PANEL Procedure

Panel Data Unit Root Tests

Unit roots are a big concern in dynamic processes because they have important implications for the stationarity of a process and hence for estimation. Using regular estimation techniques while ignoring the presence of unit roots can lead to spurious regressions and hence produce nonsensical results. Therefore, detecting unit roots in order to be able to analyze stationary processes is of vital concern for dynamic processes. One of the most widely used tests in the time series literature is the augmented Dickey-Fuller (ADF) test. This section introduces and briefly reviews the background information about the tests developed for dynamic panel data, which in most cases are enhancements of the ADF test.

Levin, Lin, and Chu Test

Levin, Lin, and Chu (2002) propose a panel data unit root test for the null hypothesis of a unit root against a hypothesis of homogeneous stationarity. The model is specified as

normal upper Delta y Subscript i t Baseline equals delta y Subscript i t minus 1 Baseline plus sigma-summation Underscript upper L equals 1 Overscript p Subscript i Baseline Endscripts theta Subscript i upper L Baseline normal upper Delta y Subscript i t minus upper L Baseline plus alpha Subscript m i Baseline d Subscript m t Baseline plus epsilon Subscript i t Baseline m equals 1 comma 2 comma 3

The panel data unit root test evaluates the null hypothesis of , for all i, against the alternative hypothesis , for all i. Three models are considered: (1) (the empty set) with no individual effects; (2) , in which the series has an individual-specific mean but no time trend; and (3) , in which the series has an individual-specific mean and a linear and individual-specific time trend. The lag order is unknown and is allowed to vary across individuals. It can be selected by the methods that are described in the section Lag Order Selection in the ADF Regression. The selected lag order is denoted as . The necessary condition for the test is that . An important assumption is that the errors, , are . In other words, cross-sectional independence is assumed. The test is implemented in the following three steps:

Step 1

The ADF regressions are implemented for each individual i, and then the orthogonalized residuals are generated and normalized. That is, the following model is estimated:

normal upper Delta y Subscript i t Baseline equals delta Subscript i Baseline y Subscript i t minus 1 Baseline plus sigma-summation Underscript upper L equals 1 Overscript ModifyingAbove p With caret Subscript i Baseline Endscripts theta Subscript i upper L Baseline normal upper Delta y Subscript i t minus upper L Baseline plus alpha Subscript m i Baseline d Subscript m t Baseline plus epsilon Subscript i t Baseline m equals 1 comma 2 comma 3

Then, two orthogonalized residuals are generated by the following two auxiliary regressions:

normal upper Delta y Subscript i t Baseline equals sigma-summation Underscript upper L equals 1 Overscript ModifyingAbove p With caret Subscript i Baseline Endscripts theta Subscript i upper L Baseline normal upper Delta y Subscript i t minus upper L Baseline plus alpha Subscript m i Baseline d Subscript m i Baseline plus e Subscript i t

y Subscript i t minus 1 Baseline equals sigma-summation Underscript upper L equals 1 Overscript ModifyingAbove p With caret Subscript i Baseline Endscripts theta Subscript i upper L Baseline normal upper Delta y Subscript i t minus upper L Baseline plus alpha Subscript m i Baseline d Subscript m i Baseline plus v Subscript i t minus 1

The residuals are then saved as and , respectively, and normalized using the regression standard error from the ADF regression in order to remove heteroscedasticity. Let denote the standard error from each of the previous ADF regressions, where . The normalized residuals are then

e overTilde Subscript i t Baseline equals StartFraction ModifyingAbove e With caret Subscript i t Baseline Over ModifyingAbove sigma With caret Subscript epsilon i Baseline EndFraction comma v overTilde Subscript i t minus 1 Baseline equals StartFraction ModifyingAbove v With caret Subscript i t minus 1 Baseline Over ModifyingAbove sigma With caret Subscript epsilon i Baseline EndFraction

Step 2

The ratios of long-run to short-run standard deviations of are estimated. Denote the ratios and the long-run variances as and , respectively. The long-run variances are estimated by the heteroscedasticity- and autocorrelation-consistent (HAC) estimators, which are described in the section Long-Run Variance Estimation. Then the ratios are estimated by . Let the average standard deviation ratio be , and let its estimator be . As the authors note in their paper (Levin, Lin, and Chu 2002), use of the long-run variance based on first differences results in lower bias in finite samples.

