RATIONAL EXPECTATIONS, BUBBLES AND MONETARY MODELS OF THE EXCHANGE RATE: THE AUSTRALIAN/US DOLLAR RATE DURING THE RECENT FLOAT*

AUSTRALIAN ECONOMIC PAPERS JUNE, 1990

RATIONAL EXPECTATIONS, BUBBLES AND MONETARY MODELS OF THE EXCHANGE RATE:

THE AUSTRALIAN1US DOLLAR RATE DURING THE RECENT FLOAT"

COLM KEARNEY and RONALD MacDONALD

University of New South Wales University of Dundee

I. INTRODUCTION

The trend towards more market-determined foreign exchange rates in Australia culminated in the free floating of the Australian dollar in December 1983. Although the Reserve Bank retained the discretion to intervene in order to test the market and to conduct smoothing operations, in practice this did not happen to any significant extent during the early years of the float. This makes the period since the beginning of 1984 in Australia an interesting testbed for investigating the appropriateness of a number of popular models of the exchange rate determination process.

This paper examines the role of fundamental economic variables in explaining movements in the AustralianIUS dollar exchange rate over the period January 1984 to December 1986. This is accomplished by first outlining in Section I1 various specifications of the monetary approach to exchange rate determination including the flexible price monetary approach (FLMA), the sticky price monetary approach (SPMA) and the real interest differential (RID) model. These models are tested in Section 111 using the recently developed cointegration methodology. Subsequent analysis in Section IV outlines the rational expectations monetary approach to exchange rate determination (REMAER) and tests the extent to which it is consistent with the data. These tests examine both the monetary model elasticity restrictions as well as the rational expectations restrictions. In

':'Previous versions of this paper have been presented at research seminars at the Universities of Adelaide, New South Wales and Western Australia and at the Reserve Bank of Australia. We are grateful to the participants whose comments led to much improvement. Financial assistance from the Reserve Bank of Australia's Economic and Financial Research Fund E/UNSW/8701 is gratefully acknowledged.

1

2 AUSTRALIAN ECONOMIC PAPERS JUNE

addition, a number of implied variance bounds restrictions of the REMAER are investigated. Section V then proceeds t o further examine the REMAER by investigating the extent to which speculative bubbles have been a feature of the Australian foreign exchange market during the recent period of floating rates. Finally, Section VI summarizes the paper and draws conclusions. Amongst the main findings are the existence of limited support for monetary models of the exchange rate determination process in Australia together with n o evidence of overshooting or of speculative bubbles.

11. REDUCED FORM MONETARY MODELS OF THE EXCHANGE RATE In this Section we present and estimate some reduced form equations which are

representative of the monetary approach to the exchange rate; namely, the FLMA, the SPMA and the RID models. All these reduced forms depend upon the following key equations:'

(mh - p*)t = By; - ai;

e e ( i - i y6) t = Ast = s ~ + ~ - st

where m denotes the money supply, p denotes the price level, y denotes real income, i denotes a nominal interest rate and s denotes the exchange rate (which is defined as the home currency price of a unit of foreign exchange). In addition, lower case letters denote natural logarithms,2 asterisks denote foreign magnitudes of the relevant variables and a n 'e' superscript denotes expectations. Equations ( l a ) and ( Ib) represent the familiar monetary equilibrium conditions for the home and foreign country while equation (2) is the uncovered interest parity (UIP) condition which obtains under the conditions of perfect capital mobility.

The Flexible Price Monetary Approach

The FLMA reduced form (see e.g. Bilson, 1978; Hodrick, 1978) may be derived in the following way. First, by subtracting ( lb) from ( l a ) we obtain a relative price equation (3) as

( p - p * ) t = (m - - P ( y - y '*) t + a(i - i")t (3)

If relative prices determine the exchange rate so that purchasing power parity (PPP) is assumed to hold,

S t = ( p - p") (4)

and using (2) with (3) in (4) we obtain the FLMA reduced form equation (5).

'Equations ( l a ) , a n d ( lb) are derived from a Cagan-type money demand function of the form M / P = YPexp-az, where the demand for money has been set equal to its supply.

'Taking logarithms of the exchange rate renders the analysis insensitive to the choice of numeraire and thus bypasses Siegel's paradox.

1990 RATIONAL EXPECTATIONS, BUBBLES AND MONETARY MODELS 3

(5) e

- P ( y - y * ) t + a Ast

This equation contains a number of well known predictions concerning the effects of variations in relative money supplies, income levels and interest rates on the exchange rate. In the next Section we consider this model further by explicitly modelling the expectations generating mechanism.

