Time Series Analysis
Definition of a Time Series process
AR, MA, ARMA, ARIMA
Vector Autoregression
Impulse Response
Forecasting
1
Four Components of a Time Series
Time Series
yt   t   t   t   t   t
Trend
 t ~ NID0,  2 
 t   t 1   t 1  t
 t   t 1   t
Season
 t   t 1  ..   t s 1   t
 j ,t   cos  j
 *    sin 
j
 j ,t  
Cycle
sin  j   j ,t 1   j ,t 

cos  j  *j ,t 1   *j ,t 
 t 
 cos c
 *     sin 
c
 t

Random
sin c   j ,t 1   t 
  *


* 
cos c    t 1   t 
 t   v t 1   t
 t ~ NID0, 2 
 t ~ NID0,  2 
 t ~ NID0,  w2 
j  1,..., s / 2
t  1,...., T
t  1,...., T
0    1
 t ~ NID0, 2 
2
Iterative Substitution in AR(1) Model
yt  y
v
t 1 t
y
 y
v
t 1
t 2
t 1
y
 y
v
t 2
t 3 t 3
y
 y
v
t 3
t 4
t 4
. …
…
yt n  y
v
t  n 1 t  n
(5)
From substitution
yt  y
v =
t 1 t

2
yt    y
v

v


yt 2

t
t 2
t  1


vt 1  vt

yt    y
v
 vt

t 3
t 2

yt   2 yt 3  vt 2   vt 1  vt   3 yt 3   2vt 2  vt 1  vt
3
AR(1) Time Series as a Function of Past Innovations (Impulses or Shocks)
yt   n yt n   n1
 ...   3 y   2v  v
t n1
t 3
t 2
t
(6)
 yt n
n
In the limit the term
n  .
becomes close to zero as
 
Rearranging (6) we can write yt in terms of current
and past values of error terms
yt  vt  v   2v   3v   4v  ...   n1 t 
t 1
t 2
t 3
t 4
(7)
4
Time Dependent Variance
What is the mean of
yt  in (7)?




2 
3 
4 
n1E 

E  yt   E  vt   E  v
   E v
   E v
   E v
  ...  


 
 t n 
 t 1 
 t 2
 t 3 
 t 4

2 



v
~
N
0
,


 ; E  yt   0
Because of assumption t
v


What is the variance of yt  ?
  








Var yt   Var  vt   v
 2 v
 3 v
 4 v
 ...   n 1  t  n


t 1
t 2
t 3
t 4


 if   1
and


if there is no autocorrelation among the random terms E  vt vt  1   0
Var yt    v2   v2   v2  .....  v2  t. v2


Thus the variance of Y term increases with time. This makes this series non
stationary.
Rule of thumb : A series is non-stationary if   1.
A series is stationary if
  1.
5
Dicky-Fuller and Augmented Dicky-Fuller Tests
yt  y
 vt
t 1
yt  1   y
 vt ;
t 1
Random Walk:
yt  y
 vt
t 1
Random Walk with a drift (intercept):
yt    y
 vt
0
t 1
Trend stationary process
Null hypotheses:
There is unit root and time
series in non-stationary
=0  (1-)=0
Alternative hypothesis:
There is no unit root and
time series is stationary
<0  (1-)<0  <1
yt     t  y
 vt
0 1
t 1
Augmented Dicky Fuller Test
yt     t  y

0 1
t 1
m
a y
 vt
i t i
i 1

6
Moving Average-MA Process
Yt    et   e
1 t 1
E yt   
var yt   var(   et   e )   e2 1   
1 t 1
1

cov(YtY )  E ( yt   )( y
  )   e21
t 1
t 1
Autocorrelation function: it tapers off after k lags
2
cov yt y 


e
t 1 

1
 
k
var yt 
 e2 1   2 
1

Some examples of MA (1) process:
Yt    et  0.8e
t 1
Yt    et  0.8e
t 1
7
Yt    et   e
1 t 1
E yt   
MA(2)
Process
 e
2
t 2


2
2
2

var yt   var(   et   e   2e )   e 1     
1 t 1
t 2
2
1


cov(YtY )  E ( yt   )( y
  )   e2     
t 1
t 1
1 2
 1
cov(YtY )  E ( yt   )( y
  )   2 e2
t 2
t 2
cov(YtY )  0
t 3
 1   
cov yt y 
t 1  1
2

