lutkepohl handout 2011[1]

8/2/2019 Lutkepohl Handout 2011[1]

1/116

Helmut Ltkepohl Time Series Econometrics 2011

Tim e Ser ies Econ om et r i cs

2 0 1 1

Helm u t L t k ep o h l


2/116


Part I Univariate Time Series

Part II Dynamic regression models

Part III Multiple Time Series

Par t I Un i v ar i at e Ti m e Ser i es1 . St a t i ona r y and i n t eg ra t ed s tochast i c p rocesses

2. ARI MA pr ocesses

3 . Es t im a t ion and speci f i cat ion o f s ta t ionar y ARMA p rocesses4. For ecast in g

5 . Est im a t ion o f I( 1 ) p ro cesse s a n d u n i t r o o t t e st s

6 . Spect r a l Ana lys is / Frequ ency Dom a in Ana lysi s

Par t I I7 . Dynam ic reg r ession m ode ls: se tu p and est im a t ion

Par t I I I Mu l t i p le Ti m e Ser i es8 . Vect o r au t o reg r ess ive m ode ls

9 . Es t im a t ion and speci f i cat ion o f VAR m ode ls

1 0 . Co i n t e gr a t i o n a n d v e ct o r er r o r c or r ect i o n m o d el s

11 . Es t im at ion and speci f i ca t ion o f VECMs


3/116


Pa r t I Un i v ar i a t e Ti m e Se r ie s



1. Sta t ionary an d in tegr a ted s tochast i c p rocesses2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

quarterly changes in U.S. fixed investment

quarterly German long-term interest rate


4/116Helmut Ltkepohl Time Series Econometrics 2011





quarterly German nominal GNP

daily log DAFOX







St ochast ic p r ocesses

sequence of random variables (stochastic process)1 2 3, , , ,y y y

observations (time series)

Notation: for stochastic process

orfor time series

sometimes

ty

ty

1 , . . . , Ty y

1 2y , y , ,yT

S i d i d h i







St a t ion ary st och ast i c p r ocesses

ty is s t a t i ona r y if

(1) ( ) for all

(2) ( )( ) for all and

=

=

t y

t y t h y h

E y t

E y y t h

ty(the first and second moments of are time invariant)

1 St t i d i t t d t h t i







quarterly German nominal GNP

Is this series stationary (generated by astationary process) ?

1 Sta t ionary an d in tegr a ted s tochast i c processes








Is this series stationary ?

P I U i i Ti S i 1 Sta t ionary an d in tegr a ted s tochast i c processes


9/116






Est im at io n o f m o m en t s

ty stationary stochastic process,

1

1

T

y t

t

y yT

=

= =

( )( )1

1T

h t t h

t h

y y y yT

= +

=

( )( )1

1

T

h t t ht h

y y y yT h

= +

=

(sample mean) estimator for y

(sample autocovariances)

estimators of h

(sample autocorrelations)

estimators of autocorrelations

0 /h h =

0/h h =

0/h h =

1 , . . . , Ty y time series (sample)


10/116

Pa r t I Un i v ar i a t e Ti m e Se r ie s 1. Sta t ionary an d in tegr a ted s tochast i c processes


11/116






autocorrelations of investment


Pa r t I Un i v ar i a t e Ti m e Se r ie s 1. Sta t ionary an d in tegr a ted s tochast i c p rocesses


12/116





y g p2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis


autocorrelations of long-term interest rate



13/116





2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

OLS estimator of in

Par t i a l au t oco r r e la t i ons

ty stationary stochastic process

h-th partial autocorrelation coefficient

( )1 1, , ,h t t h t t ha Corr y y y y +=

estimator of : h ha a h

1 1t t h t h t y y y u = + + + +



14/116







autocorrelations of investment

partial autocorrelations of investment

Pa r t I Un i v ar i a t e Ti m e Se r ie s 1. Sta t ionary an d in tegr a ted s tochast i c p rocesses2 ARIMA


15/116






partial autocorrelations of long-term interest rate

autocorrelations of long-term interest rate


Pa r t I Un i v ar i a t e Ti m e Se r ie s 1. Sta t ionary an d in tegr a ted s tochast i c p rocesses2 ARIMA


