Brooklyn College Econometrics 7020X Spring 2016 Instructor: G. Koimisis Name: Date: Practice Questions for the Final Exam Theoretical Part 1. Define dummy variable and give two examples. 2. Analyze the three different types of data (cross-sectional, time series, panel data). 3. Define R 2 and R 2. What is their important property? Show the relation between them and their differences. Analyze as much as you can. 4. (a) Analyze fully the assumption of homoscedasticity of a CLRM. Moreover, which are the differences with heteroscedasticity? (b) Analyze fully the assumption of specification bias and use an example to show your intuition. 5. State the CLRM (Classical linear Regression Model) Assumptions 5. State the Gauss-Markov Theorem and provide full definitions of the characteristics of a BLUE estimator (unbiasedness, linearity, efficiency). 6. You are given the following non-linear regression model Y = β 1 X β 2e u, where, Y: dependent variable, X: independent variable, e: exponent, u: error term, betas: coefficients. Make the necessary transformation/s so that the model can be estimated by using OLS method. 7. You are given the following two models: GPDI = 1026.5 + 0.30GDP se = (257.58) (0.04) where GDPI and GDP are measured in billions of dollars
GPDI = 0.94GDP se = (0.115) where GDPI* and GDP* are the standardized versions of the variables GDPI and GDP. Interpret the coefficient of GDP and GDP* with economic reasoning. 8. Analyze briefly the different reasons of doing hypothesis testing in a multiple regression model. 9. Discuss two types of specifications errors that we may have in a classical linear regression model. Use an example for each case. 10. Choose the correct answer. i. The α in confidence interval given by Pr (β i δ β i β i + δ) = 1 α is known as: a. Confidence coefficient b. Level of confidence c. Level of significance d. Significance coefficient ii. The 1-α in confidence interval given by Pr (β i δ β i β i + δ) = 1 α is known as: a. Confidence coefficient b. Level of confidence c. level of significance d. Significance coefficient iii. Standard error of an estimator is a measure of a. Population estimator b. Precision of the estimator c. Power of the estimator d. Confidence interval of the estimator iv. For a regression through the origin, the intercept is equal to a. 1 b. 2 c. 0 d. -1
Empirical Part Problem 1 The table contains the ACT scores and the GPA for eight college students. GPA is based on a 4- point scale and has been rounded to one digit after the decimal. Student GPA ACT 1 2.8 21 2 3.4 24 3 3.0 26 4 3.5 27 5 3.6 29 6 3.0 25 7 2.7 25 8 3.7 30 Obtain the estimates a 0 and a 1 in the linear regression model: GPA i = a 0 + a 1 ACT Problem 2 (I) The following equation is part of a nutrition-based efficiency wage model: Total calories Cals were regressed on the number of meals MG given to guests at ceremonies, the number of meals ME given to employees and the number of meals MO given to guests on other occasions: Cals = β 0 + β 1 MG + β 2 ME + β 3 MO +e (Model A) The expected sign of the coefficients are β1 > 0, β2 > 0, β3 > 0 You ran the regression in Eviews, by using OLS, for the period 1960-1999 and you obtained the following output: Dependent Variable: CALS Included observations: 40 after adjustments C 27.59394 1.584458 17.41539 0.0000 MG 0.607160 0.157120 3.864300 0.0004 ME 0.092188 0.039883 2.311452 0.0266 MO 0.244860 0.011095 22.06862 0.0000 R-squared 0.990391 Mean dependent var 50.56725 Adjusted R-squared 0.989590 S.D. dependent var 19.53879 S.E. of regression 1.993549 Akaike info criterion 4.312350 Sum squared resid 143.0726 Schwarz criterion 4.481238 Log likelihood -82.24700 Hannan-Quinn criter. 4.373414 F-statistic 1236.776 Durbin-Watson stat 0.897776
a. Some of the standard errors and t-statistics of the coefficients are missing from the output. Calculate them (be careful about the signs). Remember that the Null Hypothesis is: H 0 : β i = 0. Test in α=5% significance level. Your t-critical is given as 2.03. Do you reject or fail to reject the Null? b. Interpret the effect of MG and ME variables with economic reasoning. c. The value of R 2 is missing from the output. Calculate it. d. The standard error of regression is not visible in the above output. Calculate it by using the relevant formula: σ 2= u i 2 n k, where n is the number of observations and k is the number of parameters. e. The F value (for overall significance) you obtained is missing. Calculate it and test it for 5% significance level. The Fcritical(5%,3,36) is given as 2.87. Note that we are testing jointly for all the coefficients, excluding the intercept. f. Calculate the 95% Confidence Intervals for the coefficients of MG and MO. (II) You decide to re-specify Model A, by dropping variable MO. Your model becomes: Cals = β 0 + β 1 MG + β 2 ME +ε (Model B) You run the model and you obtain the following output: Dependent Variable: CALS Included observations: 40 after adjustments C 35.45670 5.804616 6.108363 0.0000 MG 2.567990 0.487207 5.270839 0.0000 ME 0.892676 0.062343 14.31881 0.0000 R-squared 0.860390 Mean dependent var 50.56725 Adjusted R-squared 0.852844 S.D. dependent var 19.53879 S.E. of regression 7.495264 Akaike info criterion 6.938458 Sum squared resid 2078.622 Schwarz criterion 7.065124 Log likelihood -135.7692 Hannan-Quinn criter. 6.984257 F-statistic 114.0122 Durbin-Watson stat 0.678483 a. We do not know which model is better performed between model A and model B. Which model is better in terms of model building? Make use of the Information Criteria to find out which model is better.