Step 3

The panel test statistics are calculated. To calculate the t statistic and the adjusted t statistic, the following equation is estimated:

e overTilde Subscript i t Baseline equals delta v overTilde Subscript i t minus 1 Baseline plus epsilon overTilde Subscript i t

The total number of observations is , with . The standard t statistic for testing is , with the OLS estimator and standard deviation ,

ModifyingAbove delta With caret equals StartFraction sigma-summation Underscript i equals 1 Overscript upper N Endscripts sigma-summation Underscript t equals 2 plus ModifyingAbove p With caret Subscript i Baseline Overscript upper T Endscripts e overTilde Subscript i t Baseline v overTilde Subscript i t minus 1 Baseline Over sigma-summation Underscript i equals 1 Overscript upper N Endscripts sigma-summation Underscript t equals 2 plus ModifyingAbove p With caret Subscript i Baseline Overscript upper T Endscripts v overTilde Subscript i t minus 1 Superscript 2 Baseline EndFraction

ModifyingAbove sigma With caret Subscript delta Baseline equals ModifyingAbove sigma With caret Subscript epsilon overTilde Baseline left-bracket sigma-summation Underscript i equals 1 Overscript upper N Endscripts sigma-summation Underscript t equals 2 plus ModifyingAbove p With caret Subscript i Baseline Overscript upper T Endscripts v overTilde Subscript i t minus 1 Superscript 2 Baseline right-bracket Superscript negative one-half

where is the root mean square error from the step 3 regression

ModifyingAbove sigma With caret Subscript epsilon overTilde Superscript 2 Baseline equals StartFraction 1 Over upper N upper T overTilde EndFraction sigma-summation Underscript i equals 1 Overscript upper N Endscripts sigma-summation Underscript t equals 2 plus ModifyingAbove p With caret Subscript i Baseline Overscript upper T Endscripts left-parenthesis e overTilde Subscript i t Baseline minus ModifyingAbove delta With caret v overTilde Subscript i t minus 1 Baseline right-parenthesis squared

However, the standard t statistic diverges to negative infinity for models (2) and (3). Levin, Lin, and Chu (2002) therefore propose the following adjusted t statistic:

t Subscript delta Superscript asterisk Baseline equals StartFraction t Subscript delta Baseline minus upper N upper T overTilde ModifyingAbove upper S With caret Subscript upper N Baseline ModifyingAbove sigma With caret Subscript epsilon overTilde Superscript negative 2 Baseline ModifyingAbove sigma With caret Subscript delta Baseline mu Subscript m upper T overTilde Superscript asterisk Baseline Over sigma Subscript m upper T overTilde Superscript asterisk Baseline EndFraction

The mean and standard deviation adjustments () depend on the time series dimension and model specification m, which can be found in Table 2 of Levin, Lin, and Chu (2002). The adjusted t statistic converges to the standard normal distribution. Therefore, the standard normal critical values are used in hypothesis testing.

Lag Order Selection in the ADF Regression

The methods of selecting the individual lag orders in the ADF regressions can be divided into two categories: selection based on information criteria and selection via sequential testing.

Lag Selection Based on Information Criteria

In this method, the following information criteria can be applied to lag order selection: Akaike’s information criterion (AIC), the Schwarz Bayesian criterion (SBC), the Hannan-Quinn information criterion (HQIC or HQC), and the modified AIC. As with other model selection applications, the lag order is selected from 0 to the maximum to minimize the objective function, plus a penalty term, which is a function of the number of parameters in the regression. Let k be the number of parameters and be the number of effective observations. For regression models, the objective function is , where SSR is the sum of squared residuals. For AIC, the penalty term equals . For SBC, this term is . For HQIC, it is , where c is a constant greater than 1.^[8] For MAIC, the penalty term equals , where

tau Subscript upper T Baseline left-parenthesis k right-parenthesis equals left-parenthesis normal upper S normal upper S normal upper R slash upper T Subscript o Baseline right-parenthesis Superscript negative 1 Baseline ModifyingAbove delta With caret squared sigma-summation Underscript t equals p Subscript normal m normal a normal x Baseline plus 2 Overscript upper T Endscripts y Subscript t minus 1 Superscript 2

and is the estimated coefficient of the lagged dependent variable in the ADF regression.

