Monte Carlo Results for Bootstrap Tests in Systems with Integrated-Cointegrated Variables
Sammanfattning: When we study the properties of a test procedure, two aspects are of prime importance. Firstly, we wish to know if the actual size of the test (i.e., the probability of rejecting the null when true) is close to the nominal size (used for calculating the critical values). Given that the actual size is a reasonable approximation to the nominal size, we then wish to investigate the actual power of the test (i.e., the probability of rejecting the null when false) for a number of different alternative hypotheses. When comparing different tests, we will therefore prefer those in which (a) actual size lies closest to the nominal size and, given that (a) holds, (b) have the greatest power. In most cases, however, the distributions of the test statistics we use are known only asymptotically and, unfortunately, unless the sample size is very large indeed, it is difficult to know whether asymptotic theory is sufficiently accurate to allow us to interpret our results with confidence. As a result, the tests may not have the correct size and inferential comparisons and judgements based on them might be misleading. In recent years, an approach is beginning to become popular to deal with this situation, namely to employ some variant of the Bootstrap. The basic idea of Bootstrapping test statistics is to draw a large number of “ Bootstrap samples,” which obey the null hypothesis and, as far as possible, resemble the real sample, and then compare the observed test statistic to the ones calculated from the Bootstrap samples. By using Bootstrap tests we are able to improve the critical values so that the true size of the test approaches its nominal value. It is possible to use Bootstrapping either to calculate a critical value, or to calculate the significance level, or P-value, associated with it. In this thesis, by using Monte Carlo experiments, we show a number of useful results about the small sample properties of what we shall call “ Bootstrap tests,” in systems with Integrated-Cointegrated variables. We show the ability of the Bootstrap technique to produce critical values which are much more accurate than the asymptotic ones: a) In the case of the Error Correction Model Cointegration test for a simple, single-lag, bivariate process. b) For testing Cointegration in a Vector Error Correction Model system. c) For testing Granger-causality in a Vector Autoregressive (VAR ) system. d) And for testing Granger-causality in a (VAR) system by using only the Dolado and Lütkepohl corrected test. Finally, an application of the four previous tests concludes this thesis by studying the comovement and Granger-cause effects between the French and German stock markets.
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