Essentials of Time Series Modeling
Chapter 2 in The Essentials of Time Series Modeling: An Applied Treatment with Emphasis on Topics Relevant to Financial Analysis © by Houston H. Stokes
21 July 2011. All rights reserved. Preliminary Draft
Chapter 2
Time Series Modeling Objectives
2.0 Overview
2.1 Ad Hoc Forecasting Methods
2.2 Examples
Table 2.1 Code To Test Lags 1-4 of Automatic Forecasting Methods
Table 2.2 Summary Measures of performance of Forecasts
Table 2.3 Summary of In-Sample Forecasting for Gasout and Airline Series
Figure 2.1 One Step ahead forecast using No Change Extrapolation Method
Figure 2.2 One Step ahead forecast using No Change Plus Trend Method
Figure 2.3 One Step ahead forecast using Average to Date Method
Figure 2.4 One Step ahead forecast using Moving Average Method
Figure 2.5 One Step ahead forecast using Double Moving Average Method
Figure 2.6 One Step ahead forecast using Exponential Smoothing Method
Figure 2.7 One Step ahead forecast using Double Exponential Smoothing Method
Figure 2.8 One Step ahead forecast using Holt Method
Figure 2.9 One Step ahead forecast using Winters Method
Table 2.4 Using CMAX2 to optimize the forecast
2.3 Conclusions
Time Series Modeling Objectives
2.0 Overview
Time series are analyzed with the objective of being able to filter the non white noise component. Once a filter is identified, it can be reversed to make forecasts out of sample.Define: White noise = a sequence of identically and independently distributed random disturbances with mean zero and variance .[1]
A series with no autocorrelation at any lag is first order white. There may or may not be information at higher orders interactions such as for . For a first order white series . For such a series may or may not be equal to zero for . The same holds true for higher orders. To determine if there is information at higher orders, we need nonlinearity tests. Prewhitened series or in some cases partially filtered series can be:
- Used to forecast
- Used to measure cross correlations.
- Used to diagnose the identification of a transfer function.
- Used to estimate a multiple input transfer function
- Used in specialized applications such as Baxter-King and Hodrick-Prescott to study certain frequencies of a series.
Time Series analysis helps show relationships between different forecasting techniques. Forecasting techniques that include:
- Large scale structural and econometric models.
- Smoothing approaches.
- Markov Models
- VARMA Models
- ARMA Models
- MARS, GAM, ACE, PISPLINE and other nonlinear Models
Types of Forecasts
Subjective (Intuition). Can it be replicated by others? Can it be simulated?
Structural and Econometric Models are based on Theory. Models can be thought of as special cases of transfer functions. For such models, there is no systematic method of diagnostically testing them. "Add Factors" and assumptions on exogenous variables require a seriously subjective approach. Large scale structural and econometric models are expensive to run and maintain.
Deterministic Models imply a 1 to 1 mapping of the driving (input) series and the output. There is no random component. A fast fourier transform (FFT) of a series is a deterministic model since the inverse FFT (IFFT) recovers the original series.
Stochastic Models have a random component. Many deterministic processes appear to be stochastic. For example a computer random number generator will always give the same vector of random numbers, provided the same seed is supplied. However such a vector will be seen as random.
Extrapolative Models - Future values depend on time.
With such models the question arises, why are higher order powers of t so important? In order for such a model to work k=t-i, but too many coefficients are needed. If k < t-i we need an error term and the model is no longer deterministic.
A widely used deterministic model is theexponential growth model
Deterministic models have as potential problems the assumption that disturbances will not persist in any one direction and the assumption that series is systematic and therefore predictable.
2.1 Ad Hoc Forecasting Methods
The use of Time Series models should improve on ad hoc forecasting methods. However such methods are useful in a number of situations that include modeling a large number of series. By use of optimization, the parameters of what appear to be ad hoc forecasting methods can be improved. The basic reference for these methods include Hankie (1998) for the first 9 methods listed in Table 2.1 and Gardner (2006) for the more complex last sixteen methods.
Table 2.1 Ad Hoc Forecasting Methods Supported.
