Homework Assignment #3

1. (General linear test and standardized regression) Use the log of the housing price as the dependent variable: log(price) = B ₀ + B ₁ sqrft + B ₂ bdrms + u
   1. You are interested in estimating and obtaining a confidence interval for the percentage change in price when a 150-square-foot bedroom is added to a house. In decimal form, this is ₁ = 150 B ₁ + B ₂ . Use the data in HPRICE1.RAW to estimate ₁ .
   2. Write B ₂ in terms of ₁ and B ₁ and plug this into the log(price) equation.
   3. Use part ii. To obtain a standard error for ₁ hat and use this standard error to construct a 95% confidence interval.
   4. Suppose I am interested in the following model:
      price= \({{\beta }_{0}}\) + \({{\beta }_{1}}\) sqrt + \({{\beta }_{2}}\) bdrms + u .
      Between sqrt and bdrms , which variable has bigger partial effect on housing price and why?
2. (Goodness of fit) The following three equations were estimated using the 1,534 observations in 401K.RAW:

prate hat = 80.29 + 5.44 mrate + .269 age - .00013 totemp

(.78) (.52) (.045) (.00004)

R ² = .100, R ² bar = .098

prate hat = 97.32 + 5.02 mrate + .314 age – 2.66 log (totemp)

(1.95) (.51) (.044) (.28)

R ² = .144, R ² bar = .142

prate hat = 80.62 + 5.34 mrate + .290 age - .00043totemp + .0000000039 totemp ²

(.78) (.52) (.045) (.00009) (.0000000010)

R ² = .108, R ² bar = .106

Which of these three models do you prefer. Why?

3. (Interaction term) NOTE: Use WAGE1 dataset for this exercise. Consider a model where the return to education depends upon the amount of work experience (and vice versa):

log( wage ) = B ₀ + B ₁ educ + B ₂ exper + B ₃ educ * exper + u

1. Show that the return to another year of education (in decimal form), holding exper fixed, is B ₁ + B ₃ exper.
2. State the null hypothesis that the return to education does not depend on the level of exper. What do you think is the appropriate alternative?
3. Use the data in WAGE1.RAW to test the null hypothesis in ii. against your stated alternative.
4. Let ₁ denote the return to education (in decimal form), when exper = 10: ₁ = B ₁ + 10B ₃ . Obtain ₁ hat and a 95% confidence interval for ₁ . (HINT: Write B ₁ = ₁ – 10B ₃ and plug this into the equation; then rearrange. This gives the regression for obtaining the confidence interval for ₁ .)

1. (Prediction analysis) Use the data in HPRICE1.RAW for this exercise.
   1. Estimate the model: price = B ₀ + B ₁ lotsize + B ₂ sqrft + B ₃ bdrms + u and report the results in the usual form, including the standard error of the regression. Obtain predicted price, when we plug in lotsize = 10,000, sqrft = 2,300, and bdrms = 4; round this price to the nearest dollar.
   2. Run a regression that allows you to put a 95% confidence interval around the predicted value in part i. Note that your prediction will differ somewhat due to rounding error.
   3. Let price ⁰ be the unknown future selling price of the house with the characteristics used in parts i. and ii. Find a 95% Confidence Interval for price ⁰ and comment on the width of this confidence interval.

Price: $16.13

Solution: The downloadable solution consists of 6 pages, 1013 words and 10 charts.
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