A Principal Components Analysis of Common Stochastic Trends in Heterogeneous Panel Data: Some Monte Carlo Evidence

A PRINCIPAL COMPONENTS ANALYSIS OFCOMMON STOCHASTIC TRENDS IN

HETEROGENEOUS PANEL DATA: SOME MONTECARLO EVIDENCE

Stephen Hall, Stepana Lazarova and Giovanni Urga�

I. INTRODUCTION

Over the past few years increasing attention has been paid to the presenceof non-stationarity in panel data sets. This has involved both testing for unitroots within a panel and assessing cointegration. The main contributions inthis area are Kao and Chiang (1998), McCoskey and Kao (1998a), Pedroni(1997, 1998) and Phillips and Moon (1999). McCoskey and Kao (1998b)provide a detailed survey of recent developments and Monte Carlo compari-son of the tests.

Recent research has pointed out that in dealing with cointegrated paneldata sets it is important to examine not only the relationships between thedependent variable and regressors but also the structure of the regressors.Pesaran and Smith (1995) argue that in general in a heterogeneous panelwith individually cointegrated relationships the aggregated relationship doesnot cointegrate and that any panel data estimator which imposes homogene-ity across the panel will give inconsistent estimates of the long-run effects.However, when certain cointegrating restrictions are placed on the regres-sors, aggregation bias tends to disappear asymptotically. In a recent paper,Hall and Urga (1998) have shown that if each regressor in a panel is drivenby a single common stochastic trend and each unit cointegrates then astandard panel data estimator which imposes homogeneous parametersacross the panel will give rise to consistent estimates of aggregate long-runeffects even if the true model has heterogeneous parameters.

Further, Gonzalo (1993) derives various conditions that must be met ifthe aggregation of individual non-stationary series is to be meaningful.

OXFORD BULLETIN OF ECONOMICS AND STATISTICS, SPECIAL ISSUE (1999)0305-9049

749# Blackwell Publishers Ltd, 1999. Published by Blackwell Publishers, 108 Cowley Road, Oxford

OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.

�Corresponding author: G. Urga, City University Business School, Department of Investment,Risk Management and Insurance, Frobisher Crescent, Barbican Centre, London EC2Y 8HB, UK.E-mail: [email protected]. We wish to thank the Editor, Anindya Banerjee, for helpful comments.The usual disclaimer applies. S. Lazarova and G. Urga wish to acknowledge that this work is partof an ESRC research project (Grant No. R022251032) entitled Àn Analysis of the Importance ofCommon Stochastic Trends and the Methods of Selecting Them', with Stephen Hall.

Amongst those conditions, an important role is played by the number ofcommon factors shared by the individual series. Similarly, Ghose (1995)ascertains the need to know the number of common stochastic trends whenhe evaluates the problem of aggregating a subset of non-stationary series ina cointegrated time series regression. Hence it may often be important totest a panel for the number of common stochastic trends underlying eachregressor.

All existing tests for cointegration require the underlying variables to beintegrated of order one. This means that every test for the number ofcommon stochastic trends shared by any set of variables must be precededby testing for unit root, augmenting the degree of uncertainty alreadyinvolved in the testing procedure. In this paper we propose a new approachbased on principal components which will make it possible to test for thecommon stochastic trends regardless of the presence of stationary series inthe data set.

In addition to this, the principal components approach overcomes theproblem of large dimension typical for panel data sets. When the number ofseries approaches the number of time observations, it becomes impossibleto use the existing regression methods. The principal components analysis,instead, can in principle be applied to samples of any dimension. As theproposed test is asymptotic, we assess the empirical relevance of thetheoretical advantages of the method with a small set of Monte Carlosimulations.

The organization of the paper is as follows: Section 2 shows theimportance of common stochastic trends in panel data. Section 3 brie¯yintroduces the principal components estimation. In Section 4 we describe amethod for testing the number of common stochastic trends in panel data.Section 5 reports a series of Monte Carlo experiments and Section 6concludes.

II. COMMON STOCHASTIC TRENDS IN PANELS

The main motivation of the research on common factors in panels or inlarger time series systems is to determine whether certain suf®cient ornecessary conditions for aggregations are met, as illustrated in recent papersby Gonzalo (1993), Ghose (1995), and Hall and Urga (1998). The followingexample conveniently illustrates the issue. Consider a heterogeneous paneland let us suppose that yi, t and xi, t are I(1) and that there is a cointegratingrelationship between yi, t and xi, t for each group, with the parametersvarying randomly across groups, i.e. suppose that

yi, t � bixi, t � åi, t, i � 1, . . . , N , t � 1, . . . , T , (1)

where åi, t are stationary processes and bi's are assumed to have mean b,constant covariances ù2

i and ®nite higher-order moments and cross-moments. The randomness assumption is made only for convenience and

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our results are also valid for the ®xed coef®cient case. Pesaran and Smith(1995) argue that in general the aggregated relationship does not cointegrateand that any panel data estimator that imposes homogeneity across the panelwill give inconsistent estimates of the long-run effect.

