DO NOT TURN OVER UNTIL TOLD TO BEGIN

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1 ame HEMIS o. For Internal Stdents of Royal Holloway DO OT TUR OVER UTIL TOLD TO BEGI EC5040 : ECOOMETRICS Mid-Term Examination o. Time Allowed: hor Answer All 4 qestions STATISTICAL TABLES ARE PROVIDED Silent non-rogrammable calclators may be sed PRIT YOUR AME O THE FROT OF THIS TEST PAPER WHERE IDICATED WRITE ALL YOUR ASWERS ICLUDIG ROUGH WORKIG O THIS TEST PAPER. THERE ARE ETRA BLAK SHEETS TOWARD THE BACK OF THE PAPER Royal Holloway and Bedford ew College 005

2 . Given the general linear model y B + B a Show the effect of estimating the model y B +v on the bias of the OLS estimate of B and its variance/covariance estimate assme σ is known y taking exectations E + marks Unless and are orthogonal, 0, estimates in omitted variable eqation are biased Given Using rles on artitioned matrices σ [ σ ] comare with variance estimated in case of omitted variables σ so < and omitted variable estimates have smaller variance The more highly correlated and the greater the difference

3 b Yo decide to rn a simle regression of log horly ay on years of work exerience reg lhw exer A Sorce SS df MS mber of obs F, Model Prob > F Residal R-sqared Adj R-sqared Total Root MSE lhw Coef. Std. Err. t P> t [95% Conf. Interval] exer _cons some of the regression ott has been concealed and then decide to inclde a dmmy variable for whether the individal is a gradate reg lhw exer grad B Sorce SS df MS mber of obs F, Model Prob > F Residal R-sqared Adj R-sqared Total Root MSE lhw Coef. Std. Err. t P> t [95% Conf. Interval] exer grad _cons Given the following regression of gradate stats on exerience

4 reg grad exer C Sorce SS df MS mber of obs F, Model Prob > F Residal R-sqared Adj R-sqared Total Root MSE grad Coef. Std. Err. t P> t [95% Conf. Interval] exer _cons Work ot the estimated OLS coefficient on exerience in the simle variable model A The algebra of omitted variables tells s that var mod el ex er 3 var mod el ex er E + so that the OLS estimate of exerience in the variable model eqals the OLS coefficient on exerience in the fll model ls a correction factor which is eqal to the coefficient from a regression of gradate stats on exerience,, mltilied by the OLS coefficient on gradate in the fll model Can test this by regressing gradate on exerience reg grad exer Sorce SS df MS mber of obs F, Model Prob > F Residal R-sqared Adj R-sqared Total Root MSE grad Coef. Std. Err. t P> t [95% Conf. Interval] exer _cons and sing above to give grad dislay * which is the coefficient on exerience in the variable model. also exlains why the coefficient on exerience in the nrestricted model is more ositive. a Exerience and gradate stats are negatively correlated there are relatively more gradates among yonger workers see the regression

5 coefficient on exerience in the axillary regression above b gradates earn more. The rodct of these two effects is negative. ot controlling for both these effects exerts a downward bias on exerience in the restricted model. Can test this by regressing gradate on exerience reg grad exer Sorce SS df MS mber of obs F, Model Prob > F Residal R-sqared Adj R-sqared Total Root MSE grad Coef. Std. Err. t P> t [95% Conf. Interval] exer _cons and sing above to give dislay * which is the coefficient on exerience in the variable model. also exlains why the coefficient on exerience in the nrestricted model is more ositive. a Exerience and gradate stats are negatively correlated there are relatively more gradates among yonger workers see the regression coefficient on exerience in the axillary regression above b gradates earn more. The rodct of these two effects is negative. ot controlling for both these effects exerts a downward bias on exerience in the restricted model. c LM test *R axillary ~χ no. extra variables Where R axillary is R from a regression of the residals from the restricted regression y B + on a set of additional variables If estimated χ > χ critical reject restricted regression. There are relevant variables not in the restricted model.. Given the general linear model y + yo ssect that Var i / i E i / i constant

6 a What are the conseqences for OLS estimation of the B vector and its associated variance/covariance matrix? 0 marks Heteroskedasticty exists, so y OLS + + Taking exectations OLS E so OLS estimates remain nbiased in resence of heteroskedasticity bt ] [ ] [ E E OLS Var ] [ Ω OLS Var σ σ so standard errors in OLS based on latter are biased in an nknown direction which deends on Ω and since t and F vales also calclated sing latter they are also biased id se OLS. b Given groed data and a model of the form y g g + g g,. G gros write down the form of the residal variance/covariance matrix in this case and hence the form of the Feasible GLS estimator in this case 6 marks Since Var g σ /n g then the GxG residal variance covariance matrix Ω G / / σ σ and so sing rles on inverse of diagonal matrix Ω G 0.. 0

