Econometrics Final Exam
João Valle e Azevedo Erica Marujo
28th of May, 2009 Time for completion: 2h30
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 for each question) Give a very concise answer to the following questions. Conciseness will be valued, avoid unnecessary details. 1. The acronym WLS stands for what in Econometrics? _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 2. Consider a multiple linear regression model for cross‐sectional data where the Gauss‐Markov assumptions hold. Do you need any additional assumption to conduct valid inference on the parameters of the model? Explain your answer. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________
Name:_________________________________________________ Number:_________
3. In one phrase, describe the meaning of “Bias of the OLS estimator” in a multiple linear regression context. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 4. Consider the model: Wagei = β 0 + β1 Educationi + ui , where Wagei is the monthly wage of individual i and Educationi the number of years of schooling of that individual. Give one example to show that the Zero Conditional Mean assumption most probably fails in this model. Given your example, what is the likely sign of the bias in the OLS estimator of β1 ? _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 5. Suppose the linearity, strict exogeneity and absence of multicollinearity assumptions hold in a time series regression model that includes seasonal dummies as regressors. What are the effects (in terms of bias on the estimators of the remaining regressors) of leaving the dummies out of the model? Under what conditions is the bias inexistent or negligible? (answer the question in light of the analysis of omitted variable bias) _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________
Name:_________________________________________________ Number:_________
6. Describe succinctly a test aimed at detecting AR(q) serial correlation in the error term of a multiple linear regression model. Assume the strict exogeneity assumption may fail but all the other necessary assumptions hold. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 7. Write a model aimed at testing whether the effect of education on wages is the same for men and women. Describe the variables you use and state the null hypothesis (and alternative) of the test. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 8. Which assumptions are sufficient to guarantee that the OLS estimator of a multiple linear regression model for time series data is consistent for the parameter values (describe the smallest set of assumptions). _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________
Name:_________________________________________________ Number:_________
9. Consider an F‐test of multiple exclusion restrictions in a multiple linear regression model. Can the observed test statistic be negative? Explain why or why not. _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________
Name:_________________________________________________ Number:_________
Group II (8 points) 1. Consider the following output of the model: 86
86 86
86
Dependent Variable: NARR86 Method: Least Squares Sample: 1 2725 Included observations: 2725 Variable
Coefficient
Std. Error
t-Statistic
Prob.
C PCNV AVGSEN AVGSENSQ PTIME86 QEMP86 BLACK HISPAN PTIME86*BLACK
0.578049 -0.133703 0.019469 -0.000501 -0.031646 -0.093422 0.356194 0.197789 -0.036568
0.036051 0.040478 0.009729 0.000298 0.010022 0.010383 0.046629 0.039814 0.018554
16.03439 -3.303107 2.001176 -1.680809 -3.157810 -8.997933 7.638956 4.967809 -1.970926
0.0000 0.0010 0.0455 0.0929 0.0016 0.0000 0.0000 0.0000 0.0488
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
0.067686 0.064940 0.830714 1874.275 -3356.697 1.838963
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)
0.404404 0.859077 2.470236 2.489756 24.64767 0.000000
where 86 is the number of times a man was arrested, is the proportion of prior arrests leading to conviction, is average sentence length served from past convinctions, 86 is months spent in prison prior to 1986, 86 is number of quarters in 1986 during which the man was legally employed, is a dummy variable equal to one if a man is black and zero otherwise, and is also a binary variable, equal to one if a man is hispanic and zero otherwise. a) Interpret each one of the coefficient estimates of the model, through (be careful to perform that interpretation combining the estimates in the best way possible). (0,5 points) _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________
Name:_________________________________________________ Number:_________
b) Explain how you would test the hypothesis that the number of times that a black man is arrested equals, on average, the number of times an hispanic man is arrested. Explain how you would perform that test, describing carefully every step that you would have to follow, including: the null and alternative hypothesis; the method used to perform the test; the test statistic and its distribution; the estimate for the test statistic; the decision rule and the final conclusion. (1 point)
Consider the following output: Dependent Variable: NARR86 Method: Least Squares Sample: 1 2725 Included observations: 2725 Variable
Coefficient
Std. Error
t-Statistic
Prob.