1 
var yt 

2
2
1     
1
2

cov yt y 

t

2


1
2 

; k
var yt 


2
2
1     
1
2

0
MA(2) process has tow period long memory.
8
Autoregressive Process
Yt     y  et
1 t 1

E yt   E  y   .....  E  y

 t 1 
 t k 
E Yy   E    y
e 
1 t 1 t 

     ;  
1

1 
1
2

2
e
var yt  
=>  y 
1  2
1
cov(YtY )  E ( yt  E  yt )( y
 E  yt )    2




t 1
t 1
1 y
var  y
 et 
1
t

1


Some examples:
Yt  0.8 y  et
t 1
Yt  0.8 y  et
t 1
Convergence occurs if 1  1 . The series is called
stationary.
9
ARMA(1,1) Process
Yt     y
 e  e
1 t  1 t 1 t 1
var yt  
E 

yt   
2




E 



2

  y  et   e 
1 t 1
1 t 1



2   2   2 2  2  E  y e 
e
1 0
1 e
1 1  t 1 t 1
=
10
Co-integration
If two economic variables have long-run equilibrium
relationship linear combination of these variables may be
stationary even if the individual series may be non
stationary. These two variables are said to be co-integrated
to each other.
Suppose yt is consumption and X t is disposable income.
et  Yt     X t
1 2
Even if yt and X t are I(1)
et
is I(0).
et    e  vt
0
t 1
If  is zero then series et is stationary and
I(1).
yt
and X t are
11
Error Correction Model
yt    Yt     X t  vt
1 2
1 2








The term in the parenthesis is the error term and the
coefficient  2 governs the speed of adjustment towards
long-run equilibrium.
12
Structure of a VAR Model
P
P
r
P
j 1
j 1
j 1
j 1
y1,t  a10   a11 j y1,t  j  ..   a1nj y n,t  j   b11 j x1,t  j  ..   b1mj xm,t  j  e1t
.
.
.
P
P
r
P
j 1
j 1
j 1
j 1
y n,t  an 0   an1 j y1,t  j  ..   annj y n,t  j   b11 j x1,t  j  ..   b1mj xm,t  j  ent
Simple Example
y t  a10  a11 y1t 1  a12 y 2,t  2  b11 xt 1  b12 xt  2  e1t
xt  a 20  a 21 y1t 1  a 22 y 2,t  2  b21 x1,t 1  b22 xt  2  e2t
13
Impulse Response Analysis in a VAR Model
 yt   a11 a12   y1t 1  b11 b12   x1t 1   e1t 

 
 x   a






 t   21 a22   y 2t 2  b21 b22   xt 2  e2t 
 yt   a11
 x   a
 t   21
a12 b11
a 22 b21
 y1t 1 
b12   y 2t 2   e1t 
 

b22   x1t 1  e2t 


 xt  2 
Y  I  A BX  I  A U
1
1
X  I  B  AY  I  B  U
1
0 
Y0   
1 
1
Y1  I  A
1
0 
1
 
Y2  I  A Y1  I  A
1
1
 I  A
1
0
1
 14
Stamp Program for Time Series Analysis
Estimation sample is 1971. 2 - 2000. 1. (T = 116, n = 111).
Log-Likelihood is 250.781 (-2 LogL = -501.563).
Prediction error variance is 0.0106071
Summary statistics
ER
Std.Error
0.10299
Normality
9.9490
H( 37)
0.58124
r( 1)
0.0039775
r( 9)
-0.10584
DW
1.9721
Q( 9, 6)
7.2307
Rs^2
-0.41360
ER = Trend + Trigo seasonal + Expl vars + Irregular
Eq 3 : Estimated coefficients of final state vector.
Variable Coefficient R.m.s.e.
t-value
Lvl
1.2519
0.26875
4.6583 [ 0.0000]
Slp
-0.0056233 0.0082428 -0.68221 [ 0.4965]
Sea_ 1
0.0026817 0.0081813
0.32779 [ 0.7437]
Sea_ 2
0.0017017 0.0081876
0.20784 [ 0.8357]
Sea_ 3
-0.00036460 0.0040829 -0.089298 [ 0.9290
Eq 3 : Estimated coefficients of explanatory variables.
Variable Coefficient R.m.s.e.
t-value
ER_1
0.22213 0.096664
2.298 [ 0.0234]
ER_2
-0.070019 0.099049 -0.70692 [ 0.4811]
ER_3
0.027285 0.099054
0.27545 [ 0.7835]
ER_4
0.035090 0.096723
0.36279 [ 0.7174]
Eq 3 : Seasonal analysis (at end of period).
Seasonal Chi^2( 3) test is 0.173462 [0.9818].
Seas 1 Seas 2 Seas 3 Seas 4
Value
0.0023171 0.0020663 -0.0030463 -0.0013371
15
Forecasting of the Exchange Rate
2.00
F-ER
Forecast
1.75
1.50
1.25
1985
2.00
F-ER
1990
1995
2000
2005
1990
1995
2000
2005
2000
16
2005
F-TrendX_ER
1.75
1.50
1.25
1985
F-Seas_ER
0.002
0.000
-0.002
1985
1990
1995
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