from the cointegration

vectors. We notice that they are al1 positive and decreasing, indicating that the positive

disequilibrium between investment and its determinants (GDP and interest rates) is being

eliminated very gradually over this twenty-six year period. These error-correction terms

are used in Table 17 in the estimation of error-correction models. The error-correction

terms are al1 significant and negative, as we require, since positive deviations from

equilibrium require lower investment growth to restore equilibriurn. Another notable

feature of the results in this table is the insignificance of interest rate changes in the GPDI

models. Although not reported here, higher lags for the interest rate variables proved

insignificant as well. This implies that these rates are most usehl to explain long-run

movements in the variable, not the short-run dynamics.

The general non-linearity tests are reported in Table 18. For the models in levels,

the three GPDI models show no evidence of non-linearities, while non-linearities are

mostly captured by the Tsay test for the FPI models. Meanwhile, the nul1 of linearïty is

rejected for a greater number of cases for the error-correction models. Thus, we can

conclude that the evidence of asymmetries is weak for the models in levels, but much

stronger for models of investment growth.

5.3 Threshold Models

The sequence of LM statistics for the threshold tests are presented in Figures 9, 10

and 1 1. In Figure 9 we observe that the sup-LM statistics are generally higher for the FPI

models than for the GPDI models. The patterns are also somewhat similar, as thresholds

are most Iikely at lower values of the interest rates. For the interest rate threshold

variables in the error-correction models, the sequence of statistics in Figure 10 are

noticeably larger for FPI, but virtually insignificant for GPDI. The sup-LM statistics also

occur at noticeably different values for FPI, with the peak occurring at -2.532 for the

funds rate but at 0.723 for the mortgage rate. FinaIly, in Figure 11 the statistics for the

significance of the enor-correction thresholds are ploned. The most significant thresholds

for the FPI models occur at relatively large disequilibrium levels, between 1.2 and 1.4. By

contrast, the rnost significant error-correction thresholds for GPDI are in the vicinity of

0.6.

The estimated threshold models for FPI are presented in Table 19, and in Table 20

for GPDI. No significant threshold effects appear to exist for FPI, but some striking

results are obtained for GPDI. For instance, when the funds rate is below 5.44%, a rise in

the funds rate actually leads to an increase in investment; above the threshold the

relationship between these variables is once again negative (but statistically insignificant).

The non-Iinearity tests on the investment threshold models (Table 21) reveais that

none of the models have been fûlly cleansed of their non-linearities. However, since

several of o u thresholds proved to be statistically insignificant, the evidence of non-

linearities that we do witness here is likely not caused by the relationship between interest

rates and investment. As with the strong non-linearities for the France GDP growth model

in Chapter 2, the non-linearities are likely onginating fiom other relationships within the

model, such as that between investment and GDP, or simply the fact that investment itseIf

is a univariate non-linear process.

The main findings of this section are as follows: (1) Commercial interest rates are

at Ieast as good as the funds rate at explaining the cwrent level or growth of investment.

(2) Threshold effects are more likely to be witnessed between interest rates and the level

of GPDI; at very low rates of interest investment appears to be more responsive to interest

rate movements.

6. Conclusion

The objective of this chapter is to search for potential asyrnmetries between policy

interest rates and several variables dong the monetary policy transmission mechanism,

namely cornniercial interest rates, consumption and investment. Our main findings c m be

summarized in the following manner:

There is an asymmetric relationship between the Federal funds rate and both

the prime and mortgage rates. The prime rate is roughty 12% less responsive

to the funds rate when the latter is below I 1%; the mortgage rate is 67% more

responsive to the fùnds rate when it is above 15%.

The level of consumption is more responsive to movements in the fiinds rate

than in commercial rates. Consumption is roughly half as responsive to the

funds rate when it is below 9.75%.

Asymmetries exist between changes in commercial rates and the growth of

consumption, as it is proportionately less responsive to large drops in

commercial rates.

We find that the adjustments to disequilibrium of durable consumption is

more rapid when durable consumption is above its equilibrium level than

below.

Commercial rates seem to be as good as, or better than, the b d s rate in

explaining the level of investment.

Significant threshold effects exist between the level of investment and the

Ievels of interest rates. The asymmetry is so pronounced that at low levels of

money market rates, interest rate increases appear to stimulate investment.

In light of these findings, we can state that low interest rates have smaller impacts

on consumption and investment. In a Iow interest rate regime, if policy-makers attempted

to stimulate either of these variables they would require a proportionately larger reduction

in interest rates to achieve the desired response than if they were in a high rate regime. In

terms of the funds rate thresholds, mortgage rates are less responsive at 15%; the prime

rate at 1 1 %; consumption at 10% and investment at 5%.

We can also conclude that commercial rates play a minor role in explaining the

level of consumption, but are important in explaining the growth rate of this variable.

