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Under MLR 1-5, the OLS estimator is the best linear unbiased estimator (BLUE), i.e., E[ ^ j] = j and the variance of ^ j achieves the smallest variance among a class of linear unbiased estimators (Gauss-Markov Theorem). T is said to be an unbiased estimator of if and only if E (T) = for all in the parameter space. two. Properties of Estimators BS2 Statistical Inference, Lecture 2 Michaelmas Term 2004 Steffen Lauritzen, University of Oxford; October 15, 2004 1. The Nature of the Estimation Problem. A New Way of Looking at OLS Estimators You know the OLS formula in matrix form βˆ = (X0X)−1 X0Y. Variances of OLS Estimators In these formulas σ2 is variance of population disturbances u i: The degrees of freedom are now ( n − 3) because we must first estimate the coefficients, which consume 3 df. 8 Asymptotic Properties of the OLS Estimator Assuming OLS1, OLS2, OLS3d, OLS4a or OLS4b, and OLS5 the follow-ing properties can be established for large samples. Not even predeterminedness is required. the cointegrating vector. Notation and setup X denotes sample space, typically either finite or countable, or an open subset of Rk. OLS is no longer the best linear unbiased estimator, and, in large sample, OLS does no If we assume MLR 6 in addition to MLR 1-5, the normality of U Parametric Estimation Properties 5 De nition 2 (Unbiased Estimator) Consider a statistical model. of (i) does not cause inconsistent (or biased) estimators. The X matrix is thus X = x 11 x 21 x 12 x 22 x 13 x 23 (20) This note derives the Ordinary Least Squares (OLS) coefficient estimators for the ... Properties of an Estimator. On the other hand, OLS estimators are no longer e¢ cient, in the sense that they no longer have the smallest possible variance. 1. OLS is consistent under much weaker conditions that are required for unbiasedness or asymptotic normality. This NLS estimator corresponds to an unconstrained version of Davidson, Hendry, Srba, and Yeo's (1978) estimator.3 In this section, it is shown that the NLS estimator is consistent and converges at the same rate as the OLS estimator. estimator (BLUE) of the coe cients is given by the least-squares estimator BLUE estimator Linear: It is a linear function of a random variable Unbiased: The average or expected value of ^ 2 = 2 E cient: It has minimium variance among all other estimators However, not all ten classical assumptions have to hold for the OLS estimator to be B, L or U. Consider the case of a regression with 2 variables and 3 observations. ie OLS estimates are unbiased . critical properties. An estimator possesses . Assumptions A.0 - A.6 in the course notes guarantee that OLS estimators can be obtained, and posses certain desired properties. The behavior of least squares estimators of the parameters describing the short More generally we say Tis an unbiased estimator of h( ) if and only if E (T) = h( ) for all in the parameter space. Under MLR 1-4, the OLS estimator is unbiased estimator. , the OLS estimate of the slope will be equal to the true (unknown) value . However, social … Assumption A.2 There is some variation in the regressor in the sample , is necessary to be able to obtain OLS estimators. Ordinary Least Squares (OLS) Estimation of the Simple CLRM. In particular, Gauss-Markov theorem does no longer hold, i.e. 8 2 Linear Regression Models, OLS, Assumptions and Properties 2.2.5 Data generation It is mathematically convenient to assume x i is nonstochastic, like in an agricultural experiment where y i is yield and x i is the fertilizer and water applied. Let T be a statistic. 1. 7/33 Properties of OLS Estimators An estimator is a. function only of the given sample data CONSISTENCY OF OLS, PROPERTIES OF CONVERGENCE Though this result was referred to often in class, and perhaps even proved at some point, a student has pointed out that it does not appear in the notes. There is a useful way to restate this that allows us to make a clear connection to the WLLN and the CLT. We have observed data x ∈ X which are assumed to be a

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