Numerical Distribution Functions for Unit Root and Cointegration Tests

Numerical Distribution Functions for Unit Root and Cointegration Tests

January 1995 | James G. MacKinnon
This paper presents numerical distribution functions for unit root and cointegration tests, based on response surface regressions derived from simulation experiments. The main contributions are data files containing estimated response surface coefficients and a computer program that uses them to calculate asymptotic and finite-sample critical values and P-values for various tests. The program, urcdist, is freely available via the Internet and can be used to compute critical values or P-values for any sample size. The paper also includes graphs of some tabulated distribution functions and an empirical example. The paper discusses unit root and cointegration tests, including Dickey-Fuller (DF) and Augmented Dickey-Fuller (ADF) tests, as well as Engle-Granger cointegration tests. These tests have nonstandard distributions, and the paper provides methods to estimate their asymptotic and finite-sample distributions. The simulation experiments involved 200,000 replications for each of 14 sample sizes, with 50 experiments for each of up to 14 sample sizes. The results of these experiments were used to estimate response surfaces, which allow for the calculation of critical values and P-values for any sample size. The paper also discusses the use of feasible generalized least squares (FGLS) to estimate approximating regressions and obtain approximate critical values, P-values, and even approximate densities. The results show that the approximate P-values and critical values derived from these methods are highly accurate. The paper includes an empirical example using the Canadian 91-day Treasury Bill rate to illustrate the use of the urcdist program for computing P-values for unit root tests. The results show that the P-values for z-tests are generally lower than those for tau-tests based on the same regressions. The paper concludes that the response surface coefficients provide excellent approximations to the asymptotic and finite-sample distributions of various unit root and cointegration tests.This paper presents numerical distribution functions for unit root and cointegration tests, based on response surface regressions derived from simulation experiments. The main contributions are data files containing estimated response surface coefficients and a computer program that uses them to calculate asymptotic and finite-sample critical values and P-values for various tests. The program, urcdist, is freely available via the Internet and can be used to compute critical values or P-values for any sample size. The paper also includes graphs of some tabulated distribution functions and an empirical example. The paper discusses unit root and cointegration tests, including Dickey-Fuller (DF) and Augmented Dickey-Fuller (ADF) tests, as well as Engle-Granger cointegration tests. These tests have nonstandard distributions, and the paper provides methods to estimate their asymptotic and finite-sample distributions. The simulation experiments involved 200,000 replications for each of 14 sample sizes, with 50 experiments for each of up to 14 sample sizes. The results of these experiments were used to estimate response surfaces, which allow for the calculation of critical values and P-values for any sample size. The paper also discusses the use of feasible generalized least squares (FGLS) to estimate approximating regressions and obtain approximate critical values, P-values, and even approximate densities. The results show that the approximate P-values and critical values derived from these methods are highly accurate. The paper includes an empirical example using the Canadian 91-day Treasury Bill rate to illustrate the use of the urcdist program for computing P-values for unit root tests. The results show that the P-values for z-tests are generally lower than those for tau-tests based on the same regressions. The paper concludes that the response surface coefficients provide excellent approximations to the asymptotic and finite-sample distributions of various unit root and cointegration tests.
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