Econometrics Final Exam 27thofJune,2008 João Valle e Azevedo António José Morgado Tiago Silva Vieira Timeforcompletion: 2h30min Give your answers in the space provided. Usedraftpapertoplanyouranswersbeforewritingthemontheexampaper. Unless otherwise stated, use 5% for significance level. Name: GroupI(9points,1foreachquestion) Give a concise answer to the following questions: 1. Suppose the Gauss-Markov assumptions hold in a multiple linear regression model. What(three) characteristics of the model can contribute to a low sampling variance of the OLS estimator of a particular coefficient? 2. Given a multiple linear regression model where the Gauss-Markov assumptions hold, suppose you estimate the parameters by OLS. How would you predict the expected value of the dependent variable, given particular values of the regressors? How would you construct a prediction interval for that expected value of the dependent variable? 1
3. Explain intuitively why weighted least squares estimators (WLS) have a smaller variance than the typical OLS estimators, in a model for cross-sectional data where the homoskedasticity assumption fails. 4. Consider a multiple linear regression model for cross-sectional data that analyses the impact of trade barriers on national income of countries around the world. Among other regressors, you include dummies for Africa, Europe and Asia(that equal one if the country is, respectively, in Africa, Europe and Asia and zero otherwise). Also, you include a constant in the model. What conditions must you impose in your sample so that the Absence of Multicollinearity assumption is not violated(due to the inclusion of the dummies) in the model? 5. The acronym OLS stands for what in Econometrics? 2
6. "As it happens with cross-sectional data, we can always assume random sampling in time series analysis". Is this statement true? Explain. 7. What can gowrongin a regression model if the errors follow an AR(1) process? Whatcanyoudotosolvetheproblem,andunderwhatconditionscanyoudoit? If you can t solve the problem, how can you conduct valid inference? 8. Consider the following model for time series data: y t =β 0 +β 1 x t1 +β 2 x t2 +...+β k x tk +u t wheretheerrortermfollowsanar(2)process,(u t =ρ 1 u t 1 +ρ 2 u t 2 +e t ande t is independent of the regressors) but all the other assumptions needed to guarantee unbiasedness of the OLS estimator and the validity of"typical" OLS inference are verified. How would you transform the model so that estimation by OLS of the transformedmodelisequivalenttoglsestimationoftheoriginalmodelfort>2 (don t worry about the first 2 observations)? 3
9. Suppose the linearity, strict exogeneity and absence of multicollinearity assumptions holdinatimeseriesregressionmodelthatincludesalineartrendasaregressor. Whataretheeffects(intermsofbiasontheestimatorsoftheremainingregressors) of leaving the linear trend 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) 4
Group II(8 points) 1. YouhavebeencommissionedtoperformastudyabouttheimpactoftheBSEcrises on the beef price in Portugal. You have annual data for the years 1970 through 2007 on the following variables: price t -AveragebeefpriceinPortugal(euros/100kg); keyprice t - Average beef price in Germany, the reference beef price in the EU (euro/100kg); SSR t -Portugueseself-sufficiencyratio(Production/Domesticuse); BSE t -Dummyvariable,itis1from1996on. You decide to estimate the following model using OLS: price t =β 0 +β 1 keyprice t +β 2 SSR t +β 3 BSE t +u t Obtaining the following results (standard errors in parentheses below coefficient estimates): price t =102.33 (35.22) +0.85 (0.50) keyprice t 22.32 (14.98) SSR t 35.47 (10.35) BSE t (a) Interpret each of the coefficient estimates ˆβ 1, ˆβ 2 and ˆβ 3. Do they have the expected signs? (0,5 points) 5
(b) TestifthereisashiftinthebeefpriceaftertheBSEcrisesata5%significant level. State the null and alternative hypotheses and show how you calculate the required test statistic. State the decision rule you use, and the inference you would draw from the test.(0,5 points) You remember from your econometrics course that it is important to test if your errors suffer from serial correlation. You decide to use the Durbin-Watson statistic(dw). (c) TheDWstatisticfromyourregressionisDW =0.55. Whatcanyouconclude in terms of serial correlation? (1 point) 6
(d) Imagineyourmodelhadthevariableprice t 1 asanexplanatoryvariable. Does it change your conclusions from the DW statistic? Why? (1 point) When discussing your results with your research assistant, he reminds you about the importance of including a trend in your model. After including it in your model, the only variable that is statistically significant is the trend. (e) What can be happening in your initial model? Does it affect your conclusions abouttheimpactofbseonprices? (1point) 7
2. Consider the following model, where wage is the salary of a CEO, educ is the numberofyearsofeducationandexper isthenumberofyearsofexperience. log(wage)=β 0 +β 1 educ+β 2 exper+β 3 exper 2 +u (a) What is the percentage change in wage, on average, given a caeteris paribus unitincreaseinexper? Expressyouranswerintermsoftheequationparameters. (0,5 points) Using a random sample of 523 individuals, the following results were obtained: 8
(b) Is the quadratic term significant at 1% level? Formalise your answer, stating the null and alternative hypotheses and show how you calculate the required test statistic. State the decision rule you use, and the inference you draw from the test. (0,5 points) (c) ForecasttheaveragewageofaCEOwith16yearsofeducationand10years of experience. Explain how you would proceed to obtain a standard error for your forecast based on a confidence interval of 95%. (1 point) 9
Additionally, the following output was obtained, using the residuals from the model estimation: (d) Conclude whether or not heteroskedasticity is present in the model. Formalise your answer, stating null hypothesis, test statistic and decision. (1 point) (e) How does your answer in question(2d) interfere with your answer in questions (2b)and(2c)? Explainusingnomorethan5linesoftext. (1point) 10
Group III(3 points) 1. Consider the following model for time series data: y t =β 0 +β 1 x t1 +β 2 x t2 +...+β k x tk +u t whereu t followsanma(2)process,thatis,u t =e t +ρ 1 e t 1 +ρ 2 e t 2 where{e t }is ani.i.d. sequencewithmean0andvarianceσ 2 e,uncorrelatedwiththeregressors. (a) What assumption for time series models is necessarily violated in this model? (0,5 points) (b) Whatis themeanandvarianceofu t (Lookonlyattheunconditionalmean andvariance,e[u t ]andvar[u t ])? (0,5points) 11
(c) Theautocorrelationfunctionoftheprocess{u t }isgivenby: ρ(k)=cov(u t,u t k )/Var[u t ], wherekisaninteger. Derivetheautocorrelationfunctionoftheprocess{u t } forallintegersk 1.(ifyoudidnotanswertopartb),itisenoughtoderive Cov(u t,u t k )forallintegersk 1). (1point) 12
(d) Supposenowthatu t =e t +ρ 1 e t 1 +ρ 12 e t 12 where{e t }isani.i.d. sequence withmean0andvarianceσ 2 e.derivetheautocorrelationfunctionoftheprocess {u t } forallintegersk 1.(ifyoudidnotanswertopartb),itisenoughto derivecov(u t,u t k )forallintegersk 1). (1point) 13