Lag Selection via Sequential Testing

In this method, the lag order estimation is based on the statistical significance of the estimated AR coefficients. Hall (1994) proposed general-to-specific (GS) and specific-to-general (SG) modeling strategies. Levin, Lin, and Chu (2002) recommend the GS strategy, following Campbell and Perron (1991). In the GS modeling strategy, starting with the maximum lag order , the t test for the largest lag order in is performed to determine whether a smaller lag order is preferred. Specifically, when the null of is not rejected given the significance level (), a smaller lag order is preferred. This procedure continues until a statistically significant lag order is reached. On the other hand, the SG modeling strategy starts with lag order 0 and moves toward the maximum lag order, .

Long-Run Variance Estimation

The long-run variance of is estimated by an HAC-type estimator. For model (1), given the lag truncation parameter and kernel weights , the formula is

ModifyingAbove sigma With caret Subscript y i Superscript 2 Baseline equals StartFraction 1 Over upper T minus 1 EndFraction sigma-summation Underscript t equals 2 Overscript upper T Endscripts normal upper Delta y Subscript i t Superscript 2 Baseline plus 2 sigma-summation Underscript upper L equals 1 Overscript upper K overbar Endscripts w Subscript upper K overbar upper L Baseline left-bracket StartFraction 1 Over upper T minus 1 EndFraction sigma-summation Underscript t equals 2 plus upper L Overscript upper T Endscripts normal upper Delta y Subscript i t Baseline normal upper Delta y Subscript i t minus upper L Baseline right-bracket

To achieve consistency, the lag truncation parameter must satisfy and as . Levin, Lin, and Chu (2002) suggest . The weights depend on the kernel function. Andrews (1991) proposes data-driven bandwidth (lag truncation parameter + 1 if integer-valued) selection procedures to minimize the asymptotic mean square error (MSE) criterion. For more information about the kernel functions and Andrews’s (1991) data-driven bandwidth selection procedure, see the section Heteroscedasticity- and Autocorrelation-Consistent Covariance Matrices. Because Levin, Lin, and Chu (2002) truncate the bandwidth as an integer, when LLCBAND is specified as the BANDWIDTH option, it corresponds to . Furthermore, kernel weights with kernel function .

For model (2), first the series is de-meaned individual by individual. Therefore, is replaced by , where is the mean of for individual i. For model (3) with individual fixed effects and time trend, both the individual mean and trend should be removed before the long-run variance is estimated. That is, first you regress on for each individual and save the residual , and then you replace with the residual.

Cross-Sectional Dependence via Time-Specific Aggregate Effects

The Levin, Lin, and Chu (2002) testing procedure is based on the assumption of cross-sectional independence. It is possible to relax this assumption and allow for a limited degree of dependence via time-specific aggregate effects. Let denote the time-specific aggregate effects; then the data generating process becomes

Two more models are considered: (4) (the empty set), with no individual effects but with time effects; and (5) , in which the series has an individual-specific mean and a time-specific mean.

By subtracting the time averages from the observed dependent variable , or equivalently, by including the time-specific intercepts in the ADF regression, the cross-sectional dependence is removed. The impact of a single aggregate common factor that has an identical impact on all individuals but changes over time can also be removed in this way. After cross-sectional dependence is removed, the three-step procedure is applied to calculate the Levin, Lin, and Chu (2002) adjusted t statistic.

Deterministic Variables

Three deterministic variables can be included in the model for the first-stage estimation: CS_FixedEffects (cross-sectional fixed effects), TS_FixedEffects (time series fixed effects), and TimeTrend (individual linear time trend). When a linear time trend is included, the individual fixed effects are also included. Otherwise the time trend is not identified. Moreover, if the time series fixed effects are included, the time trend is not identified either. Therefore, there are five identified models: model (1), no deterministic variables; model (2), CS_FixedEffects; model (3), CS_FixedEffects and TimeTrend; model (4), TS_FixedEffects; and model (5), CS_FixedEffects and TS_FixedEffects. PROC PANEL outputs the test results for all five model specifications.

