180 câu trắc nghiệm Kinh tế lượng – Phần 6

Chapter 18: The Theory of Multiple Regression

KTL_001_C18_1: The multiple regression model can be written in matrix form as follows:

○ \(Y = X\beta \)
○ \(Y = X + U\)
○ \(Y = \beta X + U\)
● \(Y = X\beta + U\)

KTL_001_C18_2: One of the properties of the OLS estimator is

○ \(X\hat \beta = {{\rm O}_{k + 1}}\)
○ that the coefficient vector \({\hat \beta }\) has full rank.
● \(X’\left( {Y – X\hat \beta } \right) = {{\rm O}_{k + 1}}\)
○ \({\left( {X’X} \right)^{ – 1}} = X’Y\)

KTL_001_C18_3: \(\hat \beta – \beta \)

○ cannot be calculated since the population parameter is unknown.
● \( = {\left( {X’X} \right)^{ – 1}}X’U \)
○ \( = Y – \hat Y\)
○ \( = \beta + {\left( {X’X} \right)^{ – 1}}X’U\)

KTL_001_C18_4: The formulation \(R\beta = r\) to test a hypotheses

● allows for restrictions involving both multiple regression coefficients and single regression coefficients.
○ is F-distributed in large samples.
○ allows only for restrictions involving multiple regression coefficients.
○ allows for testing linear as well as nonlinear hypotheses.

KTL_001_C18_5: The GLS estimator

○ is always the more efficient estimator when compared to OLS.
● is the OLS estimator of the coefficients in a transformed model, where the errors of the transformed model satisfy the Gauss-Markov conditions.
○ cannot handle binary variables, since some of the transformations require division by one of the regressors.
○ produces identical estimates for the coefficients, but different standard errors.

KTL_001_C18_6: The extended least squares assumptions in the multiple regression model include four assumptions from Chapter 6 (ui has conditional mean zero; (Xi,Yi), i = 1,…, n are i.i.d. draws from their joint distribution; Xi and ui have nonzero finite fourth moments; there is no perfect multicollinearity). In addition, there are two further assumptions, one of which is

○ heteroskedasticity of the error term.
○ serial correlation of the error term.
● The conditional distribution of ui given Xi is normal.
○ invertibility of the matrix of regressors.

KTL_001_C18_7: The OLS estimator for the multiple regression model in matrix form is

● \({\left( {X’X} \right)^{ – 1}}X’Y\)
○ \(X{\left( {X’X} \right)^{ – 1}}X’ – {P_X}\)
○ \({\left( {X’X} \right)^{ – 1}}X’U\)
○ \({\left( {X'{\Omega ^{ – 1}}X} \right)^{ – 1}}\left( {X'{\Omega ^{ – 1}}Y} \right)\)

KTL_001_C18_8: To prove that the OLS estimator is BLUE requires the following assumption

○ (Xi,Yi) i = 1, …, n are i.i.d. draws from their joint distribution
○ Xi and ui have nonzero finite fourth moments
○ The conditional distribution of ui given Xi is normal
● None of the above

KTL_001_C18_9: The TSLS estimator is

○ \({\left( {X’X} \right)^{ – 1}}X’Y\)
● \({\left[ {\left( {X’Z} \right){{\left( {Z’Z} \right)}^{ – 1}}\left( {Z’X} \right)} \right]^{ – 1}}X’Z{\left( {Z’Z} \right)^{ – 1}}Z’Y\)
○ \({\left( {X'{\Omega ^{ – 1}}X} \right)^{ – 1}}\left( {X'{\Omega ^{ – 1}}Y} \right)\)
○ \({\left( {X'{P_Z}} \right)^{ – 1}}{P_Z}Y\)

KTL_001_C18_10: The homoskedasticity-only F-statistic is

● \(\frac{{{{\left( {R\hat \beta – r} \right)}^\prime }{{\left[ {R{{\left( {X’X} \right)}^{ – 1}}R} \right]}^{ – 1}}\left( {R\hat \beta – r} \right)/q}}{{S_{\hat u}^2}}\)
○ \(\frac{{{{\left( {R\hat \beta – r} \right)}^\prime }{{\left[ {R{{\left( {X’X} \right)}^{ – 1}}R} \right]}^{ – 1}}\left( {R\hat \beta – r} \right)}}{{S_{\hat u}^2}}\)
○ \(\frac{{{{\left( {R\hat \beta – r} \right)}^\prime }{{\left[ {R{{\hat \Sigma }_{\hat \beta }}R} \right]}^{ – 1}}\left( {R\hat \beta – r} \right)}}{q}\)
○ \(\frac{{\hat U'{P_Z}\hat U}}{{\hat U'{M_Z}\hat U}}\)