Economics 312

Spring 2012

Econometric Project #1

Due 6am, Tuesday, February 7

The first project involves two parts. You will work in teams of two students, as assigned below (except for one student who has no partner this week). While some specialization is acceptable, do not just divide these problems with one member of your team doing each one. Both members of the team must contribute to the estimation and report-writing of both problems. For more details on what is expected in terms of collaboration and your report, refer to these guidelines and this sample report.

For this assignment, your report should be in two separate parts, one for each of the problems.

1. Hill, Griffiths, and Lim, Problem 2.10

The first part of this project involves doing one of HGL's applied problems from Chapter 2. Problem 2.10 is an application of the capital-asset pricing model. The problem itself gives about two paragraphs of background on this foundational model of financial economics. For more details, you are encouraged to read Chapter 2 of Ernst Berndt's The Practice of Econometrics.

HGL applied problems have two files associated with them: a .dta file that can be opened in Stata and a .def file that is a text definition of the variables. For this problem, the files are capm4.dta and capm4.def.

2. Extension of the Monte Carlo Simulation in Class

The second part of this project involves repeating the Monte Carlo simulation that was performed in class on January 30, then modifying it to allow the error term to follow a uniform distribution rather than a normal distribution.

• Repeat the in-class Monte Carlo simulation using 10,000 replications, saving the results for both the slope coefficient (b = _b[x]) and the standard error of the slope coefficient (se = _se[x]) in a Stata data set. The data file and do file used in class can be downloaded from these links. Then analyze the following questions about the distribution of the resulting estimates:
  • Show the summary statistics for your OLS estimates of the coefficient and the standard error. Does the coefficient estimator seem unbiased?
  • Theory says that the variance of the OLS slope estimator is given by HGL's equation (2.15); you can calculate the sum of squared x deviations that you need for this formula from the sample standard deviation of x, which is reported by the summarize command: Take the sample standard deviation (printed out by summarize) and square it to get the variance, then multiply it by N - 1 to get the sum of squared deviations. (The Stata command display makes Stata behave like a calculator.) After computing this sum, use it to compute the theoretical standard deviation of the OLS slope estimator given this sample of x values using (2.15). Compare the standard deviation of your OLS slope estimates from the Monte Carlo simulation with the theoretical standard deviation from equation (2.15). Are they close? The OLS standard error attempts to measure this standard deviation. Examine the mean of the distribution of your 10,000 estimated OLS standard errors. Is the OLS standard error a good estimator of the standard deviation of the coefficient estimator in your simulation?
  • Plot a histogram of your OLS coefficient estimates. Theory says that they should follow the normal distribution. Does this seem plausible?
  • Use the Stata Help menu to find a suitable test for whether a random variable follows the normal distribution. Look up in the pdf or printed Stata manuals the procedure used for your chosen test. Describe the test both intuitively (what's the basic idea) and computationally, and justify why it is an appropriate test in this context. Based on the results of the test, are you comfortable concluding that the OLS estimator follows a normal distribution? [Students frequently "find" estimators or tests in Stata. This is OK, but, as in this problem, you need to know how it works and be able to justify its use.]
• We now re-run the simulation with an error term that follows a uniform distribution. According to theory, the distribution of the OLS estimator is asymptotically normal, so that it should converge to a normal distribution as N gets large. Your N (157) is pretty large, but not extremely large, so it is an interesting question whether asymptotic normality is valid.
  • HGL discuss the uniform distribution in Section B.3.4 of Appendix B. The two parameters of the uniform distribution are a and b, the minimum and maximum values that the variable can have. We want to make our uniform distribution as much like the normal distribution of the previous part as possible, so we want a mean of zero and variance of two. Use the formulas for the mean and variance of the uniform distribution on page 679 to calculate the values of a and b that make the mean zero and the variance two.
  • Run a Monte Carlo simulation with 10,000 replications drawing the values of e from a uniform distribution with the a and b values that you calculated above. [You can use the Stata function runiform in place of the rnormal, but be sure to look at the Stata help file to see how it is used.] As before, save the values of the coefficient estimates and their standard errors.
  • Perform the three tasks from the previous part (summary statistics, variance, histogram, and normality tests) and interpret the results.
  • What do you conclude about the validity of the asymptotic properties of the OLS estimator for uniform errors and N = 157?

Project Teams