Hall and Urga (1998) point out the case where each regressor is driven bya single common stochastic trend. This case can be demonstrated by thefollowing univariate example of a common trend model:

xi, t � ai f t � ìi, t, (2)

f t � f tÿ1 � î t, (3)

where ìi, t and î t are I(0) processes. Then f t becomes the commonstochastic trend which drives all N individual xi's. We can then express theaggregate relationship as

yt �

XN

i�1

aibi

XN

i�1

ai

xt �XN

i�1

ìi, t

XN

i�1

aibi

XN

i�1

ai

�XN

i�1

biìi, t �XN

i�1

åi, t, (4)

where

yt � 1

N

XN

i�1

yi, t and xt � 1

N

XN

i�1

xi, t:

The error term in this equation consists of a linear combination ofstationary components, so that it is itself stationary and the aggregateequation cointegrates. Then necessarily the yi's are driven by the same trendas the xi's. Moreover, the OLS estimator of the coef®cient of xt is consistentwith the rate of convergence being Op(Tÿ1). The long-run coef®cient of theaggregated relationship is therefore representative of the micro relationshipsin the sense that it corresponds to the weighted mean of the distribution ofthe micro coef®cients.

In the case of M regressors, similarly, if each variable shares a singlecommon trend, then again the aggregate relationship cointegrates and theregressand is driven by the same M trends as the regressors. From thisanalysis it can be seen that under the assumption of cointegration in eachunit, we only need to examine the common factors present among the right-hand-side variables, as the left-hand side cannot be driven by any additionaltrends.

This result can be generalized even further using suf®cient condition foraggregation given by Gonzalo (1993). In fact, if there are M regressors ineach unit and if there is at least one cointegrating vector combining theregressors and the regressand in each unit, then for the aggregate regressionto cointegrate it is suf®cient that the whole group of N 3 M variables on

PRINCIPAL COMPONENTS ANALYSIS 751


the right-hand side share no more than M common stochastic trends. Thenagain the dependent variable is driven by these M trends and, provided thereis only a single cointegrating vector in each unit, the coef®cient of thisregression can be shown to have the same representation as in the case ofone regressor: they correspond to the weighted means of the distribution ofthe micro cointegrating vector. That is to say, if there is a suf®cient amountof cointegration across units in a panel, supported by additional cointegra-tion within units, then the micro variables can be meaningfully aggregated.However, the last generalization requires the variables to be measured in thesame units (see Johnston (1984), Snell (1999), Phillips and Ouliaris (1988)).

Yet the knowledge of the number of common stochastic trends shared bya set of variables is not only relevant for the case of dimensionalityreduction, it is also of great interest per se. For instance, we may beinterested in explaining the dominant components of an economic system.Isolating common stochastic trends is potentially interesting in cases inwhich the factors can be identi®ed with underlying macro variables (seeGeroski, Lazarova and Urga, (1999)).

It is therefore not only important to know if there is cointegration amongthe series of the panel but also how many common stochastic trends mayunderlie the non-stationary panel. The following two sections describe apossible way to test for this.

III. PRINCIPAL COMPONENTS ESTIMATION

In this section we propose a procedure based on principal componentswhich can be used to ®nd the number of common stochastic trends possiblyunderlying a series in the panel. The method exploits the fact that if in thesystem of n I(1) series there are r cointegrating vectors then the ®rst rprincipal components are stationary and the remaining nÿ r are non-stationary.

The principal components method, which uses an estimate of the basis ofthe space spanned by cointegrated vectors, has several advantages over testsbased on residuals from a cointegrated regression. Unlike residual-basedprocedures, this method does not require restrictions to be placed on thedirection in which the cointegration occurs in order to identify the elementsof the cointegrating vectors. Furthermore, the optimality of the estimatorsand the resulting hypothesis tests are not dependent on these identifyingrestrictions being valid. This feature will be particularly useful in our testingprocedure, as we abstract from the left-hand-side variables and do not wantto assign priority to any of the regressors.

Our procedure generalizes the existing research in that it allows for thepresence of stationary series in the sample otherwise consisting of non-stationary series. We show that the principal components method correctlyestimates the number of independent non-stationary and stationary combi-

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nations of the mixture of I(1) and I(0) variables in the sample. Moreover,both stationary and non-stationary space are estimated consistently.

Consider the heterogeneous model (1):

yi, t � bixi, t � åi, t, i � 1, . . . , N , t � 1, . . . , T , (5)

where the coef®cient bi varies randomly across units and where the errorsåi, t are stationary, i.e. for each unit either the micro relationship aggregatesor both yi, t and xi, t are stationary. Then the common non-stationary factorsshared across xi, t will be shared across yi, t and vice versa. The attentiontherefore needs to be focused on the assessing the number of commonstochastic factors present among xi, t. Thus, the analysis in the rest of thepaper is conducted by considering xi, t only.