The Sticky Price Monetary Model

The assumption of continuous price flexibility is perhaps the FLMA’s most unappealing feature insofar as a number of researchers have demonstrated that an equation like (4 ) has not held for the recent experience with floating exchange rates (MacDonald (1988) provides further discussion of this issue). In the short run therefore, it seems more reasonable to assume that prices are sticky and do not instantly move to clear the money market so that PPP is violated. Dornbusch (1976) has presented a version of the monetary model which features short run price stickiness combined with the equilibrium PPP properties of the FLMA in the long run. This model is noted here in the briefest detail in order to derive a reduced form equation which can be estimated for Australia’s recent floating exchange rate experience.

Insofar as PPP does not hold in the SPMA model in the short run, an alternative equation is required to describe the evolution of the price level over time. The price level in the home and foreign country is accordingly assumed to evolve as:

st = ( m -

where Apt = pt+l-pt and d represents aggregate demand which is given by

Excess demand depends upon competitiveness, real income and the interest rate. By combining equations ( 6 ) and ( 7 ) with ( l ) , a relative price equation ( 8 ) can be obtained

which can be used to derive a reduced form equation which is representative of the Dornbusch model. It is assumed that exchange rate expectations evolve according to a simple regressive expectations scheme:

As: = I (C-s ) t 0 < 1 < 1 ( 9 )

where the long run equilibrium exchange rate ( S t ) is assumed to be proportional to the relative money supply term. By using equations ( 9 ) and (2) together with (6 ) - (S ) , Driskell (1981) has demonstrated that the following reduced form equation is obtained

where x‘ = (x-x:), x = m, p , y and


The first constraint simply states that PPP must hold in the long run in this model while the sign of I72 allows for the overshooting possibility of Dornbusch (1976).

The Real Interest Differential Model

Although the SPMA model constitutes a n improvement over the FLMA in describing short run adjustment, it retains a deficiency insofar as it abstracts from short run differences in inflation between countries. The RID model of Frankel (1979) constitutes areduced form which combines elements of both the FLMA (i.e., interest rates reflect inflationary expectations) andd the SPMA (i.e., interest rates reflect the real implications of monetary policy), while facilitating discrimination between these models. The RID model is derived by first assuming that PPP holds in the long run so that equation (5), written here as ( l l ) , provides a representation of the long run exchange rate (S)

where we have substituted (Ape-Ape*) for the interest differential. The regressive expectations scheme (9) is subsequently modified to include secular inflation rates:

AS:= 1 ( S - ~ ) t + (Ape-Ap*e)t (12)

Thus in long-run equilibrium, when s=S, the exchange rate is expected t o change by an amount which is equal to the expected inflation differential. By combining equation (12) with (2) we obtain

which states that in a world without short-run PPP, the current exchange rate differs from its long-run equilibrium in proportion to the real interest differential. By combining equations (1 1) and (13), we obtain the estimable reduced form

st = (m-rn")t + GO(y-y*)t + S1(Ape-Ap*e)t + 62 (i-i'*)t - (Ape-Ap*e)t (14) 1 1 which is identical to the FLMAreduced form (5) except for the addition of the real interest rate term. If the FLMA is correct, the coefficient 61 should be positive and 62 should be zero. Alternatively, if the SPMA is correct, 62 should be negative and 61 should equal zero. If the RID model is correct, however, both terms should be significant.

111. REDUCED FORM ESTIMATES We turn now to examine some empirical estimates of equations (5), (10) and (14) for the

recent Australian experience with floating exchange rates. The data set which is used in our study is described in the Data Appendix. A novel feature of our approach is that we utilise the recently developed cointegration methodology to test the validity of the reduced form


monetary equations. Before discussing the results, therefore, it is appropriate to give a brief account of the cointegration methodology.

The cointegration technique was pioneered by Granger (1986) and Engle and Granger (1987). A variable z is said to be integrated of order d , i.e., z - I(d) , if it has a stationary, invertable, non-deterministic ARMA representation after differencing d times. Consider two variables x and y , which are both I(1) processes (this is the most relevant from our viewpoint since, as we shall see below, the series we consider are all I(1)). Following Granger (1986), if there exists some constant a, such that

is I(O), then we say that x and y are cointegrated of order zero with the coefficient ‘a’being the cointegrating parameter.

