698—Chapter 38. Cointegration Testing

Date: 05/11/09 Time: 16:01

Series: P_T S_T PSTAR_T

Sample: 1973M01 1989M10 Included observations: 202

Null hypothesis: Series are not cointegrated Cointegrating equation deterministics: C Additional regressor deterministics: @TREND

Long-run variance estimate (Bartlett kernel, User bandwidth = 13.0000)

No d.f. adjustment for variances

**Number of stochastic trends in asymptotic distribution

In contrast with the Engle-Granger tests, the results are quite similar for all six of the tests with the Phillips-Ouliaris test not rejecting the null hypothesis that the series are not cointegrated. As before, the bottom portion of the output displays intermediate results for the test associated with each dependent variable.

Panel Cointegration Testing

The extensive interest in and the availability of panel data has led to an emphasis on extending various statistical tests to panel data. Recent literature has focused on tests of cointegration in a panel setting. EViews will compute one of the following types of panel cointegration tests: Pedroni (1999), Pedroni (2004), Kao (1999) and a Fisher-type test using an underlying Johansen methodology (Maddala and Wu 1999).

Performing Panel Cointegration Tests in EViews

You may perform a cointegration test using either a Pool object or a Group in a panel workfile setting. We focus here on the panel setting; conducting a cointegration test using a Pool

700—Chapter 38. Cointegration Testing

int(min((Ti – k) § 3, 12) (Ti § 100)1 § 4)

where Ti is the length of the cross-section i . Alternatively, you may provide your own value by selecting User specified, and entering a value in the edit field.

The Pedroni test employs both parametric and non-parametric kernel estimation of the long run variance. You may use the Variance calculation and Lag length sections to control the computation of the parametric variance estimators. The Spectral estimation portion of the dialog allows you to specify settings for the non-parametric estimation. You may select from a number of kernel types (Bartlett, Parzen, Quadratic spectral) and specify how the bandwidth is to be selected (Newey-West automatic, Newey-West fixed, User specified). The Newey-West fixed bandwidth is given by 4(T i § 100)2 § 9 . The Kao test uses the Lag length and the Spectral estimation portion of the dialog settings as described below.

Here, we see the options for the Fisher test selection. These options are similar to the options available in the Johansen cointegration test (“Johansen Cointegration Test,” beginning on

page 685).

The Deterministic trend specification section determines the type of exogenous trend to be used.

The Lag intervals section specifies the lag-pair to be used in estimation.

Panel Cointegration Details

Here, we provide a brief description of the cointegration tests supported by EViews. The Pedroni and Kao tests are based on Engle-Granger (1987) two-step (residual-based) cointegration tests. The Fisher test is a combined Johansen test.

Pedroni (Engle-Granger based) Cointegration Tests

The Engle-Granger (1987) cointegration test is based on an examination of the residuals of a spurious regression performed using I(1) variables. If the variables are cointegrated then the residuals should be I(0). On the other hand if the variables are not cointegrated then the residuals will be I(1). Pedroni (1999, 2004) and Kao (1999) extend the Engle-Granger framework to tests involving panel data.

Panel Cointegration Testing—701

Pedroni proposes several tests for cointegration that allow for heterogeneous intercepts and trend coefficients across cross-sections. Consider the following regression

for t = 1, º, T ; i = 1, º, N ; m = 1, º, M ; where y and x are assumed to be integrated of order one, e.g. I(1). The parameters ai and di are individual and trend effects which may be set to zero if desired.

Under the null hypothesis of no cointegration, the residuals ei, t will be I(1). The general approach is to obtain residuals from Equation (38.6) and then to test whether residuals are I(1) by running the auxiliary regression,

j = 1

for each cross-section. Pedroni describes various methods of constructing statistics for testing for null hypothesis of no cointegration (ri = 1 ). There are two alternative hypotheses: the homogenous alternative, (ri = r) < 1 for all i (which Pedroni terms the within-dimen- sion test or panel statistics test), and the heterogeneous alternative, ri < 1 for all i (also referred to as the between-dimension or group statistics test).

The Pedroni panel cointegration statistic ¿N, T is constructed from the residuals from either Equation (38.7) or Equation (38.8). A total of eleven statistics with varying degree of properties (size and power for different N and T ) are generated.

Pedroni shows that the standardized statistic is asymptotically normally distributed,

v

where m and v are Monte Carlo generated adjustment terms.

Details for these calculations are provided in the original papers.

Kao (Engle-Granger based) Cointegration Tests

The Kao test follows the same basic approach as the Pedroni tests, but specifies cross-sec- tion specific intercepts and homogeneous coefficients on the first-stage regressors.

In the bivariate case described in Kao (1999), we have

for

702—Chapter 38. Cointegration Testing

for t = 1, º, T ; i = 1, º, N . More generally, we may consider running the first stage regression Equation (38.6), requiring the ai to be heterogeneous, bi to be homogeneous

j = 1

is estimated as