VIEWS: 10 PAGES: 11 POSTED ON: 7/6/2012
Statistical Properties of OLS Estimators To derive statistical properties of OLS estimators 0 , 1 let’s first summarize all the assumptions that we have made so far Assumption SLR1: Our empirical model is linear in parameters yi 0 1 xi ui Assumption SLR2: We have a random sample of n observations on y and x: {( xi , yi ) : i 1,..., n} Assumption SLR3: Values of x are not all the same, xi and x j such that xi x j Assumption SLR4: The expected value of the error term given x is zero, E(ui|x)=0 b1 is unbiased We can write OLS slope estimator as n n ( x x ) y ( x x )( i i i 0 1 xi ui ) b1 i 1 n i 1 SSTx ( xi x ) 2 i 1 SST 0 x n n n 0 ( xi x ) 1 ( xi x ) xi ( xi x )ui i 1 i 1 i 1 SSTx n 1 (1 / SSTx ) ( xi x )ui E (b1 | x) 1 0 1 i 1 Which means that b1 is unbiased, conditional on x. Note that E (b1 ) E x E (b1 | x) E x 1 1 b0 is unbiased Since b0 y b1 x ( 0 1 x u ) b1 x 0 ( 1 b1 ) x u E (b0 | x) E ( 0 x ( 1 b1 ) u | x) 0 Finally, as above: E (b0 ) E x E (b0 | x) E x 0 0 And it follows that OLS intercept estimator is an unbiased estimator of population intercept. Variance of b0 and b1 To derive the variance of b0 and b1 we need to make an additional assumption: Assumption SLR5: The error u has the same variance given any value of x: V u | x 2 This is known as homoskedasticity or “constant variance” assumption. Note: This assumption is NOT needed to show unbiasedness of OLS estimators b0 and b1! Variance of b0 and b1 2 Because 2 V u | x E (u 2 | x) E (u | x) 0 E (u 2 | x) 2 E (u 2 ) E x E (u 2 | x) 2 V (u ) E (u 2 ) E (u ) 2 E (u 2 ) 2 Hence, σ2 is often called the error variance or the disturbance variance. Also, σ is the standard deviation of the error. Variance of b0 and b1 Assumptions 4 and 5 are also written in the following way: E (u | x) 0 E ( y | x) 0 1 x 2 V (u | x) V ( y | x) 2 The conditional expectation of y given x is linear The conditional variance of y given x is constant When V (u | x) 2 and it depends on x, the error term is said to be heteroskedastic (HS) or non-constant variance. Because V (u | x) V ( y | x) HS is present whenever , V(u|x) and hence V(y|x) are a function of x. Variance of b0 and b1 Under assumptions SLR.1-5 2 2 V (b1 | x) n (x x) 2 SSTx i i 1 and n x 2 i n x 2 2 i 1 2 i V (b0 | x) n i 1 n SSTx n ( xi x ) 2 i 1 Proof: HW and/or quiz Analyzing the formulas Let’s look at the formula for V(b1): 2 2 V (b1 | x) n (x x) 2 SSTx i i 1 It depends on: σ2 – the larger is the variation in the unobserved factors, the harder it is to pinpoint the line SSTx – the larger variation in the regressor, the easier it is to fit the line. As the sample size increases, so will SSTx= n =i 1 ( xi x ) 2 unless the new values of x are equal to x Problem: σ is unknown… Since σ2 is an unknown parameter, we can’t calculate V(b0) and V(b1). We need to estimate σ2 and then use that estimate in the variance formulas. The formula used to estimate σ2 is: n 1 1 s ui (n 2) SSR ˆ 2 2 (n 2) i 1 We divide by (n-2) to make sure the estimator is unbiased. Reason: there are (n-2) degrees of freedom in OLS n n residuals, since u 0, ˆ i 1 i u x 0 i 1 ˆ i i Using s2 It can be shown that s2 is an unbiased estimator of σ2 (HW …). In the literature s s is called the standard error of 2 the regression (model standard error - MSE). Now, we have an estimate of σ2, which we can use to estimate the variance and the standard errors of OLS estimators: 2 s EstV (b1 ) SSTx s2 se(b1 ) s / SSTx SSTx Regression through the Origin Sometimes we need to impose 0 0 : if x=0 => y=0 In this case the econometric model is simply: yi 1 xi ui We can use the OLS approach to find b1 from: n min b1 ( yi b1 xi ) 2 i 1 n The F.O.C. gives 2 ( y b x )x 0 solution is: i 1 i n 1 i i and the x y i i b1 i 1 n xi2 i 1