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# REAL EXAMS Econ 103: Econometrics

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Till von Wachter Economics 103 Department of Economics Introduction to Econometrics UCLA Winter 2014 PRACTICE Final Examination WITH ANSWERS March 2014 These is a three hour open book exam in which you are allowed to use your notebook your textbook and a calculator Please answer ALL questions in all parts of the exam Please choose the best answer among all available answers Only one answer is the best answer Please choose one and only one answer If more than one answer is marked the question will not be counted You are provided with a results appendix labelled STATA OUTPUTS Parts IV of the exam refers to this appendix Please type your name and your UCLA ID First Name Last Name UCLA ID Please start solving the examinations only when you are instructed to do so Please stop immediately when instructed to do so Good Luck Part I Part I Theoretical Questions each question worth 2 points Consider the following regression model vi 51 52 U2z39 K UKz 5i 1 The interpretation of the slope coefficient k in the model is as follows a a 1 change in wk is associated with a 313 change in y b a 1 change in wk is associated with a 513 change in y holding all other K 1 regressors constant c a change in wk by one unit is associated with a k change in Y d a change in sick by one unit is associated with a k change in Y holding all other K 1 regressors constant ANSWER D This follows from the de ntion of the regression coe flcients in multiple regression When we interpret the coe flcient of a speci c independent variable in a multiple regression we know that the other included independent variables are held constant 2 Suppose that we run a regression with K Variables and obtain the SSE for that regression say S SEK Suppose now that we add an additional regressor resulting in a regression with K 1 Variables and obtain the SSE for that regression say S SEK1Then a SSEK Z SSEK1 always b if adjusted R2 R2 then SSEK SSEK1 c SSEK SSEK1 always d We cannot determine the relationship between S S EK and S S E K1 based on the information provided ANSWER A When we add a new regressor to the regression the R2 always either increases or remains con stant Please also refer to the formula of R2 3 R2 is a Valid measure of the goodness of t of the regression if a R2 is the same as the adjusted R2 b the regression has a constant term c d R2 is always a Valid measure of the goodness of t the regression has more than two Variables ie K gt 2 ANSWER B The R2 is de ned as SSR SST and hence is in principle ualid no matter what the regression speci cation However without a constant term SST 75 S S RS SE In that case hence R2 does not measure the fraction of the total variance in the outcome uariable ezzzplained by the regression and is not constrained between 0 and 1 Hence it is not a sensible measure of the goodness of t In a given data set the larger is R2 a the lower is SSE b the larger is SSE c the lower is SSR ANSWER A Refer to the formula of R2 1 SSE S ST Note that SST is uniquely pinned down by the data set Which of the following statements is not true a a high R2 or R2 does not mean that the regressors are a true cause of the dependent variable c a high R2 or R2 always means that an added variable is statistically signi cant d a high R2 or R2 does not necessarily mean that you have the most appropriate set of regressors b a high R2 or R2 does not mean that there is no omitted variable bias ANSWER C R2 does not indicate anything regarding any single regressor since it is a measurement of the joint goodness of t It also does not say anything more about an added uariable beyond that some additional variance of the outcome uariable may be explained by inclusion of that uariable Choosing the right functional form for the regression is important because it allows one to capture the speci c features of the observed data it assists one in collecting the right data for the empirical analysis it permits one to use the R2 as a measure for the model speci cation it prevents one from including irrelevant variables in the regression ANSWER A In class we haue referred to the question of whether a relationship between the outcome variable and a regressor is linear or nonlinear as the question regarding functional form Hence a is the correct option Note that in a broad sense functional form is sometimes also referred to the appropriate set of regressors to be included in the regression model However this is not how the concept was used in class If you had a two regressor regression model then omitting one variable which is relevant a will have no effect on the coef cient of the included variable if the correlation between the excluded and the included variable is negative b will always bias the coef cient of the included variable upwards c can result in a negative value for the coef cient of the included variable even though the coef cient will have a signi cant positive effect on Y if the omitted variable were included d makes the sum of the product between the included variable and the tted residuals different from 0 ANSWER C Optiona is wrong since the existence of bias has nothing to do with the sign of the correlation between the endogenous and exogenous uairables it only determined by whether these two variables are correlated with each other Option b is wrong since the bias can be both upward or downward Optiond is wrong since the sum of the product is always zero independent from the regression assumptions To see why the statement in option c is correct refer to the formula of the omitted uariable bias we discussed in lab lecture or the formula in the book A pattern discussed in c could arise if either the effect of the omitted variable on the outcome is positive and the correlation of the omitted variable with the included regressor negative or vice uersa the e ect of the omitted variable on the outcome is negative and the correlation of the omitted variable with the included regressor positive 8 As the sample size say N increases with Varying data points the length of the 100 1 oz con dence interval a increases does not change c d increases in some cases and decreases in others b decreases ANSWER B If the sample size increases with uarying data points meaning that the additional values for 1 are di erent than the mean and hence add information that can be used to estimate the regres sion coe lcients then the standard error of parameters decreases But when the standard error decreases the length of the con dence interual decreases Please also refer to the formulas of the estimated uariance of OLS coe icients and interval estimation 9 Which of the following will change if you scale the dependent Variable in a simple regression model p Value 3 b c R2 lt1 31 t Value of 52 ANSWER D Rescaling of the dependent variable causes the changes of all the coe lcients on the righthand side 10 Which of the following is NOT an assumption of the Simple Linear Regression Model a The Value of y for each Value of 5c is y 51 5253 e b The Variance of the random error e is Var e 02 c d The parameter estimate of 52 is unbiased The covariance between any pair of random errors e and ej is zero 4 11 12 13 14 ANSWER D Please refer to the SR assumptions Optiond is the result of the SR assumptions but it is not an assumption itself When collinear variables are included in an econometric model coef cient estimates are a biased downward and have smaller standard errors b C d unbiased but they have larger standard errors biased upward and have larger standard errors biased