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By William H. Greene

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Collect terms to give fK = (1/λ)[xλ(lnx)K - KfK-1], which completes the proof for the general case. Now, we take the limiting value limλ→0 fi = limλ→0 [xλ(lnx)i - ifi-1]/λ. Use L'Hospital's rule once again. limλ→0 fi = limλ→0 d{[xλ(lnx)i - ifi-1]/dλ}/limλ→0 dλ/dλ. Then, limλ→0 fi = limλ→0 {[xλ(lnx)i+1 - ifi]} Just collect terms, (i+1)limλ→0 fi = limλ→0 [xλ(lnx)i+1] or limλ→0 fi = limλ→0 [xλ(lnx)i+1]/(i+1) = (lnx)i+1/(i+1). 36 Chapter 10 Nonspherical Disturbances - The Generalized Regression Model 1.

Then, the probability limits of the least squares coefficient estimators is −1  x' x / n x' d / n   x' y / n σ ∗2 + σ u2 plim =   1 d' x / n d' d / n  d' y / n  πµ ( −1 πµ1   βσ *2 + γπµ1   β / 1 + σ u2 / σ ∗2  =   π   βπµ1 + γπ   γ )   π − πµ1   βσ *2 + γπµ1     1 + π (µ ) − πµ σ *2 + σ u2   βπµ1 + γπ    β(πσ *2 + π 2 (µ1 ) 2 ) 1  . = 1 2 2 2 2 1 2 2 2 2 2 π(σ * + σ u ) + π (µ )  γ(π(σ * + σ u ) + π (µ ) ) + βπσ u  = 1 π(σ *2 + σ u2 ) 2 1 2 The second expression does reduce to plim c = γ + βπµ1σu2/[π(σ*2 + σu2) - π2(µ1)2], but the upshot is that in the presence of measurement error, the two estimators become an unredeemable hash of the underlying parameters.

472 9 . 722 3 . 472 9 . 8422     42 4. Using the data in the previous exercise, use the Oberhofer-Kmenta method to compute the maximum likelihood estimate of the common coefficient vector. The estimator must be based on maximum likelihood estimators of the two disturbance variances, so they must be recomputed first. 3333. Beginning from this point, we iterate between the estimator of the coefficient vector ∧ ∧ ∧ described above and the two variance estimators sj2 = (1/50)[(y′y)j - 2 β ′(X′y)j + β ′X′X β ].

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