Week 2 Tutorial Exercises (for the content of week 1 lecture)

Readings

Review your ECON2206 (Introductory Econometrics).

Make sure that you know the meanings of the econometrics terms mentioned in ECON2206. Read Chapter 15.1-15.2 thoroughly.

Make sure that you know the meanings of the Key Terms at the chapter end. Answer the summary questions in the lecture slides.

Question Set (these will be discussed in tutorial classes)

Q1. [smoke.dta, smoke-w2.do] To better understand the determinants of tobacco demand,

consider the following specification of a demand function for daily cigarette consumption: cigs = F0  + F1 lincome + F2 lcigpric + F3 educ + F4 age + F5 agesq + u.

This model is estimated using a sample of 807 individuals. Variable definitions and          associated summary statistics, together with the OLS estimation results, are given in the tables below. Please answer the following questions using the information in the tables.

(a) Carefully interpret all of the parameters in the regression model including their magnitudes and expected and actual signs.

(b) Comment on the statistical significance of each of the estimated coefficients.

(c)  On the basis of the estimation results, comment on the role of income and price as determinants of cigarette demand.

(d) Suppose the regression model was re-estimated under the hypothesis that F1  = F2  = 0 to yield a residual sums of squares of 145, 139. If the original model had a residual sums of squares of 144,910, test the null hypothesis that F1  = F2  = 0. Does the result of this  test affect your conclusion in (c)?

(e) On the basis of the estimation results, comment on the role of age as a determinant of cigarette demand.

(f)  Notice that there are no variables relating to government policy interventions such as

prohibition of tobacco advertising, or inclusion of health warnings on cigarette packets. Provide one justification why this omission could have been a reasonable assumption   for these data.

[Run smoke-w2.do in STATA. Try to understand the commands in smoke-w2.do.]

Q2. [Continue with Q1] Suppose the following diagnostics were associated with this model

R2  = 0.0451,   RESET = 2.03 (p-value = 0.132),   BP = 25.81 (p-value = 0.0001)

where the RESET test uses both squared and cubed predictions as additional variables, and  BP is the LM version of Breusch-Pagan test (see Ch8.3) that specifies that heteroskedasticity is a function of lincome, lcigpr, educ, age, and agesq.

(a) What is the null hypothesis RESET is testing? BP? Interpret each of the diagnostics. (b) Do the values of the diagnostics lead you to modify any of your answers in Q1?

(c)  Comment on the overall adequacy ofthe model in terms of the reported diagnostics and the results discussed in Q1.

Q3. Wooldridge C3.9. (charity.dta, charity-w2.do)

Use the data in CHARITY to answer the following questions:

(i)  Estimate the equation

gift = F0  + F1 mailsyear + F2giftlast + F3propresp + u

by OLS and report the results in the usual way, including the sample size and R-squared. How does the R-squared compare with that from the simple regression that omits giftlast and propresp?

(ii) Interpret the coefficient on mailsyear. Is it bigger or smaller than the corresponding simple regression coefficient?

(iii)Interpret the coefficient on propresp. Be careful to notice the units of measurement of propresp.

(iv) Now add the variable avggift to the equation. What happens to the estimated effect of mailsyear?

(v) In the equation from part (iv), what has happened to the coefficient on giftlast? What do you think is happening?

Q4. Wooldridge 2.8. (OLS algebra)

Consider the standard simple regression model y = F0  + F1 x + u under the Gauss-Markov Assumptions SLR.1, SLR.2, SLR.3, SLR.4 and SLR.5. The usual OLS estimators F0 and F1 are  unbiased for their respective population parameters. Let F1 be the estimator of F1  obtained by assuming the intercept is zero (see Section 2-6).

(i)  Find E(F1 ) in terms of the xi , F0 , and F1 . Verify that F1 is unbiased for F1 when the population intercept F0 is zero. Are there other cases where F1 is unbiased?

(ii) Find the variance of F1 . (Hint: The variance does not depend on F0 .)

(iii)Show that Var F1     ≤ Var(F1 ). [Hint: For any sample of data,    i(n)=1 xi(2)  ≥    i(n)=1  xi  − x  2 , with strict inequality unless x = 0.]

(iv) Comment on the tradeoff between bias and variance when choosing between F1 and F1 .

