Panel Cointegration Testing in the Presence of Linear Time Trends

We consider a class of panel tests covering tests for the null hypothesis of no cointegration as well as cointegration. All tests under investigation rely on single-equations estimated by ordinary least squares, and they may be residual-based or not. We focus on test statistics computed from regressions with intercept only (i.e. without detrending) while at least one of the regressors (integrated of order 1) is dominated by a linear time trend. In such a setting, often encountered in practice, the limiting distributions and critical values provided for and applied with the situation "with intercept only" are not correct. It is demonstrated that their usage results in size distortions growing with the panel size N. Moreover, we show which are the appropriate distributions, and how correct critical values can be obtained from the literature.


Introduction
Most panel tests for the null hypothesis of (no) cointegration rely on single-equations, notable exceptions being Larsson, Lyhagen, and Löthgren (2001), Groen and Kleibergen (2003), Breitung (2005) and Karaman Örsal and Droge (2013) who proposed panel system approaches. Recent single-equation tests by Chang and Nguyen (2012) or Demetrescu, * We thank Christoph Hanck and Joakim Westerlund for helpful comments. Hanck, and Tarcolea (2014) rely on nonlinear instrumental variable estimation, while the vast majority of such panel tests builds on (fully modied or dynamic) ordinary least squares [OLS]. Here, we study exactly this class of OLS based single-equation panel tests for the null of either cointegration or no cointegration.
We focus on the situation where the test statistics are computed from regressions with an intercept only (i.e. without detrending) while at least one of the integrated regressors displays a linear time trend on top of the stochastic trend. Such a constellation is often met in practical applications, see for instance Coe and Helpman (1995) and Westerlund (2005b) on R&D spillovers (total factor productivity and capital stock), Larsson, Lyhagen, and Löthgren (2001) on log. real consumption and income (per capita), or Hanck (2009) on prices and exchange rate series testing the weak purchasing power parity (PPP). The relevance of a linear trend in panel data has been addressed in Hansen and King (1998) when commenting on the link between health care expenditure and GDP, see McCoskey and Selden (1998); consequently, Blomqvist and Carter (1997), Gerdtham and Löthgren (2000) or Westerlund (2007) worked (partly) with detrended series, i.e. they included time as an explanatory variable in their panel regressions. Hansen (1992, p. 103), however, argues that It seems reasonable that excess detrending will reduce the test's power. At the same time detrended relationships may not be economically meaningful. Think of the analysis of income (x) and consumption (y), where one is interested in checking for cointegration and β = 1 within y t =ᾱ +βx t +ū t , which amounts to stationary saving series. Here, a detrended regression, y t = α+ δ t+ β x t + u t , does not make economic sense.
The same holds true for the empirical application with PPP in this paper, see also the discussion in Hassler (1999). There are only few situations were economic interpretation can be attached to a linear time trend as explanatory variable, e.g. as technological progress in a production function. Therefore, we study the empirically relevant case where test statistics are computed from regressions with intercept only (i.e. without detrending) while at least one of the I(1) regressors displays a linear time trend, i.e. it is I(1) with drift.
The test statistic may be constructed from pooling the data or from averaging individual statistics, see e.g. Pedroni (1999Pedroni ( , 2004 or Westerlund (2007). Much of the nonstationary panel literature relies on sequential limit theory where T → ∞ is followed by N → ∞, such that limiting normality can be established under the assumption that none of the I(1) regressors follows a deterministic time trend: √ N Z (m) −μ m /σ m ∼ N (0, 1) .
The constantsμ m andσ m required for appropriate normalization are typically tabulated for a selected number of values of m. A dierent set of such momentsμ m andσ m is typically given for detrended regressions, too, where the test statistic Z (m) stems from regressions of the type (m = k + 1) Here, it holds irrespective of an eventual linear trend in the data that  Hamilton (1994, p. 596, 597). It is more relevant in our panel framework since we illustrate numerically and analytically that the size distortions of an inappropriate normalization grow with the panel size N (to zero or one, depending on the specic test).
The rest of the paper is organized as follows. The next section xes some notation and assumptions. Section 3 establishes our asymptotic result and discusses its consequences for applied work. Section 4 considers the combination of p-values as an alternative approach to panel testing. Section 5 illustrates our asymptotic ndings by means of Monte Carlo evidence. Section 6 contains the empirical application, and the last section summarizes.
Mathematical proofs are relegated to the Appendix.

