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# Choosing the right functional form for the regression is important bec Description

##### Description: INCLUDED: 4 PRACTICE FINAL EXAMS [Covering one from Rojas, Casanova, Von Wachter, Buchinsky)
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Till von Wachter Economics 103 Department of Economics Introduction to Econometrics UCLA Winter, 2014

## Choosing the right functional form for the regression is important because of what? 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.

First Name

Last Name

UCLA ID #

## As the sample size, say n, increases, with varying data points, the length of the 100 (1 − α) confidence interval is what? We also discuss several other topics like What is hongshan jade?

Please start solving the examinations only when you are instructed to do so. Please stop immediately when instructed to do so.

Good Luck!

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Part I

Part I—Theoretical Questions (each question worth 2 points): Consider the following regression model:

yi = β1 + β2x2i + · · · + βKxKi + ei.

1. The interpretation of the slope coefficient βk in the model is as follows:

## When collinear variables are included in an econometric model, coefficient estimates are what? (a) a 1% change in xki is associated with a βk% change in yi.

(b) a 1% change in xki is associated with a βk% change in yi, holding all other K − 1 regressors constant.

(c) a change in xki by one unit is associated with a βk change in Y.

(d) a change in xki by one unit is associated with a βk change in Y, holding all other K − 1 regressors constant. We also discuss several other topics like What was an interesting aspect of the informative video by the department of health?
We also discuss several other topics like What is hamilton’s financial program?

This follows from the defintion of the regression coefficients in multiple regression. When we interpret the coefficient of a specific 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 SSEK. Suppose now that we add an additional regressor, resulting in a regression with K + 1 variables and obtain the SSE for that regression, say SSEK+1.Then

(a) SSEK ≥ SSEK+1 always.

(b) if adjusted R2 = R2then SSEK = SSEK+1.

(c) SSEK ≤ SSEK+1 always.

(d) We cannot determine the relationship between SSEK and SSEK+1 based on the information provided.

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. If you want to learn more check out What is the difference between multi-domestic and home replication strategies?

3. R2is a valid measure of the goodness of fit of the regression if

(a) R2is the same as the adjusted R2.

(b) the regression has a constant term.

(c) the regression has more than two variables, i.e., K > 2.

(d) R2is always a valid measure of the goodness of fit.

The R2is defined as SSR/SST, and hence is in principle valid no matter what the regression specification. However, without a constant term SST 6= SSR+SSE. In that case, hence R2 does not measure the fraction of the total variance in the outcome variable explained by the regression, and is not constrained between 0 and 1. Hence, it is not a sensible measure of the goodness of fit

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4. In a given data set, the larger is R2

(a) the lower is SSE.

(b) the larger is SSE. If you want to learn more check out What is legalism?

(c) the lower is SSR. We also discuss several other topics like Why was the treaty of guadalupe hidalgo so important?

Refer to the formula of R2 = 1−SSE/SST. Note that SST is uniquely pinned down by the data set.

5. Which of the following statements is not true:

(a) a high R2 or R¯2 does not mean that the regressors are a true cause of the dependent variable. (b) a high R2 or R¯2 does not mean that there is no omitted variable bias.

(c) a high R2 or R¯2 always means that an added variable is statistically significant. (d) a high R2 or R¯2 does not necessarily mean that you have the most appropriate set of regressors.

R2 does not indicate anything regarding any single regressor, since it is a measurement of the joint goodness of fit. It also does not say anything more about an added variable beyond that some additional variance of the outcome variable may be explained by inclusion of that variable.

6. Choosing the right functional form for the regression is important because

(a) it allows one to capture the specific features of the observed data.

(b) it assists one in collecting the right data for the empirical analysis.

(c) it permits one to use the R2 as a measure for the model specification.

(d) it prevents one from including irrelevant variables in the regression.

In class, we have 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.

7. If you had a two regressor regression model, then omitting one variable which is relevant

(a) will have no effect on the coefficient of the included variable if the correlation between the excluded and the included variable is negative.

(b) will always bias the coefficient of the included variable upwards.

(c) can result in a negative value for the coefficient of the included variable, even though the coefficient will have a significant positive effect on Y if the omitted variable were included.

(d) makes the sum of the product between the included variable and the fitted residuals different from 0.

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Option(a) is wrong since the existence of bias has nothing to do with the sign of the correlation between the endogenous and exogenous vairables, 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. Option(d) 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 variable 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 versa, the effect 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 − α) confidence interval

(a) increases.

(b) decreases.

(c) does not change.

(d) increases in some cases and decreases in others.

If the sample size increases with varying data points (meaning that the additional values for x are different than the mean and hence add information that can be used to estimate the regres sion coefficients), then the standard error of parameters decreases. But when the standard error decreases, the length of the confidence interval decreases. Please also refer to the formulas of the estimated variance of OLS coefficients and interval estimation.

9. Which of the following will change if you scale the dependent variable in a simple regression model?

(a) p-value

(b) t-value of β2

(c) R2

(d) β1

Rescaling of the dependent variable causes the changes of all the coefficients 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 x, is y = β1 + β2x + e.

(b) The variance of the random error e is V ar (e) = σ2.

(c) The covariance between any pair of random errors ei and ej is zero.

(d) The parameter estimate of β2 is unbiased.

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Please refer to the SR assumptions. Option(d) is the result of the SR assumptions, but it is not an assumption itself.

11. When collinear variables are included in an econometric model, coefficient estimates are

(a) biased downward and have smaller standard errors.

(b) biased upward and have larger standard errors.

(c) biased and the bias can be negative or positive.

(d) unbiased but they have larger standard errors.

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 coefficient 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 K=3.

12. If one rejects the null hypothesis that H0: βk = 0 against H1: βk 6= 0 at the significant level α, then

(a) he/she will not reject it for H0: βk = 0 against H1: βk > 0.

(b) It cannot be determined whether he/she will reject it for H0: βk = 0 against H1: βk > 0. (c) he/she will reject it for H0: βk = 0 against H1: βk > 0.

(d) he/she will reject it for H0: βk = 0 against H1: βk > 0 but will not reject it for H0: βk = 0 against H1: βk < 0.

