CHAPTER 8
Consider the population regression of log earnings [Yi, where Yi = ln(Earningsi)] against two binary variables: whether a worker is married (D1i, where D1i=1 if the ith person is married) and the worker's gender (D2i, where D2i=1 if the ith person is female), and the product of the two binary variables Yi = β0 + β1D1i + β2D2i + β3(D1i×D2i) + ui. The interaction term A. allows the population effect on log earnings of being married to depend on gender B. does not make sense since it could be zero for married males C. cannot be estimated without the presence of a continuous variable D. indicates the effect of being married on log earnings
A
For the polynomial regression model, A. the techniques for estimation and inference developed for multiple regression can be applied. B. the critical values from the normal distribution have to be changed to 1.962, 1.963, etc. C. you can still use OLS estimation techniques, but the t−statistics do not have an asymptotic normal distribution. D. you need new estimation techniques since the OLS assumptions do not apply any longer.
A
In the log−log model, the slope coefficient indicates A. the elasticity of Y with respect to X. B. ΔYΔX×YX. C. ΔY/ ΔX. D. the effect that a unit change in X has on Y.
A
Including an interaction term between two independent variables, X1 and X2, allows for the following except: A. the interaction term coefficient is the effect of a unit increase in X1 × X2. B. the interaction term lets the effect on Y of a change in X1 depend on the value of X2. C. the interaction term lets the effect on Y of a change in X2 depend on the value of X1. D. the interaction term coefficient is the effect of a unit increase in X1 and X2 above and beyond the sum of the individual effects of a unit increase in the two variables alone.
A
The binary variable interaction regression A. allows the effect of changing one of the binary independent variables to depend on the value of the other binary variable. B. cannot be used with logarithmic regression functions because ln(0) is not defined. C. can only be applied when there are two binary variables, but not three or more. D. is the same as testing for differences in means.
A
The exponential function A. is the inverse of the natural logarithm function. B. can be written as exp(e^x). C. is ex, where e is 3.1415?. D. does not play an important role in modeling nonlinear regression functions in econometrics.
A
The following interactions between binary and continuous variables are possible, with the exception of A. Yi = (β0 + Di) + β1Xi + ui. B. Yi = β0 + β1Xi + β2Di + β3(Xi × Di) + ui. C. Yi = β0 + β1Xi + β2Di + ui. D. Yi = β0 + β1Xi + β2(Xi × Di) + ui.
A
The interpretation of the slope coefficient in the model ln(Yi) = β0 + β1Xi + ui is as follows: A. a change in X by one unit is associated with a 100 β1 % change in Y. B. a 1% change in X is associated with a change in Y of 0.01 β1. C. a 1% change in X is associated with a β1 % change in Y. D. a change in X by one unit is associated with a β1 change in Y.
A
You have estimated a linear regression model relating Y to X. Your professor says, "I think that the relationship between Y and X is nonlinear." How would you test the adequacy of your linear regression? (Check all that apply) A. Compare the fit between of linear regression to the non-linear regression model. B. There is evidence in favor of a non-linear relationship if there is zero correlation between the dependent and independent variable. C. If adding a quadratic term, you could test the hypothesis that the estimated coefficient of the quadratic term is significantly different from zero. D. All of the above.
A & C
An example of the interaction term between two independent, continuous variables is A. Yi = β0 + β1Xi + β2Di + β3(Xi × Di) + ui. B. Yi = β0 + β1X1i + β2X2i + β3(X1i × X2i) + ui. C. Yi = β0 + β1X1i + β2X2i + ui. D. Yi = β0 + β1D1i + β2D2i + β3 (D1i × D2i) + ui.
B
Assume that you had estimated the following quadratic regression model TestScore = 607.3 + 3.85 Income − 0.0423 Income2. If income increased from 10 to 11 ($10,000 to $11,000), then the predicted effect on test scores would be A. 3.85 B. 2.96 C. 3.85−0.0423 D. Cannot be calculated because the function is non-linear
B
Consider the following least squares specification between test scores and the student−teacher ratio: TestScore = 557.8 + 36.42 ln (Income). According to this equation, a 1% increase income is associated with an increase in test scores of A. 36.42 points B. 0.36 points C. 557.8 points D. cannot be determined from the information given here
B
Consider the polynomial regression model of degree Yi = β0 + β1Xi + β2X2i+ ...+ βrXri + ui. According to the null hypothesis that the regression is linear and the alternative that is a polynomial of degree r corresponds to A. H0: βr = 0 vs. βr ≠ 0 B. H0:β2= 0, β3 = 0 ..., βr = 0, vs. H1: at least one βj ≠ 0, j = 2, ..., r C. H0:βr= 0 vs. β1 ≠ 0 D. H0:β3= 0, ..., βr = 0, vs. H1: all βj ≠ 0, j = 3, ..., r
B
In nonlinear models, the expected change in the dependent variable for a change in one of the explanatory variables is given by A. ΔY = f(X1 + X1, X2,..., Xk)− f(X1, X2,...Xk). B. ΔY = f(X1 + ΔX1, X2,..., Xk)− f(X1, X2,...Xk). C. ΔY = f(X1 + X1, X2,... Xk). D. ΔY = f(X1 + ΔX1, X2 + ΔX2,..., Xk+ ΔXk)− f(X1, X2,...Xk).
