Review: Stationary Time Series Models
White noise process, covariance stationary process, AR(p), MA(p) and ARIMA
processes, stationarity conditions, diagnostic checks.
White noise process xt ~ i.i.d
N (0, 2 )
A sequence {xt } is a white noise process if each value in the sequence has
1. zero-mean E ( xt )  E ( xt 1 )  ..  0
2. constant conditional variance E( x2t )  E( xt 1 )  .. 2  Var( xt i )
3. is uncorrelated with all other realizations
E ( xt xt  s )  E ( xt  j xt  j  s )  ..  0  Cov( xt  j xt  j  s )
2
Properties 1&2 : absence of serial correlation or predictability
Property 3
: Conditional homoscedasticity (constant conditional variance).
Covariance Stationarity (weakly stationarity)
A sequence {xt } is covariance stationary if the mean, var and autocov do not grow over
time, i.e. it has
1. finite mean E ( xt )  E ( xt 1 )  ..  
2. finite variance E[( x t   )2 ]  E[( xt 1   )2 ]  ..   x  Varx
3. finite autocovariance
E[( xt   )( xt s   )]  E[( xt  j   )( xt  j s   )]  ..   s  Cov( xt  j xt  j s )
2
Ex. autocovariance between xt , xt  s   s 
s
0
But white noise process does not explain macro variables characterized by persistence so
we need AR and MA features.
Application:
plot wn and lyus
Program: whitenoise.prg
workfile: usuk.wf –page 1
The program generates and plots log of US real disposable income and a white
noise process WN, based on the sample mean and variance of log US income
(nrnd=normal random variable with 0 mean and SD of 0.36)
lyus=log(uspdispid)
WN= 8.03+0.36*nrnd
9.0
8.5
8.0
7.5
7.0
1960 1965 1970 1975 1980 1985 1990 1995
WN
AR(1): xt  xt 1  et
LYUS
et ~ i.i.d (0, 2 ) ,
(random walk:   1 )
MA(1): xt  et  et 1
More generally:
AR(p): xt  1 xt 1   2 xt  2  ...   p xt  p  et
MA(q): xt  et  1et 1   2et  2  ...   q et  q
ARMA(p,q): xt  1 xt 1   2 xt  2  ...   p xt  p  et  1et 1  ..   q et  q
Using the lag operator:
AR(1): (1  L) xt  et
MA(1): xt  (1  L)et
AR(p): (1  1L  2 L2  ...   p LP ) xt   ( L) xt  et
MA(q): xt  (1  1L  2 L2  ...  q Lq )et   ( L)et
ARMA(p,q): a( L) xt  b( L)et
1. AR process
Stationarity Conditions for an AR(1) process
(1  L) xt  et with  ( L)  1  L and substituting for L:  ( z )  1  z
The process is stable if  ( z )  0 for all numbers satisfying z  1 . Then we can write
xt  (1  L)1 et  i 0  i et i

If x is stable, it is covariance stationary:
1. E ( xt )   or 0 – finite
 e2
2. Varx  E[ x t ]  E (i  0  et  i )   e i  0  
  0 -- finite
1 2
2

i
2
2

2i
3. covariances
 1  E ( xt xt 1 )  E[( xt 1  et ) xt 1 ]   x2
 2  E ( xt xt  2 )  E[(( xt  2  et 1 )  et ) xt  2 ]   2 E ( xt2 2 )   2 x2
 s  E ( xt xt  s )   s x2   sVar ( x)
Autocorrelations between xt , xt  s :

rs  s   s
0
Plot of rs over time = Autocorrelation function (ACF) or correlogram.
For stationary series, ACF should converge to 0:
lim rs  0 if   1
s 
  0  direct convergence
  0  dampened oscillatory path around 0.
Partial Autocorrelation (PAC)
Ref: Enders Ch.2
In AR(p) processes all x’s are correlated even if they don’t appear in the regression
equation.
Ex: AR(1)
 s   sVar ( xt )



r1  1   ; r2  2   2  r1  r12 ; r3  3   3  r2 r1
0
0
0
We want to see the direct autocorrelation between xt  j and xt by controlling for all x’s
between the two. For this, construct the demeaned series and form regressions to get the
PAC from the ACs.
1st PAC:
*
xt*  11xt 1  et
 11  r1
2nd PAC:
x  x
*
21 t 1
*
t
 x
*
22 t  2
 et
In general, for s  3 , sth PAC:
s
rs   j 1s 1, j rs  j
ss 
s 1
1   j 1s 1, j rj
Ex: for s=3, 33 
  22
r2  r12
.

