A Panel Unit Root Test in the Presence of a Multifactor Error Structure

Transcription

1 A Panel Unit Root est in the Presence of a Multifactor Error Structure M. Hashem Pesaran a L. Vanessa Smith b akashi Yamagata c a University of Cambridge and USC b CFAP, University of Cambridge c Department of Economics and Related Studies, University of York 6 September 29 Abstract his paper extends the cross sectionally augmented panel unit root test proposed by Pesaran (27) to the case of a multifactor error structure. he basic idea is to exploit information regarding the m unobserved factors that are shared by k other time series in addition to the variable under consideration. Initially we develop a test assuming that m, the true number of factors is known, and show that the limit distribution of the test does not depend on any nuisance parameters, so long as k m. Small sample properties of the test are investigated by Monte Carlo experiments and shown to be satisfactory. Particularly, in contrast to other existing panel unit root tests, our test has correct size and reasonable power for the case with an intercept and a linear trend as well as with an intercept only, for all combinations of cross section and time series dimensions. An illustrative application is also provided where the proposed panel unit root test is applied to Fisher s in ation parity and real equity prices. JEL-Classi cation: C2, C5, C22, C23 Keywords: Panel Unit Root ests, Cross Section Dependence, Multi-factor Residual Structure, Fisher In ation Parity, Real Equity Prices. We would like to thank Anindya Banerjee, Soren Johansen, Benoit Perron, and Joachim Westerlund for useful comments and helpful discussions.

2 Introduction here is now a sizeable literature on testing for unit roots in panels where both cross section (N) and time ( ) dimensions are relatively large. Reviews of this literature are provided in Banerjee (999), Baltagi and Kao (2), Choi (24), and more recently in Breitung and Pesaran (28). he so called rst generation panel unit root tests pioneered by Levin, Lin and Chu (22) and Im, Pesaran and Shin (23) focussed on panels where the idiosyncratic errors were cross sectionally uncorrelated. More recently, to deal with a number of applications such as testing for purchasing power parity or output convergence, the interest has shifted to the case where the errors are allowed to be cross sectionally correlated using a residual factor structure. hese second generation tests include the contributions of Moon and Perron (24), Bai and Ng (24) and Pesaran (27). he tests proposed by Moon and Perron (24) and Pesaran (27) assume that under the null of unit roots the common factor components have the same order of integration as the idiosyncratic components, whilst the test procedures of Bai and Ng (24) allow the order of integration of the factors to di er from that of the idiosyncratic components, by assuming di erent processes generating the two. A small sample comparison of some of these tests is provided in Gengenbach, Palm and Urbain (29). In the case of the panel unit root test proposed by Pesaran (27), the cross section dependence is accounted for by augmenting the individual ADF regressions of y it with cross section averages of the dependent variable (current and lagged values, y t, y t = N N j= y j;t ). hese cross section averages are used as proxies for the assumed single unobserved common factor. he panel test statistic is then based on the average of the individual t-statistics over the cross section units and is shown to be free of nuisance parameters, although it has a nonnormal limit distribution as N and!. Monte Carlo experiments show that Pesaran s test has desirable small sample properties in the presence of a single unobserved common factor but show serious size distortions if the number of common factors exceeds unity. Bai and Ng (24) consider whether the source of non-stationarity is due to the common factor and/or idiosyncratic component. heir method involves applying unit root tests to the common factors and the idiosyncratic component separately, where the unobserved factors are replaced with consistent estimates obtained by use of principal components (PC). he pooled tests they propose require an estimate of the true number of factors and the factors themselves. Moon and Perron (24) follow a similar approach in that they base their test on a principal components estimator of common factors. In particular, their test is based on de-factored observations obtained by projecting the panel data onto the space orthogonal to the (estimated) factor loadings. his paper extends Pesaran s test and proposes a simple panel unit root test that is valid in the more general case of multiple common factors. In so doing we utilise the information contained in a number of k additional variables, x it, that are assumed to share the same common factors as the original series of interest, y it. he ADF regression for y it is then augmented by the cross section averages of the dependent variable as well as the additional regressors. 2 he test assumes that there exists a number of variables that are simultaneously a ected by Other panel unit root tests include that of Chang (22) that employs a non-linear IV method to account for cross-section correlation and Phillips and Sul (23) who use an orthogonalisation procedure to deal with dependence arising from a single common factor. he former is valid for a xed N and large. 2 he idea of augmenting ADF regressions with other covariates has been investigated in the unit root literature by Hansen (995) and Elliott and Jansson (23). hese authors consider the additional covariates in order to gain power when testing the unit root hypothesis in the case of a single time series. In this paper we augment ADF regressions with cross section averages to eliminate the e ects of unobserved common factors in the case of panel unit root tests. 2

