An economist estimates the following regression model:

$$y=\beta_0+\beta_1 x_1+\beta_2 x_2+\varepsilon$$

The estimates of the parameters $\beta_1$ and $\beta_2$ are not very large compared with their respective standard errors. But the size of the coefficient of determination indicates quite a strong relationship between the dependent variable and the pair of independent variables. Having obtained these results, the economist strongly suspects the presence of multicollinearity. Since his chief interest is in the influence of $X_1$ on the dependent variable, he decides that he will avoid the problem of multicollinearity by regressing $Y$ on $X_1$ alone. Comment on this strategy.

What is the purpose of doing a multiple comparison?

An experiment is performed to test the effects of a new drug on high blood pressure. The experimenter identifies 320 people ages 35 - 50 years old with high blood pressure for participation in the experiment. The subjects are divided into equal groups according to age. Within each group, subjects are then randomly selected to be in either the treatment group or the control group. What type of experimental design is being used for this experiment?

Consider a regression analysis with n = 34 and four potential independent variables. Suppose that one of the independent variables has a correlation of 0.23 with the dependent variable. Does this imply that this independent variable will have a very small Student’s t statistic in the regression analysis with all four predictor variables?

Use the data in the file Citydatr to estimate a regression equation that can be used to determine the marginal effect of the percent commercial property on the market value per owner-occupied residence (Hseval). Include the percent of owner occupied residences (Homper), percent of industrial property (Indper), the median rooms per residence (sizehse), and per capita income (Incom 72) as additional predictor variables in your multiple regression equation. The variables are described in the appendix. Indicate which of the variables are conditionally significant. Your final equation should include only significant variables. Run a second regression with median rooms per residence excluded. Interpret the new coefficient for percent commercial property that results from the second regression. Compare the two coefficients.

Transportation Research, Inc., has asked you to prepare some multiple regression equations to estimate the effect of variables on fuel economy. The data for this study are contained in the data file Motors, and the dependent variable is miles per gallon—milpgal—as established by the Department of Transportation certification.

a. Prepare a regression equation that uses vehicle horsepower—horspwer—and vehicle weight— weight—as independent variables. Interpret the coefficients.

b. Prepare a second biased regression with vehicle weight not included. What can you conclude about the coefficient of horsepower?

Suppose that a regression relationship is given by the following:

$$Y=\beta_0+\beta_1 X_1+\beta_2 X_2+\varepsilon$$

If the simple linear regression of $Y$ on $X_1$ is estimated from a sample of $n$ observations, the resulting slope estimate is generally biased for $\beta_1$. However, in the special case where the sample correlation between $X_1$ and $X_2$ is 0 , this will not be so. In fact, in that case the same estimate results whether or not $X_2$ is included in the regression equation.

a. Explain verbally why this statement is true.

b. Show algebraically that this statement is true.

Write the model specification and define the variables for a multiple regression model to predict wages in U.S. dollars as a function of years of experience and country of employment, indicated as Germany, Great Britain, Japan, United States, and Turkey.

Suppose that the true linear model for a process was

$$Y=\beta_0+\beta_1 X_1+\beta_2 X_2+\beta_3 X_3$$

and you incorrectly estimated the model

$$Y=\alpha_0+\alpha_1 X_2$$

Interpret and contrast the coefficients for $X_2$ in the two models. Show the bias that results from using the second model.

The data file Thailand Consumption shows 29 annual observations on private consumption Y and disposable income X in Thailand. Fit the regression model

$$\log y_t=\beta_0+\beta_1 \log x_{1 f}+\gamma \log y_{t-1}+\varepsilon_t$$

and write a report on your findings.

The data file Births Australia shows annual observations on the first confinement resulting in a live birth of the current marriage $(Y)$ and the number of first marriages (for females) in the previous year $(X)$ in Australia. Estimate the model (McDonald 1981)

$$y_t=\beta_0+\beta_1 x_t+\gamma y_{t-1}+\varepsilon_t$$

The data file Income Canada shows quarterly observations on income $(Y)$ and money supply $(X)$ in Canada. Estimate the model (Hsiao 1979)

$$y_t=\beta_0+\beta_1 x_t+\gamma y_{t-1}+\varepsilon_t$$

and write a report on your findings.

The YTM on a bond is the interest rate you earn on your investment if interest rates don’t change. If you actually sell the bond before it matures, your realized return is known as the holding period yield (HPY).

a. Assume that today you buy a bond with an annual coupon rate of 7% for $1,060. The bond has 17 years to maturity. What rate of return do you expect to earn on your investment? Assume a par value of$1,000.
b. Two years from now, the YTM on your bond has decreased by 1%, and you decide to sell. What price will your bond sell for? What is the HPY on your investment? Compare this yield to the YTM when you first bought the bond. Why are they different?

The data file Pension Funds contains data on the market return $(X)$ of stocks and the percentage $(Y)$ of portfolios in common stocks at market value at the end of the year for private pension funds. Estimate the model

$$y_t=\beta_0+\beta_1 x_t+\gamma y_{t-1}+\varepsilon_t$$

and write a report on your findings.