Im, Pesaran, and Shin Test

To test for the unit root in heterogeneous panels, Im, Pesaran, and Shin (2003) propose a standardized t-bar test statistic based on averaging the (augmented) Dickey-Fuller statistics across the groups. The limiting distribution is standard normal. The stochastic process is generated by the first-order autoregressive process. If , the data generating process can be expressed as in the Levin, Lin, and Chu (LLC) test,

normal upper Delta y Subscript i t Baseline equals beta Subscript i Baseline y Subscript i t minus 1 Baseline plus sigma-summation Underscript j equals 1 Overscript p Subscript i Baseline Endscripts rho Subscript i j Baseline normal upper Delta y Subscript i comma t minus j Baseline plus alpha Subscript m i Baseline d Subscript m t Baseline plus epsilon Subscript i t Baseline m equals 1 comma 2 comma 3

where is the lag order in the ADF regression, as in the LLC test. In contrast with the data generating process in the LLC test, is allowed to differ across groups. The null hypothesis of unit roots is

upper H 0 colon beta Subscript i Baseline equals 0 for all i

against the heterogeneous alternative,

upper H 1 colon beta Subscript i Baseline less-than 0 for i equals 1 comma ellipsis comma upper N 1 comma beta Subscript i Baseline equals 0 for i equals upper N 1 plus 1 comma ellipsis comma upper N

The Im, Pesaran, and Shin (2003) test also allows for some (but not all) of the individual series to have unit roots under the alternative hypothesis. But the fraction of the individual processes that are stationary is positive, . The t-bar statistic, denoted by , is formed as a simple average of the individual t statistics for testing the null hypothesis of . If is the standard t statistic, then

t overbar Subscript upper N upper T Baseline equals upper N Superscript negative 1 Baseline sigma-summation Underscript i equals 1 Overscript upper N Endscripts t Subscript i upper T Baseline left-parenthesis p Subscript i Baseline comma beta Subscript i Baseline right-parenthesis

If , then for each i the t statistic (without time trend) converges to the Dickey-Fuller distribution, , defined by

eta Subscript i Baseline equals StartFraction one-half StartSet left-bracket upper W Subscript i Baseline left-parenthesis 1 right-parenthesis right-bracket squared minus 1 EndSet minus upper W Subscript i Baseline left-parenthesis 1 right-parenthesis integral Subscript 0 Superscript 1 Baseline upper W Subscript i Baseline left-parenthesis u right-parenthesis d u Over integral Subscript 0 Superscript 1 Baseline left-bracket upper W Subscript i Baseline left-parenthesis u right-parenthesis right-bracket squared d u minus left-bracket integral Subscript 0 Superscript 1 Baseline upper W Subscript i Baseline left-parenthesis u right-parenthesis d u right-bracket squared EndFraction

where is the standard Brownian motion. The limiting distribution is different when a time trend is included in the regression (Hamilton 1994, p. 499). The mean and variance of the limiting distributions are reported in Nabeya (1999). The standardized t-bar statistic satisfies

where the standard normal is the sequential limit with followed by . To obtain better finite sample approximations, Im, Pesaran, and Shin (2003) propose standardizing the t-bar statistic by means and variances of under the null hypothesis . The alternative standardized t-bar statistic is

Im, Pesaran, and Shin (2003) simulate the values of and for different values of T and p. The lag order in the ADF regression can be selected by the same method as in Levin, Lin, and Chu (2002). For more information, see the section Lag Order Selection in the ADF Regression.

When T is fixed, Im, Pesaran, and Shin (2003) assume serially uncorrelated errors, ; is likely to have finite second moment, which is not established in the paper. The t statistic is modified by imposing the null hypothesis of a unit root. Denote as the estimated standard error from the restricted regression (),

t overbar overTilde Subscript upper N upper T Baseline equals upper N Superscript negative 1 Baseline sigma-summation Underscript i equals 1 Overscript upper N Endscripts t overTilde Subscript i t Baseline equals upper N Superscript negative 1 Baseline sigma-summation Underscript i equals 1 Overscript upper N Endscripts left-bracket ModifyingAbove beta With caret Subscript i upper T Baseline left-parenthesis bold-italic y prime Subscript i comma negative 1 Baseline upper M Subscript tau Baseline bold-italic y Subscript i comma negative 1 Baseline right-parenthesis Superscript 1 slash 2 Baseline slash sigma overTilde Subscript i upper T Baseline right-bracket

where is the OLS estimator of (unrestricted model), , , and , where is a given initial value (fixed or random). Under the null hypothesis, the standardized -bar statistic converges to a standard normal variate,

where and are the mean and variance of , respectively. The limit is taken as , and T is fixed. Their values are simulated for finite samples without a time trend. is also likely to converge to standard normal.

When N and T are both finite, an exact test that assumes no serial correlation can be used. The critical values of and are simulated.

