excel: multiple regression

EXCEL: Multiple Regression

Regression Model

A multiple regression model is:

y = β1+ β2 x2+ β3 x3+ u

Such that: y is dependent variable x2 and x3 are independent variables

β1 is constant

β2 and β3 are regression coefficients It is assumed that the error u is independent with constant variance.

We wish to estimate the regression line:y = b1 + b2 x2 + b3 x3

Regression Analysis in Excel

We do this using the Data analysis Add-in and Regression.

Example:


The regression output has three components: Regression statistics table ANOVA table Regression coefficients table.

Interpreting Regression Statistics TableRegression Statistics

The standard error here refers to the estimated standard deviation of the error term u.

It is sometimes called the standard error of the regression. It equals sqrt(SSE/(n-k)).

It is not to be confused with the standard error of y itself (from descriptive statistics) or with the standard errors of the regression coefficients given below.

R2 = 0.8025 means that 80.25% of the variation of yi around its mean is explained by the regressors x2i and x3i.

Interpreting Regression Statistics TableRegression coefficients table

The regression output of most interest is the following table of coefficients and associated output:


Let βj denote the population coefficient of the jth regressor (intercept, HH SIZE and CUBED HH SIZE). Then

Column "Coefficient" gives the least squares estimates of βj. Column "Standard error" gives the standard errors (i.e.the estimated standard

deviation) of the least squares estimates bj of βj.

Column "t Stat" gives the computed t-statistic for H0: βj = 0 against Ha: βj ≠ 0.This is the coefficient divided by the standard error. It is compared to a t with (n-k) degrees of freedom where here n = 5 and k = 3.

Column "P-value" gives the p-value for test of H0: βj = 0 against Ha: βj ≠ 0..This equals the Pr{|t| > t-Stat}where t is a t-distributed random variable with n-k degrees of freedom and t-Stat is the computed value of the t-statistic given in the previous column. Note that this p-value is for a two-sided test. For a one-sided test divide this p-value by 2 (also checking the sign of the t-Stat).

Columns "Lower 95%” and "Upper 95%” values define a 95% confidence interval for βj.


A simple summary of the previous output is that the fitted line is:

y = 0.8966 + 0.3365x + 0.0021z

Exercise

Y X1 X2

39 15 110

44 9 65

50 10 90

64 12 100

65 3 160

55 13 130

66 15 105

12 2 20

92 20 240

81 17 95

excel: multiple regression

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