Variance Ratio Tests for Panels with Cross. Section Dependence

Transcription

1 Variance Ratio Tests for Panels with Cross Section Dependence Seongman Moon Carlos Velasco February 13, 2015 Abstract This paper develops econometric methods based on variance ratio statistics to investigate persistence properties of panels with cross section dependence. We present asymptotic results for small and large panels with the time series dimension T growing. If the cross section dimension N is allowed to diverge, a factor model is used to describe cross section dependence and a simple common correlation estimate is proposed to remove factor influence. Exploiting a panel with N increasing, we show that our pooled statistic can recover asymptotic power over long horizons even when the aggregation horizon q is growing at the same rate as T, in contrast to the criticisms raised by Richardson and Stock 1989) and Deo and Richardson 2003). The finite sample properties of panel variance ratio tests are investigated and compared with panel unit root tests of Pesaran 2007) by Monte Carlo simulations. We apply our tests to a panel of 190 European Union sectoral real exchange rates and find that they are not mean-reverting in long horizons. Department of Economics, Universidad Carlos III de Madrid, Calle Madrid 126, Getafe Madrid, SPAIN. smoon@eco.uc3m.es, Tel.: , Fax: Research support from Spanish Secretary of Education ECO ) is gratefully acknowledged. Department of Economics, Universidad Carlos III de Madrid, Calle Madrid 126, Getafe Madrid, SPAIN. Carlos.Velasco@uc3m.es, Tel.: , Fax: Research supported by Ministerio de Economia y Competitividad. Ref. ECO

2 1 Introduction Variance ratio tests have been widely used to identify slowly decaying mean reverting components in long horizon returns. 1 However, the use of those tests is limited only for a univariate time series [see, for example, Cochrane 1987), Lo and MacKinlay 1989), Choi 1999), Wright 2000), and Moon and Velasco 2013)]. 2 Consequently, those statistics have not only limited power but also limited applications. For example, Richardson and Stock 1989) have shown that when the aggregation horizon q is growing at the same rate as the time series dimension T, the limit distribution of variance ratios is a functional of standard Brownian motion under the random walk hypothesis. Further, Deo and Richardson 2003) have proved that in this set up the asymptotic theory also leads to nondegenerated limits for individual variance ratios under a wide class of alternatives: the variance ratio tests are inconsistent and do not provide high power even for large values of T. In this paper, we develop econometric methods based on variance ratio statistics to investigate persistence properties of panels with cross section dependence. We propose panel variance ratio tests for the random walk hypothesis. Since this hypothesis fully specifies the dynamics of the observations, the main flexibility is introduced in the cross section dependence. We present asymptotic results for small and large panels with T growing, which is a reasonable assumption in the applications we have in mind. If the cross section dimension N is known to be finite, cross section dependence can be left completely unrestricted and does not require further modeling except for just proper standardization. If N is allowed to diverge, a factor model is used to describe cross section dependence and a simple projection on cross section averages is proposed to remove factor influence before individual or pooled variance ratios are calculated. This leads to tests with standard null asymptotics 1 See, for example, Cochrane 1987) for US output, Porteba and Summers 1988) and Lo and MacKinlay 1988) for stock prices, Liu and He 1991) for nominal exchange rates, and Glen 1992) for real exchange rates. 2 Several studies such as Richardson and Smith 1991), Chow and Denning 1993), and Cecchetti and Lam 1993) developed multiple variance ratio tests. However, their tests use a univariate time series and consider a joint null hypothesis of no serial dependence of the series over all horizons against an alternative of a certain serial dependence pattern for some horizons. In fact, our method for the panel data set employed in this paper can incorporate multiple variance ratio tests in these studies. 2

3 under mild restrictions on T and N. Further, exploiting these panels with N, we show that the pooled statistic can recover asymptotic power over long horizons even when q is growing at the same rate as T. This result overcomes the criticisms raised by Richardson and Stock 1989) and Deo and Richardson 2003) on detecting long range predictability with variance ratios. The usual approach in the literature to investigate the persistence of the time series within a panel is by means of unit root tests. The key element in the use of typical unit root tests for panel applications is to allow for cross section dependence and heterogeneity. These features are typically introduced through a factor structure where both T and N grow in the asymptotics. For example, Bai and Ng 2004), Moon and Perron 2004), and Phillips and Sul 2003) use residual factor models to account the cross section dependence by means of orthogonalization and defactoring procedures that require both N and N/T 0. In Pesaran 2007), instead of basing the unit root tests on deviations from the estimated factors, the standard augmented) Dickey-Fuller ADF) regressions also include the cross section averages of lagged levels and first-differences of the individual series. Standard panel unit root tests are then based on the averages of the cross section augmented DF statistics. Bootstrap techniques have also been used to improve finite sample performance of panel unit root tests, e.g. Smith et al. 2004) and Chang 2004), as well as subsampling techniques to deal with cross section dependence, e.g. Choi and Chue 2007). We compare our panel variance ratio tests with the panel unit root tests. Since the random walk hypothesis is a subset of the unit root hypothesis, it is not possible to make direct comparisons. However, the alternatives against the random walk hypothesis complement the unit root tests: if the series is stationary its autocorrelations over long horizons must be negative. In other words, if we observe positive serial dependence over long horizons, the series must follow a unit root process or contain a unit root component. Using the design of Monte Carlo simulations by Pesaran 2007), we compare our tests with his panel unit root tests in finite samples: we find that our panel variance ratio tests perform as well as Pesaran s tests in terms of power. 3

