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EXCEL: Multiple Regression

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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  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 • 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

  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 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.

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

  8. 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.

  9. 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

  10. Exercise

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