Consider the following nonlinear regression model: 𝑦 𝑖 = 𝛼 + 𝑥 𝑖 𝛽 + 𝜀 𝑖 , Assume i.i.d. data and 𝔼 [ 𝜀 𝑖 | 𝑥 𝑖 ] = 0 . To estimate 𝛼 and 𝛽 by GMM, we use the two theoretical moment conditions 𝔼 [ 𝑦 𝑖 − 𝛼 − 𝑥 𝑖 𝛽 ] = 0 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝑥 𝑖 𝛽 ) 𝑥 𝑖 ] = 0 To compute the variance of the GMM estimator we need the matrices 𝛤 0 and 𝛷 0 . 单项选择题

A

The matrix 𝛷 0 is: 𝛷 0 = [ 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝑥 𝑖 𝛽 ) 2 ] 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝑥 𝑖 𝛽 ) 2 𝑥 𝑖 ] 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝑥 𝑖 𝛽 ) 2 𝑥 𝑖 ] 𝔼 [ ( 𝑦 𝑖 − 𝛼 − 𝑥 𝑖 𝛽 ) 2 𝑥 𝑖 2 ] ] .

B

The matrix 𝛷 0 is: 𝛷 0 = 𝔼 [ 𝛼 𝑥 𝑖 𝛽 𝑥 𝑖 𝛽 𝑥 𝑖 𝛽 log ( 𝑥 𝑖 ) ] .

C

The matrix 𝛷 0 is: 𝛷 0 = 𝔼 [ 1 𝑥 𝑖 𝛽 log ( 𝑥 𝑖 ) 𝑥 𝑖 𝑥 𝑖 𝛽 + 1 log ( 𝑥 𝑖 ) ] .

D

There is not enough information to compute the matrix 𝛷 0 .

E

The matrix 𝛷 0 is: 𝛷 0 = 𝔼 [ − 1 − 𝑥 𝑖 𝛽 log ( 𝑥 𝑖 ) − 𝑥 𝑖 − 𝑥 𝑖 𝛽 + 1 log ( 𝑥 𝑖 ) ] .

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