The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
The linear regression model
Ragnar Nymoen
Department of Economics, UiO
9 January 2009
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Overview
We review the linear regression model,
its interpretation, and
estimation by OLS,
and the properties of OLS estimators.
Main reference is Greene Ch 1-5.5. See teaching plan for
overlapping reference to Biørn and Kennedy
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Model and assumptions
Most of the statistical properties of the linear regression model can
usefully …rst be illustrated for the regression model with only one
explanatry variable. So we start with the simple regression model,
aka the bivariate regression model.
We use the notation in Greene’s book.
As you remember from earlier courses, it is important to
distinguish between the population and the sample. The regression
model we de…ne refer to the population. Greene writes it as
yi = β1 + β2 xi + εi , i = 1, 2, ..., n.
where yi and εi are random (stochastic) variables. xi can take two
interpretations: It can be either a deterministic variable, or a
stochastic variable.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
In the population model:
yi = β1 + β2 xi + εi , i = 1, 2, ..., n.
the parameters β1 and β2 are the coe¢ cients of the model.
Because economic theory contains hypotheses about how changes
in one variable a¤ects another, the main parameter of interest of
the model is the slope co¢ cient β1 , often called the derivative
coe¢ cient (a term that covers the mathematical derivative, the
elasticity etc, depending on functional form (see below). For the
same reason yi is referred to as the dependent variable, or
regressand, and xi as the independent variable, or regressor.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Assumptions of the regression model (Table 2.1 in Greene)
A1 Linearity
A2 Full rank (absence of perfect collinarity)
A3 Exogenity of xi . Meaning that the conditional mean
of the stochastic disturbance term εi which we write
E [εi jxi ] is zero.
A4 Homoscedasticity and non-autocorrelation
A5 Analysis is conditional on the observed xi s. (meaning
that all properties of the model estimators can be
derived as if they are deterministic, even in the
stochastic regressor case.
A6 Normal distribution of the disturbances.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Is clear that A1-A5 is a mix of assumption about the sample (A2)
and about the population (A1, A3-A6). Logically, assumptions
about the regression model should be about the population, hence
A2 could be postponed until estimation of the parameters— but its
listing among the model assumptions has become custom.
Another feature, more worth deliberating, is the close relationship
between A3 and A5: In the light of A5, A3 is implied. Conversely:
When A3 holds also A3 holds automatically.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Exogenous regressors (A3 and A5)
The case of exogeneity of deterministic xi is of course trivial
since the covariance between εi and the …xed deterministic xi
is zero. Then E [εi jxi ] = E [εi ], which is zero under the
assumption that E [εi ] = 0, which in many presentations
replaces A3 (in the case of deterministic regressor).
If both xi and yt are random variables means that they have a
joint probabily density function (PDF) which we denote
f (xi , yi ). For simplicity we assume that PDF is bivariate
normal (see Greene appendix B.9, p 1009).
All the main results below also holds for other distributions
than the normal, with an important exception of
homoscedasticity (A4).
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
The binormal case
The PDF of fxi ,yi g is
f (xi , yi ) =
1
p
σx σy 2π 1
2ρ
where
ρ2
exp[
µx ) (yi
(xi
σx
(xi µx )2
1
f
(1)
2(1 ρ2 )
σ2x
µy )
σy
+
(yi
µy )2
σ2y
g]
∞ < µx , µy < ∞, 0 < σx , σy < ∞ and
ρ=
E [(xi
µx )(yi
σx σy
µy )]
=
σxy
,
σx σy
1 < ρ < 1.
where σxy is the covariance
of xi and yi (denoted Cov [xi , yi ]).
q
p
σx = σ2x and σy = σ2y , where σ2x and σ2y are the marginal
variances of xi and yi .
µx and µy are the marginal means of xi and yi .
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
The marginal PDF of xi is a normal PDF:
f (xi ) = p
(xi µx )2
1
p exp[
]
2σ2x
2π σ2x
(2)
The conditional PDF of yi is given by
f (yi jxi ) =
f (xi , yi )
f (xi )
(3)
and (1) and (2). It it not di¢ cult to show that
f (yi j xi ) = A exp[
1
2σ2y (1
ρ2 )
fyi
where
A= p
2π
q
µy + ρ
1
σ2y (1
σy
µ
σx x
.
