Lecturer’s and Number Submitted Econometrics In an influential article, ‘‘ Hedonic Housing Prices and the Demand for Clean Air,” David
Harrison and Daniel Rubinfeld study the impact of improvements in air quality on local citizens as reflected through differences in housing prices. A simplified version of their model, which is to be studied, is
ln(MVi) = β0 + β1RMi + β2 ln(DISi) +β3NOXi + β4DCHAS; i + Ɛi , (1)
where the level of observation is the census tract. Further, MVi is the median housing price (measured in $1, 000) for a given census tract in the Boston metropolitan statistical area.
RM is the average number of rooms in owner occupied housing in the census tract; DIS is a weighted distance to five major employment centers in the Boston region, NOX measures nitrogen oxide concentrations in parts per hundred million. DCHAS is a dummy variable which signifies if the Charles River passes through the census tract, i. e. DCHAS = 1 if the Charles river passes through and 0 otherwise.
Consider the table below showing the estimation of the 3 variants of the model that has the coefficients of each variant;
Dependent variable: ln MV
(1)
(2)
(3)
RM
0. 329
0. 306
0. 294
ln(DIS)
(0. 028)
(0. 027)
(0. 027)
0. 197
0. 059
0. 055
(0. 031)
(0. 052)
(0. 049)
NOX
1. 479
1. 529
(0. 242)
(0. 238)
DCHAS
0. 234
(0. 050)
Observations
506
506
506
R2
0. 464
0. 516
0. 537
2
0. 461
0. 513
0. 533
R
BP Test
< 0: 001
< 0: 001
< 0: 001
White Test
< 0: 001
< 0: 001
< 0: 001
RESET Test
< 0: 001
< 0: 001
< 0: 001
1. Interpretation of the estimates in the model
For RM (average number of rooms in the owner occupied housing in the census tract), in the first variant, a 1 percent change in RM causes 0. 329 % change in MV (the median housing price). In the second variant, a 1 percent change in RM causes a 0. 0306 % change in the median housing price (MV). Eventually, a 1 percent changes in RM causes a 0. 294 % change in the median housing price in the third variant.
In the second variable, the weighted distance to five major employment centers in the Boston region (DIS), the order of coefficients; 0. 028, 0. 027, and 0. 027 in absolute values. In the criteria of absolutism, a one percent change in DIS causes a reduction of median housing price by 0. 028 in the first variant, a reduction of 0. 027 in the second variant, and a reduction of 0. 027 in the third variant.
In the third variable, nitrogen oxide concentrations in parts per hundred million (NOX), the coefficients are 1. 479 and 1. 529 in the second and third variants respectively. These figures suggest that 1 percent change in nitrogen oxide concentration increases the medium housing price by 1. 479 in the second variant and by 1. 529 in the third variant.
2. The estimate of the intercept of the estimate model (1) using the beta coefficients approach
Beta coefficient is the measure of the sensitivity of the estimates in influencing the median housing price. In the estimates, the beta coefficient is the slope of the model summarized into β0, β1, β4, β3, and β2.
3. 10) Estimating model (1) using beta coefficients produces estimates of 1 and 2 of 0. 566 and 0. 261, respectively. What is the interpretation of these coefficient estimates?
Normally, the coefficients would imply 1 percentage change in the estimate 1 and 2 would cause an increase of 0. 566 and 0. 0261. However, using the beta approach, the two coefficients are below, suggesting that they are below the median housing price.
4. State formally the null and alternative hypotheses that ln(DIS) does not have a positive influence on median housing values.
Null Hypothesis; Ho: The weighted distance to five major, employment centers in the Boston region (DIS) does not have a positive influence on median housing values
Alternative hypothesis, H1: the weighted distance to five major employment centers in the Boston region (DIS) have positive influence on median housing values.
5. Comment on the fact that b2 has changed sign and significance (at conventional levels) from model (1) to model (2) but b1 has not. Use sound economic reasoning.
b2 and b1 measures the changes that occurs on the variants visa-a-vie the dependent variable. At conventional levels, they changed their signs due to low or no significance.
6. Suppose in model (3) I added in the variable NOX DCHAS, resulting in ln(MVi) = β0 + β1RMi + β2 ln(DISi) +β3NOXi + β4DCHAS; i + β5 NOX DCHAS +Ɛi . How would the interpretation of Ɛ3 change in model (3) after the inclusion of this variable? What is the interpretation of Ɛ5 in this model?
Ɛ3 implies the coefficient of the nitrogen oxide concentrations
Ɛ5implies the constrained coefficient of the nitrogen oxide concentration
7. In general can we say that heteroskedasticity-robust standard errors are always larger than regular standard errors?
Using robust standard errors in determining heteroskedasticity is always larger than the regular standard errors in economic practice
8. Under the circumstances, an hypothesis test would be involved in the following model and steps:
Model: ln(MVi) = β0 + β1RMi + β2 ln(DISi) +β3NOXi + β4DCHAS; i + Ɛi
Hypothesis Testing
Doing a t-test (independent)
Analyze / Compare means / Independent-samples T-test
Test variable = interval/ratio
Grouping variable = nominal
9. Given that the BP and White tests yield the same conclusion regarding the presence of heteroskedasticity, does this imply that the BP test is as good as the White test? Explain your reasoning in detail.
Heteroskedasticity implies to the circumstance when the variability of a variable is unequal across the range of values of a second variable that predicts it. In this circumstance, it means that the Bp test is as good as the white test since in the presence of heteroskedasticity, it is expected to be different for variability, which is not the case.
10. If there was heteroskedasticity in model (3), say, is there a concern that your parameter estimates are biased? Explain in detail.
Heteroskedasticity does not necessarily imply an error, but only imply variableness, i. e. variability of a variable is unequal across the range of values of a second variable that predicts it. However, it may raise concern of biasness in the parameter estimation.
