Statistics

Use the regression information presented in the following output to answer questions.
Note that
DP St = dividends per share in year t during 1970-1990;
EP St = earnings per share in year t during 1970-1990;
DP St−1 = dividends per share in the previous year t − 1, i.e. during 1969-1989.
The regression model used in the data analysis output can be specified as
DP St = β0 + β1 EP St + β2 DP St−1 + t ,
or
\
DP
St = b0 + b1 EP St + b2 DP St−1
where b0 , b1 , b2 are the estimates of β0 , β1 , β2 , respectively.
> Data1
DPS_t
[1,] 3.40
[2,] 3.40
[3,] 4.45
[4,] 5.25
[5,] 3.40
[6,] 2.40
[7,] 5.55
[8,] 6.80
[9,] 6.00
[10,] 5.30
[11,] 2.95
[12,] 2.40
[13,] 2.40
[14,] 2.80
[15,] 4.75
[16,] 5.00
[17,] 5.00
[18,] 5.00
[19,] 5.00
[20,] 3.00
[21,] 3.00

EPS_t DPS_t-1
2.09
4.30
6.72
3.40
7.51
3.40
8.34
4.45
3.27
5.25
4.32
3.40
10.08
2.40
11.62
5.55
12.24
6.80
10.04
6.00
-2.65
5.30
1.07
2.95
3.09
2.40
11.84
2.40
14.22
2.80
12.28
4.75
8.22
5.00
10.06
5.00
13.64
5.00
6.33
5.00
-4.09
3.00

3

(1)

> summary(Data1)
DPS_t
Min.
:2.400
1st Qu.:3.000
Median :4.450
Mean
:4.155
3rd Qu.:5.000
Max.
:6.800

EPS_t
Min.
:-4.090
1st Qu.: 3.270
Median : 8.220
Mean
: 7.154
3rd Qu.:11.620
Max.
:14.220

DPS_t-1
Min.
:2.400
1st Qu.:3.000
Median :4.450
Mean
:4.217
3rd Qu.:5.000
Max.
:6.800

> cor(Data1)
DPS_t
EPS_t
DPS_t-1
DPS_t
1.0000000 0.7024325 0.5147082
EPS_t
0.7024325 1.0000000 0.2006639
DPS_t-1 0.5147082 0.2006639 1.0000000
> model1=lm(DPS_t~EPS_t+DPS_t-1)
> summary(model1)
Call:
lm(formula = DPS_t ~ EPS_t + DPS_t-1)
Residuals:
Min
1Q
Median
-1.37833 -0.38614 -0.02261

3Q
0.36437

Coefficients:
Estimate Std. Error t
(Intercept)
1.34072
0.65125
EPS_t
?
0.03684
DPS_t-1
0.39741
0.14753
–Signif. codes: 0 ‘***’ 0.001 ‘**’

Max
1.65170

value
?
4.318
2.694

Pr(>|t|)
0.054297 .
0.000414 ***
0.014844 *

0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8386 on 18 degrees of freedom
Multiple R-Squared: ?,
Adjusted R-squared: 0.5989
F-statistic: 15.93 on 2 and 18 DF, p-value: 0.0001042
> anova(model1)
4

Analysis of Variance Table
Response: DPS_t
Df Sum Sq Mean Sq F value
Pr(>F)
EPS_t
1 17.2988 17.2988 24.5999 0.0001013 ***
DPS_t-1
1 5.1031 5.1031 7.2568 0.0148442 *
Residuals 18 12.6577 0.7032
–Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> newdata=data.frame(EPS_t=10, DPS_t-1=4)
> predict(model1,newdata,interval=”prediction”,level=0.95)
fit
lwr
upr
[1,] 4.521429 2.701915 6.340943
> predict(model1,newdata,interval=”confidence”,level=0.95)
fit
lwr
upr
[1,] 4.521429 4.066708 4.97615

1. What is the regression estimate of earnings per share (EP St )? Interpret it in words.

2. Write down the fitted model with the estimated coefficients.

3. What is the t-statistic for the intercept? And what is the p-value?

4. Suppose you want to know whether dividend per share from last year (DP St−1 ) has any
statistically positive impact on the current-year dividend per share (DP St ). Write down
your null hypothesis and the alternative hypothesis. Write down the t-statistic and p-value.

5

Test the hypothesis at .05 significance level. What is your conclusion?

5. Suppose you want to know whether at least one of the explanatory variables (i.e. DP St−1
and/or EP S) has any statistically significant impact on the current-year dividend per share.
Write down your null hypothesis and the alternative hypothesis. What is your conclusion at
the .05 significance level? Justify this by referring to the relevant R analysis/output.

6. What is the total variation (SST) in the response variable (current-year dividend per share)?
What is the coefficient of determination (R2 )?

7. What percentage of the variation in the current-year dividend per share is explained by the
regression model? What percentage of the variation in the current-year dividend per share is
explained by current-year earnings per share alone?

6

8. Write down the 95% confidence interval for β1 .

9. If EP St = 15, and DP St−1 = 7, compute the predicted dividend per share in year t.

10. If EP St = 10, and DP St−1 = 4, what is 95% prediction interval (for individual response) of
dividend per share in year t?

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