1  r12
and sj  s 1, j  sss 1, s  j .
r3  (21r2  22r1 )
.
1  (21r1  22r2 )
Identification for an AR(p) process
PACF for s>p: ss  0
Hence AR(1): 22 
r2  11r r2  r12 r12  r12


0
1  11r1 1  r12
1  r12
33 
r3  (21r2  22r1 )
1  (21r1  22r2 )
To evaluate it, use the relation sj  s 1, j  sss 1, s  j :
21  1,1  2211  11  r1 , substitute it to get:
33 
r3  r1 r2
1  r1
2
=
r13  r13
0
1  r12
Stability condition for an AR(p) process
(1  1L  2 L2  ...   p LP ) xt   ( L) xt  et
The process is stable if  ( z )  0 for all z satisfying z  1 , or if the roots of the
characteristic polynomial lie outside the unit circle. Then, we can write:

xt   ( L) 1 et
 a( L)et
  j  0 ai et i .
Then we have the usual moment conditions:
1. E ( xt )   or 0 – finite
2. Varx  E[ x t ]  E(i 0 ai et i )2 , a0  1, et i et  j  0

2
 Varx   e
2
i  0 ai 

2
 e2
1  a2
  0 -- finite variance, hence time independent.
3. covariances
 s  E ( xt xt  s )  E[(et  a1et 1  a2et  2  ...)(et  s  a1et  s 1  ...)]   2 E ( xt2 2 )   2 x2
= [(as  a1a1 s  a2a2  s  ...) 2e   e2 i 0 ai ai  s finite and time

independent.

a a  a a  a3a4  ...
r1  1  1 2 2 3
0
(a1  a2  ...) 2
rs 
s

0
 aa
 a
i is
2
i
If the process is nonstationary, what do we do?
Then there is a unit root, i.e. the polynomial has a root for z=1   (1)  0 . We can thus
factor out the operator and transform the process into a first-difference stationary series:
 ( L)  (1   *1 L   *2 L2  ...   *p 1 Lp 1 )(1  L)
 (1   *1 L   *2 L2  ...   *p 1 Lp 1 )xt  et -- an AR(p-1) model.
If  * ( L)  (1   *1 L   *2 L2  ...   *p1 Lp1 ) has all its roots outside the unit circle,
xt is stationary: xt ~ I (1)
If  * ( L) still has a unit root, we must difference it further until we obtain a stationary
process: xt ~ I (d )
An integrated process = a unit root process.
 unconditional mean is still finite but
 Variance is time dependent
 Covariance is time dependent
(more later).
Application: generate an AR(1) series
Program: ARMA.prg
Workfile: USUK.wf, page 2 (undated)
Program:
smpl 1 1
genr x=0
smpl 2 200
series x=0.5*x(-1)+NRND '
'nrnd=normal random variable with 0 mean and SD of 0.36
Go to workfile, Click on: series – graph -- line
3
2
1
0
-1
-2
-3
25
50
75
100
125
150
175
200
X
On series, click on View – Correlogram, level – OK
Date: 09/04/07 Time: 19:43
Sample: 2 200
Included observations: 199
Autocorrelation
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Partial Correlation
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1
2
3
4
5
6
7
8
9
10
11
12
13
14
AC
PAC
Q-Stat
Prob
0.537
0.257
0.074
0.003
-0.078
-0.087
-0.059
-0.029
0.013
-0.081
-0.015
0.010
0.076
0.101
0.537
-0.045
-0.065
-0.002
-0.087
-0.007
0.015
-0.002
0.036
-0.156
0.113
0.000
0.071
0.051
58.316
71.715
72.832
72.834
74.074
75.654
76.369
76.549
76.586
77.970
78.017
78.039
79.294
81.484
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
ACF=0 at lag 3 and PAC=0 at lag 2 hence AR(1).
On Eviews: Q stats=Ljun-Box Q statistics and their p-values. H0: there is no autocorrelation up to
lag k, asymptotically distributed as
q2 , q=# autocorrelations.
Dotted lines: 2 SE bounds calculated approximately as
Here: T=199 hence SE bounds =0.14.
Program:
smpl 1 1
genr xa=0
smpl 2 200
series xa= -0.5*xa(-1)+NRND
 2 / T , T=# observations.
rho=-0.5  dampened oscillatory path.
smpl 1 1
genr w=0
smpl 2 200
series w=w(-1)+NRND
rho=1  random walk  unit root.
2. MA process
xt  et  1et 1   2et  2  ...   q et  q ,
e = 0 mean white noise error term.
 (1  1L  2 L2  ...  q Lq )et =  ( L)et
If  ( z )  0 for z  1 , the process is invertible, and has an AR() representation:
 ( L) 1 xt   ( L) xt =