and the bias can be negative or positive ANSWER D Unbiasedness is only related to the MR assumptions As long as the expectation of the error term is zero and the regressors are nonrandom then all the coe icient estimates are unbiased However the collinearity problem can cause the standard error of estimators to increase Please also refer to the formula of the estimated variance of estimators in the MR model when K3 If one rejects the null hypothesis that H0 y 0 against H 1 k 75 0 at the signi cant level a then a heshe will not reject it for H0 y 0 against H1 k gt 0 b It cannot be determined whether he she will reject it for H0 k 0 against H 1 k gt 0 c heshe will reject it for H0 y 0 against H1 k gt 0 d heshe will reject it for H0 y 0 against H1 k gt 0 but will not reject it for H0 k 0 against H1 k lt 0 ANSWER B To answer this question draw the rejection region of the two tests completely ouerlapping You will nd they are not One will reject the null hypothesis H0 k 0 against H 1 k gt 0 at the signi cant level er a if and only if zeros does not belong to the 100 1 ac con dence interval for y b if and only if zeros belongs to the 100 1 cu con dence interval for k c d There is no relation between con dence interval and hypothesis testing the 100 1 cu con dence interval cannot help in testing this hypothesis ANSWER C The 1001 a con dence interval is related to the testing result of the two sided test H0 5 0 H1 75 0not to the one tailed test stated in the question To decide whether Y1 50 1X u or 50 1X ui ts the data better you cannot consult the regression R2 because a lnY may be negative for 0 lt Y lt 1 b the SST are not measured in the same units between the two models 15 16 17 c the slope no longer indicates the effect of a unit change of X on Y in the log linear model d the regression R2 can be greater than one in the second model ANSWER B You can not compare the R2 of two regression models when the dependent variables are not measured in the same unit because the SST is not the same Hence the R2 do not refer to the fraction uariance ezplained of the same SST To test whether or not the population regression function is linear rather than a polynomial of order r check whether the regression R2 for the polynomial regression is higher than that of the linear regression compare the SST from both regressions look at the pattern of the coefficients if they change from positive to negative to positive etc then the polynomial regression should be used use the test of r 1 restrictions using the F statistic ANSWER D This question ezzzplicitly calls for a test Hence while the relevant F test statistic can be calculated using R2 or the SST a simple comparison of these statistics is not su flcient a test is required The binary variable interaction regression can only be applied when there are two binary variables but not three or more is the same as testing for differences in means cannot be used with logarithmic regression functions because it is not de ned allows the effect of changing one of the binary independent variables to depend on the value of the other binary variable ANSWER D Please refer to the de nition of binary uariables Also compare the marginal e ects of a binary uariable when there is an interaction term and when there is not Consider the following multiple regression models a to d below DFem 1 if the individual is a female and is zero otherwise DMale is a binary variable which takes on the value one if the individual is male and is zero otherwise DMarried is a binary variable which is unity for married individuals and is zero otherwise and DSingle is 1 DMarried Regressing weekly earnings Earn on a set of explanatory variables you will experience perfect multicollinearity in the following cases except ANSWER C Please refer to the de nition of perfect collnearality The typical case when perfect collinearity arises is when indicator variables for all values of a categorical values are included since then these indicator variables sum to 1 which is equal to the constant term 18 In the regression model Yi 50 1Xi 2Di 3XiDi ui Where Xi is a continuous Variable and Di is a binary Variable to test that the effect of Xi on Yi is identical for both Values of Di you must use a separate t test for H0 52 0 and H0 53 0 F test for the joint hypothesis that 50 0 51 0 a t test for H0 53 0 F test for the joint hypothesis that 52 0 B3 0 a b c d ANSWER C The coe lcient of the interaction term measures the di erence of the e ect of two groups If it is zero then there is no di erent e ect 19 A large company is accused of gender discrimination in wages The following model has been estimated from the company s human resource information W ln Wage 1439 00834Edu 00512Exper 01932Male Where WAGE is hourly wage Edu is years of education Exper is years of relevant experience and Male indicates the employee is male What hypothesis would you test to determine if the discrimination claim is Valid a b c d H0 1 5Maze 0 H1 5Maze 2 0 H0 1 5Maze 5Edu 5Eaper 0 H1 1 5Maze 75 0 and 5Eacpea 75 0 and 5Eacpea 75 0 H0 1 5Maze 5Edu 5Eaper 0 H1 1 5Maze 75 0 07 5Eaper 75 0 07 5Eaper 75 0 Ho 1 5Maze S 5Edu I 07 I 5Maze S 5Expea H1 1 5Maze gt 5Edu I 07 I 5Maze gt 5Eaper ANSWER A 51 ALE captures the di erence in mean log wages of males and females and hence is a a measure of discrimination in wages Since the regression controls for years of education and years of experience wage di erences between gender due to di erences in education and experience are controlled for Hence the coe lcient on MALE mean di erence in wages holding education and experience constant 20 When an exogenous instrument is used IV estimators are consistent and approximately normally distributed in large samples unbiased and BLUE in all sample sizes consistent if z is normally distributed normally distributed in all sample sizes and consistent in large sample sizes 21 22 ANSWER A Please refer to the de nition and the computation of the instrumental variables IV estimator The whole point of this approach is that the IV estimator is consistent even though the OLS estimator might not be In addition it turns out to be approximately normally distributed in large samples as well A Which of the following statements is true regarding uar 2 when estimated by IV using 2 as an instrument for x a instrumental Variable estimation leads to larger Variance of estimates compared to OLS b instrumental Variable estimation leads to smaller Variance estimates compared to OLS the Variance of the IV estimates can be larger or smaller than OLS depending on 732 c d there is no way to know which estimator leads to estimates with a larger Variance it depends on the data ANSWER A Please compare the formula of the estimated variance of the instrument uariable estimators and the OLS estimators The whole point of the IV estimator is to only use a part of the uariation in the explanatory variable that part which is not correlated with the error term This is clear from the fact that in the second stage the predicted value for the explanatory uariable is used The uariance of the predicted explanatory variable is smaller than the variance of the actual explanatory uariable Hence the uariance is higher What is the null hypothesis when performing an F test to test the strength of multiple instruments j 1 J the instruments are weak with no coefficient 63 on the instruments in the rst stage different from 0 the instruments are strong with all coefficients 63 on the instruments signi cantly different from 0 the instruments are sufficiently strong with a coefficient 63 on at least one instrument sig ni cantly different from 0 the instruments