AR- MA- och ARMA-
description
Transcript of AR- MA- och ARMA-
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20001990198019701960
14
12
10
8
6
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Year
CPIC
hnge
Yearly changes in Consumer Price Index (CPI), USA, 1960-2001
How should these data be modelled?
![Page 2: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/2.jpg)
Identification step: Look at the SAC and SPAC
1110987654321
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Lag
Auto
corr
elat
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Autocorrelation Function for CPIChnge(with 5% significance limits for the autocorrelations)
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Lag
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utoc
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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)
![Page 3: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/3.jpg)
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!!
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AR(1)
We can always ask for forecasts
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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
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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
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4035302520151051
15
10
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0
Time
CPIC
hnge
Time Series Plot for CPIChnge(with forecasts and their 95% confidence limits)
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Lag
Auto
corr
elat
ion
ACF of Residuals for CPIChnge(with 5% significance limits for the autocorrelations)
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Lag
Part
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utoc
orre
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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!
![Page 8: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/8.jpg)
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
![Page 9: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/9.jpg)
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
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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
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A new look at the SAC and SPAC of original data:
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Lag
Auto
corr
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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
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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
![Page 12: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/12.jpg)
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
![Page 13: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/13.jpg)
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!
![Page 14: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/14.jpg)
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
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![Page 16: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/16.jpg)
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
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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
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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)
![Page 19: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/19.jpg)
Try an MA(1):
![Page 20: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/20.jpg)
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!
![Page 21: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/21.jpg)
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)
![Page 22: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/22.jpg)
Still seems to be problems with residuals
Look again at ACF and PACF of original series:
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)
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
![Page 23: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/23.jpg)
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)
![Page 24: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/24.jpg)
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)
![Page 25: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/25.jpg)
Always try to keep p and q small.
Try an ARMA(1,1):
![Page 26: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/26.jpg)
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!
![Page 27: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/27.jpg)
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!
![Page 28: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/28.jpg)
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
![Page 29: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/29.jpg)
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.
![Page 30: AR- MA- och ARMA-](https://reader038.fdocuments.us/reader038/viewer/2022110215/56814c40550346895db944ca/html5/thumbnails/30.jpg)
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.