1. Testing for near integration with stationary covariates
Antonio Aznar
Mª Isabel Ayuda
University of Zaragoza
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2. Objective
The objective of this paper is to propose a test procedure to discriminate between trend stationarity and difference stationarity, using information from series related to the series whose stationary character is tested.
We derive the limiting behaviour of the procedure under both the null hypothesis and the alternative. It is shown that the test satisfies what Müller (2005) calls the “ideal asymptotic rejection profile”.
In the last section we provide Monte-Carlo simulation results to illustrate the finite sample performance of the test.
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3. Introduction - Trend-stationary processes
Since the influential paper of Nelson and Plosser (1982), much attention has been devoted to developing testing procedures to discriminate between:
- Trend-stationary processes
- Difference-stationary processes
Useful references are:
-Phillips and Xiao (1998)
-Darné and Diebolt (2005)
-Haldrup and Jansson (2006)
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4. To determine whether a particular series is trend-stationary or difference-stationary there are two broad categories of procedures:
Unit Root Tests
Consider the simple null hypothesis that the observed series is difference-stationary against a composite alternative that specifies that the series is trend-stationary.
Stationary Tests
Consider the composite null hypothesis that the series is trend-stationary against the alternative that it has a unit root.
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5. Most of the test statistics have been defined using only the values of the series whose stationary character is tested, ignoring information in related time series.
Exceptions to this rule are the proposals in Hansen (1995) and Elliot and Jansson (2003)
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6. Hansen shows that the convention of ignoring the information in related time series is quite costly because large power gains can be achieved by including correlated stationary covariates in the regression equation. He proposes a unit root test procedure with covariates that he calls the “covariate augmented Dickey-Fuller” (CADF) test.
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7. Elliot and Jansson (2003) propose a family of tests for a unit root with greater power against a point alternative when an arbitrary number of stationary covariates are modelled with the potentially integrated series
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8. Structure
The presentation is organized as follows.
Section 2: we comment on the stochastic characteristics of the model that we assume as the Data Generating Process (DGP) and we derive some preliminary results.
Section 3: the test statistic and its limiting behavior are examined.
Section 4: we assess the performance of the test with finite samples reporting the results from a Monte Carlo simulation study.
Section 5: The main conclusions are summarized.
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9. 2.- Models and some preliminary results
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13. NEYMAN-PEARSON APPROACH
Two steps:
First, we define the set of admissible tests
Second, we choose a particular test from among the set of admissible tests.
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15. SET OF ADMISSIBLE TESTS
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16. CHOOSE OF AN ADMISSIBLE TEST
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17. 3.- The PONI test and its limiting behavior
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19. Theorem 1
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20. Proof:
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22. Theorem 2
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23. Proof:
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24. 4.- Monte Carlo experiments
We provide Monte Carlo simulation results to illustrate the finite sample performance of the PONI test when covariates are taken into account, PONIcov, in comparison with the behavior of the PONI test without considering these covariates (introduced in Aznar and Ayuda (2007).
These criteria are also compared with the modified KPSS test (MKPSS) for near integration that uses prewhitened Heteroskedasticity and Autocorrelation Consistent (HAC) estimates and the boundary condition rule of Sul et al. (2005) using the Quadratic spectral window (results using the Bartlett window were similar).
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27. Table 1: Size and power of the tests when the DGP is Model 1
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28. Table 2: Size and power of the tests when the DGP is Model 2
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29. Table 3: Size and power of the tests when the DGP is Model 3
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30. Results
The PONI and PONIcov tests have lower empirical size than the MKPSS test, while there is no significant difference with respect to power.
The PONI and the MKPSS tests are robust with respect to changes in the covariance between y and x.
This is not the case for the PONIcov test; it is seen that the size of this test is a decreasing function of the covariance while the power is an increasing function of this covariance.
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31. As the sample size increases, both the PONI test and the PONIcov tend to what Müller (2005) calls the ideal asymptotic rejection profile:
The empirical size is close to zero.
The power tends to 1.
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32. 5.- Conclusions
We have derived a testing procedure to discriminate between the null hypothesis of near integration against a unit root alternative.
The limiting behavior of the test fits the ideal asymptotic rejection profile because the sizes of both Type I and Type II errors tend to zero as the sample size increases.
This limiting behavior is confirmed by the simulation results presented in Section 5.
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33. THANKS
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