4 Finally, we apply our variance ratio tests to a panel of 190 European Union sectoral real exchange rates constructed using the same data as Imbs et. al. 2005) used in their study and find that the real exchange rates are not mean-reverting in long horizons This result is in line with Gadea and Mayoral 2009) but in contrast with Imbs et. al. 2005). Imbs et. al. 2005) provide evidence that the half-life of convergence to Purchasing Power Parity PPP) is about eleven months, after controlling for biases induced by aggregation of individual goods prices which differ in the speed of the convergence to the parity. This is significantly below the consensus view of three to five years, extensively surveyed in Rogoff 1996). Using the same data and the same estimation strategy as Imbs et. al. 2005) employed in their paper, however, Gadea and Mayoral 2009) find different results that the half-life of convergence to PPP is about 3 to 5 years or even longer once small sample biases are corrected. 3 The main difference between the two studies is due to the measures of persistence computed with sectoral data: Gadea and Mayoral 2009) compute the individual impulse responses and then average them to obtain an estimate of the average sectoral impulse response, while Imbs et. al. 2005) estimate the mean value of the AR model coefficients in a panel of sectoral data and then use it to compute the impulse response function. Our method is based on panel variance ratio statistics and obtains similar results using both aggregate and sectoral real exchange rate data, supporting with the findings of Gadea and Mayoral 2009). The rest of the paper is organized as follows. Section 3 presents the new variance ratio based random walk tests for panels. Section 3 discusses a bootstrap version of the previous testing procedures and Section 4 presents a simulation study of the final sample performance of the asymptotic and bootstrap tests. Section 5 applies our tests to a panel of 190 European Union sectoral real exchange rates. 3 On the other hand, Mark and Sul 2008) argue that Imbs et. al. 2005) s calculation of half-life does not include the dynamics of the common factor in the real exchange rate which is quantitatively important and highly persistent, and thus underestimates the true value of the half-life. 4

5 2 Variance Ratio Tests for Panels with Cross Section Dependence Consider an array {y j,t } generated by the panel model y j,t = φ j y j,t 1 + ξ j,t, j = 1,..., N, t = 1,..., T, 1) where j denotes an individual cross section unit and t denotes a time period. The error term ξ j,t is given by ξ j,t = α j + π j L)ɛ j,t, 2) where L is the lag operator and π j z) = k=0 π j,k z k, for j = 1,..., N. We assume that ɛ j,t has mean zero and variance σ 2 j and is independent across t, possibly not across j. We also assume that π j 1) 0 and k=0 π 2 j,k < for all j so that π j L)ɛ j,t is a stationary and not overdifferenced series. The null hypothesis that we are interested in is that y j,t follows a random walk for all j = 1,..., N: H 0 : φ j = 0 and π j z) = 1 for all j, so that y j,t = ξ j,t = α j + ɛ j,t, against the following two alternatives, a stationary one and a unit root one H i) A : φ j < 0 for some j, H ii) A : φ j = 0 and π j z) 1 for some j, so that ξ j,t is not iid for some j. In particular, the alternative i) implies that y j,t is stationary and y j,t exhibits negative serial dependence over long horizons for some j. On the other hand, the alternative ii) implies that the series has a unit root and y j,t can display either positive or negative serial dependence. For developing variance ratio tests on panel models, we consider two cases: one 5

6 assumes that the number of cross section units, N, is fixed in the asymptotics as T ; the other assumes N in the the asymptotics together with T. We begin with the first case, which does not impose any restriction on the finite dimensional cross section dependence. 2.1 Variance Ratio Tests on Panel Models with a fixed N Under the random walk hypothesis, the differenced series y j,t for each j satisfies y j,t = ξ j,t = α j + ɛ j,t, where α j represents a fixed effect for individual unit j and ɛ j,t follows an iid distribution with mean zero and variance σ 2 j. It is natural to assume σ 2 j to be different across j. However, we can take wlog σ 2 j = 1 for all j since our variance ratio statistics standardize each series by multiplying y j,t by an estimate of 1/σ j under the null. We allow dependence across cross section units and denote Covɛ j,t, ɛ l,t ) = σ j,l. We start presenting sample variance ratios for each cross section unit and discuss their joint properties for all the panel units allowing for cross section dependence under the assumption of a fixed N. Then, we present several ways of aggregating information across cross section units, while taking into account of the cross section dependence. We begin with a general result on variance estimates constructed from each cross section unit j sample at aggregation horizon q = 1,..., q max < T where V j q) = 1 qmq) T ) 2 ξj,t + ξ j,t ξ j,t+1 q q µ j t=q and µ j = 1/T ) T t=1 ξ j,t depends only on data from unit j. The factor mq) = T q + 1) 1 q/t ) corrects the biases in the variance estimator V j q) due to the use of overlapping q-period changes in series j [see, for example, Lo and MacKinlay 1989)]. 6

7 One can easily calculate an individual variance ratio statistic for each j by V R j q) = V j q), j = 1,..., N. V j 1) Our approaches shown below incorporate these individual statistics and lead to yield an efficiency gain, similar to the panel regression analysis and panel unit root tests. To exploit information from all V R j q) available in a panel data set with N cross section units, we consider the asymptotic joint distribution of U N q) = T V R 1 q) 1,..., V R N q) 1 ). 3) 2q 1)2q 1)/3q Lemma 1. Under the null hypothesis, U N q) a N 0, Σ N q)), where the diagonal elements of Σ N q) are ones, and in general Σ N q) [j,l] = ρ 2 j,l 4) where ρ j,l = Corrξ j,t, ξ,l,t ). Lemma 1 allows many alternative ways of devising tests using the whole sample of size N T. One approach is to summarize information from variance ratios in each cross section based on order statistics of U N q), Maxq) = max U N q), Minq) = min U N q). 5) Using the max and min statistics we can perform one-sided tests, left and right ones, respectively, based on normal asymptotic critical values with a significance level α/n, invoking Bonferroni inequality. We later justify a bootstrap procedure that can approximate the actual asymptotic null distribution of these extreme statistics and can achieve better power performance. Another approach is to use a class of Wald statistics. This is summarized by 7

