By Schimansky-Geier L., Poeschel T. (eds.)
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Extra resources for Stochastic dynamics
Example text
11 Consider two estimators βˆ1 and βˆ2 of β. The following two statements are equivalent: ∆(βˆ1 , βˆ2 ) R(βˆ1 , β, A) − R(βˆ2 , β, A) = tr{A∆(βˆ1 , βˆ2 )} for all matrices of the type A = aa′ . 47) we get R(βˆ1 , β, A) − R(βˆ2 , β, A) = tr{A∆(βˆ1 , βˆ2 )}. 43 it follows that tr{A∆(βˆ1 , βˆ2 )} ≥ 0 for all matrices A = aa′ ≥ 0 if and only if ∆(βˆ1 , βˆ2 ) ≥ 0. 23) may be viewed as the decomposition of the observation y into a nonstochastic part Xβ, also called the signal, and a stochastic part ǫ, also called the noise (or error), as discussed in Rao (1989).
8) is positive definite if its determinant and the element in the first row and column of H are positive. The determinant of H is T |H| = 2 T t=1 x2t − T 2 x ¯2 n = 2T ≥ 0. , xt = c (some constant). In such a case there is no relationship between x and y in the context of regression analysis. Since Tt=1 (xt − x ¯)2 > 0, therefore |H| > 0. So H is positive definite for any (β0 , β1 ); therefore S(β0 , β1 ) has a global minimum at (b0 , b1 ). 10) and the predicted values are yˆt = b0 + b1 xt (t = 1, .
52) and ⎛ ⎞2 b − β 1⎠ ⎝ 1 σ2 SXX are also independently distributed because b∗0 and b1 are independently distributed. 52), SXX(b1 − β1 )2 T (b∗0 − β0∗ )2 + ∼ χ22 . 53) 20 2. The Simple Linear Regression Model Since SSE ∼ χ2T −2 σ2 and SSE is independently distributed of b∗0 and b1 , so the ratio ∗ 2 T (b∗ 0 −β0 ) σ2 + RSS σ2 SXX(b1 −β1 )2 σ2 /2 /(T − 2) ∼ F2,T −2 . 54), we get T −2 2 Qf RSS where T Qf = T (b0 − β0 )2 + 2 t=1 T xt (b0 − β0 )(b1 − β1 ) + t=1 x2t (b1 − β1 )2 . 56) holds true for all values of β0 and β1 , so the 100(1 − α)% confidence region for β0 and β1 is T −2 2 Qf ≤ F2,T −2;1−α .
Stochastic dynamics by Schimansky-Geier L., Poeschel T. (eds.)
by Brian
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