are weak with all coefficients 63 on the instruments different from 0 ANSWER A Please refer to the de nition of the F test and the rststep of the QSLS estimation 23 When you are implementing an instrumental Variable regression you are worried about a potential direct effect of the instrumental Variable on the outcome a weak relationship between the instrumental Variable and the endogenous Variable a remaining correlation of the instrumental Variable and the error term all of the above ANSWER D A ualid IV should be truly exogenous no correlation with the error term correlated with the endogenous variable and should only a ect the dependent variable through the endogenous uariable ie haue no direct e ect on the outcome Part II Practical and Computational Questions each question worth 2 points Consider the following regression model vi 51 525622 5K UKz39 6239 24 You have estimated the following simple regression model 3 379 144c3 1 What is the elasticity when X 849 ANSWER c Elasticity 32 f 144 4 3 4 a2 4329 at 849 gt 3 379 1448493 126022 gt Elasticity 4328492 210 25 You have estimated a two Variable model ie K 2 and your printout includes the following information 3 361400 3 1272 3 39461 SST 758912 Then the R2 for this regression model is 1 N a b c d 3939 391 L 390 3 5 5 Z39 ANSWER D 2 2 sxy In the simple regression model R rxy Z5905 rxy 072 such that rgy 052 Hence cl is the correct answer Note that in the nal either exact values will be given or it will be stated explicitly that the result is approximate 2 For the numbers of the question we get that 26 Suppose that K 3 Q 9 E2 3 b2 12 53 2 b3 13 then the estimate for 51 b1 is 27 28 29 ANSWER A For any regression model we know that evaluating the estimated equation at the means will return the mean Mathematically g b1 bgfg b3E3 gt 9 b1 123 132 gt bl 28 A company is accused of racial discrimination in wages The following model has been estimated from the company s human resource information ln WAGE 1439 0834 gtlt EDU 0512 gtlt EXPER 1932 gtlt WHITE Where WAGE is hourly Wage EDU is years of education EX PER is years of relevant experience and WHITE indicates that the employee is white How much more do white employees at the rm earn on average 121 per hour more than non whites 1932 per hour 1932 more than non whites 19320 more per year than non whites ANSWER C The loglinear regression equation can be interpreted such that a one unit increase in an ask uariable independent uariable leads to approzzzimately a 100Bk change in the y dependent uariable Randomized controlled experiments are needed to accurately measure treatment effects Without a the expense of having to treat everyone b raising public debate c d omitted variable bias exposing everyone to untested treatments ANSWER D One of the main purposes of a randomized controlled experiment is to obtain estimates of the causal e ect of a treatment that is not a ected by omitted variable bias The following economic model predicts whether a voter will vote for an incumbent school board member INCUMBENT 51 B2MALE B3PARTY B4MARRIED B5KIDS Where 10 30 31 INCUMBENT 1 if the voter votes for them 0 otherwise MALE 1 if the voter is a male PARTY 1 if the voter is registered with the same political party as the incumbent MARRIED 1 for married voters 0 otherwise KIDS is the number of school age children living in the voter s house If you believe marriage affects male and female voters differently which variable should you add to the economic model to allow you to test the hypothesis a MALE gtlt PARTY b MALE gtlt MARRIED c MARRIED gtlt KIDS d MARRIED gtlt PARTY ANSWER B Inclusion of the interaction MALE gtlt MARRIED allows marriage to have di erent e ects on men and women in terms of their likelihood of voting for an incumbent Consider the same setup as in the previous question Suppose it is claimed that a married man is less devoted to the incumbent than a married woman Then the results must have ANSWER A In the setup in the previous question the variable marriage has the same e ect on both men and women Therefore the claim simpli es to a man is less devoted to the incumbent than a woman Consider the following regression model given by Eh 11 25622 5 Uz393 6i with the variance covariance matrix of the regression coefficients 7 2 3 2 5 2 3 2 3 11 32 ANSWER A 39va7quot2b2 b3 439va7quotb2 va7 b3 4c0vb2 b3 gt 39va7 2b2 b3 45 3 4 2 15 se2b2 b3 U0I 2b2 b3 0A 122 You have estimated the following equation TestSc07quoteZ 6073 385Inc0meZ 00423Inc0meZ2 2 where TestScore is the average of the reading and math scores on the Stanford 9 standardized test administered to 5th grade students in 420 California school districts in 1998 and 1999 Income is the average annual per capita income in the school district measured in thousands of 1998 dollars The equation a suggests a positive relationship between test scores and income until a value of the income variable of approximately 45508 b is positive until a value of Income of 61081 c does not make much sense since the square of income is entered d suggests a positive relationship between test scores and income for all of the sample ANSWER A To nd the point at which a di erentiable function changes slope take the first derivative and set it equal to 0 385 00846Inc0me 0 dlncomei gt Incomek 45508 12 Part III Lab Questions each question worth 1 point The questions in this part relate directly to the Lab class 103L Consider the following regression model 33 34 35 36 vi 51 52 U2z39 K5UKi Bi The essence of the regression analysis conducted by J J Espinoza is to determine how Walt Disney Company should structure advertising in the course of the introduction of new movies evaluate the performance of the analysts at Walt Disney Company determine who to award nancial aid to from a selected minority group evaluate the performance and labor market outcome of Hispanic students ANSWER A In his guest lecture JJ described how he uses regression analysis to best time the advertisements of new movies The objective of the analysis performed by Vikas Gupta for Factual is to deliver the most relevant ads through mobile devices conduct market research for new electronic devices ANSWER C In his guest lecture Vikas described how his company processes cell phone GPS data to construct tiers of individuals with similar characteristics to which his clients can deliver adduertisements to The approach of calculating the regression coefficients in three steps discussed in the lab lecture is a way to get approximate values of the multiple regression coefficient estimates is a way to get exact values of the multiple regression coefficient estimates is a way to obtain estimates of the causal effect can only be used if assumptions MR1 MR5 hold ANSWER B The threestep method discussed in the lab lecture is equivalent to regular estimation of multiple regression ie the exact identical coe flcient estimates are obtained this is based on the so called Frisch Waugh theorem Regressing average TESTSCR on the average student teacher ratio STR in school districts in California we had found a that larger classes have a positive causal effect on testscores but the effect is moderate b that larger classes have a negative causal testscores but the effect is moderate 13 37 38 c that larger classes are associated with smaller testscores but the effect is moderate relative to the mean testscore d that larger classes are associated with smaller testscores but the size of the effect is hard to interpret ANSWER C We found a negative coe flcient of STR but concluded that a relative to the mean TESTSCR the e ect is moderate and b that this e ect is likely to at least in part