Commercial rates are correspondingly more important in explaining investment. Since

in\-estment is far more volatile than consurnption, and explains a greater proportion of the

variance in output (see, for example, Barro (1989)) then the movements of commercial

interest rates should not be negtected. Policy-makers should be aware of the stickiness of

commercial rates, in particuiar the prime, in low interest rate regimes.

Appendix 1: Data

The interest rate data used in this snidy was obtained fiom the Federal Reserve Board, and macroeconomic data fiom the Federai Reserve Bank of St- Louis FRED database. The fiequency of the interest rate data is weeidy, and the sample extends fiom March 197 1 to March 1998. Consumption and investment data are quarterly, and both begin in 197 142 and end in 199841 for consumption and 199744 for investment. The variable definitions are as follows:

RiFSPFF-N-WW: Federal funds rate (cost of borrowing immediately available h d s , primarily for one day), weekly ending Wednesday.

RIFSPBLP-Nv.WW: Prime rate (charged to preferred customers by at least 75% of 30 large banks), weekly ending Wednesday.

RMMPCCFCN.WF: Mortgage rate (fixed rate on a 30-year conventional mortgage), weekly ending Friday..

FPI: Fixed Private investment, nominal.

GPDIC: Gross Private Domestic Investment, nominal.

CONS : Total Consumption, nominal.

DURAC: Durable Consumption, nominal

GDP: Gross Domestic Product, nominal

CPIAUCSL: Consumer Pnce Index.

Appendix 2: Tables and Figures

Table 1: Unit Root Tests

VariabIe ADF(k) ADFP52) PP KPSS (Lag Parameter) 1 2 3 4

Fed Funds -3.1 O(4) -2.43 -12.15.. 7.3 13.. 4.890.. 3.678.' 2.952..

Prime -2.8 l(4) -2.4 1 -9. 15.. 7.844.. 5.237.. 3.934.. 3.153**

Mongage -1.91(3) -1.91 4-84.. 12.769'. 8.52 1 ** 6.397.' 5.124"

AFed -15.14**(3) -5.65.. -1416** 0.050 0.046 0.044 0.04 1 Funds

;\DF is ihe Augmentcd Dickcy-Fuller 1 s t . with lags k chosen by minimizing rhc Schwarz criccrion and by futing them al k=52.. PP is the Phillips-Perron tat, with lags (52) x t ui cqual the annuai fqutncy of the data KPSS is thc Kwiatkowski. Phillips, Schmidt and Shin (1992) test. wilh smionarity specificd as the nuIl hypothais and a unit root proccss as the alternative. Criticai vaiucs arc -3.41 and -3.96 for the ADF and PP tests at the 5% and f % levcls. and 0.146 and 0216 for ihc KPSS test

Frime Causes Mortgage 1 1.68** Mortgage Causes Prime 0.144 l

Table 2: Granger-Causality Tests

Fed Funds Causes Mortgage

Direction of Causality F-Stat Fed Funds Causes Prime 182.9*@

Direction of Causality F-S tat Prime Causes Fed Funds 0.034

AFed Funds Causes APrime 85-43** l @rime Causes AFed Funds 35.93-

48.34**

APrirne Causes AMortgage 34.06** 1 Mortgage Causes Mrime 57.58**

Mortgage Causes Fed Funds

Table 3: Johansen Cointegration Tests

AFed Funds Causes AMortgage 72-25..

S ystern Lags Eigenvalue Trace S tat Vector Prime 5 0.01 67 32.75 1 .O00

Fed Funds -1 .O38 Constant - 1 -649

Mortgage Causes M e d Funds lm* *

Mortgage 4 0.0 1 72 28.84 1 .O00 Prime - 1 .O87

Constant 0.263

The F-stat is used to test the signiticance of the explanatory variabtes. ( ' O ) denotes significancc of the direction of causaiity

Mort gage 3 0.0 172 29.4 1 1 .O00 Fed Funds -1 .O39 Constant -2.194

Thç lags for the VARS were chosen by minimizing the Schwarz criterion. The critical values for rcjecting the nuIl of no coiniegration are 19.99 and 24.73 at the 95% and 99% levcls rcspcctiveIy.

