A Monte Carlo study on the pitfalls in determining deterministic components in cointegrating models

Journal of Macroeconomics 27 (2005) 691–703

www.elsevier.com/locate/jmacro

A Monte Carlo study on the pitfallsin determining deterministic components

in cointegrating models

Goran Hjelm *, Martin W. Johansson

National Institute of Economic Research (NIER), P.O. Box 3116, SE-103 62 Stockholm, Sweden

Received 16 September 2002; accepted 31 March 2004Available online 8 August 2005

Abstract

Recently the use of the so-called �Pantula principle� has been suggested as a means of deter-mining deterministic components in cointegrating models (see Ahking in Journal of Macroeco-

nomics 24, 2002, and Hatemi-J in Economic Modelling 19, 2002). Moreover, the procedure issuggested in the widely used CATS in RATS program (see [Hansen, H., Juselius, K., 1995.CATS in RATS. Estima, United States]). In this paper, we examine, by means of Monte Carlosimulation, the properties of the �Pantula principle�. We investigate the five models containedwithin the Johansen methodology and find that the �Pantula principle� is heavily biasedtowards choosing the model with an unrestricted constant when the model with a restrictedtrend is the true one. We suggest a modification that reduces this bias to an important extent.� 2005 Elsevier Inc. All rights reserved.

JEL classification: C15; C32; C52

Keywords: Deterministic components; Cointegration; Monte Carlo simulation; Panutla principle

0164-0704/$ - see front matter � 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jmacro.2004.03.005

* Corresponding author. Tel.: +46 (0)8 4535900; fax: +46 (0)8 4535980.E-mail addresses: [email protected] (G. Hjelm), [email protected] (M.W. Johansson).

mailto:[email protected]

mailto:[email protected]

692 G. Hjelm, M.W. Johansson / Journal of Macroeconomics 27 (2005) 691–703

1. Introduction

Cointegration analysis has become standard practice in applied macroeconomicsduring the last decade. The most commonly used methodology is the maximum like-lihood based procedure due to Johansen (1988, 1995). In short, the Johansen proce-dure can be said to consist of three steps: (i) model specification, (ii) estimation of thevector error correction (VEC) model, and (iii) inference on the estimated VEC model.

Contrary to the Engle–Granger cointegration test (see Engle and Granger, 1987),the Johansen procedure is often heralded as a single-step method, as it determinesthe short-run adjustment parameters and the cointegrating vectors simultaneously.However, it is important to remember that for practical purposes, the Johansen pro-cedure is in fact a multistep procedure. This is due to the fact that the choices madeby the researcher in step (i) will affect estimation, inference, and interpretation insteps (ii) and (iii). Thus, selecting the wrong model specification in step (i) can leadthe researcher to draw wrong conclusions from the data.1 The implications of certaintypes of misspecification were recently raised by Ahking (2002). Using Johansen�smethodology, Ahking (2002) estimates a standard money demand model for theUS and shows that the conclusions are heavily dependent on the chosen determinis-tic components, but unfortunately many authors never explicitly make it clear to thereader why a certain deterministic specification is preferred over any other.

Five possible combinations of deterministic components are contained in theJohansen procedure (see Johansen, 1994, 1995). The most restrictive model (Model1) contains no deterministic components and the least restrictive model (Model 5)contains quadratic trends in levels. The five models are nested so that Model 1 is con-tained in Model 2 and so on. Hansen and Juselius (1995) suggest a method called the�Pantula principle� for simultaneously determining rank and deterministic compo-nents which is also included in the widely used CATS in RATS program package.2

Recent applications include Ahking (2002) and Hatemi-J (2002). The procedure isdescribed in Section 2.

The question is, however, whether the �Pantula principle� is a reliable procedure.For reasons outlined in Section 2, the �Pantula principle�, as suggested by Hansenand Juselius (1995), is flawed on theoretical grounds and in Section 3 we present aMonte Carlo study that illustrates this conclusion. More specifically, the �Pantulaprinciple� chooses Model 3 (unrestricted constant) when Model 4 (restricted trend)is the true one.3 To the best of our knowledge, noMonte Carlo study exists that inves-

1 Of course, as mentioned by an anonymous referee, all model building is an iterative process, andimposing invalid restrictions leads to problems in general.2 The principle, discussed first in Johansen (1992) for choosing only between Model 2 (restricted

constant, see Section 2) and Model 3 (unrestricted constant, see Section 2), is based on an idea put forwardin a univariate setting by Pantula (1989).3 Some material in the study by Doornik et al. (1998, p. 548) also indicates that the ability of the �Pantula

principle� to choose the correct model is less than satisfactory when considering Models 3 (unrestrictedconstant) and 4 (restricted trend). Using a data generating process (DGP) corresponding to Model 4, rank2 they evaluate the rejection frequencies of the Trace test for Models 2–4. They find that in many cases,Model 3 and rank one would be wrongly chosen.