3 a given set of unobserved common factors. his requirement seems quite plausible in the case of panel data sets from economics and nance where economic agents often face common economic environments. For example, in testing for unit roots in a panel of real outputs one would expect the unobserved common shocks to output (that originate from technology) to also manifest themselves in employment, consumption and investment. In the case of testing for unit roots in in ation across countries, one would expect the unobserved common factors that cross correlate in ation rates to also a ect short-term and long-term interest rates across markets and economies. he fundamental issue is to ascertain the nature of dependence and persistence that is observed across markets and over time. he present paper can, therefore, be viewed as a rst step in the process of developing a coherent framework for the analysis of unit roots and multiple cointegration in large panels. Initially we develop a test supposing that m, the true number of factors, is known and that all additional variables are I() and not cointegrated among themselves. We show that the limit distribution of the test does not depend on the factor loadings or other nuisance parameters so long as k m. But, in practice m is rarely known. Given an assumed maximum number of factors, m max, we suggest two strategies for dealing with uncertainty that surrounds the value of m. One is to choose the number of additional regressors as k = m max. In this case, the true number of factors are allowed to be any integer value between zero and m max. However, when m max is assumed to be large, in some situations it can be di cult to nd a su cient number of suitable additional regressors. Another possibility is to estimate m consistently using suitable selection criteria, as is followed in the literature, for example, by Bai and Ng (24) and Moon and Perron (24), amongst others. he small sample properties of the proposed test are investigated by Monte Carlo experiments. he test is shown to have the correct size in a number of di erent experiments and for relatively small samples. his contrasts the results obtained for some of the prominent existing tests in the literature such as the pooled tests of Bai and Ng (24) and Moon and Perron (24) that tend to be over-sized. 3 In terms of power, when the model contains an intercept term only, the pooled tests tend to display higher power in smaller samples as compared to the proposed test, although this could partly re ect the over-sized nature of the pooled tests in small samples. 4 In the case of models with linear trends, our experimental results show that the proposed test can perform better than the pooled tests, both in terms of size and power. Empirical applications to Fisher s in ation parity and real equity prices across di erent economies illustrate how the proposed test performs in practice. he plan of the paper is as follows. Section 2 presents the panel data model and the testing procedure and derives the asymptotic distribution of the proposed cross sectionally augmented panel unit root test. Section 3 describes the Monte Carlo experiments and reports the small sample results. Section 4 presents the empirical applications, and Section 5 provides some concluding remarks. Notation: L denotes a lag operator such that L`x t = x t `, K denotes a nite positive constant such that K <, jjajj = [tr(aa )] =2, A denotes the generalised inverse of A, I q is a q q identity matrix, q and q are q vectors of ones and zeros, respectively, qr is a N q r null matrix, =) (!) N denotes convergence in distribution (quadratic mean (q.m.) or mean square errors) with xed as N!, =) (!) denotes convergence in distribution (q.m.) with 3 Westerlund and Larsson (29) provide further theoretical results on the asymptotic validity of the pooled versions of the PANIC procedure. 4 We do not present size-corrected power comparisons, since such results are likely to have limited value in empirical applications where such size corrections are not possible. 3

4 N xed (or when there is no N-dependence) as!, N; =) denotes sequential convergence in distribution with N! rst followed by!, (N; ) j =) denotes joint convergence in distribution with N,! jointly with certain restrictions on the expansion rates of and N to be speci ed, if any. 2 Panel Data Model and ests Let y it be the observation on the i th cross section unit at time t generated as y it = i (y i;t iyd t ) + iyd t + u it, i = ; 2; :::; N; t = ; 2; :::; () where i = ( i ); d t is 2 vector consisting of an intercept and a linear trend so that d t = (; t). Without loss of generality, it is assumed that d. Consider the following multifactor error structure u it = iyf t + " iyt (2) where f t is an m vector of unobserved common e ects, iy is the associated vector of factor loadings, and " iyt is the idiosyncratic component. his set up generalises Pesaran s (27) one factor error speci cation. We assume that these error processes satisfy the following assumptions: Assumption (idiosyncratic errors): he idiosyncratic shocks, " iyt, i = ; 2; :::; N; t = ; 2; :::;, are independently distributed both across i and t, have zero means, variances < 2 i K and nite fourth-order moments. Remark his assumption, which implies that the idiosyncratic shocks are serially uncorrelated, will be relaxed in Section 2.. It is also possible to relax the assumption that the idiosyncratic errors are cross sectionally independent, and replace it by assuming that " iyts are cross sectionally weakly dependent in the sense of Chudik, Pesaran, and osetti (29). However, such an extension will not be considered in this paper. Assumption 2 (factors): he m vector f t follows a covariance stationary process, with absolute summable autocovariances, distributed independently of " iyt for all i; t and t. Specifically, we assume that f t = (L)v t, where v t IID(; I m ), which have nite fourth-order moments, (L) = P `= `L` with f` `g `= being absolute summable such that P (`) `= `j rs j being the (r; s) th element of `, and speci cally the inverse of f de ned by with (`) rs exists. f = () (3) Remark 2 Since is not restricted it can always be chosen such that E(v t vt) = I m, without loss of generality. Assumption 2 is quite general but rules out the possibility of the factors having unit roots. his seems reasonable since otherwise all series in the panel could be I() irrespective of whether i = or not. Also if iy f t is assumed to be I() and cointegarted with y it, then y it will be I() even if i =, and as noted by Hansen (995, p. 59) in a similar context, a test of i = as a unit root test will not be meaningful. 4

5 Combining () and (2) it follows that y it = i (y i;t iyd t ) + iyd t + iyf t + " iyt : (4) he hypothesis that all the series, y it, have a unit root and are not cross unit cointegrated can be expressed as H : i = for all i, (5) against the alternative H : i < for i = ; 2; :::; N ; i = for i = N + ; N + 2; ::; N where N =N! and < as N!. Note that under the null hypothesis, (4) can be solved for y it to yield where y it = y i + iyd t + iys ft + s iyt, i = ; 2; :::; N; t = ; 2; :::; (6) s ft = f + f 2 + :::: + f t ; s iyt = " yt + " 2yt + ::: + " iyt ; with y i being a given initial value. herefore, under H and Assumptions and 3, y it is composed of the initial value, y i, a common stochastic component, s ft I(); and an idiosyncratic component, s iyt I(), so that while all units of the panel share the common stochastic trends, s ft, there is no cointegration among them. Under the alternative hypothesis, i <, we have y it I(), and it is essential that f t is at most an I() process. In the case where m =, Pesaran (27) proposes a test of i = jointly with f t I(); based on DF (or ADF) regressions augmented by the current and lagged cross section averages of y it as proxies for the unobserved f t. He shows that the resultant test is asymptotically invariant to the factor loadings, iy. o deal with the case where m > we assume that in addition to y it, there exists k additional observables, say x it, which depend on at least the same set of common factors, s ft, although with di erent factor loadings. For example, in the analysis of output convergence it is reasonable to argue that output, investment, consumption, real equity prices, and oil prices have the same set of factors in common. Similarly, short term and long term interest rates and in ation across countries are likely to have a number of factors in common. More speci cally, suppose the k vector of additional regressors follow the general linear process x it = A ix d t + ix f t + " ixt, i = ; 2; :::; N; t = ; 2; :::; (7) where x it = (x it ; x i2t ; :::; x ikt ), ix = ( ix ; ix2 ; :::; ixk ), A ix = (a ix ; a ix2 ; :::; a ixk ), and " ixt is the idiosyncratic component of x it which is I() and distributed independently of " iyt for all i; t and t. he level equation can be written as x it = x i + A ix d t + ix s ft + s ixt, i = ; 2; :::; N; t = ; 2; :::; (8) where s ixt = P t s= " ixs. Combining (6) and (8) we have z it = z i + i s ft + A i d t + s it ; (9) 5

