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Transcript of Econ1 Mle Notes
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Econometrics I
Contents
1 Resuming introductory econometrics 6
1.1 The basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 OLS estimation of the vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.4 Gauss Markov theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.5 Estimating the variance parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2 The normal linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.1 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.2 Interval estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.3 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
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1.2.4 Asymptotic properties of the OLS-estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3 The general linear statistical model with unknown covariance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.4 Asymptotic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.5 A heteroskedastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.6 Autocorrelation (1. order) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.7 Testing for autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2 Stochastic Regressors 65
2.1 Properties of the OLS estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1.1 Stochastic regressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1.2 The case of independence of explanatory variables and error terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.1.3 The case of partial dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.2 A general stochastic regressor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.3 Testing for autocorrelation in the dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3.1 The hypotheses of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3.2 Alternative test statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.4 Instrumental variable estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.5 OLS under measurement errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.6 Generalized methods of moments (GMM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.6.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.6.2 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.7 Example: Consumption based CAPM (CCAPM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.7.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.7.2 GMM estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
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3 Dynamic Models 101
3.1 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 01
3.1.1 Weak stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.1.2 Stationary ARMA-processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 02
3.1.3 Empirical moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.1.4 Including deterministic components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 05
3.2 Nonstationary stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 05
3.3 Convergence of moments of I(1) variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 10
3.3.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 10
3.3.2 Stochastic orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 13
3.3.3 Sample mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 14
3.3.4 Further convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 15
3.3.5 Random walk with drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.3.6 An illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 17
3.4 Spurious regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 18
3.5 Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 21
3.6 Testing for unit roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 24
3.6.1 Dickey Fuller test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 24
3.6.2 Generalizations of the DF-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 263.6.3 Testing for cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.6.4 Testing for weak exogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 34
3.6.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 35
3.6.6 An additional note on weak exogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 36
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3.6.7 A Monte Carlo Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 39
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Reading list
Banerjee, A., J. Donaldo, J.W. Galbraith, D.F. Hendry (1993): Co-Integration, Error Correction
and the Econometric Analysis of Non-Stationary Data, Chapter 1. Oxford University Press.
Greene, W.H. (2003): Econometric Analysis, Philip Allan.
Hackl, P. (2005): Einfuhrung in die Okonometrie, Munchen.
Hamilton, J.D. (1994): Time Series Analysis, Princeton University Press, Chapters 17 - 20.
Johnston, J., J. Dinardo (1997): Econometrics Methods, McGraw Hill.
Judge, G.G., R.C. Hill, W.E. Griffiths, H.Lutkepohl, T.S. Lee (1988): Introduction to the Theory
and Practice of Econometrics, John Wiley & Sons.
Mittelhammer, R.C., G.G. Judge, D.J. Miller (2000): Econometric Foundations, Cambridge Uni-
versity Press.
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1 Resuming introductory econometrics
1.1 The basic model
1.1.1 Specification
The multiple regression model:
yt = 1 + xt22 + ... + xtKK + et = xt + et, t = 1,.....,T.
In matrix form:
y = X + e,
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where
y =
y1
y2
...
yT
(T1)
, e =
e1
e2
...
eT
(T1)
, =
1
2
...
K
(K1)
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and
X =
1 x12 x13 x1K1 x22 x23 x2K... ... ... . . . ...
1 xT2 xT3 xT K
(TK)
=
x1
x2...
xT
=
x(1) x(2) x(K)
.
Variables:
yt are observed random variablesxtk are observable nonrandom known variables1, 2,....,K are the unknown scalar parameterset
are unobservable random variables
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1.1.2 Assumptions
E[et] = 0, E[e2t ] = 2, covariance E[etes] = 0 for t = s
E[ee] = E
e1e1 e1e2 e1eTe2e1 e2e2
e2eT
... ... . . . ...
eTe1 eTe2 eTeT
=
2 0... 2
...
... ... . . . ...
0 2
= 2
1 0... 1
...
... ... . . . ...
0 1
= 2
IT.
rank(X) = K (variables in X are not perfectly linearly dependent).
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1.1.3 OLS estimation of the vector
Objective function:
S() = ee
= (y X)(y X)
= (y
X)(y
X
)= yy Xy yX + XX= yy 2Xy + XX
is to be minimized:S()
!= 0.
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Note: Xy = yX
Xy =
1 2 K
1 1 1x12 x22 xT2... ... . . . ...
x1K x2K xT K
y1
y2
...
yT
=
1x(1)y + 2x
(2)y + + Kx(K)y
= yX.
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Moreover
XX =
1 2 K
x(1)x(1) x(1)x(2) x(1)x(K)
x(2)x(1) x(2)x(2) x(2)x(K)
... ... . . . ...
x(K)x(1) x(K)x(2) x(K)x(K)
1
2
...
K
= 1x(1)x(1)1 +
1x
(1)x(2)2 + + 1x(1)x(K)K
+ 2x(2)x(1)1 +
2x
(2)x(2)2 + + 2x(2)x(K)K
+ ... ... ... ...
+ Kx(K)x(1)1 +
Kx
(K)x(2)2 +
+ Kx
(K)x(K)K.
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Minimizing the objective function
First order partial derivatives:
S()
1=
(yy)
1 (2
Xy)1
+(XX)
1= 2x(1)y + 2x(1)X,
S()
2=
(yy)
2 (2Xy)
2+
(XX)
2=
2x(2)y + 2x
(2)X,
... ... ... ... ...
S()
K=
(yy)
K (2
Xy)K
+(XX)
K= 2x(K)y + 2x(K)X,
or, in compact form
S()
= 2Xy + 2XX!
= 0.
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LS-Estimator:
(XX)1XXb = (XX)1Xy
b = (XX)1Xy.
e = y XbXe = Xy
XXb
= 0
Properties of OLS estimator: E[b] = E[(XX)1Xy]
= E[(XX)1X(X + e)] = .
Cov[b] =
2
(XX)1
.
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1.1.4 Gauss Markov theorem
b is BLUE Best Linear Unbiased Estimator
Comparing linear unbiased estimators:
Let = Ay
denote a general linear estimator with E[] = .
Then b is preferred to if:
Var(ab) Var(a) for all vectors a.
In this case Cov[] Cov[b], b, is positive semidefinite.
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1.1.5 Estimating the variance parameter
E[ee] = E[e21] + E[e22] + . . . + E[e
2T] = T
2.
