Abstract
The inferential procedures discussed in previous chapters are based on asymptotic considerations in the sense that they rely on the convergence of the distribution of the test statistics to some known limit distribution as the sample size goes to infinity. However, in order to work well, first-order asymptotic approximation requires that the asymptotic distribution is an accurate approximation to the finite sample distribution. When dealing with cointegrated VAR models, this is not generally the case. In this chapter we investigate the performance of various small sample inference procedures for cointegrated vector autoregressive models. Special attention is given to the Bartlett(1937) and the bootstrap Bartlett adjustment for the likelihood ratio test. The bootstrap p-value test and an F-type test are also considered. Throughout the chapter performance is assessed in terms of the empirical sizes and power properties of the inference procedures under consideration. An empirical application is also provided to illustrate the use of these procedures with real data. The analysis should provide some guidance to practitioners in doubt about which inference procedure to use when dealing with cointegrated VAR models.
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Notes
- 1.
Note that in Johansen (2002a) a Bartlett correction factor for \(\Lambda\) is derived under the assumption that the adjustment parameter α is known. Although theoretically interesting, the resulting test statistic is less relevant for applied work. Accordingly, in this chapter we restrict our attention to Johansen (2000) where this assumption is dropped.
- 2.
See Canepa (2012) for extensive simulation results, including the case where innovations are heteroscedastic.
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Hunter, J., Burke, S.P., Canepa, A. (2017). Testing in VECMs with Small Samples. In: Multivariate Modelling of Non-Stationary Economic Time Series. Palgrave Texts in Econometrics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-137-31303-4_6
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