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F TESTS OF GOODNESS OF FIT
Y = 1 + 2 X 2 + . + k X k + u
H 0 : 2 = . = k = 0
H 1 : at least one 0
This sequence describes two F tests of goodness of fit in a multiple regression model. The
first relates to the goodness of fit of the equation as a whole.
1
EXERCISE 3.9
3.9
e
Demonstrate that is equal to zero in multiple
regression analysis. (Note: The proof is a
generalization of the proof for the simple regression
model, given in Section 1.7.)
1
EXERCISE 3.9
Yi = 1 + 2 X 2 i + . + k X ki + ui
Yi = b1 + b2
EXERCISE 3.5
3.5
Explain why the intercept in the regression of EEARN
on ES is equal to zero.
This exercise relates to the Frisch LovellWaugh procedure for graphing the relationship
between the dependent variable and one of the explanatory variables in a
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. reg EARNINGS S EXP
Source 
SS
df
MS
+67.54
Model  22513.6473
2 11256.8237
0.0000
Residual  89496.5838
537 166.660305
0.2010
+0.1980
Total  112010.231
539 207.811189
Number of obs =
F( 2,
537
MULTICOLLINEARITY
Y = 2 + 3X2 + X3
X3 = 2X2 1
X2
X3
Y
10
19
51
11
21
56
12
23
61
13
25
66
14
27
71
15
29
76
Suppose that Y = 2 + 3X2 + X3 and that X3 = 2X2 1. There is no disturbance term in the
equation for Y, but that is not important. Suppose that we h
PRECISION OF THE MULTIPLE REGRESSION COEFFICIENTS
Y = 1 + 2 X 2 + 3 X 3 + u
Y = b1 + b2 X 2 + b3 X 3
2
2
u
u
1
1
2
b2 =
=
2
2
2
( X 2 i X 2 ) 1 rX 2 , X 3 nMSD( X 2 ) 1 rX 2 , X 3
This sequence investigates the variances and standard errors of the slope
ALLEVIATION OF MULTICOLLINEARITY
Possible measures for alleviating multicollinearity
2
2
u
u
1
1
=
=
2
2
2
( X 2 i X 2 ) 1 rX 2 , X 3 nMSD( X 2 ) 1 rX 2 , X 3
2
b2
What can you do about multicollinearity if you encounter it? We will discuss some
possible
PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS
A.1: The model is linear in parameters and correctly specified.
Y = 1 + 2 X 2 + . + k X k + u
A.2: There does not exist an exact linear relationship among
the regressors in the sample.
A.3
The disturbance
F TEST OF GOODNESS OF FIT
(Y Y )2 = (Y Y )2 + e 2
TSS = ESS + RSS
In an earlier sequence it was demonstrated that the sum of the squares of the actual values
of Y (TSS: total sum of squares) could be decomposed into the sum of the squares of the
fitted va
MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE
EARNINGS = 1 + 2S + 3EXP + u
1
EARNINGS
EXP
S
This sequence provides a geometrical interpretation of a multiple regression model with
two explanatory variables.
1
MULTIPLE REGRESSION WITH TWO EXP
TESTING A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
Model:
Y = 1 + 2X + u
Null hypothesis:
H 0 : 2 = 20
Alternative hypothesis:
H 1 : 2 20
This sequence describes the testing of a hypothesis at the 5% and 1% significance levels.
It also defines what
t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
s.d. of b2 known
discrepancy between
hypothetical value and sample
estimate, in terms of s.d.:
0
b2 2
z=
s.d.
5% significance test:
reject H0: 2 = 20 if
z
> 1.96 or
z
< 1.96
The diagram summarize
ONESIDED t TESTS
probability density
function of b2
null hypothesis:
H0 : 2 = 20
alternative hypothesis: H1 : 2 = 21
2.5%
0
0
2 2sd 2 sd
2.5%
0
2
0
0
2 +sd 2 +2sd 1
2
This sequence explains the logic behind a onesided t test.