Table 4: Interest Rates - Linear Models

Y Prime Mortgage x Fed Funds Prime a 0.2 13 0.121

(8.161) (3.185) Pi 0.898 0.967

(1 36.9) (1 35.6) h 0.099 0.023

(1 5.76) (3 -902)

Mortgage Fed Funds

O. 172 (4.403) 0.962

(1 63 -9) 0.028

(6.1 32)

Mrime AFed Funds

AMongage AFed Funds

0.000 1 (0.96 1 ) 0.3 12

(7.9 16) 0.066

(3.513) -0 -007

(-3.783)

- R' 0.997 0.998 0.998 0.452 0.172 O. 199 Obs. 1408 1408 1408 1407 1407 1407 ZN 746S 27000f 204204 10474* 26320f 1 ZA 647.2* 379.5* 382.4* 230.2* 115.7* 128.5* ZH 135.4* 52.0* 34.1 * 1 56.4* 54.4* 158.3*

AR,MA@.q) (391 ) (231) (29 1) (090) (090) (0x9

Thc hg specification of x and y in each mode1 was found by minimizing the Schwarz criterion. t-statistics are in parentheses. EC is the error-correction term constructed using che cointegration vectors in Table 3. zh: is the Jarque-Bera rcst for non-normality. asymptotically distributed as ~ ' ( 2 ) ; =,+ is an LM test for 52d-order aurocorrelation and is as!-mptotically distributed as x'(60). while :, is a modified Breusch-Pagan LM test for heteroskedasticicy. distributed as %' wiih degreees of freedorn equal to the nurnber of regressors x 2. (*) denotes significance at the 5O/& Irvel. AR!A@,q) is the order selected for correcting the autocorreIation based on minimizing the Schwarz criterion when using the fe3siblc GLS estirnator.

Table 5: Interest Rates - General Non-linearïty Tests

Mode1 Obs. Keenan RESET RESET Tsay LSTAR (x, Y ) ( F(l ,T-3) F(2,T-4) F(3,T-5) X2(3)

Prime, FF 1408 0.076 0.174 8.792** 151.0** 44.4**

Mort, Prime

Mort, FF

APrime, MF

AMort, APrime

AMort, AFF

(*) denotes significance ai the 5% levcl; (@*) at the 1% Icvel.

Table 6: lntercst Hstcs - Ttircshold Modcls

lntercst Rate Threstiold Error-Correction Tlireshold x Prime Mortgage Mortgage APritnc AMortgage AMortgage APriine AMoilgage AMortgage Y Fed Funds Prime Fed Funds A17ed Fuiids APrime AFed Funds AFed Funds APrime AFed Funds a

Pi

Pr

S

Y

R2 Obs.

Threshold sup-LM p-value

ARMA@,q)

Thc lag spcciîïcalions of .r nnd y in ench modcl wcrc foiind by minimizing ihc Schwarz critcrion. Ncwcy nnd Wcst (1987) corrcctcd t-siatistics for hcicroskcdnsticity and autocorrclation o f ordcr 52 nrc in parcnthcscs. EC is thc crror-corrcciion tcrin consiriictcd using tlic coinicgrniion vcclors in Tablc 3. tù is thc thrcshold vnrinhlc as dcfiiicd in thc pnpcr. Thrcshold is the valuc of ihc nuisnncc pnrumctcr nssocintcd wilh ~ h c siip-LM statislic, and p-valuc is ohinincd by simulating thc nsyrnptotic dislrihution (1000 rcplications) o f ihc ks i sinlistic iising ihc riicihotl proposcd hy llnnscn (1996).

Table 7: Interest Rates - Non-linearity Tests on Threshold Modeb

Mode1 Obs. Keenan RESET RESET Tsay L S T A R (x. Y ) (2") F(1,T-3) F(2,T-4) F(3,T-5) 2(3) (&l)

Prime, FF 1408 40.43** 25.22** 17.61** 154.7** 27.014* Mort, Prime 1408 0.852 0.43 7 0.335 18.52** 1 .1 56**

Mort, FF 1408 7.387** 4.585* 4.571** 45.24** 5.917**

(*) denores significance at the 5% lcvel; ( * O ) at the 1% level. A denotes an enor-correction hreshold:

Table 8: Consumption - Linear Models

CONS Total Total Total Durable Durable Durable RA TE Fed Funds Prime Mortgage Fed Funds Prime Mortgage

0.20 1 0.3 15 0.454 -0.047 -0.358 -0.143

The lag specification of x and y in each rnodel were found by minimizing the Schwarz criterion. t-statistics are in parentheses. Total is total aggregate consumption, and Durable is consumption of durable goods. 2s is the Bowman- Shcnton test for non-normality, asyrnptotically distributed as &); =A is an LM test for 4&srder aurocorrelation and is asymptotically distributed as ~'(4) . white z, is a modified Breusch-Pagan LM test for hcteroskedasticity. distrïbuted as %' with degrcees of freedom equal to the nurnber of regressors x 2. (*) denotes significance ar the 5% level. ARMA@.q) is the order selected for correcting the autocorrelation based on minimizing the Schwarz criterion when using the feasible GLS estimator.