To answer this question, draw the rejection region of the two tests. You will find they are not completely overlapping.

13. One will reject the null hypothesis H0: βk = 0 against H1: βk > 0, at the significant level α

(a) if and only if zeros does not belong to the 100 (1 − α) confidence interval for βk. (b) if and only if zeros belongs to the 100 (1 − α) confidence interval for βk. (c) the 100 (1 − α) confidence interval cannot help in testing this hypothesis. (d) There is no relation between confidence interval and hypothesis testing.

The 100(1−α) confidence interval is related to the testing result of the two sided test ( H0 : β = 0, H1 : β 6= 0),not to the one-tailed test stated in the question.

14. To decide whether Yi = β0 +β1Xi +ui or ln(Yi) = β0 +β1Xi +ui fits the data better, you cannot consult the regression R2 because

(a) ln(Y) may be negative for 0 < Y < 1.

(b) the SST are not measured in the same units between the two models.

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(c) the slope no longer indicates the effect of a unit change of X on Y in the log-linear model. (d) the regression R2can be greater than one in the second model.

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 variance explained of the same SST.

15. To test whether or not the population regression function is linear rather than a polynomial of order r,

(a) check whether the regression R2for the polynomial regression is higher than that of the linear regression.

(b) compare the SST from both regressions.

(c) look at the pattern of the coefficients: if they change from positive to negative to positive, etc., then the polynomial regression should be used.

(d) use the test of (r-1) restrictions using the F-statistic.

This question explicitly 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 sufficient, a test is required.

16. The binary variable interaction regression

(a) can only be applied when there are two binary variables, but not three or more. (b) is the same as testing for differences in means.

(c) cannot be used with logarithmic regression functions because it is not defined.

(d) allows the effect of changing one of the binary independent variables to depend on the value of the other binary variable.

Please refer to the definition of binary variables. Also compare the marginal effects of a binary variable when there is an interaction term and when there is not.

17. Consider the following multiple regression models (a) to (d) below. DF em = 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; DM arried is a binary variable which is unity for married individuals and is zero otherwise, and DSingle is (1 − DM arried). Regressing weekly earnings (Earn) on a set of explanatory variables, you will experience perfect multicollinearity in the following cases except:

(a) Earni = β0 + β1DF emi + β2DM alei + β3X3i

(b) Earni = β0 + β1DM arriedi + β2DSinglei + β3X3i

(c) Earni = β0 + β1DF emi + β3X3i

(d) Earni = β0DF emi + β1M alei + β2DM arriedi + β3DSinglei + β5X5i

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Please refer to the definition 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 = β0 +β1Xi +β2Di +β3(XiDi) +ui, where Xiis a continuous variable and Diis a binary variable, to test that the effect of Xi on Yiis identical for both values of Di, you must use

(a) a separate t-test for H0 : β2 = 0 and H0 : β3 = 0.

(b) F-test for the joint hypothesis that β0 = 0, β1 = 0.

(c) a t-test for H0 : β3 = 0.

(d) F-test for the joint hypothesis that β2 = 0, β3 = 0.

The coefficient of the interaction term measures the difference of the effect of two groups. If it is zero, then there is no different effect.

19. A large company is accused of gender discrimination in wages. The following model has been estimated from the company’s human resource information

ln (\W age) = 1.439 + 0.0834(Edu) + 0.0512(Exper) + 0.1932(M ale)

where WAGE is hourly wage, Edu is years of education, Exper is years of relevant experience, and M ale indicates the employee is male. What hypothesis would you test to determine if the discrimination claim is valid?

(a) H0 : βMale = 0; H1 : βMale ≥ 0

(b) H0 : βMale = βEdu = βExper = 0; H1 : βMale 6= 0 and βExper 6= 0 and βExper 6= 0 (c) H0 : βMale = βEdu = βExper = 0; H1 : βMale 6= 0 or βExper 6= 0 or βExper 6= 0 (d) H0 : βMale ≤ βEdu : or : βMale ≤ βExper; : H1 : βMale > βEdu : or : βMale > βExper

βMALE captures the difference 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 differences between gender due to differences in education and experience are controlled for. Hence, the coefficient on MALE mean difference in wages holding education and experience constant.

20. When an exogenous instrument is used, IV estimators are

(a) consistent and approximately normally distributed in large samples

(b) unbiased and BLUE in all sample sizes

(c) consistent if z is normally distributed

(d) normally distributed in all sample sizes and consistent in large sample sizes 7

Please refer to the definition 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.

21. Which of the following statements is true regarding var(βˆ2) when estimated by IV using z 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 (c) the variance of the IV estimates can be larger or smaller than OLS, depending on r2xz (d) there is no way to know which estimator leads to estimates with a larger variance, it depends on the data

Please compare the formula of the estimated variance of the instrument variable estimators and the OLS estimators. The whole point of the IV estimator is to only use a part of the variation 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 variable is used. The variance of the predicted explanatory variable is smaller than the variance of the actual explanatory variable. Hence the variance is higher.

22. What is the null hypothesis when performing an F-test to test the strength of multiple instruments j = 1, ..., J?

(a) the instruments are weak, with no coefficient θj on the instruments in the first stage different from 0.

(b) the instruments are strong with all coefficients θj on the instruments significantly different from 0.

(c) the instruments are sufficiently strong with a coefficient θj on at least one instrument sig nificantly different from 0.

(d) the instruments are weak with all coefficients θj on the instruments different from 0.

Please refer to the definition of the F-test and the first-step of the 2SLS estimation. 23. When you are implementing an instrumental variable regression, you are worried about

(a) a potential direct effect of the instrumental variable on the outcome

(b) a weak relationship between the instrumental variable and the endogenous variable (c) a remaining correlation of the instrumental variable and the error term

(d) all of the above

A valid IV should be truly exogenous (no correlation with the error term), correlated with the endogenous variable, and should only affect the dependent variable through the endogenous variable (i.e., have no direct effect on the outcome).

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Part II—Practical and Computational Questions (each question worth 2 points):

Consider the following regression model:

yi = β1 + β2x2i + · · · + βKxKi + ei,

24. You have estimated the following simple regression model

y = 379 + 1.44x3(1)

What is the elasticity when x = 8.49?