A number of tests have been proposed in the literature to determine if x and y are cointegrated. Here we concentrate on three tests: the Sargan-Bhargava (1983) (DW) test, the Dickey-Fuller (DF) test and the augmented Dickey-Fuller (ADF) test of residuals from the cointegrating regression. The cointegrating regression for our model has the following form:

xt = b + ayt + nt

Stock (1984) has demonstrated that when x and y are cointegrated, OLS estimates of ‘a’ are consistent and highly efficient. Given OLS estimates of the residual series nt, tests of cointegration proceed by setting up the null hypothesis that x and y are not cointegrated.

Ho : x t , y t not cointegrated

In the first test of Ho, the estimated residuals from (16) are used to construct a Durbin- Watson statistic (known as the CRDW) and this is compared with computed value given in Engle and Granger (1987). If the estimated CRDW is above the critical vjilue, the null hypothesis of non-cointegration may be rejected. The CRDW test is .reinforced ‘by constructing DF and ADF statistics. These tests are computed by first running the following regression

and computing the ratio of /3 to its estimated standard error. The order of Q is set to ensure that the estimated residual series, gt, is white noise. If e=O the estimated t ratio is known as the DF statistic and for e>O the t ratio is known as the ADF statistic.

In Table I we present estimates of /3t in equation (17) for the variables used in our empirical study. When the levels of the variables are used, the null hypothesis of a unit root cannot be rejected for any series but when the first difference of the relevant variable is used the null hypothesis of a unit root is rejected in all cases. All our series are therefore I ( 1) processes and we are encouraged to proceed to the cointegration tests. An interesting


TABLE I

Tests for a Unit Root in the Levels and Changes of the Fundamental Series

DFIADF DFIADF

LS LM 1 LM3 LY RS RL LSP LP

-1.68(3) -1.81 -0.57 0.28( 1)

-1.68 -1.86 -2.76 0.8212)

ALS ALM1 ALM3 ALY ARS ARL ALSP ALP

-5.22 -6.69 -6.65

-14.08 -4.63 -6.65 -5.86 -3.05

Note: The null hypothesis is that the series in question is I(1). Approximate critical value at the five per cent level is -2.98 for 40 observations, with rejection region [8/8<-2.98] (Fuller, 1976).

implication of the fact that our exchange rate series and fundamental variables are all I( 1) processes is that it gives a first indication of the absence of speculative bubbles in the Australian foreign exchange market (see Hamilton and Whiteman, 1985). We shall consider the concept of bubbles in more detail in Section V. Since the cointegration methodology is intended to capture long-run relationships, we use it to test the FLMA model.

Table I1 presents the cointegrating regression^.^ Consider first the regressions of the spot exchange rate on relative money supplies (both narrow and broad definitions) and relative income levels. When the M1 definition of money is used (equation (5.2) in the Table) the coefficients on money and income are both wrongly signed and the null hypothesis of non- cointegration cannot be rejected using the DF statistic, although it is just rejected using the DW statistic. When M3 is used (equation (5.6) in the Table), the money term is correctly signed and the null hypothesis of non-cointegration is rejected on the basis of both the DW and ADF statistics. The income term, however, remains incorrectly signed which may indicate the paucity of our measurement of the relevant concept rather than empirical violation of the theory. Most researchers when estimating equation (5) use short term interest rates. Insofar as interest rates are proxying expected inflation in this model, however, long rates may be more appropriate and the Table presents estimates of equation (5) using both short and long rates. The short rates are correctly signed regardless of the monetary definition used (equations (5.4) and (5.8) in theTable) and the null hypothesis of non-cointegration is rejected on the basis of both the DW and ADF statistics. (The coefficients on income remain wrongly signed in both equations but the coefficient on

'Although the non-stationary nature of our variables implies that the estimated standard errors are dubious, we nevertheless report the t-ratios since this is common practice when reporting such equations.

TA

BL

E

I1

Som

e 'F

unda

men

tals

' Coi

nteg

ratio

n R

egre

ssio

n

W

W 0

St-1

R

2 SE

R

DW

D

F A

DF

Con

stan

t L

mlt

Yt

r; 't

(5.1

) St

=

0.93

4 0.

89

0.04

3 1.

70

-0.2

76

(17.

66)

(1.2

1)

(5.2

) S

t =

-0.0

76

2.00

6 0.

32

0.11

3 0.

44

-2.4

7 -3

.973

(5.3

) S

t =

-0.1

14

2.35

1 -0

.018

0.

58

0.09

1 1.

01

- 3.6

2"

-3.7

43

(0.2

7)

(4.4

1)

(4.4

4)

(2.2

0)

(0.1

5)

(3.0

7)

(2.1

6)

(5.4

) ~

t=

0.