ρ2 )
ECON 4610: Lecture 1
ρ
σy
xi g2 ],
σx
(4)
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Next, de…ne the conditional expectations of yi , E[yi j xi ] as:
E [yi j xi ] = µy
ρ
σy
σy
µ + ρ xi
σx x
σx
(5)
and the conditional variance of yi as
Var [yi j xi ] = σ2y (1
ρ2 )
(6)
We then have the following important results:
1
2
3
The conditional PDF of yi given by (4) is a normal PDF.
If yi and xi are correlated, ρ2 6= 0, the conditional mean of yi
given by (5) is a deterministic function of xi . (5) is called the
regression function.
If yi and xi are correlated, ρ2 6= 0, the conditional variance of
yi given by (6) is less than the marginal variance σ2y :
Var [yi j xi ] < σ2y i¤ ρ2 6= 0
ECON 4610: Lecture 1
(7)
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
The regression model in the binormal case
We now de…ne the disturbance εi by
εi = yt
E[yi j xi ],
(8)
which of course is a stochastic variable with a normal distribution.
We obtain the regression model as
yi = E[yi j xi ] + εi = β1 + β2 xi + εi
(9)
where the parameters of the model is linked to the PDF of the
stochastic varaibles in the following way
σy
µ = µy
σx x
σy
σxy
= ρxy
= 2
σxi
σx
β1 = µy
β2
Finally, εi
ρxy
N (0, σ2 ) with σ2 = σ2y (1
σxy
µ
σ2x x
ρ2 ).
ECON 4610: Lecture 1
(10)
(11)
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Above we have derived the population regression model for
the case where the PDF of fxi , yi g is binormal.
The results generalize to the multivariate case where fyi ,x1i ,
x2i ,...,xk g are have a multivariate normal PDF.
It shows that A3, A5 and A6 are inherent model properties is
this case
A3 and A5 hold also for other PDFs, along with more speci…c
σ
results, such as β2 = ρxy σyx (where the moment refer to that
other PDF)
A6 may not hold for other PDFs, the variance of the
disturbance may by non-constant and may also depend on xi .
It is heteroscedastic. With time series data, we may also have
Cov [εt , εt j ] 6= 0 for j = 1, 2, .... This is called
autocorrelation.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Status of the listed assumptions (#1)
Our analysis shows that it is (only) A1, A4 and A6 that are
assumptions about the regression model (for the population).
A2 is an assumption about sample variability, not about the
population.
A3 and A5 are a properties (not assumptions) of the
regression model.
For A3 (exogeneity) this seems paradoxial, since we will spend
much time in this course …nding ways of doing valid
econometric analysis of linear relationships when E[εi j xi ]
does not hold.
The solution is that whenever the parameters of interest for
our study are parameters in the regression function (5), then
the regression model, and the associated estimateion method
OLS, is the appropriate methodology.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Status of the listed assumptions (#2)
Ideally, economic theory should imply when the linear
regression model contains the parameters of interest, and
when another econometric model is needed.
Often, the theory does not take into account the stochastic
nature of the varaibles, and alternative econometric models
then need to be considered with the aid of statistical test of
exogeneity for example. This will be covered later in the
course.
Linearity (A1) is not very restrictive. This is because linearity
of parameters can be retained eventhough the data are
non-lineary transformed prior to modelling.
Gives rise to a wide range of non-linear functional forms, that
are linear in parameters: log-log, and semilog, and reciprocal
functional forms.
It a¤ects the interpretation of the derivative coe¢ cients.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Another assumption— constancy of parameters
Constancy of the parameters of the model cannot be taken for
granted.
In the case of time-series data, when there is a natural
ordering of the variables (from t = 1 to t = T for example) it
is easy to envisage that one or more parameter of the PDF
can change a given point in time.
It is possible to formalize this by working with the PDF of the
whole sequence of variables fy1 ,y2 , ..,yT , x1 , x2 , ...,xT g,
which is called the Haavelmo distribution.
Non-contancy of parameters can be due to regime-shifts
(changes in economic behaviour), and they can invalidate the
model for policy analysis, and damage forecast accuracy. On
the other hand, regime-shifts can help indentify the direction
of causality. We will show this at a later stage.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
The multivariate regression model
yi = β1 x1i + β2 x2i + ... + βk xKi + εi , i = 1, 2, ..., n.
Greene uses K to denote the the number of explanatory variables
variables including the constant term (if x1i = 1 for all i). In
Biorn’s notation, the number of variables is K + 1.
