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Subject 8. Is R2 Related to Statistical Significance? PDF Download
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.

R2 = (Total variation - Unexplained variation)/Total variation

Recall that R2 = 1 - SSE/SST, where SST = ∑(yt - y-bar)2, and SSE = ∑(e-hatti)2. The SST remains constant when another explanatory variable is added to a model because such an addition has no effect on the sum ∑(yt - 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 R2 to increase. To this extent, then, the value of R2 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-bar2, is calculated as shown below:

Adjusted R2 = 1 - [SSE/(n - k - 1)]/[SST/(n - 1)]
= 1 - [(n-1)/(n - k - 1)] (1 - R2)

  • R2 is always ≥ adjusted R2.
  • When a new independent variable is added, adjusted R2 can decrease if adding that variable has only a small effect on R2.
  • In fact, adjusted R2 can actually be negative if the correlation between the dependent variable and the independent variables is sufficient low.

Continue with our example discussed earlier:

R2 = 1 - SSE/SST = 1 - 20.8958/(574.7042 + 20.8958) = 0.9649. Adjusted R2 = 1 - [(10 - 1)/(10 - 3)] (1 - 0.9649) = 0.9549. In fact, R2 and adjusted R2 are often presented in an ANOVA table.

Learning Outcome Statements

h. contrast and interpret the R2 and adjusted R2 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

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)]
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Colin Sampaleanu

Colin Sampaleanu

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