Nonlinear econometrics for finance HOMEWORK 1
(Review of linear econometrics and conditional expectations)
This homework consists of a review of linear econometrics and conditional expectations. Regarding conditional expectations, you will see them at work and show some of their prop- erties, including the Law of Iterated Expectations.
For the empirical part, please use Matlab to estimate and test. You should provide two files: one with answers to all questions and one with your Matlab code.
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Hint. I have posted lecture notes for Linear Econometrics on OneDrive. Chapter 1, in particular, will help you review material that you have seen last quarter. Also, the linear regression in Section 4 of Chapter 1 is very similar to the linear regression in the first problem of this assignment. The corresponding code (Chapter1-spreads.m) is, therefore, a sample code for the regression in Problem 1 below. Reading Chapter 1 – and working through the sample code – will help you tremendously.
Problem 1. (40 points.) Real estate is a key asset. Investing in real estate represents the biggest investment decision for most households over their lifetimes. A real estate company in Baltimore wishes to estimate a model to relate the house prices to several characteristics of the house. The data come from Zillow and consist of a sample of houses in the Baltimore
area for the year 2014. The data are contained in the file housing data.xslx and provide the following information:
• Zillow id of the house (id) • price in dollars (price)
• street address (street)
• postal code (zip)
• year the house was built (yearBuilt)
• size of the house measured in square feet (sqft) • number of bathrooms (bathrooms)
• number of bedrooms (bedrooms).
Given this information, you need to analyze house prices using Matlab.
(1) (2 points.) Generate a histogram of the house prices and compute descriptive statistics (mean, median, variance, standard deviation, minimum, maximum). What do you notice?
(2) (2 points.) Now, take a log transformation of the house prices. Plot the histogram of the log-prices. What do you notice?
(3) (6 points.) Run a regression of the log-prices on explanatory variables: log(pricei) = β0 + β1agei + β2sizei + β3bathroomsi + β4bedroomsi + εi
where εi is an error term.
(4) (4 points.) Give an economic interpretation of the estimated coefficients in the regres-
sion above. What does the model say about the drivers of house prices?
(5) (4 points.) Why do you think that the number of bedrooms has a negative effect on log-prices?
(6) (6 points.) We want to test whether the effect of age on log-prices is statistically significant. Test at the 5% level and interpret your result. You should test in two ways:
(a) Using a suitable test statistic (b) Using p-values
(7) (6 points.) Test at the 5% level whether the coefficients associated with age and size (i.e., β1 and β2) are jointly different from zero. You should test in two ways:
(a) Using a suitable test statistic (b) Using p-values
(8) (6 points.) Test at the 5% level whether β1 = 2β2. You should test in two ways: (a) Using a suitable test statistic
(b) Using p-values
(9) (4 points.) Using your model, predict the price of a house built in 1985 with 3 bedrooms, 3 bathrooms and a size of 3500 square feet. Explain how you computed your prediction.
Problem 2. (15 points.) Suppose you have two fair coins, so that for each coin there is a 50% chance of heads or tails. Consider tossing each coin and call X1 the random variable that describes the toss of the first coin. X2 is the random variable that describes the second coin toss. If we code heads as a 1 and tails as a 0, the random variables X1 and X2 are Bernoullis with probability of success 0.5.
P(X1 =1)=0.5; P(X1 =0)=0.5, (1) P(X2 =1)=0.5; P(X2 =0)=0.5. (2)
We are interested in the random variable that describes the number of heads when we toss both coins. We will call it Y = X1 + X2.
1. (4 points.) Compute E(Y ). Show your math.
2. (4 points.) Compute E(Y |X1 = 1). Notice that this is a conditional expectation. Show
your math.
3. (7 points.) Verify the Law of Iterated Expectations (LIE). In particular, verify that
E(Y ) = E[E(Y |X1)]. Show your math.
Problem 3. (20 points.) A financial analyst wants to predict the return on a portfolio. The portfolio gives either a return of 1 or 3 percent in each period. She knows that the joint probability of returns at time t and t + 1 is
1 0.2 0.3 (3)
Given this information, please address the following questions:
1. (4 points.) Compute the unconditional expected value E(rt). Show your math.
2. (4 points.) Compute the unconditional expected value E(rt+1). Show your math.
3. (7 points.) Compute the conditional expected value Et(rt+1) = E(rt+1|rt). Show your math.
4. (5 points.) Verify the Law of Iterated Expectations (LIE) for this example. In partic- ular, show that E(rt+1) = E[E(rt+1|rt)]. Show your math.
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