Econometrics I - New York...
Transcript of Econometrics I - New York...
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Part 4: Partial Regression and Correlation 4-1/25
Econometrics I Professor William Greene
Stern School of Business
Department of Economics
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Part 4: Partial Regression and Correlation 4-2/25
Econometrics I
Part 4 – Partial Regression
and Partial Correlation
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Part 4: Partial Regression and Correlation 4-3/25
Frisch-Waugh (1933) ‘Theorem’
Context: Model contains two sets of variables:
X = [ (1,time) : (other variables)]
= [X1 X2]
Regression model:
y = X11 + X22 + (population)
= X1b1 + X2b2 + e (sample)
Problem: Algebraic expression for the second set of least squares coefficients, b2
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Part 4: Partial Regression and Correlation 4-4/25
Partitioned Solution
Method of solution (Why did F&W care? In 1933, matrix computation was not trivial!)
Direct manipulation of normal equations produces
, ] so and
( ) =
1 1 1 2 1
1 2
2 1 2 2 2
1 1 1 2 1 1
2 1 2 2 2 2
1 1 1 1 2 2 1
2 1 1 2 2 2 2 2 2 2 2 2 1 1
X X b X y
X X X X X yX = [X X X X = X y =
X X X X X y
X X X X b X y(X X)b = =
X X X X b X y
X X b X X b X y
X X b X X b X y ==> X X b X y - X X b
= 2 1 1X (y - X b )
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Part 4: Partial Regression and Correlation 4-5/25
Partitioned Solution
Direct manipulation of normal equations produces
b2 = (X2X2)-1X2(y - X1b1)
What is this? Regression of (y - X1b1) on X2
If we knew b1, this is the solution for b2.
Important result (perhaps not fundamental). Note the result
if X2X1 = 0.
Useful in theory: Probably
Likely in practice? Not at all.
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Part 4: Partial Regression and Correlation 4-6/25
Partitioned Inverse
Use of the partitioned inverse result produces a
fundamental result: What is the southeast
element in the inverse of the moment matrix?
1 1 1 2
2 1 2 2
-1X 'X X 'X
X 'X X 'X
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Part 4: Partial Regression and Correlation 4-7/25
Partitioned Inverse
The algebraic result is:
[ ]-1(2,2) = {[X2’X2] - X2’X1(X1’X1)-1 X1’X2}
-1
= [X2’(I - X1(X1’X1)-1X1’)X2]
-1
= [X2’M1X2]-1
Note the appearance of an “M” matrix. How do we interpret this result?
Note the implication for the case in which X1 is a single variable. (Theorem, p. 37)
Note the implication for the case in which X1 is the constant term. (p. 38)
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Part 4: Partial Regression and Correlation 4-8/25
Frisch-Waugh (1933) Basic Result
Lovell (JASA, 1963) did the matrix algebra.
Continuing the algebraic manipulation: b2 = [X2’M1X2]
-1[X2’M1y]. This is Frisch and Waugh’s famous result - the “double residual
regression.” How do we interpret this? A regression of residuals on residuals. “We get the same result whether we (1) detrend the other variables
by using the residuals from a regression of them on a constant and a time trend and use the detrended data in the regression or (2) just include a constant and a time trend in the regression and not detrend the data”
“Detrend the data” means compute the residuals from the
regressions of the variables on a constant and a time trend.
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Part 4: Partial Regression and Correlation 4-9/25
Important Implications
Isolating a single coefficient in a regression. (Corollary 3.2.1, p. 37). The double residual regression.
Regression of residuals on residuals – ‘partialling’ out the effect of the other variables.
It is not necessary to ‘partial’ the other Xs out of y because M1 is idempotent. (This is a very useful result.) (i.e., X2M1’M1y = X2M1y)
(Orthogonal regression) Suppose X1 and X2 are orthogonal; X1X2 = 0. What is M1X2?
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Part 4: Partial Regression and Correlation 4-10/25
Applying Frisch-Waugh
Using gasoline data from Notes 3.
X = [1, year, PG, Y], y = G as before.
Full least squares regression of y on X.
Partitioned regression strategy:
1. Regress PG and Y on (1,Year) (detrend them) and compute
residuals PG* and Y*
2. Regress G on (1,Year) and compute residuals G*. (This step
is not actually necessary.)
3. Regress G* on PG* and Y*. (Should produce -11.9827 and
0.0478181.
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Part 4: Partial Regression and Correlation 4-11/25
Detrending the Variables – Pg
Pg* are the residuals from this regression
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Part 4: Partial Regression and Correlation 4-12/25
Regression of detrended G on detrended Pg
and detrended Y
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Part 4: Partial Regression and Correlation 4-13/25
Partial Regression
Important terms in this context: Partialing out the effect of X1. Netting out the effect … “Partial regression coefficients.” To continue belaboring the point: Note the interpretation of partial
regression as “net of the effect of …” This is the (very powerful) Frisch – Waugh Theorem. This is what is
meant by “controlling for the effect of X1.” Now, follow this through for the case in which X1 is just a constant term,
column of ones. What are the residuals in a regression on a constant. What is M1? Note that this produces the result that we can do linear regression on data
in mean deviation form. 'Partial regression coefficients' are the same as 'multiple regression
coefficients.' It follows from the Frisch-Waugh theorem.