1
ONESIDED t TESTS
pro
CONFIDENCE INTERVALS
probability density function of b2
null hypothesis
0
H0: 2 = 2
0
conditional on 2 = 2 being true
2.5%
0
21.96sd
2.5%
0
2sd
0
2
0
0
2+sd 2+1.96sd
In the sequence on hypothesis testing, we started with a given hypothesis, for exampl
THE RANDOM COMPONENTS OF THE REGRESSION COEFFICIENTS
True model
Y = 1 + 2 X + u
Fitted model
Y = b1 + b2 X
The regression coefficients are special types of random variable. We will demonstrate this
using the simple regression model in which Y depends on X
PRECISION OF THE REGRESSION COEFFICIENTS
Simple regression model: Y = 1 + 2X + u
probability density
function of b2
2
b2
We have seen that the regression coefficients b1 and b2 are random variables. They provide
point estimates of 1 and 2, respectively. I
TYPE I ERROR AND TYPE II ERROR
hypothetical distribution
0
under H0 : 2 = 2
acceptance region for b2
5% level
2.5%
2.5%
0
0
21.96sd 2sd
0
2
0
0
2+sd 2+1.96sd
b2
In the previous sequence a Type I error was defined to be the rejection of a null hypothes
UNBIASEDNESS OF THE REGRESSION COEFFICIENTS
Simple regression model: Y = 1 + 2X + u
b2
( X X )(Y Y ) = + a u
=
( X X )
i
i
2
2
i
i
i
We will now demonstrate that the ordinary least squares (OLS) estimator of the slope
coefficient in a simple regression m
INTERPRETATION OF A REGRESSION EQUATION
120
Hourly earnings ($)
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
20
Years of schooling
The scatter diagram shows hourly earnings in 2002 plotted against years of schooling,
defined as
TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A
Types of data
Crosssectional: Observations on individuals, households,
enterprises, countries, etc at one moment
in time (Chapters 110, Models A and B)
Time series:
Observations on income, consumption
SIMPLE REGRESSION MODEL
Y
Y = 1 + 2 X
1
X1
X2
X3
X4
X
Suppose that a variable Y is a linear function of another variable X, with unknown
parameters 1 and 2 that we wish to estimate.
Christopher Dougherty 19992006
1
SIMPLE REGRESSION MODEL
Y
Y = 1 + 2 X
1
DERIVING LINEAR REGRESSION COEFFICIENTS
True model : Y = 1 + 2 X + u
Fitted line : Y = b1 + b2 X
Y
6
Y3
Y2
5
4
3
Y1
2
1
0
0
1
2
3
X
This sequence shows how the regression coefficients for a simple regression model are
derived, using the least squares crit
EXERCISE 1.16
1.16
The output below shows the result of regressing weight
in 2002 on height, using EAEF Data Set 21. In 2002 the
respondents were aged 3744. Explain why R2 is lower
than in the regression reported in Exercise 1.5.
. reg WEIGHT02 HEIGHT
Sou
EXERCISE 1.17
1.17 The useful results in Box 1.2 are in general no longer valid if
the model does not contain an intercept. Demonstrate, in
particular, that e will not in general be equal to zero.
1
EXERCISE 1.17
1.17 The useful results in Box 1.2 are in
GOODNESS OF FIT
Four useful results:
e =0
Y =Y
X e
ii
=0
Y e
ii
=0
This sequence explains measures of goodness of fit in regression analysis. It is
convenient to start by demonstrating three useful results. The first is that the mean value
of the residual
EXERCISE 1.9
1.9
A researcher has international crosssection data on
aggregate wages, W, aggregate profits, P, and aggregate
income, Y, for a sample of n countries. By definition,
Y=W+P
The regressions
W = a1 + a2Y
P = b1 + b2Y
are fitted using OLS regre
EXERCISE 1.7
1.7
Derive, with a proof, the slope coefficient that would
have been obtained in Exercise 1.5 if weight and
height had been measured in metric units. (Note:
one pound is 454 grams, and one inch is 2.54 cm.)
1
EXERCISE 1.7
Y = 1 + 2 X + u
Y =
EXERCISE 1.5
1.5
The output below shows the result of regressing the
weight of the respondent in 1985, measured in
pounds, on his or her height, measured in inches.
Provide an interpretation of the coefficients.
. reg WEIGHT85 HEIGHT
Source 
SS
df
MS
+