j 0
 j xt  j  et
 xt   j 1 j xt  j  et

 xt   j 1 j xt  j  et

Stability condition for MA(1) process
xt  et  1et 1
Invertibility requires 1  1
Then the AR representation would be:

xt  (1  1L)et  (1  1L) 1 xt  et  xt  i 11i xt i  et

E ( xt )  0 finite

Var ( xt )   (0)  var( et  1et 1 )  (1  12 ) e2 finite.
 (1)
1
 (1)  1Ee2t 1  1 e2  r1 

 (0) 1  12
 (2)   (3)  0  r2  r3  0 , hence autocorrelations’ cut off point = lag

1
More generally : AC for MA(q)=0 for lag q.

PAC:
11  r1 
1
2
1  1
r  r2
 1

22  2 12 
 11 ( 11 2 )
2
2
1  r1
1  11
(1  1 )(1  1  1 )
2
33 
 22r1
1  21r1


 2211
 2211
 11 22 (?? check)

1  22
1  2111 1  (11 (1  22 ))11
 For AR:
AC depends on the AC coefficient (rho), thus tapers off
PAC depends on rs or  s , cuts of 0 at s (AR(1): cutoff at L=1)
 For MA:
AC depends on var of error terms: abrupt cutoff
PAC depends on the MA coefficient  , thus tapers off.
3. ARMA process
ARMA(p,q):  ( L) xt   ( L)et
(1  1L  2 L2  ...   p LP ) xt  (1  1L  2 L2  ...  q Lq )et
If q=0  pure AR(p) process
If p=0  pure MA(q) process
If all characteristics roots of xt  i1 i xt i i0i et i are within the unit circle, then
p
q
this is an ARMA(p,q) process. If one or more roots lie outside the unit circle, then this is
an integrated ARIMA(p,d,q) process.
Stability condition for ARMA(1,1) process
--Favero, p.37—
xt  c1 xt 1  et  a1et 1
 1  a1L 
et .
(1  c1 ) xt  (1  a1L)et  xt  
 1  c1L 
If c1  1 then we can write
xt  (1  a1L)(1  c1L  (c1L) 2  ...)et
 (1  c1L  (c1L) 2  ...  a1L  a1c1L2  ...)et
 (1  (a1  c1 ) L  c1 (c1  a1 ) L2  c12 (c1  a1 ) L3  ...)et  an MA( ) representation.