However, e is unknown. e is available from OLS-regression
e = y Xb
= y X(XX)1
Xy= [I X(XX)1X](X + e)= [I X(XX)1X]e = Me.
Properties of M:
MM = M M is idempotent, M = M symmetric.
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Then ee = eMe and
E[ee] = E[eMe] = E[tr(eMe)]
= E[tr(eeM)] = tr[2M]
= (T K)2.
2
=
ee
T K,E[2] = 2.
Note:
tr(AB) = tr(BA), tr(A + B) = tr(A) + tr(B), E[tr(A)] = tr(E[A]).
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1.2 The normal linear model
1.2.1 Maximum likelihood estimation
Up to now:
y = X + e, E[e] = 0, E[ee] = 2IT,
et (0, 2).
Additional assumption: et N(0, 2)
e N(0, 2IT), yt N(xt, 2), y N(X, 2IT).
How does additional information affect estimation of and 2?
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ML-Estimation (An Illustration)
-
6
-
6
-
6
-
6
0 0
0 0
1
2
z
f(z, 2z) f(0, 2z)
f(12
, 2z)
maxtTt=1 f(zt, z, 2z)2t known
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Density ofet or yt, respectively:
f(et|2) = f(yt|xt,, 2)= (22)
1/2exp
(yt x
t)
2
22
.
Joint density:
f(y1, . . . , yT|X, , 2) = f(y1
|x1,,
2)f(y2|x2,,
2)
f(yT|xT,,
2)
f(y|X, , 2) = ( 22)T/2 exp
(y X)(y X)
22
.
Likelihood function
L(2,
|y, X) = (22)T /2 exp
(y X)(y X)2
2 .
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Maximum Likelihood estimator: Find , 2 such that L(2,
|y, X) is maximized.
Log likelihood function
l(, 2|y, X) = l n L(2,|y, X)=
T
2
ln(2)
T
2
ln(2)
(y X)(y X)
22
.
Maximizing l(, 2|y, X) with respect to is equivalent to minimizing (y X)(y X),thus,
= b = (XX)1Xy.
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Moreover,
E[] = , Cov[] = 2(XX)1
N(, 2(XX)1).
Estimating 2
l
2 = T
22 +(y
X)(y
X)
24!
= 0
T2
=(y X)(y X)
4
2 =(y X)(y X)
T=eeT
=eeT
=(T K)2
T.
Expectation: E[2] = E eeT = (TK)2T .
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1.2.2 Interval estimation
Consider single linear combination of , R1, where R1 is 1 K vector.
Then
R1 N(R1, 2R1(XX)1R1)
andR1 R1
2R1(XX)1R1 N(0, 1)
is a so-called pivotal quantity.
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Thus
Prob
z/2 R1 R1
2R1(XX)1R1 z1/2
= 1 ,
where z is the -quantile of the Standard Normal distribution, z/2 = z1/2.
Rearranging yields:
Prob(R1 z1/2
2R1(XX)1R1 R1 R1 z/2
2R1(XX)1R1) = 1 .
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However, 2 is not known, but 2 is available and independent of .
Replace 2 by 2:
R1 R12R1(XX)1R1
=
1
2(R1 R1)
12
2R1(XX)1R1
=
R1R1
2R
1(X
X)
1R
12
2(TK)(TK)
N(0, 1)2(TK)
TK
t(T K).
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1.2.3 Hypothesis testing
Test:
Rule to decide if parameter(s) of interest lie in a subspace of admissible parameter(s).
H0 : R = r vs. H1 : R = r.
Errors:
Type I: Reject H0 although H0 is true: P(H0|H0) = ,
Type II: Accept H0 although H1 is true: P(H0|H1) = .
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Testing strategy
fix determine test-statistic with known distribution under H0 determine critical region compute test statistic
decide.
Example:
H0 : R = r vs. H1 : R = r.
Recall
R N(R, 2R(XX)1R).
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Under H0
Q1 =(R r)[R(XX)1R]1(R r)
2 2(J),
Q2 =(T K)2
2 2(T K),
and Q1 and Q2 are independent.
Therefore
=Q1/J
Q2/(T K)=
(R r)[R(XX)1R]1(R r)J2
F(J, T K).
Decision:
Reject H0 whenever is too large relative to F(J, T
K).
Note: If J = 1 t2(T K).
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1.2.4 Asymptotic properties of the OLS-estimator
BASIC Concepts:
Convergence in probability
The sequence of random variables z1, z2, . . . , zT, . . . converges in probability to the random variable
z, if for all > 0
limT
Prob[|zT z| > ] = 0.
z is called the probability limit of zT
, short plim zT
= z or zT
p
z. Plim calculus: Consider two
sequences of random variables zt and wt having plim(zt) = z and plim(wt) = w. Then plim(wtzt) = wz
and plim(g(zt)) = g(plim(zt)) = g(z) ifg is a continuous function.
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Convergence in distribution
A sequence of random variables z1, z2, . . . , zT, . . . with distribution functions
F1(z), F2(z), . . . , F T(z), . . . converges in distribution to the random variable z, if at all continuity
points ofF and for every > 0, there exists T0 such that
|FT(z) F(z)| < for T > T0.
Short zTd z.
Central limit theorem I (Lindeberg-Levy)If w1, w2, . . . , wT are independent and identically distributed random variables with mean and
variance 2, then
FZT(z)d
N(0, 1), where ZT =w
/T.
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Central limit theorem II (Lindeberg-Feller)
Let w1, w2, . . . , wT be independent random variables with finite means t and variances 2t . Then
T( wT T) d N(0, 2)
if
limT
max(t)
T = 0 and lim
T2T =
2.
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Finite sample properties of OLS-estimator:
Ifet N(0, 2): b and 2 are BUE.Ifet N(0,
2):
b is BLUE not BUE
2 is not BUE
b N(, 2(XX)1), (TK)22
2(T K).
Issues:
Consistency ofb
Inference under et
(0, 2).
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Basic assumption:
limTXX
T
= Q finite nonsingular matrix.
Counterexamples:
1. Trending explanatory variables
yt = 1 + 2t + et
Then XX = T (T + 1)(T2 )
(T + 1)(T2 ) (T + 1)(T)(2T+1
6 )
and
limTXX
T = 1 .
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2. Vanishing explanatory variables
yt = 1 + 2t + et, (|| < 1),
Note: + 2 + 3 + . . . + T + T+1 + T+2 + . . . =1
1 1.
implying + 2 + . . . + T =1
1
1 T+1 11
=1 1 + T+1
1
.