6 where z it = (y it ; x it ), i = iy ; ix, Ai = ( iy ; A ix ), and s it = (s iyt ; s ixt ). Without loss of generality we set s f = m and s i = k+. Assumption 3 (factor loadings): ka i k K and k i k K, for all i, and that E(f t ft) I m. i are set such Assumption 4 (initial conditions): Ejjs f jj K; and Ejjz i jj K; Ejjs i jj K, for all i. Remark 3 Assumption 3 imposes minimal conditions on the factor loadings. For example, it does not rule out possible dependence between the factor loadings and idiosyncratic errors. Also the normalization of f t so that its variance covariance matrix is an identity matrix is innocuous since otherwise i and f t can be suitably transformed so that Assumption 3 holds. Assumption 4 is also routine in the literature on unit roots. Averaging (9) across i we obtain z t = z + s ft + Ad t + s t, () where z t = N P N i= z it, A = N P N i= A i; and s t = N P N i= s it. 5 Writing (4), (9) and () in matrix notation, under the null for each i we have y i = F iy + D iy + " iy, () Z i = F i + DA i + E i ; (2) Z = F + D A + E; (3) where F = (f ; f 2 ; :::; f ), D = (d ; d 2 ; :::; d ), " iy = (" iy ; " iy2 ; :::; " iy ), Z i = (z i ; z i2 ; :::; z i ), E i = (" i ; " i2 ; :::; " i ) with " it = (" iyt ; " ixt ) Z = (z ; z 2 ; :::; z ) and E = N P N i= E i. From (3), if has full column rank m, it follows that F = Z D A E : (4) However, from Appendix A.2. we have that E! N for each t and hence we obtain that F Z D A N! : (5) his implies that the linear combinations of (Z; D) would be a valid approximation of F for large N. his condition on the rank of the cross section average of factor loadings is stated as an assumption below: Assumption 5 (rank condition): he (k + ) m matrix of factor loadings i is such that rank() = m k +, for any N and as N! ; (6) where = N P N i= i, and N!, where is a xed bounded matrix with rank m. 5 Weighted cross section averages could also be used with appropriate granularity restrictions on the weights. 6

7 Remark 4 From the equations (9) and (4), it is clear that our approach approximates s ft of m dimension by linear combinations of the cross section average z t = (y t ; x t) of k + dimension for large N. hus, the rank condition (6), rank() = m k +, which implies k m ; is of importance. Remark 5 It is not necessary that y it and (x it,x i2t,...,x ikt ) have the same cross section dimensions. his is illustrated in Section 4. Remark 6 Note that it is not necessary for the rank condition to hold for all cross section units individually, but that it must hold on average. For example, the rank condition holds so long as a non-zero fraction of factor loadings, i, are full rank as N!. Also, so long as Assumption 5 is satis ed, we do not necessarily require that lim N! N P N i= i i exists and is positive de nite, which is typically assumed for the identi cation of factors. See, for example, Assumption A(ii) of Bai and Ng (24) and Assumption 6 of Moon and Perron (24). In view of the above we shall base our test of the panel unit root hypothesis on the t-ratio of the ordinary least square (OLS) estimate of b i (^b i ) in the following cross sectionally augmented regression y it = b i y it + c iz t + h iz t + g id t + it. he t-ratio of ^b i in this regression is given by t i (N; ) = yi My i; ^ i yi; =2 = My i; p (2k + 5)y i My i; y i M i y i =2 y i; My i; =2 ; where y i = (y i ; y i2 ; :::; y i ), y i; = (y i ; y i ; :::; y i; ), M = I W W W W, W = ( w ; w 2 ; :::; w ), w t = z t; d t; z t, ^ 2 i = y i M i y i (2k + 5), and M i = I W i W i W i W i, with W i = W; y i;. Using (4) in () y i = Z i + D i + i i, (7) where i = iy, i = iy A i, i = " iy E i =i. It is also easily seen that E( i i ) = I + O(N ): herefore, we have From (2) and (3) we obtain My i = i M i. (8) Z i; = z i + S f; i + D A i + S i; : Also Z = z + S f; + D A + S i; (9) where S f; = ( m ; s f ; :::; s f; ), D = (; d ; :::; d ), Z i; = (z i ; z i ; :::; z i ) ; S i; = ( k+ ; s i ; :::; s i; ), Z = (z ; z ; :::; z ) and S = N P N i= S i;. 7

8 Similarly from (7) where y i; = y i + Z i + D i + i s i; ; (2) s i; = (s iy; S i )= i, (2) s iy; = (; s iy ; :::; s iy; ) and y i = y i z i: herefore, My i; = i Ms i;. (22) Using (8) and (22), t i (N; ) can be re-written as t i (N; ) = ims i; ( im i i 2k 5 )=2 s =2. (23) i; Ms i; For xed N and ; the distribution of t i (N; ) will depend on the nuisance parameters through their e ects on M i and M. However, this dependence vanishes as N!, for xed. In the case of xed however, the e ect of the initial cross section mean, z, must be eliminated in order to ensure that t i (N; ) does not depend on nuisance parameters. his can be achieved by working with the deviations, z it z. he main asymptotic results concerning the distribution of t i (N; ) are summarised in the theorems below. he proofs are given in the Appendix for the case where d t = (; ) ; t = ; ; :::;, which implies D = : he asymptotic results for the case where d t = (; t) can be derived in a similar manner. heorem 2. Suppose the series z it, for i = ; 2; :::; N, and t = ; 2; :::;, is generated under (5) according to (9); d t = with z set to a zero vector. hen under Assumptions -5, the distribution of t i (N; ) given by (23), will be free of nuisance parameters as N! for any xed > 2k + 4. In particular, we have (in quadratic mean) t i (N; ) N! " iy s iy; " iy " iy g 2 i ( 2k 4) i Q i g i ( 2k 4) 2 i q i f h i =2 s iy; s =2, iy; 2 i h 2 i f h i where q i = (2m+) f = F " p iy i " iy p i S f; " iy i C A, h i = (2m+) F S f; 3=2 S f; 3=2 F F F F S f; F 3=2 3=2 2 F s iy; i 3=2 s iy; i 3=2 S f; s iy; i 2 C A, g i = C A, Q i = f h i h i q i s iy; " i 2 i s iy; s iy; 2 i 2!!. See Appendix A.3 for a proof. 8