E ( xt )  0 finite
 (a  c ) 2 
Var ( xt )   (0)  1  1 21  e2 finite
1  c1 

Covariances --finite
 (1)  c1Var ( xt )  a1 e2
 (2)  c1 (c1Var ( xt )  a1 e2 )  c1 (1)   ( j )  c1 ( j  1),
j2
Autocov functions :
 (1)
r1 
 (0)
r2  c1r1
r3  c1r2  c12 r1
Any stationary time series can be represented with an ARMA model:
AR( p)  MA()
MA( p)  AR()
Application
Plot and ARMA(1,1) and an AR(1) model with rho=0.7 and theta=0.4, and look at the
AC and PAC functions.
Prog: ARMA.prg
File: USUK, page 2.
smpl 1 1
genr zarma=0
genr zar=0
smpl 1 200
genr u=NRND
smpl 2 200
series zarma=0.7*x(-1)+u+0.4*u(-1)
series zar=0.7*x*(-1)
plot zarma zar
zarma.correl(36)
zar.correl(36)
4
3
2
1
0
-1
-2
-3
-4
25
50
75
100
ZARMA
125
150
ZAR
175
200
Summary of results
Enders Table 2.1
ACF
WN
rs  0, s  0
Exponential Decay:
AR(1)
rs   s
  0  direct decay
  0  oscillating
Positive (negative) spike
MA(1)
at lag 1 for   0 (   0 )
rs  0 for s  2
ARMA(1,1) Exponential (oscillating) decay
at lag 1 if   0 (   0 ).
Decay at lag q for ARMA(p,q)
PACF
 ss  0, s  0
Spike at lag 1 (at p for AR(p))
11  r1 ,  ss  0 for s  2
Oscillating (geometric) decay
for 11  0 ( 11  0 )
Oscillating (exponential) decay
at lag 1, 11  1
Decay after lag p for ARMA(p,q)
Stationary Time Series II
Model Specification and Diagnostic tests
E+L&K Ch 2.5,2.6
So far we saw the theoretical properties of stationary TS. To decide about the type of
model to use, we need to decide about the order of operators (# lags), the deterministic
trends, etc. We therefore need to specify a model and then conduct tests on its
specification, whether it represents the DGP adequately.
1. Order specification criteria:
i.
The t/F-statistics approach: start from a large number of lags, and reestimate
by reducing by one lag every time. Stop when the last lag is significant.
Monthly data: look at the t-stats for the last lag, and F-stats for the last quarter.
Then check if the error term is white noise.
ii.
Information Criteria: AIC, HQ or SBC
Definition in Enders:
Akaike:
AIC=T.log(SSR)+2n
Schwarz:
SBC=T.log(SSR)+n.log(T) also called Schwartz and Risanen.
Hannan-Quinn:
HQC=T. log(SSR)+2n.log(log(T))
T=#observations, n=#parameters estimated, including the constant term.
Adding additional lags will reduce the SSR, these criteria penalize the loss of
degree of freedom that comes along with additional lags.
The goal=pick the #lag that minimizes the information criteria.
Since ln(T)>2, SBS selects a more parsimonious model than AIC.
For large sample, HQC and in particular SBC is better than AIC since they
have better large sample properties.
o If you go along with SBC, verify that the error term is white
noise. In small samples, AIC performs better.
o If you go with AIC, then verify that the t-stats are significant.
This is valid whether the processes are stationary or integrated.
Note: various software and authors use modified versions of these tests. As
long as you use the criteria consistently among themselves, any version will
give the same results.
For ex: definition by LK:
AIC=log(SSR) + 2(n/T)
SC= log(SSR) + n(logT/T)
HQ=log(SSR) + 2n.log(logT)/T)
Definition used in Eviews :
AIC=-2(l/T)+(2n/T)
SC=-2(l/T)+n(logT/T)
HQ=-2(l/T)+2n.log(logT)/T
T
(1  log(2 )  log SSR / T ) and
2
 is the constant of the log likelihood often omitted.
where l is the log likelihood given by l  
Denote the order selected by each criterion as p̂ . The following holds independent of the
size of the sample:
pˆ ( AIC )  pˆ ( HQ )  pˆ (SC)
Word of warning:
It is difficult to distinguish between AR, MA and mixed ARMA processes based on
sample information. The theoretical ACF and PACF are not usually replicated in real
economic data, which have a complicated DGP. Look into other tests that analyze the
residuals once a model is fitted:
2. Plotting the residuals:
Check for outliers, structural breaks, nonhomogenous variances.
 Look at standardized residuals (subtract the mean and divide by standard
T
uˆ  uˆ
 (uˆ  uˆ ) 2
deviation): uˆ s  t~ where ̂ u  1 t t is the standard deviation and û is
u
T
the mean. If û ~N(0,  ) then û will be in general in a  2 band around the 0
line.
Look at the AC and PAC to check the remaining serial residuals in the residuals,
and AC of squared residuals to check for conditional heteroscedasticity. If the
2

s
AC and PAC of earlier lags are not in general within  2 / T band around 0 then
there is probably left over serial dependence in the residuals or conditional
heteroscedasticity.
3. Diagnostic tests for residuals:
i. Test of whether the kth order autocorrelation is significantly different from zero:
The null hypothesis: there is no residual autocorrelation up to order s
The alternative: there is at least one nonzero autocorrelation.
H 0 : rk  0 and k=1,..s
H1 : rk  0 for at least one k=1,..s.
where rk is the k-th autocorrelation.
If the null is rejected, then at least one r is significantly different from zero.
The null is rejected for large values of Q. If there are any remaining residual
autocorrelation, must use a higher order of lag.