Table 9: Consumption - Johansen Cointegration Tests

System Lags Eigenvalues Trace Stats Vectors (1) (2) (1 (2) (1) (2)

Total Cons. 3 O. 1809 GDP

Fed Funds

Total Cons. GDP Prime

Total Cons. GDP

Mortgage

Durable Cons. GDP

Fed Funds

Durable Cons. GDP Prime

Durable Cons. GDP

iMortgage -0.003 The I q s for the VARS werr chosrn by rninirnizing the Schwarz criterion. The critical values for rejecting the nul1 of no cointegration against the aItemativc of at least one cointegraiion vector are 23.076 and 29.193 at the at the 95% and 99% levels respectivcly. For the s ip i f i cmcc of the second cointegration vector the critical values are 12-21 2 and 16.158.

Table 10: Consumption - Error-Correction Models

CONS: Total Total Total Durable Durable Durable RATE: Fed Funds Prime Mortgage Fed Funds Prime Mortgage a -0.004 -0.0005 0.002 -0.00004 -0.000 1 0.0 17

The iag specification of x and y in each mode1 were found by minimiùng the Schwarz criterion. t-statistics are in parcntheses. Tora1 is total aggregate consurnption, and Durable is consumption of durable goods. =S is the Bowman- Shrnton tesr for non-nomality. asymptoiically disuibutrd as &); z, is an LM test for #"srder autocorrelation and is risymptotically disrributed as %'(3). white ZH is a modifted Breusch-Pagan LM test for heteroskedasticity. distributed as %' with drgrcrrs of freedom equal to the number of regressors x 2. (*) denotes significance at the 5% level. ARMA@.q) is the ordcr selected for correcting the autocorrelation based on minimizing the Schwarz criterion when using the feasible GLS estimaior.

Table 11: Consumption - General Non-linearity Tests

Mode1 Obs. Keenan RESET RESET Tsay LSTAR

Total, GDP 4.977** 0.0003 19.10** 0.914 Fed Funds

Total, GDP Prime

Total, GDP Mortgage

Durable, GDP Fed Funds

Durable, G D P Prime

Durable, G D P Mortgage

ATotal, AGDP AFed Funds, EC

ATotal, AGDP APrime, EC

ATotal, AGDP, AMongage, EC

&Durable, AGDP, AFed Funds,

EC 1, EC2

ADurable, AGDP, APrime,

EC 1, ECZ

ADurable, AGDP AMortgage, EC

(*) denotes significance at the 5% level; (**) ar the 1% levcl. d is the delay parameter (the lag of the dependent variable that triggers the regirne change) and k is the number of explmatory variables.

T;ible 12: Total Consumption - ï'hrcshold Models

lnterest Rate I'lireshold Error-Correction Threshold CONS Total Total Total ATotal ATotal ATotal ATotal ATotal ATotal MTE Fed Funds Prime Mortgage AFed Funds APrime AMortgage AFed Fiinds APrimc AMortgap

QI

Pl

Pz

m S

Y

R2 Obs,

Threshold sup-LM p-value

ARMAQAq)

Thc Iag spccifications o f RA7F, CONS and INCOME in cach modcl were found by minimizing thc Schwarz criicrion. 1-statisiics arc in pnrcnlhcscs. EC is the crror-correction lcrm consirucicd using ihc cointcgraiion vcciors3. o i s the thrcshold varinbtc ns dcfincd in the pnpcr. Thrcshold is thc value o f thc nuisancc pnranictcr nssociotcd wiih tIic sup-LM siaiistic, and p-vnliic is obtniiicd by siniiilnting ihc nsynipioiic distribution (1000 rcplicntians) o f the icsi stniislic using tlic nicihod proposcd by Ilnnscn (1996).

T;iblc 13: I>urablc Consumption - Threshold Models

lntercst Rate Thrcshold Error-Correction Threshold CONS Durable Durable Durable ADurable ADurable ADurable ADurable ADurabfe ADurable RA TE Fed Funds Prime Mortgage bFed Furids APrime AMortgaage AFed Funds APrime AMortgage a

Pi

Pz

P-3

4

4

Y

Rz Obs.

Threshold sup-LM p-value

ARMfVnq)

(O. 1 73) -0.178 (- 1.475)

1.226 (2.949) -0.009

(-3.1 19) -0.00 1

(-0.157) -0.134

(-3.204) 0.03 5

(0.432)

0.223 1 O6

0.077 12.13 0.006 ( 1 90)

Thc h g spccificotions o f HATE, CONS m d INCOM in cttch rnodcl wcrc foiind hy minimizing thc Schwarz criicrion. Ncwcy and West (1987) concctcd t-statistics for hctcroskcdasticily nnd nutocorrclniion o f ordcr 4 nrc in pnrcnihcscs. f3C i s ihc crror-correction icmi constriictcd using ihc cointcgraiion vcctors3. IO is thc thrcshold variable as dcfincd in ihc pnpcr. Thrsshold is itic vnluc of ihc niiisancc parnincicr nssocinicd wiîh thc siip-LM siniisiic, nrid p-value is ohiaincd hy simulaiing ihc asympioiic distribution (1000 rcplicniions) of ihc tcsi sintisiic iising ihc iiicihod proposcd by llniiscn (1990).

