By Patrick Kent Watson, Sonja Sabita Teelucksingh
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Additional info for A Practical Introduction to Econometric Methods: Classical and Modern by Patrick Kent Watson (2002-01-31)
49 Some Issues in the Application of the General Linear Regression Model We can show the biased nature of this OLS estimator in the following manner – replace yt in this expression by its value: n ∑ x (β x 1t βˆ 1 = 1 1t + β2 x 2 t + u t ) t =1 n ∑x 2 1t t =1 n β1 = ∑x n 2 1t + β2 t =1 n ∑x x + ∑x u 1t 2t 1t t =1 n t t =1 ∑x 2 1t t =1 n = β1 + β ∑ n x1t x 2 t ∑x 1t + t =1 n 2 ∑x u t t =1 n ∑x 2 1t t =1 2 1t t =1 Therefore: n ( ) ∑x x 1t E βˆ 1 = β1 + β 2 2t t =1 n ∑x + 0 ≠ β1 2 1t t =1 implying that E( βˆ 1) differs from β1 by the amount ⎛ ⎜ ⎜β ⎜ 2 ⎜ ⎝ n ∑x x 1t t =1 n ∑x t =1 2 1t 2t ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ which is a measure of the bias.
Error T-Statistic Prob. D. E. 71113 Sum squared resid 6517534. 83 on imports). Furthermore, the R2 and F-statistics leave us in no doubt that this is a good fit. What then do we do? It is very difficult to answer this question at this stage of our study of econometrics but at least one word of warning is in order: the conclusions about the significance of this or that variable under the conditions of multicollinearity shown here are very tenuous and great caution should be exercised in using any of the results shown.
Consider a random vector X whose elements are X1, X2, ... , Xn and define E(Xi) = μi. Define E(X) = μ, then: ⎡ μ1 ⎤ ⎢μ ⎥ 2 E (X) = μ = ⎢ ⎥ ⎢%⎥ ⎢ ⎥ ⎣μ n ⎦ Define var(Xi) = σ 2i , and Cov (Xi, Xj) = σij, i ≠ j. Consider the following “covariance table” showing the moments of second order of the n random variables: X1 X2 ........ Xn X1 σ12 σ12 ............. σ1n X2 σ21 σ22 ........... σ2n Xn σn1 σn2 ......... σn2 The tableau clearly shows, in an ordered way, the variances (along the principal diagonal) and the covariances (the nondiagonal elements).