G. Hjelm, M.W. Johansson / Journal of Macroeconomics 27 (2005) 691–703 693

tigates the properties of the �Pantula principle�.4 In Section 4, we suggest a modifica-tion of the �Pantula principle� and illustrate this modification with an empirical appli-cation. We summarize the results of the paper in Section 5 and suggest an easy-to-useempirical strategy of how to simultaneously determine rank and deterministic compo-nents in cointegrating models.

2. The �Pantula principle�

Based on the Johansen approach to cointegration (see Johansen, 1992, 1994,1995), Hansen and Juselius (1995) suggest a procedure to simultaneously determinerank and deterministic components in cointegrated VAR models.5 As mentionedabove, the method is also available in the widely used CATS in RATS programpackage. In short, five models with different deterministic components are consid-ered (see Johansen, 1995, Chapter 5, for a detailed description). Following Hansenand Juselius (1995), we have

Dzt ¼Xp�1

i¼1

CiDzt�i þ ab0zt�1 þ l þ dt þ et;

where zt is an n · 1 vector of stochastic variables, Ci is n · n where n is the number ofvariables, a is a n · r matrix of short run adjustment parameters, and b is a n · r ma-trix of r cointegrating vectors. If we decompose l such that l = al1 + a?l2 and dsuch that d = ad1 + a?d2, we can write6:

Dzt ¼Xp�1

i¼1

CiDzt�i þ a

b

l1

d1

0B@

1CA

0

zt�1 þ a?l2 þ a?d2t þ et. ð1Þ

The coefficients on the deterministic components are l1 (intercepts in the r cointe-grating relations), l2 (n � r linear trends in the data), d1 (linear trends in the r coin-tegrating relations) and d2 (n � r quadratic trends in the data).

4 A number of recent studies have performed Monte Carlo experiments on the Johansen procedure ingeneral. For example, Cheung and Lai (1993), Jacobson (1995) and Doornik et al. (1998) consider thebehaviour of the Trace test in small samples. Jacobson (1995), Zhou (2000) and Gredenhoff and Jacobson(2001) consider small sample issues in tests on the cointegrating vectors. Doornik et al. (1998), Boswijk andDoornik (1999) and Rahbek and Mosconi (1999) consider the Trace test when including exogenousvariables. Cheung and Lai (1993) examine the influence of lag length on the Trace test and Jacobson (1995)investigates the performance of the Trace test given non-spherical errors.5 There exist other methods of determining rank and deterministic components in cointegrating models.

For example, Nielsen and Rahbek (2000) show that rank determination of Models 2 and 4 (see Johansen,1994) do not depend on parameters of the deterministic components. Hence, the decision on the number ofcointegrating vectors can be separated from the decision of which deterministic components to include.Our (i.e., the modified �Pantula principle�, see Section 4) approach is different as the number ofcointegrating vectors and the deterministic components to include are determined simultaneously.6 Note that a 0a? = 0.


Model 1 has no deterministic components (d1 = d2 = l1 = l2 = 0). Model 2 has a(restricted) constant within the cointegration space (d1 = d2 = l2 = 0, l1 5 0).Model 3 has a (unrestricted) constant outside the cointegration space (d1 = d2 = 0,l1 5 l2 5 0). Model 4 has a (restricted) trend within the cointegration space(d2 = 0, d1 5 0, l1 5 l2 5 0). Finally, Model 5 has a (unrestricted) trend outsidethe cointegration space (d1 5 0,d2 5 0, l1 5 l2 5 0).

According to Hansen and Juselius (1995), one should first exclude models notplausible for the data set at hand which in most applications implies exclusion ofModel 1 and Model 5. The procedure is then as follows (assuming for simplicity thatall five models are worth considering). First, using the Trace test, test the nullhypothesis of zero cointegrating vectors for Model 1 (i.e., the most restricted model).If that hypothesis is rejected, the same hypothesis is considered for Model 2 and soon. If the hypothesis is rejected for the most unrestricted model considered (hereModel 5), the procedure continues by testing the null hypothesis of at most one coin-tegrating vector for the most restricted model considered (here Model 1). If thishypothesis is rejected, the same hypothesis is tested for Model 2 and so on. The pro-cess stops when the hypothesis is not rejected for the first time.