Thus
XX =
T Tt=1 tTt=1
tT
t=1 2t
= T T+11
T+11
22(T+1)12
.and
limXXT = 1 0
0 0 (singular in the limit).
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Consistency of OLS estimator
E
XeT
= 0.
VarT
Xe
T
=
1
T2Var[Xe]
=1
T22(XX)
= 1T
2XXT 0.
Thus
plim
Xe
T
= 0.
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plim(b) = plim[(XX)1Xy]
= + plim
XX
T
1Xe
T
= + plim
XX
T
1plim
Xe
T
= + Q10 = .
plim(2) = plim
e(I X(XX)1X)e
1
T K
= plimee
T eX
T XX
T 1 Xe
T TT K= 2.
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Interval estimation and hypothesis testing:
bp is not useful since is a degenerate random variable.
Consider
Var
Xe
T
=
1
T(XX)2 = Q2 finite even if T .
Then
XeT
d N(0, 2Q),
and,
T(b ) =
XXT
1Xe
T.
Var[
T(b
)] =
XX
T 1 XX
T XX
T 1
2 = 2Q1.
Thus
T(b ) d N(0, 2Q1).
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Moreover,
T(Rb R) d N(0, 2RQ1R),
T(Rb R)[RQ1R]1(Rb R)2
d 2(J).
Replace Q1 by
XXT
1, 2 by 2 and obtain:
=(Rb R)[R(XX)1R]1(Rb R)
2d 2(J).
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1.3 The general linear statistical model with unknown covariance
matrix
In practice is often unknown. Then GLS-estimation is not feasible. Estimated GLS (EGLS):
is replaced by an estimator
= (X1X)1X1y.
Remark:
has T(T+1)2 unknown elements that cannot be estimated from T error estimates e.
Further assumption on the structure of : Elements in are functions of a small number of unknown
parameters.
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Properties of
: is not linear,
in many cases one can show that E[] = ,
Cov[] cannot be determined,
is not BLUE.
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1.4 Asymptotic results
Assume
limT
X1X
T
= V, finite, nonsingular.
Then
plim() =
plim(2g) = 2
T( ) d N(0, 2V1).
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Asymptotic properties of EGLS compared with GLS
and have the same asymptotic distribution if
plimX(1 1)X
T= 0
and plimX(1 1)e
T= 0.
Moreover,
if plime(1 1)e
T= 0
then
2g
is consistent estimator for 2.
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1.5 A heteroskedastic model
If E[e] = 0 and E[ee] = 2 = , where the diagonal elements of are not identical,
error terms are heteroskedastic.
Consider a model: yt = xt + et, E[et] = 0 and E[e
2t ] =
2t .
Then
E[ee] =
21 0 . . . . . . 0
0 22 0 . . . 0
... ... . . . . . . ...
0 . . . . . . . . . 2T
= .
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Specification of 2t :
2t = 2x2t ,
2t =
2I for t = 1, . . . , T 12II for t = T1 + 1, . . . , T 2t = exp(zt
).
GLS-Estimator:
= (X1X)1X1y
= (XX)1Xy,
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with
X =
11 0 . . . . . . 0
0 12 0 . . . 0... ... . . . . . . ...
0 . . . . . . . . . 1T
x1
x2...
xT
,
y =
11 0 . . . . . . 0
0 12 0 . . . 0... ... . . . . . . ...
0 . . . . . . . . . 1T
y1
y2
...
yT
.
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Multiplicative heteroskedasticity:
2t = exp(zt)
= exp(1 + zt2 2 + . . . + ztS S)
= exp(1)exp(z
t2
2)
exp(ztS
S
)
= 2exp(zt22) exp(ztSS)= 2exp(zt
).
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E[ee] =
=
exp(z1) 0
. . .
0 exp(zT)
= 2
exp(z1) 0
. . .
0 exp(zT)
= 2.
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The estimated GLS-estimator:
= (X1X)1X1y
= (X1X)1X1y
= (XX)1Xy
= exp(zt)xtxt
1
exp(zt)xtyt.
= (X
1X)X
1y
=
exp(zt)xtxt
1exp(zt)xtyt
.
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Estimation of () 2t = exp(zt)
ln 2t = zt
.
Consider
ln e2t + ln 2t = zt
+ ln e2t
ln e2t = zt + ln
e2t2t
= zt
+ vt.
Compactly
q = Z + v
= (ZZ)1Zq.
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Properties of
= + (ZZ)1Zv The elements in v are serially correlated, E[vt] = 0, vt are heteroskedastic.= finite sample properties of cannot be determined
Assume
lim TXX
T= Q
lim TXX
T= V0
Q and V0 are finite and nonsingular.
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Thene = (I
X(XX)1X)e, et = et
x
t
(XX)1Xe.
E[et et] = E[ xt(XX)1XE[e]] = 0E[(et et)2] = [xt(XX)1XX(XX)1xt]
= 2
1
Txt
XX
T
1XX
T
XX
T
1xt
0 for T .
Therefore etp
et etd
et. Ifet N(0, 2t ), thenvt = ln
e2t2t
d vt = ln
e2t2t
and
e2t2t
2(1), E[vt ] = 1.2704, Var[vt ] = 4.9348.
Thus plim() =
1 1.2704
2
...
S
=
1.2704d;
T(
+ 1.2704d)
d
N(0, 493481ZZ)
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Diagnosing heteroskedasticity (variance shifts)
H0 : 2I =
2II =
2 vs. H1 : 2I = 2II
F-Test
=2I2II
H0 F(T1 K, T T1 K),
where 2I and 2II are LS variance estimators obtained from two separate regressions performed for
two subsamples. Reject H0 if is either to small or to large w.r.t. F(T1 K, T T1 K).
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Lagrange Multiplier (LM) Test
Perform regression under H0 and obtain residuals et
Regress e2t on a constant and Dt, a dummy variable Dt =
1 if t I
0 if t II T R2 of the latter regression is 2(1) distributed under H0.
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1.6 Autocorrelation (1. order)
The general model with 1. order autocorrelation:
yt = xt + et, et = et1 + vt, || < 1,
where
xt = (xt1, xt2, . . . , xtK),
E[vt] = 0
E[v2t ] = 2v
E[vtvs] = 0, t=s.
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Covariance structure in matrix form (E[ee] = 2 = )
et = vt + vt1 + 2vt2 + . . .
=
i=0
ivti.
Var[et] = E[e2t ]
= 2E[e2t1
] + 2E[et1
vt] + E[v2
t]
=2v
1 2 .
E[etet1] = E[e2t1] + E[vtet1]
= 2v
1 2.