9 Remark 7 When the factors are serially uncorrelated, namely f t v t IID(; I m ), (see Assumption 2), even for a nite the limit distribution of t i (N; ) as N!, does not depend on the factor loadings and i. In the case where the factors are serially correlated the limit distribution of t i (N; ) does depend on the serial correlation patterns of f t when is nite. However, as stated in the next theorem, the dependence of t i (N; ) on the autocovariances of f t disappears in the limit when! and N!, jointly. heorem 2.2 Suppose the series z it, for i = ; 2; :::; N, t = ; 2; :::;, is generated under (5) according p to (9) and d t =. hen under Assumptions -5 and as N and! ; such that =N!, ti (N; ) given by (23) has the same sequential (N! ;! ) and joint [(N; ) j! ] limit distribution, is free of nuisance parameters, and is given by where! iv Z G v = Z CADF i = Z W i () [W v (r)] dw i (r) Z W i (r)dw i (r)! iv G v iv W 2 i (r)dr iv G v iv =2, (24) [W v (r)] dr A, iv = Z Z Z Z [W v (r)] dr W i (r)dr [W v (r)] W i (r)dr [W v (r)] [W v (r)] dr W i (r) is a scaler standard Brownian motion and W v (r) is m -dimensional standard Brownian motion de ned on [,], associated with " iyt and v t ; respectively. W i (r) and W v (r) are mutually independent. See Appendix A.4 for a proof. Remark 8 Conditional on W v (r), CADF i and CADF j are independently distributed, but unconditionally they are correlated with the same degree of dependence for all i 6= j. Having established that the limit distribution of the individual t i (N; ) statistic is free of nuisance parameters, we now focus on panel unit root tests based on the average of a suitably truncated version of t i (N; ) which we denote by t i (N; ). he truncation is carried out as in Pesaran (27) to avoid certain technical di culties concerning the existence of the moments of the non-truncated version of the individual statistics when is nite. he truncated statistics are de ned by 8 < t i (N; ), if K < t i (N; ) < K 2 ; t i (N; ) = K, if t i (N; ) K ; : K 2, if t i (N; ) K 2 ; where K and K 2 are positive constants that are su ciently large so that Pr[ K < t i (N; ) < K 2 ] is su ciently large. Using the normal approximation of t i (N; ), we would have K = E(CADF i ) ("=2) p V ar(cadf i ), and K 2 = E(CADF i ) + ("=2) p V ar(cadf i ), 9 C A ; C A,

10 where (. ) is the inverse of the cumulative standard normal distribution function, and " is a su ciently small positive constant. K and K 2 can now be obtained using simulated values of E(CADF i ) and V ar(cadf i ) with " = 6 for N = 2; and = 2. he truncation does not a ect the limit distribution and heorem 2. continues to apply to t i (N; ) so that t i (N; ) CADF i = o p (); (25) where CADF i = 8 < : CADF i, if K < CADF i < K 2 ; K, if CADF i K ; K 2, if CADF i K 2. he panel unit root tests associated with the non-truncated and truncated versions of the individual unit root tests are given by and CIP S(N; ) = N CIP S (N; ) = N N X i= N X i= : t i (N; ), (26) t i (N; ). (27) Since by construction all moments of t i (N; ) exist, using (25) it now follows (under assumptions of heorem 2.2) that CIP S (N; ) CADF = o p (), almost surely, where CADF = N N X i= CADF i. Hence, CIP S (N; ) has the same limit distribution as CADF ; almost surely. But following Pesaran (27, Section 4), it can be seen that the limit distribution of CADF exits and is free from nuisance parameters, although it is not analytically tractable. But the critical values of the distribution of CADF (or CADF = N can be obtained by stochastic simulations he Case of Serially Correlated Errors N X i= CADF i ) In this section we relax Assumption, and allow for residual serial correlation. he residual serial correlation can be modeled in a number of di erent ways, directly via the idiosyncratic components, through the common e ects or a mixture of the two. We focus on the rst speci- cation where cross section dependence is present under the multifactor error structure and residual serial correlation is modeled as u it = iyf t + iyt iyt = i iy;t + iyt ; j i j < ; for i = ; 2; :::; N; t = ; 2; :::; ; (28) 6 We only report results for the non-truncated version of the test statistics. he results for the truncated version are very similar and are available upon request.

11 where iyt is independently distributed across time, with zero mean and nite positive variance, 2 i. In what follows we con ne our attention to rst order stationary processes for expositional convenience, though the analysis readily extends to higher order processes as well as to the alternative speci cations of serial correlation mentioned above. Under the above speci cation we have y it = i (y i;t iyd t ) + iyd t + iyf t + iyt ( i ) (29) where iyt ( i ) = ( i L) iyt. We also assume the coe cients of the autoregressive process to be homogeneous across i; although this could be relaxed at the cost of more complex mathematical details. Under the null that i =, with i = and d t = ; (29) becomes or Combining (7) with (3), similarly to (2) we obtain y it = iyf t + iyt (); (3) y it = y i;t + iy(f t f t ) + iyt : (3) Z i = F i + E i ; (32) where E i = ( iy(); E ix ) with E ix = (" ix ; " ix2 ; :::; " ix ) and iy () = iy (); iy2 (); :::; iy (), with the common factors F; and factor loadings i de ned as in the previous section. aking cross section averages of (32) we have that Z = F + E; where as before E = N P N i= E i, from which it follows under rank condition (6) that F = Z E. (33) hus in testing (5) we use the following cross sectionally augmented regression y i = b i y i; + W i h i + i, (34) where W i = (y i; ; Z; Z ; ; Z ), which is a (3k + 5) matrix. he t-ratio of ^b i in regression (34) is given by t i (N; ) = yi M i y i; ^ i yi; =2 = M i y i; p (3k + 6)y i M i y i; y i M i;p y i =2 y i; M i y i; =2 ; (35) where M i = I W i ( W i W i ) W i, ^2 i = [ (3k + 6)] y i M i;p y i and M i;p = I P i (P i P i) P i ; P i = ( W i ; y i; ). Writing (3) in matrix notation and using (33) we have y i = y i; + (Z Z ) i + i i ; (36) with i = [ iy (E E ) i ]= i,