8 the following definition: W q; R) = RU N q)) R Σ N R ) 1 RUN q)), 6) where R is a full row-rank non-random r N matrix. W q; R) is asymptotically distributed as a χ 2 N variable under the null for Σ N p Σ N. Consistent estimates Σ N can be obtained by sample analogs of the expectations in ρ 2 j,l. The standard case is when R = I N, involving tests for the joint hypothesis of all individual variance ratios being equal to one. Taking R = 1/N,..., 1/N) we can test whether the average variance ratio across cross sections is equal to one see the detailed discussion below). We can also test if a weighted average ratio is equal to one by taking R = w 1,..., w N ) where N w j = 1. In particular, this weighted average ratio fits well in applications dealing with series such as an aggregate price index in macroeconomics or a portfolio of several assets in finance. Define e i by a N element vector of zeros except for the ith element which is one. Setting R = e i, we can also provide t-tests for individual variance ratio for cross section i. Finally, we can construct a pooled variance ratio statistic from the individual ones. It is constructed by V R pool q) = w V j R j q), 7) where w j 0 for each j = 1,..., N and N w j = 1. Its asymptotic distribution is described in the next lemma. Lemma 2. Under the null hypothesis, T V R pool q) 1) 2q 1)2q 1)/3q a N 0, R w R w + Ω N ), where R w = w 1,..., w N ) and Ω N = w j w l ρ 2 j,l l j 8

9 is derived from covariances between cross sections. Then, z N q) = ) T V R pool q) 1 2q 1)2q 1)/3q R wr w + Ω N ) 1 2 8) follows an asymptotic standard normal distribution. The R w R w term in the asymptotic variance is equal to 1/N when w j = 1 for N all j, while the extra term, Ω N, appears due to the correlation between different cross section units and obviously does not appear in the asymptotic distribution of individual variance ratios. 2.2 Variance Ratio Tests on Panel Models with N When we allow N to increase in the asymptotics together with T we need some restriction on the covariance matrix of ξ j,t, j = 1,..., N, while allowing some degree of heterogeneity and cross sectional dependence. We introduce this dependence through a factor structure in the disturbance under the random walk hypothesis, ξ j,t = α j + λ j f t + ɛ j,t, 9) where f t is a common factor, which is assumed to be iid with mean zero, variance σ 2 f, and finite fourth moments under the null. The unit specific factor loading λ j is iid with E [λ j ] 0 and finite variance and independent of ɛ j,t for all j and t. Both f t and λ j are independent of ɛ j,t which is an iid ) 0, σ 2 j unit specific disturbance and independent across j. We then follow Pesaran 2006) to eliminate the dependence introduced by the common factor using projection on cross section averages, instead of using standardized statistics. This could also be implemented as in Pesaran 2007) by adding such averages as additional regressors in the regressions that define each serial correlation coefficient of the series ξ j,t for every j = 1,..., N, which are the basic blocks of the variance ratio statistic, see the discussion at the end of this section. We remove the common factor f t using a projection method on the cross section 9

10 average of the series. Let ξ t be the cross section average across all j and be defined by ξt = 1 N so that ξ j,t ξt = f t 1 N λ j + 1 N = λf t + ᾱ + ɛ t. α j + 1 N ɛ j,t The average ξ t proxies f t up to scale and origin) since ɛ t is small and λ E [λ i ] 0 when N is large. Using the projection of ξ t on series ξ j,t, we obtain the defactorized series ξj,t = ξ j,t α j δ j ξt, where δ j = Ĉov ξt, ξ j,t ) / V ar ξt ) is the projection coefficient and α j = ξ j δ j ξ, ξj = 1 T T t=1 ξ j,t, ξ = 1 T T t=1 ξt = 1 N ξj. Then, we can define the corresponding variance estimates constructed from the residuals ξ j,t of this projection, Ṽ j q) = 1 qmq) T ξj,t + ξ j,t ξ ) 2 j,t+1 q, t=q which are already centered. Analogous to the case of fixed N, we can calculate an individual variance ratio statistic for each j by Ṽ R j q) = Ṽ j q), j = 1,..., N. Ṽ j 1) To exploit information from Ṽ R jq) for all j when N increases, we consider the asymptotic distribution of the standardized average variance ratio, Ũ N q) = T Ṽ Rj q) 1 ), 10) N 2q 1)2q 1)/3q 10

11 which exploits the independence of ɛ j,t across j. Lemma 3. Under the null hypothesis, for fixed q and N, T ) j j = 1,..., N,, for each T Ṽ Rj q) 1 ) a N 0, 2q 1)2q 1)/3q), and if additionally NT 1 0 then Ũ N q) a N 0, 1). 11) Note that we only require both sample size parameters to grow asymptotically, with the only restriction that the time series dimension T has to grow faster than N for the aggregated statistics, but not for the individual test statistics, whose distribution can be approximated asymptotically under any relationship between N and T, each possibly growing arbitrarily slow. Remark 1. In the definition of the average variance ratio we can introduce different weights for each cross section, Ũ w Nq) = T Ṽ N w j Rj q) 1 ), 12) wj 2 2q 1)2q 1)/3q where the sum of square weights now replaces N and the asymptotic normality of ŨNq) w requires N wj 2 and T 1 N wj 2 0. Note that the unweighted case is recovered if w j are constant. Remark 2. One can consider an alternative method for removing the common factor f t by a projection on the cross section averages for all j, after centering each cross section unit to remove the fixed effects α j, ξt ξ = 1 N = 1 N ξj,t ξ ) j { λj ft f ) + ɛ j,t ɛ j } = λ f t f ) + ɛ t ɛ, 11