Method Description
nce No change extrapolation (default)
ncept No change + trend
avetd Average to date
mave Moving average
dmave Double moving average
es Exponential smoothing
des Double exponential smoothing
holt Holt's method
winters Winters' method
N_N No Trend no seasonality
N_A No Trend Additive Seasonality
N_M No Trend Mult. Seasonality
A_N Additive Trend no seasonality
A_A Additive Trend Additive Seasonality
A_M Additive Trend Mult. Seasonality
DA_N Damped Add. Trend no seasonality
DA_A Damped Add. Trend Additive Seasonality
DA_M Damped Add. Trend Mult. Seasonality
M_N Mult. Trend no seasonality
M_A Mult. Trend Additive Seasonality
M_M Mult. Trend Mult. Seasonality
DM_N Damped Mult. Trend no seasonality
DM_A Damped Mult. Trend Additive Seasonality
DM_M Damped Mult. Trend Mult. Seasonality
CROSTONIntermittent Demand
Overview
Trend Seasonality None Additive Mult.
N N_N N_A N_M
Additive A_N A_A A_M
Damped Add. DA_N DA_A DA_M
Mult. M_N M_A M_M
Damped Mult. DM_N DM_A DM_M
Intermittent Demand CROSTON
The techniques listed in Table 2.1 are often used in "data mining" or forecasting exercises. Forecasts from these methods can be usedas benchmarks from which more complex techniques can be evaluated. The formulas behind the methods available are shown next:
nce => no change extrapolation (default) or
.(2.1-1)
This technique will miss turningpoints as will be shown later.
ncept => no change + trend or
. (2.1-2)
avetd => average to date or
. (2.1-3)
This technique does not capture any autocorrelation structure in the series.
mave => moving average. Define as n period moving average
. (2.1-4)
dmave =>double moving average or moving average of the moving average p periods
ahead
(2.1-5)
The double moving average can forecast ahead. How well it performs is an open question.
es => exponential smoothing.
.(2.1-6)
implies that no weight is given to the forecast for this period which is the same as assuming the no changeextrapolation nce (2.1-1) is the best model. Neither the nce model nor the es model canforecast ahead more than one period.
des => double exponential smoothing can forecast more than one period ahead.
(2.1-7)
holt => Holt's method
(2.1-8)
winters=> Winters' method
(2.1-9)
Remarks on the above "basic" smoothing methods:
- Double exponential smoothing (Brown's Method) can be a useful technique for a series with a trend.
- Holt's method smoothes the trend and the slope directly.
- Winters' method is useful if there is seasonality in the data.
Using the B34S, forecasts of gasout using the moving average method can be obtained with:
b34sexec options ginclude(gas.b34); b34srun;
b34sexec matrix;
call loaddata;
call smooth(gasout :method mave :print);
call tabulate(%xhatobs, %xhat, %actual, %error);
b34srun;
Edited output is:
=> CALL SMOOTH(GASOUT :METHOD MAVE :PRINT)$
Forecast Summary Measures
Residual Sum of Squares (RSS) 779.7206249999998
Mean Absolute Deviation (MAD) 395.8749999999999
Mean Squared Error (MSE) 2.670276113013698
Correlation x vs xhat 0.8669210136128668
Mean Absolute % Error (MAPE) 2.528709041023920E-02
Mean % Error (MPE) 1.953280248284950E-05
=> CALL TABULATE(%XHATOBS, %XHAT, %ACTUAL, %ERROR)$
Obs %XHATOBS %XHAT %ACTUAL %ERROR
1 5.000 53.60 53.40 -0.2000
2 6.000 53.50 53.10 -0.4000
3 7.000 53.38 52.70 -0.6750
4 8.000 53.17 52.40 -0.7750
5 9.000 52.90 52.20 -0.7000
6 10.00 52.60 52.00 -0.6000
7 11.00 52.33 52.00 -0.3250
8 12.00 52.15 52.40 0.2500
9 13.00 52.15 53.00 0.8500
10 14.00 52.35 54.00 1.650
11 15.00 52.85 54.90 2.050
12 16.00 53.58 56.00 2.425
13 17.00 54.48 56.80 2.325
14 18.00 55.42 56.80 1.375
15 19.00 56.13 56.40 0.2750
16 20.00 56.50 55.70 -0.8000
17 21.00 56.42 55.00 -1.425
18 22.00 55.97 54.30 -1.675
19 23.00 55.35 53.20 -2.150
20 24.00 54.55 52.30 -2.250
The tabulate command is used to list the %xhatobs, %xhat, %actual and %error variables that are automatically created. The matrix command test cases smooth_a, smooth_b and smooth_c illustrate various estimation methods. Graphicplots can be utilized to shows how the forecasts are doing in terms of leads and lags. The file smooth_c shows how optimization procedures can be used to determine the appropriate for the des method. These techniques are useful for cases where a great many series have to be forecasted quickly. Forecasts obtained can be used as a benchmark against which to test ARMA and other modeling techniques.