Notation and assumptions
Restricting our attention to the single-equation framework we partition the m-vector z i,t of observables into a scalar y i,t and a k-element vector x i,t , z i,t = (y i,t , x i,t ), m = k + 1. As usual, the index i stands for the cross-section, i = 1, . . . , N , while t denotes time, t = 1, . . . , T . Each sequence {z i,t }, t = 1, . . . , T , is assumed to be integrated of order 1, I(1), where we allow for a non-zero drift, and assume for simplicity a negligible starting value, z i,0 = 0. While {z i,t } may be cointegrated or not, depending on the respective null hypothesis, we rule out cointegration among {x i,t }. Technically, these assumptions translate as follows, where W i,m (·) denotes an m-dimensional standard Wiener process, x stands for the integer part of a number x, and ⇒ is the symbol for weak convergence.
The stochastic zero mean process {e i,t } is integrated of order 0 in that it satises where ω 2 i,yy > 0 and Ω i,xx is positive denite.
Similarly, panel statistics rely on pooling the data across the within dimension, i.e. summing over terms showing up in the numerator and denominator separately, Here it is assumed thatN i,T and D (m) i,T are computed from individually demeaned or detrended regressions, respectively. A typical example for the function g is g(x, y) = x/ √ y in the case of t-type statistics. We allow for group and panel statistics by introducing the generic notationZ (m) and Z (m) , and maintain for the panel the joint null where a distinction between the individual null hypotheses H i,0 of cointegration or absence of cointegration is not required.
Assumption 2 Consider linear single-equation OLS regressions (i = 1, . . . , N , t = 1, . . . , T ) y i,t =ᾱ i +β i x i,t +ū i,t and y i, or ∆y i,t =ᾱ i +γ i y i,t−1 +β i x i,t−1 +ε i,t and ∆y i, where lags of ∆z i,t−j may be required as additional regressors in (12) as T → ∞ followed by N → ∞.

