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.”
Index
$US/
SEK
11079848617386154923692461231
8.5
8.0
7.5
7.0
6.5
Time Series Plot of $US/ SEK Jan 3, 1994 - Nov 3, 1998
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).”
y_t-1
y_t
8.58.07.57.06.5
8.5
8.0
7.5
7.0
6.5
Scatterplot of $US/ SEK (y_ t vs y_ t-1), J an 3, 1994 - Nov 3, 1998Strong positive autocorrelation!
“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
Lag
Auto
corr
elat
ion
80757065605550454035302520151051
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Autocorrelation Function for $US/ SEK_1(with 5% significance limits for the autocorrelations)
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.
Index
Diffe
renc
e
11079848617386154923692461231
0.2
0.1
0.0
-0.1
-0.2
-0.3
Time Series Plot of Difference Jan 3, 1994 - Nov 3, 1998
Noisy plot
As previously plot zt vs. zt – 1
z_t-1
z_t
0.20.10.0-0.1-0.2-0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
Scatterplot of Difference vs z_ t-1 Jan 3, 1994 - Nov 3, 1998
Seems to be no autocorrelation at all
Lag
Auto
corr
elat
ion
80757065605550454035302520151051
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Autocorrelation Function for Difference_1(with 5% significance limits for the autocorrelations)
Our conclusions are verified!
“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: StatTime seriesAutocorrelation…) for the variable Discharge.m.”
Lag
Auto
corr
elat
ion
80757065605550454035302520151051
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Autocorrelation Function for Discharge.m(with 5% significance limits for the autocorrelations)
“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.”
Disc
harg
e.m
YearMonth
2096207920622045202820111994janjanjanjanjanjanjan
120
100
80
60
40
20
0
Time Series Plot of Discharge.m
Additive model for deseasonalization seems best!
DESE
1
YearMonth
2096207920622045202820111994janjanjanjanjanjanjan
120
100
80
60
40
20
0
Time Series Plot of DESE1
Plot DESE1(t) vs. DESE1(t-1)
DESE1_1
DESE
1
120100806040200
120
100
80
60
40
20
0
Scatterplot of DESE1 vs DESE1_1
Indicates positive autocorrelation
Lag
Auto
corr
elat
ion
80757065605550454035302520151051
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Autocorrelation Function for DESE1(with 5% significance limits for the autocorrelations)
Indication confirmed!
“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 1994-1998
(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)
Three new columns should be entered here!
See next slide!
Must be checked (not default)
Should always by checked for diagnostic purposes
Final Estimates of Parameters
Type Coef SE Coef T P
AR 1 0.9971 0.0026 385.44 0.000
Constant 0.021782 0.001280 17.02 0.000
Mean 7.4405 0.4371
Number of observations: 1229
Residuals: SS = 2.45718 (backforecasts excluded)
MS = 0.00200 DF = 1227
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 9.0 22.9 33.3 38.2
DF 10 22 34 46
P-Value 0.529 0.410 0.504 0.786
Significant!
Keep in mind for comparison with next model
OK!
Forecasts from period 1229
95 Percent
Limits
Period Forecast Lower Upper Actual
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
These forecasts and prediction limits are stored in columns C12, C13 and C14 (as entered in dialog box)
Time
$US/
SEK_
1
117610921008924840756672588504420336252168841
8.5
8.0
7.5
7.0
6.5
Time Series Plot for $US/ SEK_1(with forecasts and their 95% confidence limits)
Lag
Auto
corr
elat
ion
78726660544842363024181261
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
ACF of Residuals for $US/ SEK_1(with 5% significance limits for the autocorrelations)
Lag
Part
ial A
utoc
orre
latio
n
78726660544842363024181261
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
PACF of Residuals for $US/ SEK_1(with 5% significance limits for the partial autocorrelations)
Seems to be OK (as was confirmed by the Ljung-Box statistic)
The column widths_1 (C15) will later be compared with the widths from another model
Use the stored prediction limits to calculate the widths of the prediction intervals
“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
Constant 0.023161 0.001280 18.09 0.000
Mean 7.4372 0.4110
Number of observations: 1229
Residuals: SS = 2.45873 (backforecasts excluded)
MS = 0.00201 DF = 1226
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 10.2 24.0 34.3 39.3
DF 9 21 33 45
P-Value 0.337 0.292 0.403 0.710
Non-significant!
Slightly larger than in AR(1)-model
Still OK!
Time
$US/
SEK_
1
117610921008924840756672588504420336252168841
8.5
8.0
7.5
7.0
6.5
Time Series Plot for $US/ SEK_1(with forecasts and their 95% confidence limits)
Lag
Auto
corr
elat
ion
78726660544842363024181261
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
ACF of Residuals for $US/ SEK_ 1(with 5% significance limits for the autocorrelations)
Lag
Part
ial A
utoc
orre
latio
n
78726660544842363024181261
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
PACF of Residuals for $US/ SEK_1(with 5% significance limits for the partial autocorrelations)
Calculate widths for the new prediction intervals
Make a time series plot of the intervals widths from the two analyses.