(a) 263.19

(b) 311.39

(c) 2.10

(d) -24.7

Elasticity =dy

dx

dy

x y

dx = 1.44 ∗ 3 ∗ x2 = 4.32x2

x = 8.49 =⇒ y = 379 + 1.44(8.49)3 = 1260.22

=⇒ Elasticity = 4.32(8.49)2 8.49

1260.22 = 2.10

25. You have estimated a two variable model, i.e., K = 2, and your printout includes the following information

sxy = 3614.00

sx = 12.72

sy = 394.61

SST = 758, 912

Then the R2for this regression model is:

(a) .72

(b) .11

(c) .03

(d) .5

In the simple regression model, R2 = r2xy =

sxy sxsy

2. For the numbers of the question, we get that

rxy = 0.72, such that r2xy = 0.52. Hence, (d) is the correct answer. Note that in the final either exact values will be given, or it will be stated explicitly that the result is approximate.

26. Suppose that K = 3, y = 9, x2 = 3, b2 = 1.2, x3 = 2, b3 = 1.3, then the estimate for β1, b1 is 9

(a) 2.8

(b) 3.8

(c) 8.3

(d) 2.5

For any regression model we know that evaluating the estimated equation at the means will return the mean.

Mathematically, y = b1 + b2x2 + b3x3

=⇒ 9 = b1 + 1.2(3) + 1.3(2) =⇒ b1 = 2.8

27. A company is accused of racial discrimination in wages. The following model has been estimated from the company’s human resource information

ln (W AGE) = 1.439 + .0834 × EDU + .0512 × EXP ER + .1932 × W HIT E

where W AGE is hourly wage, EDU is years of education, EXP ER is years of relevant experience, and W HIT E indicates that the employee is white. How much more do white employees at the firm earn, on average?

(a) \$1.21 per hour more than non-whites.

(b) \$19.32 per hour.

(c) 19.32% more than non-whites.

(d) \$19,320 more per year than non-whites.

The log-linear regression equation can be interpreted such that a one unit increase in an xk variable (independent variable) leads to approximately a (100)βk% change in the y (dependent) variable.

28. Randomized, controlled experiments are needed to accurately measure treatment effects without

(a) the expense of having to treat everyone.

(b) raising public debate.

(c) exposing everyone to untested treatments.

(d) omitted-variable bias.

One of the main purposes of a randomized, controlled experiment is to obtain estimates of the causal effect of a treatment that is not affected by omitted-variable bias.

29. The following economic model predicts whether a voter will vote for an incumbent school board member

INCUMBENT = β1 + β2MALE + β3P ART Y + β4MARRIED + β5KIDS, where

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INCUMBENT = 1 if the voter votes for them, 0 otherwise. MALE = 1 if the voter is a male. P ART Y = 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 × P ART Y

(b) MALE × MARRIED

(c) MARRIED × KIDS

(d) MARRIED × P ART Y

Inclusion of the interaction MALE × MARRIED allows marriage to have different effects on men and women in terms of their likelihood of voting for an incumbent.

30. 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

(a) β2 < 0

(b) β2 > 0

(c) β2 + β4 > 0

(d) β2 + β4 < 0

In the setup in the previous question the variable marriage has the same effect on both men and women. Therefore, the claim simplifies to ”a man is less devoted to the incumbent than a woman”.

31. Consider the following regression model given by

yi = 1.1 + 2xi2 + .5xi3 + ei

with the variance-covariance matrix of the regression coefficients

 .

The standard error for 2b2 + b3 is

(a) 1.22

(b) 0.43

(c) 0.39

(d) 0.11

.7 .2 .3 .2 .5 −.2 .3 −.2 .3

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var(2b2 + b3) = 4var(b2) + var(b3) + 4cov(b2, b3)

=⇒ var(2b2 + b3) = 4(.5) + .3 + 4(−.2) = 1.5

se(2b2 + b3) = pvar(2b2 + b3) = √1.5 = 1.22

32. You have estimated the following equation:

T estScorei = 607.3 + 3.85Incomei − 0.0423Income2i(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 45.508.

(b) is positive until a value of Income of \$610.81.

(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.

To find the point at which a differentiable function changes slope, take the first derivative and set it equal to 0.

dT estScorei

dIncomei= 3.85 − 0.0846Income∗i = 0

=⇒ Income∗i = 45.508

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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:

yi = β1 + β2x2i + · · · + βKxKi + ei,

33. The essence of the regression analysis conducted by J.J. Espinoza is to

(a) determine how Walt Disney Company should structure advertising in the course of the introduction of new movies.

(b) evaluate the performance of the analysts at Walt Disney Company.

(c) determine who to award financial aid to from a selected minority group.

(d) evaluate the performance and labor market outcome of Hispanic students.

In his guest lecture, JJ described how he uses regression analysis to best time the advertisements of new movies.

34. The objective of the analysis performed by Vikas Gupta, for Factual, is to

(a) determine the main characteristics of individuals who search for mortgages. (b) provide an evaluation those applying for access to special internet sites.

(c) deliver the most relevant ads through mobile devices.

(d) conduct market research for new electronic devices.

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 addvertisements to.

35. The approach of calculating the regression coefficients in three steps discussed in the lab lecture

(a) is a way to get approximate values of the multiple regression coefficient estimates (b) is a way to get exact values of the multiple regression coefficient estimates (c) is a way to obtain estimates of the causal effect

(d) can only be used if assumptions MR1-MR5 hold

The three-step method discussed in the lab lecture is equivalent to regular estimation of multiple regression, i.e., the exact identical coefficient estimates are obtained (this is based on the so-called Frisch-Waugh theorem).

36. 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

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(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

We found a negative coefficient of STR, but concluded that a) relative to the mean TESTSCR the effect is moderate and b) that this effect is likely to at least in part to be due to omitted factors.

37. 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) is likely to yield a biased coefficient of the effect of class size on testscores (c) cannot be interpreted since STR has a different scale than TESTSCR

(d) cannot be interpreted because the true relationship is not linear

We had discussed how other factors are likely to be present that determine TESTSCR and are correlated with STR.

