The assumptions Underlying the CLRM An alternative assumption to 4, which is slightl stronger is that the xs are non-stochastic or fixed in repeated samples a fifth assumption is required if we want to make inferences about the population parameters(the actual a and B)from the sample parameters( a and B) Additional assumption 5. u, is normally distributed 以上为 Gauss假定或古典假定
3-26 The Assumptions Underlying the CLRM • An alternative assumption to 4., which is slightly stronger, is that the xt ’s are non-stochastic or fixed in repeated samples. • A fifth assumption is required if we want to make inferences about the population parameters (the actual and ) from the sample parameters ( and ) • Additional Assumption 5. ut is normally distributed 以上为Gauss假定或古典假定 $ $
3-27 6 Properties of the Ols Estimator If assumptions 1. through 4. hold, then the estimators a and determined by ols are known as Best Linear Unbiased Estimators(BLue) °“ Estimator” and B is an estimator of the true value of c、B ·“ Linear” are linear estimators ·“ Unbiased”- On average, the actual value of thed and be equal to their true values “Best estimator B has minimum variance among the class of linear unbiased estimators The gauss- Markov theorem proves that the ols estimator is best
3-27 6 Properties of the OLS Estimator • If assumptions 1. through 4. hold, then the estimators and determined by OLS are known as Best Linear Unbiased Estimators (BLUE). • “Estimator” - and is an estimator of the true value of 、. • “Linear” - 、 are linear estimators • “Unbiased” - On average, the actual value of the and will be equal to their true values. • “Best” - estimator has minimum variance among the class of linear unbiased estimators. The Gauss-Markov theorem proves that the OLS estimator is best. $ $ $ $ $ $ $ $ $
3-28 Consistency/Unbiasedness/Efficiency Consistent The least squares estimators a and B are consistent. That is, the estimates will converge to their true values as the sample size increases to infinit Unbiased The least squares estimates are unbiased. That is E(Q)= a and e(B)=月 Thus on average the estimated value will be equal to the true values Unbiasedness is a stronger condition than consistency. Efficiency An estimator B of parameter B is said to be efficient if it is unbiased and no other unbiased estimator has a smaller variance
3-28 Consistency/Unbiasedness/Efficiency • Consistent The least squares estimators and are consistent. That is, the estimates will converge to their true values as the sample size increases to infinity. • Unbiased The least squares estimates are unbiased. That is E( )= and E( )= Thus on average the estimated value will be equal to the true values. Unbiasedness is a stronger condition than consistency. • Efficiency An estimator of parameter is said to be efficient if it is unbiased and no other unbiased estimator has a smaller variance. $ $ $ $ $
3-29 7 Precision and standard errors Any set of regression estimates of a and B are specific to the sample used in their estimation What we need is some measure of the reliability or precision of the estimators. The precision of the estimate is given by its standard error. Given assumptions 4 above, then the standard errors can be shown to be given by se(a=s (3.19) 7∑(x1-x)2Vr∑x2-7 sE(B)=S (320) ∑(x,-x) Ix where s is the estimated standard deviation of the residuals
3-29 7 Precision and Standard Errors • Any set of regression estimates of and are specific to the sample used in their estimation. • What we need is some measure of the reliability or precision of the estimators. The precision of the estimate is given by its standard error. Given assumptions 1 - 4 above, then the standard errors can be shown to be given by ( 3.19 ) ( 3.20 ) where s is the estimated standard deviation of the residuals. $ $ − = − = − = − = 2 2 2 2 2 2 2 2 2 1 ( ) 1 ) ˆ ( , ( ) ( ˆ) x Tx s x x SE s T x T x x s T x x x SE s t t t t t t
3-30 Estimating the variance of the Disturbance term The variance of the random variable u, is given by Var(u=EI(u-E(uD12=E(u) We could estimate this using the average of uf.Unfortunately this is not workable since u, is not observable. We can use the sample counterpart to u, which is u,: 2 ∑ But this estimator is a biased estimator of g2 An unbiased estimator of o is given by 回归标准差 (326) T-2
3-30 Estimating the Variance of the Disturbance Term • The variance of the random variable ut is given by Var(ut ) = E[(ut )-E(ut )]2 = E(ut 2 ) • We could estimate this using the average of .Unfortunately this is not workable since ut is not observable. We can use the sample counterpart to ut , which is : But this estimator is a biased estimator of 2 . • An unbiased estimator of is given by 回归标准差 ( 3.26 ) 2 t u = 2 2 ˆ 1 ut T s t u ˆ 2 ˆ 2 − = T u s t