OLS and simultaneous equations bias
OLS bias due to “measurement errors”
Limitations of regression analysis
Ragnar Nymoen
Department of Economics, UiO
8 February 2009
ECON 4610: Lecture 5
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
Overview
What are the limitations to regression?
Simultaneous equations bias
Measurement errors in explanatory variables
In both cases the explanatory variable is not exogenous in the
econometric sense
Main reference is G Ch 15.1 and 15.2;. B Ch 8.1, 10.1 and
10.2;K: Ch 9.3,10.2
ECON 4610: Lecture 5
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
What are the limitations to regression analysis?
It is not linearity in variables, as we have seen
it is not linearity in parameters, although we have only
covered the linear regression model here
Remember that by …rst estimating the linear model we can use
the results to estimate parameters that are non-linear functions
of the estimated model’s parameters (the “delta method” or
its equivalent in the Bårdsen method)
If the model is non-linear in the parameter from the outset, can
use Non-Linear Least Squares to …t the best non-linear curve
to the data. Greene Ch 11, not in the syllabus to this course.
It si not con…ned to single equation, as we seen with the
SURE estimator.
The real limitation to the regression model is when the
regression function does not contain the parameter of interest
ECON 4610: Lecture 5
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
A simple Keynes model
Let Yt denote GDP in period t D 1, 2, ..., T .
Ct is “endogenous expenditure” and let Xt denote “exogenous
expenditure”.
Assume that Ct depends on GDP, then our example model is
Yt
Ct
D Ct C Xt
(1)
D b1 C b2 Yt C "t , 0 < b2 < 1
(2)
"t is a random disturbance term. We assume that it is white
noise uncorrelated with Xt . For simplicity we assume
normality "t N.0, 2" /.
The parameter of interest is the marginal propensity to
consume b2 .
ECON 4610: Lecture 5
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
The reduced form of the model
(1) and (2) de…nes a simultaneous equations model. Solution for
the two endogenous variables:
Yt
Ct
11
21
b1
1 b2
b1
D
1 b2
D
D
11
C
D
21
C
12
D
21
D
12 Xt
22 Xt
C
1t
(3)
C
2t
(4)
1
1
b2
b2
1 b2
1
1t
D
1
2t
D
1
ECON 4610: Lecture 5
1
b2
b2
"t
"t
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
The distribution of Y and C
The Reduced Form written more compactly
Yt
Ct
D
yt
C
1t
(5)
D
ct
C
2t
(6)
where
1t
2
y
N 0,
2t
cy
cy
2
c
j Xt .
The conditional distributions of the stochastic variables 1t and
are binormal with zero expectations and variance matrix:
2
y
cy
cy
2
c
j Xt .
ECON 4610: Lecture 5
(7)
2t
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
Conditional distribution of C
It follows that Yt and Ct are normally distributed with the same
0
covariance matrix as . 1t
2t / and expectations
yt
D
11
C
ct
D
21
C
12 Xt ,
22 Xt .
It also follows (Lect 1) that the conditional distribution of Ct is
normal with conditional expectation:
E [Ct
j
Yt ] D
D
21
D .
C
c
ct
y
22 Xt
c
21
y
yt
c
y
.
c
C
11
11 / C . 22
Yt
(8)
y
C
12 Xt / C
c
y
12 /Xt
ECON 4610: Lecture 5
c
Yt
y
C
c
y
Yt
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
We see that
The macro model implies (8) as the conditional expextation
for Ct .
It is the valid regression model of Ct on Yt and can be
estimated with full e¢ cency by OLS.
It will not deliver an estimate of the marginal propensity to
consume, b2 !
In sum: The regression function implied by (1) and (2) is (8),
not the regression of Ct on Yt and a constant.
And the regression function (8) is not helpful for the
estimation for the parameter of interest b1 (in fact since
c
D 1 it estimates the identity in this special case) )
y
ECON 4610: Lecture 5
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
Simultaneity bias in the macro model example
Suppose we estimate the consumption function by OLS regardless.
We will estimate “some parameter”. What is it?
