In a random sample of 23 firms in the same industry the foll

In a random sample of 23 firms in the same industry, the following quantities were obtained for each:

X₁ = research and development expenditures (millions of dollars)
X₂ = advertising expenditures on television (millions of dollars)
X₃ = all other advertising expenditures (millions of dollars)
Y = annual sales (millions of dollars)

Regression analysis yields the following results:
= -2.3 + 5.8X₁ + 4.2X₂ + 7.4X₃
...............(1.20) .. (1.31) ...(1.56) [NOTE: Dots are there for formating]

The quantities in parentheses are the standard errors of the net regression coefficients. The standard error of estimate S_Y123 was 12.4. The standard deviation of the dependent variable S_Y was 25.

Question 1:
a) Interpret the net regression coefficient b₁.
b) Test at the 1% level of significance whether each of the net regression coefficients is significantly different from zero.
c) What is the expected effect when highly correlated independent variables are included in a multiple regression equation?
d) Calculate the coefficient of multiple determination.
e) Estimate the average annual sales for a firm that has research and develeopment expenditures of $6 million, television advertising expendtures of $10 million, and all other advertising expendtirues of $7 million.

Solution

= -2.3 + 5.8X1 + 4.2X2 + 7.4X3

a) b1 = 5.8

It represents the mean change in the Annual Sales for one unit of change in the research and development expenditures while holding advertising expenditures on television and all other advertising expenditures constant

b) t = Coeff/Std Error

t stat for X1 = 5.8/1.20 = 4.833

DF = n - 2 = 23 - 3 - 1 = 19

p value = TDIST(4.833,19,2) = 0.0001

Since p value < 0.01 X1 is significant

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t stat for X2 = 4.2/1.31 = 3.206

DF = n - 2 = 23 - 3 - 1 = 19

p value = TDIST(3.206,19,2) = 0.005

Since p value < 0.01 X2 is significant

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t stat for X3 = 7.4/1.56 = 4.744

DF = n - 2 = 23 - 3 - 1 = 19

p value = TDIST(4.744,19,2) = 0.0001

Since p value < 0.01 X3 is significant

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c) If highly correlated independent variables are included in a multiple regression equation then the R² value inflates and becomes really high but the model remains insignificant. This is a case of multi collinearity.

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d) R² = standard error of estimate / standard deviation of the dependent variable = 12.4/25 = 0.496 = 49.6%

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e) Research and develeopment expenditures X1 = $6 million

Television advertising expendtures X2 = $10 million

Other advertising expendtirues X3 = $7 million

Annual Sales = -2.3 + 5.8X1 + 4.2X2 + 7.4X3

= -2.3 + 5.8*6 + 4.2*10 + 7.4*7

= 126.3 $7 million