Tutorial for solution of Assignment week 39
“A. Time series without seasonal variation
Use the data in the file 'dollar.txt'. “
“Construct a time series graph of
the fluctuations of the dollar
exchange rate, yt, for the period
1994-1998.”
Time Series Plot of $US/SEK Jan 3, 1994 - Nov 3, 1998
8.5
$US/SEK
8.0
7.5
7.0
6.5
1
123
246
369
492
615
Index
738
861
984
1107
Note! The time scale is best set to index here as the days are not consecutive in
time series (Saturdays, Sundays and other holidays are not present)
“Construct also a point plot for all pairs (yt-1 , yt) and try to visually estimate how
strong the correlation between two consecutive observations is
(=autocorrelation).”
Scatterplot of $US/SEK (y_t vs y_t-1), Jan 3, 1994 - Nov 3, 1998
Strong positive
autocorrelation!
8.5
y_t
8.0
7.5
7.0
6.5
6.5
7.0
7.5
y_t-1
8.0
8.5
“How do the estimated autocorrelations change with increasing timelags
between observations?”
To estimate the autocorrelation function, copy the relevant rows (data for
1994-1998) of column $US/SEK to a new column and use the autocorrelation
function estimation on that column
Autocorrelation Function for $US/SEK_1
(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
35
40 45
Lag
50
55
60
65
70
75
80
As was deduced from the scatter plot, the autocorrelations are strongly
positive. The autocorrelations do not change very much with increasing time
lags.
Note that this is what we see when the time series is non-stationary (has a
trend).
“Construct a time series graph of the changes zt = yt - yt-1 of the dollar exchange rate.
Then try to judge upon how the estimated autocorrelations for the series zt change
with the time lag between observations and check your judgement by estimating the
autocorrelations.”
The changes are already present in the column Difference.
The analogous procedures are applied to this column to produce the time series graph
and the estimated acf plot, i.e. by including only values where column Year is  1994.
Time Series Plot of Difference Jan 3, 1994 - Nov 3, 1998
0.2
Noisy plot 
As previously plot zt vs. zt – 1
0.0
-0.1
-0.2
-0.3
1
123
246
369
492
615
738
861
984
1107
Index
Scatterplot of Difference vs z_t-1 Jan 3, 1994 - Nov 3, 1998
0.2
Seems to be no
autocorrelation at all
0.1
0.0
z_t
Difference
0.1
-0.1
-0.2
-0.3
-0.3
-0.2
-0.1
0.0
z_t-1
0.1
0.2
Autocorrelation Function for Difference_1
(with 5% significance limits for the autocorrelations)
1.0
Our conclusions are
verified!
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
35
40 45
Lag
50
55
60
65
70
75
80
“B. Time series with seasonal variation
Use the time series of monthly discharge in the lake Hjälmaren
(‘Hjalmarenmonth.txt’), which you have used in the assignment for week 36.
Compute the autocorrelation function (Minitab: StatTime
seriesAutocorrelation…) for the variable Discharge.m.”
Autocorrelation Function for Discharge.m
(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
35
40 45
Lag
50
55
60
65
70
75
80
“Deseasonalise the time series and make a new graph of the seasonally adjusted
values. Try to visually estimate how the autocorrelations look like and check your
judgement by computing the autocorrelation function.”
Time Series Plot of Discharge.m
120
Discharge.m
100
Additive model for
deseasonalization
seems best!
80
60
40
20
0
Month jan
Year 1994
jan
2011
jan
2028
jan
2045
jan
2062
jan
2079
jan
2096
Time Series Plot of DESE1
120
100
80
DESE1
60
40
20
0
Month jan
Year 1994
jan
2011
jan
2028
jan
2045
jan
2062
jan
2079
jan
2096
Scatterplot of DESE1 vs DESE1_1
120
Plot DESE1(t) vs.
DESE1(t-1)
Indicates positive
autocorrelation
100
DESE1
80
60
40
20
0
0
20
40
60
DESE1_1
80
100
120
Autocorrelation Function for DESE1
(with 5% significance limits for the autocorrelations)
1.0
0.8
Indication
confirmed!
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
35
40 45
Lag
50
55
60
65
70
75
80
“C. Forecasting with autoregressive models
Data set: The Dollar Exchange rates
Consider again the time series of dollar exchange rates for the period 1994-1998.
