Question

Refer to the following situation: Stock Prices, Y, are assumed to be affected by the annual rate of dividend of stock, X. A simple linear regression analysis was performed on 20 observations and the results were:

Variable	Unstandardized Coefficient	Standard error	t-value	Sig.
Intercept	-7.96	3.11	-2.56	0.0166
X	12.549	1.27	9.87	0.0001

State the appropriate research question:_________________

State the appropriate null hypothesis__________________

State the appropriate alternative hypothesis__________________

Report the value of B₀ ___________________________________________________

Interpret B₀ ___________________________________________________________

Report the value of B₁___________________________________

Interpret B₁___________________________________________________

What decision should be made about the null hypothesis?

What value(s) should the decision be based on?

Answer 1

Solution

We employ the following terminology:

The linear regression model Y = β₀ + β₁X + ε, ………………………………….………………………..(1)

where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ².

Estimated Regression of Y on X is given by: Y_cap = B₀ + B₁X, ………………………….……………….(2)

where

B₁ = Sxy/Sxx = r.√(Syy/Sxx) = r.(SDy/SDx) …………………………………………………………… (3)

and

B₀ = Ybar – B₁.Xbar..…………………………………………………………………………….….…..(4)

Part (a)

The appropriate research question: Is the annual rate of dividend of stock, X an effective predictor for Stock Prices, Y? Answer 1

Part (b)

The appropriate null hypothesis: H₀: β₁ = 0 Answer 2

Part (c)

The appropriate alternative hypothesis: H₁: β₁ ≠ 0 Answer 3

Part (d)

The value of B₀: - 7.96 Answer 4

Part (e)

Interpretation of B₀ :

In the estimated regression of Y on X given by: Y = B₀ + B₁X, B₀ represents the y-intercept mathematically and physically represents the expected value of the response (dependent) variable when the predictor (independent) variable is zero i.e., the expected stock price when no dividend is paid. Answer 5

Part (f)

The value of B₁=12.549 Answer 6

Part (g)

Interpretation of B₁

In the estimated regression of Y on X given by: Y = B₀ + B₁X, B₁ represents the slope of the regression line mathematically and physically represents the expected change (increase/decrease) in value of the response (dependent) variable when the predictor (independent) variable changes (increases/decreases) by one unit,

i.e., the stock price is expected go up or go down by 12.549 monetary units when dividend pay-out is up or down by one monetary unit respectively. Answer 7

Part (g) and (h)

To address the next two questions, we will perform the significance test as detailed below:

Test Statistic:

t = B₁/SE(B₁)

= 9.87 [given under ‘t-value’ against ‘X’]

Under the null hypothesis, t ~ t_{n – 2} and hence

the p-value = P(t₁₈> 9.87) = 1.0919E-08 [given as 0.0001*² under ‘Sig’ against ‘X’ ]

Decision

Since p-value is very small, even smaller than the least perceivable significance level of 0.001, null hypothesis is rejected*¹. This implies that the annual rate of dividend of stock, X can serve as an effective predictor for Stock Prices, Y. Answer 8

To the point answer for

‘What decision should be made about the null hypothesis?’ is: *¹ Reject and

‘What value(s) should the decision be based on?’ is: *²0.0001

DONE