QUESTION 1

Consider again the following demand and supply equations ((1.1) and (1.2) below, respectively) for truffles (see Assignment 2, QUESTION 2):

Q_i = β₀ + β₁Pi + β₂PSi + β₃DI_i + u_1,i, (1.1)

Q_i = γ₀ + γ₁P_i + γ₂PF_i + u_2,i, (1.2)

where the variables are:

Q_i: quantity of truffles, in ounces, traded in a particular French market place (indexed by i);
P_i: price of truffles, in dollars per ounce ($/ounce);
PS_i: price of a truffles substitute ($/ounce);
DI_i: per capita monthly disposable income of local residents, in $000s;
PF_i: price of a factor of production (hourly rental price of truffle pigs), $; and u_1,i and u_2,i are error terms.

Note: in this QUESTION, (*) means that you are required to report in a table in Appendix A the regression results on which the test is based, not the test output itself.

Using the data in the file truffles.csv:
(a) (*) Test for instrument relevance. interpret the result.
(b) (*) Test any overidentifying restriction(s). Report the p-value for the test.
(c) interpret the result of the test performed in (b) above.
(d) (*) Estimate a regression to predict the quantity of truffles demanded at a price of $50 per ounce-with PSi and Dii set to their sample means-and obtain a 95% confidence interval for that prediction. Report the prediction and interval.

Hint
Given the PRF,
E(Yi|Xi) = β₀ + β₁Xi,
the prediction based on (say) X₀ is
E(Yi|Xi = X₀) = β₀ + β₁X₀.
Subtracting the latter from the former and rearranging slightly yields
E(Yi|Xi) = E(Yi|X_i = X₀) + β₁(Xi - X₀),
which implies that regressing Yi on (Xi - X₀) gives the desired prediction as the estimated intercept.

QUESTION 2

Recall from Assignment 2 (QUESTION 3) the reduced form equations for the simultaneously determined variables Yi and Xi:
Yi = (1 + Π)Wi + (1 + Π)Zi + v_1,i (2.1)
Xi = (1 + Π)W_i + Zi + v_2,i (2.2) where E(v_1,i|Wi, Zi) = E(v_2,i|Wi, Zi) = 0, v_1,i and v_2,i are uncorrelated and Π is a positive parameter (Π > 0).
The structural equations corresponding to (2.1) and (2.2) have the form
Yi = α₁Xi + β₁Zi + u_1,i (2.3)
Xi = α₂Yi + β₂W_i + u_2,i (2.4)
The following assumptions and settings used to perform the simulations based on 1,000 replications in Assignment 2 are to be used again:
- The true DGP is given by (2.1) and (2.2)
- The parameter of interest is α1
- (2.3) is estimated separately by OLS and 2SLS using Wi and Zi as instruments
- Wi~iid N(0, 1), Zi~iid N(0, 1), v1,i~iid N(0, 1), v2,i~iid N(0, 1)
- the sample size (n) is 100

in this QUESTION, the task is to compute by simulation the rejection frequencies for H0: α1 = 1
vs. H1: α1 ≠ 1 under 2SLS and OLS, assuming a 5% level of significance.

(a) Report the rejection frequencies in a table as formatted below (please set seed of 12):

Table Q2. Rejection frequencies for H0: α1 = 1, with varying Π (1,000 replications)

Note: Each cell entry is the proportion of times out of 1,000 that α1 = 1 is rejected at the 5% level.

(b) Comment on both sets of results (2SLS and OLS) and explain why they differ (or do not).