356

0.73

7 0.

028

0.76

0.

068

0.61

-3

.7 3

(2) ''

-5.7

72

(1.1

2)

(1.7

1)

(7.7

7)

(5.0

5)

(1.0

3)

(3.2

4)

(7.7

5)

(4.4

6)

(3.1

0)

(5.5

) S

t=

0.26

4 1.

156

0.02

4 -0

.012

0.

85

0.05

5 1.

13

-3.4

2(2)

-5

.343

(5.6

) ~

t=

2.

906

0.44

1 0.

73

0.07

2 0.

55

-3.2

9(2)

""

-14.

22

(7.1

4)

(1.1

3)

(10.

18)

(5.7

) st

=

2.35

4 0.

946

-0.0

09

0.78

0.

066

0.89

-3

.39(

3)"

0.00

3 (5

.63)

(2

.38)

(2

.85)

(0

.34)

(5

.8)

St=

1.

285

0.35

4 0.

017

0.77

0.

067

0.52

-3

.57(

2)"

-8.8

1 (1

.71)

(0

.96)

(2

.50)

(3

.49)

(5.1

0)

Ast

=

0.00

0.

042

1.74

0.

008

(5.9

) st

= 0.

186

0.94

6 0.

022

-0.0

12

0.84

0.

058

1.01

-3

.83(

2)"

-5.0

2 (0

.27)

(2

.79)

(3

.72)

(4

.01)

(2

.18)

(1.1

5)

Not

es: :: an

d ':*

deno

te si

gnifi

canc

e at

the

five

and

ten

per

cent

leve

ls r

espe

ctiv

ely.

The

DW

sta

tistic

s are

all

sign

ifica

nt a

t the

five

per

cent

le

vel o

r be

tter

(So

urce

: Eng

le a

nd G

rang

er, 1

987)

.

m

X

m

C

m

m

m r

8 AUSTRALIAN ECONOMIC PAPERS J U N E

money is correctly signed in equation (5.4)). Using long rates instead of shorts (equations (5.3) and (5.7)) results in a negative coefficient on the relative interest rate terms although the null hypothesis of no cointegration is still rejected at the ten per cent level on the basis of the DR statistic.

We also estimate reduced form monetary equations with both short and long interest rate differentials entering jointly. These are interpretable as estimates of the RID model (14) with the short rates proxying real interest rates and the long rates proxying expected i n f l a t i ~ n . ~ Interestingly, only the broad money equation (5.9) indicates cointegration in this case. The coefficient signs on rS and rL conflict with the predictions of the RID model, however, and this may reflect the inability of long rates to adequately capture inflationary expectations in a commodity exporting country like Australia.

The cointegration regressions which are reported in Table I1 allow us to draw anumber of conclusions about the applicability of the FLMA and the RID models of exchange rate determination to the recent Australian experience with floating exchange rates. First, the cointegration of most equations implies that the economic variables which appear in the FLMA and RID models are capable of explaining long run movements in the Australian/US dollar exchange rate. Second, all equations expect (5.1) and (5.10) exhibit the existence of autocorrelation as indicated by low DW statistics and there are many wrongly signed coefficients. This implies that the models and/or their dynamics are misspecified which could result from the omission of important explanatory variables and/or the imposition of inappropriate elasticity restrictions. We shall pursue the latter possibility in the next section and concentrate here upon the SPMA model which incorporates more sophisticated adjustment dynamics between the exchange rate and prices.

Given the dynamic nature of the SPMA reduced form ( l o ) , it is not appropriate to treat this equation as a cointegrating regression. Our estimate of this equation is reported in Table 111. The first thing to note about equation (10.1) in the Table is that both Durbin’s h and the Ljung Box Q statistics indicate the presence of (at least) first order autocorrelation (the five per cent critical value for h is 1.645 and the number in parentheses after the Q statistic is the marginal significance level). It follows that either the economic or dynamic specification of the equation is faulty. In a bid to account for this autocorrelation we reestimated the equation assuming a n AR1 process for the error term’ and the resulting equation (10.2) appears free from autocorrelation. Some interesting conclusions emerge. First, there is no evidence of exchange rate overshooting since the coefficient on rnt is less than unity -even combining the coefficients on the current and lagged money terms gives a joint effect of 0.72. Second,the sum of the coefficients on ~ ~ - 1 , mt-1 and p t - 1 is considerably above unity (equal to 8.15) which confirms that PPP is violated for our data period. This partly explains the poor empirical performance of the FLMA model while simultaneously explaining the superior performance of the SPMA model. The poor showing of the former may also be due to inappropriate elasticity restrictions and/or a naive treatment of the expectations generating mechanism. We turn now to consider these issues.

