alberto contreras mayo 2011
TRANSCRIPT
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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
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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.
For non stationary processes we may have
COV(Xt, Xt+) = ,t.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel Chong
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Stationary ProcessesARMA models
2nd order Stationarity
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
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S i P
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Stationary ProcessesARMA models
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)
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
St ti P
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Stationary ProcessesARMA models
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)
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).
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary Processes
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Stationary ProcessesARMA models
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)
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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Stationary Processes
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Stationary ProcessesARMA models
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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Stationary ProcessesARMA models
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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Stationary Processes
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Stationary ProcessesARMA models
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)
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary Processes
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Stationary ProcessesARMA models
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.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary Processes
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ARMA 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.
Fit a SARIMA model to this data set.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary Processes
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ARMA 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.
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
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Stationary ProcessesARMA d l
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ARMA models
{Xt} are (the logarithm in base 10 of) the average weeklyprices (10kg box).
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Stationary ProcessesARMA d l
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ARMA models
{Xt} are (the logarithm in base 10 of) the average weeklyprices (10kg box).
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
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
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ARMA models
{Xt} are (the logarithm in base 10 of) the average weeklyprices (10kg box).
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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Use the model to forecast the missing observation (September24rd to 28th of 2007).
This yields X = 338.26. mxn.
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Use the model to forecast the missing observation (September24rd to 28th of 2007).
This yields X = 338.26. mxn.
Average weekly price from the previous week is 355 mxn. andfor the week after is 320 mxn.
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Use the model to forecast the missing observation (September24rd to 28th of 2007).
This yields X = 338.26. mxn.
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
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
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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
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
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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
1. A vector process {Xt} of dimension K 1 is autoregressive oforder p if
Xt =
pi=1
iXti + VZt + + dt + wt + Et, (2)
where
2. {i}p1
are K K dimensional matrices of coefficients.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
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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
2. {i}p1
are K K dimensional matrices of coefficients.
3. Zt are exogenous variables (covariates), dt vector of seasonaldummies, wt vector of intervention dummies.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
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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
2. {i}p1
are K K dimensional matrices of coefficients.
3. Zt are exogenous variables (covariates), dt vector of seasonaldummies, wt vector of intervention dummies.
4. {Et} is a sequence of uncorrelated vectors such thatEt NK(0, ) for each t
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
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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
2. {i}p1
are K K dimensional matrices of coefficients.
3. Zt are exogenous variables (covariates), dt vector of seasonaldummies, wt vector of intervention dummies.
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
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)
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.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
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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)
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
The components of the vector Xt cointegrate if
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
The components of the vector Xt cointegrate if
each component Xi,t of Xt is I(1) : Xi,t = Xi,t Xi,t1 is astationary process i = 1, . . . , K.
There is an h (K + M) dimensional matrix
+ (h 1) such
the h components of the vector
+ Xt
Yt are stationary
processes.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
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Cointegration
The components of the vector Xt cointegrate if
each component Xi,t of Xt is I(1) : Xi,t = Xi,t Xi,t1 is astationary process i = 1, . . . , K.
There is an h (K + M) dimensional matrix
+ (h 1) such
the h components of the vector
+ 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.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
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Cointegration
The components of the vector Xt cointegrate if
each component Xi,t of Xt is I(1) : Xi,t = Xi,t Xi,t1 is astationary process i = 1, . . . , K.
There is an h (K + M) dimensional matrix
+ (h 1) such
the h components of the vector
+ 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.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
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.
As a covariate Zt = underemployment rate is included.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
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Stationary ProcessesARMA models
E ti 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.
As a covariate Zt = underemployment rate is included.
Dummies for seasonal effects are included
Assumptions as Normality, no correlation and
Heteroskedasticity were tested. The model is fine atsignificance level = 0.05.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
Error correction Model
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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.
As a covariate Zt = underemployment rate is included.
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.
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
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
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
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
X1,t 0.88X2,t + 0.29Zt 0.10X3,t = t
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
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
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
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
Some references
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Some references
Brockwell D. and Davis R. Time series : theory and
applications. Springer Verlag
Alberto Contreras Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
Stationary ProcessesARMA models
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 Cristan, Julio Lopez-Gallardo, Armando Sanchez, Miguel ChongStudying some data from the Mexican Economy with Time Se
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