alberto contreras mayo 2011

7/31/2019 Alberto Contreras Mayo 2011

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Stationary ProcessesARMA models

2nd order Stationarity

A discrete time stochastic process {Xt, t = 0, 1, . . . } isstationary (second order) if its moments of order 2 exist and

IE(Xt) = , t

COV(Xt, Xt+) = , t.

Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel Chong

Studying some data from the Mexican Economy with Time Se
http://find/http://goback/


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A discrete time stochastic process {Xt, t = 0, 1, . . . } isstationary (second order) if its moments of order 2 exist and

IE(Xt) = , t

COV(Xt, Xt+) = , t.

For non stationary processes we may have

COV(Xt, Xt+) = ,t.

Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel Chong

Studying some data from the Mexican Economy with Time Se


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0 50 100 150 200

4

2

0

2

4

6

Index

x

0 50 100 150

0

5

10

15

20

25

30

time

x

Figure: (a) Stationary (b) non-Stationary

Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se

S i P


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ARMA models

Let p, q be positive integers and let B be the operator Bxt = xt1.1. ARMA(p, q) model

p(B)Xt = q(B)t, (1)


St ti P


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ARMA models


p(B)Xt = q(B)t, (1)

2. where(z) = 1 1z 2z

2 pzp,

(z) = 1 + 1z + 2z2 + + qz

q

and {t} a sequence of uncorrelated random variables, such

that for all t, t Normal(0, 2).


Stationary Processes


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ARMA models


p(B)Xt = q(B)t, (1)

2. where(z) = 1 1z 2z

2 pzp,

(z) = 1 + 1z + 2z2 + + qz

q

and {t} a sequence of uncorrelated random variables, such

that for all t, t Normal(0, 2).

3. If data are non-stationary, we may transform these in order tofit a model of this class. As in Regression Analysis, checkingassumptions and goodness of fit is compulsory.




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Seasonality

For data with a seasonal component, models of the classSARIMA((p, d, q) (P, D, Q))s are available.

1973 1974 1975 1976 1977 1978

6

7

8

9

10

11

12

months

thousands




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example average weekly price for Hass Avocado (10k box)

Precio promedio semanal

Series1

1996 1998 2000 2002 2004 2006 2008

50

100

150

200

250

300

350

400

Figure: (a) Hass avocado (b) average weekly prices




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SARIMA models

1. Let d, D, s, be positive integers. {Xt} is aSARIMA((p, d, q) (P, D, Q))s process (with period s) if

Yt = (1 B)d(1 Bs)DXt (A)

is a stationary ARMA process satisfying

p(B)P(Bs)Yt = q(B)Q(B

s)t, (B)




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SARIMA models

1. Let d, D, s, be positive integers. {Xt} is aSARIMA((p, d, q) (P, D, Q))s process (with period s) if

Yt = (1 B)d(1 Bs)DXt (A)

is a stationary ARMA process satisfying

p(B)P(Bs)Yt = q(B)Q(B

s)t, (B)

2. where p(z) = 1 1z 2z2 pz

p,

P(z) = 1 1z 2z2 Pz

P,

q(z) = 1 + 1z + 2z2 + + qzq,

Q(z) = 1 + 1z + 2z2 + + Qz

Q.

{t} are uncorrelated and identically distributed as anormal(0, 2

) r.v.




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yARMA models

We will take data corresponding to the period from January of2004 to the last week available (September 17th to 21st of2007) before the missing observation.




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ARMA models


Fit a SARIMA model to this data set.




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ARMA models


Fit a SARIMA model to this data set.

Use the model to forecast the missing datum

Logaritmodelosdatos

2004 2005 2006 2007

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6


Stationary ProcessesARMA d l


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ARMA models

{Xt} are (the logarithm in base 10 of) the average weeklyprices (10kg box).


Stationary ProcessesARMA d l


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ARMA models


A series of trials leads to difference at lag 1 corresponding tod = 1, then difference at lag s = 52 corresponding to D = 1so that the resulting series {Yt} looks stationary




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ARMA models


A series of trials leads to difference at lag 1 corresponding tod = 1, then difference at lag s = 52 corresponding to D = 1so that the resulting series {Yt} looks stationary

Its ACF allows us to choose a SARIMA model

0 20 40 60 80 100

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

Lag

ACF




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ARMA models

The ACF points to aSARIMA(p = 0, d = 1, q = 7)(P = 0, D = 1, Q = 1)52model




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ARMA models

The ACF points to aSARIMA(p = 0, d = 1, q = 7)(P = 0, D = 1, Q = 1)52model

After estimation a residual analysis follows

Time

residuals

2005.0 2006.0 2007.0

0.0

6

0.0

0

0.0

4

0 20 60 100 140

0.2

0.2

0.6

1.0

Lag

ACF

Series residuals

2 1 0 1 2

0.0

6

0.0

0

0.0

4

Normal QQ Plot

Theoretical Quantiles

SampleQuantiles

Histogram of residuals

residuals

Frequency

0.06 0.02 0.02 0.06

0

10

20

30

40

50




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ARMA models

Use the model to forecast the missing observation (September24rd to 28th of 2007).




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This yields X = 338.26. mxn.