The Box-Pierce Q-statistics (Portemanteau test for residual autocorrelation)
s
2
Q  T  rk2 ~  (s)
k 1
T=# observations.
But not reliable for small samples and it has reduced power for large s. Instead, use

Ljung-Box Q statistics:
QLB  T (T  2)k 1 rk2 /(T  k )
s
with similar null and alternative hypotheses.
It can also be used to check if the residuals from an estimated ARMA(p,q) model are
white noise (adjust for the lags in AR(p) and MA(q)):
2
2
Q~  (s  p  q) or  (s  p  q  1) with a constant.
ii. Breusch-Godfrey (LM) test for autocorrelation for AR models for residuals:
It considers an AR(h) model for residuals.
Suppose the model you estimate is an AR(p):
yt     pj1  j yt  j  ut
You fit an auxiliary equation
p
h
uˆt   j 1 j yt  j i 1 iuˆt i  errort where ût u is the OLS residual from the
(*)
AR(p) model for y.
H 0 : 1  ..   h  0
H 1 : 1  0 or
 2  0,....
The LM statistics for the null: LMh  T .R2 ~  2 (h) , where R 2 is obtained from fitting (*).
For better small sample properties use an F version:
FMLh 
R2 T  p  h 1
~ F (h, T  p  h  1)
h
1  R2
iii. Jarque-Bera test for nonnormality
It tests if the standardized residuals are normally distributed, based on the third and fourth
moments, by measuring the difference of the skewness and the kurtosis of the series with
those from the normal distribution.
H 0 : E (uts )3  0 ( skewness) and
H1 : E (uts )3  0 or
E (uts )4  3 (kurtosis)
E (uts )4  3
JB ~  2 (2) and the null is rejected if JB is large. In this case, residuals are considered
nonnormal.
Note:


most of the asymptotic results are also valid for nonnormal residuals.
the results may be due to nonlinearities. Then you should look into ARCH
effects or structural changes.
iv. ARCH-LM test for conditional heteroscedasticity
Fit an ARCH(q) model to the estimation of the residuals
q
uˆ 2t  i 0 iuˆ 2t i  errort and test if
H 0 : 1  ..   q  0 no conditional heteroscedasticity
H1 : 1  0 or  2  0,....
ARCH  LM (q)  TR2 ~  2 (q) . Large values of ARCH-LM show that the null is rejected
and there are ARCH effects in the residuals. Then fit an ARCH or GARCH model.
v. RESET
Tests a model specification against alternatives (nonlinear).
Ex: you are estimating a model
yt  axt  ut
But the actual models is
yt  axt  b zt  ut
where z can be missing variable(s) or a multiplicative relation. The test checks if powers
of predicted values of y is significant. These consist of the powers and cross-product
terms of the explanatory variables:
zt  { yˆ 2 , yˆ 3 ,...}
H 0 : b=0 --no misspecification
The test statistics has an F(h-1,T) distribution. The null is rejected if the test statistics is
too large.
vi. Stability analysis
Recursive plot of residuals, of estimated coefficients;
CUSUM test (cumulative sum of recursive residuals): if the plot diverges significantly
from the zero line, it suggests structural instability.
CUSUMSQR (square of CUSUM) in case there are several shifts in different directions.
Chow test: for exogenous break points.
Example:
Workfile: LKEdata.wf page--Enders
Series: US PPI
Plot the data:
120
100
80
60
40
20
1960 1965 1970 1975 1980 1985 1990 1995 2000
PPI
Positive trend in the series indicates nonstationarity.
Correlogram:
Date: 09/19/07 Time: 10:00
Sample: 1960Q1 2002Q2
Included observations: 169
Autocorrelation
.|********
.|********
.|*******|
.|*******|
.|*******|
.|*******|
.|*******|
.|*******|
.|*******|
.|*******|
.|*******|
.|****** |
Partial Correlation
.|********
.|.
|
*|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|.
|
AC
1
2
3
4
5
6
7
8
9
10
11
12
0.990
0.978
0.966
0.952
0.937
0.923
0.908
0.894
0.880
0.866
0.852
0.838
PAC
Q-Stat
Prob
0.990
-0.044
-0.071
-0.056
-0.036
0.003
-0.003
-0.004
0.009
-0.003
-0.011
0.001
168.44
334.05
496.37
655.05
809.87
960.86
1108.0
1251.5
1391.2
1527.5
1660.1
1789.4
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
The confidence intervals [2 / T , 2 / T ] = [2 / 169 , 2 / 169 ]  [0.154 ,  0.154 ]
Q statistics highly significant, the residuals are not white noise. Need to respecify the
model.
ACF dies out very slowly
PACF (the autocorrelation conditional on the in-between values of time series): single
spike
Observation of the data suggests U.R. Hence we cannot use the models developed so far.
First difference dppi.
Plot: it seems to be stationary, though the variance does not look constant, increases in
the second half of the sample.
4
3
2
1
0
-1
-2
-3
-4
1960
1965
1970
1975
1980
1985
DPPI
1990
1995
2000
ACF dies out quickly after the 4th lag, and PACF has a large spike and dies out
oscillating. Note the significant correlations and PAC at lag 6 and PAC at lag 8. This
can be an AR(p) or an ARMA(p,q) process.
Included observations: 168
Autocorrelation
Partial Correlation
AC
PAC
Q-Stat
Prob
.|**** |
.|**** |
1
0.553
0.553
52.210
0.000
.|***
|
.|.
|
2
0.335
0.043
71.571
0.000
.|**
|
.|*
|
3
0.319
0.170
89.229
0.000
.|**
|
.|.
|
4
0.216
-0.042
97.389
0.000
.|*
|
*|.
|
5
0.086
-0.081
98.682
0.000
.|*
|
.|*
|
6
0.153
0.149
102.82
0.000
.|*
|
*|.
|
7
0.082
-0.096
104.02
0.000
*|.
|
*|.
|
8
-0.078
-0.149
105.10
0.000
*|.
|
.|.
|
9
-0.080
-0.007
106.23
0.000
.|.
|
.|*
|
10
0.023
0.121
106.33
0.000
.|.
|
.|.
|
11
-0.008
0.004
106.34
0.000
.|.
|
.|.
|
12
-0.006
0.007
106.35
0.000
In Eviews:
run equation dppi c AR(1)
(You can also run equation dppi c dppi(-1), but with the first specification Eviews
estimates the model with a nonlinear algorithm, which helps controlling for nonlinearities
if they exist in the DGP).
Dependent Variable: DPPI
Method: Least Squares
Date: 09/19/07 Time: 10:13
Sample (adjusted): 1960Q3 2002Q1
Included observations: 167 after adjustments
Convergence achieved after 3 iterations
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
AR(1)
0.464364
0.554760
0.133945
0.064902
3.466835
8.547697
0.0007
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.306907
0.302706
0.770679
98.00107
-192.4560
2.048637
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.466826
0.922923
2.328814
2.366155
73.06313
0.000000
Inverted AR Roots
.55
Check the optimal lag length for an AR(p) process.
Check the roots: if it is <1, stability is confirmed. View-ARMA structure-roots shows
