## Applied Econometrics

**Question 1: **(10 points)

Let *f*(*z*) be the probability density of a random variable, *X*, defined as,

*f*(*x*) = *c*(*x *− *x*^{2})^{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
*z*= 1_{t},t*,…,n*, be scalar random variables with*Cov*(*z*) = 0 for_{t},z_{s}*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 (*A*^{0})^{−1 }= (*A*^{−1})^{0}. (Hint: It suffices to show that*A*^{0}(*A*^{−1})^{0 }=*I*= (*A*^{−1})^{0}*A*^{0}*.*) - Let
*A*,*B*, and*C*be three invertible square matrices of the same dimensions. Show that (*ABC*)−1 =*C*−1*B*−1*A*−1.

**Question 3: **(15 points)

This question uses the data in the file hw1.txt. The dataset contains three variables, *y*, *x*_{1}, and *x*_{2}, and there are 100 observations.

- Regress
*y*on a constant and*x*_{1}. Regress*y*on a constant,*x*_{1}, and*x*_{2}. Report the parameter estimates. - Compare the TSS, ESS, and
*R*^{2 }from the above regressions and comment on them. - Create a variable
*x*_{4 }by*x*_{4 }= 2*x*_{1}, and regress*y*on a constant and*x*_{4}. Compare the results of this regression with the result from regressing*y*on a constant and*x*_{1}.

**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: 𝑟
_{𝑡}, the interest rate on 90-day treasury bills, 𝜋_{𝑡}, the rate of inflation, and 𝑦_{𝑡}, 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 Δ𝑟_{𝑡 }= 𝑟_{𝑡 }− 𝑟_{𝑡−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 Δ𝑟
_{𝑡}on a constant, ∆𝑦_{𝑡−1}, Δ𝑟_{𝑡−1}and Δ𝑟_{𝑡−2}. Save the residuals from this regression, and call them 𝑒̂_{𝑡}. Then regress 𝜋_{𝑡−1}on a constant, ∆𝑦_{𝑡−1}, ∆𝑟_{𝑡−1}and Δ𝑟_{𝑡−2}. Save the residuals from this regression, and call them 𝑣̂_{𝑡}. Now regress 𝑒̂_{𝑡}on 𝑣̂_{𝑡}. - How does the estimated coefficient and the residuals from this last regression related to anything that you obtained when you estimated regression (1).