Econometrics

João Valle e Azevedo

Final Exam

Erica Marujo

June 15, 2010 Time for completion: 2h Give your answers in the space provided. Use draft paper to plan your answers before writing them on the exam paper. Unless otherwise stated, use 5% for significance level. Name:_________________________________________________ Number:_________

Group I (9 points, 1.5 for each question) Give a very concise answer to the following questions. Conciseness will be valued, avoid unnecessary details. 1. The acronym BLUE stands for what in Econometrics?

_______________________________________________________________________ 2. In one phrase, describe the meaning of “Contemporaneous exogeneity” in a

multiple linear regression (for time series data) context. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 3. Explain why, in general, you would want to transform series that are not weakly

dependent before using them in a multiple linear regression model for time series data. In which cases is the transformation not needed? _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________

Name:_________________________________________________ Number:_________ 4. Write in matrix form the expression for the variance of the OLS Estimator of the

parameters of a multiple linear regression model for cross-sectional data, assuming the homoskedasticity assumption is not verified. (Assume the variance of the error term is of known form and that the necessary assumptions hold) _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 5. Describe succinctly a very parsimonious (i.e., with few variables) test aimed at

detecting heteroskedasticity in the error term of a multiple linear regression model. Assume all the necessary assumptions hold. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 6. Write a model aimed at testing whether there is any effect on GDP growth of

next quarter of an extraordinary (more than 10%) growth in public investment in the current quarter, controlling for other factors. Describe one limitation of your model. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 2

Name:_________________________________________________ Number:_________ Group II (10 points) 1. Consider the following output of the model which describes grade point

averages (GPA) for college athletes:

Dependent Variable: CUMGPA Method: Least Squares Sample: 1 732 Included observations: 732

C SAT HSPERC TOTHRS CRSGPA R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient

Std. Error

t-Statistic

Prob.

-1.046971 0.000964 -0.006849 0.010177 0.727651

0.483458 0.000205 0.001550 0.000998 0.157487

-2.165588 4.691868 -4.419186 10.19751 4.620393

0.0307 0.0000 0.0000 0.0000 0.0000

0.257227 0.253140 0.855237 531.7503 -921.6870 62.94110 0.000000

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

2.080861 0.989617 2.531932 2.563324 2.544041 2.026332

where is cumulative grade point average (GPA prior to the current semester); is SAT score, measured in points; is graduating percentile in high school class; is total credit hours prior to the semester; and is a weighted average of overall GPA in courses taken by each student. a)

Interpret the coefficient estimates of the model,  and  . What can you say about the overall quality of this model? Be complete in your answer. (1 point)

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Name:_________________________________________________ Number:_________ Now consider the following two outputs: Dependent Variable: CUMGPA Method: Least Squares Sample: 1 732 IF FEMALE=1 Included observations: 180

C SAT HSPERC TOTHRS CRSGPA R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient

Std. Error

t-Statistic

Prob.

-1.226599 0.001835 -0.007056 0.013945 0.477304

1.025719 0.000468 0.003954 0.002260 0.326148

-1.195843 3.917441 -1.784380 6.170078 1.463459

0.2334 0.0001 0.0761 0.0000 0.1451

0.375130 0.360848 0.900643 141.9525 -234.0371 26.26461 0.000000

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

2.268611 1.126549 2.655968 2.744661 2.691929 2.240975

Dependent Variable: CUMGPA Method: Least Squares Sample: 1 732 IF FEMALE=0 Included observations: 552

C SAT HSPERC TOTHRS CRSGPA R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient

Std. Error

t-Statistic

Prob.

-0.826778 0.000661 -0.006438 0.008809 0.754003

0.563308 0.000228 0.001725 0.001121 0.185269

-1.467719 2.897074 -3.731255 7.860264 4.069772

0.1428 0.0039 0.0002 0.0000 0.0001

0.210642 0.204869 0.832541 379.1392 -679.5761 36.49198 0.000000

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

2.019638 0.933655 2.480348 2.519420 2.495615 2.009564

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Name:_________________________________________________ Number:_________ and consider the following model, where female and zero otherwise.

 is a dummy variable equal to 1 if

b)

Using the information from the outputs above, compute estimates for the coefficients of this model ( , , , , , ). (2 points)

c)

Test if the expected cumulative GPA for men is statistically different from the expected cumulative GPA for women. Be precise in your answer and indicate all the necessary steps to perform that test. (2 points)

2.

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Name:_________________________________________________ Number:_________ 2. Consider the following output of the model:

log invpc t =β0 +β1 log price t +β log price

t-1 +β3 log

pop t +β t+ut

Dependent Variable: LINVPC Method: Least Squares Sample (adjusted): 2 42 Included observations: 41 after adjustments

C LPRICE LPRICE_1 LPOP T R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient

Std. Error

t-Statistic

Prob.

33.20187 2.591738 -4.835751 -2.895495 0.054741

12.86664 0.924113 0.897678 1.086491 0.015734

2.580462 2.804567 -5.386956 -2.664996 3.479188

0.0141 0.0081 0.0000 0.0115 0.0013

0.635898 0.595442 0.106840 0.410935 36.18283 15.71833 0.000000

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

-0.658846 0.167975 -1.521114 -1.312141 -1.445017 1.123386

where invpc is real per capita housing investment (in thousands of dollars), price denote a housing price index (equal to 1 in 1982), and pop denote total population in the United States, in thousands. The data are annual observations in the United States for 1947 through 1988. , ,  and  . Are they statistically significant? Justify. The included trend is of which type? (1 point)

a) Interpret each one of the coefficient estimates of the model,

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Name:_________________________________________________ Number:_________ b) Given the information in the output above, can you conclude if the errors of

this model suffer from any type of serial correlation? In order to take this conclusion, which Gauss-Markov assumption needs to be satisfied? What are the consequences for your answer in a) if this problem is present? (2 points)

c) Suppose that you know for sure that:

Corr ut , log price

t

=Corr ut , log price

t-1

=Corr ut , log pop

t

=Corr ut ,t =0

and that: Corr ut , log price t‐2 0,5. Does this change your conclusions from the previous question? Why? Are the OLS estimators unbiased and consistent in this case? Why? Is it possible to test for serial correlation in this case? How? (2 points)

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Name:_________________________________________________ Number:_________

Group III (1 point) 1. Give an example of a time series process with 2 observations (you can consider more if you want) that is “covariance-stationary” and at the same time nonstationary.

a)

a) the OLS estimator is unbiased but consistent

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