12 Then ξ t ξ proxies f t f up to scale, obtaining the following proxies of the idiosyncratic sectoral disturbance after projection ξ j,t ξj δ j ξt ξ )) = ξ j,t, where δ j is as before, obtaining again the previous defactorized series ξ j,t, which is shown to be invariant to α j due to the initial recentering. Then we can define the corresponding variance estimates constructed from the idiosyncratic component of each series. Both methods produce exactly the same results. 2.3 Size and Power of Panel Variance Ratio Tests with N Our pooled variance ratio statistic overcomes criticisms regarding the inference on long horizon returns. That is, as shown in Campbell, Lo, and McKinlay 1997), when the horizon q is large relative to sample period T, the asymptotic approximations for individual variance ratios break down since the variance ratios are bounded by zero from below. Richardson and Stock 1989) have shown that when q/t δ 0, 1) the limit distribution of the individual variance ratio V R j q) is a functional of standard Brownian motion under the random walk hypothesis, without need of normalizing by T. Further, Deo and Richardson 2003) have proved that in this set up the asymptotic theory also leads to nondegenerated limits for individual) V R j q) under a wide class of alternatives, so that the variance ratio test is inconsistent and does not provide high power even for large T. However, exploiting a panel with N, we show below that the pooled statistic can recover some asymptotic power for long horizons even when q is growing at the same rate as T. Under the same assumption of q/t δ 0, 1) as in Richardson and Stock 1989), we can rewrite the pooled statistic ŨNq) in equation 10) in the following way Ũ N q) 1 1 N 4δ/3 Ṽ Rj q) 1 ), 13) 12

13 replacing 2q 1)2q 1)/ 3T q) 4δ/3 as q/t δ. Here, we are exploiting that Ṽ R j q) 1 are approximately independent standardized random variables. For the asymptotic theory for q increasing we need the following condition, which is equivalent to Assumption 5a) in Pesaran 2006) that was used to investigate the properties of the common correlated Mean Group estimator in a panel with covariance stationary errors, see his Theorem 2. This condition requires that the residuals after factor projection have no degenerated distribution so that the sample variance cannot have mass at zero and ratio statistics behave well asymptotically. Assumption A. Ṽj1) 1 has finite second moments. Then we have the following result on the asymptotic null distribution of the averaged variance ratio after factor projection, which directly compares with the convergence 11) in Lemma 3 for q fixed, though the latter uses less stringent conditions on the relationship of N and T and a different asymptotic justification due to the need of checking that standardization works well for an increasing number of variance ratios. Lemma 2.4. Under the null hypothesis, Assumption A, q/t δ 0, 1) and NT 2 log N + N 3 T 0 as N, T ) j, Ũ N q) a N 0, 1). 14) In our panel framework the asymptotic distribution is standard Normal, instead of a functional of Brownian motion, since we are averaging asymptotically iid individual variance ratios. We now move to the power properties of the pooled variance ratio. Deo and Richardson 2003) considered alternatives given by a random walk plus independent stationary autoregressive noise. Adding the stationary component to our set up with 13

14 a fixed effect and a common factor would lead to the following process ξ j,t = α j + λ j f t + x j,t + ɛ j,t, 15) where x j,t = η j x j,t + u j,t, η j < 1, uj,t iid 0, τ 2 j ), τ 2 j > 0, though the dynamics of x j,t could be easily generalized to more complex models. Under this alternative, each individual Ṽ R jq) has now limit expectation as q/t δ given by 1 + ηj ) σ 2 j 1 + ηj ) σ 2 j + 2τ 2 j < 1 as shown by Deo and Richardson 2003), whose model does not include α j and λ j f t. Note that each individual variance is not consistent for this asymptotic expectation, but their average converges to the average of the expectations given the independence of cross sections. Then, it is immediate to show that under 15) ŨNq) diverges to at the rate N 1/2 given that as N, T ) j and q/t δ 0, 1), 1 N Ṽ Rj q) 1 ) 1 p lim N N 2τ 2 j 1 + ηj ) σ 2 j + 2τ 2 j, 0), implying that the consistency of the averaged variance ratio for long horizons is obtained from panels with increasing cross section size, in contrast to the case of a single time series. 2.4 Discussion We have developed panel variance ratio tests which apply several aggregation methods across cross section units to serial dependence tests when N is both fixed and growing. The same idea can also be used for other tests such as regression and general non-parametric tests. For example, our tests can be directly related to Pesaran 2007). Pesaran 2007) proposes to run the regressions y j,t = a j + b j y j,t 1 + c j ȳ t 1 + d j ȳ t + ɛ j,t 14

15 and compute the individual t ratios for the least squares coefficients ˆb j. In these individual DF regressions, the cross section averages ȳ t 1 and ȳ t of the usual independent and dependent variables correct for the factor in the same spirit as our projection technique. The asymptotic distribution of the DF t statistics is then approximated for increasing N for both fixed T and increasing T at the same rate as N. In fact, note that our individual variance ratio statistics can be written as Ṽ R j q) = N q i ) ρ i=1 q j i) where each sample correlation coefficient ρ j i) of ξ j,t could be approximately obtained from a stationary regression y j,t = a j + ρ j i) y j,t i + c j ȳ t i + d j ȳ t + ɛ j,t, noting that y j,t = ξ j,t and ȳ t i = ξ t i, ȳ t = ξ t, after scaling by the residual variance under ρ j i) = 0, which is implied by the random walk null hypothesis. 3 A Bootstrap Approximation In this section, we propose to use a bootstrap approximation to the finite sample distribution of our test statistics in order to improve finite sample properties of the asymptotic approximation for some statistics developed in Section 2. Although preserving the correlation among cross section units under the null of iid series can be achieved using different resampling algorithms, our proposal is especially simple. When N is finite, independent bootstrap resamples ˆξ t = {ˆξ j,t from resampling the original vectors } N are computed ˆξ t = {ˆξj,t } N, ˆξj,t = ξ j,t ˆα j, 16) for t = 1,..., T with replacement, where ˆα j = T 1 T t=1 ξ j,t. This resampling automatically preserves the cross section covariance structure and imposes the null 15