38. 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 β3 be the coefficient on SUBSLUNCH were it to be included in the regression. The omitted variable bias is

(a) β3

(b) cov(ST R, SUBSLUNCH)/var(ST R)

(c) cov(T EST SCR, SUBSLUNCH)/var(ST R)

(d) β3cov(ST R, SUBSLUNCH)/var(ST R)

We had derived the omitted-variable bias formula based on the probability limit of b2, and found that the bias (the difference between the probability limit of b2 and the true β2) is equal to β3cov(ST R, SUBSLUNCH)/var(ST R). 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 e(V)” is the same as ”estat vce”.

39. The F-statistic for H0: β2 = β3 = β4 = 0 from the first regression is (approximately):

(a) 25, 717.38

(b) 8.08

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(c) 18.12

(d) 1.70

Simply observe the F-statistic reported in the output of the first regression. This is precisely the null hypothesis that is being tested.

40. The results indicate that

(a) there is high degree of collinearity between income and prices.

(b) the coefficient on the variable income is significant, because all the others are not significant. (c) one should include in the regression the interactions between all prices.

(d) it is essential to have the variable income in the regression.

Observe that in the second regression, income is highly significant (very low p-value). Further more, the standard errors of the coefficients on prices are all substantially reduced, which is the opposite of what we would expect 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 values for the root MSE), and hence reduces estimates of the standard errors.

41. The results of the first regression indicate that the hypothesis H0: β2 = β3 = β4 = 0

(a) is true for any α ≤ .10.

(b) H1 is true for any α > .10.

(c) neither H1 nor H0 are true.

(d) We cannot conclude from the results obtained which hypothesis is true.

The output for regression 1 displays the p-value on the F-statistic as 0.1921 so for any α < 0.1921 we can not reject the null hypothesis.

42. The point estimates from the second regression indicate that

(a) Meat and vegetables and fruit are substitutes, while meat and bread and cereal are comple ments.

(b) Meat and vegetables and fruit are complements, as are meat and bread and cereal.

(c) Meat and vegetables and fruit are complements, while meat and bread and cereal are sub stitutes.

(d) Vegetables and fruit and bread and cereal are substitutes.

If x and y are complements, an increase in the price of x will lead to a decrease in the quantity demanded of y. If x and y are substitutes an increase in the price of x will lead to an increase in the quantity demanded of y. Look at the sign of the coefficients of the relevant variables to make an assessment.

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43. Based on the results of the second regression, the t-statistic for the test of H0: β3+β4 = 0 against H1: β3 + β4 6= 0 will be

(a) -0.6785

(b) 1.09

(c) -2.584

(d) -1.51

Quick hint: if λˆ is a random variable and you are testing H0 : λ = c vs. H1 : λ 6= c then t − stat =λˆ−c

se(λˆ)

se(λˆ) is always positive so the sign of λˆ − c determines the sign of the t-stat. In this case λ = β3 + β4.

se(βˆ3 + βˆ4) = qvar(βˆ3) + var(βˆ4) + 2cov(βˆ3, βˆ4)

Check the third stat output table to find the appropriate variances.

=⇒ se(βˆ3 +βˆ4) = p0.0214 + 0.0498 − 2(0.0011) = 0.2627 =⇒ t−stat =0.1590−0.3372 0.2627 = −.678

44. Consider testing the restriction in model 2 that β5 = 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 (c) this can be done using an F-test based on R2in model 1 and model 2

(d) all of the above

The slides show that a t- and F-test are equivalent if we are testing only one restriction. Note that SST =P(yi − y¯)2is the same in both models, since the restrictions under the null hypothesis do not lead to a modification 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 definition of the F-test statistic is F =SSE1−SSE2

N−K

Therefore, R21 = 1 −SSE1

SST and R22 = 1 −SSE2

SSE2

1.

=⇒ R22 − R21 =SSE1−SSE2

SST

and 1 − R22 =SSE2

SST

=⇒R22−R21

SST

1−R22=SSE1−SSE2

SSE2

The F-stat can then be computed with the correct degrees of freedom and number of restrictions. Similalry, for the SSR we have that SST = SSE+SSR and hence SSE = SST −SSR. Using the

definition of the F-statistic, we have that F =SSR2−SSR1 SST −SSR2

N−K

1. Note that deviding the numerator

and denominator by SST gives the R2version 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. (b) there is no implication regarding the impact of the other variables. (c) there is almost no variability in the consumption of meat. (d) the adjusted R2in the first regression should be smaller than R2.

16

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 coefficients.

46. The interval estimate for the coefficient on income, i.e., β5, for α = .05 is approximately

(a) [−1.566, +2.912]

(b) [+1.566, +2.912]

(c) [−0.142, +0.460]

(d) [2 × 0.142, 2 × 0.460]

Look at the 95% confidence interval in output table 2 on the coefficient for y. 47. The fact that R2in model 2 is substantially higher than in model 1 means:

(a) that income explains an important part of the variance of meat purchases (b) that income has a causal effect on meat purchases

(c) that income is exogenous

(d) that the coefficients on the variables are biased

R2is a measure of how much variation in the dependent variable is explained by the model. Hence if R2goes up substantially when income is added to the model you can then conclude that income explains an important part of the variance of meat purchases.

48. 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 (c) that income is likely heterogeneous in the population

(d) that there is a low correlation of individual income and these prices

We can’t conclude that the coefficients measure the causal effect 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 cov(pi, income) for all prices, as well as the sign and size of the effect of income on meat purchases.

49. The elasticity of meat purchases with respect to a change in price of meat (a) cannot be calculated without further information

17

(b) is -67.63

(c) is -0.6756401

(d) is economically significant

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 specific values for quantities or prices are provided, hence the answer is a).

50. The 95% confidence interval for the effect of price of meat on meat purchases

(a) is approximately [-1.1629,-0.1885]

(b) changes with the scaling of income

(c) is unaffected by the scaling of meat purchases

(d) all of the above

Scaling income doesn’t affect the other regressors. Scaling the dependent variable (meat purchases) will re-scale all the coefficients, and hence their confidence intervals.

