非定常ベクトル時系列分析

我们首先使用Dickey and Fuller(DF)或增广DF(ADF)检验来确定货币、收入、GNP平减指数和利率的单变量ARMA模型。对每个选定的ARMA回归进行两个诊断性检验。一种是残差DF检验，另一种是残差的MA单位根检验。在ARMA模型选择后，用Johansen极大似然法估计协整关系下的VAR回归。在这种估计中，每个变量的滞后长度被认为是不同的。结果发现，收入带来的是钱，而不是相反。在VAR中使用不同的滞后长度，但对所有变量保持相同的滞后长度，对因果关系进行进一步的分析。再次发现了收入与货币之间的因果关系，检验了Oritani(1979)使用的较短样本期间的因果关系。在那里，货币被发现是非固定的，但收入是固定的。格兰杰检验被修改为VAR，它既包括平稳变量，也包括非平稳变量。格兰杰检验的结果是收入与货币的因果关系，而不是货币与收入的因果关系。以下是对西姆斯(1972)使用的过滤器的评论。

We firstly specify the uni-variate ARMA models of the money, income, GNP deflator, and the rate of interest using the Dickey and Fuller (DF) or the augmented DF (ADF) tests. Two diagnostic tests are applied to each selected ARMA regression. One is the residual DF test, and the other is the MA unit root test of residuals. After the ARMA model selection, the VAR regression is estimated with co-integrated relation using Johansen's maximum likelihood method. In this estimation, the lag length of each variable is taken to be different. It is found out that the income is causing money but not the opposite. Further analyses of the causality are performed using various lag lengths in VAR but keeping the same lag length for all variables. The income to money causality is found again.The causality is examined for the shorter sample periods which are used by Oritani (1979). There, the money is found to be non-stationary but the income is stationary. The Granger test is modified to the VAR which includes both stationary and non-stationary variables. The Granger test resulted in the income to money causality, not in the money to income causality. Comments follow on the filter used by Sims (1972).