Post on 18-Dec-2015
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’).”
CPI(
gro
up))
YearMonth
1998199519921989198619831980janjanjanjanjanjanjan
200
180
160
140
120
100
Time Series Plot of CPI(group))
“Then estimate the autocorrelations and display them in a graph.”
Lag
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Autocorrelation Function for CPI(group))(with 5% significance limits for the autocorrelations)
“Is there any obvious upward or downward trend?”
Yes, upward, but turning at the endCPI(
gro
up))
YearMonth
1998199519921989198619831980janjanjanjanjanjanjan
200
180
160
140
120
100
Time Series Plot of CPI(group))
“Are there any signs of long-time oscillations in the time series?
CPI(
gro
up))
YearMonth
1998199519921989198619831980janjanjanjanjanjanjan
200
180
160
140
120
100
Time Series Plot of CPI(group))
Are there any signs of seasonal variation in the series?”
No!
Not visible!
“Do the autocorrelations cancel out quickly?”
Lag
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Autocorrelation Function for CPI(group))(with 5% significance limits for the autocorrelations)
No!
“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
u
YearMonth
1998199519921989198619831980janjanjanjanjanjanjan
3
2
1
0
-1
-2
Time Series Plot of u
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Autocorrelation Function for u(with 5% significance limits for the autocorrelations)
Diffuse pattern!
Not convincingly stationary!
vt = yt – yt - 12
v
YearMonth
1998199519921989198619831980janjanjanjanjanjanjan
15
10
5
0
-5
Time Series Plot of v
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Autocorrelation Function for v(with 5% significance limits for the autocorrelations)
Definitely non-stationary!
“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)
AR(1):
AR(2):
Type Coef SE Coef T P
AR 1 0.1170 0.0671 1.75 0.082
Constant 0.38522 0.04779 8.06 0.000
…
MS = 0.512 DF = 222
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag 12 24 36 48
…
P-Value 0.000 0.000 0.000 0.000
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
Constant 0.27053 0.04576 5.91 0.000
…
MS = 0.469 DF = 221
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag 12 24 36 48
…
P-Value 0.007 0.003 0.001 0.003
2 months forecasts:
Forecast Lower Upper
195.472 194.070 196.875
195.925 193.823 198.027
2 months forecasts:
Forecast Lower Upper
194.962 193.620 196.305
195.720 193.747 197.693
MA(1):
MA(2):
Type Coef SE Coef T P
MA 1 -0.0741 0.0675 -1.10 0.273
Constant 0.43605 0.05146 8.47 0.000
…
MS = 0.514 DF = 222
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag 12 24 36 48
…
P-Value 0.000 0.000 0.000 0.000
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
Constant 0.43664 0.06071 7.19 0.000
…
MS = 0.479 DF = 221
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag 12 24 36 48
…
P-Value 0.000 0.000 0.000 0.000
2 months forecasts:
Forecast Lower Upper
195.430 194.024 196.835
195.866 193.803 197.929
2 months forecasts:
Forecast Lower Upper
195.146 193.789 196.503
196.032 194.056 198.009
ARMA(1,1):
ARMA(2,1):
* 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
Constant -0.0117678 -0.0013602 8.65 0.000
…
MS = 0.458 DF = 221
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag 12 24 36 48
…
P-Value 0.000 0.000 0.000 0.000
Type 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
Constant 0.17191 0.03287 5.23 0.000
…
MS = 0.469 DF = 220
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag 12 24 36 48
P-Value 0.004 0.001 0.000 0.001
2 months forecasts:
Forecast Lower Upper
194.516 193.190 195.843
194.114 192.199 196.029
2 months forecasts:
Forecast Lower Upper
194.746 193.403 196.088
195.300 193.355 197.246
ARMA(1,2):
ARMA(2,2):
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
Constant 0.16753 0.03043 5.51 0.000
…
MS = 0.472 DF = 220
Modified Box-Pierce (Ljung-Box) Chi-Squ
Lag 12 24 36 48
…
P-Value 0.002 0.001 0.000 0.000
* ERROR * Model cannot be estimated with these data.
2 months forecasts:
Forecast Lower Upper
194.728 193.382 196.075
195.214 193.256 197.173
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
w
YearMonth
1998199519921989198619831980janjanjanjanjanjanjan
4
3
2
1
0
-1
-2
-3
Time Series Plot of w
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Autocorrelation Function for w(with 5% significance limits for the autocorrelations)
Clear seasonal correlation and close to non-stationary
How about first order seasonal differences on the first-order non-seasonal differences?
zt = ut – ut – 12 = (yt – yt – 1) – (yt – 12 – yt – 13)
z
YearMonth
1998199519921989198619831980janjanjanjanjanjanjan
2
1
0
-1
-2
Time Series Plot of z
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Autocorrelation Function for z(with 5% significance limits for the autocorrelations)
Much more a stationary look!
Lag
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Autocorrelation Function for z(with 5% significance limits for the autocorrelations)
Lag
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Partial Autocorrelation Function for z(with 5% significance limits for the partial autocorrelations)
Tricky to identify the correct model.
