Question

Use the following info to answer the seven questions that follow.

To determine how the number of housing starts is affected by mortgage rates, an economist recorded the average mortgage rate (%) and the number of housing starts in the area for 10 years. A simple linear regression of the data follows (partial output shown):

Years	1	2	3	4	5	6	7	8	9	10
Mortgage Rate (%)	8.5	7.8	7.6	7.5	8.0	8.4	8.8	8.9	8.5	8.0
Housing starts(thousands)	115	111	185	201	106	167	155	117	133	150

ANOVA
	df	SS	MS	F	Significance F
Regression	1	1932.0185	XXXX	1.9545	0.1996
Residual	8	XXXXX	988.4977
Total	9	9840.0000

SUMMARY OUTPUT
Regression Statistics
Multiple R	XXXX
R Square	XXXX
Adjusted R Square	0.0959
Standard Error	31.4404
Observations	10

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	389.2407	175.6998	2.2154	0.0576	-15.9237	794.4052	-15.9237	794.4052
Mortgage rate	-29.9074	21.3925	XXXX	0.1996	-79.2385	19.4237	-79.2385	19.4237

RESIDUAL OUTPUT
Observation	Predicted Housing starts	Residuals	Standardized Residuals	Leverages	Cook's D
1	135.0278	-20.0278	-0.6876	0.1417	0.0390
2	155.9630	-44.9630	-1.5736	0.1741	0.2609
3	161.9444	23.0556	0.8563	0.2667	0.1333
4	164.9352	36.0648	1.3981	0.3269	0.4746
5	149.9815	-43.9815	-1.4900	0.1185	0.1492
6	138.0185	28.9815	0.9818	0.1185	0.0648
7	XXXXX	28.9444	1.0750	0.2667	0.2101
8	123.0648	-6.0648	-0.2351	0.3269	0.0134
9	135.0278	-2.0278	-0.0696	0.1417	0.0004
10	149.9815	0.0185	0.0006	0.1185	0.0000

According to the least squares line if the mortgage rate is increased by 1%, average number of housing starts in the area should _______ by ______units.

increase, 31440

increase, 38924

decrease, 29907

decrease, 38924

increase, 29907

What is the value of the calculated t statistic to test the significance of the coefficient of the variable “Mortgage rates”?

-6.0648

0.1996

-1.398

2.2154

21.3925

What is the upper limit of the 95 % confidence interval for the Slope?

19.42

79.24

15.92

794.41

21.39

To stimulate, if the FED offered 0% mortgage rates what may be the number of Housing starts expected?

564941

389241

359334

175700

A huge number, close to infinity.

Are there any outliers in this data set?

None, because all leverage values are outside of +/- 3

None, because all standardized residual values are within +/- 2

None, because all standardized residual values are larger than 0.8

One, because exactly one standardized residual is very close to zero (0.0006)

One, because there is exactly one insignificant variable.

What percentage of variation in the number of housing starts is explained by its regression on mortgage rates?

33.4%

19.6%

25.2%

44.3%

10.9%

Answer 1

Let Y be the number of housing starts and X be the mortgage rates

The linerar regression model is

where is the intercept of regression line

is the slope of regression line

is a random disturbance

The estimated values of the intercept is

the estimated value of slope is

The estimated regression line is

The slope estimate of -29.9074 says that for a 1% increase in mortage rate, the number of housing starts is going to decrease (due to the negative value of slope) by 29.9074 (thousands) (which is 29.9074*1000=29,907)

ans: decrease, 29907

What is the value of the calculated t statistic to test the significance of the coefficient of the variable “Mortgage rates”? The t-stat is calculated as The parameter estimate and the standard error of estimate is Hence the t stat is ans: -1.398 What is the upper limit of the 95 % confidence interval for the Slope? The upper linit is given in the following output ans: 19.42 To stimulate, if the FED offered 0% mortgage rates what may be the number of Housing starts expected? We go back to the esti,mated regression equation and set X=0 and we get When the mortgage rate is 0%, the number of housing starts is 389.2407*1000=389,241 ans: 389241 Are there any outliers in this data set? To indetify the ouliers we have been given 3 measures, standardized residuals, leverage and Cook's distance. the general rules are Standardized residuals: An observation is considered outlier if the value of standardized residual is outside of Leverages: An observation is an outlier if the value of its leverage is greater than , where p=2 is the number of independent variables including the intercept, and n=10 is the number of observations Cook's D: An observation is an outlier if its value of Cook's D is freater than F(0.5, p, n-p), which is F(0.5, 2, 10-2). Using F tables we get F(0.5, 2,8) = 0.76 Using the rule for standardized residuals we can see that no observation is outside +/- 2, hence there are no outliers. ans: None, because all standardized residual values are within +/- 2 What percentage of variation in the number of housing starts is explained by its regression on mortgage rates? This is indicated by the value of We have been given the value of adjusted R-square as where n=10 is the number of observations and p=2 is the number of coefficients estimated including the intercept. The value of R-square is 0.1964. this indicates that 19.64% of variation in the number of housing starts is explained by its regression on mortgage rates ans: 19.6%