Table 14: Consumption - General Non-linearity Tests on Threshold Models

Mode1 Obs. Keenan RESET RESET =Say LSTAR (2) ( - 3 ) F(2,T-4) F(3,T-5) ( , ) ( h l )

1

[Total, GDP 1 07 3.243 4.247* Fed Funds] [Total, GDP

Prime] [Total, GDP Mortgage]

[Durable, GDP Fed Funds]

[Durable, GDP Prime]

[Durable, GDF Mongage]

[ATotal, AGDP AFed Funds, EC] [ATotal, AGDP APrirne, EC]

[ATotal, AGDP, AMortgage, EC]

[ADurable, AGDP,

AFed Funds, EC 1, EC2] [ADurable,

AGDP, APrime,

EC 1, EC21 [ADurable, AGDP AMortgage, EC]

[ATotaI, AGDP AFed Funds, EC]'

[ATotal, AGDP APrime, EC]"

[ATotal, AGDP, AiMortgage, EC]'

[ADurable, AGDP,

AFed Funds. EC 1, E C ~ ] ' [ADurable,

AGDP, APrirne,

EC 1, ~ ~ 2 1 ~ [ADurable, AGDP

U J r - - ,

(*) denotrs significance at the 5% Ievel; ( * O ) at the 1% levei. d is the delay parameter (the lag of the dependent variable that triggers the regirne change) and k is the number of explanatory variables. A denotes an error-correction threshold variable.

Table 15: Investment - Linear Models

1 . FPI FPI FPI RA TE Fed Funds Prime Mortgage a 0.908 0.982 0.987

(2.8 19) (2.928) (2.806) PI 0.948 0.933 0.932

(53 -3 5) (49.82) (49.1 1) h 0.777 0.803 1.395

(3 .470) (3 -432) (6.069) f i -0.842 -0.864 - 1 -456

(-3 -779) (-3 -722) (-6.399) P a -0.005 -0.005 -0.004

(-5.871) (-5.176) (-3.527)

GPDI GPDI GPDI Fed Funds Prime Mortgage

1.198 1.324 1.710 (3 -063) (3.453) (3 -923) 0.92 1 0.907 0.887

(43 -44) (43 -86) (41 -36) 2.3 12 2.222 2.541

(4.683) (4.632) (5.337) -2.392 -2.304 -2.652

(-4.940) (-4.886) (-5.705) -0.004 -0.005 -0.006

(-2.581) (-3.087) (-3.2 14)

Obs. 105 1 O5 105 105 105 105 ZN 8-61 1 7.435* 16.32* 26.85** 26.12** 26.10** ZA 17.48* 20.12* 20.88* 7.992 6.402 7.868 ZH t 9.70* 23.40* 23.13* 1 5-85' 14.71 1 5.47

A RMA@. q) (3 90) (3-01 ( 190) (090) (090) (090)

/:Yi' refus CO the invesunent variable, defined either as Fixed Private Investment (FPI) or Gross Private Domestic In\estrnent (GPDI). The lag specification in each mode1 was found by following a gencnl-to-specific modeling straregy. t-statistics are in parentheses. is the Bowrnan-Shcnton test for non-normality, asyrnptotically distributed as ~ ' (2 ) : =A is an LM test for 4h-order autocorrdation and is asymptoticaIly distributed as ~'(4) . while ZN is a rnodified Breusch-Pagan L M test for heteroskedasticity. distributed as X' with degreees of freedom equal to the number of regressors x 2. (*) denotes significance at the 5% levcl. ARMA@.q) is the order selected for correcting the autocorreIation based on minirnizing the Schwarz criterion when using the feasible GLS enimator.

Table 16: Investment - Johansen Cointegration Tests

System Lags Eigenvaiues Trace Stats Vectors (1) (2) (1) (2)

FPI GDP

Fed Funds

FPI GDP Prime

FPI GDP

Mortgage

GPDI GDP

Fed Funds

GPDI GDP Prime

GPDI GDP

Mort gage

The lags for the VARS were chosen by minimizing the Schwarz criterion. The critical values for rejecring the nul1 of no cointcgration against the alternative of at least one cointeption vector are 24.076 and 29.194 at the at the 95% and 99% Isvels respectively. For the significance of the second cointegration vector the critical values are 12.2 12 and 16.158.