In order to simplify our presentation, let us view the samplexi, t (t � 1, 2, . . . , T, i � 1, 2, . . . , N ) as an N -dimensional time series xt.Suppose now that xt is generated by

á9Äxt � ut, (6)

ã9xt � v t, (7)

ä9xt � wt, (8)

where

(áãä) � A G 0

0 0 D

� �, (9)

A is a full rank n 3 k matrix, G is a full rank n 3 r matrix such thatG9A � 0, and D is a full rank s 3 s matrix. Here n � k � r represents thenumber of non-stationary series in the sample and the remaining s series arestationary. If we restricted our model to equations (6) and (7) with á � Aand ã � G, we would get a sample of n non-stationary series with rcointegrating vectors and k common stochastic trends.

We assume that the N 3 1 random vector î t � (u9t, v9t, w9t)9 is a zeromean stationary time series satisfying the functional limit theorem

Tÿ12

X[Ts]

t�1

î t ) Bî(s), 0 , s < 1, (10)

where Bî is an N -dimensional Brownian motion with covariance matrix

Ùîî �X1

j�ÿ1E(î tî9tÿ j): (11)

In what follows, we assume that the number of common stochastic trends kis known. Let a matrix â comprise matrices ã and ä, â � (ã, ä), and letzt � (v9t, w9t)9. Then



â9xt � zt, (12)

so that the columns of â span the space of all stationary combinations of xt,whereas the vectors in á constitute a base of the space of all non-stationarycombinations of xt.

We base the principal components procedure on the matrix

M x �XT

t�1

xtx9t: (13)

We de®ne â to be the orthonormal N 3 (r � s) matrix of eigenvectorscorresponding to the smallest r � s eigenvalues of the matrix M x. Similarly,the orthonormal eigenvectors corresponding to the largest n eigenvalues aretaken to be á. The following lemma from Harris (1997), with a slightmodi®cation of the proof, gives asymptotic properties of the estimators áand â:

Lemma 1. Let xt be generated by (6)±(9). Then

á � á(á9á)ÿ1á9á� Op(Tÿ1), (14)

â � â(â9â)ÿ1â9â� Op(Tÿ1): (15)

Furthermore, by transposing (15) and multiplying it by zt, we directly getthe following useful result:

Lemma 2. Let xt be generated by (6)±(9). Then

zt � â9xt � â9â(â9â)ÿ1zt � Op(Tÿ12): (16)

The results in Lemma 1 show that the difference between estimate á and itsprojection into the space spanned by á asymptotically disappears and henceá is a superconsistent estimator of a basis of the space of the non-stationaryspace. Likewise, columns of â asymptotically span the stationary space.Lemma 2 states that asymptotically the smallest r � s estimated principalcomponents will be stationary.

The essence of the principal components method is that it ranks indepen-dent linear combinations of xt by their variance. Thus if there are k non-stationary combinations among the components of xt, then the variance ofthe largest k principal components will be of order Op(T ), while thevariance of the remaining N ÿ n components will behave as Op(1). Hencewe can intuitively suggest the principal components method for estimatingbases of two sub-spaces of the N -dimensional space, stationary and non-stationary, spanned respectively by á and â.

However, we cannot use the method to break the space of â further intotwo sub-spaces spanned by ã and ä. Stationary combinations of I(1)variables will have variance of the same order as linear combinations of

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I(0) variables and the principal components method will not be able toseparate them in their variance ranking. Thus in the case when there areboth I(1) and I(0) variables in the sample, we can use the method to makeinferences about the number of common stochastic trends in the sample butwe cannot test for the number of cointegrating combinations among thenon-stationary variables within the sample.

We must be careful about the interpretation of common non-stationaryseries in a sample where not all of the variables are non-stationary. Onepossible way to understand this, and one which in fact motivated part of ourresearch, was the idea that in a panel of microeconomic units, the movementof each individual series follows an autonomous path, reacting at the sametime to the macroeconomic impulses which in¯uence all the economy. Thestationarity of part of the series in the panel means that these series areinsensitive to the macroeconomic trends. In other cases, touched on forexample by Granger (1993), the autonomous part of the movement may benon-stationary, containing stochastic trends which are not shared by otherunits and which are not signi®cant in the overall analysis.

So far we have been considering only the case in which the series in thesample do not contain deterministic terms. If such terms enter the system,we have to obtain the residuals from the regression of xt on the deterministicterms. With the demeaned or detrended data we proceed exactly as before.In line with the argument of Snell (1999) and Harris (1997), it can be shownthat the conclusions made for series without deterministic components carryover to the samples in which the deterministic terms have been removed.

Further, note that when the number of series exceeds the time dimensionof the sample, N > T , the last T ÿ N principal components are by de®ni-tion zero.

