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foundations of econometrics phần 3

đó là một nguyên tắc rất chung chung của dự toán có nhiều ứng dụng trong kinh tế. Các loại cơ bản nhất của mô hình hồi quy đơn giản, mô hình hồi quy tuyến tính, có thể được thể hiện bởi phương trình sau Đối với mỗi hai DGPS và mỗi mô phỏng bộ các dữ liệu N, | 142 Hypothesis Testing in Linear Regression Models they lie in orthogonal subspaces namely the images of PX and MX. Thus even though the numerator and denominator of 4.26 both depend on y this orthogonality implies that they are independent. We therefore conclude that the t statistic 4.26 for P2 0 in the model 4.21 has the t n k distribution. Performing one-tailed and two-tailed tests based on tp2 is almost the same as performing them based on Zp2. We just have to use the t n k distribution instead of the N 0 1 distribution to compute P values or critical values. An interesting property of t statistics is explored in Exercise 14.8. Tests of Several Restrictions Economists frequently want to test more than one linear restriction. Let us suppose that there are r restrictions with r k since there cannot be more equality restrictions than there are parameters in the unrestricted model. As before there will be no loss of generality if we assume that the restrictions take the form 02 0. The alternative hypothesis is the model 4.20 which has been rewritten as Hi y X101 X202 u u N 0 a21 . 4.28 Here X1 is an n ki matrix X2 is an n X k2 matrix 01 is a ki-vector 02 is a k2-vector k k1 k2 and the number of restrictions r k2. Unless r 1 it is no longer possible to use a t test because there will be one t statistic for each element of 02 and we want to compute a single test statistic for all the restrictions at once. It is natural to base a test on a comparison of how well the model fits when the restrictions are imposed with how well it fits when they are not imposed. The null hypothesis is the regression model H0 y X101 u u N 0 a2I 4.29 in which we impose the restriction that 02 0. As we saw in Section 3.8 the restricted model 4.29 must always fit worse than the unrestricted model 4.28 in the sense that the SSR from 4.29 cannot be smaller and will almost always be larger than the SSR from 4.28 . However if the restrictions are true the reduction in SSR from adding X2 to the .