(Section 4.3) magnitude of a typical regression residual ...
Measures of Fit
(Section 4.3)
A natural question is how well the regression line ¡°fits¡± or
explains the data. There are two regression statistics that provide
complementary measures of the quality of fit:
? The regression R2 measures the fraction of the variance of Y
that is explained by X; it is unitless and ranges between zero
(no fit) and one (perfect fit)
? The standard error of the regression (SER) measures the
magnitude of a typical regression residual in the units of Y.
1
The regression R2 is the fraction of the sample variance of Yi
¡°explained¡± by the regression.
Yi = Y?i + u?i = OLS prediction + OLS residual
? sample var (Y) = sample var(Y? ) + sample var( u? ) (why?)
i
i
? total sum of squares = ¡°explained¡± SS + ¡°residual¡± SS
n
2
Definition of R :
ESS
R =
=
TSS
2
2
?
?
(
Y
?
Y
)
? i
i ?1
n
2
(
Y
?
Y
)
? i
i ?1
? R2 = 0 means ESS = 0
? R2 = 1 means ESS = TSS
? 0 ¡Ü R2 ¡Ü 1
? For regression with a single X, R2 = the square of the
correlation coefficient between X and Y
2
The Standard Error of the
Regression (SER)
The SER measures the spread of the distribution of u. The SER
is (almost) the sample standard deviation of the OLS residuals:
SER =
=
1 n
2
?
?
(
u
?
u
)
?
i
n ? 2 i ?1
1 n 2
u?i
?
n ? 2 i ?1
1 n
(the second equality holds because u? = ? u?i = 0).
n i ?1
3
SER =
1 n 2
u?i
?
n ? 2 i ?1
The SER:
? has the units of u, which are the units of Y
? measures the average ¡°size¡± of the OLS residual (the average
¡°mistake¡± made by the OLS regression line)
4
Technical note: why divide by n¨C2 instead of n¨C1?
SER =
1 n 2
u?i
?
n ? 2 i ?1
? Division by n¨C2 is a ¡°degrees of freedom¡± correction ¨C just like
division by n¨C1 in sY2 , except that for the SER, two parameters
have been estimated (?0 and ?1, by ?? and ?? ), whereas in s 2
0
1
Y
only one has been estimated (?Y, by Y ).
? When n is large, it makes negligible difference whether n, n¨C1,
or n¨C2 are used ¨C although the conventional formula uses n¨C2
when there is a single regressor.
5
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