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Hprice1.dta is a data set collected from the real estate pages of the Boston Globe during 1990. These are homes that sold in the Boston, MA area. Variables are explained in table 1.
In this problem set you will take a look at some empirical evidence on housing prices of 1990 in Boston, MA area. Note that, to do this problem set, you will need to create (generate) some new variables, which are functions of the variables in hprice1.dta.
- Preliminary data analysis:
- a) Produce the scatterplot of price v. lotsize.
- b) Produce the scatterplot of lprice v. llotsize.
- c) Produce the scatterplot of price vs. sqrft.
- d) Produce the scatterplot of price vs. lsqrft.
- e) Using the scatterplots from (a) and (b), would you suggest using the variables (i) price and lotsize or (ii) lprice and llotsize for modeling using linear regression?
- f) Using the scatterplot from (c) and (d), does the relation between price and sqrft appear to be linear or nonlinear? If nonlinear, what sort of nonlinear curve might you want to explore (briefly explain)?
- g) Regress lprice on llotsize, lsqrft, bdrms and colonial. Interpret the coefficient of (i) llotsize, (ii) lsqft and (iii) bdrms.
- h) Now regress lprice on llotsize, llotsize2, lsqrft, lsqrft2, bdrms and colonial. Interpret the coefficient of (i) llotsize and (ii) lsqrft.
- i) Compare the model specification in part (g) to the one in part (h)
- j) Regress price on lotsize, sqrft, bdrms and bdrms2. Is there an optimum number of bedrooms that maximizes (or minimizes) the price of a house? (hint: check the sign of the quadratic term)
- Estimate the regressions in Table 2 and fill in the empty entries. You may write in the entries by hand or type them using the .doc electronic version of the table on the course Web site.
- Use the results in Table 2 to answer the following questions.
- a) Using regression (1), test the hypothesis that the coefficient on lsqft is zero, against the alternative that it is nonzero, at the 5% significance level. Explain in words what the coefficient means.
- b) Using regression (3), test the hypothesis that the coefficients on lsqft and lsqft2 are both zero, against the alternative that one or the other coefficient is nonzero, at the 5% significance level.
- c) Using regression (3), is there evidence that the relationship between lprice and llotsize is nonlinear?
- d) Using regression (3), is there evidence that the relationship between lprice and lsqft is
e) Using regression (5), test the null hypothesis (at the 5% significance level) that the
coefficients on the “style dummies” (Colonial and Victorian) all are zero, against the alternative hypothesis that at least one is nonzero. What is number of restrictions q in your test? What is the critical value of your test?
price assess bdrms lotsize sqrft victorian
lprice lassess llotsize lsqft
DATA DESCRIPTION, FILE: hprice1.dta
House price, in $1000.
Assessed value in $1000. Number of bedrooms
Size of lot in square feet.
Size of house in square feet
= 1 if house is in Victorian style. = 0 otherwise.
= 1 if house is in Colonial style. = 0 otherwise.
Problem Set 3, Table 2 Determinants of Housing Prices
(1) (2) (3) (4) (5)
Dependent variable: Regressor:
(lsrqft)2 __ llotsize __
(llotsize)2 __ colonial __ victorian __
lprice lprice lprice lprice
()()()()() __ ____
()()() __ __
F–statistics testing the hypothesis that the population coefficients on the indicated regressors are all zero:
lsqrft, (lsqrft)2 __ llotsize,(llotsize)2 __
Style dummies (Colonial and Victorian) Regression summary statistics R2
()()() ( )
Notes: Heteroskedasticity-robust standard errors are given in parentheses under estimated coefficients, and p-values are given in parentheses under F– statistics. The F-statistics are heteroskedasticity-robust. Coefficients are significant at the +10%, *5%, **1% significance level
4. US states differ in the generosity of their welfare programs. We here wish to analyze which factors play a role in the level of benefits across different states. The data set TANF2.dta contains data from each of 49 states. The variables in the data set are given in the following table:
tanfreal black blue
south midwest northeast
DATA DESCRIPTION, FILE: TANF2.dta
State’s real maximum benefit for single parent with three kids. Percentage of state’s population who are African Americans. Dummy variable, equals 1 if state voted Democratic in 2004 presidential election.
State’s median income. = 1 if state is in West
= 0 otherwise
= 1 if state is in South. = 0 otherwise.
= 1 if state is in Midwest = 0 otherwise
= 1 if state is in Northeast =0 otherwise
Use data set TANF2.dta to examine whether Midwest states differ in their welfare programs from other states. To do this, we will use the following regression model:
tanfreal = β0 + β1 black + β2 blue + β3 midwest + β4 (black*midwest) + β5 (blue*midwest) + u Here, black*midwest is the product of the regressors black and midwest and so forth.
- (a) Write the null hypothesis to test whether there is a difference between the welfare programs of Midwest states and all other states, explain.
- (b) Construct new set of interaction regressors in STATA. Estimate the model above. Write your answer as a regression equation with standard errors in parenthesis underneath each coefficient. Perform the test for the null hypothesis in part (a) with a robust F-test. What is your conclusion?
- (c) Introduce a new variable nonmidwest = 1 – midwest. That is nonmidwest = 1 if a state is not in the Midwest and zero otherwise. Consider the following alternative regression model: tanfreal = γ1 nonmidwest + γ2 (black*nonmidwest) + γ3 (blue*nonmidwest) + γ4 midwest + γ5 (black*midwest) + γ6 (blue*midwest ) + u Write up the hypothesis of no differences in welfare programs in terms of γ1…. γ6
What is the relationship between the parameters γ1…. γ6 in this new model and β1…. β6
in the previous model? Estimate the model in STATA and write the result in usual regression equation form with standard errors in parentheses underneath coefficients
- (d) What happens if you include an intercept γ0 in the model in part (c)? Explain.
Following questions will not be graded, they are for you to practice and will be discussed at the recitation:
- SW Empirical Exercise 8.1
- SW Empirical Exercise 8.2