to be due to omitted factors The coefficient on STR in the regression of TESTSCR on STR a is likely to yield the causal effect of class size on test scores b C d cannot be interpreted because the true relationship is not linear is likely to yield a biased coefficient of the effect of class size on testscores cannot be interpreted since STR has a different scale than TESTSCR ANSWER B We had discussed how other factors are likely to be present that determine TESTSCR and are correlated with STR Suppose We failed to include the variable SUBSLUNCH indicating the fraction of students re ceiving a subsidized lunch in the regression of TESTSCR on STR Let 53 be the coefficient on SUBSLUNCH were it to be included in the regression The omitted variable bias is a 53 b couSTRSUBSLUNCHuarSTR c couTESTSCRSUBSLUNCHuarSTR d 3couSTRSUBSLUNCHuarSTR ANSWER D We had derived the omitted uariable bias formula based on the probability limit of b2 and found that the bias the di erence between the probability limit of b2 and the true 52 is equal to 3couSTRSUBSLUNCHuarSTR The book presents a very similar formula based on the expectation rather than the probability limit Part IV Demand for Meat each question worth 2 points This question refers to the output labelled The Demand for Meat provided in the STATA Output Note that the stata command matriX list eV is the same as estat vce 39 The F statistic for H0 52 53 54 0 from the rst regression is approximately a 25 71738 b 808 14 40 41 42 c 1812 d 170 ANSWER D Simply obserue the F statistic reported in the output of the first regression This is precisely the null hypothesis that is being tested The results indicate that a there is high degree of collinearity between income and prices b C d it is essential to have the Variable income in the regression the coefficient on the Variable income is signi cant because all the others are not signi cant one should include in the regression the interactions between all prices ANSWER D Observe that in the second regression income is highly signi cant very low p ualue Further more the standard errors of the coe lcients on prices are all substantially reduced which is the opposite of what we would e3pect if there was a high degree of collinearity between income and prices Instead including income reduces the variance of the error of the regression see the ualues for the root MSE and hence reduces estimates of the standard errors The results of the rst regression indicate that the hypothesis H0 52 53 4 0 a is true for any a 3 10 b H1 is true for any a gt 10 c d We cannot conclude from the results obtained which hypothesis is true neither H1 nor H0 are true ANSWER A The output for regression 1 displays the pualue on the F statistic as 0192 so for any at lt 01921 we can not reject the null hypothesis The point estimates from the second regression indicate that Meat and Vegetables and fruit are substitutes While meat and bread and cereal are comple ments Meat and Vegetables and fruit are complements as are meat and bread and cereal Meat and Vegetables and fruit are complements while meat and bread and cereal are sub stitutes Vegetables and fruit and bread and cereal are substitutes ANSWER A If 3 and y are complements an increase in the price of 3 will lead to a decrease in the quantity demanded of y If 3 and y are substitutes an increase in the price of 3 will lead to an increase in the quantity demanded of y Look at the sign of the coe lcients of the relevant variables to make an assessment 15 43 Based on the results of the second regression the t statistic for the test of H0 53 54 0 against H1 53 54 a 0 Will be ANSWER A Quick hintA if A is a random variable and you are testing H0 A c vs H1 A 7E c then A c t stat fa seA is always positive so the sign of A c determines the sign of the tstat In this case A 53 54 3 33 34 U0 7quot5A3 Ua7 34 4 2C0 U33 34 Check the third stat output table to nd the appropriate variances gt seB3 34 00214 00498 200011 02627 gt t stat 678 44 Consider testing the restriction in model 2 that 55 0 a this can be done based on a t test b this can be done using an F test based on SSR in model 1 and model 2 this can be done using an F test based on R2 in model 1 and model 2 c d all of the above ANSWER D The slides show that a t and F test are equivalent if we are testing only one restriction Note that SST E y 3 2 is the same in both models since the restrictions under the null hypothesis do not lead to a modi cation of the dependent variable Note that model 1 is the restricted model and model 2 is the unrestricted model in this case Hence the basic de nition of the F test statistic is F SSl 1Sl 2SE2 NIK Therefore R 1 SSE1 and R3 1 SSE SST SST gt R3 R SSE SE2 and 1 Rg gt Ii2 1 SSJEELSTESE2 The Fstat can then be computed with the correct degrees of freedom and number of restrictions Similalry for the SSR we have that SST SSESSR and hence SSE SST SSR Using the de nition of the Fstatistic we have that F 5 5 I 1 NIK Note that deviding the numerator and denominator by SST gives the R2 version of the test 45 The large constant estimate in both regressions suggests that a prices cannot affect the consumption of an essential good such as meat c d the adjusted R2 in the rst regression should be smaller than R2 b there is no implication regarding the impact of the other Variables there is almost no Variability in the consumption of meat 16 46 47 48 49 ANSWER B Note that the constant is the level of meat purchases predicted at a price of meat of 0 In this case we are very unlikely to ever observe a price of zero or near zero and hence the constant term cannot be interpreted directly Instead it sets the level of the prediction of meat purchases based on all the regressors in the model More generally the value of the constant doesn t change the interpretation of the signs or magnitudes of the slope coe flcients The interval estimate for the coefficient on income ie 55 for ct 05 is approximately a 15662912 b 15662912 c 01420460 d 2 X 0142 2 X 0460 ANSWER B Look at the 95 con dence interval in output table 2 on the coe flcient for y The fact that R2 in model 2 is substantially higher than in model 1 means a that income explains an important part of the Variance of meat purchases b c that income is exogenous that income has a causal effect on meat purchases d that the coefficients on the Variables are biased ANSWER A R2 is a measure of how much variation in the dependent variable is explained by the model Hence if R2 goes up substantially when income is added to the model you can then conclude that income ezplains an important part of the variance of meat purchases The fact that coefficients on prices in model 2 barely change as income is included as regressor implies a that these coefficients measure the causal effect of these prices on meat purchases b that these coefficients do not measure the causal effect of these prices on meat purchases Z39 c that income is likely heterogeneous in the population Z39 d that there is a low correlation of individual income and these prices ANSWER D We can t conclude that the coe flcients measure the causal e ect of these prices on meat pur chases since there still is potential omitted variable bias If there were high correlation between individual income and these prices we would expect to an omitted variable bias in the regression which does not include income The sign and magnitude of the omitted variable bias depends on covpincome for all prices as well as the sign and size of the e ect of income on meat purchases The elasticity of meat purchases with respect to a change in price of meat a cannot be calculated without further information 