In the case of the multivariate model, where

yi, t � b1ix1i, t � b2ix2i, t � . . . � bMixMi, t � åi, t, i � 1, . . . , N , t � 1, . . . , T ,

and where each unit cointegrates, if all N 3 M right-hand-side series shareno more than M common stochastic trends, then the aggregate series alsocointegrates. In order to apply our procedure in the multivariate case, weonly need to change our procedure slightly: we pool the regressors togetherin the sense that we regard them as a unique set of N 3 M series. Naturally,the number of series considered N will be replaced by N 3 M. As thetesting procedure in the multivariate case is a simple modi®cation of theunivariate case, in the next section we present only the one-variable case.

IV TEST FOR COMMON STOCHASTIC TRENDS

We now turn to the description of the use of the principal componentsanalysis to test for common stochastic trends in univariate panel data sets.In testing for k versus k � 1 common stochastic trends we adopt the methodsuggested by Snell (1999). We specify our model as a VARMA process



á9Äxt � ut � a�È(L)å1 t, (17)

â9xt � zt � b�Ö(L)ÿ1å2 t, (18)

where á and â are de®ned in (9) and (12) respectively, a is a k 3 1 vectorand b is an (r � s) 3 1 vector of ®xed coef®cients, È(L) � I �È1 L�È2 L2 � . . . is an in®nite-order invertible matrix lag polynomial,Ö(L) � I �Ö1 L�Ö2 L2 � . . . � Ö p L p is an invertible, pth order matrixlag polynomial. To ensure that this representation exists, we require that å1 t

and å2 t be vectors of identically distributed innovations with ®nite fourthmoments and full rank covariance matrix Ó. We assume that the lag order pis known and in our Monte Carlo experiments we examine the changes insize and power of the test when the researcher includes more lags in theregression than there are in the DGP. We concentrate on the case of driftlessnon-stationary processes and set a � 0 in (17). As we use demeaned datathroughout our experiments, we also set b � 0 in (18) without loss ofgenerality.

Under the null of k common stochastic trends, the (k � 1)th largest (i.e.(r � s)th smallest) principal component will be stationary, while under thealternative of k � 1 common stochastic trends this component will be non-stationary. Let z t denote the smallest r � s estimated principal components,

z t � â9xt: (19)

The regression equation for the (r � s)th stationary component is

z r�s, t � ör�s11 z1, tÿ1�ör�s

21 z2, tÿ1 � . . .� ör�sr�s,1 z r�s, tÿ1�ör�s

12 z1, tÿ2

�ör�s22 z2, tÿ2 � . . .� ör�s

r�s,2 z r�s, tÿ2 � . . .� ör�s1, p z1, tÿ p

�ör�s2, p z2, tÿ p � . . .� ör�s

r�s, p zr�s, tÿ p� år�s, t, t� p� 1, . . . , T , (20)

where ör�sij denotes the ith element of the (r � s)th row of Öj. This equation

can be written in a stacked form as

dt � Zì� e, (21)

where dt is the (T ÿ p) 3 1 vector of observations of the (r � s)th principalcomponent zr�s, t, Z is the (T ÿ p) 3 (r � s) p matrix of regressors, ì is an(r � s) p 3 1 vector of coef®cients and e is the (T ÿ p) 3 1 vector of errorsår�s, t. Taking ®rst differences gives

Äz r�s, t� ör�s11 Äz1, tÿ1�ör�s

21 Äz2, tÿ1 � . . .� ör�sr�s,1Äz r�s, tÿ1�ör�s

12 Äz1, tÿ2

� ör�s22 Äz2, tÿ2 � . . .�ör�s

r�s,2Äz r�s, tÿ2 � . . .� ör�s1, p Äz1, tÿ p

�ör�s2, p Äz2, tÿ p� . . .� ör�s

r�s, pÄz r�s, tÿ p�Äår�s, t, t� p�1, . . . , T ,

(22)

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where år�s, t are residuals from regression (20) conducted in unobservablevariables zt, and where

år�s, t � år�s, t � Op(Tÿ12), (23)

which can be shown using Lemma 2.Equation (22) is a (vector) ARMA(p, 1) model in Äz r�s, t up to an

asymptotically vanishing term. Under the null, the regression is overdiffer-enced with MA parameter è equal to unity. We use Snell's closed-formestimator of è:

è � 1ÿ 1

T

Xt

z2r�s, tX

t

Äz2r�s, t

for1

T

Xt

z2r�s, tX

t

Äz2r�s, t

< 1,

è � 0 otherwise: (24)

Snell shows that under the null,Xt

z2r�s, tX

t

Äz2r�s, t

� Op(1), (25)

whereas under the alternative this ratio is of order Op(T). ThereforeT (1ÿ è) is Op(1) under the null and Op(T ) under the alternative, so theestimator è is T -consistent.

