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Problem Set 2

EC 303: Empirical Economic Analysis

1 Theoretical Problems

Problem 1 (10 points). Suppose X has a Poisson distribution with P(X = 1) = P(X = 2). Find the probability that P(X = 1orX = 2).

Problem 2 (10 points). Under what conditions

fX,Y (x, y) = ax² + bxy + cy²

is a pdf where x, y ∈ [0, 1] and a, b, c are scalars ?

Problem 3 (10 points). Suppose a = 1, b = 0, and c = 2 for the above pdf. Compute the marginal pdf of X and marginal pdf of Y.

Problem 4 (10 points). Work on the problems from the textbook chapter 5: 8, 12, 14, 28. A subset of these questions will be graded.

Problem 5 (10 points). Work on the problems from the textbook Chapter 6: 4, 6. Both questions will be graded.

2 Empirical Problems

Problem 2.1: Working with real data set. For this problem, you need a stock return data. Visit https://www.nasdaq.com/market-activity/quotes/historical and download price data for Apple, Mi-crosoft, and Amazon stocks. In your dataset you should have information about the open and close prices of each stock everyday during the last month (09.07.2021-10.07.21)

You need to submit a Stata do file to get a full credit for this problem. When your code is run by the grader (your TA), it should work with no error.

a. (5 points) Explain your rows and columns in your data.

b. (5 points) Construct a return variable for each stock in your dataset i.e. return on Apple ‘rA’, return on Microsoft ‘rM’, and return on Amazon ‘rAM’ by calculating the price change of each stock everyday in percentage terms. Once you complete this step you should be able to tell the percentage change in the price of a stock in a given day between 09.07.21-10.07.21

c. (5 points) For all 3 stock returns, report it’s sample mean and sample standard deviation over one month period.

d. (5 points) For each pair of stock returns (3 pair total), compute the sample correlation coefficient. Interpret your results.

e. (5 points) Suppose you have a portfolio of these 3 stocks. Now that you observed the returns for a month, how would you adjust your portfolio at the end of the month? Why? No single correct answer here. Reasoning is more important.

Problem 2.2: Simulation. For this simulation exercise, you will need to draw a random sample from a population with a known PMF. The goal is to understand the distribution of sample mean.

You need to submit a Stata do file to get a full credit for this problem. When your code is run by the grader (your TA), it should work with no error.

a. (5 points) Set number of observations in your data set to 10 and number of samples to 500.

b. (5 points) Let X be a discrete RV and suppose in the population it can take values 10, 20, 30, 40, and 50 with equal probabilities i.e. P(X = 10) = P(X = 20) = P(X = 30) = P(X = 40) = P(X = 50) = .2 Get a random sample of size 10 from the population described. Do it 500 times.

c. (5 points) For each sample you get above (500 samples total) calculate the sample mean and sample variance.

d. (5 points) How does the distribution of sample mean look like? Draw a histogram. What about the distribution of sample variance? Remember we are looking at the values of sample means and sample variances over 500 samples.

e. (5 points) Follow the same steps above for sample sizes 50, 100, 500, and number of samples 50,100, 500. Interpret your results. Conceptually, sample variance and variance of sample mean are two different things. Make sure you understand this. What happens to sample variance as sample size gets larger? What happens to the variance of sample mean as sample size gets larger? Comment.