Multiple regression analysis is widely used in business research in order to for

Question

Multiple regression analysis is widely used in business research in order to forecast and predict purposes. It is also used to determine what independent variables have an influence on dependent variables, such as sales.

Sales can be attributed to quality, customer service, and location. In multiple regression analysis, we can determine which independent variable contributes the most to sales; it could be quality or customer service or location.

Now, consider the following scenario. You have been assigned the task of creating a multiple regression equation of at least three variables that explains Microsoft’s annual sales.

Use a time series of data of at least 10 years. You can search for this data using the Internet.

Submit your answers in a two- to three-page Word document.

Explanation / Answer

When the values of one variable are associated with or influenced by other variable, e.g., Sales can be attributed to quality, customer service, and location. In multiple regression analysis, we can determine which independent variable contributes most to sales, it could be quality or customer's location. Let us consider a distribution of three variable X1, X2, X3 . Then the equation of plane of regression of X1 on X2 and X3 is : X1 = a + b12.3 X2 +b13.2 X3

Without loss of generality we can assume that the variables X1, X2, X3 have been measured from their respective means, so that E(X1) = E(X2) = E(X3 ) = 0

Hence on taking expectation of both sides, we get a = 0, so the regression euqation becomes

X1 = b12.3 X2 +b13.2 X3

The coefficient b12.3 and b13.2 are known as partial regression coefficient of X1 on X2 and X3.

Using the method of least square, b12.3 and b13.2 are estimated as

b12.3 = 1/2 . ( r12 - r23 r13) and b13.2 = 1/3 (r13 - r12r23 ). We can get the regression equation after substituting these values

Multiple Correlation Coefficient of X1 on X2 and X3 usually denoted by R1.23 and is the simple correlation coefficient between X1 and the joint effect of X2 and X3 on X1 . In other words R1.23 is the correlation coefficient between X1 and its estimated value as given by the plane of regression of X1 on X2 and X3 .

Let e1.23 = b12.3 X2 +b13.2 X3, then R1.23 = Cov(X1, e1.23 ) / ((V(X1) V(e1.23 ))

= (r122+ r132- 2 r12 r13 r23) / (1 - r23 2)

Let us take an example of sales (X1) of 10 years with dependent variables of X2 and X3 whose means, standard deviation and Co-relation Coefficients are give below:

Traits Mean SD r12 r23 r31

X! 28.02 4.42 +0.80 - -

X2    4.91 1.10 - -0.56 -

X3    594 85 - - -0.40

The required regression equation is

(X1 - Mean of X1) W11/1 + (X2 - Mean of X2) W12/2 + (X3 - Mean of X3) W13/3 = 0

After calculating all these we get the equation

as (0.686 / 4.42) (X1 - 28.02) + (-0.576/1.10) ( X2 - 4.91) + ( - 0.048)/85.00 (X3 - 594) = 0

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