(+ R4 FINI-S)M+L +RO+ R ISOFI( LS 4.4 Inference under a normality assumption Assume e n Ngala Then bX C BPEXA XEX N(B.G2HXAXXHXA C NB,OXA Recall that A~N(E,A(021)4) Each element of b X is normally distributed bEIXNN(BK, 02H'XAK Consider the null hy pothesis Ho IBr. C BE The t-test for t his null hy pot hesis is defined by XAk Under the normality assumption NNIF4A Va2BX乐k In addition C Furthermore C EXA X is independent of 匝-KA2CHAPTER 4 FINITE—SAMPLE PROPERTIES OF THE LSE 4 4.4 Inference under a normality assumption (i) t−test Assume ε ∼ N (0, σ 2 I). Then b|X = β + (X ′X) −1 X ′ ε|X ∼ N  β, σ2 (X ′X) −1 X ′X (X ′X) −1  = N  β, σ2 (X ′X) −1  . Recall that Aε ∼ N  0, A  σ 2 I  A ′  . Each element of b|X is normally distributed bk|X ∼ N  βk , σ2 (X ′X) −1 kk  . Consider the null hypothesis H0 : βk = β 0 k . The t−test for this null hypothesis is defined by tk = bk − β 0 k s 2 (X′X) −1 kk . Under the normality assumption, bk − β 0 k σ 2 (X′X) −1 kk ∼ N (0, 1). In addition (n − K) s 2 σ2 = e ′ e σ2 =  ε σ ′ M  ε σ  ∼ χ 2 tr(M) = χ 2 n−K. Furthermore, b − β σ = (X ′X) −1 X ′  ε σ  is independent of (n − K) s 2 σ 2