Question

One would suspect that new home construction and sales depend on mortgage interest rates. If interest rates are high, fewer people will be able to afford to borrow the funds necessary to finance the purchase of a new home. Builders are aware of this fact, therefore when interest rates are high, they will be less inclined to build new homes. A question of interest is “If interest rates go up by 1% by how much does home construction fall?”. Data on the 30 year fixed mortgage rate, housing starts (thousands), and houses sold (thousands) are contained in the file house starts.csv. There are 184 monthly observations from January 1990 to April 2005. Answer the following questions using R & Rstudio.

(a) Estimate a linear relationship of STARTS on the FIXED RATE. Interpret the intercept and slope.

(b) Obtain a scatter plot of STARTS against the FIXED RATE. Plot the fitted regression line along with the scatter plot.

(c) Construct a 95% interval estimates for the slope. Interpret the CI for the slope. What does it mean that we are “95% confident”?

(d) Is there evidence to suggest that there is a significant relationship between the 30 year fixed rate (FIXED RATE) and the house starts (STARTS)? Use a level of significance of 5%.

     i. set up the null and alternative hypothesis

     ii. show a sketch of the rejection region

     iii. state your conclusion

     iv. calculate the p-value for this test and perform the test using the p-value approach.

(e) Test that if the interest rate increases by 1%, then house starts will fall by 150,000. Use a level of significance of 5%.

(f) Comment on the goodness of fit of your model. Make sure to discuss both R2 and adjusted R2.

Answer 1

a.
The estimated regression equation is,
STARTS = 2992.74 - 194.233 FIXED_RATE
(78.9515) (10.2061)
{37.9061} {-19.0312}

Standard errors for each coefficients are given in brackets and t-statistics for each coefficients are given in curly braces.

b.
The economic interpretation is that with one percentage increase in fixed mortgage interest rate, the number of housing starts in time period would decrease by 194233.

c.
If the housing starts decrease by 150,000, the hypothesized value of slope coefficients is -150.

Null hypothesis

Ho . 3-=-150

Alternative hypothesis

H, . 32 -150

Degree of freedom = n - 2 = 184 - 2 = 182

Critical value of t at $\alpha$ = 0.05 and df = 182 is 1.973

That is we reject the null hypothesis H0 is the test statistic, t < -1.973 or t > 1.973

Test statistic, t = (Estimated Mean - Hypothesized mean) / Standard error

= (-194.233 - (-150)) / 10.2061

= -4.333977

Since t lies the rejection region, we reject the null hypothesis H0 and conclude that there is significant evidence that .

Agメー150

The economic interpretation is that there is no significant evidence that with one percentage increase in fixed mortgage interest rate, the number of housing starts in time period would decrease by 150,000.

d.

Critical value of t at $\alpha$ = 0.05 and df = 182 is 1.973

95% confidence interval of $\beta_2$

(-194.233 - 1.973 * 10.2061, -194.233 + 1.973 * 10.2061)

(-214.3696, -174.0964)

We are 95% confident that with one percentage increase in fixed mortgage interest rate, the number of housing starts in time period would decrease between 174,096 and 214,370.

e.

The estimated regression equation is,
STARTS = 2992.74 - 194.233 FIXED_RATE

For FIXED_RATE = 6% = 0.06

STARTS = 2992.74 - 194.233 * 0.06 = 2981.086

Thus, the number of monthly housing starts = 2981.086 * 1000 = 298,1086