Table 17: Investmeat - Error-Correction Modeis

INV FPI FPI FPI GPDI GPDI GPDI RATE Fed Funds Prime Mortgage Fed Funds Prime Mort gage

0.002 (O. 153) -0.3 13

(-1 -852) 3.749

(4.1 80) 0.623

(1 -065) -0.00 1

(-0.23 3) -0.034

(-2.1 15)

0.325 1 O4

9.964** 5.721 l8.62* (090)

The lag specitïcation of x and y in each model was found by minimizing the Schwarz criterion. t-statistics are in parentheses. Total is total aggregate consumption. and Durable is consumption of durable goods. ZN is the Bowman- Shenion test for non-normality, asyrnptotically distributed as ~'(2); =A is an LM test for 4*srder autocorrelation and is asyrnptotically distributcd as %'(4), while 2, is a modified Brrusch-Pagan LM test for heteroskedanicity, distributed as %' with degreees of frsedom equal to the nurnber of regressors x 2. (*) denotes significance at the 5% Ievel. ARMA@.q) is the order seIected for conecting the autocorrelation based on rninirnizing the Schwarz criterion when using the fcasiblc GLS estimator.

Table 18: Investment - General Non-linearity Tests

Model Obs. Keenan RESET RESET Tsay LSTAR ( F 1 , - F(2,T-4) F(3,T-5) x2(r 1 , ) (+I )

FPI, GDP 105 0.954 3-40 1 * 0.656 34.73** 1.324 Fed Funds

FPI, GDP 105 0.809 2.399 0.5 15 33.08** 1.405 Prime

FPI, GDP 105 0.5 1 1 2,070 0.079 22.87* 2.003 * Mortgage

GPDI, GDP 1 OS 0.362 0.522 0.04 1 15.18 0.700 Fed Funds

GPDI, GDP 105 0.786 0.866 0.459 14.9 1 0.898 Prime

GPDI, GDP 105 1 .O5 1 1.215 0.5 14 12.51 0.578 Mortgage

AFPI, AGDP 1 04 O. 166 6.175** 6.574** 23.34 3.346** AFed Funds, EC

AFP[, AGDP 1 04 1.196 5.312** 10.71** 19.12 3.376** APrime, EC

AFPI, AGDP, 1 04 0.0 1 5 1 .O74 7.699** 23 -64 4.080** AMortgage, EC

AGPDI, AGDP, 104 2.338 3.996* 3.664* 29.86* 1.454 AFed Funds,

EC

AGPDI, AGDP, 104 1.272 3 .O77 2.340 36.39** 2.513** APrime, EC

AGPDI, AGDP 1 04 0.908 2.784 1.992 29.58* 1.398 AMortgage, EC

(*) denotes significance ai the 5% Ievel; (**) at the 1% level. d is the delay parameter (the lag of the dependent variable that triggers the regime change) and Ir is the number of explanatory variables.

Tablc 19: Fixcd Privatc Invcstnicnt - Thrcshold Modcls

Interest Rate Thrcshold ~rror-~orrectiG ~hrcshold INV FPI FPI FPI AFP1 AFP1 AFP! AFP1 AFP1 AFP1

RA TE Fed Funds Prime Mortgage AFed Funds APrime AMortgage AFed Funds APrime AMortgage

a

Pl

Pr

P,

P4

6

Y

R2 Obs.

Threshold sup-LM p-value

ARMA@,q)

'l'hc log spccificntions of RATE, INI' and INCûh11:' in cnch modcl wcrc foiind by minimiring thc Schwarz critcrion. Ncwcy and Wcst (1987) corrcclcd t-siaiistics for Iicicroskcdnsiicity and niiiacorrcldon o f ordcr 4 arc in prircnihcscs. IIC is ihc crror-correction icrm constriictcd using ihc coinicgration vcctors3. o is thc thrcshold variable as dcfincd in the popcr. 'l'hreshald is ihc vnluc of ihc niiisnncc pnrnmcicr associnicd wiih ~ h c sup-LM staiistic, nnd p-valuc is obiaincd by simulnting thc asymptotic distribution (1000 rcplicntions) of ihc iesi striiisiic iising thc iricihod proposcd by llnnscn (1996).

Table 20: Cross Priviite l>omcstic Invcstment - Threshold Modcls

lntcrcst Rate Tlircshold Error-Correction Thresliold INV GPDl GPDl GPDl AGPDI AGPDI AGPDI AGPDl AGPDl AGPDI

RATE Fed Funds Prime Mortgage AFed Funds APrime AMortgage AFed Funds APrime AMortgage a

Pi

m m P d

S

Y

R2 Obs.

Threshold sup-LM p-value

Thc lag spcciticntions of HA"E, /NI/ and INCOAK in cach modcl wcrç fotind by minimizing ihc Schwnrz critcrion. Ncwcy and Wcst (1987) corrccicd t-statisiics for hctcroskcdnsticity nnd niitocorrctntion of ordcr 4 nrc in pnrcnthcscs. EC is tlic crror-corrcction term consiriicicd using itic cointcgrniion vccton3. tù i s thc thrcshold variable 8s

dcfined in the papcr. I'hrcshold is thc vnluc of the niiisnncc pnrnrnctcr associnicd with ihc siip-LM stntistic, and p-vnluc is obtnincd hy sirniilnting thc mymptotic distrihuiion (100 rcplicntions) of ihc tcst statisiic iising tlic n~cihod proposcd hy I Innscn ( IWO).

