Econometric take home APPS_Part_8

# Econometric take home APPS_Part_8 - x y 1 0 3 We first find...

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31 3. We first find the joint distribution of the observed variables. * 0 x y x u α β 1 0 = + ε 1 0 1 so [ y , x ] have a joint normal distribution with mean vector * * 0 0 y E x μ α β 1 0 α +βμ = + 0 = 1 0 1 μ∗ and covariance matrix 2 * 2 2 * * 2 2 * * 0 0 0 0 0 0 u u y Var x 2 2 2 ε ε 2 2 σ β 1 β 1 0 β σ + σ βσ ⎤ ⎢ = σ 1 0 = ⎥ ⎢ 1 0 1 βσ σ + σ σ 0 1 , The probability limit of the slope in the linear regression of y on x is, as usual, plim b = Cov[ y , x ]/Var[ x ] = β /(1 + σ u 2 / σ * 2 ) < β . The probability limit of the intercept is plim a = E [ y ] - (plim b ) E [ x ] = α + βμ * - βμ * /(1 + σ u 2 / σ * 2 ) = α + β [ μ * σ u / ( σ * 2 + σ u 2 )] > α (assuming β > 0). If x is regressed on y instead, the slope will estimate plim[ b ] = Cov[ y , x ]/Var[ y ] = βσ * 2 /( β 2 σ * 2 + σ ε 2 ). Then,plim[1/ b ] = β + σ ε 2 / β 2 σ * 2 > β . Therefore, b and b will bracket the true parameter (at least in their probability limits). Unfortunately, without more information about σ u 2 , we have no idea how wide this bracket is. Of course, if the sample is large and the estimated bracket is narrow, the results will be strongly suggestive. 4. In the regression of y on x and d , if d and x are independent, we can invoke the familiar result for least squares regression. The results are the same as those obtained by two simple regressions. It is instructive to verify this.

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