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Average weekly price from the previous week is 355 mxn. andfor the week after is 320 mxn.




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Average weekly price from the previous week is 355 mxn. andfor the week after is 320 mxn.

Using the completed data set (January 2004 to July of2008) we can fit a SARIMA model and produce forecasts




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2004 2005 2006 2007 2008

100

150

200

250

300

400

Figure: Forecasts using aSARIMA(p = 0, d = 1, q = 7)(P = 0, D = 1, Q = 1)52 model




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VAR(p) models

1. A vector process {Xt} of dimension K 1 is autoregressive oforder p if

Xt =

pi=1

iXti + VZt + + dt + wt + Et, (2)

where




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VAR(p) models


Xt =

pi=1


where

2. {i}p1

are K K dimensional matrices of coefficients.




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VAR(p) models


Xt =

pi=1


where

2. {i}p1


3. Zt are exogenous variables (covariates), dt vector of seasonaldummies, wt vector of intervention dummies.




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VAR(p) models


Xt =

pi=1


where

2. {i}p1



4. {Et} is a sequence of uncorrelated vectors such thatEt NK(0, ) for each t




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VAR(p) models


Xt =

pi=1


where

2. {i}p1



4. {Et} is a sequence of uncorrelated vectors such thatEt NK(0, ) for each t

5. , V, , are vectors and matrices (parameters) of theappropriate dimensions.




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VAR(p) models

Gathering all covariates Zt, dt, wt and , in a M 1dimensional vector Yt we can write (2) as

Xt =

pi=1

iXti + BYt + Et, (2)




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VAR(p) models


Xt =

pi=1


Under some conditions on the matrix IK 1z pzp,|z| 1, the model

Xt =

p

i=1iXti + 0 + Et,

is suitable for vectors Xt where each component is astationary process.




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VAR(p) models


Xt =

pi=1


Under some conditions on the matrix IK 1z pzp,|z| 1, the model

Xt =

pi=1

iXti + 0 + Et,

is suitable for vectors Xt where each component is astationary process.

however the model in (2) can be used for the case where eachcomponent is not stationary




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Cointegration

The components of the vector Xt cointegrate if




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Cointegration


each component Xi,t of Xt is I(1) : Xi,t = Xi,t Xi,t1 is astationary process i = 1, . . . , K.




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Cointegration



There is an h (K + M) dimensional matrix

+ (h 1) such

the h components of the vector

+ Xt

Yt are stationary

processes.




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Cointegration




+ (h 1) such


+ Xt

Yt are stationary

processes.

write (2) in the so called error correction model (VECM orVAR in Xt)

AXt =

p1i=1

iXti

+

XtYt

+ Yt + Et.




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Cointegration




+ (h 1) such


+ XtYt are stationary

processes.

write (2) in the so called error correction model (VECM orVAR in Xt)

AXt =

p1i=1

iXti

+

XtYt

+ Yt + Et.

for K K dimensional matrices A and {i} and matrices and of dimensions K h and K M




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Assuming that the model (2) is appropriate (check residuals),we can estimate the matrix

+. The estimated parameters

furnishes us with economical interpretations




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1990 1995 2000

1

2

3

4

5

6

7MAQWNU

(1)

1990 1995 2000

2.5

5.0

7.5

10.0

12.5

15.0

IMWNU

(2)

1) maquila nominal wage 2) Manufacturing nominal wageAlberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se



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1990 1995 2000

15.0

17.5

20.0

22.5

25.0

27.5

DES

(1)

1990 1995 2000

90

100

110

120

130

140

150IMYBR

(2)

1) underemployment rate2) Gross value of production manufacturing




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E i M d l


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Error correction Model

Fit the error correction model from a a VAR(4) for Xt, where

X1,tX2,tX3

,t

=

Manufacturing nominal wage

Maquila nominal wage

Manufacturing gross value of product.



E i M d l


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X1,tX2,tX3

,t

=




As a covariate Zt = underemployment rate is included.



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E ti M d l


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X1,tX2,tX3

,t

=





Dummies for seasonal effects are included

Assumptions as Normality, no correlation and

Heteroskedasticity were tested. The model is fine atsignificance level = 0.05.





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X1,tX2,tX3,t

=





Dummies for seasonal effects are included

Assumptions as Normality, no correlation and

Heteroskedasticity were tested. The model is fine atsignificance level = 0.05.

R2 = 0.93.



Right or Wrong ?


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Right or Wrong ?

From this error correction model a cointegration (long run)relationship is estimated as the stationary process t given by



Right or Wrong ?


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Right or Wrong ?


X1,t 0.88X2,t + 0.29Zt 0.10X3,t = t



Right or Wrong ?


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Right or Wrong ?


X1,t 0.88X2,t + 0.29Zt 0.10X3,t = t

namely, in the long run we have

X1,t = 0.88X2,t 0.29Zt + 0.10X3,t



Some references


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Some references

Brockwell D. and Davis R. Time series : theory and

applications. Springer Verlag



Some references


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Some references

Brockwell D. and Davis R. Time series : theory and

applications. Springer Verlag Lutkepohl, H. New Introduction to Multiple Time Series

Analysis. Springer Verlag.


alberto contreras mayo 2011

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