16/116





Tr ans f o r m a t i ons

income log income

4 log income log income

ty

1t ty y = 4t ty y =

Pa r t I Un i v ar i a t e Ti m e Se r ie s 1. Sta t ionary an d in tegr a ted s tochast i c p rocesses2 ARIMA processes


17/116





I n t eg r at ed p r ocesses

ty nonstationary stochastic process

~ (1)ty I1:t t t y y y= stationary

ty nonstationary but

( )2 1 22t t t t t y y y y y = = + stationary ~ (2)ty I etc.

ty quarterly nonstationary process

4 4t t t y y y= stationary ty seasonally integrated

(integrated of order 1)

ty stationary ~ (0)ty I

Pa r t I Un i v ar i a t e Ti m e Se r ie s 1. Stationary and integrated stochastic processes2 ARI MA processes


18/116




2. ARI MA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

( )21

0

, ~ 0,

if 1

t t t t u

i

t i

i

y y u u WN

u

=

= +

=


19/116





0

2

0 0 0

2

2

( ) ( ) 0

( )

0, 1, 2,1

i

y t t i

i

i j j h jh t t h t i t h j u

i j j

h

u

E y E u

E y y E u u

h

=

+ = = =

= = =

= = =

= =

ty is stationary if 1 holds for

Pa r t I Un i v ar i a t e Ti m e Se r ie s 1. Stationary and integrated stochastic processes2. ARI MA processes


20/116





AR(1)-Process, =0.5 AR(1)-Process, =-0.5

Autocorrelation Function (ACF) Autocorrelation Function (ACF)

Partial Autocorrelation Function (PACF) Partial Autocorrelation Function (PACF)


d l

1. Stationary and integrated stochastic processes2. ARI MA processes


21/116




p3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

1t t t

y y u

= + +

AR(1 ) with nonzero mean

or( )

( ) ( )1 1

1

1 11 1

( )1

t t

t t t

t

L y u

y L u L uL

E y

= +

= + = +

=


P t II D i i d l



22/116




p3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

Random W a lk

AR(1 ) with 1 =

( )2

1

0

1

, ~ 0,

t t t t u

t

i

i

y y u u WN

y u

=

= +

= +

0 0

2 2

0

( ) for fixed

( ) nonstationary

t

t u t

E y y y

E y y t y

=

=

However, is stationary

~ (1)

t t

t

y u

y I

=





23/116




3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

Ran d o m W alk w i t h d r i f t

1

0

1

0( )

t t t

t

i

i

t

y y u

y t u

E y y t

=

= + +

= + +

= +

ty

1t t t y y y=

has linear trend in mean

is stationary




d f f


24/116





1 1

01

1

1

if 1 0 for 1

t t p t p t i t i

ip

p

p

y y y u u

z z z

=

= + + + + = +

AR( p) p r ocess

alternatively

( )

( )

( )

1

1

1

1

11

10

1 ( )

11

hence, 1 ( )

p

p t t t

pt p t

p

i p

i pi

L L y L y u

y L L u

L L L L

=

= = +

= +

= =



1. Stationary and integrated stochastic processes2. ARI MA processes3 E ti ti d ifi ti f t ti ARMA


25/116





1

0 0

2

0

1

( )

1

( )( )

, 0, 1, 2,

is stationary if 1 0 for 1

y t

p

h t y t h y i t i j t h j

i j

h j h j

j

p

t p

E y

E y y E u u

h

y z z z

= =

+=

= =

= =

= =

AR( p) p r ocess (contd)



1. Stationary and integrated stochastic processes2. ARI MA processes3 Estimation and specification of stationary ARMA processes


26/116





1 11 0 for = , ,p

p p z z z = AR( p) p r ocess (contd)

111 1 1

p

ppz zz z

=

( )* * 11 1 11 1 (1 ) if one 1p p p p i z z z z z

= =

( )

( )

* * 1

1 1

* * 1

1 1

1 (1 )

= 1

p

p t

p

p t t

L L L y

L L y u

= +

* * 1

1 1is stationary if 1 0 for 1p

p z z z

~ (1)ty I





27/116





( )

1 1

11 ( ) is stationary

t t t q t q

qq t q t

y u m u m u

m L m L u m L u

= + + + +

= + + + + = +

MA( q) p r ocess

is called invertible if ( ) 0 for 1t y m z z

1 1

1

( ) (1)

t t

t i t i t

i

m L y m u

y y u

=

= +

= + +or (AR representation)





