Econometrics Assignment -

QUESTION 1 -

Consider again the following demand and supply equations ((1.1) and (1.2) below, respectively) for truffles

Q_i = β₀ + β₁P_i + β₂PS_i + 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 over identifying 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 PS_i and DI_i set to their sample means-and obtain a 95% confidence interval for that prediction. Report the prediction and interval.

QUESTION 2 -

The reduced form equations for the simultaneously determined variables Y_i and X_i:

Y_i = (1 + π)W_i + (1 + π)Z_i +v_1,i (2.1)

X_i = (1 + π)W_i + X_i + v_2,i (2.2)

where E(v_1,i|W_i,Z_i) = E(v_2,i|W_i,Z_i) = 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

Y_i = α₁X_i + β₁Z_i = u_1,i (2.3)

X_i = α₂Y_i + β₂W_i + u_2,i (2.4)

The following assumptions and settings used to perform the simulations based on 1,000 replications 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 W_i and Z_i as instruments
• W_i ∼ iid N(0, 1), Z_i ∼ iid N(0, 1), v_1,i ∼ iid N(0, 1), v_2,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, comprising data on 90 counties in North Carolina from 1980 to 1987 (total sample size 630). Except for the regional dummies (west_i, central_i and urban_i), 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.

cr_i,t crimes committed per person

pa_i,t 'probability' of arrest

pc_i,t 'probability' of conviction

pp_i,t 'probability' of prison sentence

s_i,t average prison sentence, in days

police_i,t police per capita

density_i,t people per square mile

west_i =1 if county in western North Carolina, 0 otherwise

central_i = 1 if in central North Carolina, 0 otherwise

urban_i = 1 if county in metropolitan area, 0 otherwise

minority_i percentage of population in county from minority groups, 1980

wcon_i,t weekly wage, construction

wtuc_i,t weekly wage, transport, utilities and communications

wtrd_i,t weekly wage, wholesale and retail trade

wfir_i,t weekly wage, finance, insurance and real estate

wser_i,t weekly wage, services industry

wmfg_i,t weekly wage, manufacturing

wfed_i,t weekly wage, Federal employees

wsta_i,t weekly wage, state employees

wloc_i,t weekly wage, local government employees

male_i,t percentage in county that are young males

In the above list, independent variables such as pa_i,t, pc_i,t, pp_i,t, s_i,t and police_i,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 pa_i,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 pa_i,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 pa_i,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 pa_i,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 + β₁log(X_i,t) + β₂B_i + β₃(B_ilog(X_i,t)) + U_i,t,

where W_i,t is male i's wage in year t, X_i,t is his work experience in year t, and B_i =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.

(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.

Attachment:- Assignment File.rar