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variance - Class Notes for Econometrics Variance of...

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Class Notes for Econometrics Variance of Ordinary Least Squares under the Gauss Markov assumptions Jean Eid Assume that the following hold MLR-1 Linearity in the parameter MLR-2 Independence of the error term u and random sampling MLR-3 No perfect Collinearity MLR-4 Zero conditional mean MLR-5 Homoskedatsticity i.e var ( u i | x ) = σ 2 Mean Squared Error Suppose we have data x 1 , x 2 , x 3 , · · · , x N that we think are coming from a model y i = β 0 + β 1 x i + u i Suppose that MLR1-MLR5 hold. Our goal is to use the information available in the data to make guesses about β 0 , β 1 . One such guess would be the OLS estimator. Another one is the method of moment estimator, a third would be whatever the data is we guess β 0 to be 1 and β 1 to be 5. Ideally, we need some measure that tells us for example whether OLS is better than the third guess we made above. One way is to only pick up the estimators that are unbiased. In the last lecture notes we proved that OLS is unbiassed. However, suppose you have two estimators that are unbiased, how do you pick and choose which one to work with. One other property we would like is to have the estimate going in probability to the true parameter. We write this as plim n →∞ | ˆ β o,n - β 0 | < = 1 Where ˆ β o,n is the OLS estimate based on n observations. What we are trying to say here is that we want an estimator that gives us an estimate that has the following property. As we increase the sample 1
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size the probability that our estimate is the actual true value of the parameter approaches 1. In other words, we are more and more certain that our estimate is the true value in the population. So this is a very important property that we would like to have. We call this property consistency. However, we can prove this property for OLS through the variance and the expectation. This is called mean square error consistent which implies consistency and is denoted by MSE . MSE is defined in the following way MSE = E ( δ ( data ) - θ ) 2 where δ ( data ) is an estimate of θ . In the case above it was ˆ β 0 is an estimate of β 0 . The reason we write the function δ as above is to stress the fact that the estimates functions of the data eg. look at the OLS estimates, they are functions of x 1 , x 2 , x 3 , · · · , x N and y 1 , y 2 , y 3 , · · · , y N . However, to conserve space, I will from now on write δ to mean δ data We say an estimator is consistent when lim n →∞ MSE = 0 Now, look at the definition of MSE, and let μ δ denote the population expected value of δ data . E δ - β 2 = E δ - μ δ + μ δ - θ 2 = E δ - μ δ 2 + μ δ - θ 2 - 2 δ - μ δ μ δ - θ = E δ - μ δ 2 + E μ δ - θ 2 - 2 E δ - μ δ μ δ - θ = E δ - μ δ 2 + μ δ - θ 2 - 2 μ δ - θ E δ - μ δ =
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