Tutorial for solution of Assignment week 40
“Forecasting monthly values of Consumer Price Index
Data set: Swedish Consumer Price Index”
sparetime
“Construct a time series graph for the monthly values of Consumer Price Index
(Konsumentprisindex (KPI) in Swedish) for spare time occupation, amusement
and culture (fritid, nöje och kultur in Swedish) (in file ‘sparetime.txt’).”
Time Series Plot of CPI(group))
200
CPI(group))
180
160
140
120
100
Month jan
Year 1980
jan
1983
jan
1986
jan
1989
jan
1992
jan
1995
jan
1998
“Then estimate the autocorrelations and display them in a graph.”
Autocorrelation Function for CPI(group))
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
55
“Is there any obvious upward or downward trend?”
Time Series Plot of CPI(group))
200
CPI(group))
180
Yes, upward, but turning at the end
160
140
120
100
Month jan
Year 1980
jan
1983
jan
1986
jan
1989
jan
1992
jan
1995
jan
1998
“Are there any signs of long-time oscillations in the time series?
Time Series Plot of CPI(group))
200
CPI(group))
180
No!
160
140
120
100
Month jan
Year 1980
jan
1983
jan
1986
jan
1989
jan
1992
jan
1995
jan
1998
Are there any signs of seasonal variation in the series?”
Not visible!
“Do the autocorrelations cancel out quickly?”
Autocorrelation Function for CPI(group))
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
No!
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
55
“Judge upon the need for differentiation according to
ut = yt - yt-1
or
vt = yt - yt-12
to get a time series that is suitable for forecasting with ARMA-models. Construct
new graphs for the series obtained by differentiation and estimate the
autocorrelations for these series.”
ut = yt – yt - 1
Time Series Plot of u
3
2
Not convincingly
stationary!
u
1
0
-1
-2
Month jan
Year 1980
jan
1983
jan
1986
jan
1989
jan
1992
jan
1995
jan
1998
Autocorrelation Function for u
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
Diffuse pattern!
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
55
vt = yt – yt - 12
Time Series Plot of v
15
10
v
5
0
-5
Month jan
Year 1980
jan
1983
jan
1986
jan
1989
jan
1992
jan
1995
jan
1998
Autocorrelation Function for v
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
Definitely nonstationary!
“E.2. Fitting different ARMA-models
Try different combinations of ARMA-models and differentiation to forecast the
Consumer Price Index. Which model seems to give the best forecasts in this case.”
From E.1.: Seems to be best to use first-order non-seasonal differences
Chosen “design”:
AR(1), AR(2)
MA(1), MA(2)
ARMA(1,1), ARMA(2,1), ARMA(1,2), ARMA(2,2)
Fixed to 1
in all
models!
Altered from
model to
model
AR(1):
Type
AR
1
Constant
Coef
SE Coef
T
P
0.1170
0.0671
1.75
0.082
0.38522
0.04779
8.06
0.000
…
MS =
0.512
2 months forecasts:
Forecast
Lower
Upper
195.472
194.070
196.875
195.925
193.823
198.027
DF = 222
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag
12
24
36
48
0.000
0.000
0.000
0.000
…
P-Value
AR(2):
Type
Coef
SE Coef
T
P
AR
1
0.0770
0.0648
1.19
0.236
AR
2
0.3012
0.0655
4.60
0.000
0.27053
0.04576
5.91
0.000
Constant
…
MS =
0.469
DF = 221
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag
12
24
36
48
0.007
0.003
0.001
0.003
…
P-Value
2 months forecasts:
Forecast
Lower
Upper
194.962
193.620
196.305
195.720
193.747
197.693
MA(1):
Type
MA
1
Constant
Coef
SE Coef
T
P
-0.0741
0.0675
-1.10
0.273
Forecast
0.43605
0.05146
8.47
0.000
…
MS =
0.514
2 months forecasts:
Lower
Upper
195.430
194.024
196.835
195.866
193.803
197.929
DF = 222
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag
12
24
36
48
0.000
0.000
0.000
0.000
…
P-Value
MA(2):
Type
Coef
SE Coef
T
P
MA
1
-0.0592
0.0668
-0.89
0.376
MA
2
-0.2533
0.0670
-3.78
0.000
0.43664
0.06071
7.19
0.000
0.479
DF = 221
Constant
…
MS =
