Econ1 Mle Notes

8/8/2019 Econ1 Mle Notes

1/148

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

1


2/148

Econometrics I

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

2


3/148

Econometrics I

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

3


4/148

Econometrics I

3.6.7 A Monte Carlo Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 39

4


5/148

Econometrics I

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.

5


6/148

Econometrics I

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,

6


7/148

Econometrics I

where

y =

y1

y2

...

yT

(T1)

, e =

e1

e2

...

eT

(T1)

, =

1

2

...

K

(K1)

7


8/148

Econometrics I

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

8


9/148

Econometrics I

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).

9


10/148

Econometrics I

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.

10


11/148

Econometrics I

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.

11


12/148

Econometrics I

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.

12


13/148

Econometrics I

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.

13


14/148

Econometrics I

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

.

14


15/148

Econometrics I

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.

15


16/148

Econometrics I

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.

16


17/148

Econometrics I

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]).

17


18/148

Econometrics I

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?

18


19/148

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

19


21/148

Econometrics I

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.

21


22/148

Econometrics I

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 .

22


23/148

Econometrics I

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.

23


24/148

Econometrics I

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 .

24


25/148

Econometrics I

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).

25


26/148

Econometrics I

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) = .

26


27/148

Econometrics I

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).

27


28/148

Econometrics I

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).

28


29/148

Econometrics I

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.

29


30/148

Econometrics I

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.

30


31/148

Econometrics I

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.

31


32/148

Econometrics I

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).

32


33/148

Econometrics I

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 .

33


34/148

Econometrics I

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).

34


35/148

Econometrics I

Consistency of OLS estimator

E

XeT

= 0.

VarT

Xe

T

=

1

T2Var[Xe]

=1

T22(XX)

= 1T

2XXT 0.

Thus

plim

Xe

T

= 0.

35


36/148

Econometrics I

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.

36


37/148

Econometrics I

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).

37


38/148

Econometrics I

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).

38


39/148

Econometrics I

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.

39


40/148

Econometrics I

Properties of

: is not linear,

in many cases one can show that E[] = ,

Cov[] cannot be determined,

is not BLUE.

40


41/148

Econometrics I

1.4 Asymptotic results

Assume

limT

X1X

T

= V, finite, nonsingular.

Then

plim() =

plim(2g) = 2

T( ) d N(0, 2V1).

41


42/148

Econometrics I

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.

42


43/148

Econometrics I

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

= .

43


44/148

Econometrics I

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,

44


45/148

Econometrics I

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

.

45


46/148

Econometrics I

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

).

46


47/148

Econometrics I

E[ee] =

=

exp(z1) 0

. . .

0 exp(zT)

= 2

exp(z1) 0

. . .

0 exp(zT)

= 2.

47


48/148

Econometrics I

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

.

48


49/148

Econometrics I

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.

49


50/148

Econometrics I

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.

50


51/148

Econometrics I

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)

51


52/148

Econometrics I

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).

52


53/148

Econometrics I

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.

53


54/148

Econometrics I

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.

54


55/148

Econometrics I

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.

55


56/148

Econometrics I

General

E[etets] = s2v

1 2 .

Autocorrelation

E[etets]E[e2t ]E[e

2ts]

=s

2v

122v

12

= s.

56


57/148


58/148

Econometrics I

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

,

58


59/148

Econometrics I

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

.

59


60/148

Econometrics I

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

60


61/148

Econometrics I

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

.

61


62/148

Econometrics I

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

62


63/148

Econometrics I

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

63


64/148

Econometrics I

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.

64


65/148

Econometrics I

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.

65


66/148

Econometrics I

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.

66


67/148

Econometrics I

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]].

67


68/148

Econometrics I

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.

68


69/148

Econometrics I

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).

69


70/148

Econometrics I

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.

70


71/148


72/148

Econometrics I

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.

72


73/148

Econometrics I

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.

73


74/148

Econometrics I

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.

74


75/148


76/148

Econometrics I

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)

76


77/148

Econometrics I

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).

77


78/148

Econometrics I

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 .

78


79/148

Econometrics I

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).

79


80/148

Econometrics I

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 .