you the root(s) visually.
View-ARMA structure-correlogram: allows you to compare the estimated correlations
with the theoretical ones.
View-ARMA structure-impulse response: shows how the DGP changes with a shock of
the size of one standard deviation.
View-Residual tests: to check if there is any residual autocorrelation left.
Optimal lag length:
lags
1
2.328814*
2.366155*
AIC
SBC
2
2.565575
2.603069
3
2.575278
2.612925
10
2.735557
2.774324
It appears that AR(1) dominates the rest in terms of the information criteria.
Roots of AR(1):
0.55<1 –stability condition satisfied, since the root is inside the unit circle.
Inverse Roots of AR/MA Polynomial(s)
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
AR roots
But serial correlation is a problem:
Breusch-Godfrey Serial Correlation LM Test:
F-statistic
Obs*R-squared
3.478636
6.836216
Prob. F(2,163)
Prob. Chi-Square(2)
0.033159
0.032774
High Q stats (correlogram), and the Godfrey-Breusch test TR2 =6.84>  (22)  5.99 rejects
the null at the 5% confidence interval, thus there is at least one i  0 . There is serial
correlation in the error terms.
Lets try ARMA(p,q):
lags
1,1
AIC
SBC
2.3305*
2.386*
1,2
2,1
2.345
2.3366
2.3926
2.402
Information criteria favor an ARMA(1,1) specification although the marginal change is
very small. So we should consider all options.
ARMA (1,1) -- Eviews: equation dppi c ar(1) ma(1)
Included observations: 167 after adjustments
Convergence achieved after 9 iterations
Backcast: 1960Q2
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.455286
0.163873
2.778280
0.0061
AR(1)
0.730798
0.100011
7.307163
0.0000
MA(1)
-0.261703
0.139158
-1.880622
0.0618
R-squared
0.313995
Mean dependent var
0.466826
Adjusted R-squared
0.305629
S.D. dependent var
0.922923
S.E. of regression
0.769062
Akaike info criterion
2.330510
96.99884
Schwarz criterion
2.386522
F-statistic
37.53259
Prob(F-statistic)
0.000000
Sum squared resid
Log likelihood
-191.5976
Durbin-Watson stat
1.935301
Inverted AR Roots
.73
Inverted MA Roots
.26
Both roots inside the unit circle. AR and MA coefficients significant.
Some of the AC at larger lags are significant.
LM test: TR2 =5.29 <  (22)  5.99 does not reject no serial correlation in the error terms.
Notice that if you estimate an ARMA(1,2) the MA coefficient is insignificant. Together
with the information criteria, we can rule out this specification. The LM test results also
do not support this specification.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.457550
0.176196
2.596826
0.0103
AR(1)
0.807616
0.120152
6.721624
0.0000
MA(1)
-0.280755
0.148664
-1.888526
0.0607
MA(2)
-0.152639
0.116160
-1.314038
0.1907
R-squared
0.322970
Mean dependent var
0.466826
Adjusted R-squared
0.310509
S.D. dependent var
0.922923
S.E. of regression
0.766355
Akaike info criterion
2.329317
95.72980
Schwarz criterion
2.404000
F-statistic
25.91910
Prob(F-statistic)
0.000000
Sum squared resid
Log likelihood
-190.4980
Durbin-Watson stat
2.011448
Inverted AR Roots
.81
Inverted MA Roots
.56
-.27
ARMA (2,1) is better than ARMA(1,2) but does not dominate ARMA(1,1). The
estimated coefficients are significant, but there is residual serial correlation.
We can perform additional tests to check the stability of the coefficient, seasonal effects,
etc.
Seasonality
Cyclical movements observed in daily, monthly or quarterly data. Often it is useful to
remove it if it is visible in lags s, 2s, 3s, …. of the ACF and the PACF. You add an AR
or MA coefficient at the appropriate lag.
Possibilities:
Quarterly seasonality with MA at lag 4:
xt  a1 xt 1  et  b1et 1  b4et  4
Quarterly seasonality with AR at lag 4:
xt  a1 xt 1  a4 xt  4  et  b1et 1
Multiplicative seasonality
Accounts for interaction of the ARMA and seasonal effects.
MA term at lag 1 interacting with the seasonal MA term at lag 4:
(1  a1L) xt  (1  b1L)(1  b4 L4 )et
xt  a1 xt 1  et  b1et 1  b4et  4  b1et 1b4et  4
AR term at lag 1 interacting with the seasonal AR term at lag 4:
(1  a1 )(1  a4 L4 ) xt  (1  b1L)et
xt  a1 xt 1  a4 xt  4  a1a4 xt 1 xt  4  et  b1et 1
One solution to remove strong seasonality is to transform the data by seasonal
differencing. For quarterly data:
yt  (1  L4 ) xt
If the data is not stationary, then also need to first difference it:
yt  (1  L)(1  L4 ) xt
Eviews:
Several methodologies for seasonal adjustment
Series—Proc—Seasonal Adjustment
Census X12
Census 11
Moving Average Methods
Tramo-Seat method
The first two options allow additive or multiplicative specifications, control for holidays,
trading days, outliers.
Homework: due September 26, 2007
Data: Enders Question 12.