Q5. Wooldridge 2.10. (OLS algebra)

Let F0 and F1 be the OLS intercept and slope estimators, respectively, and let u be the sample average of the errors (not the residuals!).

(i)  Show that F1  can be written as F1  = F1  +    i(n)=1 wiui , where wi  = (xi  − x)/SSTx  .              (ii) Use part (i), along with     i(n)=1 wi  = 0, to show that F1 and u are uncorrelated. [Hint: You

are being asked to show that E   F1 − F1   u   = 0.]

(iii)Show that F0  can be written as F0  = F0  + u − F1  − F1   x.

(iv) Use parts (ii) and (iii) to show that Var F0     = a 2 /n + a2 x 2 /SSTx .

(v) Do the algebra to simplify the expression in part (iv) to equation (2.58). [Hint: SSTxn = n xi(2)  − x2 .]

Q6. Wooldridge 3.10. (omitted variable bias)

Suppose that you are interested in estimating the ceteris paribus relationship between y       and x1 . For this purpose, you can collect data on two control variables, x2 and x3 . (For            concreteness, you might think of y as final exam score, x1 as class attendance, x2 as GPA up  through the previous semester, and x3 as SAT or ACT score.) Let F1 be the simple regression estimate from y on x1 and let F1 be the multiple regression estimate from y on x1 , x2 , x3 .

(i)  If x1 is highly correlated with x2 and x3 in the sample, and x2 and x3 have large partial effects on y, would you expect F1 and F1 to be similar or very different? Explain.

(ii) If x1 is almost uncorrelated with x2 and x3 , but x2 and x3 are highly correlated, will F1 and F1 tend to be similar or very different? Explain.

(iii)If x1 is highly correlated with x2 and x3 , and x2 and x3 have small partial effects on y, would you expect se(F1 ) or se(F1 ) to be smaller? Explain.

(iv) If x1 is almost uncorrelated with x2 and x3 ,  and x2 and x3 have large partial effects on y, and x2 and x3 are highly correlated, would you expect se(1 ) or se(F1 ) to be smaller?      Explain.

Q7. Wooldridge 15.1. (endogeneity & IV)

Consider a simple model to estimate the effect of personal computer (PC) ownership on college grade point average (GPA) for graduating seniors at a large public university:

GPA = F0  + F1PC + u,

where PC is a binary variable indicating PC ownership.

(i)  Why might PC ownership be correlated with u?

(ii) Explain why PC is likely to be related to parents’ annual income. Does this mean parental income is a good IV for PC? Why or why not?

(iii)Suppose that, four years ago, the university gave grants to buy computers to roughly       one-half of the incoming students, and the students who received grants were randomly chosen. Carefully explain how you would use this information to construct an                     instrumental variable for PC.

Q8. Wooldridge 15.3. (IV algebra)

Consider the simple regression model

y = F0  + F1 x + u

and let Z be a binary instrumental variable for x. Use (15.10) to show that the IV estimator can be written as

F1  = (y1  − y0 )/(x1  − x0 ),

where y0  and x0 are the sample averages of yi  and xi  over the part of the sample with Zi  = 0, and y1  and x1 are the sample averages of yi  and xi  over the part of the sample with Zi  = 1.

This estimator, known as a grouping estimator, was first suggested by Wald (1940).

(The above are selected from the end-of-chapter Problems and Computer Exercises.)

Computer Exercise

All data files are in the course website, suffixed with “ .dta” (STATA data file format). The

data description is available by using STATA command “describe”.

Example STATA do-files, are also posted in the course website, suffixed with “ .do” . To carry out computations for Q1-2 and Q4, you need to read the file “Guide4STATA.pdf” if

you are not already familiar with STATA.

If you want to access STATA via myAccess, please follow the instructions in the file

“Stata_via_myAccess.pdf”.

Week 3 Tutorial Exercises (Instrumental Variables)

Readings

Read Chapter 15 thoroughly.

Make sure that you know the meanings of the Key Terms at the chapter end. Answer the summary questions in the lecture slides.

Question Set

Q1. Wooldridge 15.2. (endogeneity & IV)

Suppose that you wish to estimate the effect of class attendance on student performance, as in Example 6.3. A basic model is

stndfnl = F0  + F1 atndrte + F2priGPA + F3ACT + u,

where the variables are defined as in Chapter 6.