4These are the variables used by Frankel (1979)

’The MA1 error is actually a composite error which consists of a period t f l forecast error plusa period t error which is added to equation (4).


TABLE 111

Empirical Estimates of the SPMA Model ~ ~~ ~~ ~

Estim- Equation ation Number Method Constant st-l mt mt-1 Pt-1 Yt Yf-1

(10.1) OLS -1.446 0.844 0.464 -0.285 0.437 -0.512 -0.113 (1.10) (7.71) (1.41) (0.86) (1.41) (1.35) (0.30)

(10.2) ARI -3.826 0.560 0.201 -0.522 0.863 -0.484 -0.014 (4.16) (5.64) (0.64) (3.62) (2.74) (1.20) (0.03)

Equation Diagnostics

Equation Number R2 SER h Q(l8)

(10.1) 0.89 0.043 1.60 41.17

(10.2) 0.83 0.044 0.81 24.21 (0,001)

(0.15)

Notes

1. R2, SER and h denote respectively the correlation coefficient, the standard error of the

2. The five per cent critical value for h is 1.645. 3. The number in parenthesis after the Q statistic is the marginal significance level.

regression and Durbin’s h statistic.

Iv. TESTING T H E RAMAER O N T H E A$/US$ RATE

One of the problems with the FLMA model is that if agents have static expectations, it is to be expected that a = 0 in equation ( 5 ) . An assumption of this nature is clearly of limited interest since it implies that although the exchange rate is free to fluctuate, agents do not expect it to change! A more realistic working hypothesis is to assume that agents form their expectations rationally so that the subjective expectation in ( 5 ) is equal to the rational expectation; ie. , s:+~ = E(st+l/lt) where E is the mathematical expectations operator and I t is the available information set (henceforth we shall use Et(st+l) in place of E(st+l/lt).

Imposing rational expectations on equation (5) and rearranging, we get


where x t = (rn-rn*)t - P(y-y* ) t .

Forward iteration of this equation yields

which can be rewritten as equation (20) by invoking the law of iterated projections together with the usual convergence assumption' and denoting ( l l l+a ) by y for brevity.

An important aspect of the REMAER which is described in equation (20) is that it views the exchange rate as depending upon expected future values of the economic fundamentals as well as upon their current levels. In what follows we conduct two types of test of the REMAER; f irst ly, we examine whether the embodied restrictions of the REMAER model are satisfied by the Australian data and secondZy, we examine whether the implied variance bounds of the model are consistent with the same data set.

In order to express equation (20) in terms of observable data, we follow the procedure adopted by Hoffman and Schlagenhauf (1983b) in specifying the generating process for the exogenous variables by Box-Jenkins (1970) procedures. All processes were identified as differenced AR1, and the general form adopted is consequently given by

where 8 = m, mq, y and y':

This procedure allows us to operationalize the REMAER model by expressing the period i predictions of the X vector given currently available information as

The resulting operationalized REMAER model is given by the joint estimation of equations (21) and (23) in order to account for the cross equation restrictions.

'The law of iterated projections implies in this case that

Et('t+i(St+i+~) ) = Et(st+i+l)

while the convergence assumption is that

Huang (1981) provides more detailed discussion

1990 RATlONAL EXPECTATIONS, BUBBLES AND MONETARY MODELS 11

This equation incorporates the monetary model restrictions (equal velocities and money demand elasticities with respect to income and interest rates together with unitary monetary elasticities) together with the rational expectations restrictions and the suitability ofthe processes (21) which are used to forecast the economic fundamentals. Similar tests of rational expectations models have been reported by Sargent (1978), Revankar (1980) and by Hoffman and Schlagenhauf (1983a, b).

Two generalizations of equation (23) allow us to test the rational expectations restrictions and the monetary model elasticity restrictions separately.

st = xt + emAmt + Bm-Amy4 + e y AY t + e y" . AY;

The first of these (24) identifies three restrictions relative to equation (23) which are due to the REMAER model specification (see Hoffman and Schlagenhauf, 1983b). These aregiven in equations (26).

e , ( ~ ~ ~ : e ~ : ~ - 8 . VQ .) - e .I. (emem-em+em) = o Y"- Y Y"'

while the second ( 2 5 ) relaxes the monetary model elasticity restrictions.

