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By Marno Verbeek

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ASYMPTOTIC PROPERTIES OF THE OLS ESTIMATOR 35 √ √ we multiply by N and consider the asymptotic distribution √ of N (βˆ − β), this will usually be a non-degenerate normal distribution. In that case N is referred to as the rate of convergence and it is sometimes said that the corresponding estimator is rootN-consistent. In later chapters we shall see a few cases where the rate of convergence differs from root N. 74) where → means ‘is asymptotically distributed as’. Thus, the OLS estimator b is asymp−1 totically normally distributed with variance–covariance matrix σ 2 xx .

E{yi |xi } = xi β. 30). 2 Properties of the OLS Estimator Under assumptions (A1)–(A4), the OLS estimator b for β has several desirable properties. First of all, it is unbiased. This means that, in repeated sampling, we can expect that our estimator is on average equal to the true value β. We formulate this as E{b} = β. It is instructive to see the proof: E{b} = E{(X X)−1 X y} = E{β + (X X)−1 X ε} = β + E{(X X)−1 X ε} = β. The latter step here is essential and it follows from E{(X X)−1 X ε} = E{(X X)−1 X }E{ε} = 0, because X and ε are independent and E{ε} = 0.

When the error terms are not homoskedastic or exhibit autocorrelation, the standard error of the OLS estimator bk will have to be computed in a different way (see Chapter 4). So far, we made no assumption about the shape of the distribution of the error terms εi , except that they were mutually uncorrelated, independent of X, had zero mean and a constant variance. 6 In this case the uncorrelatedness of (A4) is equivalent to independence of all error terms. The precise assumption is as follows (A5) ε ∼ N(0, σ 2 IN ), saying that ε has an N-variate normal distribution with mean vector 0 and covariance matrix σ 2 IN .

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A Guide to Modern Econometrics (2nd Edition) by Marno Verbeek


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