Panel Cointegration Testing in the Presence of Linear Time Trends

Panel Cointegration Testing in the Presence of Linear Time Trends

Econometrics 2016, 4(4), 45; doi:10.3390/econometrics4040045

Article

Panel Cointegration Testing in the Presence of Linear Time Trends

Uwe Hassler * and Mehdi Hosseinkouchack

Department of Business and Economics, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 4, 60323 Frankfurt, Germany

*

Correspondence: Tel.: +49-69-798-34762

Academic Editor: Katarina Juselius

Received: 15 May 2016 / Accepted: 20 October 2016 / Published: 1 November 2016

Abstract:

We consider a class of panel tests of the null hypothesis of no cointegration and cointegration. All tests under investigation rely on single-equations estimated by 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) and with at least one of the regressors (integrated of order 1) being 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.

Keywords:

single-equations; large N asymptotics; integrated series with drift

JEL Classification:

C22; C23

1. Introduction

Most panel tests for the null hypothesis of (no) cointegration rely on single-equations, notable exceptions being Larsson et al. [1], Groen and Kleibergen [2], Breitung [3] and Karaman Örsal and Droge [4] who proposed panel system approaches. In particular, the more recent paper by Miller [5], building on nonlinear instrumental variable likelihood-based rank tests, allows for cross-correlation between the units. Similarly, recent single-equation tests by Chang and Nguyen [6] or Demetrescu et al. [7] also rely on nonlinear instrumental variable estimation, while the vast majority of such panel tests builds on ordinary or fully modified or dynamic least squares (LS). Here, we study exactly this class of LS-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, and with at least one of the integrated regressors displaying 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 [8] and Westerlund [9] on R&D spillovers (total factor productivity and capital stock), Larsson et al. [1] on log real consumption and income (per capita), or Hanck [10] 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 [11] when commenting on the link between health care expenditure and GDP, see McCoskey and Selden [12]; consequently, Blomqvist and Carter [13], Gerdtham and Löthgren [14] or Westerlund [15] worked (partly) with detrended series, i.e., they included time as an explanatory variable in their panel tregressions. Hansen ([16], p. 103), however, argue that “it seems reasonable that excess detrending will reduce the test’s power”. Therefore, we study the empirically relevant case where test statistics are computed from regressions with intercept only (i.e., without detrending) when at least one of the I(1) regressors displays a linear time trend.

Before becoming more technical, we want to outline our findings as a rule for empirical applications. Let there is no content denote a generic panel cointegration statistic computed from a regression with intercept only involving there is no content I(1) variables. The least squares regression may be static in levels,

yi,t=α¯i+β¯i,1xi,1,t+⋯+β¯i,kxi,k,t+u¯i,t,t=1,…,T,i=1,…,N,

where there is no content is assumed to be I(1) in the case of no cointegration, or I(0) under the null hypothesis of cointegration, see Remarks 1 and 3 below, respectively. Alternatively, there is no content may be from the error-correction regression1,

Δyi,t=κ¯i+γ¯iyi,t−1+θ¯i,1xi,1,t−1+⋯+θ¯i,kxi,k,t−1+ε¯i,t,t=1,…,T,i=1,…,N,

where contemporaneous differences there is no content or additional lags of differences may be required as additional regressors to render there is no content free of serial correlation, see Remark 2 below. The test statistic may be constructed from pooling the data or from averaging individual statistics, see e.g., Pedroni [18,19] or Westerlund [15]. Much of the nonstationary panel literature relies on sequential limit theory where there is no content is followed by there is no content, such that limiting normality can be established under the assumption that none of the I(1) regressors follows a deterministic time trend:

NZ¯(m)−μ¯m/σ¯m∼N(0,1).

The constants there is no content and there is no content required for appropriate normalization are typically tabulated for a selected number of values of m, see again Remarks 1 through 3. A different set of such moments there is no content and there is no content is also typically given for detrended regressions, where the test statistic there is no content stems from regressions of the type (there is no content)

yi,t=α˜i+δ˜it+β˜i,1xi,1,t+⋯+β˜i,kxi,k,t+u˜i,t,

or

Δyi,t=κ˜i+ψ˜it+γ˜iyi,t−1+θ˜i,1xi,1,t−1+⋯+θ˜i,kxi,k,t−1+ε˜i,t.

We call such regressions “detrended” because, in a single-equation framework, the resulting parameter estimators are equivalent to what one obtains from a two-step procedure: first, regress all variables on a linear time trend, and, second, regress the individually detrended residuals on each other. This equivalence is sometimes called Frisch-Waugh-Lovell Theorem, see e.g., Greene ([20], Theorem 3.2). For generic there is no content from, e.g., the tests mentioned in Remarks 1 through 3, it holds, irrespective of an eventual linear trend in the data, that

NZ˜(m)−μ˜m/σ˜m∼N(0,1).

