By Patrick T. Brandt
A number of Time sequence types introduces researchers and scholars to the various methods to modeling multivariate time sequence information together with simultaneous equations, ARIMA, mistakes correction versions, and vector autoregression. Authors Patrick T. Brandt and John T. Williams specialize in vector autoregression (VAR) versions as a generalization of those different methods and speak about specification, estimation, and inference utilizing those types.
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Extra info for Multiple Time Series Models (Quantitative Applications in the Social Sciences)
Denotes the lag length for the observation ‘ periods before t. A dynamic simultaneous equations system for these variables is Yt = αZt + γ 11 Yt − 1 + γ 12 Zt − 1 + u1t ; ð2:1Þ Zt = θYt + γ 21 Yt − 1 + γ 22 Zt − 1 + u2t ; ð2:2Þ 16 where '11 uit ∼ N 0; '12 '12 '22 : This is a simultaneous system because all the relationships (equations) in the model are necessary to determine the value of at least one of the endogenous variables in the model. 2. The simultaneity in the model results from the fact that each variable depends on the contemporaneous value of the other variables in the model.
The first method includes graphical summaries of the residuals plotted over time for each variable. The second involves plots of the autocorrelation and crosscorrelations of the variables over different time lags. The third method involves using Portmanteau statistics. 2. 29 To assess the possible serial correlation among the residuals, one can first use graphical methods—plotting the residuals over time. The problem with this approach is that it depends on the analyst’s ability to recognize the serial correlation, which may be hard in some cases where the data are noisy, are of high frequency, or the serial correlation pattern is across multiple variables at the same time.
There are two cases to 35 consider: too many lags and too few lags. If one includes too many lags in the VAR model, the resulting estimates will possibly be inefficient, but unbiased—just as in a linear regression model. Thus, hypothesis tests will be unbiased, but inefficient. We will then be likely to fail to reject the null when we really should. This is of minor consequence, because we would then generate the null finding of noncausality. In contrast, consider the case of too few lags. In this case, the VAR estimates will be biased and inefficient, as expected from linear regression results when there are omitted variables.