16 hypothesis. Test statistics are computed using the same expressions but based on ˆξ j,t. Then, the distribution of the bootstrap test statistics can be approximated from the empirical distribution of a large number of replications and estimated critical values are obtained. The consistency of the method follows because these estimated critical values remain valid under the alternative and because the bootstrap imposes the independence of ˆξ j,t, while the actual test statistic diverges as T increases. When N is assumed to be increasing with T, we base the resamples on the vectors ξ t = { ξj,t } N obtained after factor and mean removal, avoiding the reconstruction of the original DGP or the use of factor estimates, since the asymptotic distribution of the pooled variance ratio ŨNq) does not depend on those. 4 Monte Carlo Simulations 4.1 Econometric Framework for Panels with Cross Section Dependence In this section, we investigate finite sample properties of the test statistics developed in Section 2. To do this, we consider two classes of econometric models: nonfactor models with N fixed and factor models with N. For the class of nonfactor models, we consider the following ARMA process which generates errors ξ j,t in 2): ξ j,t = α j + ρ j ξ j,t 1 + ɛ j,t + θ j ɛ j,t 1. 17) Under the random walk hypothesis, we assume φ j = 0 for all j in equation 1), ρ j = θ j = 0 for all j in 17), and ɛ j,t is not correlated over time but can be correlated across j at a given time period t. The innovations ɛ 1,t, ɛ 2,t,, ɛ N,t ) are drawn from an N-dimensional multivariate normal distribution with mean zero and covariance matrix. We use the same cross section dependence assumption as in Chang 2004) which relies on random draws for the covariance matrix of innovations with a fixed ratio between the largest and smallest eigenvalues equal to 0.1. We refer to Chang 2004, pp ) for the detail. 16

17 Table 1: Parameter values for error term ξ j,t DGP Cases Parameters AR A1 ρ j U[0.2, 0.4] A2 ρ j U[ 0.4, 0.2] A3 ρ j U[ 0.4, 0.4] MA M1 θ j U[0.2, 0.4] M2 θ j U[ 0.4, 0.2] M3 θ j U[ 0.4, 0.4] For the class of factor models, in contrast, we assume that ɛ j,t is not correlated across j, so to model cross section dependence, we followpesaran 2007) and add an iid common factor, f t, to equation 17), ξ j,t = α j + ρ j ξ j,t 1 + λ j f t + ɛ j,t + θ j ɛ j,t 1. 18) We assume λ j U[ 1, 3]. Under this assumption the average fair correlation between ξ j,t and ξ i,t for j i is 0.5. We also assume that the variance of the innovation in the common component σ 2 f is the same as that of the innovation in the idiosyncratic component σ 2 j. For simplicity, we assume that σ 2 j is the same across j. We consider two classes of alternatives to measure the rejection probability of the tests against the random walk hypothesis: the models with nonstationary series and the models with stationary series. Specifically, we set φ j = 0 for all j for the models of nonstationary series and φ j < 0 for all j for the models of stationary series. In particular, we generate φ j randomly from a uniform distribution of U 0.05, 0) or U 0.15, 0.05) for stationary series. Setting a tighter range for φ j would help to evaluate more explicitly the performance of panel variance ratio tests regarding the detection of a very persistence stationary process. Errors ξ j,t are generated from either AR1) or M A1) processes for the alternatives against the random walk hypothesis. To compare the performance of the tests developed in our paper with that of the unit root tests developed by Pesaran 2007), we mainly follow his parameterization on the error processes as reported in Table 1. 17

18 4.2 Results In our simulations, the data generating mechanism for y t is formulated at the monthly frequency. The nonfactor models consist of equations 1) and 17) and the factor models consist of 1) and 18). For each model, we construct 4 panels by varying the length of time periods T = 60 and 180 and the size of cross sections N = 20 and 200 to show the effectiveness of our panel variance ratio statistics and to investigate robustness of our results regarding finite sample properties of the statistics. We report in the tables below the rejection rates at a 10% significant level against both left-tail and right-tail one-sided alternatives obtained from 5,000 simulations. For the simulations of bootstrap asymptotic size and power, we use the method of Giacomini, Politis and White 2013) to increase numerical efficiency. The range of aggregation values is set such that the maximum value of q is 120 months relative to a base period of one month for T =180 and 40 months relative to a base period of one month for T =60. This will allow us to investigate well known statistical difficulties with long horizon returns that stem from setting large q relative to the sample period T Size Table 2 reports the rejection probabilities of five statistics based on the asymptotic critical values for both nonfactor and factor models with φ j = ρ j = θ j =0 for all j: i) ŨNq) with the equal weight in 11); ii) z N q) with the equal weight in 8); iii) Maxq) in 5); iv) Minq) in 5); v) W q; R) with R = I N in 6). Note that the first statistic is developed assuming that N is growing in the asymptotics, while the other four statistics are developed assuming that N is fixed. We employ all these statistics for both the nonfactor and factor models since in practice the true class of models is unknown and asymptotic approximations could be appropriate for different ranges of N. Overall, we find that the size of ŨNq) is close to the nominal value over most aggregation values q for all specifications considered. For example, the rejection 18