As in the section Levin, Lin, and Chu Test, it is possible to relax this assumption of cross-sectional independence and allow for a limited degree of dependence via time-specific aggregate effects. In that section, two more models (model 4 and model 5) with time fixed effects are considered. For more information, see the section Cross-Sectional Dependence via Time-Specific Aggregate Effects.

Combination Tests

Combining the observed significance levels (p-values) from N independent tests of the unit root null hypothesis was proposed by Maddala and Wu (1999) and Choi (2001). Suppose is the test statistic to test the unit root null hypothesis for individual , and is the cumulative distribution function (CDF) of the asymptotic distribution as . Then the asymptotic p-value is defined as

p Subscript i Baseline equals upper F left-parenthesis upper G Subscript i Baseline right-parenthesis

There are different ways to combine these p-values. The first way is the inverse chi-square test (Fisher 1932); this test is referred to as the P test in Choi (2001) and the test in Maddala and Wu (1999):

upper P equals minus 2 sigma-summation Underscript i equals 1 Overscript upper N Endscripts ln left-parenthesis p Subscript i Baseline right-parenthesis

When the test statistics are continuous, are independent uniform variables. Therefore, as and N is fixed. But as , P diverges to infinity in probability. Therefore, it is not applicable for large N. To derive a nondegenerate limiting distribution, the P test (Fisher test with ) should be modified to

upper P Subscript m Baseline equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts left-parenthesis negative 2 ln left-parenthesis p Subscript i Baseline right-parenthesis minus 2 right-parenthesis slash 2 StartRoot upper N EndRoot equals minus sigma-summation Underscript i equals 1 Overscript upper N Endscripts left-parenthesis ln left-parenthesis p Subscript i Baseline right-parenthesis plus 1 right-parenthesis slash StartRoot upper N EndRoot

Under the null as ,^[9] and then , .^[10]

The second way of combining individual p-values is the inverse normal test,

upper Z equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts normal upper Phi Superscript negative 1 Baseline left-parenthesis p Subscript i Baseline right-parenthesis

where is the standard normal CDF. When , as N is fixed. When N and are both large, the sequential limit is also standard normal if first and next.

The third way of combining p-values is the logit test,

upper L Superscript asterisk Baseline equals StartRoot k EndRoot upper L equals StartRoot k EndRoot sigma-summation Underscript i equals 1 Overscript upper N Endscripts ln left-parenthesis StartFraction p Subscript i Baseline Over 1 minus p Subscript i Baseline EndFraction right-parenthesis

where . When and N is fixed, . In other words, the limiting distribution is the t distribution with degree of freedom . The sequential limit is as and then . Simulation results in Choi (2001) suggest that the Z test outperforms other combination tests. For the time series unit root test , Maddala and Wu (1999) apply the augmented Dickey-Fuller test. According to Choi (2006), the Elliott, Rothenberg, and Stock (1996) Dickey-Fuller generalized least squares (DF-GLS) test offers significant size and power advantages in finite samples.

Breitung’s Unbiased Tests

To account for the nonzero mean of the t statistic in the OLS detrending case, bias-adjusted t statistics were proposed by Levin, Lin, and Chu (2002) and Im, Pesaran, and Shin (2003). The bias corrections imply a severe loss of power. Breitung and associates take an alternative approach to avoid the bias, by using alternative estimates of the deterministic terms (Breitung and Meyer 1994; Breitung 2000; Breitung and Das 2005). The data generating process is the same as in the Im, Pesaran, and Shin (IPS) test (2003). Three models are considered, as described in the section Levin, Lin, and Chu Test. When serial correlation is absent, for model (2) with individual specific means, the constant terms are estimated by the initial values . Therefore, the series is adjusted by subtracting the initial value. The equation becomes

normal upper Delta y Subscript i t Baseline equals delta Superscript asterisk Baseline left-parenthesis y Subscript i comma t minus 1 Baseline minus y Subscript i Baseline 1 Baseline right-parenthesis plus v Subscript i t

For model (3) with individual specific means and time trends, the time trend can be estimated by . The levels can be transformed as

y overTilde Subscript i t Baseline equals y Subscript i t Baseline minus y Subscript i Baseline 1 Baseline minus ModifyingAbove beta With caret Subscript i Baseline t equals y Subscript i t Baseline minus y Subscript i Baseline 1 Baseline minus t left-parenthesis y Subscript i upper T Baseline minus y Subscript i Baseline 1 Baseline right-parenthesis slash left-parenthesis upper T minus 1 right-parenthesis