De…ning two n 1 vectors y and ε, and a n K matrix X with all
the variables, and a K 1 vector β with the coe¢ cients, we can
write the model in matrix form:
y = Xβ + ε
The regression function:
E [yi j xi ] = xi0 β = β1 x1i + β2 x2i + ... + βk xKi
where the row vector xi0 are made up of the elements in the i’th
row vector in the X matrix.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Usually, the …rst variable will be speci…ed as a vector of 1’s
(to include an intercept term).
The other K
elasticities.
1 coe¢ cients are partial derivaties or
In matrix notation the assumption/result about the variance
of ε is written as
Var [ε] = E[εε0 ] = σ2 I
where I is the n
n identity matrix.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
OLS estimators
The OLS estimators of the coe¢ cient in β are given by:
0
b = (X X)
1
X0 y
(12)
This presumes that the inverse of the information matrix (X0 X)
exists, which it does when assumption A2 holds.
Invertibility depends on det(X0 X) 6=0. As an illustration, assume
X =
a c
d b
then
det(X0 X) = ab
cd
If b = αd and c = αa then det(X0 X) = 0 and the inverste of X0 X
does not exists. This is a case of perfect multicollinearity, which is
synonymous with reduced rank of X.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Assumption A2 states that X has rank = K . We have seen
that this is an assumption about variability in the population
and sample.
In the simple regression
yi = β1 + β2 xi + εi , i = 1, 2, ..., n.
xi cannot be a constant, and generally no single variable can
be an determinstic linear function of the other variables in X.
A classic example of invadvertely creating reduced rank is
when quarterly seasonal e¤ects are attempted modelled with
four seasonal dummies in a regression model where an
intercept is already included.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Consider the case of K = 3, with an intercept (x1i = 1 for all i)
and two explanatory variables. The OLS estimators for the two
derivative coe¢ cient can be written as
s 2 syx
syx3 sx2 x3
(13)
b2 = x3 2 22
,
sx 2 sx 3 sx 2 x 3
b3 =
sx22 syx3 syx2 sx2 x3
.
sx22 sx23 sx2 x3
where
sx2k
=
1 n
(xki
n i∑
=1
x̄k )2 , k = 2, 3
sx 2 x 3
=
1 n
(x2i
n i∑
=1
x̄2 )(x3i
sx k y
=
1 n
(yi
n i∑
=1
ȳ )(xki
x̄3 )
x̄k ), k = 2, 3
ECON 4610: Lecture 1
(14)
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
(13) and (14) have the same denominator which can be written as
sx22 sx23
sx2 x3 = sx22 sx23 (1
rx22 x3 )
where rx22 x3 is the squared correlation coe¢ cient between x2 and x3 .
2 > 0,
rx22 x3 < 1 (absence of perfect collinearity) and sxk
k = 2, 3 (variability of regressors), ensures the existence of the
estimators.
rx22 x3 = 0 (no collinearity) reduces the expression for b1 and b2
to
syx
syx
b2 = 2 2 , b3 = 2 3
sx 2
sx 3
which are the OLS estimators in the two regressions between
y and x2 and y and x3 . The regressors are then said to be
orthogonal.
If rx22 x3 > 0, estimation of the partial e¤ects of x2 and x3 on y
requires multiple regression (they cannot be estimated by
simple regression).
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Frisch-Waugh theorem
First regress yi on x3i (using OLS), and then regress x2i on x3i
and then regress the residuals from the …rst regression (eyx3 ,i )
on the residuals from the second regression (ex2 x3 ,i ). Let b2
denote the OLS estimate for the derivative coe¢ cients in the
regression between the residuals.
Straight-forward algebra shows that b2
b2 .
This result is due to Frisch and Waugh (1933) and shows that
an alternative to multiple regression is to …rst “…lter out” the
e¤ects of the third variable from both the dependent variable
and from the x2 variable, and then perform simple regression
on the “…ltered variables”.
The correlation coe¢ cient between eyx3 ,i and ex2 x3 ,i is the
partial correlation coe¢ cient between y and x2 , controlling for
the in‡uence of x3 .
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Residuals and multiple correlation
Least squares residuals:
e=y
0
Xβ = My
1
M = I X(X X) X0 is dubbed the “residual maker” by Greene.
Next consider the value of y predicted by the regression, call it ^
y.