It is important to note, however, that this general procedure by Hansen and Juse-lius (1995) in which all five models are considered at the same time is not suggestedelsewhere in the literature.7 Instead, Johansen (1992, 1995) and Doornik et al. (1998)suggest that the above procedure can only be used when considering either (a) Model2 against Model 3 or, (b) Model 4 against Model 5. The reason is that the likelihoodratio test Q (i.e., the Trace test) has the following properties:

logQðModel 2ðrÞjModel 2ðnÞÞ ¼ logQðModel 2ðrÞjModel 3ðnÞÞ;logQðModel 4ðrÞjModel 4ðnÞÞ ¼ logQðModel 4ðrÞjModel 5ðnÞÞ;

while

logQðModel 3ðrÞjModel 3ðnÞÞ 6¼ logQðModel 3ðrÞjModel 4ðnÞÞ.Hence, when testing Model 3, rank 6 r, against Model 3, rank = n when Model 4,rank 6 r, is true implies that Model 3, rank = n, is a misspecified model.8 This meansthat the �Pantula principle� can not be used to choose between the (arguably) twomost realistic models in empirical macroeconomic models; Model 3 and Model 4.9

This is a severe deficiency and we will suggest a modification of the procedure in

7 Ahking (2002) and Hatemi-J (2002) incorrectly attribute the �Pantula principle� to Johansen (1992).Hansen and Juselius (1995) is the correct reference. Johansen (1992) only suggests the use of the �Pantulaprinciple� for choosing between Models 2 and 3 while it is clear from Ahking (2002, footnotes 12 and 17)that Model 4 also has been included when using the �Pantula principle�.8 We thank an anonymous referee for pointing this out, see Johansen (1995, p. 99).9 Ahking (2002) claims that Model 2 and Model 3 are the most plausible models in empirical

applications. We believe this is not the view of the profession today. Due to work by, for example,Doornik et al. (1998), Model 4 (instead of Model 3) is to be preferred in cases where the variables have alinear trend. In the application by Ahking (2002), Models 2 and 3 are considered in detail. We believe thechoice should have been between Models 3 and 4 as the included variables (real income for example) havelinear trends. Model 2 is therefore not a plausible alternative.


Section 4 to remedy this. First, however, we present a Monte Carlo study that con-firms that the original �Pantula principle�, as suggested by Hansen and Juselius(1995), does not work.

3. Monte Carlo setup

We consider the combinations of (i) three variables, rank one, and (ii) four vari-ables, rank 2, for sample sizes T = {100, 400} explicitly in the paper while other com-binations (not reported) have been tested as well.10 For each Monte Carlo replicatewe use a n-dimensional vector of zeros as an initial condition and discarded the first100 observations prior to applying the �Pantula principle�. Here follows the DGPs ofModels 2 through 5, listed in Johansen (1995).11

3.1. Model 2

In Model 2, we allow for a restricted constant:

Dzt ¼ ab0 zt�1

1

� þ et; ð2Þ

with12:

et iid Nð0;RÞ;where zt�1 ¼ x1t�1 . . . xnt�1½ �0, a ¼ 0.4�a0 and b ¼ �b l1

�0. For n = 3, we use

�b ¼ 1 �1 �1½ � and �a ¼ �1 1 1½ �. For n = 4, we use �b ¼ 1 �1 1 10 1 �1 1

�

and �a ¼ �1 0 0 00 �1 1 �1

� .13 The reason for choosing a fairly high adjustment

speed, 0.4, is to make sure that the Trace test actually finds the n series to be coin-tegrated for the DGP and sample size at hand, and hence the present study is not asmall sample study. Rather, we let the �Pantula principle� operate under the best of

