Panel Unit Root Tests in the Presence of a Multifactor Error Structure

Transcription

1 Panel Unit Root ests 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 Centre for Financial Analysis and Policy (CFAP), University of Cambridge c Department of Economics and Related Studies, University of York November 3, 28 Abstract his paper extends the cross sectionally augmented panel unit root test proposed by Pesaran (27, JAE) to the case of a multifactor error structure. he basic idea is to exploit information regarding the unobserved factors that are shared by other time series in addition to the variable under consideration. Importantly, the proposed test only requires speci cation of the maximum number of factors, in contrast to other panel unit root tests based on principal components that require in addition the estimation of the number of factors as well as the factors themselves. Small sample properties of the test is 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 the existing tests in the literature that tend to be over-sized. he power of the test is also reasonably high, although large samples are needed if a linear trend is included amongst the deterministics. Empirical applications to Fisher s in ation parity and real equity prices across di erent markets illustrate how the proposed test preforms in practice. JEL-Classi cation: C2, C5, C22, C23 Keywords: Panel unit root tests, Cross Section Dependence, Multi-factor Residual Structure, Fisher In ation Parity, Real Equity Prices. We would like to thank Joachim Westerlund for useful comments on an earlier version of the paper. his version has bene ted from comments by participants, especially Anindya Banerjee and Soren Johansen, at the conference in honour of Paul Newbold at the Univeristy of Nottingham, November 27, the ime Series and Panel Modelling Conference in Honour of M. Hashem Pesaran, Goethe University, Frankfurt, June 28, the Far Eastern and South Asian Meeting of the Econometric Society, at Singapore Management University, July 28 and the Econometric Society European Meetings, Milan 28.

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 (27). 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). 2 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 (26). he 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 non-normal limit distribution as N and!. Monte Carlo experiments show that Pesaran s test has desirable small sample properties in the presence of 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 propose 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. he test assumes that there exists a number of variables that are simultaneously a ected by 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 It is also possible to allow for cross sectionally correlated errors by adjusting the standard errors of the pooled estimate of the autoregressive coe cient. Breitung and Das (27), however, show that validity of these tests depend on whether the factors and/or the errors are non-stationary. 2 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

3 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) 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 will also a ect short run and long run interest rates across markets and economies. he fundamental issue is to ascertain the nature of dependence and persistence that are 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 unit roots and multiple cointegration in large panels. Within the above set up we show that the limit distribution of the proposed unit root test does not depend on any nuisance parameters such as the factor loadings or cross section heteroskedasticity, so long as k+ is greater or equal to the true number of factors, m. herefore, the distribution of the test statistic can be well approximated by the distribution based on the augmented regression with k additional regressors for k + m. his means that our approach does not require the knowledge of the true number of factors, in contrast to the panel unit root tests based on principal components that require, in addition to the speci cation of the maximum number of factors, the estimation of the number of factors and the factors themselves. Small sample properties of the proposed test is 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 (27) 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 the proposed test does even better than the pooled tests, both in terms of size and power, although as to be expected all panel unit root tests exhibit lower power when a linear trend is included. his is in line with the theoretical results reported in Moon, Perron and Phillips (26). Empirical applications to Fisher s in ation parity and real equity prices across di erent markets illustrate how the proposed test preforms 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. he critical values of the test are derived in Section 3. Section 4 describes the Monte Carlo experiments and reports the small sample results. Section 5 presents the empirical applications, and Section 6 provides some concluding remarks. Notation: K denotes a nite positive constant, 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 (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 N xed (or when there is no N-dependence) as!, N; =) denotes sequential convergence with N! rst followed by!, (N; ) j =) denotes joint convergence with N,! jointly with certain restrictions on the expansion rates of and N to be speci ed, if any,. 3 Westerlund and Larsson (27) provide further theoretical results in regard to the asymptotic validity of PANIC used for pooling purposes. 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. N 3

