Abstract
In the previous chapter we stressed the importance of preliminary univariate analysis of the data. This chapter deals with the multivariate analysis of the data information about “unrestricted” linear time series relationships between sets of variables of interest. “Unrestricted” should not be taken too literally. Some a priori restrictions on the “true” shape of the multivariate autocorrelation function should be appropriate in order to get some degree of precision for the analysis. At this stage we only consider so-called smoothness restrictions (in particular on the graphs of the impulse response representation). Decreasing the maximum order of the VAR can have such a smoothing effect. This order is determined by the number of observations and the number of observations per year. For quarterly data a natural choice for the minimum a priori lag length is 4. A choice for a higher a priori order depends on the number of variables of interest compared with the number of observations. Other restrictions are discussed in chapter 6.
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Notes
Note \(D_i=({\hat\beta-\hat\beta_{(i)}})^\prime({X}^\prime X)({\hat\beta-\hat\beta_{(i)}})/[\kappa\sigma^2]\) with \(\hat{\beta}\) and \(\hat{\beta}_{(i)}\) the OLS estimates using the full and the case deleted sample. Di can be seen to measure the displacement of \(\hat{\beta}\).
Cook and Weisberg (1982) also discussed \(EIC_{(i)}= \ \ \ T\partial\hat{\beta}(\omega_i)/\partial\omega_i\setminus_0=T({X}^\prime X)^{-1}xi^\prime e_i/(1-h_i)^2\), where the basic model is a gross error at observation i and local influence of introducing observation i is computed. DFBETAi can be viewed as a compromise between EICi and EIC(i).
The influence measure \(DFFITS_i=((h_i/(1-h_i))PF_i)^{(\frac{1}{2})}\) has an intermediate position.
See appendix A3.2, remark A3.14.
Barnett and Lewis (1984) still used the title “Outliers in Time Series: A Little Explored Area” for their Chapter 11.
These techniques are discussed in more detail in chapter 6.
For a number of practical reasons one can include deterministic regressors if one estimates (2.1), see appendix A4.2. In §3.3 the parameters of these regressors are assumed to be part of the dynamic parameters determining the mean. Sometimes we use the short hand notation B from (3.10) to denote these parameters. Equivalently one can extend yt with deterministic components as in Sims, Stock and Watson (1990), and allow for a singular Σ.
Perron (1991, footnote 2) discovered “errors due to numerical instability of the computations” only years later.
See remark A3.4 in §A3.1.1.
See Hannan and Kavalieris (1986, Th. 2.2) for restrictions on the number of parameters that ensures consistent estimation of the “true” disturbances in a time series context.
See op cit., §10.2.1 for applications in regression models. Note that critical values obtained in op cit. eq. (10.2.22), tabulated in Lund (1975) are for internaily studentized residual \(t_{i}^{-2}\) in §A3.1.1.
It is dangerous to trust critical values from standard computer packages for very small p- values. See Press et al. (1986) for a discussion of the approximations used in numerical applications. Those work well here.
For n=100 and an F(1,98) for Ti, the Bonferroni 5 percent critical value for Tmax is the (l-0.05/100)-quantile of an F(1,98)≈11.5, which lies between the 5 and 10 percent critical values of 12.4 and 10.7 found by Abraham and Chuang.
Ignoring the effect of lagged dependent variables and assuming normality.
Judging from some persistent errors in the cross references there seemed to exist a disturbing lack of communication between different strands in literature that analyze comparable economic time series. Johansen and Juselius (1990) referred to Box and Tiao (1977) as Box and Tiao (1981). Tsay and Tiao (1990) referred to Phillips and Durlauf (1986) as Phillips and Durlauf (1985).
Fountis and Dickey (1989) gave asymptotic approximations of some distributions of the eigenvalues, see also chapter 6.
Koopmans (1974, p. 273) defined this concept for spectral smoothers.
Johnson and Geisser (1983) used p for the number of regressors, k for the size of the subset, S for the moment matrix of the regressors, Vi for the submatrix of H (in 3.6) and a2 for the residual sum of squares. We use k, m, W, Hi and s2.
It is also called standardized residual, see e.g. Pfaffenberger and Dielman (1991).
It is also called Studentized deleted residual (op cit.).
For models with lagged dependent variables, this F-distribution is not exact, but merely an approximation in finite samples, but it can be expected to outperform asymptotic x2-approximations in retaining nominal size, see Kiviet (1985) for some Monte Carlo evidence.
Johnson and Geisser (1985) used p for the number of equations (standard multivariate notation in Anderson (1984)), q for the number of regressors, k for the number of observations of interest and N for the total number of observations. We use n, k, m, and T. For n=1 we are back in appendix A3.1, see also remark A3.10 below.
Anderson’s notation (1984, Chapter 8, appendix C) corresponds to ours as follows. He used p, N, q, n, and m, where we use n, T, k+m, T-k-m (the degrees of freedom parameter of the Lawley-Hotelling Statistic), and m. For models with lagged dependent variables, this Lawley-Hotelling distribution is not exact, but merely an approximation in finite samples, but it can be expected to outperform asymptotic x2-approximations in retaining nominal size. A small Monte Carlo experiment confirmed this expectation.
For n=1 or m=1 its distribution under standard regression assumptions is F(nm, T-k-max(n,m)). We prefer to call it a multivariate Chow test in order to avoid confusion with the generalized Chow test, which considers more than two subsamples. In the statistics literature the word generalized is often used as a synonym for multivariate, e.g. in generalized variance.
For the single equation case n=1 it is also known as the Kullback-Rosenblatt F-ratio statistic, see e.g. Cantrell et al. (1991).
Anderson’s notation should be translated as follows. For the beta version of PFi in (A3.2.12):\(g_{11}=T^{-1}A_{(i)}\), \(g_{11}+h_{11}=T^{-1}\), \(n=T-k-m\)=mm and for the beta version of CHi in (A3.2.13):\(g_{11}=T^{-1}(A_{(i)}+A_i),g_{11}+h_{11}=T^{-1}A,n=T-2k, m=k\).
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© 1994 Springer-Verlag Berlin Heidelberg
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Ooms, M. (1994). Data Analysis by Vector Autoregression. In: Empirical Vector Autoregressive Modeling. Lecture Notes in Economics and Mathematical Systems, vol 407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48792-7_3
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