Insurance and real estate weekly wage

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Consider again the following demand and supply equations ((1.1) and (1.2) below, respectively) for truffles (see Assignment 2, Question 2):
𝑄𝑖 = 𝛽0 + 𝛽1𝑃𝑖 + 𝛽2𝑃𝑆𝑖 + 𝛽3𝐷𝐼𝑖 + 𝑢1,𝑖, (1.1)

𝑄𝑖 = 𝛾0 + 𝛾1𝑃𝑖 + 𝛾2𝑃𝐹𝑖 + 𝑢2,𝑖, (1.2) where the variables are:
𝑄𝑖: quantity of truffles, in ounces, traded in a particular French market place (indexed by 𝑖); 𝑃𝑖: price of truffles, in dollars per ounce ($/ounce);
𝑃𝑆𝑖: price of a truffles substitute ($/ounce);
𝐷𝐼𝑖: per capita monthly disposable income of local residents, in $000s;
𝑃𝐹𝑖: price of a factor of production (hourly rental price of truffle pigs), $;
and 𝑢1,𝑖 and 𝑢2,𝑖 are error terms.

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|𝑋𝑖 = 𝑋0) + 𝛽1(𝑋𝑖 − 𝑋0),
which implies that regressing 𝑌𝑖 on (𝑋𝑖 − 𝑋0) gives the desired prediction as the estimated intercept.

𝑋𝑖 = (1 + 𝜋)𝑊𝑖 + 𝑍𝑖 + 𝑣2,𝑖 (2.2)

where 𝐸(𝑣1,𝑖|𝑊𝑖, 𝑍𝑖) = 𝐸(𝑣2,𝑖|𝑊𝑖, 𝑍𝑖) = 0, 𝑣1,𝑖 and 𝑣2,𝑖 are uncorrelated and 𝜋 is a positive parameter (𝜋 > 0).

•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):

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.

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.

(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.

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:

𝑊𝑖,𝑡 = 𝛼𝑖 + 𝛽1 log(𝑋𝑖,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).

APPENDIX QUESTION