4 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 id t ) + id t + u it, i = ; 2; :::; N; t = ; 2; :::; () where i = ( i ); d t is 2 vector of observed common e ects including an intercept and a linear trend so that d t = (; t). Consider the following multifactor error structure u it = if t + " it (2) where f t is an m vector of unobserved common e ects, i is the associated vector of factor loadings, and " it 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 : he idiosyncratic shocks, " it, 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 " its are cross sectionally weakly dependent in the sense of Pesaran and osetti (27). However, such an extension will not be considered in this paper. Assumption 2: he m vector f t follows a covariance stationary process, with absolute summable autocovariances, distributed independently of " it for all i; t and t. Speci cally, we assume that f t = (L)e t where the error terms e t IID(; ); with nite fourth-order moments and a positive de nite covariance matrix, and (L) = P `= ` L` where f` `g `= mm are absolute summable so that V ar(f t ) is bounded and positive de nite and [ (L)] exists. In particular, fs = E(f t f t s ) = P `= s+` ` K <, for s = ; ; 2:::; where K is a xed bounded matrix, such that jjkjj < K. Further, f = P `= `P where P is the Cholesky factor of = PP. Assumption 3: he unobserved factor loadings, i are bounded, k i k < K <, for all i. Combining () and (2) it follows that y it = i (y i;t id t ) + id t + if t + " it : (3) 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, (4) 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, (3) can be solved for y it to yield y it = ~y i + id t + is ft + s it ; (5) 4

5 where s ft = f + f 2 + :::: + f t ; s it = " t + " 2t + ::: + " it ; and ~y i = y i i d. herefore, under H and Assumptions and 3, y it is composed of a deterministic component, ~y i + i d t, a common stochastic component, s ft I(); and an idiosyncratic component, s it 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, i. 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 terms 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 = ~x i + ix s ft + A ix d t + v ixt, (6) where x it = (x it ; x i2t ; :::; x ikt ), ix = ( i ; i2 ; :::; ik ), A ix = (a i ; a i2 ; :::; a ik ), ~x i = x i A ix d and v ixt is the idiosyncratic component of x it that could be either I() or I(): Here we assume v ixt to be I(), which rules out cointegration among the x it s, and assume that v ixt follow a stationary linear process. We also assume that the k vector v ixt is distributed independently of " it for all i; t and t. Combining (5) and (6) we have z it = ~z i + i s ft + A i d t + v it ; (7) where z it = (y it ; x it ), ~z i = (~y i ; ~x i ), i = ( i ; ix ), A i = ( i ; A ix ), and v it = (s it ; v ixt ). Assumption 4: he (k + ) m matrix of factor loadings i are such that rank = m k +, for any N, and as N! ; (8) where = N P N i= i, and N!, where is a xed bounded matrix with rank m. Assumption 5: Ejjf jj K, and Ejj~z i jj K, Ejv ix j K; and Ej" i j K for all i. Remark 2 Even if the k elements of x it are cointegrated our analysis continues to be valid, so long as m k r + ; where r = max r i and r i is the number of cointegrating relations <i<n amongst the elements of x it. Averaging (7) over i now yields z t = z + s ft + Ad t + v t, (9) 5

6 where z t = N P N i= z it, A = N P N i= A i; and v t = N P N i= v it. 5 Writing (3), (7) and (9) in matrix notation, under the null for each i we have y i = F i + D i + " i, () Z i = F i + DA i + V i ; () Z = F + D A + V; (2) where F = (f ; f 2 ; :::; f ) with f = m, D = (d ; d 2 ; :::; d ), " i = (" i ; " i2 ; :::; " i ) with " i =, Z i = (z i ; z i2 ; :::; z i ), V i = (v i ; v i2 ; :::; v i ), Z = (z i ; z i2 ; :::; z i ) and V = (v ; v 2 ; :::; v ). From (2) under rank condition (8) it follows that h F = Z D A Vi : (3) However, from A.2. in Appendix A we have that V! N: as N! for each t and hence we obtain that h F Z D A Vi N! ; as N! : In view of the above we shall base our test of the panel unit root on the t-ratio of the ordinary least square (OLS) estimate of b i (^b i ) in the following cross sectionally augmented regression he t-ratio of ^b i in this regression is given by y it = b i y it + c iz t + h iz t + g id t + it. t i (N; ) = yi My i; ^ i yi; =2 = My i; p (2k + 4)y i My i; y i M i y i =2 y i; My i; =2 ; (4) 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 + 4), and M i = I W i W i W i W i, with W i = W; y i;. Using (3) in () y i = Z i + D i + i i, (5) where i = i, i = i A i, i = i = i, (6) with i = " i V i, i = ( i ; i2 ; :::; i ) and E( i i) = 2 i I + O(N ): herefore, we have My i = i M i. (7) 5 Weighted cross section averages could also be used with appropriate granularity restrictions on the weights. 6

