1. In an equation for annual data, suppose that:

unemp_t = 2.7 - .68 inf_t - .25 inf_t-1 + .33 inf_t-2 + u_t,

where unemp_t is an unemployment rate at time t and inf_t is the inflation rate. What are the impact and long-run propensities?

2. In the model:

Average Income_t  = α_o + δ_o marriage_t  + δ_o marriage_t-1  + u_t,

explain the nature of potential multicollinearity in the explanatory variables when data for marriage and income is observed each month.

 3. Massachusetts Dog Chow company is interested in seeing how many pounds of dog treats a dog consumes in a season and whether this relates to the dog being black, a retriever, or both.

A regression reveals the following results:

Treats = 43.2 + 2.5 (Spring) + 5.8 (Summer) + 4.1 (Autumn) + 2.7 (Black) +

              (11.4) (.81)               (1.7)                (2.3)                  (1.2)

3.6 (Retriever) + 9.4 (Black*Retriever)

(1.9)                    (3.2)

n = 1,110                              R² = 0.43

where Summer, Spring, and Autumn are seasonal dummies, Black is a dummy variable equal to 1 if the dog has a black coat, and Retriever is a dummy variable equal to 1 if the dog is a retriever. Treats is a variable equal to the number of pounds of dog treats eaten in a season.

a. What is the average number of dog treats consumed by a dog (neither black nor a retriever) in the summer season? What is the average number of dog treats consumed by a dog (neither black nor a retriever) in the winter season?

b. How many treats does a retriever which is not black eat (on average) in each season? (List the number for each season.)

c. How many treats does a black dog who is not a retriever eat (on average) in each season? (List the number for each season.)

 d. Which variables look insignificant here? Describe a way of testing if seasonality matters for determining how many treats a dog eats.

4. Suppose that {y_t: t = 1,2, ...} is generated by y_t = δ_o + δ₁t + e_t  where δ₁ does not equal 0 and {et: t = 1,2, ...} is an i.i.d. sequence with mean zero and variance σ².

a. Is y_t covariance stationary?

b. Is y_t - E(y_t) covariance stationary?

5. Suppose you have quarterly data and you want to test for the presence of first order or fourth order serial correlation in the error term. With strictly exogenous regressors, explain how you would proceed.

6. Suppose you want to estimate the model:

y_t = δ_o + δ₁x + u_t

but you find that in a regression of residuals u_t on u_t-1 that the coefficient on u_t-1 is .52 and highly significant. What would you do about this to remove the serial correlation and estimate the original equation under the Gauss-Markov assumptions?  Assume exogenous regressors.

7. Let educ* be actual amount of schooling, measured in years (which can be a non integer.) Let educ be the reported highest grade completed. Do you think these two variables are related by the classical errors-in-variables (CEV) models? (The CEV model indicates that measurement error would be uncorrelated with the unobserved variable (educ* in this case) and that attenuation bias may result.) Explain.

8. Suppose we are interested in the effects of campaign expenditures by incumbents on voter support. Some incumbents choose not to run for reelection. If we can only collect voting and spending outcomes on incumbents that actually do run, is there likely to be endogenous sample selection?

9. Let mort indicate the infant mortality rate at hospitals. A model to estimate the effect of per capital government health expenditures on infant mortality is:

mort = β₀ + β₁ log(expend) + β₂ log (beds) + β₃ poverty + u,

where expend measures per capital government health expenditures (in real U.S. dollars), beds is the number of beds at a hospital, and poverty is the percentage of mothers at the hospital who live in poverty.

a. Let welfare be the percentage of mothers at the hospital who receive welfare payments from the government. Why would the percentage of mothers who receive welfare payments be a good proxy variable for poverty?

 b. The table that follows contains OLS estimates, with and without welfare as an explanatory variable:

Explain why the effect of expenditures on reducing infant mortality is much lower in column (2) than in column (1). Is the effect in column (2) still statistically greater than zero?

c. Does it appear infant mortality rates are lower at larger hospitals, other factors being equal?

d. Interpret the coefficient on welfare in column (2)

e. What do you make of the increase in R-squared from column (1) to column (2)?