The next step of the procedure is to back®lter the variables in regression(22),

Äzft �

XTÿ1

j�0

è jÄz r�s, tÿ j, (26)

and estimate the parameter ì in the back®ltered regression

Äd f � Ä Z f ì� Äe f , (27)

where the notation is the same as in equation (21). The estimate ì f of theparameter ì in (27) is used to construct the series

ç � d ÿ Z ì f : (28)

Further, the Breusch±Godfrey (BG) regression of residuals on theindependent variables and lagged residuals is estimated as

ç � [ Zjçÿ1][ù9zjùe]9� í � Z�ù� í, (29)



where çÿ1 is the series ç lagged once, ùz is an (r � s) p 3 1 vector ofcoef®cients and ùe is a scalar.

The BG test is written as the Wald test of H0 : ù � 0 versus Ha : ù 6� 0:

BG � ù9[Z�9Z�]ù=ó 2, (30)

where ù is the OLS estimate of ù from (29) and ó 2 � ç9ç=T . Snell provesthat the BG statistic is asymptotically distributed as a ÷2 with one degree offreedom under the null of k common stochastic trends while it is of orderOp(T ) under the alternative of k � 1 common stochastic trends.

When N > T , the procedure has to be modi®ed. Only the non-zeroprincipal components will be included in the matrix of regressors in Z andall the dimensions of the remaining vectors change accordingly.

V. A SMALL MONTE CARLO STUDY

In this section we examine the small sample properties of the test of thenumber of common stochastic trends in the system of N variables describedby equations (6)±(9). We focus our attention on the effects on small sampleproperties of the following:

(a) the number of stationary series present in the sample of otherwisenon-stationary series;

(b) including more lags in the regression than there are in the sample;(c) modifying the è estimator;(d) the difference between the actual number of common stochastic

trends in the sample and the number being tested for;(e) the case of N > T .

To restrict the number of parameters under scrutiny and to focus on theselected issues we consider only the case in which the series ut, v t and wt

are independent standard normal random variables. In the estimationprocedure we set the number p of lags in the regression either to 0, whichre¯ects the form of the underlying series, or to 1, which will illustrate theeffect of the presence of redundant lags on the test size and power. Weassume that the absence of deterministic terms is known so we do notdetrend the data before carrying out the testing procedure. The matrix G inequation (9) is of the following form:

Gij � 1, j � i, Gij � ÿ1, j � r � 1 and Gij � 0 otherwise:

Matrix A is constructed to satisfy the condition G9A � I . Matrix D is anidentity matrix of rank s. The matrices A, G, D are then conformablycompleted with zeros to form matrices á, ã and ä. In all experiments wereplicate the simulations 2000 times. We consider values of T of 30 and 100which are representative of the range of annual and quarterly data inempirical applications. The number of variables N ranges from 5, re¯ectinga standard time series case, to 90 when T � 30, relevant for many panel

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data sets. The number of common stochastic trends among the non-stationary series in the sample goes from 0 to nÿ 1. For the modi®cation ofthe Snell estimator of the MA parameter è (see Section 5.3 below), weconsider values of ð � 0:5 and 0.1. In keeping with our aim of avoiding anypretesting of the stationarity of the series, we let the number of stationaryseries contaminating the non-stationary sample vary between 0 and 85,allowing for the case in which there are in fact few non-stationary series. Inall experiments the nominal signi®cance level is set to 5 percent.

5.1. The Presence of Stationary Series in the Sample

The ®rst question we try to answer is how the properties of the test in asmall sample change when we include stationary series in the sample. Weconducted two sets of experiments. First, we took a ®xed number of non-stationary series and added different numbers of stationary series. In thesecond set, we ®xed the total number of series in the sample and changedthe proportion of stationary and non-stationary variates. For both cases weconsidered various numbers of common stochastic trends in the non-stationary part of the sample.

Table 1 reports the results of the experiments where the number n of non-stationary series is ®xed to ®ve, which is relatively small in comparison tothe time dimension. The number of stationary processes starts from zero,which is the standard case in time series analysis, and goes up to 20, whenthe number of stationary series in fact exceeds n considerably. From the ®rstblock of Table 1 it is evident that the size of the test is quite satisfactorywhen the number of common stochastic trends k is small. Further, the sizeis better the higher the number of stationary processes. Intuitively this maybe explained by the fact that the more stationary components there are inthe original data the more likely they are to be captured in the principalcomponents by a set of highly stationary components and hence thedistinction between the stationary and non-stationary components becomessharper. We can also see that for a low number of k the size is not verysensitive to the number of stationary series.

The power of the test is highest when there are a small number ofcommon trends. Unlike size, power deteriorates with increasing number ofI(0) variables in the sample. When T increases to 100, the power improvessubstantially, while size does not change dramatically. To eliminate theeffect of increasing the number of variables, N , on the power, we conductedthe second set of experiments where N is ®xed and only the percentage ofstationary series changes. The results are reported in Table 2. Our conclu-sions agree with those from the previous experiment except for the powerwhen T � 30.