Our main contribution is twofold for the case that at least one of the I(1) regressors has a linear time trend and the regressions are run with intercept only (without detrending). First, it is shown that the normalization with there is no content and there is no content and the resulting critical values for there is no content from the regression “with intercept only” are not correct in the presence of linear time trends in the data. It is analytically (Proposition 1) and numerically demonstrated that their usage results in size distortions growing with the panel size N. Second, we characterize the appropriate limiting distributions by showing that normalization of there is no content with there is no content and there is no content results in a standard normal limit, such that the size of the tests can be controlled (Theorem 1). Put differently, Theorem 1 means in non-technical terms: The limiting distribution arising from a regression on k I(1) variables with drift and an intercept amounts to the limiting distribution in the case of a regression on there is no content I(1) variables and an intercept plus a linear time trend. Such a rule is known in a pure time series context for the special case of the residual-based Phillips-Ouliaris test for no cointegration from Hansen ([16], p. 103): “[...] deterministic trends in the data affect the limiting distribution of the test statistics whether or not we detrend the data”; see also the expositions in Hamilton ([21], p. 596, 597) and Hassler ([22], Proposition 16.6). It is even 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 (either to zero or one, depending on the specific test). Moreover, we compare our proposal to account for linear trends in the data with the more traditional method of detrending the regression. By simulation, we show that power gains of our new strategy according to Theorem 1 over detrending may be considerable. We hence recommend this strategy as being superior to detrending.

The rest of the paper is organized as follows. The next section sets some notation and assumptions. Section 3 establishes and discusses our asymptotic results and illustrates them with numerical evidence. It also compares our suggestion to account for linear trends with the conventional method of detrending. The last section discusses consequences for applied work. Mathematical proofs are relegated to the Appendix A.

2. Notation and Assumptions

Restricting our attention to the single-equation framework we partition the m-vector there is no content of observables into a scalar there is no content and a k-element vector there is no content, there is no content, there is no content. As usual, the index i stands for the cross-section, there is no content, while t denotes time, there is no content. Each sequence there is no content, there is no content, 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, there is no content. While there is no content may be cointegrated or not, depending on the respective null hypothesis, we rule out cointegration among there is no content. Technically, these assumptions translate as follows, where there is no content denotes an m-dimensional standard Wiener process, there is no content stands for the integer part of a number x, and ⇒ is the symbol for weak convergence.

Assumption 1.

With obvious partitioning according to there is no content, we assume (there is no content)

zi,t=μi,zt+∑j=1tei,j=μi,yμi,xt+∑j=1tei,y,jei,x,j,t=1,…,T.

The stochastic zero mean process there is no content is integrated of order 0 in that it satisfies

T−0.5∑t=1⌊rT⌋ei,t⇒Ωi0.5Wi,m(r)=Ωi0.5Wi,y(r)Wi,x(r),r∈[0,1],

with

Ωi=ωi,yy2ωi,xy′ωi,xyΩi,xx,

where there is no content and there is no content is positive definite.

Now, we turn to assumptions with respect to the tests. Let there is no content and there is no content stand again for generic test statistics computed from individual single-equation least squares regressions with “intercept only” and “intercept plus linear trend”, respectively. The superscript there is no content stands for the dimension of the I(1) vector entering the equations. One route to panel testing relies on so-called group statistics averaging individual statistics. We denote them as follows:

G¯(m)=1N∑i=1NS¯i(m)orG˜(m)=1N∑i=1NS˜i(m).

Similarly, panel statistics rely on pooling the data across the dimension within, i.e., summing over terms showing up in the numerator and denominator separately,

P¯(m)=g∑i=1NN¯i,T(m),∑i=1ND¯i,T(m)orP˜(m)=g∑i=1NN˜i,T(m),∑i=1ND˜i,T(m).

A typical example for the mapping g is there is no content in the case of t-type statistics. Here, it is assumed that the generic there is no content and there is no content or there is no content and there is no content are computed from individually demeaned or detrended regressions, respectively. We allow for group and panel statistics by introducing the generic notation there is no content and there is no content, and maintain for the panel the joint null hypothesis

H0=⋂i=1NHi,0.

(1)

A distinction between the individual null hypotheses there is no content of cointegration or absence of cointegration is not required, and both cases are treated in the generic assumption as follows.

Assumption 2.

Consider linear single-equation least squares regressions (there is no content, there is no content)

yi,t=α¯i+β¯i′xi,t+u¯i,tandyi,t=α˜i+δ˜it+β˜i′xi,t+u˜i,t,

(2)

or

Δyi,t=κ¯i+γ¯iyi,t−1+θ¯i′xi,t−1+ε¯i,tandΔyi,t=κ˜i+ψ˜it+γ˜iyi,t−1+θ˜i′xi,t−1+ε˜i,t,

(3)

where contemporaneous differences there is no content or lags of there is no content, there is no content, may be required as additional regressors in (3) to ensure residuals free of serial correlation. Let there is no content and there is no content stand for group statistics there is no content and there is no content or for panel statistics there is no content and there is no content computed from regressions with “intercept only” and “intercept plus linear trend”, respectively. We assume under the null hypothesis (1) that

NZ¯(m)−μ¯m⇒N(0,σ¯m2)ifμi,z=0foralli=1,…,N,NZ˜(m)−μ˜m⇒N(0,σ˜m2)forμi,zunrestricted,

as there is no content followed by there is no content.

Tests, e.g., by Kao [23], Pedroni [18,19], Westerlund [9,24] or Westerlund [15] meet Assumption 2 under different sets of restrictions, and they will be considered in the next section, see Remarks 1 through 3. In particular, these authors tabulate values of there is no content and there is no content, there is no content. Our assumption of a single equation approach is motivated by the fact that much of applied work relies on this. However, such an assumption comes at a price: In (2), we have to assume that the regressors there is no content alone are not cointegrated (there is no content is positive definite according to Assumption 1), and, in (3), we have to assume under the alternative of cointegration that there is no content adjust to deviations from the long-run equilibrium, and not there is no content.

