Mastering Econometrics II: Key Concepts and Problem Solutions

School

London School of Economics**We aren't endorsed by this school

Course

EC 2C1

Subject

Economics

Date

Dec 11, 2024

Pages

Uploaded by MateDangerArmadillo40

London School of Economics and Political ScienceMarcia SchafgansDepartment of EconomicsAutumn 2024EC2C1: Econometrics IIProblem Set 4Date due: Wednesday October 30, 20241. In the multiple linear regression model,y=Xβ+u, under Assumptions MLR.1-MLR.5, the unbiased estimator ofσ2iss2=SSR/(n-k), whereXis then×kmatrix of explanatory variables. Show (clearly define ˆuandM):SSR= ˆu0ˆu=u0Mu.2. Consider the partitioned regression modely=X1β1+X2β2+uunder MLR.1-MLR.5, whereX1isn×k1andX2isn×k2.(a) Show that the OLS estimator ofβ1iny=˜X1β1+v,(2.1)where˜X1=M2X1equals the partitioned regression estimator ofβ1:ˆβ1= (X01M2X1)-1X01M2y,whereM2=I-X2(X02X2)-1X02(b) Discuss what is wrong with this statement: “In order to obtain an estimator ofthe error variance,σ2, we need to divide the residual sum of squares from (2.1)byn-k1-k2.”Let us consider the setting whereX1andX2are orthogonal, that isX01X2= 0.(c) Show that the OLS estimator ofβ1whenX1andX2are orthogonal is equal to(X01X1)-1X01y. Interpret your result.Hint:Show thatM2X1=X1.(d) Let us consider an application: We are interested in the following regressionearningsi=β1traini+β2nontraini+uii= 1,· · ·, n(2.2)wheretrainindicates whether an individual is enrolled in a training programmeandtraini= 1 if individualireceives training andtraini= 0 otherwise. Note:nontraini= 1-traini. We assume the training is randomly assigned to peoplein a population of interest. LetE(ui) = 0.1

i. Explain why (2.2) cannot contain an intercept.ii. Provide the interpretation ofβ1.What is the usefulness of the randomassignment of training for this interpretation?iii. Are the regressors in (2.2) orthogonal (explain)?Use the result in (a) toobtain the OLS estimator ofβ1.Hint:You do not have to use linearalgebra to answer this question.3. [Wooldridge Ch. 3, Problem 10] Suppose that you are interested in estimating theceteris paribus relationship betweenyandx1. For this purpose, you can collect dataon two control variables,x2andx3. (For concreteness, you might think ofyas finalexam score,x1as class attendance,x2as GPA up through the previous semester, andx3as SAT or ACT score.) We can obtain the data by random sampling (MLR.2).y=β0+β1x1+β2x2+β3x3+uLeteβ1be the simple regression estimate fromyonx1and letbβ1be the multipleregression estimate fromyonx1,x2,x3.(In this question, “correlation” meanssample correlation.)(a) Ifx1is highly correlated withx2andx3in the sample, andx2andx3have largepartial effects ony, would you expecteβ1andbβ1to be similar or very different?Explain.(b) Ifx1is almost uncorrelated withx2andx3, butx2andx3are highly correlated,willeβ1andbβ1tend to be similar or very different? Explain.(c) Ifx1is highly correlated withx2andx3, andx2andx3have small partial effectsony, would you expect se(eβ1) or se(bβ1) to be smaller? Explain.(d) Ifx1is almost uncorrelated withx2andx3,x2andx3have large partial effectsony, andx2andx3are highly correlated, would you expect se(eβ1) or se(bβ1) tobe smaller? Explain.2