19 rates of the pooled statistic ŨNq) associated with the aggregation values q = 2, 12, 36, 60, 84, 108, and 120 months are about 11, 11, 12, 12, 10, 8, and 8% at the left-tail and 10, 11, 12, 13, 11, 8, and 7% at the right-tail, respectively, for the factor model with N = 200 and T = 180. Strikingly, the rejection rates are close to the nominal value even for a very large q relative to T. This result confirms our analysis in Section 2.3 that ŨNq) follows the standard Normal distribution even if both q and T grow at the same rate, i.e., q/t δ > 0. That is, our statistic solve the statistical difficulty raised by Richardson and Stock 1989) that the usual asymptotic approximations for univariate variance ratio tests are no longer justified when both q and T grow at the same rate. ŨNq) also performs well in nonfactor models, although it slightly overrejects the null. For example, the rejection rates of ŨNq) associated with q = 2, 12, 36, 60, 84, 108, and 120 months are about 13, 13, 14, 15, 13, 10, and 10% at the left-tail and 13, 13, 15, 15, 13, 11, and 10% at the right-tail, respectively, for the nonfactor model with N = 200 and T = 180. This suggests that the statistic can be used in practice without concerning with the identification of the true class of data generating processes. These results hold true for the other specifications considered by varying values of N and T. On the other hand, the last four statistics developed under the assumption of N fixed substantially either overreject or underreject for many specifications. This suggests that the use of those statistics in practice requires other methods which overcome these size distortions. A natural alternative is to employ a bootstrap method. Table 3 reports the rejection probabilities of the five statistics based on the critical values obtained from the bootstrap method specified in Section 3. Overall, we find that Maxq) and Minq) tests perform well in that the simulated sizes of those tests are close to the nominal value for all specifications. For example, the rejection rates of Maxq) associated with the aggregation values q = 2, 12, 36, 60, 84, 108, and 120 months are about 9, 11, 10, 10, 10, 10, and 10% at the right-tail and those of Minq) are about 10, 9, 9, 11, 9, 10, and 11% at the left-tail, respectively, for the nonfactor model with N = 200 and T = 180. On the other hand, the performance of z N q) and W q; R) is sensitive to a 19

20 particular class of models: The rejection rates of both statistics are close to the nominal value for the factor models but are smaller than the nominal value for the nonfactor models, in particular for larger values of N. ŨNq) also underrejects the null when N is large for both nonfactor and factor models. From the results regarding the size properties of the five statistics, we focus on investigating the power performance of ŨNq) based on the asymptotic critical values and of Maxq) and Minq) based on the critical values from the bootstrap method in 16) Power We consider various specifications by changing parameter values in the error term as well as the value of φ j to investigate the power properties of the three statistics: ŨNq), Maxq), and Minq). Note that ŨNq) measures the average of the deviations of variance ratios from the null value of one in the panel, while Maxq) and Minq) are the maximum and minimum of the distribution of the deviations. In this sense, these three statistics provide information about the distribution of individual variance ratios, which can be used to better understand the persistence properties of the panel. The summary of our findings follows: All three test statistics become more powerful for a larger N. ŨNq) is more powerful than Maxq) and Minq) for each specification. The power of ŨNq) does not decrease much even for a large q relative to T. The first result confirms one of our objectives of constructing panel variance ratio statistics: They produce higher power than the univariate variance ratio statistics. The second result is interesting in that the mean measure ŨNq) performs more powerfully than the extreme statistics for all the alternatives considered against the random walk hypothesis. The main reason is that ŨNq) is constructed using the residuals from the projection method which eliminates the iid common factor in the original series, while the other two statistics are constructed using the original 20

21 series which contains more noise. The third result illustrates the power advantage of applying ŨNq) for the investigation of the behavior of long horizon returns which can be more informative than their short horizon counterparts for some alternatives to the random walk. Specifically, Table 4 reports the rejection probabilities of the three statistics for both nonfactor and factor models with a stationary series φ j U[ 0.05, 0] for each j and ρ j = θ j = 0 for all j). The rejection probabilities of ŨNq) associated with q = 2, 60, and 120 months are 50, 100, and 100% at the left-tail for the nonfactor model with N = 200 and T = 180, while those are 15, 65, and 34% for the same model with N = 20 and T = 180, confirming the first result. On the other hand, those of Maxq) are 12, 37, and 28% and those of Minq) are 7, 12, and 8% for the same model with N = 200 and T = 180, confirming the second result. Further, the power of ŨNq) increases with q, reaches at maximum at a certain value of q q = 24 months in this case), and does not decreases even beyond it, confirming the third results. This is noticeable in that it demonstrates the power advantage of incorporating information across cross section and confirms our analysis in Section 2.3. Finally, all three statistics reject at the left-tail, consistent with the prediction of the specification with an AR1) stationary process. In addition, the rejection probabilities increase up to a certain aggregation value q and then gradually decrease beyond it or stays), consistent with the power pattern of the typical variance ratio tests for capturing the serial dependence behavior of the stationary series. We find that the above results regarding the power properties of the three statistics hold true for both nonfactor and factor models. From now on, to conserve space, we only report the results from the factor model and relegate the results from the nonfactor model to the Technical Appendix. To further conserve space, we only report the results from the specifications with N = 200 and T = 180 and relegate those from other panels to the Technical Appendix. Table 5 reports the rejection probabilities of the statistics for the factor model with another stationary series generated from the class of ARM A1, 1) models. There is one difference between the results from the AR1) model in Table 4 and those from the ARMA1, 1) models in Table 5: the power pattern over q is different 21

22 for some parameter values in the error term. For example, the rejections occur at the right-tail initially and then eventually at the left-tail beyond a certain q for several specifications see, for example, Panel A and D). In particular, when rejections occur at the left-tail, its power increases with q up to a certain q and then gradually decreases or stays at maximum). Nevertheless, this is also consistent with the power pattern of the typical variance ratio tests for a univariate stationary series. Table 6 reports the rejection probabilities of the statistics for the factor model with nonstationary series that have common parameter values of φ j = 0 for all j. For all the specifications considered, the rejection probabilities of both M axq) and Minq) are largest at q = 2 and then gradually decrease with q, consistent with a univariate nonstationary series See for example Lo and MacKinlay 1988)). For the case of ŨNq), the rejection probabilities are 100% for all q for several cases Panel A, B, D, and E) or largest at q = 2 and then gradually decrease with q Panel F). In contrast to the stationary series, the direction of the rejections occur at either the left- or right-tail, depending on the parameter values in the error term. Overall, we find that the direction of the rejections and the power pattern of the three statistics are different between stationary and nonstationary series in the following ways: For stationary series, rejections mainly occur at the left-tail in particular over long horizons, while they occur at either the left- or right-tail for nonstationary series. For stationary series, the power of Maxq) and Minq) gradually increases with q and then decreases beyond a certain q, while their power is largest at q = 2 and then gradually decreases with q for nonstationary series. These results are directly related to the mean reverting behavior of the stationary series. That is, if a series is stationary, its serial dependence over long horizon should be negative. Conversely, if we observe positive serial dependence of a series over long horizon, the series is likely to be nonstationary. Further, if a stationary series is highly persistent, then its first difference behave less random walk 22