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Part 4: Partial Regression and Correlation 4-14/25
Partial Correlation
Working definition. Correlation between sets of residuals. Some results on computation: Based on the M matrices. Some important considerations: Partial correlations and coefficients can have signs and magnitudes that differ greatly from gross correlations and
simple regression coefficients. Compare the simple (gross) correlation of G and PG with the partial
correlation, net of the time effect. (Could you have predicted the negative partial correlation?)
CALC;list;Cor(g,pg)$ Result = .7696572 CALC;list;cor(gstar,pgstar)$ Result = -.6589938
G
1. 00
1. 50
2. 00
2. 50
3. 00
3. 50
4. 00
4. 50
. 50
80 90 100 110 12070PG
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Part 4: Partial Regression and Correlation 4-15/25
Partial Correlation
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Part 4: Partial Regression and Correlation 4-16/25
THE Most Famous Application of
Frisch-Waugh: The Fixed Effects Model
A regression model with a dummy variable for
each individual in the sample, each observed Ti times.
There is a constant term for each individual.
yi = Xi + diαi + εi, for each individual
1 1
2 2
N
=
=
N
1
2
N
y X d 0 0 0
y X 0 d 0 0 βε
α
y X 0 0 0 d
β [X,D] ε
α
Zδ ε
N may be thousands. I.e., the regression has thousands of variables (coefficients).
N columns
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Part 4: Partial Regression and Correlation 4-17/25
Application – Health and Income
German Health Care Usage Data, 7,293 Individuals,
Varying Numbers of Periods
Variables in the file are
Data downloaded from Journal of Applied Econometrics Archive. This is an unbalanced
panel with N = 7,293 individuals. There are altogether n = 27,326 observations. The
number of observations ranges from 1 to 7 per family.
(Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987).
The dependent variable of interest is
DOCVIS = number of visits to the doctor in the observation period
HHNINC = household nominal monthly net income in German marks / 10000.
(4 observations with income=0 were dropped)
HHKIDS = children under age 16 in the household = 1; otherwise = 0
EDUC = years of schooling
AGE = age in years
We desire also to include a separate family effect (7293 of them) for each family. This
requires 7293 dummy variables in addition to the four regressors.
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Part 4: Partial Regression and Correlation 4-18/25
Estimating the Fixed Effects Model
The FE model is a linear regression model with many independent variables
1
1
Using the Frisch-Waugh theorem
=[ ]
D D
b X X X D X y
a D X D D D y
b X M X X M y
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Part 4: Partial Regression and Correlation 4-19/25
Fixed Effects Estimator (cont.)
1 ( )
...
...
D
1
2
N
1 1
2 2
N N
M I D D D D
d 0 0
0 d 0D
0 0 d
d d 0 0
0 d d 0DD
0 0 d d
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Part 4: Partial Regression and Correlation 4-20/25
Fixed Effects Estimator (cont.)
i i i
i i i
i i i
1
1
i
1 1 1T T T
1 1 1T T T
1 1 1T T T
( )
(The dummy variables are orthogonal)
( ) (1/T )
1 - - ... -
- 1 - ... -
... ... ... ...
- - ... 1 -
i i
D
1
D
2
DD
N
D
i
D T i i i T i
M I D D D D
M 0 0
0 M 0M
0 0 M
M I d d d d = I d d
=
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Part 4: Partial Regression and Correlation 4-21/25
‘Within’ Transformations
1
i i i i
i i i i i i i i i ii
i i
N
i=1 i i
T
i i t=1k,l
( ) (1/T )
(1/T ) (1/T ) (T )
(T K) (T 1) * (1 K)
,
i i
i
D T i i i T i
i i
i
D D
i
D
M X I d d d d X = I d d X
X d d X X d x X d x
X M X = XM X
XM X
i
i
it,k i.,k it,l i.,l
N
i=1 i i
T
i i t=1 it,k i.,k it i.k
(x -x )(x -x ) (k,l element)
,
(x -x )(y -y )
i
D D
i
D
X M y = XM y
XM y
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Part 4: Partial Regression and Correlation 4-22/25
Least Squares Dummy Variable Estimator
b is obtained by ‘within’ groups least squares (group
mean deviations)
Normal equations for a are D’Xb+D’Da=D’y
a = (D’D)-1D’(y – Xb) (see slide 5)
iT
i i t=1 it it ia=(1/T)Σ (y - )=ex b
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Part 4: Partial Regression and Correlation 4-23/25
A Fixed Effects Regression Constant terms
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Part 4: Partial Regression and Correlation 4-24/25
What happens to a time invariant
variable such as gender?
iTn n 2
i=1 i i i=1 t=1 it i.
it i it i. it i.
= a variable that is the same in every period (FEMALE)
(f -f )
but f = f , so f f and (f -f ) 0 for all i and t
This is a simple case of multicollinearity.
=
i
D D
f
f M f fM f
fi
diag(f )*D
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Part 4: Partial Regression and Correlation 4-25/25