^
y=y
0
P = X(X X)
1
e = (I
M)y = Py
X0 is the projection matrix. Note
y =^
y + e = projection + residal
If the variable x1 is a constant then
∑ ei = 0 and ȳ = ŷ
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
We can therefore write (yi
ȳ ) as
= ŷi ei ŷ = xi0 b x0 b+ei
ȳ = (xi x)0 b+ei
yi
ȳ
yi
for the K = 2 case we know ∑ni ei (x2i x2 ) = 0, by virtue of OLS
estimation (1oc) and this generalizes to the multivariate case
∑ni (xi x)0 bei = 0, hence
n
∑ (yi
i =1
motivating
|
ȳ )2 =
{z
SST
}
R2 =
n
∑ (ybi
i =1
|
{z
SSR
SSR
=1
SST
n
ŷ )2 + ∑ ei2
}
i =1
| {z }
SSE
SSE
SST
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Properties of the coe¢ cient of determination
0
R2
1.
R 2 > any simple or partial correlation coe¢ cients.
R 2 is increasing in K , so is not useful for determining K .
For that purpose: use adjusted, R 2
2
R =1
n
n
1
(1
k
R2)
or
do a formal test of signi…cance.
R 2 is not invariant to “trivial changes in the model”.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Unbiasedness
Conditional unbiasedness
For the K variable regression model we have
b = (X0 X )
1
X0 y = (X0 X )
1
X0 [Xβ + ε] = β + (X0 X)
1
X0 ε
meaning that each element in b is the sum of the true coe¢ cient
and a weighted sum of the n disturbances. The weights are
constructs of the 2nd order moments of the explanatory variables in
X. Since ε is random, b is also a stochastic variable— an estimator.
If we condition on a particular set of observations (a particular
realization of all the possible X’s), the conditional expectation of b
is β since
0
E [b j X] = β+E [(X X)
1
X0 ε j X ] = β
Note that we can show the conditional results by thinking as if the
x 0 s were deterministically generated!
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Unbiasedness
Unconditionally
b has expectation β for any given set of observations. Therefore,
when we average over all possible realizations (all possible data
sets) the mean of all the conditional expectations must be β.
Formally we show (unconditional) unbiasedness by the law of
iterated expectations:
E [b] = EX fE [b j X]g = EX [ β] = β
here EX denotes that we take the average over variations of X.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Conditional distribution
If assumption A5 holds, ε has a multivariate normal PDF, a mean
of 0 and a variance equal to σ2 I:
ε
N (0,σ2 I)
Since each bk in b is a linear combination of εi0 s it follows that
0
1
b j X =N ( β,σ2 (X X)
)
The variance is derived from
E [(b
β)(b
β)0
j
X] = E [((X0 X)
0
0
= E [εε ](X X)
1
1
X0 ε)(ε0 X(X0 X)
2
0
= σ (X X)
ECON 4610: Lecture 1
1
.
1
)j X]
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
OLS variance and consistency
We have for bk :
Var [bk ] =
σ2
nsk2 (1
rx22 x3 )
, k = 2, 3
(15)
The higher the degree of multicollinearity, the larger the
variances.
The variances are decreasing functions of n.
If Var [bk ] ! 0 if n ! ∞.
Together with unbiasedness, this implies consistency: plim
b = β.
These results generalize to any number of variables.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
A (…rst) Monte Carlo experiment
We let the computer generate 1000 data series according to
Ci = 54, 5 + 0, 8 Ii + εi , i = 1, 2, ....., n.
C symbolizes consumption, and I income.
We generate the 1000 replications by “drawing from” a
normal PSD for the disturbance εi .
In the experiment we vary sample length, n,and the sample
variance of Ii , and σ2 .
Ii is …rst treated as deterministic, so we …rst look at
conditional properties of OLS.
Then random It .
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Determinstic It
MC-1
n = 47
σ2 = 3
σ̂2X = “high”
MC-2
n = 28
σ2 = 3
σ̂2X = “high”
MC-3
n = 47
σ2 = 1, 5
σ̂2X = “high”
MC-4
n = 47
σ2 = 3
σ̂2X = “low”
E [ β̂2 ]
0.79940
0.80601
0.79958
0.80595
q
0.0616
0.072909
0.04356
0.18622
Var [ β̂2 ]
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
It is stochastic
MC-1
n = 47
σ2 = 3
σ̂2X = “high”
MC-2
n = 28
σ2 = 3
σ̂2X = “high”
MC-3
n = 47
σ2 = 1, 5
σ̂2X = “high”
MC-4
n = 47
σ2 = 3
σ̂2X = “low”
E [ β̂2 ]
0.80192
0.80601
0.80136
0.79731
Var [ β̂2 ]
0.066575
0.072909
0.047076
0.16021
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Equipped with the regression model and a data sample, OLS
estimates are obtained and statistical testing of hypotheses
about the population parameters can be done.