18

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:

1

Unless specified otherwise, use a level of significance of 0.05 for hypothesis testing. Critical values at 0.05 significance level:

Standard Normal, two-sided test: 1.96

Standard Normal, one-sided test: 1.64

F1,1 = 3.84  F2,1 = 3.00  F3,1 = 2.60  F4,1 = 2.37  F5,1 = 2.21 21 = 3.84  22 = 5.99  23 = 7.81  24 = 9.49  25 = 11.07

BEGINNING ECON103 EXAM

PART 1: TRUE/FALSE/EXPLAIN

YOU ARE GRADED ON YOUR EXPLANATION - USE BOTH MATH AND INTUITION (5 points each, 50 points total)

1. The significance level of a hypothesis test is best defined as the probability of rejecting H0 when H0 is not true.

FALSE: the significance level is the probability of rejecting H0 when H0 is true.

2. We decide to measure the e↵ect 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 e↵ect of class size using the di↵erences estimator, we only need to observe students’ test scores after the treatment was introduced. On the other hand, to compute the di↵erences-in-di↵erences estimator we need to observe test scores both before and after the treatment.

TRUE: The di↵erences estimator is equal to Y¯ treated,after  Y¯ control,after, while the di↵erences in-di↵erences estimator is equal to Y¯ treated,after  Y¯ treated,beforeY¯ control,after  Y¯ control,before.

2

3. 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 e↵ect of a one-unit change in income on the probability of smoking is equal to a constant.

TRUE: In the linear probability model Y = ↵+X +", the marginal e↵ect of a unit-change in X on the variable Y is constant and equal to  (that is, dY

dX = ).

4. (**) 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).

5. In the linear regression model with only one regressor, Y = 0+1X +u, the overall regression F-test and the t-test on 1 reported by Stata are testing the same hypothesis.

TRUE: The t-test reported by Stata tests the hypothesis that the coecient is equal to zero, that is, H0 : 1 = 0. The overall regression F-test tests the hypothesis that all slope coecients are equal to 0. In this case there is just one slope coecient, so the null hypothesis is also H0 : 1 = 0.

6. You use observations from 1,000 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..

7. 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 = ↵ +  ln P + ", the price elasticity is /Q (Notice that the price elasticity is defined as QPPQ and that, in the linear-log model,  = Q

ln P = Q

P/P = QP P).

3

8. For ˆ2 to be an ecient estimator of 2, it must be an unbiased estimator of 2.

TRUE: According to the definition given in Lecture Notes 2, an estimator is ecient if it has the lowest variance among all unbiased estimators.

9. Fifth-graders’ test scores (T S) are positively influenced by both time spent studying (T) and by their parents’ income (I). We regress T S on T, but omit I from the regression. If studying time is inversely related to parents’ income, the coecient on studying time may turn out to be negative.

TRUE: The true model is T S = ↵+1T +2I +", but instead we estimate T S = ↵+1T +". Our estimate ˆ1 will have omitted variable bias because the two necessary conditions are fulfilled in this case: the omitted variable I has an e↵ect on T S, and it is correlated with T. Therefore: E(ˆ1) = 1 + bias.

Recall that the sign for omitted variable bias is given by 2Cov(T,I). In this case you are told that 2 is positive and Cov(T,I) is negative, so the bias will be negative. Since 1 is positive but the bias is negative, if the bias is large enough, the sum of the two may turn out to be negative.

10. (**) To determine the e↵ect of a new drug against the flu, 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 = 0 + 1X + 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 classifies 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 e↵ect of the drug will be too large.

FALSE: In the presence of random measurement error the coecient will always be biased towards 0, that is, the estimate of the e↵ect of the drug will be too small, not too large.

4

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:

Hours d i = 19.794 (3.89)

0.004 (0.0018)

Pi + 1.387 (0.44)

Wagei  0.102 (0.05)

Wage2i

Ci

where:

+ 1.281 (0.91)

Mi + 7.801 (2.04)

Ki + 3.210 (1.62)

Hours is the number of hours worked per week

P is pension income (in \$ per week)

W age 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 coecient 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, 1.281 more hours per week than single men.

3. Suppose you create a dummy variable S equal to 1 if the individual is single (i.e. not married). You re-run the regression above including the variable S instead of M. What will the coecient on S measure? (5 points)

5

How many less hours single men work, compared to married men.

4. What will the coecient on S be equal to? (5 points)

Since it measures the opposite to the coecient on M, it has to be equal to its opposite, that is, to -1.281.

5. What will be the new value for the constant? (5 points)

In the first 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 19.794+1.281=21.075.

6. 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)

Hours d = 19.794  0.004 ⇥ 100 + 1.387 ⇥ 4  0.102 ⇥ 16 + 1.281 ⇥ 1+7.801 ⇥ 1+3.210 ⇥ 1

7. Derive a formula for the e↵ect of wages on hours in this model. (5 points)

@Hours

@W age= 1.387  2 ⇥ 0.102 ⇥ W age

8. Which regressor would you have to add to the model to allow the e↵ect 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 W age (i.e., C ⇥ W age. You would estimate one extra coecient (let’s call it ) for this interaction. The marginal e↵ect of wages on hours would be:

@Hours

@W age= 1.387  2 ⇥ 0.102 ⇥ W age + C

9. Another researcher uses the same sample to regress Hours on a di↵erent set of regressors and obtains an R2 equal to 0.49. Does her regression fit the data better than the one above? (use R2, rather than R¯2, to compare the regressions) (10 points).

6

The R2 for the regression above is 1  400

1000 = 0.6. The regression above fits the data better.

10. A third researcher uses the same sample to regress Hours on another set of regressors and finds that the explained sum of squares is equal to 300. Does his regression fit the data better? (Again, use the R2 to compare regressions) (5 points).

The R2 for the third regression is 300

1000 = 0.3. (Notice that the total sum of squares does not

depend on the regressors included in the model. Since the third researcher is using the same dependent variables, the value of the TSS does not change). It fits the data worse than the first one.

11. What does it mean for an estimator to be unbiased? Be specific. (10 points)

If we drew infinitely 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(ˆ) = . (Notice that stating the formal definition was enough).