Much of the earlier panel cointegration literature assumed independent units invoking a central limit theorem to establish Assumption 2, see e.g., Pedroni [18,19] and Kao [23]. Cross-sectional independence, however, is not maintained in our Assumption 2. Westerlund [15,24] e.g., allows for cross-correlation driven by a common factor. To account for this, he suggests replacing there is no content and there is no content by the cross-sectionally demeaned time series,

xi,t−x¯tandyi,t−y¯t,wherex¯t=1N∑i=1Nxi,t,y¯t=1N∑i=1Nyi,t.

This way, he establishes that the limiting results maintained under Assumption 2 are met under cross-sectional correlation (subject to some restrictions).

3. Results

3.1. Asymptotic Theory

The first paper allowing for linear time trends in a panel cointegration context was by Kao [23]. He considers a residual-based unit root test for the null hypothesis of no cointegration in the tradition of Phillips and Ouliaris [25]. His test builds on pooling the data while allowing for a individual-specific intercept. Kao [23] does not consider regressions containing a linear time trend as additional regressors, but allows for a linear drift in the data when performing a regression with a fixed effect intercept. In the case of the there is no content regressor (i.e., there is no content), Kao ([23], Equation (15)) observed that the linear time trend dominates the I(1) component; hence, the limiting distribution amounts to that of the panel unit root test by Levin et al. [26] upon detrending. To be precise: let there is no content and there is no content denote the normalizing constants provided by Levin et al. [26] for detrended panel unit root tests; then, one should use them for the pooled residual-based panel cointegration statistic there is no content in a bivariate regression if the regressor is I(1) with drift, see Kao ([23], Theorem 4):

NP¯(2)−μ˜1⇒N(0,σ˜12)underμi,x≠0.

(4)

In Theorem 1, we extend Kao’s result for any panel or group statistics from static or dynamic regressions with there is no content computed from regressions with intercept only in the presence of linear time trends.

Theorem 1.

Let the data satisfy Assumption 1, and the generic test statistic there is no content meets Assumption 2 for there is no content. Furthermore, assume that there is no content for all there is no content. Under the null hypothesis (1), it then holds true that

as there is no content, where there is no content are from Assumption 2.

For proof, see Appendix A.

Note that Assumption 2 does not impose any restriction on there is no content. As is shown in the proof, Theorem 1 holds irrespective of whether there is no content displays a linear trend or not (there is no content or there is no content).

Two research strategies can be employed in the presence of linear time trends when dealing with statistics resulting from regressions with intercept only. The first one simply ignores the linear time trends in the data and standardizes there is no content with there is no content and there is no content. The second strategy accounts for the drift in the data according to Theorem 1; in other words, it applies there is no content upon standardizing with there is no content and there is no content. We summarize as follows:

Strategy SI:

When there is no content is computed from panel regressions without detrending, then compare there is no content with quantiles from the standard normal distribution, i.e., ignore the presence of linear trends in the data.

Strategy SA:

When there is no content is computed from panel regressions without detrending, then compare there is no content with quantiles from the standard normal distribution, i.e., account for the presence of linear trends in the data.

For the rest of the paper, we assume that an applied econometrician is able to distinguish between the two cases, whether a linear time trend underlies the variables (e.g., log income or log prices) or not (e.g., interest or inflation rates). Hence, we maintain the assumption behind Theorem 1: the researcher knows that at least one regressor is I(1) with drift (there is no content). We assume that strategy there is no content is only employed when linear trends are truly present and thus refrain from the discussion of misspecification: what happens if there are no linear time trends in the data, but one erroneously accounts for trends.

The situation analyzed in Theorem 1 has not been considered in the previous panel cointegration literature, with the notable exception of Kao [23]. Consequently, all applied papers that we are aware of standardize there is no content, with there is no content and there is no content ignoring the effect of deterministic trends in the series, which amounts to strategy there is no content. The effect of strategy there is no content under linear time trends is discussed for growing N in the following proposition. The resulting size distortions depend on whether the test is right-tailed or left-tailed (null hypothesis is rejected for too large or too small values, respectively).

Proposition 1.

Let the assumptions from Theorem 1 hold true. Furthermore, assume

μ˜m−1<μ¯m.

(5)

Under the null hypothesis, one has the following results for strategy there is no content:

(a) 

For a left-tailed test, the probability to reject according to strategy there is no content increases with growing N to 1;

(b) 

for a right-tailed test, the probability to reject according to strategy there is no content decreases with growing N to 0.

For proof, see Appendix A.

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 [18,19] build on static regressions as in (2). The null hypothesis (1) is rejected for too negative values of the test statistic (of there is no content in our generic notation). The expected values and standard deviations there is no content and there is no content showing up in Assumption 2 are available from Pedroni ([18], Table 2) for there is no content and from Pedroni ([19], Corollary 1) for there is no content. In order to apply Theorem 1 (strategy there is no content) for there is no content, one requires there is no content and there is no content. These values stem from the detrended Dickey-Fuller distribution in the case of group statistics and have been tabulated by Nabeya ([27], Table 4): there is no content and there is no content. Throughout this, we observe there is no content. Hence, Proposition 1(a) applies. If strategy there is no content is employed under linear trends, and then the probability to reject the true null hypothesis converges into one with growing panel size N. Alternatively, Westerlund [24] suggested group and panel variance ratio type tests along the lines of Breitung [28]. The null hypothesis of no cointegration is rejected again for too small values of the variance ratio statistic, and there is no content and there is no content showing up in Assumption 2 are given in Westerlund ([24], Table 1) for there is no content. To apply Theorem 1 with there is no content, we need there is no content and there is no content. For the detrended Breitung distribution we obtain by simulation there is no content and there is no content, which are the values corresponding to the case of group statistics. Again, we observe there is no content, so that (5) holds. Consequently, Proposition 1(a) applies, and the probability to reject the true null hypothesis under strategy there is no content grows with N as long as there is a linear trend in the data. To sum up: in the case of residual-based tests for no cointegration, strategy there is no content results in massive size distortions; numerical evidence for finite N is given in Table 1 below.

