… Variance and the Combination of Least Squares Estimators 297 1989). Asymptotic Normality. For ex-ample, could be the population mean (traditionally called µ) or the popu-lation variance (traditionally called 2). The bias for the estimate ˆp2, in this case 0.0085, is subtracted to give the unbiased estimate pb2 u. Bias Bias If ^ = T(X) is an estimator of , then the bias of ^ is the di erence between its expectation and the ’true’ value: i.e. correct specification of the regression function or the propensity score for consistency. We say that an estimate ϕˆ is consistent if ϕˆ ϕ0 in probability as n →, where ϕ0 is the ’true’ unknown parameter of the distribution of the sample. Omitted variable bias: violation of consistency From the omitted variable bias formula b 1!p 1 + 2 Cov (X i;W i) Var (X i) we can infer the direction of the bias of b 1 that persists in large samples Suppose W i has a positive effect on Y i, then 2 >0 Suppose X i and W … random sample from a Poisson distribution with parameter . 1. Bias and Consistency in Three-way Gravity Models ... intervals in fixed-T panels are not correctly centered at the true point estimates, and cluster-robust variance estimates used to construct standard errors are generally biased as well. This is in contrast to optimality properties such as efficiency which state that the estimator is “best”. When appropriately used, the reduction in variance from using the ratio estimator will o set the presence of bias. 2. bias( ^) = E ( ^) : An estimator T(X) is unbiased for if E T(X) = for all , otherwise it is biased. In the above example, E (T) = so T is unbiased for . Consistency is a relatively weak property and is considered necessary of all reasonable estimators. Example: Suppose X 1;X 2; ;X n is an i.i.d. 5.1.2 Bias and MSE of Ratio Estimators The ratio estimators are biased. • The bias of an estimator is an inverse measure of its average accuracy. j βˆ • Thus, an unbiased estimator for which Bias(ˆ) 0 βj = -- that is, for which E(βˆ j) =βj-- is on average a The bias and variance of the combined estimator can be simply The bias occurs in ratio estimation because E(y=x) 6= E(y)=E(x) (i.e., the expected value of the ratio 6= the ratio of the expected values. As the bias correction does not affect the variance, the bias corrected matching estimators still do not reach the semiparametric efficiency bound with a fixed number of matches. An estimator is consistent if ˆθn →P θ 0 (alternatively, θˆn a.s.→ θ 0) for any θ0 ∈ Θ, where θ0 is the true parameter being estimated. In the more typical case where this distribution is unkown, one may resort to other schemes such as least-squares fitting for the parameter vector b = {bl , ... bK}. We will prove that MLE satisfies (usually) the following two properties called consistency and asymptotic normality. The first way is using the law 2. Theorem 4. (van der Vaart, 1998, Theorem 5.7, p. 45) Let Mn be random functions and M be Estimation and bias 2.2. Relative e ciency: If ^ 1 and ^ 2 are both unbiased estimators of a parameter we say that ^ 1 is relatively more e cient if var(^ 1)

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