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag
12
24
36
48
0.000
0.000
0.000
0.000
…
P-Value
2 months forecasts:
Forecast
Lower
Upper
195.146
193.789
196.503
196.032
194.056
198.009
ARMA(1,1):
* WARNING * Back forecasts not dying out rapidly
Type
Coef
SE Coef
T
P
AR
1
1.0186
0.0238
42.85
0.000
MA
1
0.9769
0.0006
1560.10
0.000
-0.0117678
-0.0013602
8.65
0.000
Constant
…
MS =
0.458
DF = 221
2 months forecasts:
Modified Box-Pierce (Ljung-Box) Chi-Squ
Forecast
Lag
12
24
36
48
…
P-Value
Type
ARMA(2,1):
0.000
0.000
0.000
Lower
Upper
194.516
193.190
195.843
194.114
192.199
196.029
0.000
Coef
SE Coef
T
P
AR
1
0.3311
0.2045
1.62
0.107
AR
2
0.2711
0.0764
3.55
0.000
MA
1
0.2821
0.2129
1.33
0.186
Forecast
0.17191
0.03287
5.23
0.000
Constant
…
MS =
0.469
DF = 220
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag
P-Value
12
24
36
48
0.004
0.001
0.000
0.001
2 months forecasts:
Lower
Upper
194.746
193.403
196.088
195.300
193.355
197.246
ARMA(1,2):
Type
Coef
SE Coef
T
P
AR
1
0.6136
0.1635
3.75
0.000
MA
1
0.5577
0.1679
3.32
0.001
MA
2
-0.2202
0.0763
-2.89
0.004
0.16753
0.03043
5.51
0.000
0.472
DF = 220
Constant
…
MS =
2 months forecasts:
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag
12
24
36
48
…
P-Value
ARMA(2,2):
0.002
0.001
0.000
Forecast
Lower
Upper
194.728
193.382
196.075
195.214
193.256
197.173
0.000
* ERROR * Model cannot be estimated with these data.
None of the models are satisfactory in goodness-of-fit and prediction intervals are
quite similar (slightly more narrow for the more complex models).
Maybe second-order non-seasonal differences would work?
wt = ut – ut – 1 = (yt – yt – 1) – (yt – 1 – yt – 2) = yt – 2yt – 1 + yt – 2
Time Series Plot of w
4
3
2
w
1
0
-1
-2
-3
Month jan
Year 1980
jan
1983
jan
1986
jan
1989
jan
1992
jan
1995
jan
1998
Autocorrelation Function for w
(with 5% significance limits for the autocorrelations)
Clear seasonal correlation
and close to non-stationary
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
55
How about first order seasonal differences on the first-order non-seasonal
differences?
zt = ut – ut – 12 = (yt – yt – 1) – (yt – 12 – yt – 13)
Time Series Plot of z
2
1
z
0
-1
-2
Much more a stationary look!
Month jan
Year 1980
jan
1983
jan
1986
jan
1989
jan
1992
jan
1995
jan
1998
Autocorrelation Function for z
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
Autocorrelation Function for z
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
Tricky to identify the correct
model.
0.4
0.2
0.0
-0.2
Clearly a seasonal model must
be used, most probably with at
least one MA –term
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
Non-seasonal part more
difficult. ARMA(1,1) ?
Partial Autocorrelation Function for z
(with 5% significance limits for the partial autocorrelations)
1.0
Partial Autocorrelation
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
Try ARIMA(1,1,1,0,1,1)12
Type
Coef
SE Coef
T
P
AR
1
-0.6368
1.2524
-0.51
0.612
MA
1
-0.6085
1.2902
-0.47
0.638
SMA
12
0.8961
0.0484
18.51
0.000
-0.07528
0.01129
-6.67
0.000
Constant
Differencing: 1 regular, 1 seasonal of order 12
Number of observations:
Residuals:
Original series 225, after differencing 212
SS =
77.9325 (backforecasts excluded)
MS =
0.3747
DF = 208
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
DF
P-Value
12
24
36
48
12.9
22.8
28.9
36.0
8
20
32
44
0.115
0.300
0.626
0.800
2 months forecasts:
Forecast
Lower
Upper
194.971
193.771
196.171
195.580
193.907
197.253
Compare with ARIMA(0,1,0,0,1,1)12
Type
SMA
Coef
SE Coef
T
P
0.9039
0.0472
19.15
0.000
-0.045964
0.006893
-6.67
0.000
12
Constant
Differencing: 1 regular, 1 seasonal of order 12
Number of observations:
Residuals:
Original series 225, after differencing 212
SS =
78.0539 (backforecasts excluded)
MS =
0.3717
DF = 210
Slightly smaller MS!