b. The output is problematic. The standard errors and the t-statistics are missing. Use the Wald Test to do Restriction Testing between models A and B. The Null Hypothesis is: H 0 : β 3 = 0. The Fcritical is given as 4.11 and the X 1 2 critical value is given as 3.841 (both for a 5% significance level).. What can you conclude regarding the dropped variable MO? Problem 3 lnip t+1 = β 0 + β 1 lnol t + β 2 lnrsr t + β 3 empl t + u t The above linear regression model states that the industrial productivity (lnip) is positively affected by the stock market returns (lnrsr) and the employment ratio (empl) and negatively affected by the oil prices (lnlo). We are examining the USA market growth during the period 1960-1999. In 1980 we have the introduction of the personal computer and we want to test whether there is any structural change after the year of 1980: Ho: There was no structural break (or change) after 1980 H1: There was a structural break (or change) after 1980 You get the following three outputs by doing the Chow Test. Dependent Variable: LNIP Sample: 1960 1999 Included observations: 40 C 27.59394 1.584458 17.41539 0.0000 LNOL -0.607160 0.157120-3.864300 0.0004 LNRSR 0.092188 0.039883 2.311452 0.0266 EMPL 0.244860 0.011095 22.06862 0.0000 R-squared 0.990391 Mean dependent var 50.56725 Adjusted R-squared 0.989590 S.D. dependent var 19.53879 S.E. of regression 1.993549 Akaike info criterion 4.312350 Sum squared resid 143.0726 Schwarz criterion 4.481238 Log likelihood -82.24700 Hannan-Quinn criter. 4.373414 F-statistic 1236.776 Durbin-Watson stat 0.897776 Prob(F-statistic) 0.000000
Dependent Variable: LNLIP Date: 03/10/14 Time: 13:50 Sample: 1960 1979 Included observations: 20 C 27.59882 2.433883 11.33942 0.0000 LNOL -0.899693 0.297873-3.020394 0.0081 LNRSR 0.181932 0.098121 1.854171 0.0822 EMPL 0.265328 0.058970 4.499342 0.0004 R-squared 0.913357 Mean dependent var 34.28700 Adjusted R-squared 0.897112 S.D. dependent var 6.199594 S.E. of regression 1.988596 Akaike info criterion 4.389592 Sum squared resid 63.27225 Schwarz criterion 4.588738 Log likelihood -39.89592 Hannan-Quinn criter. 4.428467 F-statistic 56.22198 Durbin-Watson stat 1.116410 Prob(F-statistic) 0.000000 Dependent Variable: LNLIP Date: 03/10/14 Time: 13:51 Sample: 1980 1999 Included observations: 20 C 16.18376 3.874379 4.177124 0.0007 LNOL -0.345689 0.136746-2.527964 0.0224 LNRSR 0.151866 0.046860 3.240840 0.0051 EMPL 0.272712 0.008611 31.67100 0.0000 R-squared 0.993168 Mean dependent var 66.84750 Adjusted R-squared 0.991887 S.D. dependent var 13.68186 S.E. of regression 1.232329 Akaike info criterion 3.432546 Sum squared resid 24.29817 Schwarz criterion 3.631692 Log likelihood -30.32546 Hannan-Quinn criter. 3.471421 F-statistic 775.3400 Durbin-Watson stat 1.665783 Prob(F-statistic) 0.000000 a. Explain the procedure of the Chow Tests, i.e the steps you have to take in order to make conclusions about the structural stability. b. Calculate the Chow F-Statistic. According to the F statistic you just calculated do you reject or fail to reject the Null Hypothesis? What that means for your data? You are given that Fcritical(0.05,4,32)=2.69.
Formulas For Joint Hypothesis Testing: F test R 2 k 1 1 R 2 n k For Restriction Testing: Wald Test (F variant) RSS R RSS UR m (RSS UR )/(n k) or R 2 UR R2 R m 1 R 2 UR n k or R 2 NEW R2 OLD # of new regressors 1 R 2 NEW n # of parameters in the new model For Restriction Testing: Wald Test (X 2 variant) W = nm n k F For Restriction Testing: LM Test LM = n(rss R RSS UR ) RSS R For Structural Break Test Chow RSS N RSS N1 RSSN2 k (RSS N1 +RSS N2 )/(N 2k) AIC (Akaike), SIC (Schwarz) Information Criteria lnaic = ( 2k n ) + ln (RSS n ) lnsic = (k ) ln (n) + ln (RSS ) n n