Student-t distributed test statistics: Hypotheses about single
parameters, con…dence/prediction intervals.
χ2 and F distributed test statistics: Joint hypotheses, i.e. H0 :
Rβ = q.(Greene Ch 5.3)
These statistics (and modi…cations of them) also play a role in
testing for functional form and structural break, a point we
shall return to because of its importance for modelling.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Robustness of OLS inference (#1)
Important premises for the above procedure:
That OLS estimators are conditionally normally distributed
The sum of squared and standardized OLS residuals
ei = yi
xi0 b = εi
xi0 (b
β)
has a conditional χ2 distribution:
∑ni=1 ei2
j X χ2 (n K ),
σ2
That all inference is conditional on X is not entirely
convincing— after all the explanatory variables are a mix of
deterministic and random variables, and the sample only give one
realization of the random variables. Inference seeks to generalize
from the sample results to the population and should not depend
on the observations of the x’s that the data generation happened
to deliver.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Robustness of OLS inference (#2)
Heuristically this problem is not so serious. For example, the
t distibution used for testing H0 : βk = β0k
β̂
β0k
qk
Var ( β̂k )
t (n
K)
depends on the degrees of freedom, but not on X.
In this sense, OLS inference is unconditional.
ECON 4610: Lecture 1
(16)
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Robustness of OLS inference— asymptotics
Even if the disturbances are not normally distributed
(departure from A6), or if they are heteroscedastic (departure
from A4), inference based on OLS remains approximately
correct,
and the statistical quality of the inference gets better as the
sample size n increases. Also, b is asymptotically normally
distributed even though it may depart from the normal
distribution for …nite samples.
This builds on the insight of the central limit theorem:
Sequences of averages from non-normal but independent
variables converges in distribution to a normal PDF.
Since OLS estimates takes the form of averages of a similar
nature, the result holds in many cases.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Limitations of the regression model and OLS estimation.
Despite the above, improved inference may still be achieved in
small samples by modifying the estimators and/or the
test-statistics (e.g., correction for heteroscedasticity or
autocorrelation). You will learn about that in this course.
The …rst serious limitation is that the parameters of interest is
not always “in” the regression model.
Then need a di¤erent econometric model. This is also covered
by this course
A second limitation lies in the assumption that the random
variables xk 1 , xk 2 , ...., xkn are independent. This too is not so
serious as it …rst seems, cf Greene’s discussion of “well
behaved data” on page 65:
Moreover OLS perform relatively well even in the case with
“lagged dependent variable” mentioned by Greene at the
bottom of page 73.
ECON 4610: Lecture 1
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Lagged endogenous regressor (#1)
The essence of the model discussed by Greene is:
yt = β2 yt
1
+ εt ,
εt
N (0, 1), t = 1, . . . T .
The problematic feature here is that yt
and all “older” εt ’s.
1 is
correlated with εt
(17)
1
This creates a …nite sample bias in the OLS estimate (the
Hurwicz-bias).
For the simplest
Function Asymptotic Finite sample
E [b2 ]
Var [b2 ]
β2
0
β2
(1
2β2 /(T
β22 )/T
ECON 4610: Lecture 1
1)
The interpretation of the linear regression model
Estimation of the regression model
Properties of the OLS estimator
Inference in the regression model
Lagged endogenous regressor (#2)
Can illustrate these results by Monte Carlo simulation using
PcNaive, which is part of the PcGive family of programs.
The limiting case, when this extension of the OLS inference
breaks down, is β2 = 1. We are then in the realm of
non-stationarity random variables, ie. assumption AD5 on
page 74 in Greene does not hold.
One of the most noted pitfalls of non-stationary is when we
estimate:
yt = β1 + β2 xt + εt .
The standard OLS inference “always” leads to rejection of H0 :
β2 = 0 when H0 is true. Spurious regression.
Solved by another model (not in this course)–cointegration.
ECON 4610: Lecture 1