7 >From () and (2) we obtain Also Z i; = ~z i + S f; i + D A i + V i; : Z = z + S f; + D A + V i; (8) where S f; = ( m ; s f ; :::; s f; ), D = (d ; d ; :::; d ), with d = (; ), Z i; = (~z i ; z i ; :::; z i ) ; V i; = (v i ; v i ; :::; v i; ), Z = (z ; z ; :::; z ) and V = (v ; v ; :::; v ). Similarly from (5) where y i; = y i + Z i + D i + i s i; ; (9) s i; = (s i; V i )= i, (2) s i; = (; s i ; :::; s i; ) and y i = y i i(z + v ) i d : herefore, My i; = i Ms i;. (2) Using (7) and (2), t i (N; ) can be re-written as ims t i (N; ) = i; ( im i i 2k 4 )=2 s =2. (22) 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 a 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 deviation z it z. Before proceeding further, for notational convenience, we set Also S vxi; = V xi; ; with S vxi; = ( k ; s vxi ; :::; s vxi; ), s = N S vi; = V i; so that S vi; = (s i; ; S vxi; ). N X i= X N s i;, S vx; = N S vxi;, S v; = (s ; S vx; ) (23) and so (2) becomes s i; = s i; S v; i =i : he main asymptotic results concerning the distribution of t i (N; ) are summarized in the theorems below. he proofs are given in Appendix for the case where d t = ; t = ; ; :::;, which implies D = and D 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; :::;, are generated under (4) according to (7); d s =, s = ; ; :::;, with z i set to a zero vector. hen under Assumptions -5, the distribution of t i (N; ) given by (22), will be free of nuisance parameters as N! for any xed > 2k + 4. In particular, we have (in quadratic mean) i= t i (N; ) N! " i s i; " i " i g 2 i ( 2k 4) i Q i g i ( 2k 4) 2 i q i f h i =2 s i; s =2, i; 2 i h 2 i f h i 7

8 where q i = (2m+) f = F p " i i " i p i S f; " i i See Appendix A.3 for a proof. 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 i; i 3=2 s i; i 3=2 S f; s i; i 2 C A, g i = C A, Q i = f h i h i q i s i; " i 2 i s i; s i; 2 i 2 Remark 3 he limit distribution of t i does not depend on the factor loadings and i but only on f t and the standardised errors " it = i. heorem 2.2 Suppose the series z it, for i = ; 2; :::; N, t = ; 2; :::;, are generated under (4) according to (7) and d s =, s = ; ; :::;. hen under Assumptions -5 and as N and! ; such that p =N!, t i (N; ) given by (22) has the same sequential (N! ;! ) and joint [(N; ) j! ] limit distribution, is free of nuisance parameters and is given by where! if Z W i (r)dw i (r)! if G f if CADF i = Z =2, (24) Wi 2(r)dr if G if Z G f = W i () [W f (r)] dw i (r) Z [W f (r)] dr A, if = Z Z f Z Z [W f (r)] dr W i (r)dr [W f (r)] W i (r) [W f (r)] [W f (r)] dr See Appendix A.4 for a proof. Note that CADF i does not depend on f as de ned in Assumption 2. Remark 4 CADF i and CADF j are dependently distributed 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 ; 8 C A : C A,!!.