(i)  Let dist be the distance from the students’ living quarters to the lecture hall. Do you think dist is uncorrelated with u?

(ii) Assuming that dist and u are uncorrelated, what other assumption must dist satisfy to be a valid IV for atndrte?

(iii)Suppose, as in equation (6.18), we add the interaction term priGPA·atndrte:

stndfnl = F0  + F1 atndrte + F2priGPA + F3ACT + F4 priGPA · atndrte + u.                     If atndrte is correlated with u, then, in general, so is priGPA·atndrte. What might be a good IV for priGPA·atndrte? [Hint: If , as happens when priGPA, ACT, and dist are all exogenous, then  any function of priGPA and dist is uncorrelated with u.]

Q2. Wooldridge 15.8. (endogeneity & IV)

Suppose you want to test whether girls who attend a girls’ high school do better in math  than girls who attend coed schools. You have a random sample of senior high school girls from a state in the United States, and score is the score on a standardized math test. Let girlhs be a dummy variable indicating whether a student attends a girls’ high school.

(i)  What other factors would you control for in the equation? (You should be able to reasonably collect data on these factors.)

(ii) Write an equation relating score to girlhs and the other factors you listed in part (i).

(iii)Suppose that parental support and motivation are unmeasured factors in the error term in part (ii). Are these likely to be correlated with girlhs? Explain.

(iv) Discuss the assumptions needed for the number of girls’ high schools within a 20-mile radius of a girl’s home to be a valid IV for girlhs.

Suppose that, when you estimate the reduced form for girlshs, you find that the coefficient on numghs (the number of girls’ high schools within a 20-mile radius) is negative and         statistically significant. Would you feel comfortable proceeding with IV estimation where numghs is used as an IV for girlshs? Explain.

Q3. Wooldridge 15.11. (measurement error, time series, lags as IV)

Consider a simple time series model where the explanatory variable has classical measurement error:

yt  = F0  + F1 xt(*) + ut ,

xt  = xt(*)  + et ,

where ut  has zero mean and is uncorrelated with x and et . We observe yt  and xt  only.        Assume that et  has zero mean and is uncorrelated with x and that x also has a zero mean (this last assumption is only to simplify the algebra).

(i)  Write x = xt  − et  and plug this into (15.58). Show that the error term in the new         equation, say, vt , is negatively correlated with xt  if F1  > 0. What does this imply about the OLS estimator of F1 from the regression of yt  on xt?

(ii) In addition to the previous assumptions, assume that ut  and et  are uncorrelated with all past values of x and et; in particular, with x−1 and et−1 . Show that E  xt−1vt     = 0           where vt  is the error term in the model from part (i).

(iii)Are xt  and xt−1 likely to be correlated? Explain.

(iv) What do parts (ii) and (iii) suggest as a useful strategy for consistently estimating F0 and F1 ?

Q4. Wooldridge 15.C2. (FERTIL2.dta, FERTIL2-w3.do)

The data in FERTIL2 include, for women in Botswana during 1988, information on number of children, years of education, age, and religious and economic status variables.

(i)  Estimate the model

children = F0  + F1 educ + F2 age + F3 age2  + u

by OLS and interpret the estimates. In particular, holding age fixed, what is the                  estimated effect of another year of education on fertility? If 100 women receive another year of education, how many fewer children are they expected to have?

(ii) The variable frsthalf is a dummy variable equal to one if the woman was born during the first six months of the year. Assuming that frsthalf is uncorrelated with the error term     from part (i), show thatfrsthalf is a reasonable IV candidate for educ. (Hint: You need to do a regression.)

(iii)Estimate the model from part (i) by using frsthalf as an IV for educ. Compare the estimated effect of education with the OLS estimate from part (i).

(iv) Add the binary variables electric, tv, and bicycle to the model and assume these are            exogenous. Estimate the equation by OLS and 2SLS and compare the estimated                   coefficients on educ. Interpret the coefficient on tv and explain why television ownership has a negative effect on fertility.

Q5. Wooldridge 15.C3. (CARD.dta, CARD-w3.do)

Use the data in CARD for this exercise.