17 50 b is 6763 C is 06756401 d is economically signi cant ANSWER A We need to know the quantity of meat for which we will evaluate the elasticity as well as a level of the price One approach would be to take the mean of the quantity of meat purchased and the mean of the price Neither the means nor speci c values for quantities or prices are provided hence the answer is a The 95 con dence interval for the effect of price of meat on meat purchases a is approximately 11629 01885 b c is unaffected by the scaling of meat purchases d all of the above changes with the scaling of income ANSWER A Scaling income doesn t a ect the other regressors Scaling the dependent variable meat purchases will rescale all the coe lcients and hence their con dence intervals 18 at Raxldall R jas EG 11 I39I11iCE 103 DEprartme11t of Ecnnumics I tFDdUC139riD 11 EcnI1 metricE UCLA Fa1iI 2U12 Fi11a1 Exam quotDtember I 2012 F131 E11111 l 4E39391it 1111 1 E39I a I l JlE39l1l you 11tEE391 to sham all mur ciJurk and t13 ft1rm11 a15j 11591 F39i rst Name qk quot UCLA ID E f ectim11 P102159 fZIif mmij ata11 39t tlim ec1j1I1 111i1t 4i4I19tr1T1Tt3d tn Th 50 Part 1 Short Answer Pl bl lll 1 Let 3151 113 d31101E L119 pe139nenmjgE emf t mliJ11 gmd1s at E3 liigh h301 EirfiEiK 3911g a pai95i11g arc39r3 mm 1 5tm1d1139 cliEed I11at11E111a1i3 E i l S11 I1 E39E we wizslil 151 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ehteiI1ed e 1elue 2f pk fer the heturhel predictor e nr e value eff 45 fer the ee1ree1ed predieter Wllet ie the reepeetiw 1331118 ef the eetimete1 errer e tencL1rd de1e391etieu A FR 7 034 id e8 l V2M 4 e 2355 C 2250 L500 Economics 103 Introduction to Econometrics Spring 2011 Professor Maria Casanova FINAL EXAM Version D TIME ALLOWED 3 hours Instructions DO NOT USE A CALCULATOR For complicated math simply set up the problem but do not solve it You will be graded on your setup SOME CRITICAL VALUES THAT YOU MAY FIND USEFUL ARE PROVIDED ON THE NEXT PAGE POINTS ARE NOTED NEXT TO EACH QUESTION Be sure to allocate time appropriately GOOD LUCK NAME STUDENT ID TA SECTION Unless speci ed otherwise use a level of signi cance of 005 for hypothesis testing Critical values at 005 signi cance level Standard Normal tWo sided test 196 Standard Normal one sided test 164 F1O384 F2300 F3oO260 F4O237 F5oO221 X 384 X 599 X 781 X3949 X 1107 BEGINNING ECON103 EXAM PART 1 TRUEFALSEEXPLAIN YOU ARE GRADED ON YOUR EXPLANATION USE BOTH MATH AND INTUITION 5 points each 50 points total 1 The signi cance level of a hypothesis test is best de ned as the probability of rejecting H0 when H0 is not true FALSE the signi cance level is the probability of rejecting H0 when H0 is true 2 We decide to measure the effect of class size on students test scores by running the following experiment 50 of 3rd graders in the Los Angeles School District are assigned to classes with 30 students large classes and the other 50 are assigned to classes with 15 students small classes In order to estimate the effect of class size using the differences estimator we only need to observe students test scores after the treatment was introduced On the other hand to compute the differences in differences estimator We need to observe test scores both before and after the treatment TRUE The differences estimator is equal to 17quot39e ed f 39 1C quot39 l f 39 While the differences indi 39eIenCeS estimator is equal to Yt739eatedafte739 1t739eatedbef0739e Ycont739olafte739 1cont739olbefo739e Y is a binary variable equal to 1 if an individual smokes and X measures income If we model the relationship between the two variables using a linear probability model the marginal effect of a one unit change in income on the probability of smoking is equal to a constant TRUE In the linear probability model Y oz X 5 the marginal effect of a unit change in X on the variable Y is constant and equal to that is 3 B All of the following can cause the OLS estimator to be biased a the omission of an important explanatory variable b measurement error in the dependent variable and c a small sample size FALSE While a can cause cause the OLS estimator to be biased measurement error in the dependent variable and a small sample size do not the latter will just cause the estimator to be imprecise In the linear regression model with only one regressor Y 30 1X u the overall regression F test and the t test on B1 reported by Stata are testing the same hypothesis TRUE The t test reported by Stata tests the hypothesis that the coefficient is equal to zero that is H0 31 0 The overall regression F test tests the hypothesis that all slope coefficients are equal to 0 In this case there is just one slope coef cient so the null hypothesis is also H0 31 0 You use observations from 1000 households to run a regression of food expenditure F on income I You are concerned that the variance of the error term increases with income and therefore decide to use the weighted least squares WLS estimator In this case WLS will give more weight to observations from low income households TRUE WLS gives more weight to observations with smaller variance In this case since the variance increases with income observations from low income households will have a smaller variance and therefore receive more weight If we estimate a demand function with a linear log model the price elasticity of demand will not be constant TRUE For the price elasticity to be constant we have to use a log log model In the linear log model Q oz B lnP 5 the price elasticity is 662 Notice that the price elasticity is de ned as g and that in the linear log model g P 3 8 10 For B2 to be an ef cient estimator of S2 it must be an unbiased estimator of S2 TRUE According to the de nition given in Lecture Notes 2 an estimator is ef cient if it has the lowest variance among all unbiased estimators Fifth graders test scores TS are positively in uenced by both time spent studying T and by their parents income I We regress TS on T but omit I from the regression If studying time is inversely related to parents income the coef cient on studying time may turn out to be negative TRUE The true model is TS oz 1T 2I 5 but instead we estimate TS oz y1T 5 Our estimate 1 will have omitted variable bias because the two necessary conditions are ful lled in this case the omitted variable I has an effect on TS and it is correlated with T Therefore E 31 bias Recall that the sign for omitted variable bias is given by 62C039vT I In this case you are told that S2 is positive and C0vT I is negative so the bias will be negative Since S1 is positive but the bias is negative if the bias is large enough the sum of the two may turn out to be negative To determine the effect of a new drug against the u we randomly assign patients into a treatment group which receives the drug or a control group which receives a placebo and then measure the number of days until their symptoms subside A young researcher runs the following regression Y 30 31X u where Y is the number of days that the patient was sick and X is equal to 1 if they were part of the treatment group and 0 otherwise However the researcher makes a coding mistake and he erroneously classi es some patients from the treatment group as belonging to the control group and vice versa Since he ends up with an imprecise measure of X his estimate of the effect of the drug will be too large FALSE In the presence of random measurement error the coef cient will always be biased towards 0 that is the estimate of the effect of the drug will be too small not too large PART 2 80 points total A researcher has data from a sample of men aged 60 to 70 who have not yet retired She obtains the following