Table 1. Approximate effective size of the group t-test by Pedroni [18,19] computed from (A2) at nominal level α under strategy there is no content for there is no content.

| there is no content | 10 | 20 | 30 | 40 | 50
| there is no content | 0.030 | 0.053 | 0.079 | 0.107 | 0.137
there is no content | there is no content | 0.126 | 0.190 | 0.249 | 0.307 | 0.361
| there is no content | 0.227 | 0.314 | 0.389 | 0.455 | 0.515
| there is no content | 0.017 | 0.024 | 0.030 | 0.036 | 0.043
there is no content | there is no content | 0.080 | 0.102 | 0.122 | 0.141 | 0.159
| there is no content | 0.154 | 0.188 | 0.217 | 0.243 | 0.268
| there is no content | 0.014 | 0.017 | 0.020 | 0.022 | 0.025
there is no content | there is no content | 0.067 | 0.078 | 0.087 | 0.096 | 0.104
| there is no content | 0.130 | 0.148 | 0.162 | 0.175 | 0.187

Remark 2.

The error-correction tests by Westerlund [15] relies on regressions of type (3). It is again a left-tailed test: The null hypothesis of no cointegration is rejected for too negative t-values associated with γ. Values of there is no content and there is no content are tabulated in Westerlund ([15], Table 1) for there is no content. In case of there is no content (i.e., no there is no content on the right-hand side), the limiting distributions are of the usual Dickey-Fuller type. Hence, there is no content and there is no content for group statistics are again from detrended Dickey-Fuller-type distributions and given in Nabeya ([27], Table 4) (see above). Comparing there is no content with there is no content, we find there is no content meeting (5) again. Consequently, strategy there is no content is increasingly liberal in the presence of linear time trends, and the probability to reject the true null hypothesis approaches 1 in the limit as long as the series display a linear time trend. For numerical evidence, see Table 2 below.

Table 2. Approximate effective size of the group t-test by Westerlund [15] computed from (A2) at nominal level α under strategy there is no content for there is no content

| there is no content | 10 | 20 | 30 | 40 | 50
| there is no content | 0.139 | 0.352 | 0.564 | 0.732 | 0.846
there is no content | there is no content | 0.394 | 0.669 | 0.836 | 0.924 | 0.967
| there is no content | 0.566 | 0.808 | 0.921 | 0.969 | 0.988
| there is no content | 0.089 | 0.208 | 0.344 | 0.478 | 0.598
there is no content | there is no content | 0.283 | 0.484 | 0.644 | 0.763 | 0.846
| there is no content | 0.436 | 0.645 | 0.783 | 0.870 | 0.924
| there is no content | 0.067 | 0.150 | 0.247 | 0.349 | 0.450
there is no content | there is no content | 0.232 | 0.392 | 0.531 | 0.647 | 0.738
| there is no content | 0.372 | 0.553 | 0.687 | 0.783 | 0.852

Remark 3.

We now flip the null and the alternative hypotheses. Westerlund [9] suggested testing the null hypothesis of cointegration. He proposed a CUSUM group test statistic for this null hypothesis to be applied with tabulated values there is no content and there is no content, there is no content. To apply Theorem 1 for there is no content, we provide as moments of the univariate, detrended distribution by simulation: there is no content and there is no content2. This test is right-tailed and in accordance with Westerlund ([9], Table 1) there is no content. Thus, this time Proposition 1(b) comes in. Under strategy there is no content in the presence of linear trends, the test is increasingly undersized with growing N. Such a conservative behaviour implies low power under the alternative hypothesis.

3.2. Numerical Evidence

The statements obtained from Proposition 1 may be quantified more precisely by means of Equations (A2) and (A3) given in the Appendix. These rejection probabilities apply approximately (for large N) under the null hypothesis at nominal significance level α. We report results for the group t-tests by Pedroni [18,19] and by Westerlund [15] in Table 1 and Table 2.

Generally, the size distortions in Table 1 and Table 2 grow with N, while decreasing with there is no content at the same time. The fact that there is no content is too liberal is characteristic for these tests where we reject for too negative values (of there is no content in our generic notation). Overrejection is not the general case, however, as we see when reversing the null and alternative hypotheses. To quantify distortions for the CUSUM test discussed in Remark 3, we use Equation (A3) from the Appendix. When evaluating there is no content under there is no content, we observe rejection probabilities equal to zero up to three digits for there is no content; this strongly supports the limiting result from Proposition 1 (b).

3.3. Regressions with a Linear Time Trend

For regressions with intercept only, strategy there is no content has been used in the literature and applied with the tests mentioned in the remarks above. We have illustrated its failure to control size under the null hypothesis in the presence of a linear time trend. In practice, one may use two strategies to account for linear time trends. The first one is the new there is no content according to Theorem 1 from regressions without detrending. The second one consists of detrending the series, or equivalently running detrended regressions, i.e., including there is no content and there is no content in (2) and (3), respectively. The empirical strategy then becomes the following:

Strategy SD:

Compute there is no content from detrended panel regressions and compare the normalization there is no content with quantiles from the standard normal distribution.

