Abstract
In this paper the theory on the estimation of vector autoregressive (VAR) models for I(2) processes is extended to the case of long VAR approximation of more general processes. Hereby the order of the autoregression is allowed to tend to infinity at a certain rate depending on the sample size. We deal with unrestricted OLS estimators (in the model formulated in levels as well as in vector error correction form) as well as with two stage estimation (2SI2) in the vector error correction model (VECM) formulation. Our main results are analogous to the I(1) case: We show that the long VAR approximation leads to consistent estimates of the long and short run dynamics. Furthermore, tests on the autoregressive coefficients follow standard asymptotics. The pseudo likelihood ratio tests on the cointegrating ranks (using the Gaussian likelihood) used in the 2SI2 algorithm show under the null hypothesis the same distributions as in the case of data generating processes following finite order VARs. The same holds true for the asymptotic distribution of the long run dynamics both in the unrestricted VECM estimation and the reduced rank regression in the 2SI2 algorithm. Building on these results we show that if the data is generated by an invertible VARMA process, the VAR approximation can be used in order to derive a consistent initial estimator for subsequent pseudo likelihood optimization in the VARMA model.
Highlights
Many macroeconomic variables have been found to exhibit trend-like behaviour that can be modelled by using vector autoregressions (VARs)
Building on these results we show that if the data is generated by an invertible vector autoregressive moving average (VARMA) process, the VAR approximation can be used in order to derive a consistent initial estimator for subsequent pseudo likelihood optimization in the VARMA model
In the asymptotic distribution of the estimation error Brownian motions occur relating to the processt∈Z : Under Assumption 1 we have ut ⇒ B(r ) = [ B1 (r )0, Bc (r )0 ]0 = [ B1 (r )0, B2 (r )0, B3 (r )]0 t =1 where B(r ), 0 ≤ r ≤ 1, denotes a Brownian motion with corresponding variance
Summary
Many macroeconomic variables have been found to exhibit trend-like behaviour that can be modelled by using vector autoregressions (VARs). Assuming the data generating process to be a VAR of known finite order, the rank of matrices can be tested using (pseudo) likelihood ratio tests. For the I(2) case no such extensions are currently known This is the research gap this paper tries to fill: First we establish consistency and asymptotic normality of estimated autoregressive coefficients (in the sense of Lewis and Reinsel) for unrestricted ordinary least squares (OLS) estimation in the VECM representation. This can be used in order to derive Wald type tests of linear restrictions on the autoregressive parameters.
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