(a) Consider the classical linear regression model: y = Xβ + u under the Gauss Markov assumptions, such that E uu = σ 2 I . Consider the estimator for σ 2 given by s2∗ = e e/(T − 10) where e is the OLS residual vector: is this an unbiased estimator? If not, how can we correct it to get an unbiased estimator of σ 2 .
Question
(a) Consider the classical linear regression model: y = Xβ + u under the Gauss Markov assumptions, such that E uu = σ 2 I . Consider the estimator for σ 2 given by s2∗ = e e/(T − 10) where e is the OLS residual vector: is this an unbiased estimator? If not, how can we correct it to get an unbiased estimator of σ 2 .
Solution
(a) The estimator for σ^2 given by s2∗ = e'e/(T − 10) is not an unbiased estimator. This is because the denominator (T - 10) does not correctly account for the degrees of freedom in the model.
In a classical linear regression model with k predictors, the unbiased estimator for σ^2 is given by s^2 = e'e/(T - k), where e is the vector of residuals from the ordinary least squares (OLS) regression, T is the number of observations, and k is the number of predictors (including the intercept).
The denominator (T - k) represents the degrees of freedom in the model, which is the number of observations minus the number of estimated parameters. By using (T - 10) in the denominator, the estimator s2∗ incorrectly assumes that there are 10 estimated parameters in the model, regardless of the actual number of predictors.
To correct this and obtain an unbiased estimator of σ^2, we should replace 10 with k in the denominator, giving the corrected estimator s^2 = e'e/(T - k).
Similar Questions
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