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ECON3208
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 = β0 + β1lincome + β2lcigpTic + β3educ + β4age + β5agesq + 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 β1 = β2 = 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 β1 = β2 = 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 of the model in terms of the reported diagnostics and the results discussed in Q1.
Q3. 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 examscore, x1 as class attendance, x2 as GPA up through the previous semester, and x3
estimate from y on x1 and let β1 be the multiple regression estimate from yon x1, x2, x3 . (i) If x1 is highly correlated with x2 and x3 in the sample, and x2 and x3 have large partial
effectsony, would you expect β1 and β1 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 β1
and β1 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 effectsony,
(iv) If x1 is almost uncorrelated with x2 and x3, and x2 and x3 have large partial effectsony,
and x2 and x3 are highly correlated, would you expectse(β1) or se(β1) to be smaller? Explain.
Q4. Wooldridge C3.9. (charity.dta, charity-w2.do)
Use the data in CHARITY to answer the following questions:
(i) Estimate the equation
gift = β0 + β1 mailsyear + β2giftlast + β3propresp + u
by OLS and report the results in the usual way, including the sample size and R-squared. How doesthe 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?
Q5. Wooldridge 2.8. (OLS algebra)
Consider the standard simple regression model y = β0 + β1x + u under the Gauss-Markov
Assumptions SLR.1, SLR.2, SLR.3, SLR.4 and SLR.5. The usual OLS estimators β0 and β1 are
unbiased for theirrespective population parameters. Let β1 be the estimator of β1 obtained by assuming the intercept is zero (see Section 2-6).
(i) population intercept β0 1 is unbiased?
(ii) Find the variance of β1. (Hint: The variance does not depend on β0.)
(iii)Show that Var1 ) ≤ Var(1). [Hint: For any sample of data, 1 xi(2) ≥ 1 (xi ? )2,
with strict inequality unless = 0.]
(iv) Comment on the tradeoff between bias and variance when choosing between β1 and β1 .
Q6. Wooldridge 2.10. (OLS algebra)
Let β0 and β1 be the OLS intercept and slope estimators, respectively, and letu(-) be the sample average of the errors (not the residuals!).
(i) Show that β1 can be written as β1 = β1 + 1 wiui, where wi = (xi ? )/SSTx .
(ii) Use part (i), along with
are being asked to show that E[(1 ? β1 )u(-)] = 0.]
(iii)Show that 0 can be written as 0 = β0 +u(-) ? (1 ? β1 ) .
(iv) Use parts (ii) and (iii) to show that Var(0) = σ 2 /n + σ 22 /SSTx .
(v) Do the algebra to simplify the expression in part (iv) to equation (2.58). [Hint: SSTxn = n ? 1
Q7. Wooldridge 15.1. (endogeneity & IV)
Consider asimple 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 = β0 + β1PC + 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.