If an additional independent variable, however irrelevant,...

Name: Problem 86, Chapter 12: Statistics for Business and Economics: Global Edition
Uploaded: 2022-01-17T03:03:15-08:00
Duration: 3 min 4 s
Channel: Prashant Bana
Description: If an additional independent variable, however irrelevant, is added to a multiple regression model, a smaller sum-of-squared errors will result. Explain why this is so, and discuss the consequences for the interpretation of the coefficient of determination.

If an additional independent variable, however irrelevant, is added to a multiple regression model, a smaller sum-of-squared errors will result. Explain why this is so, and discuss the consequences for the interpretation of the coefficient of determination.

Key Concepts

Coefficient of Determination (R-squared)

R-squared is a statistic that quantifies the proportion of variance in the dependent variable that is explained by the model, ranging from 0 to 1. A key property of R-squared is that it never decreases with the inclusion of additional variables, even if they are irrelevant, leading to potentially misleading interpretations of the model’s explanatory power.

Adjusted R-squared

Adjusted R-squared is a modified version of R-squared that accounts for the number of predictors in the model. Unlike raw R-squared, adjusted R-squared can decrease when an irrelevant independent variable is added, thereby providing a more accurate reflection of the model's quality by penalizing unnecessary complexity.

Sum-of-Squared Errors (SSE)

The sum-of-squared errors measures the overall discrepancy between the observed values and the values predicted by a model. In a regression context, adding any additional independent variable—even one that is irrelevant—provides the model with more flexibility to capture small variances, which typically results in a lower SSE by reducing the residual errors, albeit sometimes only marginally.

Multiple Regression

Multiple regression is a statistical technique used to model the relationship between a dependent variable and two or more independent variables. When additional predictors are included, the model aims to better explain variations in the dependent variable, though it may sometimes capture noise rather than true signals if the additional variables are irrelevant.

Transcript

00:01 Here the answer.

00:02 Given that a regression model has three explanatory variables, the coefficient of determination r squared is given as 0 .653 based on a sample of size n is equal to 25 observations.

00:19 Now for the part a, r square of adjacent is equal to 1 minus in bracket n minus 1 divided by n minus k minus 1 in bracket 1 minus r square by substituting the values for the next step we get 1 minus in bracket 25 minus divided by 25 minus 3 minus in bracket 0 .653 or 1 minus in bracket 24 divided by 21 in bracket 0 .347 or 1 minus 1 .1429 multiply by 0 .347 or 1 minus 1 .1429 multiply by 0 .347 or 1 minus in bracket 0 .3965863 which is equal to 0 .603.

01:03 The answer for the part a.

01:06 Now for the part b, the f statistic is given by that f is equal to r squared divided by 1 minus r square multiplied by n minus in bracket k plus 1 divided by k.

01:19 By substituting the values for the next step, we get 0 .653 divided by 1 minus 0 .653 multiply by 25 in bracket 3 plus 1 divided by 3 or 0 .653 divided by 0 .347 multiply by 21 divided by 3 which is equal to 1 .881818 multiplied by 7 which is equal to 13 .1729 the answer for the part b.

01:48 Now for the part c suppose one additional explanatory variable is added then r square will be equal to 0 .665 then the adjusted r square is that r square adjacent is equal to 1 minus n minus 1 .m .2 divided by n minus k minus 1 in bracket 1 minus r square.

02:09 By substituting the values for the next step, we get 1 minus in bracket 25 minus 1 divided by 25 minus 4 minus 1 in bracket 1 minus in bracket 0 .665, which is equal to 1 minus 24 divided by 20 in bracket 0 .335, which is equal to 1 minus in bracket 0 .66...

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