10 We have run an exhaustive set of combinations of: n = {2, 3, 4}; r = {1, 2}; T = {50, 100, 200, 400}where n, r and T are the number of variables, cointegrating vectors, and observations, respectively. Thecombinations not shown in the paper show no important differences in results and are not presented tospare the reader from too many tables. They can, however, be received from the authors on request.11 In our experiments, we exclude Model 1 (no deterministic components) from the selection process. Thisis due to the fact that this specification is too restrictive in empirical macroeconomics. Moreover, we foundthat the performance of the �Pantula principle� was severely degraded when Model 1 was included and,hence, it is highly recommendable that the practitioner excludes Model 1 from the selection process.12 We set R = In. We have also applied non-zero values for the off-diagonal elements with no importantdifferences in the results.13 In our four variable system, we let some variables be weakly exogenous for simplicity (i.e. having a zeroa coefficient). This is common in the literature, see, among others, Benerjee et al. (1986), Engle andGranger (1987), Gonzalo (1994), Jacobson (1995), Haug (1996), Ostermark and Hoglund (1999), andZhou (2000).


circumstances (i.e., when the rank is clearly 1 or 2) to see if the �Pantula principle�works in a consistent manner in selecting correct deterministic components. Weuse two values for the restricted constant throughout the Monte Carlo experiments:l1 = {3,30}.

3.2. Model 3

Model 3 allows for an unrestricted constant:

Dzt ¼ a?l2 þ ab0 zt�1

1

� þ et; ð3Þ

with a?l2 = {0.3i,0.6i}, where i denotes a n · 1 vector of ones.

3.3. Model 4

Model 4 allows for a restricted trend:

Dzt ¼ a?l2 þ ab0zt�1

1

t

264

375þ et; ð4Þ

where b ¼ �b l1 d1

�0. We use two parameter values for the restricted trend:

d1 = {0.1,0.2}.

3.4. Model 5

Finally, Model 5 allows for unrestricted quadratic trends in the levels of the series,which implies an unrestricted trend in first differences:

Dz0t ¼ a?l2 þ a?d2t þ ab0zt�1

1

t

264

375þ et; ð5Þ

with a?d2 = 0.05i.14

3.5. Monte Carlo results

In this section we present the results of evaluating Models 2–5 using variousparameter combinations. As mentioned, we generate data under the null of one coin-tegrating vector when considering three variables, and under the null of two cointe-grating vectors when considering four variables. For each replicate, we perform the�Pantula principle� as described in Section 2. We thereby get information on boththe rank and model chosen. More specifically we focus on two issues: (i) how oftenthe �Pantula principle� chooses correct rank and correct model and, (ii) the most fre-

14 We have also tested a?d2 = {0.01i,0.1i} with no important differences in the presented results.


quent model actually chosen. The 5% significance level is used throughout the paper.The critical values for the Trace test are taken from Hansen and Juselius (1995), p. 80.

Without foregoing the results too much, the �Pantula principle� is heavily biasedtowards Model 3 when the DGP is Model 4. Due to this fact, we propose an exten-sion of the �Pantula principle� in Section 4.

3.5.1. Model 2

When examining Model 2, the sample size (T) as well as the restricted constant(l1) are varied, see Table 1. As the value of l1 goes to zero, Model 2 limits to Model1 (i.e. without any deterministic components). The �Pantula principle� chooses thecorrect rank and model in over 90% of our eight specifications, see the �p[correct]�-column. The model chosen most frequently is displayed in the �Most freq.�-columnand it is clear that the �Pantula principle� works well in our experimental design—it chooses correctly rank 1, Model 2 in specifications 1–4 and rank 2, Model 2 inspecifications 5–8.

3.5.2. Model 3

When examining Model 3, the sample size (T), the restricted constant (l1), and theunrestricted constant (a?l2) are varied. Our parameter settings do not play anyimportant part for the model chosen and the �Pantula principle� gets it right mostof the time, see the �p[correct]-column�. The reader may have noted that Table 2 in-cludes two more columns. These correspond to our modification of the �Pantulaprinciple� and will be discussed in Section 4.

3.5.3. Model 4

Contrary to the DGPs of Models 2 and 3 above, when examining Model 4, the�Pantula principle� chooses the incorrect deterministic component almost every time

Table 1Monte Carlo results for Model 2

Spec. T l1 p[correct] Most freq. (r:mod)

n = 3; r = 11 100 3 0.94 1:22 400 3 0.94 1:23 100 30 0.94 1:24 400 30 0.94 1:2

n = 4; r = 25 100 3 0.93 2:26 400 3 0.94 2:27 100 30 0.94 2:28 400 30 0.94 2:2

Note: 10,000 replicates for each specification (�spec.�) of the DGP shown in (2). �n� denotes the number ofvariables, �r� shows the simulated rank. The first column of the table is a simple counter. �T � denotes thenumber of observations, l1 is the restricted constant, �p[correct]� is the proportion of correct choices madeby the �Pantula principle�, �Most freq.� (r:mod) denotes the most frequent combination of rank (r) andmodel (mod) chosen by the �Pantula principle�.