The exponentially weighted smoothing model has to be estimated but can be made to forecast ahead. Given is the forecast of made in period t. Assume:
(2.1-10)
Note that the weights sum to 1.0
(2.1-11)
Forecasts can be made for periods > 1 ahead if we extend the model to use successive forecast horizons
(2.1-12)
The new forecast is times the current period plus times the old forecast for that period made in the prior period.
Gardner (2006) discusses more advanced methods that use a slightly altered notation and were listed in Table 2.1. Thesewill be discussed in turn:
The no trend-no seasonality or n_n method defines
(2.1-13)
where = the series in period t, = smoothed series and = smoothing coefficient. forecasted X value. If the n_n method is the no change extrapolation.
The no trend-additive seasonality or n_a method defines
(2.1-14)
where is the seasonality in period t and is the seasonality coefficient. This method forecasts based on the seasonality observed in the data p periods back.
The no trend-multiplicative or n_m method, in contrast defines,
(2.1-15)
The additive trend-no seasonality or a_n method modifies the n_n method to
(2.1-16)
where = the estimated trend and the trend adjustment coefficient. In contrast to the
n_n method that forecasts a constant for all future periods, the a_n method will forecast a positive of negative increase equal to .
The additive trend-additive seasonality or a_a method modifies the n_a method (2.1-14) to add an additive trend or alternatively could be thought of as modifying the a_n method (2.1-16) to add an additive seasonality,
(2.1-17)
The additive trend-multiplicative seasonality method a_m modifies the n_m method (2.1-15) to add an additive trend.
(2.1-18)
The damped additive trend-no seasonality da_n method modifies the a_n method (2.1-16) by dampening the trend with the coefficient where .
(2.1-19)
The damped additive trend-additive seasonality da_a method modifies the a_a method
(2.1-17) by dampening the trend adjustment.
(2.1-20)
The damped additive trend-multiplicative seasonality da_m method modifies the a_m method (2.1-18) by dampening the trend adjustment.
(2.1-21)
The multiplicative trend-no seasonality m_n method modifies the a_nmodel (2.1-16) by adding a multiplicative trend in placed of the additive trend .
(2.1-22)
The multiplicative trend-additive seasonality model m_a modifies the a_a model (2.1-17) by adding a multiplicative trend in placed of the additive trend .
(2.1-23)
The multiplicative trend-multiplicative seasonality model m_m modifies the a_m model (2.1-18) by adding a multiplicative trend in placed of the additive trend .
(2.1-24)
The damped multiplicative-no seasonality model dm_n modifies the m_n model (2.1-22)
by dampening the trend adjustment
(2.1-25)
The damped multiplicative-additive seasonality model dm_a modifies the m_a model (2.1-23) by dampening the trend adjustment
(2-1-26)
The damped multiplicative-multiplicative seasonality model dm_m modifies the m_m model (2.1-24) by dampening the trend adjustment
(2.1-27)
The Croston (1972) intermittent demand model and its two variants, Syntetos-Boylan (2001) and Teuner-Sani (2009), correct for sporadic zero demand values in the series. If the series to be smoothed has non-zero demand at most every point, then the three Croston methods are the same as the n_n method discussed above. Using the Croston notation define D(t) as the observed series, Q(t) as the inter-arrival time of transactions, Z(t) as the smoothed nonzero demand and P(t) as the smoothed inter-arrival time then the Croston (1972) method defines
(2.1-28)
although in many cases . The Syntetos-Botlan (2001) variant adjusts as
(2.1-29)
while the Teuner-Sani (2009) method makes a further adjustment
(2.1-30)
In the next section graphics and summary measures are utilized you give an idea how these various methods compare.