12 and E( i i ) = I + O(N ): From (36) it follows that where y i; = iy + y i + Z i + i s i; s i; = s i; S i =i, s i; = (; s i ; :::; s i; ) with s it = P t s= iys(), S = (s ; ; S x; ) with s ; = N P N and y i = y i z i. he test statistic (35) then becomes i= s i; t i (N; ) = i M i;p i 3k 6 im i s i; =2 s =2. (37) i; M i s i; heorem 2.3 Suppose the series z it, for i = ; 2; :::; N, t = ; 2; :::;, is generated under (5) according to (32) and jj <. hen under Assumptions -5 and as N and!, t i (N; ) in (37) has the same sequential (N! ;! ) and joint [(N; ) j! ] limit distribution given by (24) obtained for =. Proof: See Appendix A.5. For an AR(p) error speci cation in (28), the relevant t i (N; ) statistic will be given by the OLS t-ratio of b i in the following p th order augmented regression: y i = b i y i; + W ip h ip + i, where W ip = (y i; ; y i; 2 ; :::; y i; p ; Z; Z ; :::; Z p ; ; Z ), which is a [(k + 2)(p + 2) ] matrix. However, it is easily seen that the limit distribution of t i (N; ) with N! for a xed depends on the augmentation order, p. hus, we will obtain critical values of t i (N; ) for di erent choices of p. 2.2 Uncertainty about the Number of Factors So far we have considered the case in which the true number of unobserved factors, m, is given. In practice m is not known, although it is reasonable to assume that it is bounded by a su ciently large integer value, m max. In the case of the proposed test there are two possible ways that one could proceed when m is not known. One possibility would be to set k = m max, if there exists m max I() and not cointegrated additional regressors for augmenting the ADF regressions. In this case, the true number of factors are allowed to be any integer value between zero and m max. However, when m max is assumed to be large, it can be di cult to nd m max such regressors. Alternatively, m can be estimated consistently using suitable selection criteria, as is followed in the literature, for example, by Bai and Ng (24) and Moon and Perron (24), amongst others. Since typically m is estimated to be around 2 to 4 in most economic applications, it may not be particularly di cult to identify suitable additional variables for augmentation. 7 7 One could follow the bounds test approach proposed by Pesaran et al. (2) when there is uncertainty in integration and/or cointegration properties of k additional regressors. his route is not pursued in this paper. 2

13 2.3 Critical Values he critical values of CADF i and CADF = N P N i= CADF i for di erent values of k, N, and lag-augmentation p, are obtained by stochastic simulation. Based on the results in Section 2 the limit distribution of CADF does not depend on the factor loadings i or i. his implies that the distribution of the test statistic is invariant to the choice of i and i so long as m k +. hus, without loss of generality we set i = =, and i = = in the simulation exercise. o be more precise, the y it process is generated as y it = y it + " iyt, i = ; 2; :::; N; t = ; 2; :::;, where " iyt iidn(; ) with y i =. he j th element of the k vector of the additional regressors x it, is generated as x ijt = x ij;t + " ixjt, i = ; 2; :::; N; j = ; 2; :::; k; t = ; 2; :::;, (38) with " ixjt iidn(; ) and x ij =. he CADF i test statistic is calculated as the t-ratio of the coe cient on y it of the regression of y it on y it, z t, (z i:t ; :::; z i:t p ), (z t ; :::; z t p) where the following cases for the deterministics are entertained Case I: Case II: Case III: no deterministics, intercept only, an intercept and a linear trend, and E(CADF i ) and V ar(cadf i ) are obtained as an average over all replications of CADF and the square of the standard deviation of CADF respectively, for N; = 2. he % critical values of the CADF and CADF statistics are computed for N; = 2; 3; 5; 7; ; 2, k = ; 2; 3 and p = ; ; :::; 4; as the quantiles of CADF and CADF for = :; :5; :. 8 he critical values of the CADF statistic for case II and III are reported in ables and 2, respectively. Critical values for the CADF statistic for case I as well as for the individual statistics CADF i are available upon request. All stochastic simulations are based on, replications. 9 3 Small Sample Performance: Monte Carlo Evidence In what follows we investigate the small sample properties of the CIPS test de ned by (26) and compare its performance to the pooled tests by Bai and Ng (24), and the t b and t# tests by Moon and Perron (24), by means of Monte Carlo experiments. he t b test statistic is for the case with an intercept only, and the t # test statistic is for the case with an intercept and a linear trend. he pooled test statistics proposed by Bai and Ng (24), using our notation as set out in Section 2, are computed as follows. Firstly we de ne the transformed y it, yit ; for the case with an intercept y it = (39) y it y i ; for the case with an intercept and a linear trend 8 he critical values for k = are tabulated in Pesaran (27). 9 It is also possible to simulate the critical values directly using (24) by replacing the integrals of the Brownian motions with their simulated counterparts. Our analysis suggests that the critical values obtained from this procedure closely matches the ones tabulated in the paper. 3