9 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 ), 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. It is clear that the truncation does not a ect the limit distribution and heorem continues to apply to t i (N; ) so that where CADF i = t i (N; ) 8 < : CADF i = o p (); (25) 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 is easily 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 N X i= CADF i ) can be obtained easily by stochastic simulations. Note that these critical values need to be tabulated for di erent values of k (see Section 3 for details). In what follows we only report the results for non-truncated version of the test statistics. he results for the truncated version are very similar and are available upon request. One could think of the t i (N; ) statistic to be analogous to the common Dickey-Fuller test statistic in the sense that the latter has a non-standard distribution, is free of nuisance parameters and the critical values depend on whether a trend term is included in the regression or not. he same applies to the CIP S(N; ) and CIP S (N; ) statistics which are averages of the t i (N; ) statistic, and similarly their critical values depend on the nature of the deterministics and the number of x s in the ADF type regressions as a proxy for the unobserved components. 9

10 2. he Case of Serially Correlated Errors 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 u it = if t + it and residual serial correlation is modeled as it = i i;t + it ; j i j < ; for i = ; 2; :::; N (28) where it (; 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 id t ) + id t + if t + it ( i ) (29) where it ( i ) = ( i L) it. 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 y it = if t + it (); (3) y it = y i;t + i(f t f t ) + it : (3) aking the rst-di erence of (6) and combining it with (3) we obtain in matrix notation Z i = F i + V i ; (32) where V i = ( i(); V ix ) and i () = ( i (); i2 (); :::; i ()), 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 + V; where V = N P N i= V i, from which it follows under rank the condition (8) that F = Z V. (33) hus in testing (4) 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)

11 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 with y i = y i; + (Z Z ) i + i i ; (36) i = i = i, where i = [ i ( V V ) i ], i = ( i ; i2 ; :::; i ) and E( i i) = 2 i I + O(N ): From (36) it follows that y i; = y i + Z i + i s i; where s i; = s i; S v; i =i, s i; = (; s i ; :::; s i; ) with s it = P t j= ij(), S v; = (s ; ; S vx; ) with s ; = N P N i= s i; and y i = y i i(z + v ): he test statistic (35) then becomes 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 (4) 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 xed depends on the augmentation order, p. hus, we will obtain critical values of t i (N; ) for di erent choices of p. 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 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 + u it, i = ; 2; :::; N; t = ; 2; :::;,

12 where u it 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 + v ijt, i = ; 2; :::; N; j = ; 2; :::; k; t = ; 2; :::;, (38) with v ijt 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; 4 and p = ; ; :::; 4; as the quantiles of CADF and CADF for = :; :5; :. Results for the 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. 4 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 test by Moon and Perron (24), by means of Monte Carlo experiments. he pooled test statistics proposed by Bai and Ng (24), using our notation as set out in Section 2, is obtained as follows. Firstly transform y it : ȳ it = y it if y it has individual e ects, or ȳ it = y it y i with y i = ( ) P t=2 y it, if y it has a linear trend. 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 = (ȳ ; ȳ ; :::; ȳ ) and ȳ 2 N i = (ȳ i2 ;ȳ i3 ; :::; ȳ ). Under the normalisation i ^F ^F=( ) = I m, the estimates of the factor loadings are given by ^ i = ^F i ȳ =( ), which i yield the residuals e it =ȳ it ^ i^f t. Now set ^u it = P t s=2 e is, and compute the ADF statistic for the ADF(p) regressions in ^u it 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 2 P N P c^u = i= ln(pvc i ) 2N p, 4N and P ^u = 2 P N i= ln(pv i ) 2N p 4N, 2

13 where pv c i and pv i are the p-values of t c BN;i and t BN;i statistics, respectively.6 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 = (ȳ it ; it x ), rather than ȳ it only. he variable it is constructed from x it in a manner x similar to the construction of ȳ it described above. hese variants of P c^u and P ^u are denoted by P c^u;z and P ^u;z, respectively. he t b test statistic is obtained following Moon and Perron (24),7 he test is 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 defactored data where the long-run variance is estimated by employing Andrews and Monahan s (992) estimator based on the quadratic spectral kernel and prewhitening. he null hypothesis is rejected if the t b test statistic is less than (5% level test). For further details on the above statistics see Bai and Ng (24) and Moon and Perron (24). For the principal components based tests, we consider experiments where the number of factors is treated as known. 4. Monte Carlo Design Initially we consider dynamic panel models with xed e ects and a two-factor error structure. he data generating process (DGP) is given by y it = ( i ) i + i y i;t + i f t + i2 f 2t + " it ; i = ; 2; :::; N; t = 49; :::; (39) with y i; 5 =, where i iidn(; ), fj = for j = ; 2; and f jt = fj f fj;t + $ jt ; $ jt iidn(; ); f j; 5 = " it = i" " it + it ; it iidn(; 2 i ); " i; 5 = (4) i" = " =, 2 i iidu[:5; :5]. he x it process is generated as x it = i + ix f t + ix2 f 2t + v ixt ; i = ; 2; :::; N; t = 49; :::; (4) with x i; 5 =, where i iidn(; ), and v ixt = v ixt + e ivxt ; e ivxt = ivx e ivx;t + % it ; % it iidn(; ) with v ix; 5 =, e ivx; 5 =, and ivx iidu[:2; :4]. he factor loadings are generated as i iidu[; 3], i2 iidu[; 2], ix iidu[; 2], ix2 iidu[; 3], so that 2 E( i ) = ; 2 6 In our experiment p-values of t c BN;i and t BN;i are obtained using a linear interpolation and discrete quantiles. he latter are provided by Serena Ng in the les adfnc.asc and lm.asc. 7 he t a test, which is also proposed by Moon and Perron, is not listed in the simulation result tables since it displayed inferior performance relative the t b test. 3