Table IV provides the results of testing these models together with the forecast-generating equations (21) for the exogenous variables on the Austra1ianKJ.S. dollar exchange rate during the recent floating rate period. The data was first differenced following the finding of a unit root in the relevant variables which was reported in Section 111. All equations have been estimated using the full information maximum likelihood (FIML) procedure. Equation (25) provides the estimates for the general unconstrained REMAER model while equation (24) provides the estimates of the monetary model with the relevant elasticity restrictions imposed. In each case, the figures in parentheses below the coefficient estimates are t-statistics while the equation diagnostics include the regression standard errors (SER), the Durbin-Watson statistics (DW), the log of the likelihood functions (LL), and the likelihood ratio (LR) together with the Wald (W) test statistics.

The central focus of the present exercise rests upon the extent to which the REMAER is consistent with the data for the Australian/U.S. dollar exchange rate. We first perform an LR test on equations (24) and (25) to examine the extent to which the elasticity restrictions


TABLE IV

Tests of the Rational Expectations and Elasticity Restrictions implied by the REMAER

Narrow money Broad money

Variable Equation Equation Equation Equation (11) (12) (11) (12)

-0.460 (1.91) -3.230 (1.50)

(0.27) 1.870

(1.29)

-0.176

-

-0.691 (0.59) -

-0.233 (0.63)

(2.74)

(0.66) 1.547

(0.88)

(0.77)

(0.87)

-4.134

-0.307

-0.419

-0.827

-

0.994 (1.05)

(0.48) -1.767

-1.332 (1.73)

-11.880 (2.77) -0.151 (0.24) 1.779

(1.16) -

-0.589 (0.50) -

-1.477 (1.61) 23.117 (4.69)

(0.81) 2.469

(1.41) -1.039 (0.64) -3.747 (1.24)

-0.565

-

1.690 (1.25) -0.784 (0.21)

Equation Diagnositcs

SER 0.050 0.048 1.43 0.081 DW 1.55 1.60 1.43 1.55 LL 494.364 498.523 561.632 565.208 LR 8.318 3.576 W 3.035 1.049

of the monetary model are sustained by the data. This is followed by conducting a W test to discover whether, in addition, the rational expectations restrictions are also consistent with the data. Both of these tests are asymptotically equivalent and are possessed of comparable power in small samples, so the choice in this context is based upon computational efficiency. In addition, both test statistics are distributed asX2(r) with rbeing the number of restrictions. With a five per cent critical value ofz2(3) = 7.81, it is clear from the Table that the REMAER cannot be rejected at this probability level for the broad monetary aggregate although the narrow money version of the model narrowly rejects the elasticity restrictions at the five per cent level of statistical significance (it accepts these restrictions at the six per cent level).


Having examined the extent to which the restrictions implicit in the REMAER model are consistent with the Australian/U.S. data, it is interesting to examine whether the same data set is consistent with a number of variance bounds tests which are derivable from the REMAER model. The original work in this area was performed on bond and stock price volatilityby Shiller (1979), Singleton (1980) and Le Royand Porter (1981). Huang (1981), Vander I<rats and Booth (1983) and Diba (1987) examine the extent to which various exchange rate models generate variance bounds tests which are consistent with the data. In what follows we implement the variance bounds tests of Huang (1981) as amended by Diba (1987) to ensure consistency between the measure of interest rates and the data periodicity. We shall not derive these tests here since they are readily available in the cited works. Suffice it to remark that Huang (1981) examines three variance bounds tests of the REMAER as given by equation (20). They are:

Var(st-xt) 2 a2Var(Axt)

where all variables retain their previous meanings and y = a / ( l+a ) .

Given the degree of uncertainty which surrounds empirical estimates of the relevant elasticities, we have conducted sensitivity tests to determine the extent to which variations in these magnitudes have implications for the variance bounds of the REMAER model. The results are presented in Table V for tests (27a), (27b) and (27c) respectively. Variance bounds Test 1 is upheld for both narrow and broad definitions of money for all possible elasticity combinations except for income and interest rate elasticites of the demand for money which are less than 0.50 and 0.05 respectively. Test 2 is upheld for narrow money for all possible elasticity combinations as long as the interest elasticity exceeds zero. This is not the case for broad money, however, which requires that the interest rate elasticity exceeds 0.10 if the income elasticity does not exceed 0.5. A similar pattern emerges for variance bounds Test 3 when narrow money is used as emerged for Test 1, although slightly greater restrictions on the elasticities are required to pass the test when broad money is used.