2.2 Examples
Table 2.1 contains a command file to perform a number of simple tests of the capability of the automatic forecasting methods using the utility subroutine tsmooth that is listed in staging2.mac. Default settings have been used and no optimization has been attempted to obtain better parameters, The file tsmooth in staging.mac contains the source. The analysis will first proceed to look at graphs of the 1 step ahead forecast for the nine methods. Next, a table of summaryperformance measures described in Table 2.2 will be displayed. Finally graphics will be generated and briefly discussed.
Table 2.1 Code To Test Lags 1-4 of Automatic Forecasting Methods
/;
/; Illustrates "Automatic Methods"
/; Results can be graphed
/;
/; Uses
/;
/; subroutine tsmooth(x,minlag,maxlag,ismooth,
/; iprintdt,igraph,iprint,sum_tab,igrid);
/;
/; using defaults look at performance of "Automatic Methods"
/; in in-sample "forecasting"
/;
/; x - series
/; minlag - minumum lag considered
/; maxlag - maximum lag considered
/; ismooth - = 0 use Hankie models
/; = 1 use Gardner models
/; iprintdt - ne 0 print %actual and %xhat / %s
/; igraph - ne 0 => draw graphs. Use minlag=maxlag
/; iprint - ne 0 show smooth output
/; sum_tab - ne 0 print a summary table
/; igrid - ne 0 show a grid in graphs.
/;
/; Built 21 July 2011
/;
b34sexec options ginclude('gas.b34'); b34srun;
b34sexec matrix;
call loaddata;
call load(tsmooth :staging);
call echooff;
x=gasout;
/; Tests for lag=1-4 forecasts
minlag=1;
maxlag=4;
iprintdt=0;
igraph=0;
iprint=0;
igrid=1;
sum_tab=1;
ismooth=0;
call tsmooth(x,minlag,maxlag,ismooth,iprintdt,igraph,iprint,
sum_tab,igrid);
ismooth=1;
call tsmooth(x,minlag,maxlag,ismooth,iprintdt,igraph,iprint,
sum_tab,igrid);
b34srun;
b34sexec options ginclude('b34sdata.mac') member(bj_g); b34srun;
b34sexec matrix;
call loaddata;
call load(tsmooth :staging);
call echooff;
x=airline;
/; Tests for lag=1 forecasts
minlag=1;
maxlag=4;
iprintdt=0;
igraph=0;
iprint=0;
igrid=1;
sum_tab=1;
ismooth=0;
call tsmooth(x,minlag,maxlag,ismooth,iprintdt,igraph,iprint,
sum_tab,igrid);
ismooth=1;
call tsmooth(x,minlag,maxlag,ismooth,iprintdt,igraph,iprint,
sum_tab,igrid);
b34srun;
Table 2.2 Summary Measures of performance of Forecasts
%rss => residual sum of squares for nlags.
%mad => sum(dabs(x-xhat))/n for n lags.
%mse => rss/n
%mape => sum(dabs(x-xhat)/X)/n to measure how large forecast
is in relation to series.
%mpe => sum((x-xhat)/x)/n + and - tests if over or under.
%corr => Correlation between x and xhat using usual formula.
Table 2.3 Summary of In-Sample Forecasting for Gasout and Airline Series
______
Gasout Series
Summary Measures for insample Forecast for Lag 1
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 NCE 164.6 176.8 0.5581 0.1120E-01 0.9836E-04 0.9727
