Nonlinear econometrics for finance HOMEWORK 1
(Review of linear econometrics)
金融非线性计量经济学代写 The homework consists of a theoretical part and an empirical part. For Question 3, please copy-paste the matlab code at the end of the file.
The homework consists of a theoretical part and an empirical part. For Question 3, please copy-paste the matlab code at the end of the file.
Problem 1. (36 points.) 金融非线性计量经济学代写
Consider the linear regression model Y = Xβ +ε with pre-determined (i.e., non-stochastic) regressors X. All of the usual as- sumptions hold except, instead of V ar(ε) = σ2In, we have V ar(ε) = E(εε‘) = Σε, where Σε is a generic symmetric positive definite matrix. In other words,we are assuming that the errors are not necessarily homoskedastic and un- correlated.
(1)(4 points) The traditional least-squares estimator is 
bLS = (X‘ X)−1X‘ Y .Defifine, now, a new estimator bGLS = (X‘ Σε−1X)−1X‘ Σε−1Y . Call it “generalized least-squares” (GLS) estimator. Show that E(βbGLS) = β,i.e., bGLS is unbiased.
(2)(6points) Show that the variance of 
LS under the new assumption is (X‘X)−1X‘ΣεX(X‘X)−1.
(3)(6 points) Show that the variance of βttLSunder the new assumption is (X‘Σ−ε 1X)−1.
(8points) Show that the variance of ttLS is not larger than the vari-ance of the traditional least-squares estimator β^LS . In other words,show that (X‘X)−1X‘ΣεX(X‘X)−1 − (X‘Σ−ε 1X)−1 ≥ 0.
[Hint: This is the same as showing (X‘Σ−ε 1X) (X‘X)(X‘ΣεX)−1(X‘X)0.Write the expression in terms of a suitable “quadratic form” using asuitable“idempotent and symmetric matrix” and the proof is almost complete. The proof is very similar to the one showing that LS is BLUE in the standard regression model.] 金融非线性计量经济学代写
Assume, now, that the regression Y = Xβ+ε is in a time-series context. The error terms εt are independent of xt and modeled as an MA(1) process. Specifically, write
εt = ut − θut−1,
where ut is a white noise process with mean zero and variance σ2u.
(5)(6 points) Find the variance-covariance matrix of the regression’sresid- uals (E(εε‘) = Σε) as a function of the MA parameters.
[Hint: Compute V ar(εt) for all t and Cov(εt, εt+j) for all t and j as a function of the M As parameters and then use these values to fill in Σε.]
- (6points) We have learned that under the assumed structure on the error terms the GLS estimator is more efficient than the least-squares estimator. Using your response in (5) above, how would you make the GLS estimator “feasible”? In other words, how would you estimate the matrix Σε (i.e., obtain Σε) to define an implementable version of βttLS = (XjΣ−ε 1X)−1Xj Σ−ε 1Y ? Be as precise as possible.
Problem 2. (32 points.)
Consider the regression model Y = Xβ + ε. Assume X is stochastic and ε is such that E(X‘ε) ƒ= 0. However, there is a matrix of variables Z such that E(Z‘ε) = 0 and E(Z‘X) ƒ= 0. The dimension of the matrix X is T ×k (T is the number of observations and k is the number of regressors) whereas the dimension of the matrix Z is T × q with q > k.
(1)(4 points) Is from a regression of Y on Xconsistent for ?
(2)(4 points) Regress the matrix X on the matrix Z (i.e., you want to regress each column of X on the matrix Z). Express the fitted values compactly as a function of X andZ.
(3)(4 points) Regress the observations Y on the fitted values from the previous regression (a T k matrix). Express compactly the new es- timator as a function of X, Z, and Y . (Note: you could use a very specific idempotent matrixhere). 金融非线性计量经济学代写
(4)(6points) Is the new estimator consistent for β?
(5)(6 points) Assume k = q. Does the form of the estimatorsimplify?
(6)(8points) Interpret all of your previous results from an applied stand- point. Why are they useful?
Problem 3. (32 points) 金融非线性计量经济学代写
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 wants to estimate a model to relate the house prices to several characteristics of the house. The data come from Zillow and consists 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)
- pricein dollars (price)
- street address(street)
- postal code(zip)
- year the house was built(yearBuilt)
- sizeof the house measured in square feet (sqft) 金融非线性计量经济学代写
- number of bathrooms(bathrooms)
- number of bedrooms(bedrooms).
Given this information, you need to run a linear regression for the price of houses using Matlab.
(1)(2point) Generate an histogram of the house prices and compute de- scriptive statistics (mean, median, variance, standard deviation, mini- mum, maximum). What do you notice?
(2)(2 point) Now take a log transformation of the house prices. Plot the histogram of the log-prices. What do younotice?
(3)(4points) Run a regression of the log-prices on the explanatory vari- ables:
log(pricei) = β0 + β1agei + β2sizei + β3bathroomsi + β4bedroomsi + ui
where ui is an error term.
(4)(4 points) Give an economic interpretation of the estimated coefficients in the regression above. What does the model say about the house prices?
(5)(4 points) Why do you think β4is negative?
(6)(4 points) We want to test whether the coefficient β3for the number of bathrooms is statistically significant. What test would you use? Compute the test statistic and interpret the result. 金融非线性计量经济学代写
(7)(4points) Test whether the coefficients for the number of bathrooms and the number of bedrooms (i.e., β3 and β4) are jointly different than zero.
(8)(4 points) Test whether β3= −β4.
(9)(4points) Using your model, predict the price of a house with 4 bed- rooms, 3 bathrooms, size of 2500 square feet and built in Explain how you compute your prediction.
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