The Helmert transformation is applied to the dependent variable to remove the mean of the differenced variable:

normal upper Delta y Subscript i t Superscript asterisk Baseline equals StartRoot StartFraction upper T minus t Over upper T minus t plus 1 EndFraction EndRoot left-parenthesis normal upper Delta y Subscript i t Baseline minus StartFraction normal upper Delta y Subscript i comma t plus 1 Baseline plus midline-horizontal-ellipsis plus normal upper Delta y Subscript i upper T Baseline Over upper T minus t EndFraction right-parenthesis

The transformed model is

normal upper Delta y Subscript i t Superscript asterisk Baseline equals delta Superscript asterisk Baseline y overTilde Subscript i comma t minus 1 Baseline plus v Subscript i t

The pooled t statistic has a standard normal distribution. Therefore, no adjustment is needed for the t statistic. To adjust for heteroscedasticity across cross sections, Breitung (2000) proposes a UB (unbiased) statistic based on the transformed data,

upper U upper B equals StartFraction sigma-summation Underscript i equals 1 Overscript upper N Endscripts sigma-summation Underscript t equals 2 Overscript upper T Endscripts normal upper Delta y Subscript i t Superscript asterisk Baseline y overTilde Subscript i comma t minus 1 Baseline slash sigma Subscript i Superscript 2 Baseline Over left-parenthesis sigma-summation Underscript i equals 1 Overscript upper N Endscripts sigma-summation Underscript t equals 2 Overscript upper T Endscripts y overTilde Subscript i comma t minus 1 Superscript 2 Baseline slash sigma Subscript i Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline EndFraction

where . When is unknown, it can be estimated as

ModifyingAbove sigma With caret Subscript i Superscript 2 Baseline equals sigma-summation Underscript t equals 2 Overscript upper T Endscripts left-parenthesis normal upper Delta y Subscript i t Baseline minus StartFraction sigma-summation Underscript t equals 2 Overscript upper T Endscripts normal upper Delta y Subscript i t Baseline Over upper T minus 1 EndFraction right-parenthesis squared slash left-parenthesis upper T minus 2 right-parenthesis

The UB statistic has a standard normal limiting distribution as followed by sequentially.
To account for the short-run dynamics, Breitung and Das (2005) suggest applying the test to the prewhitened series, . For model (1) and model (2) (constant-only case), they suggest the same method as in step 1 of the Levin, Lin, and Chu (LLC) test (2002).^[11] For model (3) (with a constant and linear time trend), the prewhitened series can be obtained by running the following restricted ADF regression under the null hypothesis of a unit root ( ) and no intercept and linear time trend (),

where is a consistent estimator of the true lag order and can be estimated by the procedures listed in the section Lag Order Selection in the ADF Regression. For the LLC and IPS tests, the lag orders are selected by running the ADF regressions. But for Breitung and his associates’ tests, the restricted ADF regressions are used to be consistent with the prewhitening method. Let be the estimated coefficients.^[12] The prewhitened series can be obtained by

normal upper Delta ModifyingAbove y With caret Subscript i t Baseline equals normal upper Delta y Subscript i t Baseline minus sigma-summation Underscript upper L equals 1 Overscript ModifyingAbove p With caret Subscript i Baseline Endscripts ModifyingAbove theta With caret Subscript i upper L Baseline normal upper Delta y Subscript i t minus upper L

and

ModifyingAbove y With caret Subscript i t Baseline equals y Subscript i t Baseline minus sigma-summation Underscript upper L equals 1 Overscript ModifyingAbove p With caret Subscript i Baseline Endscripts ModifyingAbove theta With caret Subscript i upper L Baseline y Subscript i t minus upper L

The transformed series are random walks under the null hypothesis,

normal upper Delta ModifyingAbove y With caret Subscript i t Baseline equals delta ModifyingAbove y With caret Subscript i comma t minus 1 Baseline plus v Subscript i t

where for . When the cross-sectional units are independent, the t statistic converges to standard normal under the null, as followed by ,

where with the OLS estimator .
To account for cross-sectional dependence, Breitung and Das (2005) propose the robust t statistic and a GLS version of the test statistic. Let be the error vector for time t, and let be a positive definite matrix with eigenvalues . Let and . The model can be written as a system of equations of the seemingly unrelated regressions (SUR) type:

normal upper Delta y Subscript t Baseline equals delta y Subscript t minus 1 Baseline plus v Subscript t