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
DF
P-Value
12
24
36
48
12.5
21.8
28.0
34.8
10
22
34
46
0.254
0.475
0.757
0.887
2 months forecasts:
Forecast
Lower
Upper
194.987
193.792
196.182
195.587
193.897
197.277
“E.3. Residual analysis
Construct a graph for the residuals (the one-step-ahead prediction errors) and
examine visually if there is any pattern in the residuals indicating that the selected
forecasting model is not optimal.”
Residual plots for
ARIMA(1,1,1,0,0,0)
ARIMA(1,1,1,0,1,1)12
ARIMA(0,1,0,0,1,1)12
ARIMA(1,1,1,0,0,0):
Residual Plots for CPI(group))
ACF of Residuals for CPI(group))
(with 5% significance limits for the autocorrelations)
Normal Probability Plot of the Residuals
1.0
2
99
Percent
0.6
0.4
0.2
90
Residual
0.8
50
10
0.0
1
-0.2
0.1
-2
-1
-0.4
-0.6
0
Residual
1
1
0
-1
-2
2
Histogram of the Residuals
-0.8
1
5
10
15
20
25
30
Lag
35
40
45
50
55
Frequency
-1.0
(with 5% significance limits for the partial autocorrelations)
125
150
Fitted Value
10
0
-1
-1.2
-0.6
0.0
0.6
Residual
1.2
1.8
-2
1
20
40
60
80 100 120 140 160 180 200 220
Observation Order
Partial Autocorrelation
0.8
0.6
0.4
0.2
0.0
Non-satisfactory
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
Lag
35
40
45
50
55
200
1
1.0
-0.2
175
2
20
0
PACF of Residuals for CPI(group))
100
Residuals Versus the Order of the Data
30
Residual
Autocorrelation
Residuals Versus the Fitted Values
99.9
ARIMA(1,1,1,0,1,1)12
ACF of Residuals for CPI(group))
(with 5% significance limits for the autocorrelations)
Residual Plots for CPI(group))
1.0
Normal Probability Plot of the Residuals
0.8
99
Percent
0.4
0.2
0.0
-0.2
1
90
Residual
0.6
Autocorrelation
Residuals Versus the Fitted Values
99.9
50
10
0
-1
1
0.1
-0.4
-2
-1
0
Residual
-0.6
-0.8
1
-2
2
Histogram of the Residuals
100
125
150
175
Fitted Value
200
Residuals Versus the Order of the Data
-1.0
12
18
24
30
36
42
48
Lag
PACF of Residuals for CPI(group))
30
20
10
0
(with 5% significance limits for the partial autocorrelations)
1
Residual
6
Frequency
1
-1.2
-0.6
0.0
Residual
0.6
1.0
Partial Autocorrelation
0.8
0.6
0.4
0.2
0.0
Satisfactory!
-0.2
-0.4
-0.6
-0.8
-1.0
1
6
12
18
24
30
Lag
36
42
48
1.2
0
-1
-2
1
20
40
60
80 100 120 140 160 180 200 220
Observation Order
ARIMA(0,1,0,0,1,1)12
ACF of Residuals for CPI(group))
(with 5% significance limits for the autocorrelations)
Residual Plots for CPI(group))
1.0
Normal Probability Plot of the Residuals
0.8
2
99
0.2
0.0
-0.2
1
90
Residual
0.4
Percent
50
10
0
-1
1
-0.4
0.1
-0.6
-0.8
-2
-1
0
Residual
1
-2
2
Histogram of the Residuals
-1.0
1
6
12
18
24
30
36
42
48
Frequency
Lag
PACF of Residuals for CPI(group))
(with 5% significance limits for the partial autocorrelations)
2
30
1
20
10
0
1.0
-1.2
-0.6
0.0
Residual
0.6
1.2
0.8
0.6
0.4
0.2
0.0
-0.2
Satisfactory!
