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The AUTOREG Procedure |
For nonlinear time series models, the portmanteau test statistic based on squared residuals is used to test for independence of the series (McLeod and Li 1983):
where
This Q statistic is used to test the nonlinear effects (for example, GARCH effects) present in the residuals. The GARCH(p,q) process can be considered as an ARMA(max(p,q),p) process. See the section "Predicting the Conditional Variance" later in this chapter. Therefore, the Q statistic calculated from the squared residuals can be used to identify the order of the GARCH process.
Engle (1982) proposed a Lagrange multiplier test for ARCH disturbances. The test statistic is asymptotically equivalent to the test used by Breusch and Pagan (1979). Engle's Lagrange multiplier test for the qth order ARCH process is written
where
The presample values ( ,..., ) have been set to 0. Note that the LM(q) tests may have different finite sample properties depending on the presample values, though they are asymptotically equivalent regardless of the presample values. The LM and Q statistics are computed from the OLS residuals assuming that disturbances are white noise. The Q and LM statistics have an approximate distribution under the white-noise null hypothesis.
where
The ^{2}(2)-distribution gives an approximation to the normality test T_{N}.
When the GARCH model is estimated, the normality test is obtained using the standardized residuals . The normality test can be used to detect misspecification of the family of ARCH models.
where the parameter vector contains k elements.
Split the observations for this model into two subsets at the break point specified by the CHOW= option, so that y = ( y'_{1}, y'_{2})',
X = ( X^{'}_{1}, X^{'}_{2})^{'}, and
u = ( u'_{1}, u'_{2})'.
Now consider the two linear regressions for the two subsets of the data modeled separately,
where the number of observations from the first set is n_{1} and the number of observations from the second set is n_{2}.
The Chow test statistic is used to test the null hypothesis conditional on the same error variance V(u_{1}) = V(u_{2}). The Chow test is computed using three sums of square errors.
where is the regression residual vector from the full set model, is the regression residual vector from the first set model, and is the regression residual vector from the second set model. Under the null hypothesis, the Chow test statistic has an F-distribution with k and (n_{1}+n_{2}-2k) degrees of freedom, where k is the number of elements in .
Chow (1960) suggested another test statistic that tests the hypothesis that the mean of prediction errors is 0. The predictive Chow test can also be used when n_{2} < k.
The PCHOW= option computes the predictive Chow test statistic
The predictive Chow test has an F-distribution with n_{2} and (n_{1}-k) degrees of freedom.
where the disturbances might be serially correlated with possible heteroscedasticity. Phillips and Perron (1988) proposed the unit root test of the OLS regression model.
Let and let be the variance estimate of the OLS estimator , where is the OLS residual. You can estimate the asymptotic variance of using the truncation lag l.
where , for j>0, and .
Then the Phillips-Perron Z() test (zero mean case) is written
and has the following limiting distribution:
where B(·) is a standard Brownian motion. Note that the realization Z(x) from the the stochastic process B(·) is distributed as N(0,x) and thus .
Therefore, you can observe that as , which shows that the limiting distribution is skewed to the left.
Let t be the t-test statistic for . The Phillips-Perron test is written
and its limiting distribution is derived as
When you test the regression model for the true random walk process (single mean case), the limiting distribution of the statistic is written
Finally, the limiting distribution of the Phillips-Perron test for the random walk with drift process (trend case) can be derived as
where c=1 for and for ,
When several variables z_{t} = (z_{1t}, ... ,z_{kt})' are cointegrated, there exists
a (k×1) cointegrating vector c such that c'z_{t} is stationary and c is a nonzero vector. The residual based cointegration test is based on the following regression model:
where y_{t} = z_{1t}, x_{t} = (z_{2t}, ... ,z_{kt})', and = (_{2},...,_{k})'. You can estimate the consistent cointegrating vector using OLS if all variables are difference stationary, that is, I(1). The Phillips-Ouliaris test is computed using the OLS residuals from the preceding regression model, and it performs the test for the null hypothesis of no cointegration. The estimated cointegrating vector is .
Since the AUTOREG procedure does not produce the p-value of the cointegration test, you need to refer to the tables by Phillips and Ouliaris (1990). Before you apply the cointegration test, you might perform the unit root test for each variable.
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