23 We now compare the power performance of ŨNq) to that of one of the test statistics CIP SN, T ) developed by Pesaran 2007, p. 276). To be completed.) 5 Serial Dependence of Sectoral Real Exchange Rates In this section, we apply the panel variance ratio tests to a panel of 187 European Union sectoral real exchange rates 19 sectoral real exchange rates for each of 10 European Union countries). 4 We use the same Eurostat price data as Imbs et. al. 2005) used in their study to directly compare our results with theirs. 5 The data frequency is monthly and the sample period is between 1981:01 and 1994:09 [see Imbs et. al. 2005) for the detailed description of the data]. We consider three statistics, ŨNq), Minz N q), and Maxz N q) and investigate the serial dependence of the sectoral real exchange rates in EU countries. The three statistics provide the summary statistics of the joint distribution of the deviations of individual variance ratios from the null value of one. ŨNq) measures the average of the deviations of variance ratios from the null value of one in the panel, while both Minz N q), and Maxz N q) are the largest deviations of variance ratios from the null value. Therefore, by looking into the direction of the rejections of and the power pattern of these statistics, we can obtain information about stationarity and nonstationarity of the series in the panel. That is, if the real exchange rates are stationary, then they must exhibit negative serial dependence over long horizons. Conversely, if the real exchange rates exhibit positive serial dependence over long horizons, then they are likely to be nonstationary. Figure?? draws the three test statistics over aggregation values q = 2 to 120 months. In the first box, two dotted lines are asymptotic critical values at 5 and 95% quantiles, respectively, while in the second and third boxes, the two dotted lines are critical values at 5 and 95% quantiles of the empirical bootstrap distribution 4 We drop three price series coded it1140, pt1310, and pt1710 because of some missing observations. 5 The data is taken from the webpage of Morten O. Ravn uctpmo0/). 23

24 constructed using 5,000 bootstrap samples. We find that real exchange rates exhibit significant positive serial dependence over the entire 10 year horizon, suggesting that they are not likely to follow stationary processes. Specifically, ŨNq) and Maxq) are all positive and much greater than the critical values at the 95% quantiles over all aggregation values q. On the other hand, Minq) are negative at short horizons and are not statistically significant beyond some aggregation value q. The combination of the two results suggests that the EU sectoral real exchange rates are not likely to be mean reverting in long horizons. This result contrasts to Imbs et. al. 2005) but is in line with Gadea and Mayoral 2009). We now do the same exercise using 10 EU aggregate real exchange rates constructed using each country s consumer price index CPI) data and investigate if there is difference in the persistence behavior between sectoral and aggregate real exchange rates. As shown in Figure 2, we reach the same conclusion as the sectoral real exchange rates are used: the aggregate real exchange rates are not likely mean reverting. In fact, the evidence is stronger than the case of sectoral real exchange rates. Again, ŨNq) and Maxq) are all positive and much greater than the critical values at the 95% quantiles over all aggregation values q. In contrast to the case of using sectoral real rates, M inq) are also positive over all aggregate values q. One key argument raised by Imbs et. al. 2005) is that previous panel methods failed to control for a dynamic aggregation bias. However our methods are not affected by this problems since we do not estimate any parameter or an "average" effect, but rather test a given hypothesis so that our tests control size asymptotically under the null and are consistent against a relevant class of alternatrives. 6 Conclusion We develop econometric methods based on variance ratio statistics to investigate persistence properties of panels with cross section dependence. We present asymptotic results for small and large panels with time series dimension T growing in both situations. If the cross section dimension N is allowed to diverge, a factor model is used to describe cross section dependence and a simple common correlation es- 24

25 timate is proposed to remove factor influence. If N is fixed, we justify a bootstrap procedure to improve finite sample properties of our tests. Exploiting a panel with N, we show that the pooled statistic can recover some asymptotic power even for long horizons when the aggregation horizon q is growing at the same rate as T, in constrast to the criticisms raised by Richardson and Stock 1989) and Deo and Richardson 2003). The finite sample properties of the statistics developed in this paper are investigated by Monte Carlo simulations. The followings are the summary of our findings. First, all three statistics improve their power performance for larger N. Second, the power of ŨNq) does not decrease much even for long horizons. Third, ŨNq) developed based on the assumption of N growing performs best in all specifications considered. Fourth, the use of the three statistics, ŨNq), Minz N q), and Maxz N q) gives us information about the distribution of variance ratios across cross section. Appendix A Proofs We first state and prove an instrumental lemma that is used throughout the Appendix. First introduce this notation: T i A j i) = T 1 t=1 T i B j i) = T 1 t=1 T i Ci) = T 1 t=1 T i Di) = T 1 t=1 T i Ei) = T 1 t=1 ɛ j,t ɛ j ) ɛ t+i ɛ), 19) ɛ j,t ɛ j ) f t+i f ), 20) ɛ t ɛ) f t+i f ), 21) ft+i f ) f t f ), 22) ɛ t ɛ) ɛ t+i ɛ). 23) 25

