If the disturbances were observable an unbiased

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(PRF), the corresponding disturbances are still unobservable. If the disturbances were observable, an unbiased estimator of 2 would be obtained as (Erlat, 1992, pp.32-33): 2 2 1 1 T t t u s T However as it has emphasized above, t u ’s are unobservable and we need sample counterparts for them, which are called residuals ( ˆ t u ). Thus, we will substitute ˆ t u ’s for t u ’s and we will calculate the variance of these ˆ t u ’s. However, this is a bit problematic. Note th at ˆ t u ’s are OLS residuals . Hence, they must obey the OLS estimator rules of: (1) 1 ˆ 0 T t t u , and (2) 1 ˆ 0 T t t t u X Hence, to call a series of observations as OLS residuals, these two conditions must be met. Now, the question here is: how many of ˆ t u ’s are random or how many ˆ t u are free to vary ? This is an important question since in statistics, the variance is calculated by dividing the sum of squared deviations from mean to the number of observations that are free to vary . To digest this crucial issue let us give an example. Suppose we have a sample size of 5 (i.e., T=5):
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 19 t=1 t=2 t=3 t=4 t=5 1 ˆ u may happen randomly 2 ˆ u may happen randomly 3 ˆ u may happen randomly If 4 ˆ u happen randomly, there is no guarantee that 1 ˆ 0 T t t t u X holds. If 5 ˆ u happen randomly, there is no guarantee that 1 ˆ 0 T t t u holds. As can be seen from this example that the number of values of ˆ t u that are free to vary is not 5 but instead 3! In other words, in this example we are free to choose the first three numbers at random, but the fourth and fifth must be chosen so that 1 ˆ 0 T t t t u X and 1 ˆ 0 T t t u conditions hold, thus the number of values of ˆ t u that are free to vary is not 5 but instead 3. We call the number of observations that are free to vary as degrees of freedom 2 . For a regression model with k independent variables and 1 intercept term, we can generalize the degrees of freedom as df=T-k-1. Hence, an unbiased estimator of 2 can be calculated as: 2 2 1 ˆ ˆ ( ) ˆ 1 T t t t u u T k Note that ˆ 0 t u since 1 ˆ 0 T t t u condition of OLS estimation. Hence, the unbiased estimator of 2 can be rewritten as: 2 The concept of degrees of freedom is central to the principle of estimating statistics of populations from samples of them.
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 20 2 2 1 ˆ ˆ 1 T t t u T k where ˆ t u is the OLS residual, and it is given by ˆ ˆ t t t u Y Y .
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