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General
E[etets] = s2v
1 2 .
Autocorrelation
E[etets]E[e2t ]E[e
2ts]
=s
2v
122v
12
= s.
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1 =
1 0 . . . . . . 0 1 + 2 0 . . . 0
0 1 + 2 0 ...... ... . . . . . . . . . ...
..
.
..
.
..
. 1 + 2
0 . . . . . . . . . 1
= PP,
where P =
1 2 0 0 . . . 0 1 0 . . . 0
0
1 . . . 0
... . . . . . . ... ...
0 . . . . . . 1
,
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Transformation of the model:
y =
1 2y1
y2 y1...
yT yT1
X =
1 2
1 2x12 . . .
1 2x1K
1 x22 x12 . . . x2K x1K... ... . . . ...
1 xT2 x(T1),2 . . . xT K x(T1),K
.
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Estimation of
Ifet were known could be estimated from the autoregression
et = et1 + vt.
Since plim(et) = et a possible estimator for is
= Tt=2 etet1Tt=2 e
2t1
.
Iterative EGLS: Cochrane-Orcutt Procedure
0
0
e0
1
1
e1
. . . until the convergence of i or
i
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1.7 Testing for autocorrelation
Durbin-Watson Test for 1. order autocorrelation:
d =
T2 (et et1)2T
1 e2t
=eAeee
=eMAMeeMe
,
A =
1 1 0 . . . . . . 01 2 1 0 . . . 00 1 2 1 0 ...... ... . . . . . . . . . ...
.
.....
.
.. 1 2 10 . . . . . . . . . 1 1
.
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Relation between d and :
d =
T2 e
2t 2
T2 etet1 +
T2 e
2t1T
1 e2t
=
T1 e
2t +
T1 e
2t e21 e2T 2
T2 etet1
T1 e
2t
= 2 e21 + e
2TT
1 e2t
2T2 etet1T1 e
2t
d 2 2, 1 12
d
The distribution of d under H0 : = 0 is difficult to determine:
MAM is not idempotent eMe is not independent ofeMAMe
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The distribution ofd is between a known upper (dU) and lower (dL) distribution which depend only
on T and K.
H0 : = 0 H1 : > 0
reject H0, ifd < dL, Prob(dL < dL) = 0.05, accept H0, ifd > dU, Prob(dU > dU) = 0.05,
the test is inconclusive if dL < d < dU.DW-test requires:
et normally distributed X fixed, including a constant
only first order autocorrelation
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Reading list
Banerjee, A., J. Donaldo, J.W. Galbraith, D.F. Hendry (1993): Co-Integration, Error Correction
and the Econometric Analysis of Non-Stationary Data, Chapter 1. Oxford University Press.
Greene, W.H. (2003): Econometric Analysis, Chapter 18. Philip Allan.
Hackl, P. (2005): Einfuhrung in die Okonometrie, Munchen.
Hamilton, J.D. (1994): Time Series Analysis, Princeton University Press, Chapter 17, 18, 19 and
20.
Judge, G.G., R.C. Hill, W.E. Griffiths, H.Lutkepohl, T.S. Lee (1988): Introduction to the Theory
and Practice of Econometrics, Chapter 13, 14, 15 and 19. John Wiley & Sons.
Johnston, J., J. Dinardo (1997): Econometrics Methods, Chapter 7, 8 and 9. McGraw Hill.
Mittelhammer, R.C., G.G. Judge, D.J. Miller (2000): Econometric Foundations, Cambridge Uni-
versity Press.
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2 Stochastic Regressors
2.1 Properties of the OLS estimator
2.1.1 Stochastic regressors
Till now, the regressors are assumed to be fixed or nonstochastic in repeated samples.
In economics, these variables are often not perfectly controlled i.e. they may be stochastic.
There is a need to check to which extend the properties of the OLS estimator change
if the regressors are stochastic.
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Consider the model:
y = Z + e,
where Z contains stochastic explanatory variables, E[e] = 0, E[ee] = 2IT.
OLS-estimator:
b = (ZZ)1Zy may not be unbiased
E[b] = + E[(ZZ)1Ze].
Note:
E[(ZZ)1Ze] = (ZZ)1Z E[e]
in general.
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2.1.2 The case of independence of explanatory variables and error terms
LS estimator is unbiased:
E[b] = + E[(ZZ)1Ze]
= + E[(ZZ)1Z]E[e]
= .
Independence of Z and e implies
E[e|Z] = E[e] = 0E[ee|Z] = E[ee] = 2IT.
Moreover: If w and v are random variables then E[w] = Ev[E[w|v]].
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Therefore,
Cov[b] = E[(ZZ)1ZeeZ(ZZ)1]
= E[E[(ZZ)1ZeeZ(ZZ)1] |Z]= E[(ZZ)1Z2ITZ(ZZ)1]
= 2
E[(ZZ)1
].
E[2] = E[E[2 |Z]] = E[E[(T K)1 ee |Z]]= E[(T K)1E[ee |Z]]= E[(T K)1(T K) 2]= 2.
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Finally
E[2(ZZ)1] = E[E[2(ZZ)1 |Z]]= E[2(ZZ)1]
= 2E[(ZZ)1].
Summary:
b, 2, 2(ZZ)1 are unbiased estimators b is no longer BLUE b is still efficient (conditional on Z).
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2.1.3 The case of partial dependence
Consider:
yt = xt + yt1 + et
yt1 = xt1 + yt2 + et1, t = 1, . . . , T, y0 given.
Compactly:
y = Z + e.
Assumptions:
plimeeT
= 2, plimZZ
T= zz, plim
ZeT
= 0.
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2 is consistent:
plim 2 = plim(T K)1ee= plim[T(T K)1](2)= plim[T(T
K)1]plim(2)
= 2.
Summary:
In the case of partial dependence the classical OLS properties hold asymptotically.
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2.2 A general stochastic regressor model
Example: The AR(1) model with an autocorrelated error:
yt = yt1 + et,
et = et1 + vt, || < 1,
where by assumption E[vt] = 0, E[v2t ] =
2v, E[vtvs] = 0 for t = s .
Then
E[et] = 0 and E[e2t ] =
2v(1 2).
Note: vt is not correlated with yt1 but et and the stochastic regressor yt1 are dependent.
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Then,
E[etyt1] = E[(et1 + vt)yt1]
= E[et1yt1] + E[vtyt1]
= = [2v /(1 2)]
i=0ii
= 0.
plim(b) = + plim
ZZ
T
1plim
ZeT
= .