P
P
N
N
Ob2 D PCt .Yt Y / D P Ct .Yt Y /
.Yt YN /2
Yt .Yt YN /
P
where YN D 1/T
Yt .
X
1
fb1 C b2 Yt C "t gt .Yt YN / (9)
bO 2 D P
Yt .Yt YN /
P
"t .Yt YN /
D b2 C P
.Yt YN /2
We must evaluate the term
P
in the light of the model.
" t .Yt YN /
P
.Yt YN /2
ECON 4610: Lecture 5
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
Since Yt depends on the shocks "t to consumption, and Ct
depends on Yt , then "t and Yt are correlated.
This correlation will not go away as T grows. Using the RF
expression for Yt , the denominator can be written as
1 X
1 X
2
XN / C . 1t N 1 /
.Yt YN /2 D
12 .Xt
T
T
Take probability limits:
1 X
plim
.Yt YN /2 D
T
1 X 2
XN /2
D plim
12 .Xt
T
1 X
C 2 12 plim
.Xt XN /. 1t N 1 /
T
1 X
. 1t N 1 /2
C plim
T
D 212 Var .Xt / C 2y
ECON 4610: Lecture 5
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
bO 2
plim
b2
D
plim T1
P
"t .Yt YN /
P
plim T1 .Yt YN /2
Cov ."t , Yt /
2
2
12 Var .Xt / C y
D
From the Reduced Form we also have
Cov ."t , Yt / D E ["t
D
2
12 Var .Xt / C
2
y
D
yt ]
D E [" t
1
1
b2
"t ] D
2
"
1
b2
1
1 b2
2
Var .Xt / C
2
"
ECON 4610: Lecture 5
1
1
b2
Var ["t ]
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
The inconsistency of OLS
,
2
"
plim bO 2
b2
D
D
1 b2
1
1 b2
2
2
"
Var .Xt / C
.1 b2 /
.1 b2 / 2"
D Var .X /
2
t
Var .Xt / C "
C1
2
"
The bias is positive
Large variance in Xt relative to "t reduces the biases. But it
does not kill the bias.
The reason is that OLS “assumes the wrong model for Ct ”,
one with Cov .Yt , "t / D 0. It is not here.
ECON 4610: Lecture 5
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
Example with an expectations variable
Assume the simple regression model (in Greene’s notation
again):
yi D 1 C 2 xi C " i , i D 1, 2, ..., n.
(10)
with all the classical assumptions holding.
If xi is an expectations variable that we as econometricians
cannot observe or cannot measure without error, we can still
try to estimate 1 and 2 using the observable (actual) where
xi .
We then need to make assumptions about the properties of
the di¤erence
ui D xi
xi .
ECON 4610: Lecture 5
(11)
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
Assumptions:
ui is random, zero mean, variance
2
u
Cov .ui , "i / D 0
Cov .ui , xi / D 0
Both ui and "i have the classical properties
The model that we estimate becomes:
yi
i
D
1
D "i
2 xi
C
2 ui
C
(12)
i
(13)
But with
E [xi i ] D E [.xi C ui /."i
2 ui /]
D
ECON 4610: Lecture 5
2
2
u
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
OLS gives
O2
and we have plim
P
x/
N
i .xi
b2 D 2 C P
.xi x/
N 2
plim O 2
2
D
plim T1
plim T1
P
P
i .xi
.xi
x/
N
x/
N 2
2
we already have that
2 u goes into the numerator. The
denominator is more work (like in the sim eq case) but intuitively it
must boil down to the sum of the variances of xi and ui , hence
plim ( O 2
2/ D
2
2
u
Var .xi / C
2
u
ECON 4610: Lecture 5
OLS and simultaneous equations bias
OLS bias due to “measurement errors”
plim O 2 D
2
1C
2
u
<
2
if
2
is positive.
Var .xi /
It can be shown that by taking the “inverse regression”, xi on
yi , gives an overestimation, so OLS de…nes a bound around
the true parameter.
Measurement errors in yi : No bias problem, but potential for
heteroscedasticity.
Solution to both classes of bias problems exempli…ed here:
Replace OLS with other estimators. IV, 2SLS as we shall see.
ECON 4610: Lecture 5