(i)  The equation we estimated in Example 15.4 can be written as

log(wage) = F0  + F1 educ + F2 exper + … + u,

where the other explanatory variables are listed in Table 15.1. In order for IV to be consistent, the IV for educ, nearc4, must be uncorrelated with u. Could nearc4 be     correlated with things in the error term, such as unobserved ability? Explain.

(ii) For a subsample of the men in the data set, an IQ score is available. Regress IQ on nearc4 to check whether average IQ scores vary by whether the man grew up near a four-year   college. What do you conclude?

(iii)Now, regress IQ on nearc4, smsa66, and the 1966 regional dummy variables reg662, …, reg669. Are IQ and nearc4 related after the geographic dummy variables have been       partialled out? Reconcile this with your findings from part (ii).

(iv) From parts (ii) and (iii), what do you conclude about the importance of controlling for smsa66 and the 1966 regional dummies in the log(wage) equation?

Q6. Wooldridge 15.C8. (401KSUBS.dta, 401KSUBS -w3.do)

Use the data in 401KSUBS for this exercise. The equation of interest is a linear probability model:

pira = F0  + F1p401k + F2 inc + F3 inc2  + F4 age + F5 age2  + u.

The goal is to test whether there is a tradeoff between participating in a 401(k) plan and having an individual retirement account (IRA). Therefore, we want to estimate F1 .

(i)  Estimate the equation by OLS and discuss the estimated effect of p401k.

(ii) For the purposes of estimating the ceteris paribus tradeoff between participation in two different types of retirement savings plans, what might be a problem with ordinary least squares?

(iii)The variable e401k is a binary variable equal to one if a worker is eligible to participate in a 401(k) plan. Explain what is required for e401k to be a valid IV for p401k. Do these assumptions seem reasonable?

(iv) Estimate the reduced form for p401k and verify that e401k has significant partial           correlation with p401k. Since the reduced form is also a linear probability model, use a heteroskedasticity-robust standard error.

(v) Now, estimate the structural equation by IV and compare the estimate of F1 with the OLS estimate. Again, you should obtain heteroskedasticity-robust standard errors.

(vi) Test the null hypothesis that p401k is in fact exogenous, using a heteroskedasticity- robust test.

Q7. Wooldridge 15.C10. (HTV.dta, HTV-w3.do)

Use the data in HTV for this exercise.

(i)  Run a simple OLS regression of log(wage) on educ. Without controlling for other factors, what is the 95% confidence interval for the return to another year of education?

(ii) The variable ctuit, in thousands of dollars, is the change in college tuition facing students from age 17 to age 18. Show that educ and ctuit are essentially uncorrelated. What does  this say about ctuit as a possible IV for educ in a simple regression analysis?

(iii)Now, add to the simple regression model in part (i) a quadratic in experience and a full set of regional dummy variables for current residence and residence at age 18. Also      include the urban indicators for current and age 18 residences. What is the estimated   return to a year of education?

(iv) Again using ctuit as a potential IV for educ, estimate the reduced form for educ.           [Naturally, the reduced form for educ now includes the explanatory variables in part (iii).] Show that ctuit is now statistically significant in the reduced form for educ.

(v) Estimate the model from part (iii) by IV, using ctuit as an IV for educ. How does the         confidence interval for the return to education compare with the OLS CI from part (iii)?

(vi) Do you think the IV procedure from part (v) is convincing?

Week 4 Tutorial Exercises (Simultaneous Equations)

Readings

Read Chapter 16.1 to 16.4.

Make sure that you know the meanings of the Key Terms at the chapter end. Answer the summary questions in the lecture slides.

Question Set

Q1. Wooldridge 16. 1. (reduced form equations)

Write a two-equation system in “supply and demand form,” that is, with the same variable (typically, “quantity”) appearing on the left-hand side:

y1  = a1 y2  + F1 z1  + u1 ,

y1  = a2 y2  + F2 z2  + u2 .

(i)  If a1  = 0 or a2  = 0, explain why a reduced form exists for y1 . (Remember, a reduced form expresses y1  as a linear function of the exogenous variables and the structural  errors.) If a1  ≠ 0 or a2  = 0, find the reduced form for y2 .

(ii) If a1  ≠ 0, a2  ≠ 0, and a1  ≠ a2 , find the reduced form for y1 . Does y2 have a reduced form in this case?

(iii)Is the condition a1  ≠ a2 likely to be met in supply and demand examples? Explain.