The OLS estimator is biased.
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2.3.2 Alternative test statistics
1. The Durbin Watson test
d =
Tt=2(et et1)2T
t=1 e2t
is invalid due to lagged dependent variables.
2. Likelihood-ratio test (LR)
=L(, , = 0|.)
L(, , |.)
=(2)T /2(2e )
T /2 exp(0.5ee/2e)(2)T /2(2v)T /2 exp(0.5vv/2v)
=
2e2v
T /2,
LR = 2ln() = T(ln(ee) ln(vv))d
2
(1).
3. Lagrange Multiplier test (LM)
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Denote = (,,) and d() = ln L(|.) .
Then:
E[d() ] = 0
Cov[d()] = E[d()d()] = E
2 ln L(|.)
= I() (Informationmatrix).
Denote further d(0) = d()|=0 and I(0) = I()|=0.Then,
LM = d(0)I(0)1d(0)
d 2(1).
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Note: ln L = lt(, 2 | y Z) Max!
lt = ln
1
22
1
22(v2t )
= ln
1
22
1
22
yt ( + )yt1 + yt2 xt + xt1
2.
First order partial derivatives
lt
= 12
(yt ( + )yt1 + yt2 xt + xt1)(yt1 + yt2 + xt1)
=1
2vt(yt1 yt2 xt1) = 1
2vtet1,
lt =
1
2 vt(yt1 + yt2) =1
2 vtyt1,lt
= 12
vt(xt + xt1) = 12
vtxt .
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Second order partial derivatives
2lt2
= 12
et1(yt1 + yt2 + xt1) = 12
et1et1,
2lt2
=1
2yt1(yt1 + yt2) =
1
2yt1y
t1,
2lt2
=1
2xt (xt + xt1) =
1
2xt x
t ,
2lt
=1
2yt1(xt + xt1) =
1
2yt1x
t ,
2lt
=1
2{et1(yt1 + yt2) vtyt2} = 1
2(et1yt1 + vtyt2),
2lt
=1
2{et1(xt + xt1) vtxt1} = 1
2(et1xt + vtxt1).
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In matrix form:
l
=
1
2ve,
l
= 12
(Z Z)v = 12
Zv,
2l
2= 1
2ee,
2l
= 1
2(eZ
+ vZ), Z = Z Z,
2l
= 12 ZZ.Compactly,
l
=
1
2 ev
Zv , 2l
= 1
2 ee e
Z
+ vZ
Ze + Zv ZZ .
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Then,
LM =1
2
evZv
2 E[ee] E[eZ + vZ]E[Ze + Zv] E[ZZ]
1 12
evZv
.
Under H0: = 0 or et = vt
LM = 12 ee
Ze
ee eZZe ZZ
1 eeZe
d 2(1)
with
T explained variation
observed variation = T 1
2 ee
Ze = e(e Z)
ee T = T R2
e,[Ze].
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An equivalent version of the LM-statistic:
Perform an OLS regression under H0 and obtain residuals et Run an auxiliary regression of et on xt, yt1 and et1 LM is (numerically) equal to T-times the R2 of the auxiliary regression.
4. Durbins h
h = T
1 T 2
, =T
t=2 etet1Tt=2 e
2t1
,
d N(0, 1)
where 2
is the usual variance estimator for in the dynamic model yt = xt + yt1 + et.
It can be shown that h2 approaches LM asymptotically h2d
LM.
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6
-
ln L()d ln L() |d
c(
)ln L
ln LR
0
Likelihoodratio
R MLE
Maximum Likelihood Estimation
ln L()
LangrangeMultiplier
6
?
6
?
d ln L() |d
c ()
6
?
Wald
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2.4 Instrumental variable estimation
Consider a general stochastic linear regression model
y = Z + e.
Assume a (T K)-matrix X of instrumental variables such that:
(i) plimXe
T
= 0, (ii) plimXZ
T
= xz, (iii) plimXy
T
= xy.
Then
Xy = XZ + Xe Xy
T=
XZT
+Xe
T.
Taking the probability limit xy = xz + 0.The instrumental variable estimator is therefore biv = (X
Z)1Xy.
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Properties of instrumental variable estimator:
biv is consistent because
plim biv = + 1xz 0 =
If XeT
N0, 2plim(XXT ) thenT(biv )
d
N0, 2plimXZT 1XXT ZXT
1 .Remarks:
biv is mostly not efficient cov[biv] is the smaller the more X and Z are correlated.
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2.5 OLS under measurement errors
Let z, y be vectors of scalar unobservable variables. Then, the observed or measured values of these
variables can be given as z = z + u
y = y + v,
where u and v are (T 1) error vectors.Assumptions:
(i) E[u] = E[v] = 0, (ii) E[uu] = 2uI, (iii) E[vv] = 2vI, (iv) E[uv
] = 0
Economic relationship (scalar case) y = z
or in terms of observed variables y = z+ v u= z+ e.
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Properties of the stochastic vector e:
E[e] = 0 E[ee] = 2vI + 22uI.
The explanatory variables in z and the error terms in e are not independent:
E[(z E[z])(e E[e]) ] = E[u(v u)]= E [uu]= 2uIT.
Bias of OLS-estimator
plim b = 2
u2z
, 2z = Var[zt].
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2.6 Generalized methods of moments (GMM)
2.6.1 Estimation
The model:
y = Z + e, yt = zt + et,
and E[et|xt] = 0, where xt is L 1, L K vector of instruments.Note:
The nonlinear case E[yt|zt] = f(zt,) is analogous.
No assumptions are made for the covariance structure ofe, heteroskedastic and/or autocorrelated
error terms are not excluded.
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As special cases the model covers:
Instrumental variable case, E[ee] = I2
Standard regression model E[ee] = I2, xt = zt.
E[et|xt] = 0 implies E[xt(yt zt)] = 0 and, thus,
E1
T
X(y
Z) = E1
T
T
t=1 xt(yt zt) = E[ m()] = 0.Empirical moment conditions
1
TX(y Z) =
1
T
Tt=1
xt(yt zt)
= [ m()] = 0 in case L = K.
A sensible Method of Moments estimator will choose to minimize
q = m()m().
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Consider three cases:
1. Underidentification: L < K, The number of unknowns exceeds the number of orthogonality
conditions. A solution for does not exist.
2. Exact identification: L = K, The number of unknowns is equal to the number of orthogonality
conditions MM = (XZ)1Xy, m(MM) = 0.