Then use the Minitab time series module ARIMA (see further below) to estimate
the parameters in an AR(1)-model (1 nonseasonal autoregressive parameter) and
plot the observed values together with forecasts for a period of 20 days after the last
observed time-point.”
Use the already created column of
$US/SEK exchange rates from 19941998
(there is no opportunity in Minitab’s
ARIMA module to just analyze a
subset of a column like in the
graphing modules)
Forecasts for a 20 days
period are requested.
(Origin field is left blank
analogously to previous
modules)
See next slide!
Three new columns should
be entered here!
Must be checked (not
default)
Should always by
checked for
diagnostic purposes
Final Estimates of Parameters
Type
AR
Coef
SE Coef
T
P
0.9971
0.0026
385.44
0.000
0.021782
0.001280
17.02
0.000
7.4405
0.4371
1
Constant
Mean
Number of observations:
Residuals:
1229
SS =
2.45718 (backforecasts excluded)
MS =
0.00200
DF = 1227
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
DF
P-Value
Significant!
12
24
36
48
9.0
22.9
33.3
38.2
10
22
34
46
0.529
0.410
0.504
0.786
Keep in mind for
comparison with next
model
OK!
Forecasts from period 1229
95 Percent
Limits
Period
Forecast
Lower
Upper
1230
7.79895
7.71122
7.88668
1231
7.79790
7.67401
7.92178
1232
7.79685
7.64535
7.94836
1233
7.79581
7.62112
7.97050
1234
7.79477
7.59974
7.98979
1235
7.79373
7.58040
8.00706
1236
7.79270
7.56261
8.02278
1237
7.79167
7.54605
8.03728
1238
7.79064
7.53051
8.05077
1239
7.78961
7.51581
8.06342
1240
7.78859
7.50184
8.07534
1241
7.78757
7.48850
8.08664
1242
7.78655
7.47572
8.09739
1243
7.78554
7.46344
8.10764
1244
7.78453
7.45161
8.11745
1245
7.78352
7.44018
8.12687
1246
7.78252
7.42912
8.13592
1247
7.78152
7.41839
8.14464
1248
7.78052
7.40798
8.15306
1249
7.77952
7.39786
8.16119
Actual
These forecasts and prediction
limits are stored in columns
C12, C13 and C14 (as entered
in dialog box)
Time Series Plot for $US/SEK_1
(with forecasts and their 95% confidence limits)
8.5
$US/SEK_1
8.0
7.5
Seems to be OK (as
was confirmed by the
Ljung-Box statistic)
7.0
6.5
1
84
168
252
336
420
504
588
672
Time
756
840
924 1008 1092 1176
PACF of Residuals for $US/SEK_1
ACF of Residuals for $US/SEK_1
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
(with 5% significance limits for the autocorrelations)
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
1
6
12
18
24
30
36
42
Lag
48
54
60
66
72
78
1
6
12
18
24
30
36
42
Lag
48
54
60
66
72
78
Use the stored
prediction limits to
calculate the widths
of the prediction
intervals
The column
widths_1 (C15) will
later be compared
with the widths from
another model
“Investigate also if the forecasts can improve by instead using an AR(2)-model.”
Don’t forget to enter new
columns here!
Final Estimates of Parameters
Type
Coef
SE Coef
T
P
AR
1
1.0107
0.0286
35.35
0.000
AR
2
-0.0138
0.0285
-0.48
0.629
0.023161
0.001280
18.09
0.000
7.4372
0.4110
Constant
Mean
Number of observations:
Residuals:
Non-significant!
1229
SS =
2.45873 (backforecasts excluded)
MS =
0.00201
DF = 1226
Slightly larger than in
AR(1)-model
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
DF
P-Value
12
24
36
48
10.2
24.0
34.3
39.3
9
21
33
45
0.337
0.292
0.403
0.710
Still OK!
Time Series Plot for $US/SEK_1
(with forecasts and their 95% confidence limits)
8.5
7.5
7.0
6.5
1
84
168
252
336
420
504
588
672
Time
756
840
924 1008 1092 1176
PACF of Residuals for $US/SEK_1
ACF of Residuals for $US/SEK_1
(with 5% significance limits for the partial autocorrelations)
(with 5% significance limits for the autocorrelations)
1.0
1.0
0.8
Partial Autocorrelation
0.8
0.6
Autocorrelation
$US/SEK_1
8.0
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
1
1
6
12
18
24
30
36
42
Lag
48
54
60
66
72
78
6
12
18
24
30
36
42
Lag
48
54
60
66
72
78
Calculate widths for
the new prediction
intervals
Make a time series plot of the intervals widths from the two analyses.