14 and the rank condition (8) is satis ed. We consider three combinations of serial dependence in the errors: (A) serially uncorrelated " it and f jt ( i" = " = and f = f2 = ); (B) serially correlated " it ( i" iidu[:2; :4] whilst f = f2 = ); (C) serially correlated f jt ( i" = " = whilst f = f2 = :3). v ixt are generated as serially correlated processes in all experiments ( ivx iidu[:2; :4]). Similar sets of experiments are carried out for the model with a linear time trend. he DGP corresponding to (39) and (4) are y it = i + ( i ) i t + i y i;t + i f t + i2 f 2t + " it ; i = ; 2; :::; N; t = 49; :::; x it = i + i t + ix f t + ix2 f 2t + v ixt ; i = ; 2; :::; N; t = 49; :::; respectively, where i iid[:; :2] and i iid[:; :2]. he rest of the variables are de ned as above. he parameters i, i, i2, fj, i, i, ix, ix2, ivx, i and i are drawn once and xed over the replications. For size, i = =, and for power i iidu[:9; :99], where i is de ned by (4). 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. 4.2 Results Size and power of the tests in the case of the experiments with an intercept only are summarised in ables 3-6. Recall that for all experiments the rst di erence of the augmented variables, x it, are serially correlated and the models contain two factors. Also note that in the case of serially correlated idiosyncratic errors and/or factors, lag augmentation is required for the validity of the CIPS test and the pooled tests of Bai and Ng (24), while the t b test of Moon and Perron (24) is based on an estimate of the long-run variance. able 3 provides the results for the model where the factors, f t and f 2t, and the idiosyncratic components, " it, are serially uncorrelated. 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 Moon-Perron test, t b, shows signi cant size distortions unless is much larger than N. 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 considered, for smaller values of (which could be partly due to the over-sized nature of the pooled tests), while in general it tends to be more powerful for larger N and. ables 4 and 5 present the results for the case where " it are serially correlated but f t and f 2t are not. 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 and P c^u;z tests display a higher tendency of overrejecting the null relative to the case where the idiosyncratic errors are serially uncorrelated. he t b test shows similar size distortions to that in able 3 for positive serial correlation, though somewhat more exaggerated in the negative serially correlated case. ables 6 provides the results for the model where f t and f 2t are serially correlated but " it is not. In this case all tests exhibit size distortions unless is su ciently large relative to N, as predicted by the theory for the CIPS test. he extent of over-rejection of the CIPS test is less than that of the P c^u and P c^u;z tests, and is reduced with time series augmentation. he performance of the t b test is similar to that reported in the previous experiments. Monte Carlo results for the case of an intercept and a linear trend are summarised in ables 7-, the experimental designs of which correspond to those underlying the results in ables 4