5.2. Higher Number of Lags in the Regression than in the DGP

Since in practice the number of lags in the underlying AR process is neverknown, we want to assess the effect of including a higher number of lags inthe regression. For this purpose we repeated the experiment in Table 1 withone lag in our regression ( p � 1). The results are reported in Table 3. Theoverall picture is that there are big losses in both size and power involved.This indicates that it is important to be parsimonious in the number of lags.

5.3. Modi®cation of the Snell Estimator

The Snell estimator of T (1ÿ è) is of order Op(1) under the null and oforder Op(T ) under the alternative. If we divide the ratio in the estimator byT ð instead of T (0 ,ð < 1), the test becomes only T ð-consistent but thereis a potential gain in power. As ð decreases, both size and power of the testfor a given T increase. In small samples, there may be a region in which

TABLE 1Size and Power of the Test of the Number of Common Stochastic Trends for a FixedNumber of Non-Stationary Series and Increasing Number of Stationary Series in

the Sample

Size Power

T � 30, n � 5, ð � 1, p � 0

s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 40 6.8 11.5 17.5 39.3 62.6 41.4 40.1 52.31 7.3 11.7 12.5 21.4 59.4 30.0 27.3 29.32 7.0 10.0 11.5 16.4 56.8 27.2 23.4 23.35 6.6 9.8 8.0 9.5 47.2 21.5 14.8 12.210 7.1 9.1 8.5 7.3 40.1 17.8 11.8 8.820 7.5 8.7 7.3 6.5 35.0 13.1 9.9 7.8

T � 100, n � 5, ð � 1, p � 0

s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 40 9.9 9.2 21.0 41.8 100.0 99.7 96.7 100.01 6.0 7.7 14.8 21.9 100.0 98.2 85.4 74.32 5.7 5.4 10.2 24.1 100.0 99.0 95.3 83.05 5.7 6.0 9.0 11.3 100.0 98.7 96.5 88.710 5.4 4.9 6.0 9.1 99.8 98.3 93.3 82.220 4.3 5.5 6.9 10.7 99.8 96.8 88.3 70.5

Notes:T � number of time observations; n � number of non-stationary series in the sample; s � number ofstationary series in the sample; k � tested number of common stochastic trends among the non-stationary series in the sample (for the power calculation the true number of trends is k ÿ 1);p � number of lags in the VAR regression; ð � exponent of the modi®ed Snell statistics (see Section5.3). Nominal level of signi®cance is 5 percent.

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TABLE 2Size and Power of the Test of the Number of Common Stochastic Trends for a FixedNumber of Series and Variable Proportion of Non-Stationary and Stationary Series

in the Sample

Size Power

T � 30, N � 30, ð � 1, p � 0

n s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 4

5 25 6.0 7.7 7.5 6.3 29.6 12.6 8.6 7.2

k � 1 k � 3 k � 6 k � 9 k � 1 k � 3 k � 6 k � 9

10 20 6.6 8.5 6.3 5.8 32.5 8.1 5.8 6.3

k � 1 k � 5 k � 10 k � 14 k � 1 k � 5 k � 10 k � 14

15 15 7.0 7.5 7.5 13.6 31.1 7.0 7.5 14.3

k � 1 k � 7 k � 13 k � 19 k � 1 k � 7 k � 13 k � 19

20 10 7.3 7.4 12.1 37.6 30.8 4.8 11.7 39.4

k � 1 k � 8 k � 16 k � 24 k � 1 k � 8 k � 16 k � 24

25 5 5.6 5.8 19.3 68.8 32.0 6.3 22.1 70.8

k � 1 k � 10 k � 20 k � 29 k � 1 k � 10 k � 20 k � 29

30 0 7.3 8.0 44.1 74.0 29.8 8.3 44.2 81.7

T � 100, N � 30, ð � 1, p � 0

n s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 4

5 25 4.7 5.2 8.3 11.7 99.0 94.5 84.3 66.2

k � 1 k � 3 k � 6 k � 9 k � 1 k � 3 k � 6 k � 9

10 20 5.0 8.5 16.1 17.5 99.5 84.3 36.9 24.3

k � 1 k � 5 k � 10 k � 14 k � 1 k � 5 k � 10 k � 14

15 15 6.0 15.2 17.3 16.0 99.0 48.0 21.8 18.6

k � 1 k � 7 k � 13 k � 19 k � 1 k � 7 k � 13 k � 19

20 10 5.0 17.9 17.1 18.8 99.1 29.0 18.1 22.7

k � 1 k � 8 k � 16 k � 24 k � 1 k � 8 k � 16 k � 24

25 5 5.1 17.3 14.6 21.9 99.3 26.5 17.9 24.0

k � 1 k � 10 k � 20 k � 29 k � 1 k � 10 k � 20 k � 29

30 0 5.1 15.8 22.0 25.8 99.5 23.2 18.1 29.5

Notes:See Table 1. In addition, N � number of units in the sample.