28/116


y g



MA(1)-Process, m=0.8 MA(1)-Process, m=-0.8

Autocorrelation Function (ACF) Autocorrelation Function (ACF)

Partial Autocorrelation Function (PACF) Partial Autocorrelation Function (PACF)



1. Stationary and integrated stochastic processes2. ARI MA processes3. Estimation and specification of stationary ARMA processes


29/116


y g



1 1 1 -1 t t p t p t t q t q y y y u m u m u = + + + + + + +

ARMA( p ,q) p r ocess

or

is stationary and invertible

has MA representation

has AR representation

( ) 0 for 1

( ) 0 for 1

z z

m z z

with

( ) ( )t tL y m L u = +

1( ) ( )(1)

t t y L m L u

= +

1( ) ( )(1)

t tm L L y um

= +



1. Stationary and integrated stochastic processes2. ARI MA processes3. Estimation and specification of stationary ARMA processes


30/116



p y p4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

ARI MA( p,d,q ) p r ocess

( ) ( ) is ( )dt t

L y m L u I d = +



1. Stationary and integrated stochastic processes2. ARIMA processes3. Es t im at ion an d spec i f i ca t ion o f s ta t ionary ARMA processes


31/116



p y p4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

If is stationary, the estimators have the usual asymptoticproperties.

ty

Est im a t i on o f AR m ode ls

1Estimate , ,..., by OLS.p

AR( p)

( )21 1 ... , ~ 0,t t p t p t t u y y y u u WN = + + + +

E.g., AR(4) investment series

1 2 3 40.82 0.51 0.10 0.06 0.22t t t t t t y y y y y u = + + +(2.76) (4.86) (-0.83) (0.54) (-2.02)



1. Stationary and integrated stochastic processes2. ARIMA processes3. Es t im at ion an d spec i f i ca t ion o f s ta t ionary ARMA processes


32/116


Part III Multiple Time Series 4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

Est im a t i on o f ARMA m ode ls

1 1

1

( ,..., , ,..., ) ( )T

p q t

t

l m m l

=

=

ARMA( p ,q)

( )2( ) ( ) , ~ 0, normally distributedt t t uL y m L u u WN =

log-likelihood

with

( )2

2 1 21 1( ) log 2 log ( ) ( ) 22 2

t u t ul m L L y =

ML est im a t ion



1. Stationary and integrated stochastic processes2. ARIMA processes3. Es t im at ion an d spec i f i ca t ion o f s ta t ionary ARMA processes4 F i


33/116



Model speci f i cat ion

max0,...,n p=

Specification of AR order

2

2

2

2( ) log ( ) (Akaike)

2loglog( ) log ( ) (Hannan-Quinn)

log( ) log ( ) (Schwarz, Rissanen)

u

u

u

AIC n n nT

T HQ n n n

T

TSC n n n

T

= +

= +

= +

Estimate AR(n) model for

Choose such that it minimizes a criterion such asp



P III M l i l Ti S i

1. Stationary and integrated stochastic processes2. ARIMA processes3. Es t im at ion an d spec i f i ca t ion o f s ta t ionary ARMA processes4 F ti


34/116



( ) ( ) ( ) (if 16) p SC p HQ p AIC T

n

0 1 2 3 4 5 6 7 8 9 10

AIC(n) 2.170 1.935 1.942 1.950 1.942 1.963 1.990 2.018 1.999 1.997 2.032

HQ(n) 2.180 1.956 1.974 1.997 1.995 2.027 2.065 2.104 2.097 2.107 2.153

SC(n) 2.195 1.987 2.020 2.059 2.073 2.122 2.176 2.231 2.241 2.268 2.331

Order selection criteria for U.S. investment series


35/116

Autocorrelation Function (ACF)


36/116


Partial Autocorrelation Function (PACF)

Autocorrelations of AR(1) Autocorrelations of AR(1)

Partial Autocorrelations of AR(1) Partial Autocorrelations of AR(1)




1. Stationary and integrated stochastic processes2. ARIMA processes3. Es t im at ion an d spec i f i ca t ion o f s ta t ionary ARMA processes4 Forecasting