15 3-6. he P ^u and P ^u;z tests severely over-reject the null hypothesis in all experiments, except for = 2 and N = 2. he size distortion of the t b test is worse in the linear trend case in almost all experiments. he performance of the CIPS test for the experiments with an intercept and a linear trend is very similar to that with the intercept only case. he CIPS test controls the size very well in almost all cases considered, and its power is reasonably high particularly for large and N. 5 Empirical Applications As an illustration of the proposed test we consider two applications. One to the real equity prices across N = 26 markets, and another application to the Fisher parity across N = 32 economies. Under the Fisher parity hypothesis the di erence between the nominal short term interest rate and in ation rate (both measured over a given period) is stationary. For both applications we employ quarterly observations over the period 979Q2 23Q4 (i.e. = 99). Existing evidence on the validity of the Fisher parity is rather mixed. he second application is chosen since it is 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. he table below shows the dependent variable, y it ; and the additional regressors, x it, considered for both applications: where y it Fisher Parity (N = 32) r S it it (gdp it ; eq it ; ep it ; r L it ; poil t) Real Equity Prices (N = 26) eq it (gdp it ; it ; ep it ; r S it ; rl it ; poil t) (42) x it gdp it = ln (GDP it =CP I it ) ; p it = ln(cp I it ); eq it = ln(eq it =CP I it ); e it = ln(e it ); r S it = :25 ln( + RS it =); rl it = :25 ln( + RL it =); poil t = ln(p OIL t ) with GDP it the nominal Gross Domestic Product of country i during period t in domestic currency, CP I it the consumer price index in country i at time t, EQ it the nominal equity price index, E it the nominal exchange rate of country i in terms of U:S: dollars, Rit S is the short rate of interest per annum in per cent (typically a three month rate), Rit L the long rate of interest per annum in per cent (typically the yields on ten year government bonds), and P OIL t is the price of Brent Crude oil, so that rit S is the quarterly short-term interest rate, it = (p it p i;t ) the quarterly in ation rate, ep it = e it p it is the real exchange rate, eq it is the real equity price index, gdp it is real (log) output, rit L is the quarterly long-term interest rate and poil t is the nominal oil price. 8 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 x it variables are available for all countries due to data constraints. In particular, there are 32 series for gdp it, 26 series for eq it, 3 series for ep it (excluding the US), and 8 series for rit L. 8 For a detailed description of the data and sources see Supplement A of Dees, di Mauro, Pesaran and Smith (27). 5

16 o begin with we applied the information criteria of Bai and Ng (22) to estimate the number of common factors in eq it and rit S it. For the Fisher parity, the Bai-Ng information criteria suggested di erent number of factors depending on the maximum number of factors assumed. We decided to set m max = 4. For the real equity series applying the information criteria of Bai and Ng (22) was not very helpful as the number of factors always turned out to be the maximum set. We therefore also set m max = 4 for this application. However, not knowing the true number of factors, we present results for m = ; 2; 3 and 4. Results are presented for all tests, namely CIPS, P^u ; P^u;z and t b ; considering up to four lags for the underlying DF regressions, in the case of the rst three statistics. In what follows we use, directly, the x it series as de ned in (42). he panel unit root regressions are augmented with the cross section averages of y it and x it, namely z t = (y t ; x t) to compute the CIPS test statistics as described in Section 2. For the P^u;z test, the principle component approach is applied to z it = (y it ; x it ) in order to estimate the common factors. ables and 2 present the results for the Fisher parity and the real equity prices, respectively, for all panel unit root tests at the 5% signi cance level. Note that for the Fisher equation a constant only is included in the regressions, while for real equity prices both a constant and a linear trend are included. Results for the Fisher parity across 32 countries show that the CIPS test rejects the joint null hypothesis of a unit root and no cross section cointegration at the 5% level. his nding holds for all choices of the number of common factors, m, and autoregressive lag orders, p, considered. he use of P c^u ; P c^u;z and t b statistics also yield the same test outcomes. hese results suggest that for a signi cant number of countries the Fisher parity holds and are in line with recent ndings from the panel cointegration tests applied to the Fisher equation. See, for example, Westerlund (27). urning to the test results applied to the real equity prices, the CIPS test does not reject the joint null hypothesis at the 5% level for all combinations of m and p. his is also the case for the t b statistic. Contrary to these results, the P ^u ; P ^u;z tests reject the joint null hypothesis at the 5% level for almost all combinations of m and p; with the exception of p = where the test results are rather mixed. his outcome does not accord with the generally accepted view that real equity prices approximately follow random walks with a drift. 6