2 NCEPT 56.03 96.30 0.1906 0.6134E-02 0.1561E-03 0.9926
3 AVETD 3059. 764.9 10.37 0.4782E-01 0.1454E-01 0.2805
4 MAVE 779.7 395.9 2.670 0.2529E-01 0.1953E-04 0.8669
5 DMAVE 597.9 341.2 2.069 0.2201E-01 0.2411E-03 0.9275
6 ES 783.6 393.7 2.656 0.2489E-01 -0.5614E-04 0.8615
7 DES 503.3 314.2 1.706 0.1984E-01 0.1434E-03 0.9282
8 HOLT 1007. 446.8 3.414 0.2824E-01 0.1450E-03 0.8571
9 WINTERS 976.9 439.0 3.311 0.2773E-01 0.2109E-03 0.8617
Summary Measures for in sample Forecast for Lag 2
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 NCE 602.3 344.7 2.049 0.2188E-01 0.6666E-04 0.9000
2 NCEPT 323.0 239.2 1.102 0.1526E-01 0.2008E-03 0.9505
3 AVETD 3126. 773.8 10.63 0.4856E-01 0.1454E-01 0.2362
4 MAVE 1359. 522.1 4.670 0.3343E-01 -0.4203E-04 0.7668
5 DMAVE 1550. 552.9 5.382 0.3574E-01 0.2251E-03 0.8327
6 ES 1284. 502.1 4.367 0.3185E-01 -0.1101E-03 0.7688
7 DES 1211. 488.2 4.118 0.3091E-01 0.1384E-03 0.8410
8 HOLT 1793. 598.0 6.100 0.3791E-01 0.1525E-03 0.7664
9 WINTERS 1736. 587.1 5.905 0.3720E-01 0.2328E-03 0.7736
Summary Measures for in sample Forecast for Lag 3
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 NCE 1210. 490.2 4.129 0.3120E-01 -0.2877E-04 0.7986
2 NCEPT 879.7 406.2 3.013 0.2598E-01 0.1042E-03 0.8626
3 AVETD 3189. 782.0 10.89 0.4926E-01 0.1455E-01 0.1939
4 MAVE 1957. 627.5 6.748 0.4031E-01 -0.7542E-04 0.6627
5 DMAVE 2999. 769.1 10.45 0.4982E-01 0.1413E-03 0.7117
6 ES 1811. 594.7 6.180 0.3785E-01 -0.1673E-03 0.6705
7 DES 2202. 659.6 7.517 0.4188E-01 0.9727E-04 0.7323
8 HOLT 2804. 750.2 9.570 0.4773E-01 0.1553E-03 0.6638
9 WINTERS 2703. 734.9 9.225 0.4672E-01 0.2522E-03 0.6737
Summary Measures for in sample Forecast for Lag 4
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 NCE 1868. 612.0 6.398 0.3905E-01 -0.1235E-03 0.6878
2 NCEPT 1583. 553.5 5.438 0.3547E-01 -0.2371E-04 0.7510
3 AVETD 3248. 789.5 11.12 0.4991E-01 0.1456E-01 0.1551
4 MAVE 2502. 704.7 8.657 0.4541E-01 -0.4819E-04 0.5668
5 DMAVE 4854. 979.2 16.97 0.6358E-01 0.9729E-04 0.5838
6 ES 2303. 667.3 7.888 0.4261E-01 -0.2083E-03 0.5780
7 DES 3382. 821.7 11.58 0.5232E-01 0.8040E-04 0.6187
8 HOLT 3959. 891.8 13.56 0.5696E-01 0.1888E-03 0.5628
9 WINTERS 3793. 871.9 12.99 0.5566E-01 0.3132E-03 0.5751
Summary Measures for in sample Forecast for Lag 1
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 N_N 783.6 393.3 2.674 0.2504E-01 -0.3747E-04 0.8615
2 N_A 641.6 360.2 2.182 0.2286E-01 -0.1387E-03 0.8884
3 N_M 639.4 357.9 2.175 0.2271E-01 0.1658E-04 0.8888
4 A_N 1035. 458.6 3.521 0.2910E-01 -0.9273E-03 0.8532
5 A_A 803.2 405.3 2.732 0.2570E-01 -0.9272E-03 0.8832
6 A_M 799.0 404.2 2.718 0.2563E-01 -0.9495E-03 0.8838
7 DA_N 705.2 378.0 2.399 0.2398E-01 -0.1675E-03 0.8770
8 DA_A 584.3 346.0 1.987 0.2195E-01 -0.2486E-03 0.8995