37/116



Autocorrelations of AR(2) Autocorrelations of AR(2)

Partial Autocorrelations of AR(2) Partial Autocorrelations of AR(2)




1. Stationary and integrated stochastic processes2. ARIMA processes3. Es t im at ion an d spec i f i ca t ion o f s ta t ionary ARMA processes4. Forecasting


38/116



Autocorrelations of ARMA(1,1) Autocorrelations of ARMA(1,1)

Partial Autocorrelations of ARMA(1,1) Partial Autocorrelations of ARMA(1,1)


39/116


40/116




1. Stationary and integrated stochastic processes2. ARIMA processes3. Es t im at ion an d spec i f i ca t ion o f s ta t ionary ARMA processes4. Forecasting


41/116


p5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

Po r t m an t eau t est f o rr esidu a l au t oco r r e la t i on

0 ,1 ,2 ,: ... 0 where ( , )u u u i t t i H Corr u u = = = =

{ }1 ,: 0 for at least one 1,2,...u iH i

vs.

or

test statistic

2

,

1

h

h u j

j

Q T =

= ,1

1

Ts s

u j t t j

t j

u uT

= +

=

* 2 2 2

,

1

1 ( )

h

h u j

j

Q T h p q

T j

=

=

standardized residuals

tu




1. Stationary and integrated stochastic processes2. ARIMA processes3. Es t im at ion an d spec i f i ca t ion o f s ta t ionary ARMA processes4. Forecasting5 E ti ti f I (1) d it t t t


42/116


p5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

LM t est for residual autocorrelation in AR models

0 1: 0hH = = = 1 1: 0 or ...or 0hH vs.

estimate auxiliary model

Breusch-Godfrey test

1 1t t h t h t

u u u error

= + + +

1 1 1 1

t t p t p t h t h t u y y u u e = + + + + + + +

2 2 ( )h LM TR h=

or 2

2

1( , 1)

1h

R T p hFLM F h T p h

R h

=




1. Stationary and integrated stochastic processes2. ARIMA processes3. Es t im at ion an d spec i f i ca t ion o f s ta t ionary ARMA processes4. Forecasting5 Estimation of I (1) p ocesses and nit oot tests


43/116


5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

Ot he r m odel check s

test for nonnormality

ARCH test

nonlinearity test (RESET)

stability analysis:

Chow tests, CUSUM tests,

recursive analysis etc.




1. Stationary and integrated stochastic processes2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecast in g5 Estimation of I (1) processes and unit root tests


44/116


5. Estimation ofI(1) processes and unit root tests6. Spectral Analysis / Frequency Domain Analysis

Forecast ing

1 1t t p t p t y y y u = + + +

1-step ahead forecast at forecast origin T:

1 11 T p T pT T y y y + + = + +

h-step ahead forecast at forecast origin T:

1 1 pT h T T h T T h p T y y y + + + = + +

(compute forecasts recursively for h=1,2,)

DGP


45/116




1. Stationary and integrated stochastic processes2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5 Es t im a t i on o f I ( 1 ) p rocesses and un i t roo t t ests


46/116


5 . Es t im a t i on o f I( 1 ) p rocesses and un i t roo t t ests6. Spectral Analysis / Frequency Domain Analysis

Un i t r o o t t est s

1 1

1has unit root if 1 0

t t p t p t

p

y y y u

= + + + =

* *

1 1 1 1 1

*1 1where (1 ), ( )

t t t p t p t

p j j p

y y y y u ++

= + + + +

= = + +

Reparameterize process:

DGP:

Augm ent ed D ick ey -Fu l l e r ( ADF) t est

0 : 0H = 1 : 0H


47/116


5 . Es t im a t i on o f I( 1 ) p rocesses and un i t roo t t ests6. Spectral Analysis / Frequency Domain Analysis

estimate model by OLS

test statistic: -statistic fort t =

reject if is small0

critical values depend on deterministic terms in the

model

H t




1. Stationary and integrated stochastic processes2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5 . Es t im a t i on o f I( 1 ) p rocesses and un i t roo t t ests


48/116


( ) p6. Spectral Analysis / Frequency Domain Analysis

KPSS t est(Kwiatkowski, Phillips, Schmidt, Shin)

