tailieunhanh - Class Notes in Statistics and Econometrics Part 25

CHAPTER 49 Distributed Lags. In the simplest case of one explanatory variable only, the model is () y t = α + β0 xt + β1 xt−1 + · · · + βN xt−N + εt This can be written in the form 1 1 X = . . . x1 x2 . . . x0 x1 xn−1 ··· ··· . . . ··· x1−N x2−N . xn−N | CHAPTER 49 Distributed Lags In the simplest case of one explanatory variable only the model is yt a Pox p- _xt-i Pn Xt-N t This can be written in the form y Xp e where X 1 1 X1 X2 Xo X1 X1-N X2-N 1 Xn Xn-1 Xn-N Note that X contains presample values. 1051 1052 49. DISTRIBUTED LAGS Two problems lag length often not known and X matrix often highly multicollinear. How to determine lag length Sometimes it is done by the adjusted R2. Mad88 p. 357 says this will lead to too long lags and proposes remedies. Assume we know for sure that lag length is not greater than M. JHG 88 pp. 723-727 recommends the following general-to-specific specification procedure for finding the lag length First run the regression with M lags if the t-test for the parameter of the Mth lag is significant we say the lag length is M. If it is insignificant run the regression with M 1 lags and test again for the last coefficient If the t-test for the parameter of the M 1st coefficient is significant we say the lag length is M 1 etc. The significance level of this test depends on M and on the true lag length. Since we never know the true lag length for sure we will never know the true significance level for sure. The calculation which follows now allows us to compute this significance level under the assumption that the N given by the test is the correct N. Furthermore this calculation only gives us the one-sided significance level the null hypothesis is not that the true lag length is N but that the true lag length is N. Assume the null hypothesis is true . that the true lag length is N. Since we assume we know for sure that the true lag length is M the null hypothesis 49. DISTRIBUTED LAGS 1053 is equivalent to pN 1 pN 2 Pm 0. Now assume that we apply the above procedure and the null hypothesis holds. The significance level of our test is the probability that our procedure rejects the null although the null is true. In other words it is the probability that either the first t-test .