Spec. T l1 a?l2 p[correct] Most freq. (r:mod) Modified �Pantula principle�

p[correct] (r:mod)

n = 3; r = 11 100 3 0.3 0.85 1:3 0.79 1:32 400 3 0.3 0.95 1:3 0.90 1:33 100 30 0.3 0.85 1:3 0.80 1:34 400 30 0.3 0.95 1:3 0.90 1:35 100 3 0.6 0.95 1:3 0.89 1:36 400 3 0.6 0.94 1:3 0.89 1:37 100 30 0.6 0.94 1:3 0.88 1:38 400 30 0.6 0.94 1:3 0.89 1:3

n = 4; r = 29 100 3 0.3 0.79 2:3 0.73 2:310 400 3 0.3 0.94 2:3 0.90 2:311 100 30 0.3 0.78 2:3 0.72 2:312 400 30 0.3 0.94 2:3 0.89 2:313 100 3 0.6 0.93 2:3 0.86 2:314 400 3 0.6 0.94 2:3 0.89 2:315 100 30 0.6 0.93 2:3 0.86 2:316 400 30 0.6 0.94 2:3 0.89 2:3

Note: See Table 1. a?l2 is the unrestricted constant. The last two columns concern our modification to the�Pantula principle� and will be dealt with in detail in Section 4.


(see the �p[correct]�-column in 3) and different parameter settings do not alter thisfinding. The most conspicuous item in Table 3 is that the �Pantula principle� is in-clined to pick Model 3 (see the �most freq.�-column). Simply put, the �Pantula prin-ciple� is not consistent when the DGP is given by Model 4. In Section 4 we thereforepropose a modification to the �Pantula principle� and there we will comment on thelast two columns in Table 3.

3.5.4. Model 5As can be seen in the last column of Table 4 the �Pantula principle� picks the right

model most of the time. Hence, analogous to Models 2 and 3 the �Pantula principle�works well given our parameter settings.

4. Modification of the �Pantula principle�

As was shown above, the �Pantula principle� is inclined to select Model 3 whenModel 4 serves as the DGP. Considering that Models 3 and 4 are likely to be the trueDGP in empirical macroeconomics, this is a severe drawback. It would have beenpreferable if the �Pantula principle� had mistaken Model 3 for Model 4 since thiswould only have entailed a loss of degrees of freedom. However, the situationencountered by us constitutes an �omitted variable� problem, which is more serious.As it stands, the �Pantula principle� is therefore, in our view, of limited use if this flawis not dealt with.


Spec. T l1 a?l2 d1 p[correct] Most freq. (r:mod) Modified �Pantula principle�

p[correct] (r:mod)

n = 3; r = 11 100 3 0.3 0.1 0.04 1:3 0.67 1:42 400 3 0.3 0.1 0.04 1:3 0.98 1:43 100 30 0.3 0.1 0.04 1:3 0.67 1:44 400 30 0.3 0.1 0.04 1:3 0.98 1:45 100 30 0.6 0.1 0.04 1:3 0.37 1:36 400 30 0.6 0.1 0.03 1:3 0.97 1:47 100 3 0.6 0.1 0.04 1:3 0.37 1:38 400 3 0.6 0.1 0.04 1:3 0.97 1:4

9 100 3 0.3 0.2 0.03 1:3 0.88 1:410 400 3 0.3 0.2 0.03 1:3 0.98 1:411 100 30 0.3 0.2 0.03 1:3 0.87 1:412 400 30 0.3 0.2 0.03 1:3 0.98 1:413 100 30 0.6 0.2 0.03 1:3 0.77 1:314 400 30 0.6 0.2 0.03 1:3 0.98 1:415 100 3 0.6 0.2 0.03 1:3 0.77 1:316 400 3 0.6 0.2 0.03 1:3 0.98 1:4

n = 4; r = 217 100 3 0.3 0.1 0.04 2:3 0.61 2:418 400 3 0.3 0.1 0.03 2:3 0.98 2:419 100 30 0.3 0.1 0.04 2:3 0.60 2:420 400 30 0.3 0.1 0.04 2:3 0.98 2:421 100 30 0.6 0.1 0.04 2:3 0.36 2:322 400 30 0.6 0.1 0.04 2:3 0.97 2:423 100 3 0.6 0.1 0.04 2:3 0.37 2:324 400 3 0.6 0.1 0.03 2:3 0.97 2:4