-0.4
-0.6
-0.8
-1.0
1
6
12
18
24
30
Lag
36
42
48
100
125
150
175
Fitted Value
200
Residuals Versus the Order of the Data
40
Residual
Autocorrelation
0.6
Partial Autocorrelation
Residuals Versus the Fitted Values
99.9
0
-1
-2
1
20
40
60
80 100 120 140 160 180 200 220
Observation Order
“F. ARMA-models and exponential smoothing
Data set: The Dollar-Danish Crowns Exchange rates
Consider the time series of monthly exchange rates US$/DKK.”
“At first, calculate forecasts by using exponential smoothing and note the
prediction formula.”
Time Series Plot of Exchange rate
Change scale so that y-axis
starts at 0 (and ends at 10)
6.5
6.0
5.5
Month jan
Year 1991
jan
1992
jan
1993
jan
1994
jan
1995
jan
1996
jan
1997
jan
1998
Time Series Plot of Exchange rate
10
8
Single exponential
smoothing will
probably work well.
Optimize 
Exchange rate
Exchange rate
7.0
6
4
2
0
Month jan
Year 1991
jan
1992
jan
1993
jan
1994
jan
1995
jan
1996
jan
1997
jan
1998
Calculate forecasts for 6 months (an arbitrarily
chosen value)
Single Exponential Smoothing Plot for Exchange rate
Variable
Actual
Fits
Forecasts
95.0% PI
Exchange rate
7.0
Smoothing Constant
Alpha
0.995540
6.5
Accuracy
MAPE
MAD
MSD
6.0
Measures
2.41996
0.14983
0.03784
5.5
Forecasts
1
10
20
30
40
50
60
Index
70
80
90
100
Period
Prediction formula:
yˆT    T    yT  1      T 1 
 0.9955  yT  0.0045   T 1
Forecast
Lower
Upper
96
6.31118
5.94410
6.67825
97
6.31118
5.94410
6.67825
98
6.31118
5.94410
6.67825
99
6.31118
5.94410
6.67825
100
6.31118
5.94410
6.67825
101
6.31118
5.94410
6.67825
“Then calculate forecasts by fitting a MA(1)-model to first differences of the
original series (i.e. you must differentiate the series once).”
Time Series Plot for Exchange rate
(with forecasts and their 95% confidence limits)
Final Estimates of Parameters
7.5
Type
Exchange rate
7.0
MA
6.5
Coef
SE Coef
0.0052
0.1043
0.00537
0.02027
1
Constant
6.0
5.5
1
10
20
30
40
50
Time
60
70
80
90
100Forecasts
from period 95
95 Percent
“How does the prediction formula look like
in this case?”
Limits
Period
Forecast
Lower
Upper
96
6.31652
5.92916
6.70387
97
6.32189
5.77550
6.86828
98
6.32726
5.65864
6.99587
99
6.33263
5.56091
7.10434
100
6.33800
5.47542
7.20057
101
6.34336
5.39862
7.28811
“How do the forecasts differ between the two different methods of forecasting?”
SES
ARIMA(0,1,1)
Single Exponential Smoothing Plot for Exchange rate
(with forecasts and their 95% confidence limits)
7.5
Smoothing Constant
Alpha
0.995540
6.5
Accuracy
MAPE
MAD
MSD
6.0
Measures
2.41996
0.14983
0.03784
7.0
Exchange rate
Exchange rate
Time Series Plot for Exchange rate
Variable
Actual
Fits
Forecasts
95.0% PI
7.0
6.5
6.0
5.5
5.5
1
10
20
30
40
50
60
Index
70
80
90
Forecasts from period 95
100
1
10
20
30
40
50
Time
60
70
80
90
100
95 Percent
Forecasts
Limits
Forecast
Lower
Upper
Period
Forecast
Lower
Upper
96
6.31118
5.94410
6.67825
96
6.31652
5.92916
6.70387
97
6.31118
5.94410
6.67825
97
6.32189
5.77550
6.86828
98
6.31118
5.94410
6.67825
98
6.32726
5.65864
6.99587
99
6.31118
5.94410
6.67825
99
6.33263
5.56091
7.10434
100
6.31118
5.94410
6.67825
100
6.33800
5.47542
7.20057
101
6.31118
5.94410
6.67825
101
6.34336
5.39862
7.28811
Period