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EXCEL: Multiple Regression. Regression Model. A multiple regression model is: y = β 1 + β 2 x 2 + β 3  x 3 + u Such that: y is dependent variable x 2 and x 3 are independent variables β 1 is constant β 2 and β 3 are regression coefficients

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EXCEL: Multiple Regression

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EXCEL: Multiple Regression

Regression Model

  • 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

Regression Analysis in Excel

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

  • Example:

Regression Analysis in Excel

Regression Analysis in Excel

  • 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:

Interpreting Regression Statistics TableRegression coefficients table

  • 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.

Interpreting Regression Statistics TableRegression coefficients table

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

    y = 0.8966 + 0.3365x + 0.0021z


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