12. Considering that this model excludes men who have already retired, are you worried that your estimate of the e↵ect 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.

7

PART 3

-------------+------------------------------ F( 5, 801) = 9.17   Model | 10.3345153 5 2.06690305 Prob > F = 0.0000   Residual | 180.582461 801 .225446269 R-squared = 0.0541  -------------+------------------------------ Adj R-squared = 0.0482   Total | 190.916976 806 .236869698 Root MSE = .47481

------------------------------------------------------------------------------   smoke | Coef. Std. Err. t P>|t| [95% Conf. Interval]  -------------+----------------------------------------------------------------

(50 points total)

Y | .0288442 .0059515 4.85 0.000 .0171617 .0405267   wage | 3.59e-08 2.01e-06 0.02 0.986 -3.90e-06 3.98e-06

You decide to study the labor supply of older men using the same dataset as in part 3, but this

wage2 | .0204584 .0056403 3.63 0.000 .009387 .0315299   M | -.0002657 .0000615 -4.32 0.000 -.0003864 -.0001451

time use as a dependent variable an indicator that is equal to 1 if the man is working, and equal to  K | -.1335984 .2020691 -0.66 0.509 -.5302458 .2630491  0 if he has retired. You call this variable W ork. You run a logit regression and obtain the following

own | .1224413 .0738685 1.66 0.098 -.0225977 .2674802

results:

_cons | .9755488 .8320104 1.17 0.241 -.6576293 2.608727  ------------------------------------------------------------------------------

. logit work P wage wage2 M K C

Iteration 0: log likelihood = -537.50555

Iteration 1: log likelihood = -514.1408

Iteration 2: log likelihood = -513.4642

Iteration 3: log likelihood = -513.46068

Logit estimates Number of obs = 1000   LR chi2(6) = 48.09   Prob > chi2 = 0.0000  Log likelihood = -513.46068 Pseudo R2 = 0.0447

------------------------------------------------------------------------------   smoke | Coef. Std. Err. z P>|z| [95% Conf. Interval]  -------------+----------------------------------------------------------------   P | -.1341957 .0278469 4.82 0.000 -.1887747 -.0796167   wage | 0.045672 0.008942

wage2 | 0.001967 0.000815

M | .0013956 .0003171

K | .6435501 .8991412

C | 10.420281 2.365785

_cons | 2.164421 3.698581 0.59 0.558 -5.084664 9.413506  ------------------------------------------------------------------------------

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 e↵ects are not constant, so potentially more flexible.

2. What is the interpretation of the coecient on P? (5 points)

Coecient tells us the sign of the marginal e↵ect of pension income. Negative coecient indicates that a higher pension income decreases the probability of work.

8

3. How does the coecient on P compare to the coecient 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 coecient and the one obtained from a LPM, since they are measuring di↵erent things. In the LPM, the coecient measures the marginal e↵ect. In this case, the marginal e↵ect is equal to the coecient times the logistic distribution evaluated at (2.16  0.1341 ⇥ P + ....)

4. How would you compute the p-value associated to W age2? (10 points) (**) p-value = 2(|t|)

5. Why doesn’t Stata report the R2 in this case? (10 points)

(**) The R2 is not an appropriate measure of fit 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.

9

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:

ln Hi = ↵0 + ↵1 ln Ii + "i (1)

You run the regression in Stata and obtain the following output:

. reg H I

Source | SS df MS Number of obs = 1000  -------------+------------------------------ F( 1, 998) = 11.54   Model | 36166.3086 1 36166.3086 Prob > F = 0.0007   Residual | 3126869.07 998 3133.13535 R-squared = 0.0114  -------------+------------------------------ Adj R-squared = 0.0104   Total | 3163035.38 999 3166.20158 Root MSE = 55.974

------------------------------------------------------------------------------   H | Coef. Std. Err. t P>|t| [95% Conf. Interval]  -------------+----------------------------------------------------------------   I | 5.920467 1.742583 3.40 0.001 2.50092 9.340014   _cons | -5.015161 8.970182 -0.56 0.576 -22.61774 12.58742  ------------------------------------------------------------------------------

Next you draw a plot with the fitted values from the regression and the residuals:

0

0

3

0

0

2

0

)

e

s

u

o

H

(

H

0

1

0

0

0

1

-

0

0

2

-

2 4 6 8 I (Income) Fitted values 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 ↵ˆ1 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 coecient.

3. According to the Stata output on the previous page, does Income have a significant e↵ect on House value? Based on your answer to 1. above, are you concerned about this conclusion? (10 points)

Income has a positive and significant e↵ect on house value, as can be seen from the t-statistic, which is greater than 1.96 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 I2 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 NR2 = 1000 ⇥ 0.1360 = 136 (based on output below). Compare with critical value from Chi squared with 2 degrees of freedom, given on page 2 (equal to 5.99). Reject null. Conclude that model is heteroskedastic.

11

. reg res2 I I2

Source | SS df MS Number of obs = 1000  -------------+------------------------------ F( 2, 997) = 78.44   Model | 4.9051e+09 2 2.4526e+09 Prob > F = 0.0000   Residual | 3.1172e+10 997 31266158.8 R-squared = 0.1360  -------------+------------------------------ Adj R-squared = 0.1342   Total | 3.6078e+10 999 36113623.2 Root MSE = 5591.6

------------------------------------------------------------------------------   res2 | Coef. Std. Err. t P>|t| [95% Conf. Interval]  -------------+----------------------------------------------------------------   I | -2126.172 1280.6 -1.66 0.097 -4639.152 386.8078   I2 | 417.8111 125.3756 3.33 0.001 171.7807 663.8415   _cons | 2785.206 3207.932 0.87 0.385 -3509.867 9080.278  ------------------------------------------------------------------------------

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 V ar(" | I) = I2 ⇥ 2. Propose a transformation of the model that would have an error term whose variance does not depend on income. (10 points)

ln Hi

I = ↵0I + ↵1ln Ii

I +"iI

The variance of the new error term  = "I is constant.