Results
The rst paper allowing for linear time trends in a panel cointegration context was by Kao (1999). He considered a residual-based unit root test for the null hypothesis of no cointegration in the tradition of Phillips and Ouliaris (1990), see Remark 1 below. His test builds on pooling the data while allowing for a individual-specic intercept, which amounts to a least-squares dummy variable estimation. Kao (1999) did not consider regressions containing a linear time trend, but he does allow for a linear drift in the data when performing a regression with a xed eect intercept. In the case of k = 1 regressor (i.e. m = 2), Kao (1999, eq. (15))) observed that the linear time trend dominates the I(1) component; hence, the limiting distribution amounts to that by Levin, Lin, and Chu (2002) upon detrending. To become precise: Let µ 1 and σ 1 denote the normalizing constants provided by Levin et al. (2002) for detrended panel unit root tests; then one should use them for pooled residual-based panel cointegration testing in a bivariate regression if the regressor is I(1) with drift, see Kao (1999, Theo. 4): The claim that (4) continues to hold in the case of m > 2 (see Kao (1999, Remark 12)), however, is not correct. Instead, we can prove Theorem 1 for panel or group statistics computed from regressions with intercept only in the presence of linear time trends.
Theorem 1 Let Assumption 1 hold true for m ≥ 2 with µ i,x = 0. Under Assumption 2 it holds under the null hypothesis that The situation analyzed in Theorem 1 has not been considered in the previous panel cointegration literature, with the notable exception of Kao (1999). Consequently, all applied papers we are aware of standardizeZ (m) withμ m andσ m ignoring the eect of eventual trends in the series, which amounts to strategy S I . The eect of both strategies, S I and S A , is discussed for growing N in the following proposition. In the absence of a linear trend, S I performs without size distortion, while S A provides correct inference under linear trends. Here we study the asymptotic consequences of inappropriate use of these strategies.
Proposition 1 Let Assumptions 1 and 2 hold true for m ≥ 2. Further assume Under the null hypothesis one has the following.
a) For a test rejecting for too negative values, the probability to reject ...
for a test rejecting for too large values, the probability to reject ...
We now discuss a couple of panel tests satisfying Assumption 2 and (5), such that Theorem 1 and Proposition 1 apply.
Remark 1 The residual-based unit root tests for the null hypothesis of no cointegration proposed by Pedroni (1999Pedroni ( , 2004) build on static regressions. In particular, the suggested group statistics are constructed from individual regression residuals: The null hypothesis (1) Breitung (2002). The null hypothesis of no cointegration is rejected again for too small values. To apply Theorem 1 with m = 2, we need µ 1 and σ 1 . For the detrended Breitung distribution we obtain by simulation µ 1 = 0.0110 and σ 1 = 0.005197, which are value corresponding to the case of group statistics. Here, 0 < µ m−1 <μ m , so that (5) holds, see Westerlund (2005a, Table   1). Consequently, Proposition 1 a) applies, and the probability to reject the true null hypothesis under strategy S I grows with N as long as there is a linear trend in the data. The other way round, strategy S A results in increasingly conservative tests in the absence of linear trend, which will of course be accompanied by a loss in power.
Remark 2 The error-correction test by Westerlund (2007) where lags of ∆z i,t−j may be required as additional regressors to ensure errors free of serial correlation. The null hypothesis of no cointegration is rejected for too negative t-values associated with γ. In case of m = 1 (i.e. no x i,t on the right-hand side), the limiting distributions are of the usual Dickey-Fuller type. Hence, µ 1 and σ 1 for group statistics are again from detrended Dickey-Fuller-type distributions and given in Nabeya (1999, Pedroni (1999Pedroni ( , 2004  Remark 3 Westerlund (2005b) (22) given in the Appendix. The rejection probabilities apply approximately under the null hypothesis at nominal signicance level α. We report results for the group t-tests by Pedroni (1999Pedroni ( , 2004 and by Westerlund (2007) in Tables 1 through   4, and for the group CUSUM test by Westerlund (2005a) in Table 5.
Generally, the size distortions in Tables 1 through 4 (20) and (22) from the Appendix. When evaluating S I under µ i,x = 0, we observe rejection probabilities equal to zero up to three digits. For that reason we only report a Table 5 for strategy S A in the absence of linear time trends, where this strategy is very liberal under the null.
1 The univariate distribution is the supremum over the absolute value of a so-called second-level Brownian bridge, which shows up with the detrended KPSS test, too; see Kwiatkowski, Phillips, Schmidt, and Shin (1992).  Pedroni (1999Pedroni ( , 2004 (2007) (2007) (2005b) and established the limiting results from Assumption 2 when allowing for cross-correlation driven by a common factor. Consequently, Theorem 1 continues to hold.
Strategy S I is clearly predominant in the literature and applied with the tests mentioned in the remarks above. The performance of strategies S I or S A under the null hypothesis depends of course on the presence or absence of a linear time trend, and on the specic test. In view of Proposition 1, we nd neither strategy S I nor strategy S A acceptable for group or panel statistics.