14 with y i = P t= y it. Apply principal components to the transformed series to estimate F, denoted as ^F, which is p times the m eigenvectors corresponding to the rst m largest eigenvalues of the matrix Y Y, where Y = (y ; y 2 ; :::; y N ) with y i = (y i ; y i2 ; :::; y i ). Under the normalisation ^F ^F= = I m, the estimates of the factor loadings are given by ^ iy = ^F i y i =, which yield the residuals ^" iyt = y it ^ iy^f t. Now set e iyt = P t s= ^" iys, and compute the ADF statistic for the ADF(p) regressions in e iyt without deterministics for each cross section unit. Denoting this statistic by t c BN;i if y it has individual e ects, and by t BN;i if y it has a linear trend, the pooled test statistics are then de ned as P c^u = 2 P N i= ln(pvc i ) 2N p and P ^u = 4N 2 P N i= ln(pv i ) 2N p 4N, where pv c i and pv i are the p-values of the tc BN;i and t BN;i statistics, respectively. hese statistics are asymptotically distributed as standard normal so that the null hypothesis is rejected if P c^u (or P ^u ) is larger than.645 (at the 5% level). We also consider variants of P c^u and P ^u that make use of all the available variables, y it and x it, to estimate the common factors. his version is more directly comparable to the test proposed in this paper which makes use of the additional variables, x it. he procedure is similar to that described above with the principal component estimator of F now computed using z it = (y it ; x it), wherex it is constructed from x it in a manner similar to that speci ed by (39) fory it. hese variants of P c^u and P ^u are denoted by P c^u;z and P ^u;z, respectively. he t b and t# test statistics are as de ned by Moon and Perron (24). he t b test is for the case with an intercept only, and the t # test is for the intercept and a linear trend case. he tests are based on de-factored panel data obtained by projecting the panel data onto the space orthogonal to the (estimated) factor loadings. he nuisance parameters are de ned on the residuals of the de-factored data where the long-run variance is estimated by employing Andrews and Monahan s (992) estimator based on the quadratic spectral kernel and pre-whitening. he null hypothesis is rejected if the test statistics are less than (5% level test). For further details on the above statistics see Bai and Ng (24) and Moon and Perron (24). We consider experiments where the number of factors is treated as known as well as unknown. 3. Monte Carlo Design Initially we consider dynamic panel models with xed e ects and a two-factor (m = 2) error structure. he data generating process (DGP) is given by y it = ( i ) iy + i y i;t + iy f t + iy2 f 2t + " iyt ; i = ; 2; :::; N; t = 49; :::; (4) with y i; 5 =, where iy iidn(; ), iy iidu[; 2], iy2 iidu[; 2], f`t = f`f f`;t + v`t ; v`t iidn(; 2 f`); f`; 5 = In our experiments the P^u statistics are computed by a GAUSS code which is a translation of the Matlab programme provided by Serena Ng. p-values of t c BN;i and t BN;i are obtained using the tables adfnc.asc and lm.asc, respectively, also provided by Serena Ng. he t a test, which is also proposed by Moon and Perron (24), is not considered in our simulations since the t b test is preferred in their paper. 4

15 for ` = ; 2; and " iyt = iy" " iyt + iyt ; iyt iidn(; ( 2 iy") 2 i ); " iy; 5 = ; (4) 2 i iidu[:5; :5]. We include at most two additional regressors, x it and x i2t in the experiments. he DGPs are x ijt = x ijt + ixj f t + ixj2 f 2t + " ixjt for j = ; 2; (42) i = ; 2; :::; N; t = 49; :::; with x ij; 5 =, " ixjt = ixj " ixjt + $ ixjt ;, $ ixjt iidn(; 2 ixj), (43) with " ixj; 5 =, and ixj iidu[:2; :4] for j = ; 2: he rst set of experiments assumes that m = 2 is given and, hence, k is equated to m =. We use only one additional regressor, x t. he factor loadings in (42) are generated as ix iidu[; 2] and ix2 =, so that E( i ) = ; (44) of which the rank condition (6) is satis ed. 2 Note that x t contains s ft only under this design. As discussed in section 2.2, this is enough for augmenting the CADF regression to asymptotically eliminate two factors in the y it equation. 3 We consider three combinations of serial dependence in the errors: (A) serially uncorrelated " iyt and f jt ( iy" = y" = and f = f2 = ); (B) serially correlated " iyt ( iy" iidu[:2; :4] and f = f2 = ); (C) serially correlated f`t ( iy" = y" = and f = f2 = :3). Note that x ixjt are serially correlated in all experiments for j = ; 2, as speci ed above. In addition, we consider spatially correlated factor loadings generated as ir NX c r = s ij jr c r + 'ir ; ' ir iidn(; 2 'i); r = y; y2; x; x2 j= where s ij is the (i; j) element of an (N N) spatial weighting matrix, S = fs ij g, which is row standardised with s ij = if units i and j are adjacent and s ij = otherwise. We set = :8. he parameter 2 'i is chosen so that var( ir) = =3, and we set c y =, c y2 =, c x =, c x2 =, for the results to be comparable to our other experimental designs. 4 2 Another experiment relating to the speci cation of the factor loadings is considered, where iidu[; 2] iidu[; 2] i = ( iy ; ix ) = for i = ; 2; :::; N=2 iidu[; 2] but j = ( jy ; jx ) = for j = N=2 + ; :::; N so that the rank condition is satis ed. he results are very similar to those using (44). 3 We have also implemented the experiments with ix2 replaced by non-zero values, generated as ix2 iidu[ :5; :5]. he results are very similar to those with ix2 =, and are available upon request from the authors. 4 PN q We further generated bounded factor loadings where iy = iy =q j= 2 jy and ix2 = ix2 = PN j= 2 jx2 and iy and ix2 are draws from di erent uniform distributions, iidu(; ): he factor loadings iy2 and ix are generated as in the spatially correlated case with zero expected values. Results were similar to the spatially correlated case. 5

