What did housing prices look like in the good old days? The median sale prices f

Question

What did housing prices look like in the good old days? The median sale prices for new single-family houses are given in the accompanying table for the years 1972 through 1979. Letting Y denote the median sales price and x the year (using integers 1, 2, ..., 8), fit the model cap y = cap B_0 + cap beta_1x + epsilon. Assume that E(epsilon) = 0 and that V (epsilon) = sigma^2. a. Find the least square estimator cap beta_0 and cap beta_1. b. Estimate sigma^2. c. Find SSE, and estimate the variance of cap beta_0, cap beta_1. d. Find a 95% confidence interval for the slope of the line.

Explanation / Answer

Solution:

Here, first of all we have to run the regression analysis for the dependent variable median sales price and independent time variable year. The required least square regression output is given as below:

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.990841021

R Square

0.981765929

Adjusted R Square

0.978726917

Standard Error

1.745748805

Observations

8

ANOVA

df

SS

MS

F

Significance F

Regression

1

984.5529167

984.5529167

323.0543226

1.90763E-06

Residual

6

18.28583333

3.047638889

Total

7

1002.83875

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

21.575

1.36027651

15.86074585

3.9855E-06

18.24652329

24.90347671

Year

4.841666667

0.269374889

17.97371199

1.90763E-06

4.18253006

5.500803273

Part a

The least square estimator 0 and 1 are given as below:

0 = 21.575 and 1 = 4.8417

Where, 0 is the y-intercept of the least square regression equation and 1 is the slope for the regression equation or least square equation.

Part b

From the given regression output, the estimate for 2 is given as square of standard error.

Estimate of 2 = 1.746^2 = 3.048516

(We know that the estimate for the population standard deviation is the standard error. So, estimate of the population variance is nothing but the square of the standard error.)

Part c

The value for SSE and estimates for slope coefficients are given as below:

SSE = 18.28583

Estimate for variance of 0 = 1.850352183 and

Estimate for variance of 1 = 0.072562831

(We know that the estimate for the population standard deviation is the standard error. So, estimate of the population variance is nothing but the square of the standard error.)

Part d

The 95% confidence interval for the slope of the regression line is given as below:

Confidence interval = (4.18253, 5.500803) (From the given regression output)

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.990841021

R Square

0.981765929

Adjusted R Square

0.978726917

Standard Error

1.745748805

Observations

8

ANOVA

df

SS

MS

F

Significance F

Regression

1

984.5529167

984.5529167

323.0543226

1.90763E-06

Residual

6

18.28583333

3.047638889

Total

7

1002.83875

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

21.575

1.36027651

15.86074585

3.9855E-06

18.24652329

24.90347671

Year

4.841666667

0.269374889

17.97371199

1.90763E-06

4.18253006

5.500803273

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