Combination of p-values
The idea to combine p-values from individual units to obtain panel signicance on unit roots was put forward by Maddala and Wu (1999) and Choi (2001), see also Hanck (2009) for cointegration testing. With p-values p i , i = 1, . . . , N , Maddala and Wu (1999) proposed Fisher's classical test statistic, which follows a χ 2 (2N ) distribution under the null hypothesis and for independent units.
Under the same assumptions the so-called inverse normal method discussed by Choi (2001) relies on comparing with a standard normal distribution, where Φ −1 denotes the inverse of the standard normal distribution function. Clearly, this is a left-sided test rejecting the null hypothesis for too small values of IN M . Hartung (1999) suggested to robustify the inverse normal method against certain forms of cross-correlation, see also the discussion in Demetrescu, Hassler, and Tarcolea (2006).
We now maintain the following assumption, where for notational simplicity the index i is suppressed.
There are plenty of tests satisfying (8) and (9). We briey mention three tests for which (10) has been established as well, so that they meet Assumption 3. Note that (10) is the time series analogue to our panel result from Theorem 1.

Phillips-Ouliaris-Hansen test Building on the static regressions
Phillips and Ouliaris (1990) suggested residual-based unit root test statistics for the null hypothesis of no cointegration. Under some additional assumptions the corresponding statistics satisfy Assumption (8) and (9). Hansen (1992, Theo. 7) proved (10) for the so-called Z statistics introduced by Phillips (1987) and Phillips and Perron (1988). For m = 2, the limit L (1) is simply the detrended Dickey-Fuller distribution. Critical values for these tests are most often taken from MacKinnon (1991) or MacKinnon (1996), who also provided p-values.
KPSS test Under the null hypothesis of cointegration residual-based tests relying on the static regression (11) have been proposed. They rely on the KPSS statistic, see Kwiatkowski, Phillips, Schmidt, and Shin (1992). To circumvent the problem of endogeneity, Shin (1994) considered so-called ecient variants of least squares estimators, see also Harris and Inder (1994), and established (8) and (9). Independence of u t and ∆x s is sucient to render OLS ecient, too; hence, in such an environment (8) and (9) holds for OLS residuals, see also Leybourne and McCabe (1994). Hassler (2001) proved (10).
Again, L (1) is simply the detrended univariate distribution characterized in Kwiatkowski et al. (1992). To the best of our knowledge, however, p-values are not available for the KPSS cointegration test.
Error-correction test A test for the absence of an error correction mechanism has been proposed by Banerjee, Dolado, and Mestre (1998) suggesting the t-statistic for γ = 0 in ∆y t+1 =ᾱ+γ y t +β x t +di.s+ε t+1 or ∆y t+1 = α+ δ t+ γ y t + β x t +di.s+ ε t+1 . (12) Under appropriate exogeneity restrictions (8) and (9)  Strategy S I is predominant in the literature. An exception is the residual-based Phillips-Ouliaris test, because strategy S A has been advocated by Hansen (1992) for this test. In fact, strategy S A has been adopted for the Phillips-Ouliaris test in inuential textbooks such as Stock and Watson (2003, Table 14.2). The reason for this is: S A has correct size if µ x = 0, while it is only mildly conservative under the null hypothesis for µ x = 0. The latter statement is true because the tails of the distributions L (m−1) andL (m) happen to be very close for residual-based Phillips-Ouliaris tests, with critical values from L (m−1) being a bit more negative than those fromL (m) , see (13) below; the dierence decreases with growing m.
We now consider the strategies S I and S A with and without linear trends, respectively, which parallels Proposition 1. LetF (m) and F (m−1) denote the distribution functions of L (m) and L (m−1) . We assume the following rst-order stochastic dominance: Proposition 2 Let Assumptions 1 and 3 hold true for m ≥ 2. Further assume (13).
Under the null hypothesis (1) one has the following.
Using the codes accompanying MacKinnon (1996), we have veried (13) numerically for a ne grid covering the range of the two distributions with m = 2, . . . , 9 for the Phillips-Ouliaris t-type test. Similarly, we did with the codes accompanying Ericsson and MacKinnon (2002). Hence, Proposition 2 a) applies for the Phillips-Ouliaris residual test and the error correction test as well. We learn that strategy S A recommended in a time series context by Hansen (1992) or Stock and Watson (2003)  α from the distributionsL (m) and L (m−1) . From Shin (1994) and Kwiatkowski, Phillips, Schmidt, and Shin (1992) we observe c (m−1) α <c (m) α for nominal level of 1%, 5% and 10%, which supports (13) However, we must admit not to have the numerical means to verify (13) on a ner grid.
Still, we conjecture that Proposition 2 b) holds for p-values from the KPSS cointegration test.