Time Series Plot of widths_1; widths_2
0.8
Variable
widths_1
widths_2
0.7
Data
0.6
0.5
0.4
0.3
0.2
0.1
2
4
6
8
10
12
Index
14
16
18
20
Slightly wider prediction intervals with AR(2)-model (widths_2)
 Forecasts do not improve with AR(2)-model
“Finally perform a residual analysis of the errors in the one-step-ahead forecasts
(can be asked for under the “Graph” button in the dialog box. By residuals we mean
here the errors in the one-step-ahead forecasts).
Are there any signs of serial correlations in the residuals?”
PACF of Residuals for $US/SEK_1
ACF of Residuals for $US/SEK_1
(with 5% significance limits for the partial autocorrelations)
(with 5% significance limits for the autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
AR(1):
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
1
6
12
18
24
30
36
42
Lag
48
54
60
66
72
1
78
6
12
30
36
42
Lag
48
54
60
66
72
78
72
78
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
24
PACF of Residuals for $US/SEK_1
ACF of Residuals for $US/SEK_1
(with 5% significance limits for the autocorrelations)
AR(2):
18
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
1
6
12
18
24
30
36
42
Lag
48
54
60
66
72
78
1
6
12
No signs of serial correlations in resiaduals in any of the models
18
24
30
36
42
Lag
48
54
60
66
“D. ARIMA models and differentiation
In this task you will first have to judge upon whether you need to differentiate the
current time series ( zt = yt - yt-1 ) before forecasting with an ARMA-model can be
applied. Then you shall try different models with a number of parameters to find
the model that gives the least one-step-ahead prediction errors on the average.
Finally you shall make some residual plots to investigate if the selected model of
forecasting can be improved.”
“Forecasting monthly dollar exchange rates in Danish crowns (DKK)
Data set: The Dollar-Danish Crowns Exchange rates”
“D.1. The need for differentiation
Construct a time series graph for the monthly means of dollar exchange rates in
Danish crowns (file ‘DKK.txt’). Then estimate the autocorrelations and display
them in a graph. Does the time series show any obvious upward or downward
trend?”
Time Series Plot of Exchange rate
Exchange rate
7.0
Note that the yaxis do not
start at zero!
6.5
6.0
5.5
Month jan
Year 1991
jan
1992
jan
1993
jan
1994
jan
1995
jan
1996
jan
1997
jan
1998
A slight upward trend may be concluded
“Are there any signs of long-time oscillations in the time series (that can be seen
from the time series graph)?”
Yes, there seem to be a cyclical variation with cycle periods longer than a year.
Autocorrelation Function for Exchange rate
(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
2
4
6
8
10
12
14
Lag
16
18
20
22
24
“Is there a fast cancel-out in the autocorrelations?”
No, the cancel-out is not fast (although the spikes come quickly within the red
limits)
“Is there need for differentiation to get a time series suitable for ARMAmodelling?”
Probably, but not certainly!
“D.2 Fitting different ARMA-models
Calculate the estimated autocorrelations possibly after differentiation of the original
series and display these estimates in a graph.”
Without differentiation:
Partial Autocorrelation Function for Exchange rate
Autocorrelation Function for Exchange rate
(with 5% significance limits for the partial autocorrelations)
(with 5% significance limits for the autocorrelations)
1.0
1.0
0.8
Partial Autocorrelation
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
2
4
6
8
10
12
14
Lag
16
18
20
22
24
2
4
6
8
10
12
14
Lag
16
18
20
22
24
(Slowly) decreasing postive autocorrelations. One positive spike (at lag 1) in SPAC

Either this is a non-stationary time series or an AR(1)-time series with a  close to 1.
With first-order differentiation
(use the ready series of
differences):
Partial Autocorrelation Function for Difference in exchange rate
Autocorrelation Function for Difference in exchange rate
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
(with 5% significance limits for the autocorrelations)
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
2
4
6
8
10
12
14
Lag
16
18
20
22
24
2
No obvious pattern in any of these two plots.
 The differentiated series may be an ARMA-series
4
6
8
10
12
14
Lag
16
18
20
22
24
“Then try to predict the dollar exchange rate by combining differentiation with
ARMA-models of different orders.”