26 Lemma A. With the definitions in 19)-23) and i, i = 1, 2,..., T 1, E A j i) 2 = O NT ) 1) E [A j i)a j i )] = O N 2 T ) 1 1 {i = i } + NT 2 ) 1 1 {j = j } + NT ) 2) E [B j i)b j i )] = O T 1 1 {i = i } + T 2) E [Ci)Ci )] = 0 + O NT ) 1) 1 {i = i } E [ Di) ] = O T 1/2) E [Ei)Ei )] = O NT ) 2 + N 2 T ) 1 1 {i = i } ) Proof of Lemma A. The proof follows at once exploiting the independence properties of ɛ it and f t, mutually and across i and t. Proof of Lemma 2.1. Define by ρ j,l = Cov ) ξ j,t, ξ l,t / σj σ l ) = E [ ξj,t ξl,t ] / σj σ l ), ξj,t = ξ j,t α j, the contemporaneous correlation between series specific innovations. Then, under the iid assumption of ξ j,t, only these correlations show up in the asymptotic covariance ACov ˆρ j i), ˆρ l i ) ) = = 1 Cov ξj,t ξj,t+i, T σ 2 jσ ξ ) l,t ξl,t+i 2 l 1 E ξj,t ξl,t ) 2 = ρ 2 T σ 2 jσ 2 j,l l if i = i ; but is 0 otherwise. Then ACov V R j q), V R l q) ) = 4ρ 2 j,l q 1 i ) 2 = ρ2 j,l2q 1)2q 1), i=1 q 3q so Σ N has ones in the diagonal, the off diagonal terms being ρ 2 j,l. Proof of Lemma 2.2. We can write V R pool q 1 q) 1 = w j 2 1 i ) ˆρ i=1 q j i) 26

27 so that using the analysis in Lemma 2.1 we obtain )) AV ar T 1/2 V R pool q) q 1 = 4 w j w l ρ 2 j,l 1 i ) 2 l=1 i=1 q = R wr 2 q 1) 2q 1) w + w j w l ρ 2 2 q 1) 2q 1) j,l 3q l j 3q and the result follows at once. Proof of Lemma 2.3. Proof of asymptotic distribution for individual Ṽjq). Using the usual procedures, Ṽ j q) = 2 q 1 i=1 1 i/q) ρ j i) depends on the sample covariances of ξ j,t 1) from lags 1 up to q 1, up to negligible end effects in the summations, e.g. ρ j i) = 1 T T i ξj,t ξj,t+i, t=1 where ξj,t = ɛ j,t ɛ j η j λ ft f ) δ j ɛ t ɛ) 24) with η j = δ j λ j λ = T 1 T t=1 ξt ξ ) { ɛ j,t λ j ɛ } λ t T 1 T t=1 ξt ξ ) 2. Then, using the properties of ξ j,t under the null T 1 T t=1 while the numerator of η j satisfies T 1 T t=1 ξt ξ ) 2 p E [λ i ] 2 σ 2 f > 0 ξt ξ ) { ɛ j,t λ } j λ ɛ t = O p N 1 + T 1/2), so that η j = O p N 1 + T 1/2) for each j. Then we can show using a standard CLT for the autocovariances of iid data 27

28 that T 1/2 ρ j i) = 1 T i ɛ T 1/2 j,t ɛ j ) ɛ j,t+i ɛ j ) + o p 1) a N ) 0, σ 2 j = Op 1), t=1 since under the null from Lemma A ηj λ) 2 T 1/2 D i) = O p N 2 + T 1) O p 1) = o p 1) δ2 jt 1/2 E i) = O p 1) O p N 1 ) = o p 1) η j λ δj T 1/2 C i) = O p N 1 + T 1/2) O p N 1/2 ) = o p 1) η j λt 1/2 B j i) = O p N 1 + T 1/2) O p 1) = o p 1) δj T 1/2 A j i) = O p 1) O p N 1/2 ) = o p 1) as N and T both tend to infinity jointly and f t is serially uncorrelated. Proof of asymptotic distribution of ŨNq) for fixed q. We can write Ũ N q) = 1 1 N 2T q 1)2q 1)/3q Ṽ j q) Ṽj1) Ṽ j 1) where Ṽj1) p σ 2 j for each j using the following arguments and using the notation introduced in the Appendix, q 1 Ṽ j q) Ṽj1) = 2 1 i ) q 1 ρ q j i) = 2 1 i ) ρ ɛ j i) 25) q T 1/2 N 1/2 i=1 q i=1 q 1 i=1 1 i q 1 i q We have to show that for each i ρ j i) = 1 T 1/2 N 1/2 T i t=1 i=1 ) [ δj {A j i) + A j i)} + η j λ {Bj i) + B j i)} ] ) [ηj δj λ {Cj i) + C j i)} + η 2 j λ 2 Di) + δ 2 jei) ]. ɛ j,t ɛ j ) ɛ j,t+i ɛ j )+o p 1) a N 0, 1) = O p 1), and jointly and independently for i = 1,..., q 1, where the convergence in distribution follows applying joint asymptotics in N and T under the null exploiting the 28

29 independence of ɛ j,t across j. To show the first equality we can use that 1 N η 2 j = T 1 T t=1 ξt ξ ) 2) 2 1 NT 2 = O p 1) O p N 2 + T 1) T ξt ξ ) { ɛ j,t λ }) 2 j λ ɛ t t=1 by calculating the expectation of the squared term, so that also 1 N δ2 j = O p 1), and then using 24) and the following bounds derived from Lemma A, T 1/2 N 1/2 T 1/2 N 1/2 2 ηj λ) D j) = Op T 1/2 N 1/2 N 2 + T 1)) O ) p T 1/2 = o p 1) T 1/2 N 1/2 δ2 je i) = O p T 1/2 N 1/2) O p T 1/2 N 1) = o p 1) η j λ δj C j i) = O p T 1/2 N 1/2 N 1 + T 1/2)) O p T N) 1/2 ) = o p 1) and, using NT 1 0, T 1/2 N 1/2 η j λbj i) = O p N 1/2 1 N 1/2 η 2 T j N 1/2 Bj 2 i) = O p N 1/2 N 1 + T 1/2)) O p 1) = o p 1) 29