5
Monte Carlo evidence

Evidence on Proposition 2
The rst set of experiments assesses the eect of the absence or presence of a linear time trend on the combination of independent p-values from Phillips-Ouliaris t-type tests for the null hypothesis H 0 of no cointegration. This illustrates Proposition 2 a). We now add the subscript P O to test statistics and related entities. Using the numerical distribution functions by MacKinnon (1996) we generate N pseudo random numbers from the limiting distribution L (1) P O resulting from a bivariate regression with intercept under drifts, see (10).

WithF
(2) P O being the distribution function ofL conservative dropping below 1% for N = 20 already. In practice, this will come with a power loss, of course. The distortions do not occur when working with the appropriate limit in the absence of drifts: WithF (2) P O being the distribution function ofL (2) P O used to determine the p-values, the 5% level is achieved for all N , see Figure 2.
The experiments show that the very mild distortions of the liberal or conservative strategies with the Phillips-Ouliaris tests are severely aggravated when combining signicance of a panel with growing N .

Evidence on Proposition 1
The next set of experiments refers to the error-correction test by Westerlund (2007). As data-generating process (DGP) we consider y i,t = x i,1,t + x i,2,t + · · · + x i,k,t + r i,0,t , t = 1, 2, ..., T = 250 , i = 1, 2, ..., N , (14) x i,j,t = 1 + x i,j,t−1 + v i,j,t , j = 1, 2, ..., k , where {v i,j,t } are standard normal iid sequences independent of each other. Using the regression we computed the group t-statistic proposed by Westerlund (2007), and repeated this experiment 10.000 times. Tables 6 and 7 report the frequencies of rejection. In particular, Table 6 is for the same constellation as Table 3 containing approximate gures; the correspondence between the experimental sizes and the approximate sizes is rather close notwithstanding the fairly small panel dimension N . Table 7 illustrates how well the rule from (10) works: The experimental sizes are close to the nominal ones under strategy S A in the presence of linear trends. Of course, applying strategy S A in the case of no linear trends would yield to empirical size distortions analogous to those reported in Table 4.

Evidence on detrending
Since neither strategy S I nor strategy S A can be recommended when we are unsure about the presence or absence of a linear time trend in the data, one might always run detrended regressions. While this may not always be economically meaningful, it is a statistically  (2007) (2007) (2007)  First, we generate again data under (14), and observe that the experimental size from detrended regressions is close to the nominal one under the null hypothesis of no cointegration, see Table 8. Second, the DGP under the alternative becomes: where x i,j,t and v i,0,t are generated as before. We now report rejection frequencies under detrending and when applying strategy S A , where the true DGP still contains linear time trends, see (15). The results from Table 9 are very clear: As a rule of thumb we observe that S A is twice as powerful as detrending in that the rejection frequencies are roughly twice as large (for N ≤ 30).