17 able. Panel Unit Root est Statistics for esting the Null Hypothesis of a Unit Root in the Fisher Parity Across 32 Countries Over the Period 979Q2 23Q4 m = m = 2 p CIPS -5. y y -3.8 y -3.7 y -5. y y -3.8 y -3.7 y P c^u 2.4 y 5. y 5.89 y 7.52 y 9.4 y 3.63 y 6.62 y 7.8 y P c^u;z 6.2 y.44 y 5.3 y 6.2 y.97 y 8.37 y 5.75 y 6.4 y t b -7.9 y y m = 3 m = 4 p CIPS -5. y y -3.8 y -3.7 y -5. y y -3.8 y -3.7 y P c^u 7.33 y.7 y 4.8 y 6.26 y 9.42 y 4.7 y 6.93 y 7.98 y P c^u;z. y 7.74 y 4.5 y 5.47 y 9.29 y 7.42 y 4.29 y 5.8 y t b y y Note: y denotes rejection at the 5% signi cance level. P c^u and P c^u;z are the Bai and Ng (24) statistic and its augmented variant, respectively, and t b is that of Moon and Perron (24). m is the number of common factors and p is the lag order of the ADF regressions. P c^u and P c^u;z reject the null hypothesis of a unit root if they are greater than.645, and t b if it is less than he latter adopts automatic lag-order selection for the estimation of long-run variances, which explains why only one value is reported for this test. he critical values of the CIPS test for values of p = to 4 lie in the range [-2.5,-2.9] for m =, [-2.4,-2.3] for m = 2, [-2.6,-2.47] for m = 3, [-2.78,-2.6] for m = 4, as given in able, corresponding to k = ; ; 2 and 3, respectively. k denotes the number of variables included as additional regressors in the regressions for obtaining the CIPS statistic, and included in the estimation of the unobserved common factors in the case of the P c^u;z test. 7

18 able 2. Panel Unit Root est Statistics for esting Null Hypothesis of a Unit Root in Real Equity Prices Across 26 Countries Over the Period 979Q2 23Q4 m = m = 2 p CIPS P ^u y 3.37 y y 4.43 y 4.6 y P ^u;z.78 y 2. y 3.7 y 4.33 y y 4.5 y 4.25 y t b m = 3 m = 4 p CIPS P ^u y 4. y 4.42 y.89 y 3.3 y 5.5 y 5.88 y P ^u;z y 4.34 y 4.5 y y 3.6 y 3.8 y t b Note: y denotes rejection at the 5% signi cance level. P ^u and P ^u;z is the Bai and Ng (24) statistic and its augmented variant, respectively, and t b is that of Moon and Perron (24). he variable m is the number of common factors and p is the lag order of the ADF regressions. P ^u and P ^u;z reject the null hypothesis of a unit root if they are greater than.645, and t b if it is less than he latter adopts automatic lag-order selection for the estimation of long-run variances, which explains why only one value is reported for this test. he critical values of the CIPS test for values of p = to 4 lie in the range [-2.65,-2.6] for m =, [-2.85,-2.75] for m = 2, [-3.,-2.85] for m = 3, [-3.5,-2.94] for m = 4; as given in able 2, corresponding to k = ; ; 2 and 3, respectively. k denotes the number of variables included as additional regressors in the regressions for obtaining the CIPS statistic, and included in the estimation of the unobserved common factors in the case of the P ^u;z test. 6 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. It only requires speci cation of the maximum number of factors, in contrast to other panel unit root tests based on principal components that require in addition the estimation of the number of factors as well as the factors themselves. So long as k + m, the asymptotic distribution of our proposed panel unit root test does not depend on any nuisance parameters and though non-standard, its critical values can be calculated easily by simulation. Small sample properties of the proposed test are investigated by Monte Carlo experiments, which suggest the test has the correct size and reasonable power for larger values of and N. In comparing the performance of the proposed test with that of Moon and Perron (24) and Bai and Ng (24) when a constant only is included in the data generating process, we nd that the CIPS test is somewhat less powerful than the pooled tests of Bai and Ng (24), which partly could be due to the over-sized nature of the latter tests. However, when both an intercept and trend are included, the CIPS test has the correct size for all combination of sample sizes compared to the 8

