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EXCEL: Multiple Regression

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EXCEL: Multiple Regression

- A multiple regression model is:
y = β1+ β2 x2+ β3 x3+ u

Such that:

- y is dependent variable
- x2and x3are independent variables
- β1 is constant
- β2and β3are 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

- 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.

- 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.

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

- Let βjdenote 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