16 In another set of experiments, we consider the case in which m = 2 is not known but the maximum number of factors is assumed to be three, i.e., m max = 3. Here the value of m is estimated (denoted by ^m ) based on the information criterion IC, proposed by Bai and Ng (22) and used in the simulation exercises of Bai and Ng (24). Accordingly, ^m factors are extracted from y it for the P^u and t b (t# ) statistics and from (y it ; x it ; x i2t ) for the P^u;z statistic. For the CADF regressions k = ^m additional regressors are included in the augmentations. At most we need m max = 2 additional regressors. In this experiment we consider uncertainty about the integration properties of the two needed additional regressors. he CIPS test is implemented assuming x it and x i2t are I() and not cointegrated, but in the DGP x it and x i2t are generated as cointegaretd variables. he factor loadings in (42) are generated as ix iidu[; 2], ix2 iidu[; 2], ix2 = ix22 =, with " ixjt replaced by its rst di erence " ixjt so that the cumulative sum of the idiosyncratic errors in x ijt becomes " ixjt I(), and E( i ) = Under this design x it I() and x 2it I() but they are cointegrated. When ^m = 2, only one regressor is required by CADF augmentation, thus, x it is included in the experiment. If ^m = or, no additional regressors are included. For this set of experiments we con ne our attention to case (A) with regard to serial dependence in the errors. 5 Similar sets of experiments are carried out for the model with a linear time trend. he DGPs corresponding to (4) and (42) are y it = iy + ( i ) i t + i y i;t + iy f t + iy2 f 2t + " iyt ; i = ; 2; :::; N; t = 49; :::; : x ijt = x ijt + ixj + ixj f t + ixj2 f 2t + " ixjt ; for j = ; 2 (45) respectively, where iy iidu[:; :2], i iidu[:; :2], ixj iidu[:; :2] for j = ; 2. he rest of the variables are de ned as above. he parameters i, iy", iy`, f`, i, ixj`, ixj, iy, i, and ixj are drawn once and xed over the replications. For size the DGP is given by (4) with i = =, and for power with i iidu[:9; :99]. All tests are conducted at the 5% signi cance level. All combinations of N; = 2; 3; 5; 7; ; 2 are considered, and all experiments are based on 2, replications each. 3.2 Results Size and power of the tests are summarized in ables 3 to 8. Recall that for all experiments, the models contain two factors, m = 2, and the idiosyncratic errors of additional regressors, v ixjt, can be I() or I() and are generated as serially correlated variables. Also note that in the case of serially correlated idiosyncratic errors, lag augmentation is required for the asymptotic validity of the CIPS test and the pooled tests of Bai and Ng (24), while the t b and t# tests of Moon and Perron (24) correct for the residual serial correlation in a non-parametric manner. he results reported in ables 3 to 7 are obtained assuming that m = 2 is known and that the one additional regressor to augment the CIPS test statistic (k = ) is known to be I(). able 3 provides the results for the model where the factors, f t and f 2t, and the idiosyncratic components, " iyt, are serially uncorrelated. Panel A of the table reports the results for the case 5 Another experiment, in which x it and x i2t are generated as I() and non-cointegrated, is considered, but the results are very similar and will not be included below to save space. 6

17 of an intercept only. he P c^u and P c^u;z tests of Bai and Ng (24) tend to over-reject the null moderately, with the extent of over-rejection rising as N increases. he same applies to the Moon-Perron test, t b. hese results are consistent to those reported in Gengenbach, Palm and Urbain (29). In contrast, the CIPS test has the correct size for all combinations of sample sizes, even when is small relative to N. In terms of power, the CIPS test seems less powerful than the other tests for small values of (which could partly be due to the over-sized nature of the other tests), while in general it tends to be more powerful for larger N and. In panel B of able 3 the results for the case with an intercept and a linear trend are reported. Now the P ^u and P ^u;z tests severely over-reject the null hypothesis in all experiments. Even when = 2 and N = 2, the size of these tests is around 3%. he size distortion of the t # test is even worse for all experiments. On the other hand, the CIPS test has the correct size for all combinations of sample sizes. Not surprisingly, the power of the CIPS test in the linear trend case is lower than the intercept only case. his is a feature common to all unit root tests in the literature. able 4 (able 5) presents the results for the case where " iyt are positively (negatively) serially correlated but f t and f 2t are serially uncorrelated. With time series augmentation the size and the power properties of the CIPS test are similar to those reported in able 3. he P c^u, P c^u;z, P ^u and P ^u;z tests display a higher tendency to over-reject the null relative to the case where the idiosyncratic errors are serially uncorrelated. he t b and t# tests show slightly less (more) size distortions as compared to the results given in able 3 when " iyts are positively (negatively) serially correlated. able 6 provides the results for the experiments where f t and f 2t are serially correlated, but " iyt is not. In this case all the tests exhibit size distortions unless is su ciently large relative to N. However, the extent of over-rejection of the CIPS test is less than that of the P c^u and P c^u;z tests. he performance of the t b test is similar to that reported in the previous experiments. able 7 displays the size results for the case of spatially correlated factor loadings when the factors, f t and f 2t, and the idiosyncratic components, " iyt, are serially uncorrelated. he results are similar to those in able 3. able 8 gives the results for the case where m (= 2) is unknown, and is estimated using the selection criterion IC of Bai and Ng (24), with m max = 3. Recall also that in these experiments x i2t and x it are I() and cointegrated. 6 he results are similar to those in able 3, in that the CIPS test has the correct size in all designs considered, maintaining reasonable power. hus, cointegration between the additional regressors might not be a problem if the there is a su cient number of I() regressors amongst the additional regressors under consideration. 4 Empirical Applications As an illustration of the proposed test we consider two applications. One to the real interest rates across N = 32 economies, and another to the real equity prices across N = 26 markets. Under the Fisher parity hypothesis, the real interest rates, the di erence between the nominal short-term interest rate and in ation rate, are stationary. For both applications we employ quarterly observations over the period 979Q2 23Q4 (i.e. 99 data points). Existing evidence on the validity of the Fisher parity is rather mixed. he second application is chosen since it is 6 We found that the results for ^m matched those of m in most cases except when or N were small. 7