12

PART 5

(70 points total)

Consider the following wage equation that has been estimated from a random sample of 20-65 year old male workers:

ln Wi = 0 + 1Edui + ui, (1)

where ln Wi is the log-wage (in dollars per hour) of individual i, Edui is the number of years of education, and ui is the error term. The estimates you obtain for 0 and 1 are:

ˆ0 = 0.954(0.025)

ˆ1 = 0.0743(0.0052)

Numbers in parentheses are standard errors.

(a) Interpret these estimates (5 points)

The log wage for an individual with no education would be 0.954. Each extra year of education increases wages by 7.43%.

(b) Can you reject the hypothesis that education has no e↵ect on wages? (5 points)

YES. z = (0.07430)/0.0052 > 1.96. Another way of seeing this (obtained by arranging terms) is by noticing that that the coecient is more than 1.96 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 = 0.0743  0.05

0.0052 = 0.0243

0.052 > 1.96

You could also have tested the hypothesis by constructing the confidence interval. (d) What is the confidence interval for the returns to education? (5 points)

0.0743 ± 1.96 ⇥ 0.0052, or [0.0743  1.96 ⇥ 0.0052, 0.0743 + 1.96 ⇥ 0.0052] 13

(e) Under which assumptions can ˆ1 be interpreted as the causal e↵ect of edu cation on wages? (10 points)

(**) OLS assumptions (SR1 to SR5)

(f) 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 e↵ect on wages independently of education, the coecient on Edu in the regression above is biased upwards.

(g) 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:

ln Wi = 0 + 1Edui + 2IQ + ui

The estimates you obtain for 1 and 2 are (standard errors in parentheses):

ˆ1 = 0.060(0.0050)

ˆ2 = 0.097(0.010)

What do these numbers tell you about the correlation between education and IQ? Do these estimates support your concern that ˆ1 cannot be interpreted as a causal e↵ect? (5 points)

They do support the concern. After including the additional variable the estimate for the e↵ect of education has decreased, indicating that the previous estimate (ˆ1) was likely biased upwards. The upward bias tells you that 2 ⇥ (Cov(IQ, Edu)) > 0. A hypothesis test based on ˆ2 indicates that 2 is positive. Hence the covariance between IQ and Edu must also be positive for the OVB to be greater than 0.

14

(h) You re-estimate the first 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 e↵ect 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.

(i) Note: this question is challenging. Do not attempt to answer it until you have finished 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 20-65 year old male workers, the average number of years of education is 10.5 for workers whose fathers do not have a college degree, and 11.6 for workers whose fathers did graduate from university. The average log-wage of workers whose fathers do not have a college degree is 2.52. The average log-wage of workers whose fathers did graduate from college is 2.63. 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 = ⇡0 + ⇡1F atherDegree + "

F atherDegree can only take on the values 0 and 1. Hence:

Edu d = 10.5=ˆ⇡0 + ˆ⇡1 ⇥ 0=ˆ⇡0

Edu d = 11.6=ˆ⇡0 + ˆ⇡1 ⇥ 1=ˆ⇡0 + ˆ⇡1

Hence ˆ⇡0 = 10.5, ˆ⇡1 = 11.6  10.5=1.1

The second regression is:

ln Wi = 0 + 1Edu di + ui

Notice that 1 = W

ˆ1 = 2.632.52

11.610.5 = 0.11

Edu d . Using the information provided, you can obtain an estimate

1.1 = 0.1. An extra year of education increases wages by 10%. 15

Moshe A. Buchinsky Economics 103 Department of Economics Introduction to Econometrics UCLA Fall, 2011

Final Examination

December 6, 2011

STATA OUTPUTS

1

Boca Raton ñOutput 2

Wage Regression Analysis Output

The following economic model predicts individualís wage based on education and region. E (W AGE) =  1 +  2EDU +  3M idW est +  4NorthEast +  5South +  6W est

where M idW est = 1 if Region == ìmidwest,îNorthEast = 1 if Region == ìnortheast,îSouth = 1 if Region == ìsouth,îW est = 1 if Region == ìwestî

STATA Command: table region

----------------------

region | Freq.

----------+-----------

midwest | 337

northeast | 57

south | 248

west | 178

----------------------

STATA Command: reg wage edu NorthEast MidWest South West

Source | SS df MS Number of obs = 820 -------------+------------------------------ F( 4, 815) = 16.07 Model | 13416.8521 4 3354.21302 Prob > F = 0.0000 Residual | 170118.196 815 208.733983 R-squared = 0.0731 -------------+------------------------------ Adj R-squared = 0.0686 Total | 183535.048 819 224.096518 Root MSE = 14.448

------------------------------------------------------------------------------ wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- edu | 1.206153 .16092 7.50 0.000 .8902869 1.52202 NorthEast | (dropped)

MidWest | -4.232238 2.069481 -2.05 0.041 -8.294378 -.1700976 South | -4.99543 2.122288 -2.35 0.019 -9.161225 -.8296364 West | -5.825008 2.198793 -2.65 0.008 -10.14097 -1.509043 _cons | 2.472763 2.818565 0.88 0.381 -3.05974 8.005266 ------------------------------------------------------------------------------

3

STATA Command: test (_b[MidWest]=0)(_b[South]=0)(_b[West]=0)

( 1) MidWest = 0

( 2) South = 0

( 3) West = 0

F( 3, 815) = 2.48

Prob > F = 0.0601

4

Corn Production Output

Consider the following model

ln(corn) =  1 +  2capital +  3labor +  4land +  5(labor   land) + 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. (lab_lan stands for labor   land as usual).

