- CFA Exams
- 2023 Level I
- Topic 1. Quantitative Methods
- Learning Module 2. Multiple Regression
- Subject 8. Is R
^{2}Related to Statistical Significance?

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Subject 8. Is R^{2} Related to Statistical Significance?
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Just as in the simple linear regression model, we can decompose the total variation in the dependent variation into two components: explained variation and unexplained variation.R Adjusted R

= 1 - [(n-1)/(n - k - 1)] (1 - R

^{2}= (Total variation - Unexplained variation)/Total variation

Recall that R

^{2}= 1 - SSE/SST, where SST = ∑(y_{t}- y-bar)2, and SSE = ∑(e-hat_{ti})^{2}. The SST remains constant when another explanatory variable is added to a model because such an addition has no effect on the sum ∑(y_{t}- y-bar)^{2}. On the other hand, adding an additional explanatory variable to the model causes SSE to decline provided the estimated coefficient of the new variable is not exactly 0 and thus causes the value of R^{2}to increase. To this extent, then, the value of R^{2}depends on the number of explanatory variables included in the model. This causes a problem when we try to compare the goodness of fit of two models that have the same dependent variable but different number of explanatory variables.Another measure of goodness-of-fit takes into account the number of explanatory variables included in an equation. The measure, called the adjusted R square and denoted R-bar

^{2}, is calculated as shown below:

^{2}= 1 - [SSE/(n - k - 1)]/[SST/(n - 1)]

= 1 - [(n-1)/(n - k - 1)] (1 - R

^{2})

- R
^{2}is always ≥ adjusted R^{2}. - When a new independent variable is added, adjusted R
^{2}can decrease if adding that variable has only a small effect on R^{2}. - In fact, adjusted R
^{2}can actually be negative if the correlation between the dependent variable and the independent variables is sufficient low.

Continue with our example discussed earlier:

R

^{2}= 1 - SSE/SST = 1 - 20.8958/(574.7042 + 20.8958) = 0.9649. Adjusted R^{2}= 1 - [(10 - 1)/(10 - 3)] (1 - 0.9649) = 0.9549. In fact, R^{2}and adjusted R^{2}are often presented in an ANOVA table.

**Learning Outcome Statements**

^{2}and adjusted R

^{2}in multiple regression;

i. evaluate how well a regression model explains the dependent variable by analyzing the output of the regression equation and an ANOVA table;

CFA® 2023 Level I Curriculum, Volume 1, Module 2

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**User Contributed Comments**
4

User |
Comment |
---|---|

danlan2 |
Adjusted R^2=1-[(n-1)/(n-k)]*(1-R^2) is right, but k is the number of all variables (including dependant and independant), or it is the number of independant variables + 1. |

JimM |
I just googled adjusted R2 and most sites gave (n-k-1) in the formula. |

arudkov |
2 danlan - k - is the number of indep variables. +1 means interciept |

Adi8232 |
Which Textbook? In L2 Vol1, its n-k-1, not n-k. Guess they corrected it. or maybe AnalystNotes is putting down the book for those who haven't read it, hehe. [i haven't, had to check, but Cmon guys, don't say bad things about CFA (textbook)] |

You have a wonderful website and definitely should take some credit for your members' outstanding grades.