Post on 11-Jan-2016
First try: AR(1)
ttt ayy 1
Use MINITAB’s ARIMA-procedure
ARIMA Model
ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
AR 1 0,8247 0,1048 7,87 0,000
Constant 0,7634 0,3347 2,28 0,030
Mean 4,354 1,909
Number of observations: 33
Residuals: SS = 111,236 (backforecasts excluded)
MS = 3,588 DF = 31
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 25,9 32,2 * *
DF 10 22 * *
P-Value 0,004 0,075 * *
ARIMA Model
ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
AR 1 0,8247 0,1048 7,87 0,000
Constant 0,7634 0,3347 2,28 0,030
Mean 4,354 1,909
Number of observations: 33
Residuals: SS = 111,236 (backforecasts excluded)
MS = 3,588 DF = 31
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 25,9 32,2 * *
DF 10 22 * *
P-Value 0,004 0,075 * *
Ljung-Box statistic:
where
n is the sample size
d is the degree of nonseasonal differencing used to tranform original series to be stationary. Nonseasonal means taking
differences at lags nearby, which can be written (1–B)d
rl2(â) is the sample autocorrelation at lag l for the residuals of the estimated model.
K is a number of lags covering multiples of seasonal cycles, e.g. 12, 24, 36,… for monthly data
K
ll arldndndnKQ
1
2* )ˆ(2
Under the assumption of no correlation left in the residuals the Ljung-Box statistic is chi-square distributed with K – nC degrees of freedom, where nC is the number of estimated parameters in model except for the constant
A low P-value for any K should be taken as evidence for correlated residuals, and thus the estimated model must be revised.
ARIMA Model
ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
AR 1 0,8247 0,1048 7,87 0,000
Constant 0,7634 0,3347 2,28 0,030
Mean 4,354 1,909
Number of observations: 33
Residuals: SS = 111,236 (backforecasts excluded)
MS = 3,588 DF = 31
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 25,9 32,2 * *
DF 10 22 * *
P-Value 0,004 0,075 * *
Low P-value for K=12.
Problems with residuals at nonseasonal level
Study SAC and SPAC for the original series:
PACF look not fully consistent with AR(1)
More than one significant spike (2 it seems)
If an AR(p)-model is correct, the ACF should decrease exponentially (montone or oscillating)
and PACF should have exactly p significant spikes
Try an AR(2)
ARIMA Model
ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
AR 1 1,1872 0,1625 7,31 0,000
AR 2 -0,4657 0,1624 -2,87 0,007
Constant 1,3270 0,2996 4,43 0,000
Mean 4,765 1,076
Number of observations: 33
Residuals: SS = 88,6206 (backforecasts excluded)
MS = 2,9540 DF = 30
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 19,8 25,4 * *
DF 9 21 * *
P-Value 0,019 0,231 * *
OK!
Still not OK
Might still be problematic.
Could it be the case of an Moving Average (MA) model?
MA(1):
1 ttt aay
ta are still assumed to be uncorrelated and identically distributed with mean zero and constant variance
MA(q):
qtqttt aaay 11
• always stationary
• mean=
• is in effect a moving average with weights
q ,,,1 ,21
for the (unobserved) values qttt aaa ,,, 1
Index
AR(1
)_0.2
200180160140120100806040201
5
4
3
2
1
0
Time Series Plot of AR(1)_0.2
Index
AR(1
)_0.8
200180160140120100806040201
14
13
12
11
10
9
8
7
6
5
Time Series Plot of AR(1)_0.8
Index
MA
(1)_
0.2
3002702402101801501209060301
3
2
1
0
-1
-2
-3
Time Series Plot of MA(1)_0.2
Index
MA
(1)_
0.8
3002702402101801501209060301
4
3
2
1
0
-1
-2
-3
-4
Time Series Plot of MA(1)_0.8
Index
MA
(1)_
(-0.5
)
3002702402101801501209060301
4
3
2
1
0
-1
-2
-3
Time Series Plot of MA(1)_ (-0.5)
Index
AR(1
)_(-
0.5
)
200180160140120100806040201
5
4
3
2
1
0
-1
-2
-3
Time Series Plot of AR(1)_ (-0.5)
Try an MA(1):
ARIMA Model
ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
MA 1 -0,9649 0,1044 -9,24 0,000
Constant 4,8018 0,5940 8,08 0,000
Mean 4,8018 0,5940
Number of observations: 33
Residuals: SS = 104,185 (backforecasts excluded)
MS = 3,361 DF = 31
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 33,8 67,6 * *
DF 10 22 * *
P-Value 0,000 0,000 * *
Still seems to be problems with residuals
Look again at ACF and PACF of original series:
The pattern corresponds neither with AR(p), nor with MA(q)
Could it be a combination of these two?
Auto Regressive Moving Average (ARMA) model
ARMA(p,q):
qtqttptptt aaayyy 1111
• stationarity conditions harder to define
• mean value calculations more difficult
• identification patterns exist, but might be complex: exponentially decreasing patterns or sinusoidal decreasing patterns in both ACF and PACF (no cutting of at a certain lag)
Index
ARM
A(1
,1)_
(0.2
)(0.2
)
3002702402101801501209060301
3
2
1
0
-1
-2
-3
Time Series Plot of ARMA(1,1)_ (0.2)(0.2)
Index
ARM
A(1
,1)_
(-0.2
)(-0
.2)
3002702402101801501209060301
3
2
1
0
-1
-2
-3
Time Series Plot of ARMA(1,1)_ (-0.2)(-0.2)
Index
ARM
A(2
,1)_
(0.1
)(0.1
)_(-
0.1
)
3002702402101801501209060301
3
2
1
0
-1
-2
-3
-4
Time Series Plot of ARMA(2,1)_ (0.1)(0.1)_ (-0.1)
Always try to keep p and q small.
Try an ARMA(1,1):
ARIMA Model
ARIMA model for CPIChnge
Unable to reduce sum of squares any further
Final Estimates of Parameters
Type Coef StDev T P
AR 1 0,6513 0,1434 4,54 0,000
MA 1 -0,9857 0,0516 -19,11 0,000
Constant 1,5385 0,4894 3,14 0,004
Mean 4,412 1,403
Number of observations: 33
Residuals: SS = 61,8375 (backforecasts excluded)
MS = 2,0613 DF = 30
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 9,6 17,0 * *
DF 9 21 * *
P-Value 0,386 0,713 * *
Calculating forecasts
For AR(p) models quite simple:
1)1(211
)2(2)1(1
)2(2112
)1(1211
ˆˆˆˆˆˆˆˆ
ˆˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆ
tpptptpt
tpptptpt
ptpttt
ptpttt
yyyy
yyyy
yyyy
yyyy
kta is set to 0 for all values of k
For MA(q) ??
MA(1):
1ˆˆˆ ttt aay
If we e.g. would set and equal to 0
the forecast would constantly be .
which is not desirable.
ta 1ta
Note that
ˆ)ˆ1(ˆˆ
)1(
0
1
1
2
1
211
ttt
ttt
t
ttt
ttt
yya
yya
a
aay
aay
Similar investigations for ARMA-models.