19 pooled tests of Bai and Ng (24) and Moon and Perron (24) that over-reject, in some cases substantially, the null hypothesis. he various panel unit root tests are applied to real interest rates (Fisher s in ation parity) and real equity prices across countries. All tests reject a unit root in real interest rates which is in line with panel cointegration tests of the Fisher equation. However, for real equity prices, the pooled test of Bai and Ng (24) rejects the unit root hypothesis in real equity prices for almost all combinations of the number of unobserved common factors and lag lengths considered. his is in contrast to the results of the other panel unit root tests considered in this paper, and does not accord with the generally accepted view that real equity prices approximately follow random walks with a drift. Appendix A: Mathematical Proofs A.2 Preliminary Order Results he results shown below are for the serially uncorrelated case. For the serially correlated case, analogous order results are obtained. A.2. Order Results A Under Assumptions -5, A.2.2 Order Results B Under Assumptions -5, " i" " F " S v; " " " i s i; S v; 2 = Op = Op = Op = Op p, " i V N = O p( p ), N p, V = O p p, N N " V, V V = O p, N N p, F V = O p p. N N S v; V = O p p, N = O p N S = O p p, v; F N, = O p p, N r! s = O p, r! S v; = O p, N N S v; S v; S = O 2 p, v; s = O N 2 p, N = O p p, S f; S v; = O N 2 p p, N s i; V = O p p, S f; V = O p p. N N 9

20 A.2.3 Order Results C Recall that i = (" i V i)= i and i; = s i; S v; i =i. hus, from (2) and (8) we have and Z = F + V; (A.) Z i p Z = z + S f; + S v; : (A.2) Using (A.), (A.2), Order Results A and B, under Assumptions - 5, we obtain the following expressions p p + O p, N N Z i = F i p + V i p = F " i i p = z i = z " i i i p + O p = " i p + O p i + S f; i p, N + S v; i r + S f; " i + O p p + O p i N N Z i; = F i; + V i; = F s i; 3=2 3=2 3=2 i Z i; 2 = z i; = s i; 3=2 + i 3=2 Op i; 2 = z s i; i 2 3=2 + Op p ; N + S f; i; + S f; s i; + O i 2 p p, N S + v; i; 2 2 p, N!, Z Z Z Z Z = z 3=2 F F = z Z 3=2 = F F + F V = F F + O p = F + V 3=2 + V 3=2 3=2 + S f; F + S f; F + O 3=2 p + V F + V V p + O p N N, = F + O p p, N + S f; V S + v; F S + v; V 3=2 3=2 3=2 3=2 p ; N = p z + S f; S + v; = p z + S f; + O 3=2 3=2 3=2 p p ; N Z Z = zz 2 + S f; S f; S + v; S v; S + v; S f; + S f; S v; z S f; 2 + S f; 2 z +z S v; 2 + S v; 2 z = zz + z S f; + S f; 2 + S f; S f; + O p 2 z 2 p + O p p, N N 2

21 i; A.3 Proof of heorem 2.: A.3. xed and N! Recall equation (22) that can be written as i = s i; " i + 3=2 Op 3=2 2 i i; i; = s i; s i; 2 2 i + O 2 p t i(n; ) = i M i i 2k 4 ims i; p ; N p : N =2 s i; Ms i; 2 =2. Expanding i M i; = in (22) gives i M i; = i i; iwb B W B W WB i;! ; where and B W i p B = I k+2 Z i= p i= p Z i= B W WB = A, B W i; Z Z Z Z Z 3=2 I k+ X Z 3=2 Z i; = 3=2 i; = 3=2 Z i; = 2 Z Z 3=2 Z Z 3=2 Z 2 Next, note that M i i are the residuals from the regression of i on W i = ( W; y i; ), but from equation (9) y i; has components (Z, ; i; ). As ( ; Z ) W, but i; is not contained in W, by regression theory C A. A, M i i = M i i where with H i = ( W; i; ). hus i M i i = i i M i = I H i H i H i H i; i H ie E H i H ie E H i i, where Also, E H i i = B W i i= i; B E =., E H ih ie B W WB B W = i; = i; WB= i; i; = 2 Under Assumptions -5, using the order results in (A.2) and assuming z = or re-de ning z it as the deviation from z, as N! with xed,. where B W i N! q i, B W i; N! h i, B W WB N! ' ; 2