Q2. Wooldridge 16.2. (demand & supply equations)

Let corn denote per capita consumption of corn in bushels at the county level, let price be       the price per bushel of corn, let income denote per capita county income, and let rainfall be    inches of rainfall during the last corn-growing season. The following simultaneous equations model imposes the equilibrium condition that supply equals demand:

corn = a1price + F1 income + u1 ,

corn = a2price + F2 rainfall + y2 rainfall2  + u2 .

Which is the supply equation, and which is the demand equation? Explain.

Q3. Wooldridge 16.4. (order & rank conditions)

Suppose that annual earnings and alcohol consumption are determined by the SEM log(earnings) = F0  + F1 alcohol + F2 educ + u1 ,

alcohol = y0  + y1 log(earnings) + y2 educ + y3 log(price) + u2 ,

where price is a local price index for alcohol, which includes state and local taxes. Assume that educ and price are exogenous. If F1 , F2 , y1 , y2 , and y3 are all different from zero, which equation is identified? How would you estimate that equation?

Q4. Wooldridge 16.5. (simultaneity bias & IV)

A simple model to determine the effectiveness of condom usage on reducing sexually transmitted diseases among sexually active high school students is

infrate = F0  + F1 conuse + F2percmale + F3 avginc + F4 city + u1

where

infrate = the percentage of sexually active students who have contracted venereal disease.

conuse = the percentage of boys who claim to regularly use condoms.

percmale = boy percentage in a school.

avginc = average family income.

city = a dummy variable indicating whether a school is a city.

The model is at the school level.

(i)  Interpreting the preceding equation in a causal, ceteris paribus fashion, what should be the sign of F1 ?

(ii) Why might infrate and conuse be jointly determined?

(iii)If condom usage increases with the rate of venereal disease, so that y1  > 0 in the

equation

conuse = y0  + y1 infrate + other factors,

what is the likely bias in estimating F1 by OLS?

(iv) Let condis be a binary variable equal to unity if a school has a program to distribute        condoms. Explain how this can be used to estimate F1  (and the other betas) by IV. What do we have to assume about condis in each equation?

Q5. Wooldridge 16.6. (simultaneity & IV validity)

Consider a linear probability model for whether employers offer a pension plan based on the percentage of workers belonging to a union, as well as other factors:

pension=β0 +β1percunion+ β2avgage+ β3avgeduc+ β4percmale+ β5percmarr+u1.

(i)  Why might percunion be jointly determined with pension?

(ii) Suppose that you can survey workers at firms and collect information on workers’          families. Can you think of information that can be used to construct an IV for percunion?

(iii)How would you test whether your variable is at least a reasonable IV candidate for percunion?

Q6. Wooldridge 16.C1. (SMOKE.dta, SMOKE-w4.do)

Use SMOKE for this exercise.

(i)  A model to estimate the effects of smoking on annual income (perhaps through lost work days due to illness, or productivity effects) is

log  income   = F0  + F1 cigs + F2 educ + F3 age + F4 age2  + u1,

where cigs is number of cigarettes smoked per day, on average. How do you interpret F1 ? (ii) To reflect the fact that cigarette consumption might be jointly determined with income,

a demand for cigarettes equation is

cig = y0  + y1 log  income   + y2 educ + y3 age + y4 age2  + y5 log  cigpric + y6 restaurn + u2,

where cigpric is the price of a pack of cigarettes (in cents) and restaurn is a binary             variable equal to unity if the person lives in a state with restaurant smoking restrictions. Assuming these are exogenous to the individual, what signs would you expect for y5 and y6 ?

(iii)Under what assumption is the income equation from part (i) identified? (iv) Estimate the income equation by OLS and discuss the estimate of F1 .

(v) Estimate the reduced form for cigs. (Recall that this entails regressing cigs on all       exogenous variables.) Are log(cigpric) and restaurn significant in the reduced form?

(vi) Now, estimate the income equation by 2SLS. Discuss how the estimate of F1  compares with the OLS estimate.

(vii)              Do you think that cigarette prices and restaurant smoking restrictions are

exogenous in the income equation?

Week 5 Tutorial Exercises (Finish SEM + Start Logit and Probit Models)

Readings

Finish reading Chapter 16. Read Chapter 17.1 to 17.2.

