1. EXCEL: Multiple Regression

2. 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

3. Regression Analysis in Excel
We do this using the Data analysis Add-in and Regression.
Example:

4. Regression Analysis in Excel

5. Regression Analysis in Excel
The regression output has three components:
Regression statistics table
ANOVA table
Regression coefficients table.

6. Interpreting Regression Statistics Table Regression 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 = means that 80.25% of the variation of yi around its mean is explained by the regressors x2i and x3i.

7. Interpreting Regression Statistics Table Regression coefficients table
The regression output of most interest is the following table of coefficients and associated output:

8. Interpreting Regression Statistics Table Regression coefficients table
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.

9. Interpreting Regression Statistics Table Regression coefficients table
A simple summary of the previous output is that the fitted line is:
y = x z

10. Exercise Y X1 X2 39 15
110 44 9 65 50 10 90 64 12
100 3
160 55 13
130 66
105 2 20 92
240 81 17 95