3. Overidentification: L > K, The number of unknowns is smaller than the number of orthogonalityconditions. Then,
q
= 2
m(MM)
m(MM) = 2G(MM)m(MM)
= 2
1
TZX
1
TXy 1
TXZMM
= 0
MM = [(ZX)(XZ)]1
(ZX)Xy, m(MM) = 0.
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The MM estimator is consistent if
plim m() = 0 (convergence) plim G() = plim m()
= plim (1TX
Z) = xz exists
and has rank greater or equal to K (identification).
The MM estimator is asymptotically normal if
Tm()
d N(0, V),
and V denotes the asymptotic covariance of
Tm().
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Covariance of MM:
Cov[] = 1T
[xzxz]1(xzVxz)[
xzxz]
1
= T[(ZX)(XZ)]1(ZX)V(XZ)[(ZX)(XZ)]1, where
standard regression V = 2 X
XT
heteroskedasticity V = 1TT
t=1 e2t xtx
t
general V = 1TTt=1 e2t xtxt +Pp=1Tt=p+1 1 pP+1 etetp(xtxtp + xtpxt).
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General minimum distance estimators (MDE)
min
q = m()Wm(),
where W is positive definite matrix.
MDE = [(ZX)W(XZ)]1(ZX)W Xy
Cov[MDE] =
1
T[xzWxz]1xzW V Wxz[xzWxz]1= T
[ZXW XZ]1ZX W V W X Z[ZXW XZ]1
.
The MM estimator discussed above uses W = I. In case of exact identification the choice of W is
irrelevant. In case of overidentification the efficient MDE is obtained when using W = V1. This
estimator is the GMM estimator:
min
q = m()V1m()
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GMM = [(ZX)V1(XZ)]1[(ZX)V1(Xy)]
Cov[GMM] = 1T[xzV1xz]1
= T[(ZX)V1(XZ)]1.
In practice GMM will be obtained in two steps: In the first step W = I is used to obtain (initial)
estimates for V.
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2.6.2 Testing
1. Testing overidentifying restrictions (model checking in case L > K).
Since
Tm()d N(0, V)
= T q
= Tm()V1m()
d 2(L K)
under the null hypothesis of correct specification.
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2. Testing parameter restrictions
H0 : = 0 R = r vs. H1 : R = r.
LR: SinceT q0 = Tm
(0)V1m(0)
d 2(L (K J))
= T(q0 q) d 2(J)
under H0.
Wald:
= T(RGMM r)[R[(ZX)V1(XZ)]1R]1(RGMM r)d
2(J)
under H0.
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LM: Define
g0(0) =q
=0 = 2G0(0)V1m(0)Note:
Cov[g0(0)] =4
TG0(0)
V1G0(0)
Let m0, G0 be short for m(0), G0(0).
LM-statistic
= Tm0V1G0[G0V
1G0]1G0V1m0
d 2(J)
under H0.
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2.7 Example: Consumption based CAPM (CCAPM)
2.7.1 The model
In a single good economy, a representative agent, who has endowment Wt in period t and may invest in
a single asset with price Pt and dividend Dt+1 at time t+1, maximizes his utility flow from consumption
(Ct) by choosing the optimal investments t in the asset in time t. Formally,
maxt U(Ct) + Et[U(Ct+1)]
s.t Ct = Wt Ptt and Ct+1 = Wt+1 + (Pt+1 + Dt+1)t,
where denotes the (subjective) time discount factor.
From FOC, PtU(Ct) = Et[U(Ct+1)(Pt+1 + Dt+1)],
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the fundamental asset pricing equation is obtained as: 1 = Et[Mt+1Rt+1], where
Rt+1 = Pt+1 + Dt+1
Pt, Mt+1 = U
(Ct+1)U(Ct)
.
With a log utility function:
U(Ct) = ln(Ct), Mt+1 = Ct
Ct+1.
2.7.2 GMM estimation
Data: Quarterly data from 1965:1 to 2002:1 Ct, Germany real consumption; Pt, DAX price index.Lucas tree model:
et+1 = Ct
Ct+1
Pt+1 + Dt+1Pt
1.
Instruments: a constant, lag 1 of consumption, lag 1 of DAX.
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2 Dynamic models
2.1 Stochastic processes
2.2 Nonstationary stochastic processes
2.3 Convergence of moments of I(1) variables
2.4 Spurious regression
2.5 Cointegration
2.6 Testing for unit roots
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3 Dynamic Models
3.1 Stochastic processes
3.1.1 Weak stationarity
In econometrics an observed time series yt, t = 1, . . . , T , is regarded as one realization of a finite part
of a stochastic process yt(), t = 0, 1, 2, . . .. A process is called weakly stationary if first andsecond order moments of the process exist and are invariant through time. Thus,
E[y1] = E[y2] = E[y3] = . . . = E[yT] = ,
Var[y1] = E[(y1 )(y1 )] = Var[y2] = . . . = Var[yT] = 0,Cov[y1+k, y1] = E[(y1+k )(y1 )] = Cov[y2+k, y2] = . . . = Cov[yT, yTk] = k.
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3.1.2 Stationary ARMA-processes
1. Autoregressive models:
yt = et + yt1, || < 1, (AR(1)) (1 L)yt = et, Lyt = yt1,
yt = (1 L)1et.yt = et + 1yt1 + . . . + pytp, (AR(p)),
where
(z) = ( 1 1z . . . pzp)= (1
1z)(1
2z)
. . .
(1
pz)
= 0 for
|z
|< 1.
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Moments of an AR(1)-process:
E[yt] = 0,
Var[yt] = 2(1 + 2 + 4 + . . .) = 0,
Cov[yt, ytk] = k
= k1, k > 0.
2. Moving average model:
yt = et + met1 (MA(1))
= (1 + mL)et.
3. ARMA(p,q) model: (L)yt = m(L)et.
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3.1.3 Empirical moments
Mean = y =
1
T
Tt=1
yt
Variance0 =
1
T
T
t=1(yt y)2
Covarianceh =
1
T
Tt=h+1
(yt y)(yth y), h = 1, 2, . . .
Autocorrelations h = h/0, h = 1, 2, . . .
Assume h = 0 for h > l . Then Var(h)
T1(1 + 221
+ 22
2
+ . . . + 22
l
).
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Moments ofyt
E[yt] = y0
Var[yt] = E[yt y0]2
= E[(e1 + e2 + . . . + et)2] = t 2,
Cov[yt, ytk] = E[(e1 + e2 + . . . + et)(e1 + e2 + . . . + etk)]
= (t k)2.