Make sure that you know the meanings of the Key Terms at the chapter end. Answer the summary questions in the lecture slides.

Question Set

Q1. Wooldridge 16.7. (SEM, time series context)

For a large university, you are asked to estimate the demand for tickets to women’s             basketball games. You can collect time series data over 10 seasons, for a total of about 150 observations. One possible model is

lATTENDt = β0 + β1lPRICEt + β2 WINPERCt + β3RIVALt + β4 WEEKENDt + β5 t + ut, where

PRICEt = the price of admission, probably measured in real terms—say, deflating by a regional consumer price index.

WINPERCt = the team’s current winning percentage.

RIVALt = a dummy variable indicating a game against a rival.

WEEKENDt = a dummy variable indicating whether the game is on a weekend.

The l denotes natural logarithm, so that the demand function has a constant price elasticity.

(i)  Why is it a good idea to have a time trend t in the equation?

(ii) The supply of tickets is fixed by the stadium capacity; assume this has not changed over  the 10 years. This means that quantity supplied does not vary with price. Does this mean that price is necessarily exogenous in the demand equation? (Hint: The answer is no.)

(iii)Suppose that the nominal price of admission changes slowly—say, at the beginning of each season. The athletic office chooses price based partly on last season’s average      attendance, as well as last season’s team success. Under what assumptions is last         season’s winning percentage (SEASPERCt- 1) a valid instrumental variable for lPRICEt?

(iv) Does it seem reasonable to include the (log of the) real price of men’s basketball games  in the equation? Explain. What sign does economic theory predict for its coefficient? Can you think of another variable related to men’s basketball that might belong in the             women’s attendance equation?

(v) If you are worried that some of the series, particularly lATTEND and lPRICE, have unit roots, how might you change the estimated equation?

(vi) If some games are sold out, what problems does this cause for estimating the demand function? (Hint: If a game is sold out, do you necessarily observe the true demand?)

Q2. Wooldridge 16.C8. (SEM, time series, FISH.dta, FISH-w5.do)

Use the data set in FISH, which comes from Graddy (1995), to do this exercise. The data set is also used in Computer Exercise C9 in Chapter 12. Now, we will use it to estimate a             demand function for fish.

(i)  Assume that the demand equation can be written, in equilibrium for each time period, as log  totqtyt     = a1 log  avgprct    + b10  + b11 mont  + b12 tuest  + b13 wedt

+b14 thurst  + ut1,

so that demand is allowed to differ across days of the week. Treating the price variable as endogenous, what additional information do we need to estimate the demand-          equation parameters consistently?

(ii) The variables wave2t and wave3t are measures of ocean wave heights over the past  several days. What two assumptions do we need to make in order to use wave2t and wave3t as IVs for log(avgprct) in estimating the demand equation?

(iii)Regress log(avgprct) on the day-of-the-week dummies and the two wave measures. Are wave2t and wave3t jointly significant? What is the p-value of the test?

(iv) Now, estimate the demand equation by 2SLS. What is the 95% confidence interval for the price elasticity of demand? Is the estimated elasticity reasonable?

(v) Obtain the 2SLS residuals, ut1 . Add a single lag ut−1, 1 ,  in estimating the demand                   equation by 2SLS. Remember, use ut−1, 1 as its own instrument. Is there evidence of AR(1) serial correlation in the demand equation errors?

(vi) Given that the supply equation evidently depends on the wave variables, what two         assumptions would we need to make in order to estimate the price elasticity of supply?

(vii)  In the reduced form equation for log(avgprct), are the day-of-the-week dummies jointly significant? What do you conclude about being able to estimate the supply elasticity?

Discrete Choice Models

Q3. Wooldridge 17.1. (binary response algebra)

(i)  For a binary response y, let y be the proportion of ones in the sample (which is equal to the sample average of the yi). Let q0 be the percent correctly predicted for the outcome y = 0 and let q1 be the percent correctly predicted for the outcome y = 1. If p is the       overall percent correctly predicted, show that p is a weighted average of q0 and q1 :

p =   1 − y  q0  + yq1 .

(ii) In a sample of 300, suppose that y = .7, so that there are 210 outcomes with yi  = 1 and 90 with yi  = 0. Suppose that the percent correctly predicted when y = 0 is 80, and the  percent