Question 1: (10 points)
Let f(z) be the probability density of a random variable, X, defined as,
f(x) = c(x − x2)2 if 0 ≤ x ≤ 1 f(x) = 0 if x < 0 or x > 1.
- What must the value of the constant c be? Why?
- Compute Pr(X ≤ 0.5), E(X), and V ar(X).
Question 2: (25 points)
- Let zt,t = 1,…,n, be scalar random variables with Cov(zt,zs) = 0 for t 6= s. Show that
- Let b be a k × 1 random vector. Show that V ar(b) is (i) symmetric and (ii) positive semidefinite.
- Let A be an invertible square matrix. Show that (A0)−1 = (A−1)0. (Hint: It suffices to show that A0(A−1)0 = I = (A−1)0A0.)
- Let A, B, and C be three invertible square matrices of the same dimensions. Show that (ABC)−1 = C−1B−1A−1.
Question 3: (15 points)
This question uses the data in the file hw1.txt. The dataset contains three variables, y, x1, and x2, and there are 100 observations.
- Regress y on a constant and x1. Regress y on a constant, x1, and x2. Report the parameter estimates.
- Compare the TSS, ESS, and R2 from the above regressions and comment on them.
- Create a variable x4 by x4 = 2x1, and regress y on a constant and x4. Compare the results of this regression with the result from regressing y on a constant and x1.
Question 4: (20 points)
The file ceosal2.dta contains data on 177 chief executive officers and can be used to examine the effects of firm performance on CEO salary.
- Estimate a model relating annual salary to firm sales and market value. Make the model of the constant elasticity variety for both independent variables. Write the results out in equation form.
- Add profits to the model from part (1). Why can this variable not be included in logarithmic form? Would you say that these firm performance variables explain most of the variation in CEO salaries?
- Add the variable ceoten to the model in part (2). What is the estimated percentage return for another year of CEO tenure, holding other factors fixed?
Question 5: (30 points)
- The file tbrate.data contains data for 1950 :1 to 1996 :4 for three series: 𝑟𝑡, the interest rate on 90-day treasury bills, 𝜋𝑡, the rate of inflation, and 𝑦𝑡, the logarithm of real GDP.
- For the period 1950 :4 to 1996 :4, run the regression
Δ𝑟𝑡 = 𝛽1 + 𝛽2𝜋𝑡−1 + 𝛽3∆𝑦𝑡−1 + 𝛽4∆𝑟𝑡−1 + 𝛽5∆𝑟𝑡−2 + 𝑢𝑡 (1)
where Δ is the first-difference operator, defined so that Δ𝑟𝑡 = 𝑟𝑡 − 𝑟𝑡−1.
- Plot the residuals and fitted values against time.
- Regress the residual on the fitted values and on a constant. What do learn from this second regression.
- Now regress the fitted values on the residuals and on a constant. What do learn from this third regression ?
- For the same sample, regress Δ𝑟𝑡 on a constant, ∆𝑦𝑡−1, Δ𝑟𝑡−1 and Δ𝑟𝑡−2. Save the residuals from this regression, and call them 𝑒̂𝑡. Then regress 𝜋𝑡−1 on a constant, ∆𝑦𝑡−1, ∆𝑟𝑡−1 and Δ𝑟𝑡−2. Save the residuals from this regression, and call them 𝑣̂𝑡. Now regress 𝑒̂𝑡 on 𝑣̂𝑡.
- How does the estimated coefficient and the residuals from this last regression related to anything that you obtained when you estimated regression (1).