"Frontmatter". In: Analysis of Financial Time Series

NONLINEARITY TESTS 159• Step 2: Compute the predictive residualâ (m+1)+d = x (m+1)+d − ˆβ 0,m −p∑i=1ˆβ i,m x (m+1)+d−iand its standard error. Let ê (m+1)+d be the standardized predictive residual.• Step 3: Use the recursive least squares method to update the least squares estimatesto ˆβ i,m+1 by incorporating the new data point x (m+1)+d .• Step 4: Repeat Steps 2 and 3 until all data points are processed.• Step 5: Consider the linear regression of the standardized predictive residualê (m+ j)+d = α 0 +p∑α i x (m+ j)+d−i + v t , j = 1,...,T − d − m (4.46)i=1and compute the usual F statistic for testing α i = 0 in Eq. (4.46) for i =0,...,p. Under the null hypothesis that x t follows a linear AR(p) model, theF ratio has a limiting F distribution with degrees of freedom p + 1andT −d − m − p.We refer to the earlier F test as a Tar-F test. The idea behind the test is that underthe null hypothesis there is no model change in the arranged autoregression inEq. (4.45) so that the standardized predictive residuals should be close to iid withmean zero and variance 1. In this case, they should have no correlations with theregressors x (m+ j)+d−i . For further details including formulas for a recursive leastsquares method and some simulation study on performance of the Tar-F test, seeTsay (1989). The Tar-F test avoids the problem of nuisance parameters encounteredby the likelihood ratio test. It does not require knowing the threshold r 1 . It simplytests that the predictive residuals have no correlations with regressors if the nullhypothesis holds. Therefore, the test does not depend on knowing the number ofregimes in the alternative model. Yet the Tar-F test is not as powerful as the likelihoodratio test if the true model is indeed a two-regime SETAR model with a knowninnovational distribution.4.2.3 ApplicationsIn this subsection, we apply some of the nonlinearity tests discussed previously tofive time series. For a real financial time series, an AR model is used to remove anyserial correlation in the data, and the tests apply to the residual series of the model.The five series employed are as follows:1. r 1t : A simulated series of iid N(0, 1) with 500 observations.2. r 2t : A simulated series of iid Student-t distribution with 6 degrees of freedom.The sample size is 500.