Thus,
limt Var[yt] =
,lim
t
Cov[yt, ytk] =
, for finite k
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and
t,k = Cov[yt, ytk]Var[yt]Var[ytk]
=2(t k)2t2(t k)
=
t k
t.
Properties of processes with stochastic trends:
No tendency to return to unconditional mean Variance increases with time
Persistent autocorrelation.
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Random walk with drift yt = + yt1 + et so
= y0 + t +t
i=1
ei
E[yt] = y0 + t.
Obtaining a stationary residual process: yt
yt
1 = et so
(1 L)yt = etyt = et.
A process yt is called integrated of order d (I(d)) if yt is nonstationary and differencing yt d-times
obtains a stationary residual series.
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White noise sequence, stationary AR(1) processes, Random Walks (with drift)
tt
et (0, 1) yt = 0.9yt1 + et
yt = yt1 + et yt = 0.5 + yt1 + et
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3.3 Convergence of moments of I(1) variables
3.3.1 Brownian motion
Let yt = yt1 + et, et (0, 2), y0 = 0, t = 1, 2, . . . , T . Furthermore [T r], 0 r 1 is the largestinteger smaller or equal to T r. Define
XT(r) =
0/
T for 0
r < 1/T
e1/T for 1/T r < 2/T(e1 + e2)/
T for 2/T r < 3/T
(e1 + e2 + e3)/
T for 3/T r < 4/T... ...
(e1 + e2 + . . . + eT)/T for r = 1.
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Then
XT(r) = y[T r]/T , 0 r 1XT(r)
d W(r) XT(r)/ d W(r)f(XT(r))
d f(W(r)), iffcontinuous
Standard Brownian Motion (Wiener process):
Initialization W(0) = 0
Continuity: W(t) is continuous in t with probability 1
Independent increments: For all time points 0 t1 < t2 < . . . tk 1 (W(t2) W(t1)), (W(t3)
W(t2)), . . . (W(tk)W(tk1)), are independent multivariate normally distributed with Var[W(s)W(t)] = s t.
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Convergence of a Random Walk, yt = yt1 + et, et (o, 2), to a standard Brownian motion
T = 10
yt yt/ yt/
T y2t
/2T
T = 100
T = 1000
W(r) W2(r)
t t r r
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3.3.2 Stochastic orders
Let XT denote sequence of random variables
(a) XT converges in probability if Problim T
(|XT x| < ) = 1 or plim XT = x, e.g.
plim
1
T
et
= 0.
(b) XT is of smaller order in probability than qT if plimXTqT = 0, short XT = op(qT), e.g.et = op(T).
(c) XT is of order in probability qT if M exists and for every > 0 Problim T
XTqT > M < , shortXT = Op(qT), e.g.
et = Op(T1/2)
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3.3.3 Sample mean
10
XT(s)ds =T
t=1
t/T(t1)/T
XT(s)ds
=T
t=1
T1XT
t 1
T
= T1T
t=1 yt1/
T
= T3/2T
t=1
yt1.
Thus
y1/
T =
1
0
XT(s)dsd
1
0
W(r)dr.
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3.3.4 Further convergence results
T2T
t=1
y2t1d 2
10
(W(r))2dr
T1/2T
t=1
etd W(1)
T3/2T
t=1
tetd W(1) 1
0
W(r)dr
T5/2T
t=1
tyt1d
10
rW(r)dr
T2T
t=1 ytztd v
1
0
We(r)Wv(r)dr,
where zt = zt1 + vt.
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3.4 Spurious regression
Consider
yt = yt1 + et, et iid(0, 2e ),zt = zt1 + vt, vt iid(0, 2v), E[etvs] = 0 for all t,s.
yt = zt+ t.
OLS estimator
b = b =
T2T
t=1
ytzt
T2
Tt=1
z2t
1
d ve 10 We(r)Wv(r)dr2v
10 W
2v (r)dr
= 0.118
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Remarks:
Typical economic models delivering versions of the autoregressive distributed lag model (ADL) aree.g. partial adjustment specifications, models with adaptive expectations, etc.
The interpretation of the model parameters and the properties of its estimators depend crucially
on the time series properties ofxt and yt.
The upper case ofxt being a scalar variable is easily generalized towards the vector case.
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3.5 Cointegration
Definition:
Two stochastic processes yt, zt which are I(1) are cointegrated if a linear combination of yt and zt
exists which is stationary,
wt = yt zt I(0).Here measures the long-run impact of zt on yt.
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Integrated and cointegrated processes, cointegrating relationship
I(1), no cointegration I(1), cointegration
I(1), cointegration
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Properties of cointegrated variables:
Individual series are dominated by stochastic trends
Cointegrated series evolve somehow similar (parallel)
Some linear combination(s) of cointegrated series show properties of stationary processes.
Ifyt and zt are cointegrated and zt is weakly exogenous the dynamics ofyt might be specified according
to an ADL model, the ADL(1,1) say.
Then the parameter = (1 1) is known as the error correction coefficient. A reparameterizationof the ADL model has become popular as the so called error correction model. The ECM model
provides sensible error correcting dynamics if < 0 and the ECM is stable if 0 > > 2.
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3.6 Testing for unit roots
3.6.1 Dickey Fuller test
The Dickey Fuller statistic is the standard t-ratio for the OLS estimator in the DF regression
yt = yt
1 + et.
The pair of hypotheses is
H0 : = 0 vs. H1 : < 0.
H0 : = 1 H1 : < 1
in yt = yt1 + et in yt = yt1 + et.
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Then,
DF = 1Var[]
=
Var[]
=(yy)1ye
2(yy)1, y = (y0, y1, . . . , yT1)
=ye
(yy)d 1
2
2
W(1)2 12 W(r)2dr
under H0
=1
2
W(1)2 1W(r)2dr
Remarks: The asymptotic distribution is called Dickey Fuller distribution It is not symmetric with negative expectation Critical values are determined by simulation.
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3.6.2 Generalizations of the DF-test
1. Augmented Dickey-Fuller (ADF) Test:
An AR(p) process is integrated of order one if(1) = 1 1 p = 0.ADF regression
yt = yt1 +p1
j=1j ytj + et,
where = (1) and j = (j+1 + + p).Hypotheses of interest:
H0 : = 0 versus H1 : < 0.
The standard tratio of obeys a DF-distribution asymptotically.
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2. Another view at the ADF-statistic:
Consider the ADF regression in compact form
y = y + + e
y = y + + e
where contains lagged values of y.