By Assumption 2, this strategy will provide asymptotically correct size. However, tests from detrended regressions will be prone to power losses relative to strategy there is no content, which is more parsimonious. For this reason, we next investigate the price of strategy there is no content relative to there is no content in terms of power.

In Monte Carlo experiments, we study in particular the error-correction test (group t-statistic) by Westerlund [15]. Before turning to a power analysis, we make sure that size is under control. For the data-generating process (DGP), we consider hence the null hypothesis of no cointegration under linear time trends:

yi,t=δit+xi,1,t+xi,2,t+⋯+xi,k,t+ri,0,t,t=1,2,…,T,i=1,2,…,N,

(6)

xi,j,t=μi,j,x+xi,j,t−1+vi,j,t,j=1,2,...,k,

(7)

where there is no content are normal iid sequences, there is no content, independent of each other. Finally, there is no content is an independent random walk entering (6). The DGP under the alternative of cointegration becomes

Δyi,t=−0.02yi,t−1−δi(t−1)−xi,1,t−1−xi,2,t−1−⋯−xi,k,t−1+vi,0,t,

(8)

where there is no content and there is no content are generated as before. Using the regression

Δyi,t=κ¯i+γ¯iyi,t−1+θ¯i′xi,t−1+ϕ¯i′Δxi,t+ε¯i,t,

(9)

we computed the group t-statistic proposed by Westerlund [15]. Strategy there is no content is employed with

Δyi,t=κ˜i+ψ˜it+γ˜iyi,t−1+θ˜i′xi,t−1+ϕ˜i′Δxi,t+ε˜i,t.

(10)

All reported rejection frequencies rely on 10,000 replications.

The leading case consists of the following parameterization, where only the first component of the regressors there is no content is driven by a linear time trend:

T=250,μi,x′=(1,0,…,0),σi,02=σi,12=⋯=σi,k2=1.

(11)

This mimics with there is no content or there is no content a typical macro panel with monthly data and e.g., income, interest rates and inflation rates as regressors. Table 3 reports the frequencies of rejection for different values of there is no content from (6), and rejection is based on strategy there is no content according to Theorem 1. It illustrates how well the rule of Theorem 1 works: the experimental sizes are close to the nominal ones. This is particularly true for there is no content, while the test is mildly conservative for there is no content, and a bit more conservative for there is no content, in particular for N large relative to there is no content. Next, we consider strategy there is no content with the same data. The rejection frequencies are given in Table 4. We observe that the experimental size from detrended regressions is close to the nominal one under the null hypothesis of no cointegration, irrespective of there is no content.

Table 3. Experimental size at nominal level α under there is no content according to Theorem 1 for (11); data-generating process (DGP): (6) and (7).

| there is no content | 10 | 20 | 30 | 40 | 50
| there is no content | 0.010 | 0.009 | 0.008 | 0.008 | 0.008
there is no content | there is no content | 0.048 | 0.048 | 0.043 | 0.040 | 0.041
| there is no content | 0.093 | 0.095 | 0.087 | 0.080 | 0.083
| there is no content | 0.008 | 0.009 | 0.007 | 0.009 | 0.006
there is no content | there is no content | 0.046 | 0.040 | 0.039 | 0.040 | 0.034
| there is no content | 0.093 | 0.082 | 0.083 | 0.082 | 0.073
| there is no content | 0.010 | 0.010 | 0.012 | 0.010 | 0.011
there is no content | there is no content | 0.051 | 0.052 | 0.052 | 0.049 | 0.052
| there is no content | 0.101 | 0.101 | 0.102 | 0.098 | 0.098
| there is no content | 0.009 | 0.010 | 0.009 | 0.008 | 0.008
there is no content | there is no content | 0.050 | 0.048 | 0.048 | 0.044 | 0.043
| there is no content | 0.096 | 0.095 | 0.096 | 0.089 | 0.088
| there is no content | 0.009 | 0.009 | 0.008 | 0.009 | 0.007
there is no content | there is no content | 0.044 | 0.047 | 0.045 | 0.046 | 0.040
| there is no content | 0.093 | 0.094 | 0.090 | 0.092 | 0.085
| there is no content | 0.013 | 0.013 | 0.013 | 0.013 | 0.011
there is no content | there is no content | 0.055 | 0.056 | 0.058 | 0.056 | 0.054
| there is no content | 0.107 | 0.112 | 0.110 | 0.107 | 0.108
| there is no content | 0.009 | 0.010 | 0.008 | 0.009 | 0.009
there is no content | there is no content | 0.047 | 0.045 | 0.045 | 0.042 | 0.044
| there is no content | 0.095 | 0.092 | 0.095 | 0.090 | 0.091
| there is no content | 0.011 | 0.009 | 0.008 | 0.010 | 0.009
there is no content | there is no content | 0.051 | 0.046 | 0.045 | 0.044 | 0.042
| there is no content | 0.097 | 0.093 | 0.090 | 0.086 | 0.084
| there is no content | 0.011 | 0.013 | 0.011 | 0.013 | 0.013
there is no content | there is no content | 0.056 | 0.054 | 0.055 | 0.055 | 0.056
| there is no content | 0.105 | 0.108 | 0.108 | 0.107 | 0.107

Table 4. Experimental size at nominal level α under there is no content (detrending) for (11); DGP: (6) and (7).