STATA Command: reg ln_corn capital labor land lab_lan

Source | SS df MS Number of obs = 1000 -------------+------------------------------ F( 4, 995) =26509.48 Model | 145016.524 4 36254.1309 Prob > F = 0.0000 Residual | 1360.75347 995 1.36759142 R-squared = 0.9907 -------------+------------------------------ Adj R-squared = 0.9907 Total | 146377.277 999 146.523801 Root MSE = 1.1694

------------------------------------------------------------------------------ ln_corn | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- capital | 2.654115 .0098252 270.13 0.000 2.634835 2.673396 labor | -.0045532 .3219485 -0.01 0.989 -.6363292 .6272228 land | .4869289 .21929 2.22 0.027 .056605 .9172528 lab_lan | .4458096 .0547902 8.14 0.000 .338292 .5533272 _cons | 2.946969 1.284514 2.29 .426302 5.467636 ------------------------------------------------------------------------------

------------------------------------------------------------------------------ STATA Command: test land = labor

. test land = labor

( 1) - labor + land = 0

F( 1, 995) = 8.08

Prob > F = 0.0046

5

Wage Regression for Rich and Poor Output

Consider the following model

ln(wage) =  1 +  2im_poor +  3im_rich + e;

where wage denotes the hourly wage in dollars, im_poor takes the value of 1 if the person is an immigrant from a poor country (0 otherwise), im_rich takes the value of 1 if the person is an immigrant from a rich country (0 otherwise). Consider also the variable native deÖned as

native = 1  im_poor  im_rich;

. After estimating the aforementioned model in Stata, the following results were obtained:

. reg ln_wage im_poor im_rich

Source | SS df MS Number of obs = 6770 -------------+------------------------------ F( 2, 6767) = 28.88 Model | 31.2933892 2 15.6466946 Prob > F = 0.0000 Residual | 3666.08138 6767 .541758738 R-squared = 0.0085 -------------+------------------------------ Adj R-squared = 0.0082 Total | 3697.37477 6769 .546221712 Root MSE = .73604

------------------------------------------------------------------------------ ln_wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- im_poor | -.1009196 .0210986 -4.78 0.000 -.1422795 -.0595597 im_rich | .1188251 .0260645 4.56 0.000 .0677303 .1699198 _cons | 2.719635 .0115901 234.65 0.000 2.696915 2.742356 ------------------------------------------------------------------------------

6

Moshe A. Buchinsky Economics 103 Department of Economics Introduction to Econometrics UCLA Fall, 2011

Final Examination

December 6, 2011

STATA OUTPUTS

1

Boca Raton ñOutput 2

Wage Regression Analysis Output

The following economic model predicts individualís wage based on education and region. E (W AGE) =  1 +  2EDU +  3M idW est +  4NorthEast +  5South +  6W est

where M idW est = 1 if Region == ìmidwest,îNorthEast = 1 if Region == ìnortheast,îSouth = 1 if Region == ìsouth,îW est = 1 if Region == ìwestî

STATA Command: table region

----------------------

region | Freq.

----------+-----------

midwest | 337

northeast | 57

south | 248

west | 178

----------------------

STATA Command: reg wage edu NorthEast MidWest South West

Source | SS df MS Number of obs = 820 -------------+------------------------------ F( 4, 815) = 16.07 Model | 13416.8521 4 3354.21302 Prob > F = 0.0000 Residual | 170118.196 815 208.733983 R-squared = 0.0731 -------------+------------------------------ Adj R-squared = 0.0686 Total | 183535.048 819 224.096518 Root MSE = 14.448

------------------------------------------------------------------------------ wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- edu | 1.206153 .16092 7.50 0.000 .8902869 1.52202 NorthEast | (dropped)

MidWest | -4.232238 2.069481 -2.05 0.041 -8.294378 -.1700976 South | -4.99543 2.122288 -2.35 0.019 -9.161225 -.8296364 West | -5.825008 2.198793 -2.65 0.008 -10.14097 -1.509043 _cons | 2.472763 2.818565 0.88 0.381 -3.05974 8.005266 ------------------------------------------------------------------------------

3

STATA Command: test (_b[MidWest]=0)(_b[South]=0)(_b[West]=0)

( 1) MidWest = 0

( 2) South = 0

( 3) West = 0

F( 3, 815) = 2.48

Prob > F = 0.0601

4

Corn Production Output

Consider the following model

ln(corn) =  1 +  2capital +  3labor +  4land +  5(labor   land) + 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. (lab_lan stands for labor   land as usual).

STATA Command: reg ln_corn capital labor land lab_lan

Source | SS df MS Number of obs = 1000 -------------+------------------------------ F( 4, 995) =26509.48 Model | 145016.524 4 36254.1309 Prob > F = 0.0000 Residual | 1360.75347 995 1.36759142 R-squared = 0.9907 -------------+------------------------------ Adj R-squared = 0.9907 Total | 146377.277 999 146.523801 Root MSE = 1.1694

------------------------------------------------------------------------------ ln_corn | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- capital | 2.654115 .0098252 270.13 0.000 2.634835 2.673396 labor | -.0045532 .3219485 -0.01 0.989 -.6363292 .6272228 land | .4869289 .21929 2.22 0.027 .056605 .9172528 lab_lan | .4458096 .0547902 8.14 0.000 .338292 .5533272 _cons | 2.946969 1.284514 2.29 .426302 5.467636 ------------------------------------------------------------------------------

------------------------------------------------------------------------------ STATA Command: test land = labor

. test land = labor

( 1) - labor + land = 0

F( 1, 995) = 8.08

Prob > F = 0.0046

5

Wage Regression for Rich and Poor Output

Consider the following model

ln(wage) =  1 +  2im_poor +  3im_rich + e;

where wage denotes the hourly wage in dollars, im_poor takes the value of 1 if the person is an immigrant from a poor country (0 otherwise), im_rich takes the value of 1 if the person is an immigrant from a rich country (0 otherwise). Consider also the variable native deÖned as

native = 1  im_poor  im_rich;

. After estimating the aforementioned model in Stata, the following results were obtained:

. reg ln_wage im_poor im_rich

Source | SS df MS Number of obs = 6770 -------------+------------------------------ F( 2, 6767) = 28.88 Model | 31.2933892 2 15.6466946 Prob > F = 0.0000 Residual | 3666.08138 6767 .541758738 R-squared = 0.0085 -------------+------------------------------ Adj R-squared = 0.0082 Total | 3697.37477 6769 .546221712 Root MSE = .73604

------------------------------------------------------------------------------ ln_wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- im_poor | -.1009196 .0210986 -4.78 0.000 -.1422795 -.0595597 im_rich | .1188251 .0260645 4.56 0.000 .0677303 .1699198 _cons | 2.719635 .0115901 234.65 0.000 2.696915 2.742356 ------------------------------------------------------------------------------

6

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