Strategy:
On original series, try AR(1)
On differentiated series, try AR(1), AR(2), MA(1), MA(2), ARMA(1,1), ARMA(1,2),
ARMA(2,1) and ARMA(2,2)
Compare the values of MS from each model. This measure corresponds with onestep-ahead prediction errors on the average.
Model
Original
MS
Differentiated
AR(1)
0.03682
AR(1)
0.03904
AR(2)
0.03914
MA(1)
0.03904
MA(2)
0.03916
ARMA(1,1)
0.03905
ARMA(2,1)
0.03889
ARMA(1,2)
0.03869
ARMA(2,2)
0.03807
None of the models on the differentiated series produces better MS value than the
AR(1) on original series, but MS seems to decrease with larger complexity.
“What happens if one tries to fit a very complex model with a lot of parameters to
the observations?”
Study e.g. ARMA(3,3) and ARMA(4,4) on the differentiated series:
Final Estimates of Parameters
Type
Coef
SE Coef
T
P
AR
1
-0.1113
0.3369
-0.33
0.742
AR
2
0.4786
0.2274
2.10
0.038
AR
3
0.3689
0.3237
1.14
0.258
MA
1
-0.1098
0.2941
-0.37
0.710
MA
2
0.4351
0.2136
2.04
0.045
MA
3
0.6165
0.2846
2.17
0.033
0.000924
0.001931
0.48
0.634
Constant
ARMA(3,3)
Differencing: 1 regular difference
Number of observations:
Residuals:
Original series 95, after differencing 94
SS =
3.25649 (backforecasts excluded)
MS =
0.03743
DF = 87
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag
Chi-Square
DF
P-Value
12
24
36
48
4.9
17.3
26.4
39.5
5
17
29
41
0.425
0.431
0.606
0.537
Even lower than in
ARMA(2,2)
No severe problems but
not all parameters are
significant!
ACF of Residuals for Exchange rate
(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
2
4
6
8
10
12
Lag
14
16
18
20
22
PACF of Residuals for Exchange rate
(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
2
4
6
8
10
12
Lag
14
16
18
20
22
No severe problems here either,
but spikes seem to increase with
lag!
Unable to reduce sum of squares any further
Estimation problems!
ARMA(4,4)
Final Estimates of Parameters
Type
Coef
SE Coef
T
P
AR
1
0.4196
2.3514
0.18
0.859
AR
2
0.4329
0.4304
1.01
0.317
AR
3
0.0536
1.2079
0.04
0.965
AR
4
-0.0652
0.7425
-0.09
0.930
MA
1
0.4119
2.3452
0.18
0.861
MA
2
0.3871
0.4030
0.96
0.340
MA
3
0.3397
1.0707
0.32
0.752
MA
4
-0.1736
1.2715
-0.14
0.892
0.000597
0.001779
0.34
0.738
Constant
None of the parameters are
significant!
Estimation problems and an
increase in MS.
Differencing: 1 regular difference
Number of observations:
Residuals:
Original series 95, after differencing 94
SS =
3.26434 (backforecasts excluded)
MS =
0.03840
DF = 85
Increased!
The conclusion must be that an AR(1)-model on original data seems to be the best.
“D.3. Residual analysis
Construct a graph for the residuals (the one-step-ahead prediction errors) and
examine visually if anything points to a possible improvement of the model.”
PACF of Residuals for Exchange rate
ACF of Residuals for Exchange rate
(with 5% significance limits for the partial autocorrelations)
(with 5% significance limits for the autocorrelations)
1.0
1.0
0.8
Partial Autocorrelation
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.6
-1.0
-0.8
2
-1.0
2
4
6
8
10
12
14
Lag
16
18
20
22
4
6
8
10
12
14
Lag
16
18
20
22
24
24
SAC and SPAC of residuals do not indicate that another ARIMA-model should
be used.
Residual Plots for Exchange rate
Normal Probability Plot of the Residuals
Residuals Versus the Fitted Values
99
0.50
90
0.25
Residual
Percent
99.9
50
10
0.00
-0.25
1
0.1
-0.50
-0.25
0.00
Residual
0.25
-0.50
0.50
20
0.50
15
0.25
10
7.0
0.00
-0.25
5
0
6.0
6.5
Fitted Value
Residuals Versus the Order of the Data
Residual
Frequency
Histogram of the Residuals
5.5
-0.4
-0.2
0.0
0.2
Residual
0.4
0.6
-0.50
1
10
20
30 40 50 60 70
Observation Order
80
90
There do not seem to be any violations of the assumption of normal distribution
and constant variance either.