Housing Price Analysis: Impact of Features and Square Footage
EJERCICIO 2
Intercept:The expected value of the price of housing will be **39,954 Thousand US Dollars** when there is no pool, no living room, and no fire place in the house, there are no bathrooms, and the log of square feet is 0, which doesn’t make economic sense.
The house has a pool: This is a dummy variable with 2 options: having or not having a pool. The difference effect in the price of housing is **5,198 thousand US dollars** on average between having or not having a pool in the house, keeping constant the number of square feet, the living room, and the number of bathrooms. t-ratio = 5.94 > 1 \).
Individual hypothesis testing
5%: H0: B_3 = 0 → the house having or not having a pool doesn’t affect its price. ; HA:B3≠0
p-value=0.0046
Decision rule: – p-value > alpha → large p-value → H0. ; – p-value<alpha → small p-value→HA
Conclusion: \( 0.0046 < 5\% \), \( \Rightarrow H_A \)
We conclude, on a 95% confidence, that having or not having a pool in the house is statistically significant for explaining the price of housing.
The house has a living room.: The difference effect in the price of housing is **22,374 thousand USD** on average between a house with a living room and a house without it, keeping constant the log of square feet, number of bathrooms, and the house having or not a pool. T-ratio = 0.85 < 1
Living room hypothesis testing alpha = 5%
H0: B2 = 0 → the house having or not a living room doesn’t affect its price.
HA: B2≠ 0
P-value = 0.6338
Decision rule:
– \( p\text{-value} > \alpha \) → large \( p\text{-value} \) → \( H_0 \)
– \( p\text{-value} < \alpha \) → small \( p\text{-value} \) → \( H_A \)
Conclusion: \( 0.6338 > 5\% \), \( \Rightarrow H_0 \)
We conclude, on a 95% confidence, that having or not having a living room in the house does not affect its price.
N. Of bathrooms: If we increase in one bathroom in the house, the price of it will decrease in **0.136 thousand US$ on average**, keeping constant the rest of indep variables.
IHT alpha = 5%:
\( H0: B6 = 0 \) ; ( HA: B6 ≠0 \) ; p-value} = 0.995 ; 0.995 > 5\% \Rightarrow H0
26,483 thousand US Dollars is the difference effect in the price of a house with a fireplace and a house without one, on average, keeping constant the number of square feet, the number of rooms, the number of bathrooms, and the fact of having or not having a pool and a living room.
We test for the hypothesis: alpha = 5%:
– H0:b3 = 0 → A house with or without a fireplace does not affect its price.
– H1:b3 ≠0 → A house with or without a fireplace affects its price.
P-value = 0.644
Decision rule: If p-value > alpha , fail to reject H0
Conclusion: 0.644 > 5% , so H0 is not rejected.
We conclude with 95% confidence that having or not having a fireplace in the house does not affect its price.
Number of Square Feet in the House:
If we increase the square footage of the house, the price of the house will increase by 0.4655 thousand US Dollars on average, keeping constant the number of rooms, bathrooms, and the fact of having or not having a pool, a living room, and a fireplace.
We test for the hypothesis:\alpha = 5%:
– H0: b4 = 0 → The number of square feet does not affect the price of the house.
– H1: b4 0 → The number of square feet affects the price of the house.
P-value = 0.0083
Decision rule: If p-value < alpha, reject H0 .
Conclusion: 0.0083 < 5%, so H0 is rejected.
We conclude with 95% confidence that the number of square feet in the house affects its price.
Number of Rooms in the House:
If we increase the number of rooms in the house, the price of the house will decrease by \( 7.055 \) thousand US Dollars on average, keeping constant the number of square feet, bathrooms, and the fact of having a pool, living room, and fireplace.
We test for the hypothesis: alpha = 5%:
– H0: b5 = 0 → The number of rooms does not affect the price of the house.
– H1: b5 ≠ 0 → The number of rooms affects the price of the house.
P-value = 0.8134
Decision rule: If p-value >alpha, fail to reject H_0.
Conclusion: 0.8134 >5%, so H0 is not rejected.
We conclude with 95% confidence that the number of rooms does not affect the price of the house.
B) H0: B2 = B3 = B4 = B5 = B6 = 0
Having a pool, a living room, a fire place, the n° of square feet, and bathrooms doesn’t explain globally the price of the house.
\( H_A: B_2 \neq 0, B_3 \neq 0, B_4 \neq 0, B_5 \neq 0, B_6 \neq 0 \)
Having … Affects the price of the house.
F-value = 0.9415 (1-alpha), f-value} = 12.02
F-value = 14.14
Decision rule: – f-value} > f^* \Rightarrow HA ; – f-value} < f^* \Rightarrow H_0 \)
Conclusion: f = 14.14 > f^* = 12.02 \Rightarrow H_A \)
We conclude on a 99% confidence that all the independent variables explain globally the price of housing.
c) The ultimate goal in econometrics is to obtain the best possible model. The best possible model is the one with the highest \( R^2 \) and adjusted \( R^2 \). In this case, we have built a model with different independent variables, so we need to look at the absolute \( t\text{-values} \) in order to know if the model can be simplified by subtracting those independent variables with an \( |t\text{-value}| < 1 \). If we take them out, the adjusted \( R^2 \) will increase, which means a simpler and a better model. We talk about adjusted \( R^2 \) because we compare models with different n° of independent variables.
In the actual model, the variables with an \( |t\text{-value}| < 1 \) are: The house has a living room, the house has a fire place, the n° of rooms, and the n° of bathrooms.
Since we cannot take them out at the same time, we choose the n° of bathrooms because it has the lowest \( t\text{-value} \). After that, we would calculate the model without this independent variable, and then we would see if another IV could be subtracted.