9 DA_M 580.4 343.9 1.974 0.2182E-01 -0.1260E-03 0.9001
10 M_N 1014. 448.0 3.450 0.2840E-01 -0.8746E-03 0.8563
11 M_A 782.8 396.2 2.663 0.2510E-01 -0.6312E-03 0.8861
12 M_M 780.2 394.5 2.654 0.2500E-01 -0.6539E-03 0.8865
13 DM_N 702.4 375.4 2.389 0.2382E-01 -0.6358E-04 0.8776
14 DM_A 580.2 343.7 1.974 0.2181E-01 -0.1509E-03 0.9002
15 DM_M 578.3 341.6 1.967 0.2168E-01 -0.2757E-04 0.9005
Summary Measures for in sample Forecast for Lag 2
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 N_N 1284. 501.7 4.397 0.3204E-01 -0.8532E-04 0.7689
2 N_A 1046. 457.9 3.571 0.2914E-01 -0.1675E-03 0.8154
3 N_M 1044. 455.8 3.564 0.2901E-01 -0.6442E-05 0.8158
4 A_N 1843. 613.1 6.290 0.3902E-01 -0.1253E-02 0.7605
5 A_A 1421. 539.3 4.850 0.3431E-01 -0.1198E-02 0.8062
6 A_M 1417. 538.6 4.837 0.3426E-01 -0.1262E-02 0.8066
7 DA_N 1203. 491.9 4.106 0.3130E-01 -0.2409E-03 0.7881
8 DA_A 990.8 448.8 3.382 0.2856E-01 -0.2926E-03 0.8282
9 DA_M 986.9 447.0 3.368 0.2845E-01 -0.1704E-03 0.8288
10 M_N 1807. 599.4 6.168 0.3812E-01 -0.1235E-02 0.7650
11 M_A 1388. 527.7 4.738 0.3353E-01 -0.8503E-03 0.8103
12 M_M 1385. 526.3 4.727 0.3345E-01 -0.9154E-03 0.8107
13 DM_N 1199. 488.9 4.092 0.3111E-01 -0.1214E-03 0.7889
14 DM_A 985.5 446.4 3.363 0.2841E-01 -0.1836E-03 0.8291
15 DM_M 983.7 444.5 3.357 0.2829E-01 -0.6071E-04 0.8294
Summary Measures for in sample Forecast for Lag 3
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 N_N 1811. 594.2 6.223 0.3807E-01 -0.1364E-03 0.6705
2 N_A 1464. 540.0 5.013 0.3448E-01 -0.2125E-03 0.7385
3 N_M 1462. 538.0 5.007 0.3435E-01 -0.2709E-04 0.7389
4 A_N 2882. 768.0 9.868 0.4906E-01 -0.1593E-02 0.6563
5 A_A 2201. 673.4 7.539 0.4298E-01 -0.1497E-02 0.7163
6 A_M 2198. 673.1 7.529 0.4296E-01 -0.1571E-02 0.7167
7 DA_N 1757. 592.5 6.016 0.3782E-01 -0.3157E-03 0.6888
8 DA_A 1435. 538.8 4.913 0.3439E-01 -0.3486E-03 0.7487
9 DA_M 1431. 537.1 4.900 0.3429E-01 -0.2053E-03 0.7493
10 M_N 2829. 752.9 9.687 0.4804E-01 -0.1660E-02 0.6617
11 M_A 2154. 660.5 7.378 0.4210E-01 -0.1130E-02 0.7215
12 M_M 2150. 659.1 7.362 0.4202E-01 -0.1206E-02 0.7219
13 DM_N 1751. 589.3 5.996 0.3762E-01 -0.1877E-03 0.6898
14 DM_A 1428. 536.3 4.891 0.3423E-01 -0.2346E-03 0.7498
15 DM_M 1427. 534.6 4.886 0.3412E-01 -0.9072E-04 0.7500
Summary Measures for in sample Forecast for Lag 4
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 N_N 2303. 666.4 7.941 0.4284E-01 -0.1515E-03 0.5781
2 N_A 1841. 602.0 6.328 0.3856E-01 -0.2389E-03 0.6666
3 N_M 1840. 600.2 6.322 0.3844E-01 -0.1565E-04 0.6669
4 A_N 4068. 911.8 13.98 0.5848E-01 -0.1896E-02 0.5543
5 A_A 3072. 796.2 10.56 0.5102E-01 -0.1774E-02 0.6247
6 A_M 3070. 796.1 10.55 0.5102E-01 -0.1829E-02 0.6250
7 DA_N 2288. 671.9 7.864 0.4303E-01 -0.3595E-03 0.5926
8 DA_A 1849. 608.9 6.355 0.3899E-01 -0.3818E-03 0.6719
