Problem Set # 3 - Introduction to Econometrics
Due: June 21st, 5:00PM
1. Measurement Errors in the Data (Again!)
Suppose the true model is given by
yi = β0 + β1xi + ui
and we are interested in estimating β1 . Suppose our sample is i.i.d and E[ui|xi] = 0. Unfortu- nately, we only have a noisy measure of xi. That is, we observe
xi(*) = xi + εi
Assume E[εi] = 0 and that εi is independent from everything. In this context, εi is our mea- surement error for observation i. This measurement error is not systematic, and is not related to any of the parameters we are interested in estimating. (So, it has zero covariance with any of the variables we are interested in). As we don’t have access to the “real” value of xi, we run the following regression:
yi = β0(*) + β1(*)xi(*) + ui(*)
We now know that in this case β(ˆ)1(*) is a biased and inconsistent estimator for β1 . Specifically,
What is happening here is that, from the true model,
as long as β1 ≠ 0 it is as if we are omitting a variable, εi, that correlates with both xi(*) and yi. (Notice that Cov(xi(*),εi) = Var(εi))
Now, suppose we have a separate measurement for xi:
˜(x)i = xi + ηi
where ηi is independent from xi , ui and ηi. So, we have a separate measurement of xi that is still measured with error, but the two measurement errors η and ε is independent.
(a) Verify that ˜(x)i is a valid instrument for xi(*) . (Hint: Two conditions to show.)
(b) Conclude that β1(TSLS) is an unbiased estimator for β1 in this case. How would you calculate β1(TSLS)? (Hint: After Part (a) you don’t need any additional math for this part.)
2. The Fulton Fish Market
In her 2006 paper, Graddy studies the Fulton fish market in Manhattan. She collected data for prices and quantities of fish sold. The very first thing to think about is that this is a market, so supply and demand curves are involved. We will dive deeper into this problem, focusing on the estimation of the demand curve.
First, let’s take a look at a simple supply and demand system of equations:
Where the first equation is a demand curve, the second is a supply curve, and the last equation is the market equilibrium condition. Qd and Qs are the demanded and supplied quantities,p is the price, ud and us are error terms and (α0 ,α 1 ,β0 ,β1 ) are parameters.
(a) Show that Cov(ud , p) ≠ 0. In other words, show that you can’t estimate the demand curve as is with ordinary least squares.
We will use the instrumental variables approach to proceed with the estimation. In her re- search about the market, Graddy concluded that the majority of clients were restaurants, and fishermen went to the sea with their own boats. Her proposed instrument for prices is the storminess of the weather, which acts as a shifter of the supply curve.
(b) Describe the conditions under which this is a valid instrument for prices, and discuss whether they likely hold.
With the proposed instrument, this is the regression she runs with OLS and IV:
log(q) = β0 + β1 log(p) + β2 Monday + β3Tuesday + β4 Wednesday+ +β5Thursday + β6Weather + β7Rain + ε
Where q is the quantity of fish traded, p is the price at which they’re traded. The extra con- trols are Monday , Tuesday , Wednesday, and Thursday , dummies for each day of the week, Weather is a variable that captures weather conditions in that specific day, and Rain is the precipitation in that specific day.
The table in the next page details the results of her regressions (standard errors are reported in parentheses). The first two columns show the results for the OLS estimation, without extra controls (regression 1) and with extra controls (regression 2), and using storminess as an in- strument for price (regressions 3 and 4 exclude and include the extra controls, respectively).
(c) Assuming all the controls are exogenous, write down the first stage regression to be used in the IV estimation.
(d) Interpret the coefficient associated with log(p) in regression (4).
(e) Is there (informal) evidence that prices are endogenous? You can calculate the confidence interval of the OLS coefficient in regression (2) and compare it to the same coefficient in regres- sion (4).
3. Identifying Cigarette Demand
You are given a dataset, “cigarettes1995.csv” which has information on quantity and price for cigarettes in 1995 for 48 states in the United States. In the data, you observe:
• state: Factor indicating state.
• year: Factor indicating year.
• cpi: Consumer price index.
• population: State population.
• packs: Number of packs per capita.
• income: State personal income (total, nominal).
• salestax: Sales tax for state in fiscal year.
• price: Average (real) prices for fiscal year.
You are interested in identifying the demand curve for cigarettes in the U.S.
(a) Regress log(packs) on log(price). Interpret the coefficients.
(b) Do you think you identified the demand curve? Do you expect log(price) to be exogenous? Why?
(c) To identify the demand curve, a researcher suggests using salestax as an instrument for log(prices). Is this a valid instrument? Why/Why not?
(d) Regress log(price) on salestax. Using the fitted values of log(price), create a new variable in your dataset called fitted logprice.
(e) Regress log(packs) on fitted logprice. Report the regressions from (a), (d) and (e) on the same table using Stargazer.
(f) Using the ivreg function in R, run an IV regression of log(packs) on log(price) where salestax is used as an instrument for log(price). Report the models from (a), (e) and (f) on the same table using Stargazer.
Models (e) and (f) should have the same coefficients but different standard errors. The stan- dard errors from model (e) are invalid, because the procedure does not take into account that we are using fitted values in the regression. The ivreg function in R corrects this problem, thus the standard errors in the model at (f) are the correct ones.
(g) Worried that income might be moving the demand curve, you decide to add it as a control variable. Regress log(packs) on log(price) using log(income) as a control variable.
(h) Regress log(price) on salestax using log(income) as a control variable. Save the fitted values as a new variable in your dataset, and name it fitted logprice income.
(i) Regress log(packs) on fitted log price and log(income). Also use the ivreg function in R to regress log(packs) on log(price) and log(income) where log(price) is instrumented using salestax and log(income). Report the 3 models from parts (h) and (i) using Stargazer.