Table 21: Investment - Non-tinearity Tests on Investment Threshold Models

Mode1 Obs. Keenan RESET RESET Tsay LSTAR ( F 1 - 3 F(2, T-4) F(3, T-5) X'(x t ) (& 1 )

[FPI, GDP 105 0.824 3.353* 0.508 39.8g8* 1.102 Fed Funds] [FPI, GDP

Prime] [FPI, GDP Mortgage]

[GPDI, GDP Fed Funds]

[GPDI, GDP Prime]

[GPDI, GDP Mort gage]

[AFPI, AGDP AFed Funds, EC]

[AFPI, AGDP APrime, EC]

[AFPI, AGDP, AMortgage, EC] [AGPDI, AGDP,

AFed Funds, ECI

[AGPDI, AGDP, APrime, EC]

[AGPDI, AGDP AMongage, EC]

[AFPI, AGDP AFed Funds, E C ] ~

[AFPI, AGDP APrime, E C ] ~ [AFPI, AGDP,

AMongage, E C ] ~ [AGPDI, AGDP,

AFed Funds, EC j6

[AGPDI, AGDP, APrime, E C ] ~

[AGPDI, AGDP AMortgage, E C ] ~

(*) denotes significance at the 5% Ievei; (**) at the 1% levcl. d is the delay parameter (the iag of the dependent variablc thar triggers the regime change) and k is the nurnber of explanatory variables. A (&) denotes an error- correction threshold variable.

Figure 1: Interest Rates, 1971-1998

' C r ) . r -

Q,

Figure 2: Interest Rates - Error-Correction Terms

Figure 3: Interest Rates - Sequence of LM Statistics for Threshold Test

Figure 4: Consumption - Error-Correction Terms

Figure 5: Consumption Levels - Sequence of LM Statistics for Thresbold Test

Figure 6: Consumption Growth - Sequence of LM Statistics for Threshold Test, Interest Rate Threshold

Figure 7: Consumption Growth - Sequence of LM Statistics for Threshold Test, Error-Correction Threshold

Figure 8: Investrnent - Error-Correction Terms

Figure 9: Investment LeveIs - Sequence of LM Statistics for Tbreshold Test

Figure 10: Investment Growth - Sequence of LM Statistics for Thresbold Test, Interest Rate Thresbold

Figure 11: Investment Growth - Sequence of LM Statistics for Threshold Test, Error-Correction Threshold

GENERAL CONCLUSION

In this dissertation our objective is to test for possible asymmetries between

selected Iinks dong the path of the monetary policy transmission mechanism. If

asymmetries appeared to exist, we then attempted to model them. We relied on threshold,

non-parametric and neural network models.

The fust essay is concemed with testing for asymmetries between interest rate

spreads and inflation changes in the United States. Relaxing the assumption of constant

real interest rates in the model of Mishkin (1990), we show how non-linear relationships

can arise in theory. In our empirical work we confirm that a positive relationship exists

between long-short yield spreads and inflation changes, horizons that can br given a

monetary policy interpretation. However, non-linearity tests reveal that the relationship

displays marked evidence of asymmetries. When constructing threshold, neural network

and non-parametric models, we find that the relationship intensifies for negative values of

long-short yield spreads. Since such horizons occur when the yield curve inverts, which

can ofien be attributrd to a tightening of monetary policy, it is irnperative that policy-

makers be made aware of such non-linear relationships. Should policy-makers operate

under linear models, they would in fact be under-estimating the effect of a tightening in

policy on reductions in inflation. Although our models capture some of the apparent non-

linearities, our diagnostic tests reveal that the underlying non-linearities may in fact be

more complex than what our models can allow for. The innovations in this chapter are al1

related to the inference and estimation regarding non-linearities in this relationship. AI1

previous studies have dealt solely with linear models.

The second essay is largely concemed with testing for threshold effects between

hg-short yield spreads and output growth for the G-7 countries. Several authors have

found strong empirical relationships between the yield spread and output growth. in the

United States (Estrella and Hardouvetis ( 199 1 )), Canada (Cozier and Tkacz ( 1994),

Harvey (1997)) and other industrial countries. As an indicator of economic activity the

yield spread is certainly impressive, with several authors attributing its success to the fact

that i t can conveniently capture the position of monetary policy. Therefore in conjunction

with the recent literature that has uncovered asymmetries between monetary policy and

output growth (e-g. Cover (1 992) and Karras (1 996)), we test for possible non-linearities

between the yield spread and output growh as captured by a threshold effect.