QUESTION 3

This QUESTION uses crime.csv, which is an abridged version of the dataset used by Baltagi (2006) to analyse determinants of the crime rate,1 comprising data on 90 counties in North Carolina from 1980 to 1987 (total sample size 630). Except for the regional dummies (westi,
centrali and Urbani), the variables in the dataset (listed below) are all in natural logs. in the datafile, the cross-section identifier is "county" and the time-series identifier is "year". in the list of variables below, the first is the dependent variable; all others are independent variables. The i and t subscripts denote county and year, respectively.2
cri,t crimes committed per person
pai,t ‘probability' of arrest
pci,t ‘probability' of conviction
ppi,t ‘probability' of prison sentence
si,t average prison sentence, in days
policei,t police per capita
densityi,t people per square mile
westi = 1 if county in western North Carolina, 0 otherwise
centrali = 1 if in central North Carolina, 0 otherwise
Urbani = 1 if county in metropolitan area, 0 otherwise
minorityi percentage of population in county from minority groups, 1980
wconi,t weekly wage, construction
wtuci,t weekly wage, transport, utilities and communications
wtrdi,t weekly wage, wholesale and retail trade
wfiri,t weekly wage, finance, insurance and real estate
wseri,t weekly wage, services industry
wmfgi,t weekly wage, manufacturing
wfedi,t weekly wage, Federal employees
wstai,t weekly wage, state employees
wloci,t weekly wage, local government employees
malei,t percentage in county that are young males

In the above list, independent variables such as pai,t, pci,t, ppi,t, si,t and policei,t are thought of as ‘deterrence' variables (for obvious reasons) and the weekly wage by industry variables are thought of as representing returns to legal opportunities (hereafter, ‘opportunity' variables); i.e., getting a wage rather than committing crime! Others are miscellaneous control variables.

(a) Regress the dependent variable on all the other variables using the pooled OLS estimator. Report the estimated coefficient on pai,t and its standard error.

(b) Regress the dependent variable on all the other variables using the within estimator with cross-section fixed effects only. Report the estimated coefficient on pai,t and its standard error.

(c) Regress the dependent variable on all the other variables using the within estimator with time fixed effects only. Report the estimated coefficient on pai,t and its standard error.

(d) Regress the dependent variable on all the other variables using the within estimator with two-way fixed effects. Report the estimated coefficient on pai,t and its standard error.

(e) What does a comparison of the coefficient estimates cited in (a), (b), (c) and (d) above suggest to you about the relative importance of unobserved cross-section fixed effects and unobserved time fixed effects, and why?

Base your answers to the remaining parts on the two-way fixed effects results.

(f) interpret the estimated coefficient of any ‘deterrence' variable that is statistically significant at 1% and has the expected sign.

(g) interpret the estimated coefficient of any ‘opportunity' variable that is statistically significant at 1% and has the expected sign.

(h) Name any ‘deterrence' variable that has an estimated coefficient that is statistically significant at 1% and has the opposite of the expected sign. Suggest a reason for the unexpected sign.

(i) Explain what might be done econometrically to resolve the incorrect sign problem referred to in (h) above. Describe any proposed additional variables(s) needed to implement this strategy in practice.

(j) Suggest a relevant variable that might have been excluded from the model and explain why it should be included.

QUESTION 4

This question uses panel data on the wages (in dollars per hour in real terms), work experience (in years) and skin colour of 545 males for each of the years 1980-1987 to estimate the following one-way fixed effects model using the within estimator:

W_i,t = α_i + β₁ log(X_i,t) + β₂Bi + β3(Bi log(Xi,t)) + Ui,t,

where Wi,t is male i's wage in year t, Xi,t is his work experience in year t, and Bi = 1 if he is black (0 otherwise).

The results are shown in Table 4.1 below, in which heteroskedasticity-consistent standard errors are in brackets.

Table 4.1: Estimation results
Dependent variable: W_i,t
log(Xi,t)               2.0544***
                          (0.1273)
Bi log(Xi,t)             -0.8008**
                          (0.2619)
R²                        0.1796
Adj. R²                  0.0621
Num. obs.              4358
***p < 0.001, **p < 0.01, *p < 0.05

(a) Why are there only two estimated coefficients in Table 4.1? Also provide some rationale for logging the experience variable.
(b) interpret the coefficient estimates in Table 4.1.

APPENDIX QUESTION

(a) Present neatly tabulated regression results for all parts above marked (*) in Appendix A.
(b) Present your R-code in Appendix B.