| there is no content | 10 | 20 | 30 | 40 | 50
| there is no content | 0.010 | 0.010 | 0.008 | 0.010 | 0.010
there is no content | there is no content | 0.047 | 0.047 | 0.045 | 0.048 | 0.044
| there is no content | 0.098 | 0.095 | 0.091 | 0.091 | 0.087
| there is no content | 0.012 | 0.012 | 0.013 | 0.012 | 0.012
there is no content | there is no content | 0.057 | 0.057 | 0.056 | 0.059 | 0.057
| there is no content | 0.112 | 0.108 | 0.106 | 0.108 | 0.111
| there is no content | 0.013 | 0.008 | 0.012 | 0.013 | 0.010
there is no content | there is no content | 0.055 | 0.050 | 0.050 | 0.055 | 0.049
| there is no content | 0.104 | 0.099 | 0.101 | 0.098 | 0.099
| there is no content | 0.009 | 0.009 | 0.010 | 0.009 | 0.010
there is no content | there is no content | 0.051 | 0.050 | 0.049 | 0.051 | 0.048
| there is no content | 0.097 | 0.103 | 0.100 | 0.100 | 0.096
| there is no content | 0.012 | 0.013 | 0.013 | 0.015 | 0.014
there is no content | there is no content | 0.059 | 0.062 | 0.061 | 0.058 | 0.063
| there is no content | 0.116 | 0.117 | 0.115 | 0.113 | 0.117
| there is no content | 0.011 | 0.012 | 0.012 | 0.012 | 0.013
there is no content | there is no content | 0.054 | 0.052 | 0.052 | 0.052 | 0.055
| there is no content | 0.107 | 0.104 | 0.104 | 0.104 | 0.107
| there is no content | 0.011 | 0.010 | 0.013 | 0.011 | 0.010
there is no content | there is no content | 0.052 | 0.047 | 0.048 | 0.050 | 0.050
| there is no content | 0.104 | 0.095 | 0.096 | 0.098 | 0.095
| there is no content | 0.015 | 0.011 | 0.014 | 0.014 | 0.012
there is no content | there is no content | 0.062 | 0.061 | 0.060 | 0.062 | 0.058
| there is no content | 0.119 | 0.119 | 0.116 | 0.119 | 0.115
| there is no content | 0.013 | 0.015 | 0.012 | 0.014 | 0.013
there is no content | there is no content | 0.053 | 0.056 | 0.053 | 0.053 | 0.054
| there is no content | 0.110 | 0.106 | 0.101 | 0.112 | 0.105

Since strategies there is no content and there is no content both hold the nominal size, the question of which one is more powerful naturally arises. The results contained in Table 5 are very clear: first, the power increases with there is no content; second, strategy there is no content always outperforms there is no content considerably, and has, e.g., rejection frequencies more than twice as large for there is no content or there is no content. In particular, detrending becomes all the more costly; relative to strategy there is no content, the larger N is, which is intuitively clear: including a linear time trend in a regression requires the estimation of an additional parameter; in a panel of N units, detrending thus involves the estimation of N additional parameters compared to strategy there is no content. At the same time, these estimated trends can be spuriously correlated with the stochastic trends in the data, and, therefore, incorrectly lead to support for cointegration, in particular when the time dimension is relatively short.

Table 5. Experimental power at nominal level 5% for (11); DGP: (8) and (7).

there is no content | 10 | 20 | 30 | 40 | 50
there is no content: Theorem 1 for there is no content
there is no content | 0.486 | 0.727 | 0.859 | 0.933 | 0.965
there is no content | 0.284 | 0.433 | 0.559 | 0.672 | 0.751
there is no content | 0.155 | 0.234 | 0.288 | 0.336 | 0.387
there is no content: Detrended regression for there is no content
there is no content | 0.235 | 0.379 | 0.497 | 0.615 | 0.707
there is no content | 0.163 | 0.247 | 0.331 | 0.399 | 0.452
there is no content | 0.087 | 0.118 | 0.133 | 0.156 | 0.173
there is no content: Theorem 1 for there is no content
there is no content | 0.536 | 0.772 | 0.904 | 0.953 | 0.981
there is no content | 0.311 | 0.485 | 0.622 | 0.726 | 0.802
there is no content | 0.174 | 0.252 | 0.319 | 0.384 | 0.439
there is no content: Detrended regression for there is no content
there is no content | 0.262 | 0.402 | 0.531 | 0.636 | 0.716
there is no content | 0.170 | 0.261 | 0.337 | 0.417 | 0.476
there is no content | 0.090 | 0.127 | 0.143 | 0.168 | 0.186
there is no content: Theorem 1 for there is no content
there is no content | 0.824 | 0.974 | 0.997 | 1.000 | 1.000
there is no content | 0.526 | 0.759 | 0.890 | 0.950 | 0.979
there is no content | 0.292 | 0.419 | 0.553 | 0.661 | 0.734
there is no content: Detrended regression for there is no content
there is no content | 0.395 | 0.605 | 0.757 | 0.852 | 0.911
there is no content | 0.219 | 0.330 | 0.433 | 0.524 | 0.595
there is no content | 0.101 | 0.129 | 0.145 | 0.170 | 0.189

We have varied the leading case with the parameterization from (11). First, we allowed for more and stronger trends in the regressors,

μi,x′=(1,1,…,1),orμi,x′=(1,2,…,k),

with all other parameters fixed. This corrects the mild undersizedness of strategy there is no content reported in Table 3 yielding empirical sizes very close to the nominal one. At the same, time power relative to Table 5 is increased, with strategy there is no content still dominating there is no content. Second, we have increased the magnitude of the random walks, namely there is no content, while the other parameters are from (11) and there is no content (see Table 6). Here, the linear trends are less pronounced, such that there is no content results in slightly more conservative tests (compared to the first panel in Table 3), and similarly, power is reduced (compared to the first panel in Table 5). Still, there is no content clearly dominates there is no content in Table 6. Third, we simulated shorter panels, there is no content. This makes both strategies, there is no content and there is no content, conservative under there is no content, which is accompanied by a loss of power.