7 Hence given G G GLS Ω Ω y g xg xg g xg yg g g where x g is the g th row of and y g is the g th element of the y vector c What does this imly abot how to transform the data in the original model in order to obtain this Feasible GLS estimator 3 marks To carry ot GLS estimation all have to do is mltily each element of the x g row of the matrix and the y g element by the sqare root of g weighted least sqares d What is the White robstness correction? 3 marks Since exact form of heteroskedasticity is often nknown may be better to fix OLS standard errors to resence of heteroskedasticity Given OLS residals then can show S i i xixi σ Ω σ ixixi, so relace i Var [ Ω ] OLS σ with Var robst S OLS σ is a consistent estimator of 3. Given the following model, in mean deviation form, yo ssect that the variable, x, is measred with error and so the OLS estimate of b sffers from attenation bias y i bx + i Given a ossible instrment x, let W [y : x : x ] and W W

8 the samle size is 00 a Find the OLS and IV estimates of the coefficient on x 6 marks OLS estimates given by x x - x y where x [x ] IV estimates given by z x - z y where x [x ] and z [ x ] so in this case Given y y W W x y x y y x x x x x y x x x x x So b ols 9/3 3 And b iv 0/ 5 as wold exect in resence of measrement error OLS estimates sffer from attenation bias closer to zero b The estimated residal variance from the IV estimation and hence the standard error and statistical significance of the IV estimate of b 0 marks eed s IV IV IV / n y x b IV y x b IV / n y y - b IV y x + b IV x x b IV / n From matrix and answer to above s IV 75 *5* / 00 60/00.6 So Varb IV s IV - - s IV x x - x x x x -.6/5/ So SEb IV.4 Hence IV estimate of b is statistically significant at 5% level, since t5/ c What do yo nderstand by the term reliability ratio?

9 The reliability ratio σ xt σ σ xt + σ σ σ xt + 3 marks which can be derived from the eqation for the degree of attenation bias of OLS in the resence of measrement error is the roortion of the variation in the nobserved variable that can be exlained by the variation in the observed variables. The closer this ratio to the less measrement error noise and the greater the signal in the observed variable. d Show how yo cold estimate the reliability ratio if yo had indeendent measres of the variable x 6 marks Given two alternative roxies for x : x a and x b measred with error sch that x a x + e and x x + v Then the correlation coefficient r x a, x b Cov x a, x b Var x a Var x b Covx + e, x + v Varx + e Varx + v which since e and v are indeendent imlies r xa, xb Varx Varx + var e + Varx + var v σ xt σ σ xt+ and identical if varevarvσ so correlation coefficient between the two mismeasred roxies eqals the reliability ratio

10 4. a Show that the IV estimator is a consistent estimator of in y + if [ /n ] 0 0 marks P y P y IV ] [ where P z - Hence + P P P y P IV + P P IV since P all these terms samle averages are finite ie will not converge to zero as samle size increases However P and final term will converge to zero as by assmtion needed for instrmental variables Hence + 0 * P IV

11 and IV is a consistent estimator b What are the roblems cased by weak instrments for IV estimation? In variable model zy z / z IV + + zx zx / zx so Cov z Corr z * se IV + + Cov zx Corr zx * se x 6 marks weak instrment means low correlation and hence IV estimate may be long way from vale even in large samles comare with Cov x Corr x * se OLS + + Var x se x so not necessarily better to se IV rather than OLS if Corr z Corr zx > Corr x c How cold yo test for the resence of a weak instrment in yor data? 4 marks st stage of sls will atomatically give significance of instrment. In case of single endogenos variable can show + OLS IV E F where EF is the F vale of goodness of fit in the st the sls regression stage of so if EF 0 then size of bias of IV wrt OLS is /9 which is deemed small enogh to be accetable and comensate for higher standard errors in IV

12 d Otline the form of a test of the validity of the overidentifying restrictions in yor instrment set. 5 marks either Hasman test of sbset of instrments against fll instrment set or. Estimate model by SLS and save the residals. Regress these residals on all the exogenos variables inclding those variables in the original eqation that are not ssect sls d 0 + b + d + d +. d l l + v and save the R 3. Comte *R 4. Under the nll that all the instrments are ncorrelated then *R ~ χ with L-k degrees of freedom L is the nmber of instrments and k is the nmber of endogenos right hand side variables in the original eqation