This finding sits in slight contrast to that which emerged from Table IV where the REMAER model with broad money accepted the elasticity restrictions with greater ease than when a narrow measure of money was used. Overall, however, the differing results are marginal and the REMAER model seems to be consistent with the Australian data for reasonable values of the relevant elasticities. It is interesting to note here that the superior empirical performance of the REMAER model relative to its FLMA counterpart may stem from its more sophisticated modelling of the expectations generating mechanism. It may also stem from instability in the money demand equations due to financial innovation which has made the relevant elasticities vary over time in a manner which is detrimental to the empirical performance of the FLMA model.

V. SPECULATIVE BUBBLES AND THE AUSTRALIAN FOREIGN EXCHANGE MARKET

In this Section we examine the REMAER model from a somewhat different perspective. More specifically, we turn to an explicit examination of the existence of rational speculative


TABLE V Variance Bounds Tests

Income Test 1 elasticity LHS RHS

interest rate elasticity Narrow .05 .10 .20 money

.50 ,0296 . 0 2 02 :x ,0809 ,3234 1.00 .0359 ,0362 .1446 ,5785 1.50 ,0432 .0626 ,2506 1.0023

Broad money

.50 .0133 0073" ,0291 ,1163 1.00 .0171 .0228 ,0910 ,3642 1.50 ,0221 ,0488 .1952 ,7809

Income Test 2 elasticitv LHS R H S

Narrow money

.50 .0018 1 .oo .0018 1.50 .OO 18

Broad money

.50 ,0018 1.00 ,0018 1.50 ,0018

interest rate elasticity .05 .10 .20

,0027 ,0047 ,0086 .0049 ,0084 ,0154 ,0084 0145 .0266

.0010" .0017" ,0031 ,0031 .0053 ,0097 ,0066 .0113 .0207

Income elasticity

Test 3 Narrow Money

.05 .50 LHS = .0036"

RHS = .0025 1.00 LHS = .0039

RHS = .0045 1.50 LHS = .0042

RHS = ,0079 Income

elasticity

interest rate elasticity .10 .20 .0024 ,0021 .0045 ,0084 .0025 ,0021 .0080 ,0150 ,0026 .002 1 ,0139 .0260

Broad Money

interest rate elasticity .05 .10 .20

.50 LHS = .0028* ,0022 .0020 RHS = .0009 ,0016 .0030

1.00 LHS = ,0030" ,0023 ,0020 RHS = ,0029 ,0023 ,0020

1.50 LHS = .0033 .0023 .0020 RHS = .0061 .0108 ,0203


bubbles in the Australian dollar over the recent period of floating rates. Consider again the rational expectations solution to the monetary model which is denoted here in first differences

In obtaining (20’) the no-bubbles or transversatility condition (28) has been imposed

On using the differenced ARI prediction formula of the previous section, this gives the no- bubbles solution as

where Act denotes the ‘no-bubbles market fundamentals’ solution. In the presence of a speculative bubble some extraneous event affects the exchange rate because everyone expects it to do so, the transversality condition will be violated and, as Blanchard and Watson (1982) point out, this results in a family of rational expectations solutions to equation (28). Denoting ct as a speculative bubble, any s t which satisfies

will also give a solution to equation (28). The purpose of this Section is to test st=Stagainst st = St + ct for some non-trivial value of c.

Following Meese (1986) and West (1986), this is conducted by first noting that in the presence of a speculative bubble, maximum likelihood estimates of equation (29) will produce an inconsistent estimate of y if a bubble term is correlated with the regressors in (29a). West (1986) has demonstrated that consistent estimates of y may nevertheless be obtained by estimating equation (5) using instrumental variables. More particularly, the McCallum (1976) technique of substituting st+l for E(st+l) and using an appropriate set of instruments provides a valid estimation strategy. Thus, on using st+l=sF+l+ut+l we get

where qt is a composite MA1 distance term. Given our specification, an appropriate instrument list consists of Ast-i and Axt-1, i2 l . Under the alternative hypothesis of bubbles, the McCallum technique ensures that the parameter estimates are consistent. The test is conducted by obtaining an OLS estimate of y from equation (29) together with an IV


estimate of y , specification test statistic (M)'

from equation (31) and using these estimates to construct the Hausman

Our estimation strategy proceeds as follows. First, we estimate equation (29) by FIML and equation (31) by OLS correcting the standard errors for the implied MA1 error term.' Table VI presents the results. Three sets of estimates of equations (29) and (31) have accordingly been computed using the data-determined income elasticity 0.26 (equation system 2) and two other values which bracket this estimate (namely 0.2 and 0.3). It is interesting t o observe that the estimated value for equation (29) results in a positive interest rate semi-elasticity, whilst that from equation (3 1) gives a negative value. This finding concurs with that of Meese (1986). Since the derivation of the m test statistic requires y to be less than unity, we use the value from equation (31) in calculating the rn statistic. In addition to the estimate of y from equation (31), we use estimates of u E from equation (29a) and of ex and u: from (29b). Our computed rn statistics for the three different specifications which are reported as the last row of the Table are all insignificant which implies that we cannot reject the null hypothesis of n o bubbles. This finding confirms our inference from the order of integration of the fundamental economic variables and exchange rates which was reported in Section 111.

