Post on 19-Mar-2016
description
20001990198019701960
14
12
10
8
6
4
2
0
Year
CPIC
hnge
Yearly changes in Consumer Price Index (CPI), USA, 1960-2001
How should these data be modelled?
Identification step: Look at the SAC and SPAC
1110987654321
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Lag
Auto
corr
elat
ion
Autocorrelation Function for CPIChnge(with 5% significance limits for the autocorrelations)
1110987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Part
ial A
utoc
orre
latio
n
Partial Autocorrelation Function for CPIChnge(with 5% significance limits for the partial autocorrelations)
Looks like an AR(1)-process. (Spikes are clearly decreasing in SAC and there is maybe only one sign. spike in SPAC)
Then we should try to fit the model
The parameters to be estimated are and .
One possibility might be to uses Least-Squares estimation (like for ordinary regression analysis)
Not so wise, as both response and explanatory variable are randomly varying.
Maximum Likelihood better So-called Conditional Least-Squares method can be derived
ttt ayy 1
Use MINITAB’s ARIMA-procedure!!
AR(1)
We can always ask for forecasts
MTB > ARIMA 1 0 0 'CPIChnge';
SUBC> Constant;
SUBC> Forecast 2 ;
SUBC> GSeries;
SUBC> GACF;
SUBC> GPACF;
SUBC> Brief 2.
ARIMA Model: CPIChnge
Estimates at each iteration
Iteration SSE Parameters
0 316.054 0.100 4.048
1 245.915 0.250 3.358
2 191.627 0.400 2.669
3 153.195 0.550 1.980
4 130.623 0.700 1.292
5 123.976 0.820 0.739
6 123.786 0.833 0.645
7 123.779 0.836 0.626
8 123.778 0.837 0.622
9 123.778 0.837 0.621
Relative change in each estimate less than 0.0010
Final Estimates of Parameters
Type Coef SE Coef T P
AR 1 0.8369 0.0916 9.13 0.000
Constant 0.6211 0.2761 2.25 0.030
Mean 3.809 1.693
Number of observations: 42
Residuals: SS = 122.845 (backforecasts excluded)
MS = 3.071 DF = 40
4035302520151051
15
10
5
0
Time
CPIC
hnge
Time Series Plot for CPIChnge(with forecasts and their 95% confidence limits)
10987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Auto
corr
elat
ion
ACF of Residuals for CPIChnge(with 5% significance limits for the autocorrelations)
10987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Part
ial A
utoc
orre
latio
n
PACF of Residuals for CPIChnge(with 5% significance limits for the partial autocorrelations)
All spikes should be within red limits here, i.e. no correlation should be left in the residuals!
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 26.0 35.3 39.8 *
DF 10 22 34 *
P-Value 0.004 0.036 0.227 *
Forecasts from period 42
95% Limits
Period Forecast Lower Upper Actual
43 1.54176 -1.89376 4.97727
44 1.91148 -2.56850 6.39146
Ljung-Box statistic:
where
n is the sample size
d is the degree of non-seasonal differencing used to transform original series to be stationary. Non-seasonal means taking differences at lags nearby.
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.
In this example:Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 26.0 35.3 39.8 *
DF 10 22 34 *
P-Value 0.004 0.036 0.227 *
Here, data is not supposed to possess seasonal variation so interest is mostly paid to K = 12.
P – value for K =12 is lower than 0.05 Model needs revision!
K
A new look at the SAC and SPAC of original data:
1110987654321
1.00.80.60.40.20.0
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Lag
Auto
corr
elat
ion
Autocorrelation Function for CPIChnge(with 5% significance limits for the autocorrelations)
1110987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Part
ial A
utoc
orre
latio
n
Partial Autocorrelation Function for CPIChnge(with 5% significance limits for the partial autocorrelations)
The second spike in SPAC might be considered crucial!