Clearly a seasonal model must be used, most probably with at least one MA –term
Non-seasonal part more difficult. ARMA(1,1) ?
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
Constant -0.07528 0.01129 -6.67 0.000
Differencing: 1 regular, 1 seasonal of order 12
Number of observations: Original series 225, after differencing 212
Residuals: SS = 77.9325 (backforecasts excluded)
MS = 0.3747 DF = 208
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 12.9 22.8 28.9 36.0
DF 8 20 32 44
P-Value 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 Coef SE Coef T P
SMA 12 0.9039 0.0472 19.15 0.000
Constant -0.045964 0.006893 -6.67 0.000
Differencing: 1 regular, 1 seasonal of order 12
Number of observations: Original series 225, after differencing 212
Residuals: SS = 78.0539 (backforecasts excluded)
MS = 0.3717 DF = 210
Modified Box-Pierce (Ljung-Box) Chi-Square statistic
Lag 12 24 36 48
Chi-Square 12.5 21.8 28.0 34.8
DF 10 22 34 46
P-Value 0.254 0.475 0.757 0.887
Slightly smaller MS!
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):
Lag
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ACF of Residuals for CPI(group))(with 5% significance limits for the autocorrelations)
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PACF of Residuals for CPI(group))(with 5% significance limits for the partial autocorrelations)
Residual
Perc
ent
210-1-2
99.9
99
90
50
10
1
0.1
Fitted Value
Resi
dual
200175150125100
2
1
0
-1
-2
Residual
Fre
quency
1.81.20.60.0-0.6-1.2
30
20
10
0
Observation Order
Resi
dual
220200180160140120100806040201
2
1
0
-1
-2
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 CPI(group))
Non-satisfactory
ARIMA(1,1,1,0,1,1)12
Lag
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ACF of Residuals for CPI(group))(with 5% significance limits for the autocorrelations)
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PACF of Residuals for CPI(group))(with 5% significance limits for the partial autocorrelations)
Residual
Perc
ent
210-1-2
99.9
99
90
50
10
1
0.1
Fitted Value
Resi
dual
200175150125100
1
0
-1
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ResidualFre
quency
1.20.60.0-0.6-1.2
30
20
10
0
Observation Order
Resi
dual
220200180160140120100806040201
1
0
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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 CPI(group))
Satisfactory!
ARIMA(0,1,0,0,1,1)12
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ACF of Residuals for CPI(group))(with 5% significance limits for the autocorrelations)
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4842363024181261
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PACF of Residuals for CPI(group))(with 5% significance limits for the partial autocorrelations)
Residual
Perc
ent
210-1-2
99.9
99
90
50
10
1
0.1
Fitted Value
Resi
dual
200175150125100
2
1
0
-1
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ResidualFre
quency
1.20.60.0-0.6-1.2
40
30
20
10
0
Observation Order
Resi
dual
220200180160140120100806040201
2
1
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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 CPI(group))
Satisfactory!
“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.”
Exch
ange r
ate
YearMonth
19981997199619951994199319921991janjanjanjanjanjanjanjan
10
8
6
4
2
0
Time Series Plot of Exchange rate
Exch
ange r
ate
YearMonth
19981997199619951994199319921991janjanjanjanjanjanjanjan
7.0
6.5
6.0
5.5
Time Series Plot of Exchange rate
Change scale so that y-axis starts at 0 (and ends at 10)
Single exponential smoothing will probably work well.
Optimize
Index
Exch
ange r
ate
1009080706050403020101
7.0
6.5
6.0
5.5
Alpha 0.995540Smoothing Constant
MAPE 2.41996MAD 0.14983MSD 0.03784
Accuracy Measures
ActualFitsForecasts95.0% PI
Variable
Single Exponential Smoothing Plot for Exchange rate
Prediction formula:
1
1
0045.09955.0
1ˆ
TT
TTTT
y
yy
Forecasts
Period 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
Exch
ange r
ate
1009080706050403020101
7.5
7.0
6.5
6.0
5.5
Time Series Plot for Exchange rate(with forecasts and their 95% confidence limits)
Forecasts from period 95
95 Percent
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 does the prediction formula look like in this case?”
Final Estimates of Parameters
Type Coef SE Coef
MA 1 0.0052 0.1043
Constant 0.00537 0.02027
“How do the forecasts differ between the two different methods of forecasting?”
Forecasts
Period 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
Forecasts from period 95
95 Percent
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
SES ARIMA(0,1,1)
Index
Exch
ange r
ate
1009080706050403020101
7.0
6.5
6.0
5.5
Alpha 0.995540Smoothing Constant
MAPE 2.41996MAD 0.14983MSD 0.03784
Accuracy Measures
ActualFitsForecasts95.0% PI
Variable
Single Exponential Smoothing Plot for Exchange rate
Time
Exch
ange r
ate
1009080706050403020101
7.5
7.0
6.5
6.0
5.5
Time Series Plot for Exchange rate(with forecasts and their 95% confidence limits)