Empirical application
We now turn to an analysis of the weak purchasing power parity (PPP) hypothesis. Let p i,t and p U S,t denote the logarithms of domestic and US price levels (consumer price indices), respectively, and s i,t is the corresponding log exchange rate. According to weak PPP the relationship p i,t − β 1,i p U S,t − β 2,i s i,t should be stationary, where the individual series are assumed to be I(1). With annual OECD data from 1973 until 2013 we run the following Table 9: Experimental power of the group t-test at 5% by Westerlund (2007) for the N = 11 currencies reported in Table 10; Germany stands for the Euro. We deleted variables with coecients not signicantly dierent from zero (typically ∆s i,t , ∆p U S,t−1 and ∆s i,t−1 ). Note that there are m = 3 I(1) variables where the mean growth of p U S,t−1 may be approximated by a linear time trend. Moreover, the 11 equations are clearly dependent. Sectorally demeaning to account for cross-sectional dependence as proposed by Westerlund (2007) is not feasible in our framework, since the identical US series enters each equation. Therefore, we combine p-values from individual regressions obtained from programmes accompanying Ericsson and MacKinnon (2002). These nite sample values computed according to strategies S I and S A are denoted byp(3) andp(2), respectively.
The results are given in Table 10.
Clearly, there is mixed evidence with respect to cointegration (validity of weak PPP We now combine the p-values from Table 10 by means of the inverse normal method  Hartung (1999) against correlation ρ which is assumed to be constant: where consistent estimationρ is discussed in Hartung (1999). This estimate is 0.6863 and 0.6299 under strategies S I and S A . The resulting test statistic IN M (ρ) is listed in Table  10, too. The combined signicance is given by the corresponding p-values computed from a one-sided standard normal distribution: 0.0623 and 0.0996 for S I and S A , respectively. Hence, the conventional procedure S I results in a too optimistic view, rejecting the panel null hypothesis of no cointegration at a 6% level, while the procedure S A accounting for a linear time trend in the price series yields signicance just at the 10% level.

Concluding remarks
In time series econometrics it has been known for a long time that the deterministic trends in the data aect the limiting distributions of the test statistics whether or not we detrend the data. (Hansen, 1992, p. 103). This has been shown for the residualbased Phillips-Ouliaris (or Engle-Granger) cointegration test by Hansen (1992), see also the exposition in Hamilton (1994, p. 596, 597). Analogous results have been given for other cointegration tests by Hassler (2000) and Hassler (2001). In this paper these ndings are carried to the panel framework, and they are shown to continue to hold for To avoid such size distortions one might always work with detrended regressions. Such a strategy, however, has two drawbacks: First, a regression with intercept only will be more powerful (see e.g. Hamilton, 1994, p. 598); second, a detrended regression may not be meaningful from an economic point of view.
A conservative strategy rejecting with a probability smaller than or equal to the nominal size irrespective of whether the data display a linear trend or not has been proposed by Hansen (1992) and adopted by e.g. Stock and Watson (2003, Ch. 14). It may be adequate for certain tests in a pure time series setting, but it is not acceptable in a panel framework where it becomes increasingly conservative as N grows. In panel applications it is hence crucial to recognize the presence of linear time trends in the data when testing from regressions with intercept only, in order to apply Theorem 1. In practice this requires pretesting for the presence of a linear time trend, which introduces the problems of multiple testing of course. Alternatively, one may consider the combination of the evidence from two cases: assuming a linear trend and assuming no linear trend.
One could follow the lines by Harvey et al. (2009) Hence, statistics computed from regressions including the term c + β x t = c + β λ 1 τ t + θ ξ t amount to statistics from regressions with c + δ t + θ ξ t asymptotically, where δ = β λ 1 √ µ x µ x . This completes the proof.

Proof of Proposition 1
According to Theorem 1 the statisticZ Analogously, when rejecting for too large values, the rejection probability of strategy S I becomes under µ i,x = 0 according to Theorem 1 with growing N : In the same way, one may analyze the eect of strategy S A in the absence of linear time trends: For N → ∞ ones gets the limits given in Proposition 1 from (19) through (22).

Proof of Proposition 2
We turn to the case of left-sided tests under a) rst. By (13) it holds for p-values from the laws characterized in Assumption 3: p (m) < p (m−1) .
When employing strategy S I , one feedsp (m) into IN M , while p (m−1) would be correct in the presence of linear time trends according to (10). Consequently for S I : where Z is a well dened random variable under the null hypothesis. Consequently, IN M diverges to −∞. This establishes the rst statement. The second one refers to S A where IN M is computed from p (m−1) , althoughp (m) would be correct: which goes o to ∞. This establishes a). Second, we turn to case b), where by (13) p (m) = 1 −F (m) (x) > 1 − F (m−1) (x) = p (m−1) , x ∈ R .
Repeating the above arguments, it is straightforward to complete the proof.