results when estimating a model of their labor supply H31Trs 19794 0004 P 1387 Wage 0102 Wage 389 00018 044 005 1281 M 7801 K 3210 0 091 204 162 Where Hours is the number of hours Worked per week P is pension income in per Week Wage is the hourly Wage M is a dummy variable indicating Whether the individual is married K is a dummy variable indicating Whether there are children in the household C is a dummy variable indicating Whether the individual lives in a city The total sum of squares equals 1000 The residual sum of squares equals 400 Numbers in parentheses are standard errors 1 What is the excluded group in this regression 5 points The excluded group is that for which all the dummy variables are equal to zero In this case men who are not married who live in households with no children and who do not live in a city 2 What does the coefficient on the variable M measure 5 points How many more hours married men Work compared to single men In this case married men Work on average 1281 more hours per week than single men 3 Suppose you create a dummy variable S equal to 1 if the individual is single ie not married You rerun the regression above including the variable S instead of M What will the coefficient on S measure 5 points 5 How many less hours single men Work compared to married men What will the coefficient on S be equal to 5 points Since it measures the opposite to the coefficient on M it has to be equal to its opposite that is to 1281 What will be the new value for the constant 5 points In the rst model the constant measured the average Wage for a single men for Whom all other variables are equal to 0 In the new model it measures the average Wage for a married men for Whom all other variables are equal to 0 It is equal to 19794 128121075 Consider a married man living in a household with children living in a city with an hourly wage of 4 and pension income of 100 per week How many hours do you predict he will work 5 points Iohms 19794 0004 X 100 1387 X 4 0102 X 16 1281 X 1 7801 X 1 3210 X 1 Derive a formula for the effect of wages on hours in this model 5 points 91ours T 1387 2 0102 W 5 Wage X X age Which regressor would you have to add to the model to allow the effect of wages on hours to depend on whether there are children in the household 10 points You would have to add an interaction between C and Wage ie C X Wage You would estimate one extra coefficient let s call it 6 for this interaction The marginal effect of wages on hours would be 91ours T 1387 2 0102 W 6C 8Wage X X age Another researcher uses the same sample to regress H ours on a different set of regressors and obtains an R2 equal to 049 Does her regression t the data better than the one above use R2 rather than R2 to compare the regressions 10 points 10 11 12 The R2 for the regression above is 1 Li 1000 06 The regression above ts the data better A third researcher uses the same sample to regress H ours on another set of regressors and nds that the explained sum of squares is equal to 300 Does his regression t the data better Again use the R2 to compare regressions 5 points 300 The R2 for the third regression is 1000 depend on the regressors included in the model Since the third researcher is using the same 03 Notice that the total sum of squares does not dependent variables the value of the TSS does not change It ts the data worse than the rst one What does it mean for an estimator to be unbiased Be speci c 10 points If we drew in nitely many samples and computed an estimate for each sample the average of all these estimates would give the true value of the parameter Formally E 3 3 Notice that stating the formal de nition was enough Considering that this model excludes men who have already retired are you worried that your estimate of the effect of wages on hours might be biased Explain your answer 10 points There may be selectivity bias if for example those who have lower wages are more likely to retire PART 3 50 points total You decide to study the labor supply of older men using the same dataset as in part 3 but this time use as a dependent Variable an indicator that is equal to 1 if the man is Working and equal to 0 if he has retired You call this Variable Work You run a logit regression and obtain the following results logit work P wage wage2 M K C Iteration 0 log likelihood 53750555 Iteration 1 log likelihood 5l4l408 Iteration 2 log likelihood 5l34642 Iteration 3 log likelihood 5l346068 Logit estimates Number of obs 1000 LR chi26 4809 Prob gt chi2 00000 Log likelihood 5l346068 Pseudo R2 00447 smoke Coef Std Err z Pgtz 95 Conf Interval P l34l957 0278469 482 0000 l887747 0796l67 wage 0045672 0008942 wage2 0001967 0000815 M 00l3956 0003l7l K 643550l 899l4l2 C 10420281 2365785 cons 2164421 3698581 059 0558 5084664 9413506 1 What are the advantages of using a logit rather than a linear probability model to run this regression 5 points Predicted probabilities are between 0 and 1 Marginal effects are not constant so potentially more exible 2 What is the interpretation of the coefficient on P 5 points Coefficient tells us the sign of the marginal effect of pension income Negative coef cient indicates that a higher pension income decreases the probability of Work 3 How does the coefficient on P compare to the coefficient you would have obtained in a linear probability model Would you expect them to be close Are they measuring the same thing 10 points There need not be any relationship between this coefficient and the one obtained from a LPM since they are measuring different things In the LPM the coefficient measures the marginal effect In this case the marginal effect is equal to the coefficient times the logistic distribution evaluated at 216 01341 X P 4 How would you compute the pvalue associated to Wage2 10 points p value 2CD t 5 Why doesn t Stata report the R2 in this case 10 points The R2 is not an appropriate measure of t for nonlinear models 6 How has the pseudo R2 reported in the regression been computed 10 points It has been computed from the likelihood function It measures the improvement in the value of the log likelihood relative to having no X s PART 4 50 points total You have information on 1000 randomly chosen people living in LA You observe the Value of their houses H and their income I and decide to study the relationship between these two Variables using the following regression lI1Hi 0z0 0z1 lI1IZ39I82 You run the regression in Stata and obtain the following output reg H I Source I SS df MS Number of obs 1000 F 1 998 1154 Model 36l663086 1 36l663086 Prob gt F 00007 Residual 312686907 998 313313535 R squared 00114 Adj R squared 00104 Total 316303538 999 316620158 Root MSE 55974 H Coef Std Err t Pgtt 95 Conf Interval I 5920467 1742583 340 0001 250092 9340014 Cons I 5015161 8970182 056 0576 2261774 1258742 Next you draw a plot with the tted Values from the regression and the residuals 300 I 200 I H House 0 100 I 100 I 200 I I Income Fitted values 0 Residuals 10 1 Based on the scatterplot shown above do you detect any problem with these residuals 5 points Their magnitude increases linearly with income suggesting heteroskedasticity 2 Based on your answer to point 1 above are you concerned about the estimate oh reported by Stata 5 points No The OLS estimators are unbiased in the presence of heteroskedasticity You would be concerned about the standard error not the coefficient 3 According to the Stata output on the previous page does Income have a signi cant effect on House value Based on your answer to 1 above are you concerned about this conclusion 10 points Income has a positive and signi cant effect on house value as can be seen from the t statistic