If an AR(p)-model is correct, the ACF should decrease exponentially (monotonically or oscillating)
and PACF should have exactly p significant spikes
Try an AR(2)
i.e.
tttt ayyy 1211
Type Coef SE Coef T P
AR 1 1.1684 0.1509 7.74 0.000
AR 2 -0.4120 0.1508 -2.73 0.009
Constant 1.0079 0.2531 3.98 0.000
Mean 4.137 1.039
Number of observations: 42
Residuals: SS = 103.852 (backforecasts excluded)
MS = 2.663 DF = 39
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 18.6 30.6 36.8 *
DF 9 21 33 *
P-Value 0.029 0.081 0.297 *
Forecasts from period 42
95% Limits
Period Forecast Lower Upper Actual
43 0.76866 -2.43037 3.96769
44 1.45276 -3.46705 6.37257
PREVIOUS MODEL:
Residuals: SS = 122.845 (backforecasts excluded)
MS = 3.071 DF = 40
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 26.0 35.3 39.8 *
DF 10 22 34 *
P-Value 0.004 0.036 0.227 *
Forecasts from period 42
95% Limits
Period Forecast Lower Upper Actual
43 1.54176 -1.89376 4.97727
44 1.91148 -2.56850 6.39146
10987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Auto
corr
elat
ion
ACF of Residuals for CPIChnge(with 5% significance limits for the autocorrelations)
10987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Part
ial A
utoc
orre
latio
n
PACF of Residuals for CPIChnge(with 5% significance limits for the partial autocorrelations)
Might still be problematic!
Could it be the case of an Moving Average (MA) model?
MA(1):
1 ttt aay
{at } 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 at, at – 1, … , at – q
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):
Final Estimates of Parameters
Type Coef SE Coef T P
MA 1 -1.0459 0.0205 -51.08 0.000
Constant 4.5995 0.3438 13.38 0.000
Mean 4.5995 0.3438
Number of observations: 42
Residuals: SS = 115.337 (backforecasts excluded)
MS = 2.883 DF = 40
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 38.3 92.0 102.2 *
DF 10 22 34 *
P-Value 0.000 0.000 0.000 *
Forecasts from period 42
95% Limits
Period Forecast Lower Upper Actual
43 1.27305 -2.05583 4.60194
44 4.59948 -0.21761 9.41656
Not at all good!
Much wider!
4035302520151051
15
10
5
0
Time
CPIC
hnge
Time Series Plot for CPIChnge(with forecasts and their 95% confidence limits)
10987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Auto
corr
elat
ion
ACF of Residuals for CPIChnge(with 5% significance limits for the autocorrelations)
10987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Part
ial A
utoc
orre
latio
n
PACF of Residuals for CPIChnge(with 5% significance limits for the partial autocorrelations)
Still seems to be problems with residuals
Look again at ACF and PACF of original series:
1110987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Auto
corr
elat
ion
Autocorrelation Function for CPIChnge(with 5% significance limits for the autocorrelations)
1110987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Part
ial A
utoc
orre
latio
n
Partial Autocorrelation Function for CPIChnge(with 5% significance limits for the partial autocorrelations)
The pattern corresponds neither with pure AR(p), nor with pure 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):
Type Coef SE Coef T P
AR 1 0.6558 0.1330 4.93 0.000
MA 1 -0.9324 0.0878 -10.62 0.000
Constant 1.3778 0.4232 3.26 0.002
Mean 4.003 1.230
Number of observations: 42
Residuals: SS = 77.6457 (backforecasts excluded)
MS = 1.9909 DF = 39
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 8.4 21.5 28.3 *
DF 9 21 33 *
P-Value 0.492 0.429 0.699 *
Forecasts from period 42
95% Limits
Period Forecast Lower Upper Actual
43 -1.01290 -3.77902 1.75321
44 0.71356 -4.47782 5.90494
Much better!
4035302520151051
15
10
5
0
-5
Time
CPIC
hnge
Time Series Plot for CPIChnge(with forecasts and their 95% confidence limits)
10987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Auto
corr
elat
ion
ACF of Residuals for CPIChnge(with 5% significance limits for the autocorrelations)
10987654321
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Lag
Part
ial A
utoc
orre
latio
n
PACF of Residuals for CPIChnge(with 5% significance limits for the partial autocorrelations)
Now OK!
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
at + k is set to 0 for all values of k
For MA(q) ??
MA(1):
1ˆˆˆ ttt aay
If we e.g. would set at and at – 1 equal to 0
the forecast would constantly be
which is not desirable.
Note that
ˆ)ˆ1(ˆˆ
)1(0
1
1
2
1
211
ttt
ttt
t
ttt
ttt
yya
yyaa
aayaay
Similar investigations for ARMA-models.