IE-Lecture7-Mulitiple regression

t 4.31 tcritical 1.96
5%
and t 4.31 tcritical 2.576
1%
Overall fit of regression
(R2 and
R
2
)
ˆ EAR N INGS 26 . 93 2 . 674 S 0 . 594 X R
2
Constant term
Years Years Schooling Experience (+ve) (+ve)
We expect the sign of β2 and β3 to be positive. Why?
Estimation using Microfit
Using data on 540 individuals we estimate the model E i 1 2 S i 3 X i u i and obtain the following regression output
b1 Y ( b 2 X 2 b3 X 3 )
b2
Cov ( X 2 Y ) Var ( X 3 ) Cov ( X 3Y ) Cov ( X 2 X 3 ) Var ( X 2 ) Var ( X 3 ) Cov ( X 2 X 3 )
2
Cov ( X 3Y ) Var ( X 2 ) Cov ( X 2 Y ) Cov ( X 2 X 3 ) b3 2 Var ( X 3 ) Var ( X 2 ) Cov ( X 2 X 3 )
Note: Often used to compare models. Evaluating specific model use R2
Writing estimated regression:
^ E i 26 . 93 2 . 674 S i 0 . 594 X i
Interpretation
^ E i 26 . 93 2 . 674 S i 0 . 594 X i
Interpreting slope coefficient b2 = 2.674: Schooling and Earnings are positively related. For every grade completed, hourly earnings will increase by $2.674 per hour on average, holding experience constant. Interpreting slope coefficient b3 = 0.594: Experience and Earnings are positively related. An extra year of experience, will on average induce a $0.594 increase in hourly earnings, ceteris paribus Hourly Earnings for someone with 12 years of schooling and 10 years of experience:
Derivation of OLS estimators b1 and b2 same as for bi-variate case: Minimising the residual sum of squares,
e
2 i
OLS Estimators in Multiple Regression
Lower correlation between regressors
s .e ( b 2 )
S
2

1 1 rX 2 X 3
2
n var( X 2 )
The variance or s.e is smaller for lower correlation among regressors
ˆ EAR N INGS 26 . 93 2 . 674 S 0 . 594 X
E i 26 . 93 2 . 674 (12 ) 0 . 594 (10 ) 11 . 098
^
Inference on Partial Coefficients
Bivariate regression ˆ Y b b X
1 2
Multiple regression
ˆ Y b1 b 2 X 2 b3 X 3
Adjusted
2( R2) R
Adding new variables always increases R2 Not legitimate to compare R2 from models containing different numbers of regressors.
Use R 2 which adjusts for the number of explanatory variables
s .e ( b 2 ) S
2

1 1 rX 2 X 3
2
n var( X 2 )
As in simple regression, it is desirable for n and Var (X2) to be large and S 2 to be small But in multiple regression, also desirable for correlation between X2 and X3 to be low.
• However, can have more than one explanatory variable:
‘unobservable’ model: ‘fitted’ regression:
Y 1 2 X 2 3 X 3 u
ˆ Yi b1 b 2 X 2 b3 X 3
INTRODUCTORY ECONOMETRICS - P12205
Lecture 7 Multiple Regression Analysis: Estimation & Testing
• So far bivariate regression model
Y 1 2 X 2 u
Highly correlated regressors reduces the precision of estimation and increases the standard error
Shape of the sampling distribution Positively related to correlation coefficient
( 4.5234) (0.232) (0.13792)
H 0 : 3 0 Experience has no effect on Earnings H1 : 3 0 Experience has an effect on Earnings
Does experience have an effect on earnings?
OLS Estimators in Multiple Regression
b2 Cov ( X 2 Y ) Var ( X 2 )
b2 Cov ( X 2 Y ) Var ( X 3 ) Cov ( X 3Y ) Cov ( X 2 X 3 ) Var ( X 2 ) Var ( X 3 ) Cov ( X 2 X 3 )
2
Differ only if: Cov(X2, X3) ≠ 0
But when regressors are uncorrelated: Cov(X2, X3) = 0
b2 Cov ( X 2 Y ) Var ( X 3 ) Var ( X 2 ) Var ( X 3 ) Cov ( X 2 Y ) Var ( X 2 )
0 . 1991
R
2
0 . 1961
R2 gives the proportion of the variation in the dependent variable Y explained by the explanatory variables X2 and X3
R2 = 0.1991 implies model accounts around 20% of the variation in earnings
R
2
R
2

k 1 nk
(1 R )
2Leabharlann observationsparameters
Overall fit of regression
ˆ EAR N INGS 26 . 93 2 . 674 S 0 . 594 X R
2
0 . 1991
R
2
0 . 1961
The R tells that after accounting for the number of regressors, the model explains around 20% of the variation in Earnings
b3 3 s.e.(b3 )
t
~ t ( n k )
t
0.594 0 0.13792
,
where 3 3 0
0
t
Critical Values
0.594 0.13792
4.31
We can reject H0, at 1% level and conclude that we are 99% confident that experience has a statistically significant effect on earnings.