B
In the model Yi = β0 + β1X1 + β2X2 + β3(X1 × X2) + ui, the expected effect ΔY/ΔX1 is A. β1 + β3X1. B. β1 + β3X2. C. β1 + β3. D. β1.
B
In the regression model Yi = β0 + β1Xi + β2Di + β3(Xi × Di) + ui, where X is a continuous variable and D is a binary variable, β3 A. indicates the slope of the regression when D=1. B. indicates the difference in the slopes of the two regressions. C. has a standard error that is not normally distributed even in large samples since D is not a normally distributed variable. D. has no meaning since (Xi × Di) = 0 when Di = 0.
B
The interpretation of the slope coefficient in the model ln(Yi) = β0 + β1 ln(Xi)+ ui is as follows: A. a change in X by one unit is associated with a β1 change in Y. B. a 1% change in X is associated with a β1 % change in Y. C. a 1% change in X is associated with a change in Y of 0.01 β1. D. a change in X by one unit is associated with a 100 β1 % change in Y.
B
A nonlinear function A. can be adequately described by a straight line between the dependent variable and one of the explanatory variables. B. makes little sense, because variables in the real world are related linearly. C. is a function with a slope that is not constant. D. is a concept that only applies to the case of a single or two explanatory variables since you cannot draw a line in four dimensions.
C
An example of a quadratic regression model is A. Yi = β0 + β1ln(X) + ui. B. Yi = β0 + β1X + β2Y2 + ui. C. Yi = β0 + β1X + β2X2 + ui. D. Y2i= β0 + β1X +
C
A "Cobb-Douglas" production function relates production (Q) to factors of production, capital (K), labor (L), raw materials (M), and an error term u using the equation Q=λKβ1Lβ2Mβ3eu, where λ, β1, β2, and β3 are production parameters. Suppose that you have data on production and the factors of production from a random sample of firms with the same Cobb-Douglas production function. Which of the following regression functions provides the most useful transformation to estimate the model? A. A linear regression function. B. An exponential regression function. C. A logarithmic regression function. D. A quadratic regression function.
C
In the case of regression with interactions, the coefficient of a binary variable should be interpreted as follows: A. first set all explanatory variables to one, with the exception of the binary variables. Then allow for each of the binary variables to take on the value of one sequentially. The resulting predicted value indicates the effect of the binary variable. B. for the case of interacted regressors, the binary variable coefficient represents the various intercepts for the case when the binary variable equals one. C. first compute the expected values of Y for each possible case described by the set of binary variables. Next compare these expected values. Each coefficient can then be expressed either as an expected value or as the difference between two or more expected values. D. there are really problems in interpreting these, since the ln(0) is not defined.
C
In the regression model Yi = β0 + β1Xi + β2Di + β3(Xi × Di) + ui, where X is a continuous variable and D is a binary variable, to test that the two regressions are identical, you must use the A. t−statistic separately for β2 = 0, β2 = 0. B. t−statistic separately for β3 = 0. C. F−statistic for the joint hypothesis that β2 = 0, β3= 0. C D. F−statistic for the joint hypothesis that β0 = 0, β1 = 0.
C
In the regression model Yi = β0 + β1Xi + β2Di + β3(Xi × Di) + ui, where X is a continuous variable and D is a binary variable, β2 A. is usually positive. B. indicates the difference in the slopes of the two regressions. C. indicates the difference in the intercepts of the two regressions. D. is the difference in means in Y between the two categories.
C
The following are properties of the logarithm function with the exception of A. ln(xa) = a ln(x). B. ln(1/ x) = −ln(x). C. ln(a + x) = ln(a) + ln(x). D. ln(ax)= ln(a) + ln(x).
C
To decide whether Yi = β0 + β1X + ui or ln(Yi) = β0 + β1X + ui fits the data better, you cannot consult the regression R2 because A. the slope no longer indicates the effect of a unit change of X on Y in the log−linear model. B. the regression R2 can be greater than one in the second model. C. the TSS are not measured in the same units between the two models. D. ln(Y) may be negative for 0<Y<1.
C
A polynomial regression model is specified as: A. Yi = β0 + β1X1i + β2X2 + β3 (X1i × X2i) + ui. B. Yi = β0 + β1Xi + β21Xi + ••• + βr1Xi + ui. C. Yi = β0 + β1Xi + β2Y2i+ ••• + βrYri + ui. D. Yi = β0 + β1Xi + β2X2i+ ••• + βrXri +
D
The best way to interpret polynomial regressions is to A. take a derivative of Y with respect to the relevant X. B. look at the t−statistics for the relevant coefficients. C. analyze the standard error of estimated effect. D. plot the estimated regression function and to calculate the estimated effect on Y associated with a change in X for one or more values of X.
D
The interpretation of the slope coefficient in the model Yi = β0 + β1 ln(Xi) + ui is as follows: A. a 1% change in X is associated with a β1 % change in Y. B. a change in X by one unit is associated with a β1 100% change in Y. C. a change in X by one unit is associated with a β1 change in Y. D. a 1% change in X is associated with a change in Y of 0.01 β1.
D
To test whether or not the population regression function is linear rather than a polynomial of order r, A. check whether the regression R2 for the polynomial regression is higher than that of the linear regression. B. look at the pattern of the coefficients: if they change from positive to negative to positive, etc., then the polynomial regression should be used. C. compare the TSS from both regressions. D. use the test of (r−1) restrictions using the F−statistic.
D