lowering ð will bring improvement in power without a worsening of size.We explore this possibility by repeating the experiments summarized inTables 1±3 for the values of ð � 0:5 and 0.1. For the number of lags p � 0the small sample size and power virtually did not change with movementsin ð and we do not report the results. The outcome for the case withredundant lag in the regression ( p � 1) is given in Table 4. For T � 100and ð � 0:5, the improvement in power was quite remarkable while the sizedid not worsen very much. This result suggests that for moderate samplesizes the modi®cation of the Snell estimator can make up for a lack ofinformation on the correct number of lags in the regression.

5.4. The Difference between the Actual Number of Common StochasticTrends in the Sample and the Number being Tested

The test is formulated in terms of the null of k common stochastic trendsagainst the alternative of k � 1 trends but it is also consistent against thealternatives of more than k � 1 trends. In practice, the number of commonstochastic trends is unknown and the procedure is employed in the sequenceof tests of k � 1 against k � 2, k � 2 against k � 3 and so on until the nullis not rejected. It is therefore likely that at the beginning of the sequence,the number of underlying stochastic trends exceeds the maintained number

TABLE 3Size and Power of the Test of the Number of Common Stochastic Trends for p � 1

Size Power

T � 30, n � 5, ð � 1, p � 1

s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 40 8.5 8.9 7.2 10.9 10.7 10.5 10.6 15.61 10.7 8.8 7.8 8.4 10.8 9.8 9.9 10.52 10.0 11.3 10.9 8.2 11.4 11.6 8.7 9.25 15.0 13.7 11.8 11.2 16.8 13.8 11.9 12.6

10 27.3 24.2 22.3 21.2 28.8 24.7 23.1 21.320 81.5 74.6 68.3 62.9 80.8 74.8 69.3 63.5

T � 100, n � 5, ð � 1, p � 1

s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 40 7.6 8.0 14.6 24.4 17.7 7.8 11.5 19.71 6.8 5.7 8.9 12.9 18.5 8.2 11.0 10.82 6.6 6.5 7.6 11.9 17.3 7.4 8.6 12.15 7.3 6.3 7.0 8.0 18.3 8.6 7.6 9.2

10 8.0 9.2 8.5 7.8 20.2 9.4 9.5 7.920 11.9 11.0 9.9 10.4 22.5 12.4 10.2 9.1

Note:See Table 1.

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by more than one. To see how the power of the test re¯ects this fact, weconducted experiments in which the difference between the actual (k0) andhypothesized (k) number of trends varied between one and four. Theconclusion was quite intuitive: the further the tested principal componentfrom the stationary ones, the easier the test rejects the null about itsstationarity. We report a representative experiment in Table 5. It can be seen

TABLE 4Size and Power of the Modi®ed Test of the Number of Common Stochastic Trends

Size Power

T � 30, n � 5, ð � 0:5, p � 1s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 40 8.8 8.9 8.9 20.1 29.8 12.8 14.6 27.81 10.3 10.3 8.8 13.1 31.1 12.8 12.5 15.32 12.2 9.7 8.5 11.7 27.7 13.8 11.0 12.95 14.8 13.8 12.5 12.1 30.0 16.3 13.8 12.510 29.0 26.6 21.2 20.0 42.0 26.4 22.3 21.820 82.3 76.8 71.5 64.6 85.8 79.2 70.4 65.5

T � 100, n � 5, ð � 0:5, p � 1

s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 40 9.3 9.0 20.5 37.7 97.3 88.0 69.3 65.21 6.3 7.0 11.4 16.0 96.8 84.2 58.8 37.62 7.2 7.2 9.0 17.8 96.9 85.7 62.9 37.75 7.0 7.5 7.3 9.5 95.8 83.0 57.0 31.710 8.6 7.0 7.5 8.7 94.2 79.4 50.6 27.220 13.1 11.8 10.5 11.7 93.3 74.3 45.9 22.5

T � 30, n � 5, ð � 0:1, p � 1

s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 40 21.6 25.0 28.8 45.9 68.5 49.8 46.5 55.11 23.2 25.3 26.3 30.6 67.3 40.9 36.4 39.22 25.2 27.0 24.8 29.2 66.8 38.9 36.2 34.05 37.2 33.0 28.7 26.8 68.8 46.2 37.6 30.910 60.7 44.8 44.7 43.0 76.8 60.7 50.3 41.620 98.0 95.3 92.0 88.0 98.5 96.3 93.3 88.1

T � 100, n � 5, ð � 0:1, p � 1

s k � 1 k � 2 k � 3 k � 4 k � 1 k � 2 k � 3 k � 40 45.2 37.2 43.6 53.1 100.0 100.0 97.8 100.01 39.3 33.7 41.3 43.3 99.9 99.0 92.1 84.42 38.8 35.0 34.4 47.7 100.0 99.7 98.0 91.75 42.3 35.9 33.4 42.2 100.0 99.5 97.7 93.510 53.5 46.5 44.6 44.6 99.9 99.5 97.2 90.720 73.8 69.0 65.5 64.3 99.9 99.3 96.4 91.3

Note:See Table 1.