The unknown covariance matrix can be estimated by its sample counterpart,

ModifyingAbove normal upper Omega With caret equals sigma-summation Underscript t equals 2 Overscript upper T Endscripts left-parenthesis normal upper Delta y Subscript t Baseline minus ModifyingAbove delta With caret y Subscript t minus 1 Baseline right-parenthesis left-parenthesis normal upper Delta y Subscript t Baseline minus ModifyingAbove delta With caret y Subscript t minus 1 Baseline right-parenthesis Superscript prime Baseline slash left-parenthesis upper T minus 1 right-parenthesis

The sequential limit followed by of the standard t statistic is normal with mean 0 and variance . The variance can be consistently estimated by . Thus the robust t statistic can be calculated as

as followed by under the null hypothesis of random walk. Because the finite sample distribution can be quite different, Breitung and Das (2005) list the , , and critical values for different N’s.

When , a (feasible) GLS estimator is applied; it is asymptotically more efficient than the OLS estimator. The data are transformed by multiplying as defined before, . Thus the model is transformed into

normal upper Delta ModifyingAbove z With caret Subscript t Baseline equals delta ModifyingAbove z With caret Subscript t minus 1 Baseline plus e Subscript t

The feasible GLS (FGLS) estimator of and the corresponding t statistic are obtained by estimating the transformed model by OLS and denoted by and , respectively:

Hadri Stationarity Test

Hadri (2000) adopts a component representation where an individual time series is written as a sum of a deterministic trend, a random walk, and a white-noise disturbance term. Under the null hypothesis of stationarity, the variance of the random walk equals 0. Specifically, two models are considered:

For model (1), the time series is stationary around a level ,
For model (2), is trend stationary,

where is the random walk component,

The initial values of the random walks, , are assumed to be fixed unknowns and can be considered as heterogeneous intercepts. The errors and satisfy , and are mutually independent.

The null hypothesis of stationarity is against the alternative random walk hypothesis .

In matrix form, the models can be written as

y Subscript i Baseline equals upper X Subscript i Baseline beta Subscript i Baseline plus e Subscript i

where ; , where ; and , where is a vector of ones, , and .

Let be the residuals from the regression of on ; then the LM statistic is

upper L upper M equals StartStartFraction sigma-summation Underscript i equals 1 Overscript upper N Endscripts StartFraction 1 Over upper T squared EndFraction sigma-summation Underscript t equals 1 Overscript upper T Endscripts upper S Subscript i t Superscript 2 Baseline OverOver upper N ModifyingAbove sigma With caret Subscript epsilon Superscript 2 Baseline EndEndFraction

where is the partial sum of the residuals and is a consistent estimator of under the null hypothesis of stationarity. With some regularity conditions,

where is a standard Brownian bridge in model (1) and a second-level Brownian bridge in model (2). Let be a standard Wiener process (Brownian motion),

upper V left-parenthesis r right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column upper W left-parenthesis r right-parenthesis minus r upper W left-parenthesis 1 right-parenthesis 2nd Column for model left-parenthesis 1 right-parenthesis 2nd Row 1st Column upper W left-parenthesis r right-parenthesis plus left-parenthesis 2 r minus 3 r squared right-parenthesis upper W left-parenthesis 1 right-parenthesis plus 6 r left-parenthesis r minus 1 right-parenthesis integral Subscript 0 Superscript 1 Baseline upper W left-parenthesis s right-parenthesis d s 2nd Column for model left-parenthesis 2 right-parenthesis EndLayout

The mean and variance of the random variable can be calculated by using the characteristic functions,

xi equals upper E left-bracket integral Subscript 0 Superscript 1 Baseline upper V squared left-parenthesis r right-parenthesis d r right-bracket equals StartLayout Enlarged left-brace 1st Row 1st Column one-sixth 2nd Column for model left-parenthesis 1 right-parenthesis 2nd Row 1st Column one-fifteenth 2nd Column for model left-parenthesis 2 right-parenthesis EndLayout

and

zeta squared equals normal upper V normal a normal r left-bracket integral Subscript 0 Superscript 1 Baseline upper V squared left-parenthesis r right-parenthesis d r right-bracket equals StartLayout Enlarged left-brace 1st Row 1st Column one-forty-fifth 2nd Column for model left-parenthesis 1 right-parenthesis 2nd Row 1st Column StartFraction 11 Over 6300 EndFraction 2nd Column for model left-parenthesis 2 right-parenthesis EndLayout

