1. AR- MA- och ARMA-

2. First try: AR(1)
Use MINITAB’s ARIMA-procedure

4. ARIMA Model
ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
AR , , , ,000
Constant 0, , , ,030
Mean , ,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
Chi-Square , , * *
DF * *
P-Value , , * *

6. ARIMA Model
ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
AR , , , ,000
Constant 0, , , ,030
Mean , ,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
Chi-Square , , * *
DF * *
P-Value , , * *

7. 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

8. 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.

9.  Problems with residuals at nonseasonal level
ARIMA Model
ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
AR , , , ,000
Constant 0, , , ,030
Mean , ,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
Chi-Square , , * *
DF * *
P-Value , , * *
Low P-value for K=12.
 Problems with residuals at nonseasonal level

10. Study SAC and SPAC for the original series:

11. 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)

12. OK! Still not OK ARIMA Model ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
AR , , , ,000
AR , , , ,007
Constant 1, , , ,000
Mean , ,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
Chi-Square , , * *
DF * *
P-Value , , * *
OK!
Still not OK

14. Might still be problematic.
Could it be the case of an Moving Average (MA) model?
MA(1):
are still assumed to be uncorrelated and identically distributed with mean zero and constant variance

16. is in effect a moving average with weights
MA(q):
always stationary
mean=
is in effect a moving average with weights
for the (unobserved) values

19. Try an MA(1):

20. ARIMA Model
ARIMA model for CPIChnge
Final Estimates of Parameters
Type Coef StDev T P
MA , , , ,000
Constant 4, , , ,000
Mean , ,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
Chi-Square , , * *
DF * *
P-Value , , * *

22. Still seems to be problems with residuals
Look again at ACF and PACF of original series:

23. 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

24. mean value calculations more difficult
ARMA(p,q):
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)

26. Always try to keep p and q small.
Try an ARMA(1,1):

27. ARIMA Model
ARIMA model for CPIChnge
Unable to reduce sum of squares any further
Final Estimates of Parameters
Type Coef StDev T P
AR , , , ,000
MA , , , ,000
Constant 1, , , ,004
Mean , ,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
Chi-Square , , * *
DF * *
P-Value , , * *

29. Calculating forecasts
For AR(p) models quite simple:
is set to 0 for all values of k

30. For MA(q) ??
MA(1):
If we e.g. would set and equal to 0
the forecast would constantly be
which is not desirable.

31. Note that
Similar investigations for ARMA-models.