Interestingly. we only find significant evidence of a threshold effect for the United States,

as the relationship appears to differ for large positive values of the long-short yield

spread. These instances can occur when monetary policy is highly expansionary, as short

rates are low relative to long rates. We find that when the spread is large, its effect on

output g rou~h is mitigated. consistent with arguments espousing that monetary policy can

slow the economy, but has difficulty stimulating it (the "pushing versus pulling on a

string" argument). In this chapter we also test more general non-lineanties, which may

not necessarily originate from the spread-output relationship. We find that with the

esception of Canada and Germany, non-linearities are quite apparent. Using neural

net~vork rnodels we can successfully capture these non-linearities for al1 countries, with

the esception of France for which our attempts at fully capturing the asymmetries proved

unsuccessful. More elaborate models, that can account more fully for the particularities of

the French economy, would need to be constructed.

The third essay is concemed with testing for asymmetries between money market

rates and commercial rates, consumption and investment. The objective is to determine

whether the links between the policy variable and intermediate variables in the

transmission are responsible for the asymmetries uncovered in the h t two chapters. We

find that asyrnmetries exist between al1 these variables, although the level of consumption

is more responsive to movements in money market rates than commercial rates.

Combining our results we c m arrive at several conclusions. The finding of

asymmetries between the Fed h d s rate and the intermediate variables in the

transmission mechanism is consistent with our finding of a threshold effect between the

yieid spread and output growth. In al1 instances we find that commercial rates.

consumption, investment and output are al1 less responsive (in terms of their expected

direction) to h d s rate movements when the fùnds rate is low. In these cases,

espansionary policy, as depicted by either a low funds rate or a large long-short yield

spread, has a lorver impact on output growth than a contractionary policy. This follows

since when policy expands and the hnds rate falls, commercial rates fa11 less rapidly,

consumption expands more slowly, and investment spending is virtually unaffected by

interest rate movements at low levels. In these cases we can therefore expect output

growth to expand less rapidly, which is consistent with our finding in Chapter 2.

Our inflation change mode1 finds that when policy is expansionary, inflation

changes are less pronounced than when policy is contractionary. Again, this is consistent

with the above resdts. If output growth is relatively lower when policy is expansionary,

then it will take longer for the output gap to widen and therefore for inflation to

accelerate. When policy is contractionary, we expect output growth to fa11 relatively more

quickly, perhaps because consumptior! and investment become more sensitive to interest

rate changes, and therefore the output gap to close more quickly, and hence for inflation

to decelerate more quickly in such a regime. The following Table sumrnarizes our results,

and indicates what our models predict (or, in italics, what we expect them to predict)

given an expansionary or contractionary policy initiative:

Summarizing the Policy Predictions for the United States

Mode1 Monetary Commercial Consurnption Output Inflation Policy Rates & Investment Growth

Intermediare ExpanSionary less I ~ S S Z ~ S S [ ~ S S Variables (low Fed fiuids) responsive responsive responsive responsive (Essay 3)

Contractionary more more more more (high Fed funds) responsive responsive responsive responsive

Output Expansionary -- Growth (high spread) (Essay 2)

Contractionary -- (low spread)

Inflation Expansionary -- Changes (high spread) (Essay 1)

Contractionary - (Io w spread)

less Iess responsive responsive

more more responsive responsive

less responsive

more responsive

The tetms "less" and "more responsive" refer to the absolute magnitude of the

actual (or anticipated) parameters for the variables in the monetary policy transmission

mechanism, in relation to expansionary and contractionary policies, They Say nothing

about the signs of the parameters, since we would expect these to be the same regardless

of the policy stance. For example, if the Fed fùnds rate is low, consumption will be less

responsive relative to the responsiveness of consumption when the funds rate is high. As

such, we would then expect output to be less responsive, since consumption has not

changed as much relative to the consumption change that would occur if the funds rate

were high. As can be seen, the results in al1 three papers are consistent. This therefore

leads us to conclude that, empirically, the effects of monetary policy on key

macroeconornic variables are asymmetrïc.

As stressed in the introduction of this dissertation, we have confined our analysis

to selected Iinks along the monetary policy transmission mechanism. It is hoped that our

findings will bring us closer to a better understanding of the relationship between key

macroeconomic variables and interest rates, which are a central tool of monetary policy.

However, there is still much work to be undertaken in order to more fiilly understand the

entire transmission mechanism, and potential non-linearities that lie therein. We still need

to gain a better understanding of non-linearities in the link between output growth and

inflation (e.g, Laxton, Rose and Tetlow (1993a,b)), non-linearities in exchange rates (e.g.

Meese and Rose (1991)) and the entire exchange rate channel, and also non-linearities

arnong interest rates of varying maturities along the yield c u v e (e-g. Pfann, Schotman

and Tschemig (1996)). As the tools of non-linear econometrics continue to be developed,

they will be applied to the aforementioned issues in an attempt IO M e r our

understanding of the macroeconomy.

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