1

t t t

t t t

y x z

x x v

= +

= +

DGP:

0 1: ~ (0) vs. : ~ (1)t t H y I H y I

2 2

0 1: =0 vs. : 0v vH H >

RW

test




1. Stationary and integrated stochastic processes2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5 . Es t im a t i on o f I( 1 ) p rocesses and un i t roo t t ests


49/116


6. Spectral Analysis / Frequency Domain Analysis

2 2

21

1

1

( )

T

t

t

t

t j

j

KPSS ST

S y y

=

=

=

=

2 2 1

1

is estimator of lim VarT

tT

t

T z =

=

e.g.,2 2

1 1 1

1 1 = ( ) 2 ( )( )

where 1 and lag truncation parameter 1

qlT T

t j t t j

t j t j

j q

q

y y y y y yT T

j ll

= = = +

+

= +

e.g.,

( )

14/100

q

l q T=


50/116

Pa r t I Un i v ar i a t e Ti m e Se r ie sPart II Dynamic regression models


1. Stationary and integrated stochastic processes2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6 S t l A l i / F D i A l i


51/116


6. Spect ra l Ana lys is / Frequ ency Domain Ana lys is

Spect r a l Ana lysi s / Fr equency Dom a in Ana lys is

Let a and h be independent random variables such that

2

~ (0, ) and ~ (0, )aa h U

Define

cos( )t y a t h= +

[ ]( ) ( ) cos( ) 0tE y E a E t h = + =2

2( , ) ( ) cos( )at t j t t jCov y y E y y j

+ += =

N o t e : yt

is deterministic

yt is stationary



1. Stationary and integrated stochastic processes2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6 Spect ra l Ana lys is / Frequ ency Domain Ana lys is


52/116



Te r m ino logy :

amplitude

frequency (number of cycles per unit measured in radians)

phase

wavelength, period

cos( )t y a t h= +

a

2

h





53/116



More genera l l y :

21 1,..., , ~ (0, ), ,..., ~ (0, ) M j j M a a a h h U

mutually independent random variables

1

cos( )

M

t k k k

k

y a t h=

= +

( ) 0tE y =

2

21

( , ) cos( )kM

t t j k

k

Cov y y j

+=

=

yt is stationary

but deterministic





54/116



Judge et al. (1985)


55/116



1. Stationary and integrated stochastic processes2. ARIMA processes3. Estimation and specification of stationary ARMA processes4. Forecasting5. Estimation ofI(1) processes and unit root tests6. Spect ra l Ana lys is / Frequ ency Domain Ana lys is


56/116


6 Spect a a ys s / equ e cy o a a ys s

Spect r a l Repr esen t a t ion Th eor em

then there exist random variables

1 2 3 4and for any 0 , < < < <

= >

= =

use Johansens LR tests or other systems cointegration tests

choose rsuch that is the first null hypothesis

which cannot be rejected

( )0 : rkH r=



Par t I I I M u l t i p l e Tim e Se r i es

8. Vector autoregressive models

9. Estimation and specification of VAR models10. Cointegration and vector error correction models11 . Est im at i on an d spec i f i ca t ion o f VECMs

The critical values of the tests depend on the deterministic

terms in the VECMMain cases:


114/116


10

1 1

(1)

1

t t

t

t t t

y x

y y y u

= +

= + + +

0 1

1

1 1

(2)

1

t t

t

t t t

y t x

y y y u

t

+ = + +

= + + + +

0 1 1

1 1 1

(3) , 0t t

t t t t

y t x

y y y u

= + + =

= + + + +

(linear trend in variables but notin cointegration relations)

(constant)

(unrestricted linear trend)






Sp eci f i cat io n of VECM

y1t,,yKt time series of interest


115/116


1t t

St ep 4 : Model checking

plot residuals

test for residual autocorrelation test for nonnormality

test for ARCH

stability analysis

St ep 1 : Determine orders of integration ofykts

St ep 2 : Determine cointegration relations between all integratedvariables, also considering subsystems of variables

St ep 3 : Set up, estimate and simplify an overall

VECM for yt=(y1t,,yKt)






Extens ions


116/116


structural breaks in deterministic term

I(2) variables

seasonal cointegration

lutkepohl handout 2011[1]

Documents