Normal equations:
yy = y
y + y + y
e,
yy = y
y + y + y
e,
y = y + + e,
y = y + + e.
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Thus,
yy( ) + y( ) = yey( ) + ( ) = e
Now observe:
The estimator is superconsistent and independent of the estimates in . The estimates converge at the usual rate T and exhibit a standard asymptotic distribution.
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Proof
(i) Determine stochastic order of ( )
= (yy)1ye (yy)1y( )
Thus
(
) = ()1
e
()1
y
[(y
y
)1y
e
(y
y
)1y
(
)]
[I ()1 Op(T1)
yOp(T)
(yy)
1 Op(T2)
y
Op(T)
Op(T1)]( ) = ()1
Op(T1)
eOp(T1/2)
Op(T1/2) ()1
Op(T1)
yOp(T)
(yy)
1 Op(T2)
ye
Op(T)
Op(T1)
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T( ) = T 1
e
T
+ Op(T1/2) = T 1
e
T
+ op(1)
so, converges at rate
T to .
(ii) Determine stochastic order and asymptotic distribution of ( ) since ( ) = Op(T1/2)
( ) = (yy)1 Op(T2) yeOp(T)
Op(T1)
(yy)1 Op(T2) yOp(T) Op(T
1/2)
Op(T3/2)
T( ) =yy
T2
1ye
T+ Op(T
1/2) =yy
T2
1ye
T+ op(1)
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3. Deterministic components in DF regression:
yt = yt1 + et yt = c + yt1 + et yt = c + dt + yt1 + et
H0 yt RW yt RW yt RW with drift( = c = 0) ( = d = 0, c = 0)
H1 yt I(0), E[yt] = 0 yt I(0), E[yt] = 0 yt trend stationary( < 0, c = 0) ( < 0, d = 0)
true model td
DF t
d
DF not sensible
yt = yt1 + et (no constant, no trend) (constant, no trend)
yt = + yt1 + et not td N(0, 1) t
d DF= t+ t1 + et sensible Note: Test regression looks like (constant, trend)
yt = c + [(t 1)+ t1] + et
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4. Structural breaks in deterministic terms:
Unit root processes may show similar patterns as trend stationary processes with changing trend
parameters.
Distinguish scenarios where time point of structural change is known (e.g. German unification)
or unknown. In any case the asymptotic distribution of the (A)DF-statistic will depend on the
specification of (changing) deterministic parts and the point of structural change.
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3.6.3 Testing for cointegration
The hypotheses
H0 : No cointegration vs. H1 : yt and zt are cointegrated.
Note: wt is integrated of order one under H0.
Test statistics:
Durbin Watson statistic obtained from the static regression.
ADF test applied to wt. Remark: The DF distribution is different when testing residuals of thestatic regression.
Standard t
ratio for in ECM. Remark: Standard asymptotic theory does not hold under the
null hypothesis.
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3.6.4 Testing for weak exogeneity
zt = 2(yt1 zt1) + 1zt1 + 2yt1 + t, t (0, 2)
H0 : 2 = 0 vs. H1 : 2 = 0.
The t-ratio for 2 will be asymptotically normally distributed (N(0,1)) under H0 in case yt and zt are
cointegrated.
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3.6.5 Estimation
1. Two step LS
Run a superconsistent static regression
yt = zt + wt
and use first step error estimates wt to implement the dynamic model, which in turn is estimated
by OLS:
yt = wt1 + 1zt + et.
2. Nonlinear LS
yt = yt1 zt1 + 1zt + et.
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3.6.6 An additional note on weak exogeneity
Consider ytxt
N(t, ), t = 1, . . . , T , t = t1t2
, = 11 1212 22
.For example
yt = xt + 1,t, xt = 1xt1 + 2yt1 + 2,t.
An alternative as representation is the conditional model for yt given xt
yt = xt1 + xt12 + 1yt1 + et, xt = 1xt1 + 2yt1 + 2,t,
where1 = +
1222
, 2 = 1 1222
, 1 = 2 1222
.
andet = 1,t +
12
222,t, E[e
2t ] = 11
212/22 =
2
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Then
= (, 1, 2, 12, 11, 22)
denotes the parameters of the bivariate model and
1 = (1, 2, 1, 2) or 2 = (1, 2, 22)
the parameters of the conditional and marginal process, respectively.
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Definitions:
xt is weakly exogenous for one (or more) parameters in if first this (these) is (are) only afunction of parameters of the conditional model (1) and second the parameters in 1 and 2
do not obey any cross restrictions. Then 1 and 2 are called variation free.
xt is strongly exogenous for one (or more) parameters in if first xt is weakly exogenous andsecond yt is not Granger causal for xt. yt is not Granger causal for xt if an information set
x,yt = {xt, xt1, . . . , yt, yt1, . . .} will not improve forecasts of xt+h, h = 1, 2, . . . relative to aninformation set xt = {xt, xt1, . . .} .
xt is super exogenous iffirst xt is weakly exogenous and the parameters of the conditional model1 stay invariant if the parameters of the marginal processes 2 change.
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3.6.7 A Monte Carlo Experiment
Issues: -Spurious regression-Static regression
-ECM cointegration test
-Superconsistency
Data generating process: et = et1et2
0, 1 1
, zt = zt1 + et2, 3 Cases (i) yt = yt1 + et1, = 0; (ii) yt = zt + et1, = 0; (iii) yt = zt + et1, = 0.8
Regression models: (a) yt = zt + ut
(b) yt = zt+ (yt1 zt1) + ut.
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CASE (i:)
T = 1000
T = 500
T = 100
T = 50
T = 10
b tb
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CASE (ii:)
T = 1000
T = 500
T = 100
T = 50
T = 10
b tb
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CASE (iii:)
T = 1000
T = 500
T = 100
T = 50
T = 10
b tb
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CASE (i:) Static ECM ECM ECM
T = 1000
T = 500
T = 100
T = 50
T = 10
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CASE (i:) Static ECM ECM ECM
T = 1000
T = 500
T = 100
T = 50
T = 10t t t t
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CASE (ii:) Static ECM ECM ECM
T = 1000
T = 500
T = 100
T = 50
T = 10
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CASE (ii:) Static ECM ECM ECM
T = 1000
T = 500
T = 100
T = 50
T = 10t t t t
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CASE (iii:) Static ECM ECM ECM
T = 1000
T = 500
T = 100
T = 50
T = 10
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CASE (iii:) Static ECM ECM ECM
T = 1000
T = 500
T = 100
T = 50
T = 10t t t t