25 100 3 0.3 0.2 0.04 2:3 0.79 2:426 400 3 0.3 0.2 0.03 2:3 0.97 2:427 100 30 0.3 0.2 0.04 2:3 0.80 2:428 400 30 0.3 0.2 0.04 2:3 0.98 2:429 100 30 0.6 0.2 0.04 2:3 0.73 2:430 400 30 0.6 0.2 0.04 2:3 0.98 2:431 100 3 0.6 0.2 0.04 2:3 0.73 2:432 400 3 0.6 0.2 0.03 2:3 0.98 2:4

Note: See Tables 2 and 3. d1 is the parameter on the restricted trend.


We suggest the following simple modification.15 If the �Pantula principle� selectsModels 2, 4 or 5, accept the result. If Model 3 is chosen, estimate the VEC modelincluding a restricted trend (i.e. Model 4). Then perform a LR test for the signifi-cance of the parameter on the restricted trend. If the null of no trend is rejected,

15 A MATLAB program that performs our modified �Pantula principle� is available on request.


Spec. T l1 a?l2 d1 p[correct] (r:mod) Most freq.

n = 3; r = 11 100 3 0.3 0.1 0.94 1:52 400 3 0.3 0.1 0.95 1:53 100 30 0.3 0.1 0.94 1:54 400 30 0.3 0.1 0.95 1:55 100 30 0.6 0.1 0.94 1:56 400 30 0.6 0.1 0.94 1:57 100 3 0.6 0.1 0.94 1:58 400 3 0.6 0.1 0.95 1:5

9 100 3 0.3 0.2 0.94 1:510 400 3 0.3 0.2 0.95 1:511 100 30 0.3 0.2 0.94 1:512 400 30 0.3 0.2 0.95 1:513 100 30 0.6 0.2 0.94 1:514 400 30 0.6 0.2 0.95 1:515 100 3 0.6 0.2 0.94 1:516 400 3 0.6 0.2 0.95 1:5

n = 4; r = 217 100 3 0.3 0.1 0.93 2:518 400 3 0.3 0.1 0.94 2:519 100 30 0.3 0.1 0.93 2:520 400 30 0.3 0.1 0.94 2:521 100 30 0.6 0.1 0.93 2:522 400 30 0.6 0.1 0.94 2:523 100 3 0.6 0.1 0.93 2:524 400 3 0.6 0.1 0.95 2:5

25 100 3 0.3 0.2 0.70 2:526 400 3 0.3 0.2 0.95 2:527 100 30 0.3 0.2 0.93 2:528 400 30 0.3 0.2 0.95 2:529 100 30 0.6 0.2 0.93 2:530 400 30 0.6 0.2 0.94 2:531 100 3 0.6 0.2 0.93 2:532 400 3 0.6 0.2 0.94 2:5

Note: See Table 3. a?d2 is the parameter for the unrestricted trend, fixed and equal to 0.05.


select Model 4, otherwise select Model 3.16 In the case when the DGP is Model 3, thismodification can only deteriorate the performance of the original �Pantula principle�.As will be shown, however, the improvement of the results when the DGP is Model 4is greater.

16 This is in line with Nielsen and Rahbek (2000) who shows that the Trace (rank) test for Model 4 isasymptotically similar with respect to (among others) the parameter on the trend within the cointegrationspace (d1 in Eq. (4)). This implies that one can first test for the number of cointegrating relations in Model4 and then test restrictions on the parameters—i.e., the operation we carry out in our modification to the�Pantula principle�.


We begin by examining the outcome of our modification for Model 4, see the lasttwo columns of Table 3. For example, Specification 2 implies that the �Pantula prin-ciple� selects the correct model and rank in 4% of the replicates (see the �p[correct]�-column) which is a dismal result. However the LR test rejects the null of no trend inthe cointegration space in most of the cases where Model 3 was previously chosen.This implies that many previous replicates deemed to be generated by Model 3 turn(correctly) out to be those of Model 4 when applying our modification. According toour modified �Pantula principle� the correct rank and model are chosen in 98% (seethe second �p[correct]�-column, Spec. 3) of the replicates and Model 4, rank one, isobviously the most frequent model chosen, see the �r:mod�-column. Occasionally,when T = 100 the modified �Pantula principle� fails to identify the correct model(see Table 3, Spec. 5, 7, 13, 15, 21 and 23). This is of course due to the power ofthe LR test. Yet, by comparing the two �p[correct]�-columns in Table 3, it is clear thatour modification improves the result to an important extent.