C PCNV AVGSEN AVGSENSQ PTIME86 QEMP86
0.703225 -0.153880 0.024430 -0.000595 -0.037824 -0.102643
0.033182 0.040866 0.009813 0.000301 0.008792 0.010397
21.19301 -3.765506 2.489596 -1.975934 -4.302036 -9.872399
0.0000 0.0002 0.0128 0.0483 0.0000 0.0000
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
0.043567 0.041808 0.840927 1922.762 -3391.496 1.836123
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)
0.404404 0.859077 2.493575 2.506588 24.77107 0.000000
Name:_________________________________________________ Number:_________
c) Which would you prefer? This one or the initial model? Justify your answer. (0,5 points) _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ The following output was reported, using the residuals of the first model estimation: F-statistic Obs*R-squared
5.549573 70.63723
Probability Probability
0.000000 0.000000
Test Equation: Dependent Variable: RESID^2 Method: Least Squares Sample: 1 2725 Included observations: 2725
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C PCNV PCNV^2 AVGSEN AVGSEN^2 AVGSENSQ^2 PTIME86 PTIME86^2 QEMP86 QEMP86^2 BLACK HISPAN PTIME86*BLACK (PTIME86*BLACK)^2
0.917044 2.051029 -2.140770 -0.074075 0.003277 -7.54E-07 0.825814 -0.079244 -0.157009 -0.017861 0.746270 0.267765 -0.779970 0.061203
0.189874 0.734108 0.742687 0.074223 0.003734 9.11E-07 0.257678 0.022395 0.195056 0.045671 0.218562 0.186838 0.442773 0.039562
4.829750 2.793909 -2.882465 -0.998015 0.877656 -0.828227 3.204824 -3.538479 -0.804944 -0.391083 3.414453 1.433139 -1.761556 1.547036
0.0000 0.0052 0.0040 0.3184 0.3802 0.4076 0.0014 0.0004 0.4209 0.6958 0.0006 0.1519 0.0783 0.1220
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
0.025922 0.021251 3.859331 40378.81 -7539.685 1.987515
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)
0.687807 3.901003 5.543989 5.574353 5.549573 0.000000
Name:_________________________________________________ Number:_________
d) What can you conclude from this regression? Which one of the Gauss‐Markov assumptions is being violated? What are the consequences for the OLS estimators of the first model? Does your answer to this question affect your answers to questions b) and c)? Why? (1 point) e) Describe succinctly a method to overcome the problem detected in the previous question, so that all Gauss‐Markov assumptions are satisfied. (1 point) f)
Name:_________________________________________________ Number:_________
2. Consider the following output of the model: log
log log
log 6
log
Dependent Variable: LCHNIMP Method: Least Squares Sample(adjusted): 2 131 Included observations: 130 after adjusting endpoints Variable
Coefficient
Std. Error
t-Statistic
Prob.
C LCHEMPI LCHEMPI_1 LGAS LRTWEX AFDEC6
-23.21921 -9.079570 12.38285 0.406122 0.933382 -0.559091
20.32619 3.530387 3.557363 0.872305 0.359006 0.264842
-1.142329 -2.571834 3.480908 0.465574 2.599907 -2.111037
0.2555 0.0113 0.0007 0.6423 0.0105 0.0368
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
0.360217 0.334419 0.570318 40.33254 -108.3876 1.507991
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)
6.180590 0.699063 1.759809 1.892157 13.96312 0.000000
Where log is the logarithm of the volume of imports of barium chloride from China, log is the logarithm of an index of chemical production, log is the logarithm of the volume of gasoline production, log is the logarithm of an exchange rate index, and 6 is a dummy equal to 1 during the six months after the positive decision of the International Trade Commission (ITC) in favor of the US barium chloride industry about dumping behaviour of China in this sector. The regression was estimated using monthly data from February 1978 through December 1988. a) Interpret each of the coefficient estimates , and . Are they statistically significant? Why? (0,5 points) _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________
Name:_________________________________________________ Number:_________
b) What can you conclude from the Durbin‐Watson (DW) statistic of this regression? What are the consequences of your conclusions over the OLS estimators of this model? And what about your conclusions in question a)? State ALL the necessary steps which lead to your conclusion and be FORMAL when interpreting the value of the DW statistic. (1 point)
The following output of the same model but including dummy variables for the months from February through December as well as a linear time trend was also reported: Dependent Variable: LCHNIMP Method: Least Squares Sample(adjusted): 2 131 Included observations: 130 after adjusting endpoints Variable
Coefficient
Std. Error
t-Statistic
Prob.
C LCHEMPI LCHEMPI_1 LGAS LRTWEX AFDEC6 FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC T
16.77820 -11.00896 11.12070 -0.508752 0.001624 -0.385276 -0.388018 -0.058696 -0.476671 -0.128246 -0.310414 -0.045021 -0.181036 -0.145297 -0.089654 -0.356714 0.007950 0.010949
30.15189 3.708719 3.650904 1.298227 0.466131 0.273688 0.289971 0.248826 0.248516 0.252003 0.249018 0.255617 0.254768 0.246592 0.249990 0.245860 0.253586 0.003739
0.556456 -2.968401 3.046013 -0.391882 0.003485 -1.407718 -1.338128 -0.235891 -1.918069 -0.508904 -1.246550 -0.176129 -0.710594 -0.589218 -0.358630 -1.450882 0.031350 2.928486
0.5790 0.0037 0.0029 0.6959 0.9972 0.1620 0.1836 0.8139 0.0576 0.6118 0.2152 0.8605 0.4788 0.5569 0.7205 0.1496 0.9750 0.0041
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
0.447292 0.363399 0.557763 34.84320 -98.87804 1.511222
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)
6.180590 0.699063 1.798124 2.195167 5.331696 0.000000
Name:_________________________________________________ Number:_________
c) Can you conclude that there is seasonality in this model? Why? How would you test the joint statistical significance of the coefficients of the seasonal dummies? State the null and the alternative hypothesis and show how you would calculate the required test statistic. State the decision rule you use, and the inference you would draw from the test. (1 point) d) In the regression presented above, the included trend is of which type? Interpret the coefficient on the time trend. (0,5 points)
Name:_________________________________________________ Number:_________
e) Can you guarantee that there is a spurious relationship between log and the other regressors of the initial model? Explain. What additional information would you need to answer this question? (1 point)
Name:_________________________________________________ Number:_________
Group III (3 points) 1. Consider the following multiple linear regression model for time series data , where | | , where
1
0,
is an i.i.d. sequence such that
0 |
,
,…
0
a) What kind of process does the error term ut follow? (1 point)
b) What standard assumption is necessarily violated in this model? (1 point)
Name:_________________________________________________ Number:_________
c) How would you write the model so that none of the standard assumptions is violated? Clarify the relation between the parameters of the original model and those of the rewritten model. (1 point)