which is greater than 196 in absolute value You would be concerned about this conclusions however because in the presence of heteroskedasticity the standard errors obtained from an OLS regression that doesn t use the robust option are biased 4 You decide to test formally whether the variance of the residuals is related to Income Which test do you run Explain how you proceed with the test and what you conclude To answer this question you may need to use the Stata output shown on the top of the next page where the variable res2 is the square of the residuals from the regression of H on I and I 2 is income squared 10 points White s heteroskedasticity test Null hypothesis model is homoskedastic Alternative hy pothesis model is heteroskedastic Generate residuals from the model above compute square of residuals regress on income and income squared Compute test statistic N R2 1000 x 01360 136 based on output below Compare with critical value from Chi squared with 2 degrees of freedom given on page 2 equal to 599 Reject null Conclude that model is heteroskedastic 11 reg res2 I I2 Source I SS df MS Number of obs 1000 F 2 997 7844 Model I 49051e09 2 24526e09 Prob gt F 00000 Residual I 31172e10 997 312661588 R squared 01360 Adj R squared 01342 Total I 36078e10 999 361136232 Root MSE 55916 res2 I Coef Std Err t PgtItI 95 Conf Interval I I 2126172 12806 166 0097 4639152 3868078 I2 I 4178111 1253756 333 0001 1717807 6638415 cons I 2785206 3207932 087 0385 3509867 9080278 5 Suppose that you don t know how the variance of the residuals depends on Income What could you do to improve your estimate of the standard errors of the model 10 points Estimate model by OLS using formula for heteroskedasticy robust standard errors 6 Someone points out to you that the Variance of the residuals is known and given by Va7 5 I I 2 X 02 Propose a transformation of the model that would have an error term whose Variance does not depend on income 10 points InI20z0 lI1IZ39 z39 1 1 11 1 The Variance of the new error term 391 is constant 12 PART 5 70 points total Consider the following Wage equation that has been estimated from a random sample of 2065 year old male Workers 111 Wi 50 51EdUz39 1 Mi 1 Where ln Wi is the log Wage in dollars per hour of individual 0 Edui is the number of years of education and ui is the error term The estimates you obtain for 60 and B1 are 80 09540025 B1 0074300052 Numbers in parentheses are standard errors a Interpret these estimates 5 points The log Wage for an individual with no education would be 0954 Each extra year of education increases wages by 743 b Can you reject the hypothesis that education has no effect on wages 5 points YES 2 00743 0 00052 gt 196 Another way of seeing this obtained by arranging terms is by noticing that that the coefficient is more than 196 times larger than the standard error You could also have constructed the CI c Can you reject the hypothesis that the return to education is 5 5 points YES The test statistic is Z 00743 005 00243 00052 0052 gt 196 You could also have tested the hypothesis by constructing the con dence interval d What is the con dence interval for the returns to education 5 points 00743 196 X 00052 or 00743 196 X 00052 00743 196 X 00052 13 6 Under which assumptions can 31 be interpreted as the causal effect of edu cation on wages 10 points OLS assumptions SR1 to SR5 Are these assumptions reasonable in this case Give one reason why they may fail 5 points Assumption SR2 would fail in the presence of an omitted variable that is correlated with Edu For example ability If more able individuals tend to have more years of education and ability has an effect on wages independently of education the coefficient on Edu in the regression above is biased upwards In order to get an idea whether these assumptions are violated you include the stan dardized IQ score as an additional regressor and run the following wage regression 111 Wi 70 71Ed uz39 72IQ U2 The estimates you obtain for 71 and 72 are standard errors in parentheses 71 006000050 72 00970010 What do these numbers tell you about the correlation between education and IQ Do these estimates support your concern that 31 cannot be interpreted as a causal effect 5 points They do support the concern After including the additional variable the estimate for the effect of education has decreased indicating that the previous estimate T was likely biased upwards The upward bias tells you that 72 X C039vIQEdu gt 0 A hypothesis test based on 72 indicates that 72 is positive Hence the covariance between IQ and Edu must also be positive for the OVB to be greater than 0 14 h You reestimate the rst equation equation 1 above using an instrumental variable Z Which assumptions do you need to make on the variable Z in order to interpret the IV estimate as the causal effect of education on wages 10 points Z is a relevant instrument that is the correlation between Z and Edu is not 0 Z is exogenous that is Z is not correlated with the error term u Note this question is challenging Do not attempt to answer it until you have nished the rest of the exam You choose as an instrument for the number of years of education a variable Z that is equal to 1 if the individual s father obtained a university degree and equal to 0 if he did not get a university degree In the sample of 2065 year old male workers the average number of years of education is 105 for workers whose fathers do not have a college degree and 116 for workers whose fathers did graduate from university The average log wage of workers whose fathers do not have a college degree is 252 The average log wage of workers whose fathers did graduate from college is 263 Based on this information compute the IV estimate of the return to education 20 points This part was not required for a correct answer but notice that the information provided is enough to guess the value of the parameters of the first stage Edu 7m 7r1Fathe7 Deg7 ee 5 F athe7quotDeg7quotee can only take on the values 0 and 1 Hence 7czu105707 r1x07 r0 Hence fro 105 frl 116 105 11 The second regression is 1I1lVi 0 1 CELiui AW 2s Notice that 31 Am B1 23963 23952 03911 01 An extra year of education increases wages by 10 116 105 11 Using the information provided you can obtain an estimate 15 Moshe A Buchinsky Economics 103 Department of Economics Introduction to Econometrics UCLA Fall 2011 Final Examination December 6 2011 STATA OUTPUTS Boca Raton Output reg price sqft bedrooms bath pool traditional waterfront summarize Variable Obs Mean Std Dev Min Max price 1080 1543632 1229123 22000 1530000 sqft 1030 2325933 1003093 662 7397 bedrooms 1080 317963 7094959 1 8 baths 1030 1973143 6120669 1 5 age 1030 1957407 1719425 1 30 owner 1030 4383339 5001031 0 1 pool 1080 0796296 2703444 0 1 traditional 1030 5333339 4937163 0 1 fireplace 1030 562963 4962496 0 1 waterfront 1080 0722222 2589754 0 1 Source SS df MS Number of obs 1030 F 6 1073 30933 Model 10330e13 6 17217e12 Prob gt F 00000 Residual 59711e12 1073 55649e09 R squared 06337 Adj R squared 06317 Total 16301e13 1079 15108e10 Root MSE 74593 price Coef Std Err t Pgtt 95 Conf Interval sqft 3133231 3756303 2173 0000 7446129 8920433 bedrooms 2202313 4561115 483 0000 3097285 1307342 baths 4311362 5439773 793 0000 3244431 5379243 pool 4364432 3706576 056 0576 2194833 1221936 traditional 1307622 4675475 337 0000 2725033 390211 waterfront 5706061 9142954 624 0000 3912052 7500071 cons 4452149 1131474 393 0000 6672302 2231997 4 5 matrix list eV symmetric eV77 sqft bedrooms baths pool traditional waterfront cons sqft 14113603 bedrooms 3352335 20803771 baths 10239019 45513546 29591179 pool 52963361 13643761 16703772 75304463 