80


81/148

Econometrics I

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].

81


82/148

Econometrics I

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.

82


83/148

Econometrics I

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

83


84/148

Econometrics I

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.

84


85/148

Econometrics I

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.

85


86/148

Econometrics I

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.

86


87/148

Econometrics I

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].

87


88/148

Econometrics I

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.

88


89/148

Econometrics I

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().

89


90/148

Econometrics I

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.

90


91/148

Econometrics I

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().

91


92/148

Econometrics I

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).

92


93/148

Econometrics I

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()

93


94/148

Econometrics I

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.

94


95/148

Econometrics I

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.

95


96/148

Econometrics I

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.

96


97/148

Econometrics I

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.

97


98/148

Econometrics I

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)],

98


99/148

Econometrics I

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.

99


100/148

Econometrics I

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

100


101/148

Econometrics I

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.

101


102/148

Econometrics I

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.

102


103/148

Econometrics I

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.

103

E I


104/148

Econometrics I

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

).

104


105/148

E i I


106/148

Econometrics I

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

106

E t i I


107/148

Econometrics I

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.

107

E t i I


108/148

Econometrics I

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.

108

E t i I


109/148

Econometrics I

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

109

E t i I


110/148

Econometrics I

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.

110

E t i I


111/148

Econometrics I

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.

111

Econometrics I


112/148

Econometrics I

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

112

Econometrics I


113/148

Econometrics I

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)

113

Econometrics I


114/148

Econometrics I

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.

114

Econometrics I


115/148

Econometrics I

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.

115


116/148


117/148

Econometrics I


118/148

Econometrics I

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


119/148

Econometrics I


120/148

Econometrics I

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.

120

Econometrics I


121/148

Econometrics I

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.

121

Econometrics I


122/148

Econometrics I

Integrated and cointegrated processes, cointegrating relationship

I(1), no cointegration I(1), cointegration

I(1), cointegration

122

Econometrics I


123/148

Econometrics I

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.

123

Econometrics I


124/148

Econometrics I

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.

124

Econometrics I


125/148

Econometrics I

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.

125

Econometrics I


126/148

Econometrics I

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.

126

Econometrics I


127/148

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.

127

Econometrics I


128/148

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.

128

Econometrics I


129/148

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)

129

Econometrics I


130/148

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)

130

Econometrics I


131/148

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

131

Econometrics I


132/148

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.

132

Econometrics I


133/148

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.

133

Econometrics I


134/148

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.

134

Econometrics I


135/148

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.

135

Econometrics I


136/148

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

136

Econometrics I


137/148

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.

137

Econometrics I


138/148

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.

138

Econometrics I


139/148

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.

139

Econometrics I


140/148

CASE (i:)

T = 1000

T = 500

T = 100

T = 50

T = 10

b tb

140

Econometrics I


141/148

CASE (ii:)

T = 1000

T = 500

T = 100

T = 50

T = 10

b tb

141

Econometrics I


142/148

CASE (iii:)

T = 1000

T = 500

T = 100

T = 50

T = 10

b tb

142

Econometrics I


143/148

CASE (i:) Static ECM ECM ECM

T = 1000

T = 500

T = 100

T = 50

T = 10

143

Econometrics I


144/148

CASE (i:) Static ECM ECM ECM

T = 1000

T = 500

T = 100

T = 50

T = 10t t t t

144

Econometrics I


145/148

CASE (ii:) Static ECM ECM ECM

T = 1000

T = 500

T = 100

T = 50

T = 10

145

Econometrics I


146/148

CASE (ii:) Static ECM ECM ECM

T = 1000

T = 500

T = 100

T = 50

T = 10t t t t

146

Econometrics I


147/148

CASE (iii:) Static ECM ECM ECM

T = 1000

T = 500

T = 100

T = 50

T = 10

147

Econometrics I


148/148

CASE (iii:) Static ECM ECM ECM

T = 1000

T = 500

T = 100

T = 50

T = 10t t t t

Date post:	10-Apr-2018
Category:	Documents
Upload:	astarkweather3488
View:	223 times
Download:	0 times

Econ1 Mle Notes

Documents