Table 6. Experimental size and power at nominal level 5% for there is no content, μi,x′=(1,0,…,0), there is no content, and there is no content; DGP: (6) or (8) and (7).

there is no content | 10 | 20 | 30 | 40 | 50
there is no content: size
there is no content | 0.043 | 0.039 | 0.037 | 0.036 | 0.034
there is no content | 0.043 | 0.041 | 0.037 | 0.034 | 0.031
there is no content | 0.050 | 0.049 | 0.045 | 0.045 | 0.042
there is no content: power
there is no content | 0.410 | 0.622 | 0.771 | 0.869 | 0.920
there is no content | 0.252 | 0.383 | 0.501 | 0.604 | 0.683
there is no content | 0.147 | 0.205 | 0.257 | 0.304 | 0.364
there is no content: size
there is no content | 0.048 | 0.047 | 0.049 | 0.048 | 0.049
there is no content | 0.056 | 0.063 | 0.061 | 0.062 | 0.062
there is no content | 0.058 | 0.052 | 0.053 | 0.051 | 0.057
there is no content: power
there is no content | 0.236 | 0.387 | 0.522 | 0.622 | 0.699
there is no content | 0.178 | 0.273 | 0.350 | 0.437 | 0.493
there is no content | 0.102 | 0.136 | 0.161 | 0.192 | 0.213

4. Conclusions

In time series econometrics, it has been known for a long time that “the deterministic trends in the data affect the limiting distributions of the test statistics whether or not we detrend the data” (Hansen [16], p. 103). This has been shown for the residual-based Phillips-Ouliaris (or Engle-Granger) cointegration test by Hansen [16], see also the exposition in Hamilton ([21], p. 596, 597). Analogous results have been given for other cointegration tests by Hassler [30,31], see also the summary by Hassler ([22], Proposition 16.6). In this paper, these findings are carried over to the panel framework, and they are shown to continue to hold for single-equation tests relying on least squares, no matter whether the null hypothesis is absence or presence of cointegration. In a regression involving there is no content variables, much of the panel cointegration theory relies on normalization with suitable constants there is no content and there is no content and letting the panel dimension N go to infinity to obtain a standard normal distribution. The numbers there is no content and there is no content are tabulated for the case of regressions with intercept only. Different figures there is no content and there is no content are tabulated for regressions with intercept and linear time trend. We show the following: when statistics are computed from regressions with m integrated variables with intercept only, but one of the integrated regressors is dominated by a linear time trend, then normalization with there is no content and there is no content is required to achieve asymptotically valid inference under the null hypothesis (Theorem 1). Normalization with there is no content and there is no content, however, which has been the conventional strategy so far, results in a loss of size control under the null hypothesis. In fact, employing there is no content and there is no content in the presence of linear time trends gives rejection probabilities converging with N to 1 or 0, depending on whether the null hypothesis is no cointegration or cointegration, respectively (see Proposition 1). To avoid such size distortions, one may employ the strategy following Theorem 1, or one may work with detrended regressions. Detrending, however, comes at a price: a regression with intercept only will provide more powerful tests (see e.g., Hamilton [21], p. 598); according to our simulations, power gains of our new strategy over detrending may be considerable and growing with N, and this also holds true if there is a linear trend superimposing the level relation. Our Monte Carlo evidence, however, is limited to the case of testing for the null hypothesis of no cointegration.

Hence, we propose the following empirical strategy if at least one of the integrated regressors is driven by a linear time trend when testing for no cointegration. First, test the null hypothesis of no cointegration with our new strategy there is no content from Theorem 1, since it is more powerful than tests relying on detrending. If the null hypothesis of no cointegration is rejected according to Theorem 1, then one may test, in a second step, whether a linear time trend is present, superimposing the level relation between there is no content and there is no content. If strategy there is no content does not reject the null hypothesis of no cointegration, then one may, of course, try a test building on detrending, although it will tend to be less powerful, since it requires the estimation of N additional parameters.

Acknowledgments

We thank Matei Demetrescu, Christoph Hanck and Joakim Westerlund for helpful comments. Moreover, comments by Katarina Juselius and three anonymous referees that improved the paper are gratefully acknowledged.

Author Contributions

The authors contributed equally to the paper.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Appendix A.1. Proof of Theorem 1

Since group statistics there is no content are computed from individual regressions and since panel statistics there is no content build on pooled regressions, we suppress the index i and consider the generic case with there is no content and there is no content satisfying Assumption 1. Furthermore, we focus on the stochastic regressors and ignore the constant intercept without loss of generality. We will proceed in four steps.

Step 1: First, note that the regressors may be rotated in such a way that all linear trends are concentrated in one scalar component. To that end, define the k-element vector there is no content of unit length with

τt:=λ1′xt=μx′μxt+λ1′∑j=1tex,j.