TABLE 5Effect of the Difference between Actual and Tested Number of Common Stochastic Trends on the Power of the Test of the Number of

Common Stochastic Trends

Power

T � 30, N � 5, ð � 1, p � 0s k0 k � 1 k � 2 k � 3 k � 4 s k0 k � 1 k � 2 k � 3 k � 40 k � 1 63.0 38.8 38.3 51.9 5 k � 1 48.3 21.4 15.5 12.9

k � 2 91.9 76.3 67.4 k � 2 81.4 44.8 23.5k � 3 98.8 97.8 k � 3 93.8 68.5k � 4 100.0 k � 4 98.8

1 k � 1 56.7 32.3 28.1 30.8 10 k � 1 42.0 18.2 12.6 8.7k � 2 87.8 63.8 41.9 k � 2 74.3 35.3 18.7k � 3 98.3 87.2 k � 3 90.8 54.6k � 4 100.0 k � 4 96.7

2 k � 1 55.6 27.9 21.9 24.1 20 k � 1 33.0 12.3 10.5 9.0k � 2 85.6 56.4 35.2 k � 2 64.2 24.0 13.6k � 3 97.3 80.5 k � 3 82.5 41.3k � 4 99.7 k � 4 94.0

Notes:See Tables 1 and 2. In addition, k0 � number of the common stochastic trends in the DGP.

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that the increase in power with increasing difference between k and k0 isquite substantial. All other experiments have brought the same evidence,even though for p � 1 the increase in power was considerably smaller.

5.5. The Case of N > T

Practitioners in the ®eld of panel data econometrics are very often con-fronted by the case in which the number of units in the sample exceeds thesample size in a given unit. We carried out two experiments with short timeseries, T � 30. In one of them the number of variables equals the timedimension, in the other one there are three times more variates in the samplethan there are observations on them. The results are given in Table 6.Overall, the size of the test in these samples tends to equal the power. Theconclusion in this case must be that the test is unsuitable for panels whereN the number of units, exceeds or approaches the number of observationson them.

TABLE 6Size and Power of the Test of the Number of Common Stochastic Trends for the

Case N > T

Size Power

T � 30, N � 30, ð � 1, p � 0

n s k � 1 k � 3 k � 4 k � 1 k � 3 k � 4

5 25 7.3 7.6 7.5 28.8 9.7 7.8

k � 5 k � 10 k � 15 k � 5 k � 10 k � 15

20 10 6.2 7.4 16.3 6.0 7.5 16.3

k � 5 k � 15 k � 25 k � 5 k � 15 k � 25

30 0 6.4 16.0 73.8 6.1 17.0 74.3

T � 30, N � 90, ð � 1, p � 0

n s k � 1 k � 3 k � 4 k � 1 k � 3 k � 4

5 25 7.3 6.0 5.8 18.6 6.5 5.2

k � 5 k � 10 k � 15 k � 5 k � 10 k � 15

20 10 5.5 6.3 12.3 5.8 7.3 13.1

k � 5 k � 15 k � 25 k � 5 k � 15 k � 25

30 0 6.0 11.5 44.6 5.5 12.3 46.1

Note:See Tables 1 and 2.



VI. CONCLUSION

In this paper we propose a new approach to testing for the number ofcommon stochastic trends driving the non-stationary series in a panel dataset. The procedure enables us to carry out the testing even if we have amixture of I(0) and I(1) series in the sample.

With a set of Monte Carlo experiments we assess the empirical relevanceof the testing procedure. The test is shown to have reasonable size andpower when the sample size T is larger than the number of series N . Thetest performs best when there are relatively few stochastic trends underlyingthe data. The size of the test improves with increasing numbers of stationaryseries present in the sample while the power deteriorates.

The principal components approach allows us to carry out the test evenwhen N is equal to or greater than T . However, from the ®rst simpleexperiments reported in this paper, the power of the test gets poorer. Moreappropriate experiments need to be designed and explored. Further, theestimation of the number of common stochastic trends is done in order tovalidate the aggregate relationship. Therefore, it would be of interest toinvestigate the behaviour of the aggregate estimator when there is a lowenough number of common trends. We leave this for future research.

Imperial College, LondonLondon Business SchoolCity University Business School, London

Date of Receipt of Final Manuscript: July 1999

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A Principal Components Analysis of Common Stochastic Trends in Heterogeneous Panel Data: Some Monte Carlo Evidence