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Panel-data cointegration tests

Highlights

• Kao panel-data cointegration test
• Dickey–Fuller
• Modified Dickey–Fuller
• Augmented Dickey–Fuller
• Pedroni panel-data cointegration test
• Phillips–Perron
• Modified Phillips–Perron
• Augmented Dickey–Fuller
• Westerlund panel-data cointegration test
• Variance ratio

Researchers perform cointegration tests when time series are nonstationary to determine whether they have a stable, long-run relationship. xtcointtest implements a variety of tests for data containing many long panels, known as the large-N large-T case. Think of a long series on supermarket purchases for a large number of buyers. Or think of repeated visits to a website by the site's subscribers.

Time series are said to be nonstationary when they have a mean or variance that varies over time. Some nonstationary time series are stationary if you first difference them. Nonstationary time series tend to wander. Cointegration says that they wander together, meaning that there is a long-run equilibrium relationship among the series. And in Stata, we can test for cointegration using the xtcointtest command.

xtcointtest tests for the presence of this long-run cointegration relationship. Three tests are available: Kao, Pedroni, and Westerlund.

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We have real-estate data and want to estimate the long-run relationship between prices of houses, neighboring house prices, and average house prices in each area. We believe that the housing market is volatile—that it is not stable—that the time series have a unit root—but we also believe that the prices have a stable long-run relationship.

Here is a graph that seems to substantiate our belief:

To test our intuition, we perform an Im-Pesaran-Shin unit-root test for the series. This is not part of the xtcointtest that we want to show you, but the unit-root test should be performed first. We find that we cannot reject the null hypothesis that the panels have unit roots.

. xtunitroot ips hprice if house

H0: All panels contain unit roots           Number of panels  =    700
Ha: Some panels are stationary              Number of periods =    500

AR parameter: Panel-specific                Asymptotics: T,N -> Infinity
Panel means:  Included                                        sequentially
Time trend:   Not included

Fixed-N exact critical values
Statistic      p-value         1%      5%      10%

t-bar               -0.7598                     -1.730  -1.670  -1.640
t-tilde-bar         -0.7582
Z-t-tilde-bar       24.2109        1.0000



We have three prices in our data: hprice, nprice, and aprice. We showed you only the test for hprice, but we tested all three prices and obtained similar results.

Unit roots in hand, we now perform the Kao cointegration test:

. xtcointtest kao hprice aprice nprice

Kao test for cointegration

H0: No cointegration                        Number of panels       =    700
Ha: All panels are cointegrated             Number of periods      =    498

Cointegrating vector: Same
Panel means:          Included              Kernel:           Bartlett
Time trend:           Not included          Lags:             3.54 (Newey–West)
AR parameter:         Same                  Augmented lags:   1

Statistic         p-value

Modified Dickey–Fuller t                   -1.7e+03          0.0000
Dickey–Fuller t                            -5.3e+02          0.0000
Augmented Dickey–Fuller t                  -3.6e+02          0.0000
Unadjusted modified Dickey–Fuller t        -4.1e+03          0.0000


The null hypothesis of no cointegration is rejected. This is true for the five tests statistics reported in the table and provides strong evidence that all panels in the data are cointegrated.

Not ones to give up, let's do some more tests. We want to show you the other xtcointtest features, and anyway, a reviewer might ask whether the above results are robust. xtcointtest just reported five statistics, but they are all related.

. xtcointtest pedroni hprice aprice nprice

Pedroni test for cointegration

H0: No cointegration                        Number of panels       =    700
Ha: All panels are cointegrated             Number of periods      =    499

Cointegrating vector: Panel specific
Panel means:          Included              Kernel:           Bartlett
Time trend:           Not included          Lags:             5.00 (Newey–West)
AR parameter:         Panel specific        Augmented lags:   1

Statistic         p-value

Modified Phillips–Perron t               -1779.0263          0.0000
Phillips–Perron t                         -675.1867          0.0000
Augmented Dickey–Fuller t                 -674.2714          0.0000



These related tests of cointegration work differently but allow us to come to the same conclusion: the panels are cointegrated.

The Westerlund test uses yet another approach, one that imposes fewer restrictions. It tests the same null hypothesis, but the alternative hypothesis is different, namely, that some (not necessarily all) of the panels are cointegrated.

. xtcointtest westerlund hprice aprice nprice

Westerlund test for cointegration

H0: No cointegration                        Number of panels       =    700
Ha: Some panels are cointegrated            Number of periods      =    500

Cointegrating vector: Panel specific
Panel means:          Included
Time trend:           Not included
AR parameter:         Panel specific

Statistic         p-value

Variance ratio                             -30.6619          0.0000


This test also rejects the null.