At the same time, there exist there is no content linearly independent k-element columns collected in the there is no content-matrix there is no content. Due to the Gram-Schmidt orthogonalization, one may assume that the invertible matrix Λ:=(λ1,Λ2) is orthogonal: there is no content. All columns of there is no content eliminate the linear trend in there is no content:

ξt:=Λ2′xt=Λ2′∑j=1tex,j.

Hence, there is no content is a there is no content-vector integrated of order 1 without drift. Now, we are able to write

Λ′xt=τtξt.

Step 2: Second, we show that the deterministic term in there is no content dominates the I(1) component, which is clear from

T−1τ⌊rT⌋=μx′μx⌊rT⌋T+Op(T−0.5)⇒μx′μxr,0≤r≤1.

More precisely, we can show that empirical moments involving there is no content equal those with there is no content up to there is no content. We have from Park and Phillips ([32], Lemma 2.1) that the row vector

1T2∑t=1Tτt,1T3∑t=1Tτt2,1T2.5∑t=1Tτtξt′,1T2∑t=1TτtΔxt−j′

equals

μx′μx1T2∑t=1Tt,μx′μxT3∑t=1Tt2,1T2.5∑t=1Ttξt′,1T2∑t=1TtΔxt−j′+OpT−0.5;

furthermore, if there is no content,

1T3∑t=1Tτtyt,1T2∑t=1TτtΔyt=μx′μx1T3∑t=1Ttyt,1T2∑t=1TtΔyt+OpT−0.5,

or, if there is no content

1T2.5∑t=1Tτtyt,1T1.5∑t=1TτtΔyt=μx′μx1T2.5∑t=1Ttyt,1T1.5∑t=1TtΔyt+OpT−0.5.

Now, we are equipped to deal with the two cases: residual-based tests from (2) and t-type tests from (3).

Step 3: Consider the least squares regression (without intercept for brevity)

with

β^=∑t=1Txtxt′−1ΛΛ′∑t=1Txtyt=∑t=1TΛ′xtxt′−1∑t=1TΛ′xtyt=∑t=1Tτtξtxt′−1∑t=1Tτtξtyt.

Similarly,

β^′Λ=∑t=1Tτt,ξt′yt∑t=1Tτtξtτt,ξt′−1.

Consequently, the empirical residuals are

u^t=yt−β^′ΛΛ′xt=yt−∑t=1Tτt,ξt′yt∑t=1Tτtξtτt,ξt′−1τtξt.

(A1)

For there is no content, we have by assumption

yt=δt+β′xt+ut=δt+β′ΛΛ′xt+ut=δt+β′λ1τt+θ′ξt+ut,θ:=Λ2′β.

If there is no content is I(1), then the series are not cointegrated. If there is no content is I(0), then there is cointegration, where a linear time trend may superimpose the cointegrating relation (there is no content) or not (there is no content). In any case, there is no content is composed of the there is no content-vector there is no content, which is I(1), and a linear time trend asymptotically (since δt+β′λ1τt≈(δ+β′λ1μx′μx)t in the sense of Step 2). Therefore, the residuals there is no content behave asymptotically as if they were computed from a regression on there is no content I(1) regressors and on a linear trend. This establishes Theorem 1 for the case of residual-based tests.

Step 4: Consider the dynamic least squares regression (without intercept and without (lagged) differences as further regressors for brevity):

Δyt=γ^yt−1+θ^′xt−1+ε^t.

In order to investigate error-correction tests relying on the t statistic there is no content, we employ what is sometimes called the Frisch-Waugh-Lovell theorem. In the first stage, regress both there is no content and there is no content on there is no content, and denote the fitted values as there is no content and there is no content, respectively. In the second stage, the regression of there is no content on there is no content produces a slope estimator that is numerically identical to there is no content, and so are the residuals, while the t-statistics differ negligibly due to differences in degrees of freedom. As in Step 3, Equation (A1), one can argue that both there is no content and there is no content behave asymptotically, as if they were computed from a regression on there is no content I(1) regressors and on a linear trend. Hence, because of the Frisch-Waugh-Lovell theorem, there is no content behaves as if there is no content in (3) had been replaced by a linear time trend as regressor plus there is no content regressors that are I(1). This establishes Theorem 1 for the case of error-correction tests, and the proof is complete. ☐

Appendix A.2. Proof of Proposition 1

According to Theorem 1, the statistic there is no content requires under there is no content normalization with there is no content and there is no content, in order to result in a standard normal distribution under there is no content. Let there is no content denote a quantile from the standard normal distribution. In the case where the panel tests are left-tailed, the rejection probability of strategy there is no content under the null hypothesis becomes approximately for large N (at nominal level α):

PNZ¯(m)−μ¯mσ¯m<−z1−α=ΦNμ¯m−μ˜m−1σ˜m−1−σ¯mσ˜m−1z1−α.

(A2)

Analogously for right-tailed tests, the rejection probability of strategy there is no content becomes under there is no content according to Theorem 1 with growing N:

PNZ¯(m)−μ¯mσ¯m>z1−α=1−ΦNμ¯m−μ˜m−1σ˜m−1+σ¯mσ˜m−1z1−α.

(A3)

For there is no content, one gets the limits given in Proposition 1 from (A2) and (A3). ☐

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1.Under the alternative of cointegration, the intercept κ from the error correction form can be decomposed in a rather complicated way, see Juselius ([17], Section 6.2)); below; however, we will maintain the assumption of no cointegration.

2.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 et al. [29].

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