ECON 3150/4150 (Introductory Econometrics) Problem sets

1 ECON 3150/4150 ( introductory Econometrics) Problem setsSpring 2004 This set consists of 11 Problem sets , one for each seminar. Notice that some of theproblem sets consist of more than one Problem . Thefirst 3 Problem sets should be preparedby all students. Some of you will be asked to present your solution to each of the the remaining Problem sets , a group of students will prepare a solution that will behanded out to the other students a few days before the set 1 Problem 1(i)LetX,Ybe stochastic variables and consider the realizations(xi,yi),i=1,.., that the following relations must hold:(a)nXi=1(xi x)=0(b)nXi=1(xi x)(yi y)=nXi=1(xi x)yi=nXi=1xi(yi y)(c)nXi=1(xi x)(yi y)=nXi=1xiyi nx yThe empirical correlation coefficient betweenXandYis given by:(d)r=Pni=1(xi x)(yi y)qPni=1(xi x)2 Pni=1(yi y)2(ii)Use(c)and(d)to calculaterwhenx=2y=72n=6nXi=1x2i=106nXi= 1y2i=103nXi=1xiyi=50(iii)Let us defineexi=xi xsxandeyi=yi ysywheresxandsyare the standard deviationsofxandy.

2 Find the empirical mean and variance of theexiandeyi. Show that the empiricalcorrelation coefficient betweenexiandeyiwill always be in the interval[ 1,1](Hint: You canuse the fact thatPni=1z2i 0is true for any variablez,and thus holds forzi=exi+eyiandzi=exi eyirespectively). This proof extends directly to the correlation coefficient betweenany pair of un-standardized variablesxiandyi. Why must this be true?2 Problem 2 LetXandYbe two stochastic variables.(i)Show that if at least one ofXandYhas expectation equal to zero, thencov(X, Y)=E(XY)(1)(ii)Use the result under(i)to calculate an expression forE(X2)whenE(X)=0.

3 (iii)Use the definition of covariance to show thatcov(AX+a, BY+b)=ABcov(X, Y)whenA, a, B, bare constants.(iv)Use the definition of variance to show thatvar(X+Y)=var(X)+var(Y)+2 Cov(X, Y)var(X Y)=var(X)+var(Y) 2 Cov(X, Y)(v)The theoretical correlation coefficient betweenXandYis defined by: =cov(X,Y)pvar(X)pvar(Y)Under Problem 1 you established that the empirical correlation coefficient between tovariablesxiandyiwasalwaysintheinterval [ 1,1]. Use the result under(iv)to showthatthesamemustbetrueforthetheoretic alcorrelationcoefficient between any pair ofstochastic set 2 Problem 1 Consider a population of households consisting of either one, two or three householdmembers above the age of 18 and assume that none of these households own more thanthree cars in total.

4 Let the variableXdenote the number of household members and thevariableYthe number of cars owned by the household. The bivariate distribution of(X, Y)is depicted a random draw from this population and definetheeventsA:"Thenumberofcars in the household is one" and B: "The number of household members is three".(a)FindP(A),P(B),P(A|B)andP(B|A). (b)CalculateE(Y),E(Y|X=1),E(Y|X=2)andE(Y |X=3).(c)CalculateVAR(Y|X=1)andVAR(Y).(d )Confirm thatEx(E(Y|X)) =E(Y). Can you show that this will hold for anybivariate distribution(X,Y)? Problem 2(a) of bivariate distributed stochastic variablesXandY.

5 Explain how you could proceed tofitastraightlinetothepointsinthisdia-g ram. Do youfind that the concept of "best linearfit" is well defined without any furtherspecifications?(b)If ordinary least squares (OLS) is used tofit a straight line to the data points(Xi,Yi) X i=1,2, .., nthen thefitted constant and slope parameters are given by:b =Y b Xb =P Xi X Yi Y P Xi X 2 Show that if OLS instead is applied tofit a straight line to the data points(xi,Yi)i=1,2,..,nwherexi=Xi X,then thefitted constant and slope parameters are givenby:e =b +b Xe =b (c)Suppose that the reason for your interest in the relationship betweenXandYis yourconfidence in an economic theory that suggests that we can writeYi= + Xi+uiwhere and are unknown constants anduiis an unobservable stochastic variable withexpectation zero and a constant variance.

6 Moreover, assume thatXican be treated as "fixedin repeated sampling". Why should we turn to the particular method of OLS if the abovepresumptions accurately describes the relation betweenXandYand our interest is to drawinference about the unknown parameters and ?5(d)SupposeX= that we estimateb = = you estimateE(Y)andE(Y|X=10)?(e)How isb related to the empirical coefficient of correlation betweenXandY?(f)How would you characterize the difference between correlation analysis and regressionanalysis?6 Problem set 3(a)Use the "File->Open"menuinGiveWin. Choosetoopenthedatafile ,which is located in "c:\Programfiles\Givewin2\".

7 Use the "Tools-Graphics" menu to drawa "Scatter plot" and an "Actual values plot" using the variables CONS (consumption) andINC (income).(b)Click the "Modules->Start-PcGive" menu to start the PcGive module. Use "Package->Descriptive statistics" to tell PcGive that we want to do descriptive statistics. Choose"Model->Formulate" to select variables. Calculate the coefficient of correlation betweenCONS and INC. In the Descriptive statistics check box you can choose the option "selectsample". Use this facility calculate the coefficient of correlation using data from 1953(1) to1972(4) only.

8 (c)Choose "Calculator" from the "Tools" menu in GiveWin. Use this facility to generatea new variable representing thefirst difference of the variable CONS (you can use the functiondiff(CONS,1). You must assign a name to the new variable). Repeat the procedure for thevariable INC. Repeat the procedures under(b)using thefirst differences of CONS and INCinstead of the levels. Try to compare the results.(d)Choose "Package->Econometric modelling" in PcGive to tell PcGive that we want torun regressions. Then choose "Model->Cross section regression" as we want to use a staticmodel.

9 Specify consumption to be the dependentvariableandIncometobeexogenous( ex-planatory variable) and estimate the model with OLS. Do youfind the estimated coefficientsin line with a priori expectations? In PcGive, choose "Test->Graphic analysis", and select"Actual andfitted values", "Residuals" and "Residual density". Give a description of thegraphics you obtain. Repeat the analysis using the differenced variables. Do youfind theestimated coefficients in line with a priori expectations?(e)Choose the calculator from the "Tools" menu in GiveWin and use the "dummy"option to create a variable that takes the value zero in the period 1953(1)-1971(4) and thevalue one 1972(1)-1992(4).

10 Run the same regression as above except that now you add thedummy as a second exogenous variable and interpret the results. Turn back to the calculatorand make a new variable that equals the product of Income and the dummy variable andadd this variable too to the list of explanatory variables. Run this regression and try tointerpret the (f)Repeat the analysis under(f)using the log of income and the log of consumption(construct these variables using the calculator). Interpret the set 4 Problem 1In a study of wage differences between native and non-native workers of similar age andtraining the following equation is estimated:(1)Wi= + Di+uii=1,2.