Index
Data
2018161412108642
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
widths_1widths_2
Variable
Time Series Plot of widths_1; widths_2
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?”
Lag
Auto
corr
elat
ion
78726660544842363024181261
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
ACF of Residuals for $US/ SEK_1(with 5% significance limits for the autocorrelations)
Lag
Part
ial A
utoc
orre
latio
n
78726660544842363024181261
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
PACF of Residuals for $US/ SEK_1(with 5% significance limits for the partial autocorrelations)
AR(1):
Lag
Auto
corr
elat
ion
78726660544842363024181261
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
ACF of Residuals for $US/ SEK_1(with 5% significance limits for the autocorrelations)
LagPa
rtia
l Aut
ocor
rela
tion
78726660544842363024181261
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
PACF of Residuals for $US/ SEK_1(with 5% significance limits for the partial autocorrelations)
AR(2):
No signs of serial correlations in resiaduals in any of the models
“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?”
Exch
ange
rate
YearMonth
19981997199619951994199319921991janjanjanjanjanjanjanjan
7.0
6.5
6.0
5.5
Time Series Plot of Exchange rate
Note that the y-axis do not start at zero!
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.
Lag
Auto
corr
elat
ion
24222018161412108642
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Autocorrelation Function for Exchange rate(with 5% significance limits for the autocorrelations)
“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 ARMA-modelling?”
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.”
Lag
Auto
corr
elat
ion
24222018161412108642
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Autocorrelation Function for Exchange rate(with 5% significance limits for the autocorrelations)
Without differentiation:
Lag
Part
ial A
utoc
orre
latio
n24222018161412108642
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Partial Autocorrelation Function for Exchange rate(with 5% significance limits for the partial autocorrelations)
(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):
Lag
Auto
corr
elat
ion
24222018161412108642
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Autocorrelation Function for Difference in exchange rate(with 5% significance limits for the autocorrelations)
Lag
Part
ial A
utoc
orre
latio
n
24222018161412108642
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
Partial Autocorrelation Function for Difference in exchange rate(with 5% significance limits for the partial autocorrelations)
No obvious pattern in any of these two plots.
The differentiated series may be an ARMA-series
“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 one-step-ahead prediction errors on the average.
Model MSOriginal DifferentiatedAR(1) 0.03682
AR(1) 0.03904AR(2) 0.03914MA(1) 0.03904MA(2) 0.03916ARMA(1,1) 0.03905ARMA(2,1) 0.03889ARMA(1,2) 0.03869ARMA(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
Constant 0.000924 0.001931 0.48 0.634
Differencing: 1 regular difference
Number of observations: Original series 95, after differencing 94
Residuals: SS = 3.25649 (backforecasts excluded)
MS = 0.03743 DF = 87
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 4.9 17.3 26.4 39.5
DF 5 17 29 41
P-Value 0.425 0.431 0.606 0.537
No severe problems but not all parameters are significant!
ARMA(3,3)
Even lower than in ARMA(2,2)
Lag
Part
ial A
utoc
orre
latio
n
222018161412108642
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
PACF of Residuals for Exchange rate(with 5% significance limits for the partial autocorrelations)
Lag
Auto
corr
elat
ion
222018161412108642
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
ACF of Residuals for Exchange rate(with 5% significance limits for the autocorrelations)
No severe problems here either, but spikes seem to increase with lag!
Unable to reduce sum of squares any further
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
Constant 0.000597 0.001779 0.34 0.738
Differencing: 1 regular difference
Number of observations: Original series 95, after differencing 94
Residuals: SS = 3.26434 (backforecasts excluded)
MS = 0.03840 DF = 85
Estimation problems!
ARMA(4,4)
Increased!
None of the parameters are significant!
Estimation problems and an increase in MS.
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.”
Lag
Auto
corr
elat
ion
24222018161412108642
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
ACF of Residuals for Exchange rate(with 5% significance limits for the autocorrelations)
LagPa
rtia
l Aut
ocor
rela
tion
24222018161412108642
1.00.80.60.40.20.0
-0.2-0.4-0.6-0.8-1.0
PACF of Residuals for Exchange rate(with 5% significance limits for the partial autocorrelations)
SAC and SPAC of residuals do not indicate that another ARIMA-model should be used.
Residual
Perc
ent
0.500.250.00-0.25-0.50
99.999
90
50
10
10.1
Fitted Value
Resi
dual
7.06.56.05.5
0.50
0.25
0.00
-0.25
-0.50
Residual
Freq
uenc
y
0.60.40.20.0-0.2-0.4
20
15
10
5
0
Observation Order
Resi
dual
9080706050403020101
0.50
0.25
0.00
-0.25
-0.50
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Histogram of the Residuals Residuals Versus the Order of the Data
Residual Plots for Exchange rate
There do not seem to be any violations of the assumption of normal distribution and constant variance either.