It is important to note that a drawback of our modification is that it also tests forthe presence of a restricted trend when in fact there is no trend present, i.e. the DGPis Model 3. This means that the precision of our modified �Pantula principle� will beadversely affected as some replicates will incorrectly be relabelled as Model 4. InTable 2 we can see that the cost of this �distortion� is acceptable in relation to thecorresponding benefit in Table 3. In Table 2 we see that the proportion of correctcases is generally about five percentage points lower, using the our modified versioncompared to the original one. Most importantly, however, the most frequent rankand model chosen are still the correct combination. It is evident that the ability ofour modified �Pantula principle� to find the correct model depends on the propertiesof the data set at hand and the power of the LR test. If the trend is weak and/or thesample size is small, this will of course have an adverse effect on our modification.Hence, it could be advisable for a practitioner to estimate the VEC model underModel 4, and then conduct a power study on the LR test for the presence of therestricted trend.

4.1. An empirical application

In this section we will apply our modified �Pantula principle� to a real world dataset and compare the result of rank and model selection with that of the original �Pan-tula principle�. Binner and Elger (2002) investigate the demand for Divisia money inthe UK from 1980Q1 to 1999Q4 using a five-variate VEC model. The variablesincluded in their system are (log) Divisia money, (log) private final consumptionexpenditures in 1995 prices, an annualized 30-day interest rate, an annualized10-year bond rate and annualized inflation.17 They select a lag length of three inthe VEC model since this assures non-serially correlated errors. Calculating theTrace test for the data (excluding Model 1), yields Table 5.

17 The Divisia money index is a weighted average of notes/coins, non-interest-bearing bank deposits,interest-bearing bank sight deposits, interest-bearing bank time deposits, building society deposits, andnational savings investment accounts. See Binner and Elger (2002) for closer description of the data set.

Table 5Binner–Elger (2002) data set

Rank Model

2 3 4 5

r = 0 107.9* 89.8* 106.9* 105.0*

r 6 1 67.6* 50.9* 67.8* 66.2*

r 6 2 39.7* 26.5 37.2 35.8r 6 3 17.7 7.9 17.5 16.44r 6 4 6.2 1.7 6.0 5.6

Note: A starred entry, *, denotes rejection at the 5% level using critical values from Hansen and Juselius(1995).


As can be seen from Table 5 the �Pantula principle� picks Model 3. However, esti-mating the system, including a linear trend in the cointegration space and then per-forming an exclusion test on the trend gives a p-value of 0.042. Thus, we concludethat the trend should be included and that the correct specification is Model 4 andrank 2, rather than Model 3 and rank 2 as suggested by the �Pantula principle�.

5. Conclusions

This paper analyses the properties of the so called �Pantula principle� for thesimultaneous determination of cointegrating rank and deterministic component incointegrating systems. This procedure is, for example, included in the widely usedCATS in RATS cointegration package (see Hansen and Juselius, 1995) and has re-cently been used in applied work by Ahking (2002) and Hatemi-J (2002). We exam-ine the �Pantula principle� in detail and find that it performs well when the DGP isModels 2, 3 or 5. It is heavily biased, though, towards choosing Model 3 whenthe actual DGP is Model 4. We therefore suggest a modification of the method thatimproves the probability of choosing the correct model to a important extent (seeTable 3). More specifically, we propose the following strategy:

1. Exclude in advance models that are not compatible with economic theory and/orthe data set at hand. This especially concerns Model 1.

2. Use the �Pantula principle� on the specifications that are deemed plausible (i.e.,those not excluded in step one).

3. If the �Pantula principle� chooses Models 2, 4 or 5, accept the result.4. If the �Pantula principle� chooses Model 3, test for the presence of a linear trend in

the cointegrating space. If the null of no trend is rejected, choose Model 4. If not,choose Model 3.

Acknowledgements

Comments from Michael Bergman, David Edgerton, seminar participants atLund university, and an anonymous referee are gratefully acknowledged. The usualdisclaimer applies.


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Documents

A Monte Carlo study on the pitfalls in determining deterministic components in cointegrating models