2

VI. SUMMARY A N D CONCLUSIONS The purpose of this paper has been to examine the role of fundamental economic

variables in explaining movements in the Australian/US dollar exchange rate during the recent period of floating rates. The relationships which exist between the relevant variables were formulated in a manner which is representative of various specifications of the monetary approach to exchange rate determination. More specifically, both the flexible price and the real interest differential models were examined along with the sticky price model and the rational expectations version of the monetary approach to the exchange rate determination process.

The former two versions of the monetary model were confronted with the data utilizing the recently developed cointegration methodology. The results obtained indicate that although these models cointegrate, their specification is suspect and their explanatory power is low. The results from estimating the sticky price version indicate that exchange rate overshooting has not been a prominent feature of the recent Australian experience with floating rates. The monetary model was then tested along with its elasticity restrictions and

'This test statistic has a x 2 distribution with one degree of freedom:

T(hv - P i 2 111 -

y 2 ( I + Q , ) ~ 2( 1+ex)2( 1 - y ~ ~ ) ~ [ ( l - b ~ ) ~ + 2 b 2 ( 1-c)] + L7;

QX 2 0; 2 (1-c2)c2

Meese (1986) notes that the advantage of this version of the Hausman statistic is that the denominator must be positive, (us ingVAR(61~) and VAR(6) in the denominator would not necessarily produce an estimate of the difference which is positive).

*We have employed Hansen's (1987) method of moments procedure to correct the standard errors for conditional heteroscedasticity and the implied MA1 error process.

ID

ID 0

TA

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6.52

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0.19

4E-2

-

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28

2.58

9 29

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31

(0.2

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-2.0

67

28

4.30

4 29

-

31

(0.2

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-1.9

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67

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09

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2 U

0.

195E

-2

0.19

5E-2

2 0.

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-3

0.66

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-

0.70

7E-3

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3 z U

2 17 U

0.

154E

-2

0.00

3 (0

.007

) 0.

24

1.73

27.4

7

0.15

3E-2

0.00

4 (0

.007

) 0.

24

1.72

27.5

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-

0.15

1E-2

-

0.01

3 -0

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003

0.04

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06

0.26

1.63

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89

1.71

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20.8

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) 0.

03

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25.8

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R2

Q(1

5)

m=X(

1)

DW


those which are implied by the existence of rational expectations. Both sets of restrictions were found to be consistent with the data for broad and (to a slightly lesser extent) narrow monetary measures. The final sections of the paper provided further tests of the rational expectations monetary model by examining a number of implied variance bounds and by investigating the extent to which speculative bubbles have been a feature of the floating exchange rate experience. The results of these tests are supportive of the model for reasonable ranges of the money demand elasticities, while indicating the absence of speculative bubbles during the recent float. Overall, the results reported in this paper suggest that popular northern hemispherical models of the exchange rate determination process may require modification in the context of the Australian economy. This is a topic for further research.

DATA APPENDIX The variables which are used in this study, along with their definitions and data sources, are

presented in alphabetical order below. The sources are abbreviated as follows: International Financial Statistics (IFS) and Reserue Bank of Australia Bulletin (RBAB). All variables were sampled over the period 1984 (January) to 1986 (December).

M1, Ml" : Australian and US M1 measures of money, IFS. M 3 , M3::: Australian and US M 3 measures of money, IFS.

p , p " : Australian and US wholesale price indices, IFS. rS, r l : Australian and US interest rates on 3-month treasury bills, IFS re, r i : Australian and US yields on 5-year government bonds, IFS. S: AustralianIUS dollar exchange rate, RBAB.

y, y": Australian and US indices of industrial production, IFS


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Documents

RATIONAL EXPECTATIONS, BUBBLES AND MONETARY MODELS OF THE EXCHANGE RATE: THE AUSTRALIAN/US DOLLAR RATE DURING THE RECENT FLOAT*