Estimate the parameters of nonlinear equation

Question

Assignment:

1) For the following models, label each as either I) linear in both parameters and variables, II) linear in parameters but not linear in variables, III) linear in variables but not linear in parameters, or IV) neither linear in parameters nor linear in variables. By linear in variables, the question is referring to the variables X_2i and X_3i .

a) Y_i = β₁ + β₂ log(X_2i) + β₃ log(X_3i) + u_i

b) Yi = β1 + β₂² log(X_2i) + β₃ log(X_3i) + u_i

c) Yi = β₁ + β₂(X_2i √ X_3i) + β₃X²_3i + u_i

d) Yi = β₁ + β₂X2i + β₃X_3i + u_i

e) Yi = β₁ + e β₂X_2i + β₃X_3i + u_i

f) Yi = β₁ (1 - β₂) + β₂X_2i + β₃X_3i + u_i

2) Consider the following model of the relationship between a dependent variable Yi and two independent variables, X_2i and X_3i :

Yi = e ^β1X ^β2_2i e ^β3X3ivi

a) How would you estimate the parameters of this nonlinear equation?

b) Give an interpretation of the parameter β₂.

3) Stock market investors give considerable attention to a firms quarterly earnings announcement, in which the firm releases its quarterly accounting earnings. Suppose we are interested in estimating a model to measure the stock markets response to the information contained in the earnings announcement. We settle on the following model:

R_i = β₁ + β₂S_i + β₃Loss_i + β₄Beat_i + β₅I^B_i + β₆I^C_i + β₇I^D_i + u_i

where:

R_i = the percentage change in the stock price of company i in a short time-period (24 hours) around the time of the earnings announcement.

S_i = the earnings "surprise," measured as the deviation of the firms accounting earnings from the average expectations of Wall Street analysts. This is measured in cents per share of stock outstanding.

Lossi = a dummy variable that is 1 if firm i had negative accounting earnings (a loss) and is 0 otherwise.

Beati = a dummy variable that is 1 if firm i had accounting earnings in excess of the average expectations of Wall Street analysts (i.e. S_i is positive), and is zero otherwise.

I^B_i = a dummy variable that is 1 if firm i is in industry B, and is zero otherwise.

I^C_i = a dummy variable that is 1 if firm i is in industry C, and is zero otherwise.

I^Di = a dummy variable that is 1 if firm i is in industry D, and is zero otherwise.

All firms in the sample are in either industry A, B, C, or D.

Estimation of this model yields the following fitted regression line:

ˆR_i = 0.1 + 0.5S_i - 2Loss_i + 0.4Beat_i + 0.1I^B_i + 0.2I^C_i - 0.3I^D_i

a) What is the reference category for this regression?

b) What is the effect on Ri of moving from industry A to industry B, all else held equal?

c) What is the effect on Ri of moving from industry C to industry B, all else held equal?

d) Define a new variable called I^A_i , which is 1 if firm i is in industry A, and is zero otherwise. Using this variable and the other variables defined above, write down a regression that would fall into the Dummy Variable Trap.

4) Data was collected from a random sample of 220 home sales from a community in 2017.

Let P denote the selling price (in $1000), Bdr denote the number of bedrooms, Hsize denote the size of the house (in square feet), Age denote the age of the house (in years), P oor denote a dummy variable that is equal to 1 if the condition of the house is reported as "poor" and is zero otherwise, and V iew denote a dummy variable that is 1 if the house has a view of a nearby mountain range and is zero otherwise. An estimated regression yields the following fitted regression line:

ˆP_i = 119.2 + 0.485Bdr_i + 0.156H size_i + 0.090Age_i - 48.8P oor_i + 25.5V iew_i -0.005(Hsize_i ∗ Poor_i) + 0.005(Hsize_i ∗ View_i)

a) Suppose that a homeowner of a 2500 square foot house removes a row of tall trees that is blocking the view of the mountains from the house. What is the regression's prediction for the increase in the value of the house?

b) Consider a house that has a view of the mountains and is in poor condition. Suppose the homeowner adds 100 square feet to the house. What is the regressions prediction for the increase in the value of the house?