The LM statistics can be standardized to obtain the standard normal limiting distribution,

Consistent Estimator of

Hadri’s (2000) test can be applied to the general case of heteroscedasticity and serially correlated disturbance errors. Under homoscedasticity and serially uncorrelated errors, can be estimated as

ModifyingAbove sigma With caret Subscript epsilon Superscript 2 Baseline equals StartFraction sigma-summation Underscript i equals 1 Overscript upper N Endscripts sigma-summation Underscript t equals 1 Overscript upper T Endscripts ModifyingAbove epsilon With caret Subscript i t Superscript 2 Baseline Over upper N left-parenthesis upper T minus k right-parenthesis EndFraction

where k is the number of regressors. Therefore, formodel (1) and for model (2).

When errors are heteroscedastic across individuals, the standard errors can be estimated by for each individual i and the LM statistic needs to be modified to

upper L upper M equals StartFraction 1 Over upper N EndFraction sigma-summation Underscript i equals 1 Overscript upper N Endscripts left-parenthesis StartStartFraction StartFraction 1 Over upper T squared EndFraction sigma-summation Underscript t equals 1 Overscript upper T Endscripts upper S Subscript i t Superscript 2 Baseline OverOver ModifyingAbove sigma With caret Subscript epsilon comma i Superscript 2 Baseline EndEndFraction right-parenthesis

To allow for temporal dependence over t, has to be replaced by the long-run variance of , which is defined as . An HAC estimator can be used to consistently estimate the long-run variance . For more information, see the section Long-Run Variance Estimation.

Harris and Tzavalis Panel Unit Root Test

Harris and Tzavalis (1999) derive the panel unit root test under fixed T and large N. Five models are considered, as in Levin, Lin, and Chu (2002). Model (1) is the homogeneous panel,

y Subscript i t Baseline equals phi y Subscript i t minus 1 Baseline plus v Subscript i t

Under the null hypothesis, . For model (2), each series is a unit root process with a heterogeneous drift,

y Subscript i t Baseline equals alpha Subscript i Baseline plus phi y Subscript i t minus 1 Baseline plus v Subscript i t

Model (3) includes heterogeneous drifts and linear time trends,

y Subscript i t Baseline equals alpha Subscript i Baseline plus beta Subscript i Baseline t plus phi y Subscript i t minus 1 Baseline plus v Subscript i t

Let be the OLS estimator of ; then

ModifyingAbove phi With caret minus 1 equals left-bracket sigma-summation Underscript i equals 1 Overscript upper N Endscripts y prime Subscript i comma negative 1 Baseline upper Q Subscript upper T Baseline y Subscript i comma negative 1 Baseline right-bracket Superscript negative 1 Baseline dot left-bracket sigma-summation Underscript i equals 1 Overscript upper N Endscripts y prime Subscript i comma negative 1 Baseline upper Q Subscript upper T Baseline v Subscript i Baseline right-bracket

where , , and is the projection matrix. Formodel (1), there are no regressors other than the lagged dependent value, so is the identity matrix . For model (2), a constant is included, so , where is a column of ones. For model (3), a constant and time trend are included. Thus , where and .

When in model (1) under the null hypothesis, as

As , it becomes .

When the drift is absent in model (2), , under the null hypothesis, as

As , .

When the time trend is absent in model (3), , under the null hypothesis, as

When , .

^[8]In practice c is set to 1, following the literature (Hannan and Quinn 1979; Hall 1994).

^[9]The time series length T is subindexed by because the panel can be unbalanced.

^[10]Choi (2001) also points out that the joint limit result where N and go to infinity simultaneously is the same as the sequential limit, but it requires more moment conditions.

^[11]For more information, see the section Levin, Lin, and Chu Test. The only difference is the standard error estimate . Breitung suggests using instead of as in the LLC test to normalize the standard error.

^[12]Breitung (2000) suggests the approach in step 1 of Levin, Lin, and Chu (2002), whereas Breitung and Das (2005) suggest the prewhitening method as described in this section. In Breitung’s code, to be consistent with the papers, different approaches are adopted for model (2) and model (3). Meanwhile, for the order of variable transformation and prewhitening, in model (2), the initial values are deducted (variable transformation) first, and then the prewhitening is applied. In model (3), the order is reversed: the series is prewhitened and then transformed to remove the mean and linear time trend.

Last updated: June 19, 2025