Table11-7

TABLE 11-7

REAL STOCK RETURNS, OUTPUT GROWTH, AND INFLATION,

UNITED STATES 1954 TO 1981

Questions for you (regression output below, ANSWERS ON THE NEXT TAB)

(i) Given the SRF, what is the first derivative with respect to inflation?

(ii) Our hypothesis is that the output growth parameter is six (6). What is the p value of the test? Interpret the p value.

(iii) Conduct a joint hypothesis test: are both partial slope coefficients statistically significant?

(iv) What is the adjusted R^2 and why is it better than the unadjusted R^2? 

Stock

Output

Return

Growth

Inflation

Dynamic output

SUMMARY OUTPUT

Year

Y

X2

X3

X3

X2

1954

53

6.7

-0.4

Inflation

Output

Regression Statistics

1955

31.2

2.1

0.4

 (2.499)

 3.943 

 3.532 

Multiple R

0.756554

1956

3.7

1.8

2.9

 1.082 

 1.293 

 8.111 

R Square

0.572374

1957

-13.8

-0.4

3

 0.572 

 14.202 

 - 0 

Adjusted R Square

0.538164

1958

41.7

6

1.7

 16.731 

 25.000 

 - 0 

Standard Error

14.202228

1959

10.5

2.1

1.5

 6,749.451 

 5,042.582 

 - 0 

Observations

28

1960

-1.3

2.6

1.8

1961

26.1

5.8

0.8

b3

b2

b1

ANOVA

1962

-10.5

4

1.8

se(b3)

se(b2)

se(b1)

df

SS

MS

F

Significance F

1963

21.2

5.3

1.6

R^2

se(y)

Regression

2

6749

3374.7253

16.7311

0.0000

1964

15.5

6

1

F

d.f.

Residual

25

5043

201.7033

1965

10.2

6

2.3

ESS

RSS

Total

27

11792

1966

-13.3

2.7

3.2

1967

21.3

4.6

2.7

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

1968

6.8

2.8

4.3

Intercept

3.5318

8.1114

0.4354

0.6670

-13.1739

20.2375

-13.1739

20.2375

1969

-13.5

-0.2

5

X Variable 1

3.9433

1.2934

3.0487

0.0054

1.2794

6.6072

1.2794

6.6072

1970

-0.4

3.4

4.4

X Variable 2

-2.4994

1.0821

-2.3098

0.0294

-4.7281

-0.2708

-4.7281

-0.2708

1971

10.5

5.7

3.8

1972

15.4

5.8

3.6

1973

-22.6

-0.6

7.9

1974

-37.3

-1.2

10.8

1975

31.2

5.4

6

1976

19.1

5.5

4.7

1977

-13.1

5

5.9

1978

-1.3

2.8

7.9

1979

8.6

-0.3

9.8

1980

-22.2

2.6

10.2

1981

-12.2

-1.9

7.3

Answers:

Highlighted cells use formulas below from Gujarati

(i) Given the SRF, what is the first derivative with respect to inflation?

Because the regression is Y = B1 + B2X2 + B3X3, the first (partial) derivative with respect to X2 is simply B3 = -2.5

(ii) Our hypothesis is that the output growth parameter is six (6). What is the p value of the test?

First we need the calculated t (absolute value):

 1.59 

G6: calculated t stat (with three d.f.)

The we can use TDIST():

12.4%

This means, we can reject the null with only (1-12.4%) confidence

To understand this,…

Note if confidence is 87% we (barely) reject null

 1.56 

Note that if confidence is 88% we "accept" (fail to reject) null

 1.61 

(iii) Conduct a joint hypothesis test: are both partial slope coefficients statistically significant?

The F is given for us:

 16.73 

But it is good to know:

 16.73 

G14: F stat

Which exceeds the critical F, so we reject the null:

 5.57 

Or, the p value is

0.00245%

So, we can reject with confidence of:

99.99755%

(iv) What is the adjusted R^2 and why is it better than the unadjusted R^2

We are given the adjusted R^2

 0.538 

But we NEED TO KNOW:

 0.538 

G20: Adjusted R^2, k = 3 because we have TWO partial slope coefficients Plus the intercept

The adjusted R^2 accounts for degrees of freedom (i.e., the additional explanatory variable)

Stock

Output

Return

Growth

Inflation

Dynamic output

SUMMARY OUTPUT

Year

Y

X2

X3

X3

X2

1954

53

6.7

-0.4

Inflation

Output

Regression Statistics

1955

31.2

2.1

0.4

 (2.499)

 3.943 

 3.532 

Multiple R

0.7566

1956

3.7

1.8

2.9

 1.082 

 1.293 

 8.111 

R Square

0.5724

1957

-13.8

-0.4

3

 0.572 

 14.202 

 - 0 

Adjusted R Square

0.5382

1958

41.7

6

1.7

 16.731 

 25.000 

 - 0 

Standard Error

14.2022

1959

10.5

2.1

1.5

 6,749.451 

 5,042.582 

 - 0 

Observations

28

1960

-1.3

2.6

1.8

1961

26.1

5.8

0.8

b3

b2

b1

ANOVA

1962

-10.5

4

1.8

se(b3)

se(b2)

se(b1)

df

SS

MS

F

Significance F

1963

21.2

5.3

1.6

R^2

se(y)

Regression

2

6749

3374.7253

16.7311

0.0000

1964

15.5

6

1

F

d.f.

Residual

25

5043

201.7033

1965

10.2

6

2.3

ESS

RSS

Total

27

11792

1966

-13.3

2.7

3.2

1967

21.3

4.6

2.7

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

1968

6.8

2.8

4.3

Intercept

3.5318

8.1114

0.4354

0.6670

-13.1739

20.2375

-13.1739

20.2375

1969

-13.5

-0.2

5

X Variable 1

3.9433

1.2934

3.0487

0.0054

1.2794

6.6072

1.2794

6.6072

1970

-0.4

3.4

4.4

X Variable 2

-2.4994

1.0821

-2.3098

0.0294

-4.7281

-0.2708

-4.7281

-0.2708

1971

10.5

5.7

3.8

1972

15.4

5.8

3.6

1973

-22.6

-0.6

7.9

1974

-37.3

-1.2

10.8

1975

31.2

5.4

6

1976

19.1

5.5

4.7

1977

-13.1

5

5.9

1978

-1.3

2.8

7.9

1979

8.6

-0.3

9.8

1980

-22.2

2.6

10.2

1981

-12.2

-1.9

7.3