traditional 13115597 37078142 43462991 19414549 21860062 waterfront 36334658 34016601 57745514 17549714 50155303 33593614 cons 13912195 36134537 19572605 27294739 36433643 42385946 1230e03 Wage Regression Analysis Output The following economic model predicts indiVidual s wage based on education and region E WAGE 61 I 2EDU I 3MidWest I 4N07quotthEast I 5S0uth I 6West where M 7IdWest 1 if Region midwest N 07quotthEast 1 if Region northeast South 1 if Region south West 1 if Region west STATA Command table region region I Freq midwest I 337 northeast I 57 south I 248 west I 178 STATA Command reg wage edu NorthEast Midwest South West Source I SS df MS Number of obs 820 F 4 815 1607 Model I 134168521 4 335421302 Prob gt F 00000 Residual I 170118196 815 208733983 Rsquared 00731 Adj Rsquared 00686 Total I 183535048 819 224096518 Root MSE 14448 wage I Coef Std Err t PgtItI 95 Conf Interval edu I 1206153 16092 750 0000 8902869 152202 NorthEast I dropped Midwest I 4232238 2069481 205 0041 8294378 1700976 South I 499543 2122288 235 0019 9161225 8296364 West I 5825008 2198793 265 0008 1014097 1509043 ConS I 2472763 2818565 088 0381 305974 8005266 STATA Command test bMidWest0bSouth0bWest0 1 Midwest 0 2 South 0 3 West 0 248 00601 F 3 815 Prob gt F Corn Production Output Consider the following model lnc07quotn 61 I zcapital I 3lab07quot I 4lcmd I 5lab07quot gtlt land I e where corn denotes the production of corn in pounds while the regressors have the obvious meaning Based on the next STATA output answer the questions below lablan stands for labor gtlt land as usual STATA Command reg lncorn capital labor land lablan Source I SS df MS Number of obs 1000 F 4 995 2650948 Model I 145016524 4 362541309 Prob gt F 00000 Residual I 136075347 995 136759142 Rsquared 09907 Adj Rsquared 09907 Total I 146377277 999 146523801 Root MSE 11694 lncorn I Coef Std Err t PgtIt 95 Conf Interval Capital I 2654115 0098252 27013 0000 2634835 2673396 labor I 0045532 3219485 001 0989 6363292 6272228 land I 4869289 21929 222 0027 056605 9172528 lablan I 4458096 0547902 814 0000 338292 5533272 ConS I 2946969 1284514 229 426302 5467636 test land labor 1 lamrlam10 F 1 995 Prob gt F 808 00046 Wage Regression for Rich and Poor Output Consider the following model lnwage 61 I 2imp007quot I 37Im7quot7Ich I e where wage denotes the hourly wage in dollars impoor takes the Value of 1 if the person is an immigrant from a poor country 0 otherwise imrich takes the Value of 1 if the person is an immigrant from a rich country 0 otherwise Consider also the Variable native de ned as native 1 impoor imrich After estimating the aforementioned model in Stata the following results were obtained reg lnwage impoor imrich Source I SS df MS Number of obs 6770 F 2 6767 2888 Model I 312933892 2 156466946 Prob gt F 00000 Residual I 366608138 6767 541758738 Rsquared 00085 Adj Rsquared 00082 Total I 369737477 6769 546221712 Root MSE 73604 ln wage I Coef Std Err t PgtIt 95 Conf Interval impoor I 1009196 0210986 478 0000 1422795 0595597 imrich I 1188251 0260645 456 0000 0677303 1699198 ConS I 2719635 0115901 23465 0000 2696915 2742356 Moshe A Buchinsky Economics 103 Department of Economics Introduction to Econometrics UCLA Fall 2011 Final Examination December 6 2011 STATA OUTPUTS Boca Raton Output reg price sqft bedrooms bath pool traditional waterfront summarize Variable Obs Mean Std Dev Min Max price 1080 1543632 1229123 22000 1530000 sqft 1030 2325933 1003093 662 7397 bedrooms 1080 317963 7094959 1 8 baths 1030 1973143 6120669 1 5 age 1030 1957407 1719425 1 30 owner 1030 4383339 5001031 0 1 pool 1080 0796296 2703444 0 1 traditional 1030 5333339 4937163 0 1 fireplace 1030 562963 4962496 0 1 waterfront 1080 0722222 2589754 0 1 Source SS df MS Number of obs 1030 F 6 1073 30933 Model 10330e13 6 17217e12 Prob gt F 00000 Residual 59711e12 1073 55649e09 R squared 06337 Adj R squared 06317 Total 16301e13 1079 15108e10 Root MSE 74593 price Coef Std Err t Pgtt 95 Conf Interval sqft 3133231 3756303 2173 0000 7446129 8920433 bedrooms 2202313 4561115 483 0000 3097285 1307342 baths 4311362 5439773 793 0000 3244431 5379243 pool 4364432 3706576 056 0576 2194833 1221936 traditional 1307622 4675475 337 0000 2725033 390211 waterfront 5706061 9142954 624 0000 3912052 7500071 cons 4452149 1131474 393 0000 6672302 2231997 4 5 matrix list eV symmetric eV77 sqft bedrooms baths pool traditional waterfront cons sqft 14113603 bedrooms 3352335 20803771 baths 10239019 45513546 29591179 pool 52963361 13643761 16703772 75304463 traditional 13115597 37078142 43462991 19414549 21860062 waterfront 36334658 34016601 57745514 17549714 50155303 33593614 cons 13912195 36134537 19572605 27294739 36433643 42385946 1230e03 Wage Regression Analysis Output The following economic model predicts indiVidual s wage based on education and region E WAGE 61 I 2EDU I 3MidWest I 4N07quotthEast I 5S0uth I 6West where M 7IdWest 1 if Region midwest N 07quotthEast 1 if Region northeast South 1 if Region south West 1 if Region west STATA Command table region region I Freq midwest I 337 northeast I 57 south I 248 west I 178 STATA Command reg wage edu NorthEast Midwest South West Source I SS df MS Number of obs 820 F 4 815 1607 Model I 134168521 4 335421302 Prob gt F 00000 Residual I 170118196 815 208733983 Rsquared 00731 Adj Rsquared 00686 Total I 183535048 819 224096518 Root MSE 14448 wage I Coef Std Err t PgtItI 95 Conf Interval edu I 1206153 16092 750 0000 8902869 152202 NorthEast I dropped Midwest I 4232238 2069481 205 0041 8294378 1700976 South I 499543 2122288 235 0019 9161225 8296364 West I 5825008 2198793 265 0008 1014097 1509043 ConS I 2472763 2818565 088 0381 305974 8005266 STATA Command test bMidWest0bSouth0bWest0 1 Midwest 0 2 South 0 3 West 0 248 00601 F 3 815 Prob gt F Corn Production Output Consider the following model lnc07quotn 61 I zcapital I 3lab07quot I 4lcmd I 5lab07quot gtlt land I e where corn denotes the production of corn in pounds while the regressors have the obvious meaning Based on the next STATA output answer the questions below lablan stands for labor gtlt land as usual STATA Command reg lncorn capital labor land lablan Source I SS df MS Number of obs 1000 F 4 995 2650948 Model I 145016524 4 362541309 Prob gt F 00000 Residual I 136075347 995 136759142 Rsquared 09907 Adj Rsquared 09907 Total I 146377277 999 146523801 Root MSE 11694 lncorn I Coef Std Err t PgtIt 95 Conf Interval Capital I 2654115 0098252 27013 0000 2634835 2673396 labor I 0045532 3219485 001 0989 6363292 6272228 land I 4869289 21929 222 0027 056605 9172528 lablan I 4458096 0547902 814 0000 338292 5533272 ConS I 2946969 1284514 229 426302 5467636 test land labor 1 lamrlam10 F 1 995 Prob gt F 808 00046 Wage Regression for Rich and Poor Output Consider the following model lnwage 61 I 2imp007quot I 37Im7quot7Ich I e where wage denotes the hourly wage in dollars impoor takes the Value of 1 if the person is an immigrant from a poor country 0 otherwise imrich takes the Value of 1 if the person is an immigrant from a rich country 0 otherwise Consider also the Variable native de ned as native 1 impoor imrich After estimating the aforementioned model in Stata the following results were obtained reg lnwage impoor imrich Source I SS df MS Number of obs 6770 F 2 6767 2888 Model I 312933892 2 156466946 Prob gt F 00000 Residual I 366608138 6767 541758738 Rsquared 00085 Adj Rsquared 00082 Total I 369737477 6769 546221712 Root MSE 73604 ln wage I Coef Std Err t PgtIt 95 Conf Interval impoor I 1009196 0210986 478 0000 1422795 0595597 imrich I 1188251 0260645 456 0000 0677303 1699198 ConS I 2719635 0115901 23465 0000 2696915 2742356

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