MAE256 T2 Sample Assignment

Therefore, our estimated model is:

The adjusted R² of the model is 0.866, which is a measure of goodness of fit. The intercept of our model is the base price of the sale price. It is the average price level which is not dependent on any of the explanatory variables. The estimates of slope parameters of size, age and proximity are 3.8901, -0.6016 and 195.8438 respectively. The estimate of slope coefficient of size can be interpreted as the sale price is increased by $3.89 thousand if size increases by 1 square meters, keeping other variables constant. Similarly, if the age of the house is increased by 1 year the sale prices are decreased by $-0.6015 thousand while keeping the effect of size and proximity constant. If the house is near major business district then the proximity variable will take the value of one, therefore, the prices of house which are near business districts are on 195.8438 thousand more as compared to the houses which are not near the business district keeping other factors constant. The results are expected as, the houses with more size and which are near business districts will costs more. The age of house is also very important factor, the more old the age of the house, the prices of house will decrease. It is also important to note that at 5% only size, age and proximity estimates are significant.

Q.5 After adding two more variables of swimming pool and fire place into our previous model, the following model is obtained.

Both the variables that are introduced into the model are dummy variable. Pool variable takes the value of 1 if house has swimming pool or 0 otherwise. Similarly, fireplace takes the value of 1 if house has fireplace or 0 otherwise. The regression was performed using excel, the output is represented in Table 3.

The R² of model before introduction of pool and fireplace variable was 0.8666 and the R² after the introduction of the said variable is 0.8686. As it can be seen that R² of the model is increased very little. Therefore, the introduction of the model did not increase the ‘goodness of fit’ of the model. It is better to also consider the value of adjusted R² as it penalize the model for the introduction of new variables. The adjusted R² before introducing new variable is 0.8662 and after introducing new variable the value of adjusted R² is 0.8679. Therefore, introduction of adding new variable has increased the goodness of fit of the model but with a very modest amount. Nevertheless, the variation explained by the model has increased by introduction of two new variables.

Table 3 Regression output

Q6. After transforming the price and size variables into natural logarithms. The following model is obtained:

The regression was performed using excel and the output is presented in Table 4. The estimated model is following;

Table 4 Regression output

The slope coefficient of log(size), represents the percentage change in percentage change in price due to one percent change in size. Therefore, it can be interpreted as the sale price of houses increased by 84.72% or nearly 85% when there is one percent increase in size of the house. The estimate of  represents the percentage difference in house of sale prices when there is availability of swimming pool in the house keeping size, proximity, age and fireplace constant. Therefore, it can be interpreted as the sale prices of houses with swimming pool availability will be nearly 1.86% more as compared to prices of house which does not have swimming pool.

Q7. Null Hypothesis: H₀: Fireplace does not influence the house prices: H₀:

Alternate hypothesis: H₁ Fireplace influence the house prices: H_a: .

It can be seen from Table 4 that p-value of the estimate of fireplace is 0.08 which is far greater than the 0.01 level of significance. Therefore, at 1% level of significance, cannot reject the null hypothesis as p-value is greater than significance level. Hence, it can be stated that there are no enough evidence to reject the claim that fireplace does not influence the house prices. However, at 10% level of significance, the p-value is smaller than 0.10, therefore, we can state that there are enough evidences to reject the null hypothesis. Hence, it can be stated at 10% level of significance that fireplace does influence the sale prices of house.

Q.8 Null Hypothesis: all the variables in the model are not significant.

Alternate hypothesis: at least one coefficient is different from zero.

According to the output generated in Table 4, it can be seen that p-value of F statistic is given by 0 which is lower than the 0.05 level of significance. Hence, there are enough evidences to reject the null hypothesis. Therefore, it can be concluded that the variables are jointly significant. Hence at least one variable from size, proximity, age, pool and fireplace affects the sale prices of house.