18 generally believed that real equity prices are non-stationary, and it would be interesting to see if the outcomes of the tests considered in this paper are in line with this belief. As discussed in Section 2.2, we begin with a choice of the maximum number of factors, m max, as with other panel unit root tests that are based on principal components. We believe that it is reasonable to suppose that both the real interest rates and the real equity prices contain at most six unobserved common factors. As we set m max = 6, our test requires at most ve additional regressors (k = m max = 5); with their cross section averages being I() and not cointegrated. he set of regressors that are likely to have common factors with real interest rates, rit S it, and real equity prices, eq it, are as follows: where y it candidates of x it Real Interest Rates (N = 32) rit S it Real Equity Prices (N = 26) eq it (poil t ; rit L; eq it; ep it ; gdp it ) (poil t ; rit L; it; ep it ; gdp it ) (46) r S it = :25 ln( + RS it =); it = p it p it with p it = ln(cp I it ); poil t = ln(p OIL t ), r L it = :25 ln( + RL it =); ep it = e it p it with e it = ln(e it ); eq it = ln(eq it =CP I it ); gdp it = ln(gdp it =CP I it ) with Rit S the short rate of interest per annum in per cent (chosen to be a three month rate) in country i at time t, CP I it the consumer price index, P OIL t the price of Brent Crude oil, Rit L the long rate of interest per annum in per cent (typically the yields on ten year government bonds), E it the nominal exchange rate of country i in terms of U:S: dollars, EQ it the nominal equity price index, and GDP it the nominal Gross Domestic Product of country i during period t in domestic currency, so that rit S is the quarterly short-term interest rate, it is the quarterly in ation rate, poil t is the logarithm of the nominal oil price, rit L is the quarterly long-term interest rate, eq it is the logarithm of the real equity price index, ep it is the logarithm of the real exchange rate and gdp it is real log output. 7 he 32 countries considered are: Argentina, Australia, Austria, Belgium, Brazil, Canada, Chile, China, France, Finland, Germany, Indonesia, India, Italy, Japan, Korea, Malaysia, Mexico, Netherlands, New Zealand, Norway, Peru, Philippines, Spain, Sweden, Switzerland, Singapore, South Africa, hailand, urkey, UK. Note that not all candidates of x it variables are available for all countries due to data constraints. In particular, there are 26 series for eq it, 3 series for ep it (excluding the US), and 8 series for rit L. For m max = 6, we estimated the true number of common factors in rit S it and eq it, using the Bai-Ng information criterion IC, since it performs well in the Monte Carlo exercises reported by Bai and Ng (24). According to IC, ^m = 2 for the real interest rates and ^m = 3 for the real equity prices. herefore, to apply the CIPS test we require only one additional regressor for testing the unit root hypothesis in the real interests, and two additional regressors for the real equity prices. o check the robustness of the test outcomes to the choice of the additional regressors used in augmentation we present the CIPS test results for all possible combinations of candidate regressors. We consider lag orders p = ; 2; 3; 4. he test results are reported in able 9. Panel A of this table reports the results for the real interest rates. As can be seen, the null hypothesis of a panel unit root is strongly rejected at the 5% level for all cases considered across di erent choices of x t and the lag-augmentation orders, p. 7 For a detailed description of the data and sources see Supplement A of Dees, di Mauro, Pesaran and Smith (27). 8

19 hese results suggest that for a signi cant number of countries the Fisher parity holds and are in line with recent ndings reported in Westerlund (28) using panel cointegration tests. Panel B of able 9, summarises the test results for the real equity prices. For all the ten combinations of additional regressors and all the values of p, the null hypothesis of panel unit root cannot be rejected at the 5% level. his result is in line with the generally accepted view that real equity prices approximately follow random walks with a drift. We also applied the tests proposed by Bai and Ng (24) and Moon and Perron (24). Speci cally, the tests discussed in section 3, P^u ; P^u;z and t b or t# are computed for the real interest rates and the real equity prices, using the same estimates of ^m as above. In the case of the P^u and P^u;z tests, up to four lags are considered for the underlying ADF regressions. he test results are summarised in able. he results for the real interest rates are summarised in Panel A, and show that the P c^u ; P c^u;z and t b tests reject the null hypothesis of a panel unit root at the 5% level for all autoregressive lag orders, p, considered, which accord with the results of the CIPS test. Panel B in able reports the test results for the real equity prices. he results of the P ^u and P ^u;z tests are sensitive to the choice of lag orders. When p =, they do not reject the null of panel unit root. However, when p >, the null is rejected. his is in contrast to the results of our CIPS test, which could not reject the null for all lag augmentation orders and for all combinations of additional regressors considered. he t # test also does not reject the null hypothesis. But since t # lacks power when the regressions include a linear trend, the test outcome might not be reliable. 8 As a way of dealing with the sampling uncertainty associated with the choice of ^m, we also consider the application of the CIPS test assuming m max = 6, allowing m to take any value between and 6. Panel A of able reports the results for the real interest rates, and shows that for all lag orders considered, all the panel unit root tests point to a clear rejection of the null hypothesis. his is in line with the previous results obtained with an estimated value of m. he test results for the real equity prices are given in Panel B of the table. For all lag-orders considered, the CIPS test does not reject the null, but as before the results of the P ^u and P ^u;z tests are sensitive to the choice of lag orders. But now t# tests reject the null hypothesis, indicating the sensitivity of this test to the choice of the number of factors. 5 Concluding Remarks his paper considers a simple panel unit root test that is valid in the presence of cross section dependence induced possibly by m common factors. he proposed test is based on the average of t-ratios from ADF regressions of the variables of interest augmented by the cross section averages of the dependent variable as well as k additional regressors with similar common factor features. Initially we develop a test supposing that m, the true number of factors is known, and show that the limit distribution of the test does not depend on any nuisance parameters, so long as k + m. However, in practice m is not known. Given an assumed maximum number of factors, m max, we suggest two strategies for dealing with uncertainty that surrounds the value of m. One is to choose the number of additional regressors as k = m max, which avoids having to estimate m. In this case, the true number of factors are allowed to be any integer value between zero and m max. However, for large values of m max, in some situations it can be di cult to nd a su cient number of additional regressors. Another strategy is to 8 he t # test has the asymptotic power within a N =6 -neighbourhood of the null hypothesis of a unit root. Moon, Perron and Phillips (27) show that a full bias correction, rather than just a correction to the numerator of t #, is required to achieve power in N =4 neighbourhood of the null. 9