9 DA_M 1846. 607.4 6.343 0.3889E-01 -0.2022E-03 0.6724
10 M_N 3997. 894.7 13.74 0.5731E-01 -0.2098E-02 0.5600
11 M_A 3010. 781.9 10.34 0.5004E-01 -0.1418E-02 0.6303
12 M_M 3002. 780.2 10.32 0.4993E-01 -0.1478E-02 0.6309
13 DM_N 2281. 668.8 7.840 0.4284E-01 -0.2262E-03 0.5939
14 DM_A 1842. 606.4 6.330 0.3883E-01 -0.2643E-03 0.6731
15 DM_M 1841. 604.9 6.326 0.3873E-01 -0.8426E-04 0.6734
Airline Data
Summary Measures for in sample Forecast for Lag 1
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 NCE 0.1625E+06 3698. 1136. 0.9019E-01 0.3784E-02 0.9602
2 NCEPT 0.2240E+06 4314. 1577. 0.1057 0.2917E-02 0.9505
3 AVETD 0.2079E+07 0.1339E+05 0.1453E+05 0.2810 0.2706 0.9249
4 MAVE 0.3871E+06 5515. 2765. 0.1325 0.5221E-02 0.8995
5 DMAVE 0.4971E+06 6150. 3629. 0.1588 -0.1214E-01 0.8844
6 ES 0.3010E+06 4832. 2105. 0.1132 0.1664E-01 0.9255
7 DES 0.3075E+06 4775. 2150. 0.1148 -0.1128E-01 0.9254
8 HOLT 0.3902E+06 5653. 2728. 0.1337 -0.1618E-01 0.9024
9 WINTERS 0.3931E+06 5406. 2749. 0.1306 -0.2360E-01 0.9018
Summary Measures for in sample Forecast for Lag 2
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 NCE 0.4224E+06 5820. 2975. 0.1406 0.3988E-02 0.8957
2 NCEPT 0.5041E+06 6565. 3575. 0.1639 0.1757E-03 0.8852
3 AVETD 0.2118E+07 0.1355E+05 0.1491E+05 0.2873 0.2766 0.9208
4 MAVE 0.5595E+06 6694. 4025. 0.1615 0.1005E-01 0.8526
5 DMAVE 0.1029E+07 9095. 7566. 0.2329 -0.2392E-01 0.7797
6 ES 0.4501E+06 5884. 3170. 0.1391 0.2195E-01 0.8865
7 DES 0.6453E+06 7175. 4545. 0.1727 -0.1997E-01 0.8463
8 HOLT 0.6368E+06 7181. 4485. 0.1712 -0.2213E-01 0.8408
9 WINTERS 0.6526E+06 6974. 4596. 0.1688 -0.3152E-01 0.8373
Summary Measures for in sample Forecast for Lag 3
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 NCE 0.6514E+06 7059. 4620. 0.1714 0.5708E-02 0.8374
2 NCEPT 0.8100E+06 7854. 5786. 0.1957 -0.1199E-02 0.8123
3 AVETD 0.2154E+07 0.1369E+05 0.1528E+05 0.2927 0.2821 0.9188
4 MAVE 0.6638E+06 7618. 4810. 0.1829 0.1725E-01 0.8227
5 DMAVE 0.1646E+07 0.1171E+05 0.1219E+05 0.2979 -0.3301E-01 0.6724
6 ES 0.5469E+06 6707. 3879. 0.1584 0.2878E-01 0.8608
7 DES 0.9840E+06 8833. 6978. 0.2148 -0.2773E-01 0.7687
8 HOLT 0.8509E+06 8394. 6035. 0.2019 -0.2662E-01 0.7867
9 WINTERS 0.8863E+06 8379. 6286. 0.2033 -0.3702E-01 0.7784
Summary Measures for in sample Forecast for Lag 4
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 NCE 0.7991E+06 8212. 5708. 0.1989 0.1081E-01 0.7977
2 NCEPT 0.1058E+07 8893. 7613. 0.2217 -0.1425E-02 0.7519
3 AVETD 0.2188E+07 0.1383E+05 0.1563E+05 0.2980 0.2880 0.9183
4 MAVE 0.7068E+06 7759. 5159. 0.1850 0.2609E-01 0.8092
5 DMAVE 0.2286E+07 0.1408E+05 0.1706E+05 0.3570 -0.3824E-01 0.5683
6 ES 0.5857E+06 6923. 4184. 0.1627 0.3801E-01 0.8507
7 DES 0.1251E+07 0.1048E+05 8939. 0.2542 -0.3151E-01 0.7065
8 HOLT 0.9868E+06 9162. 7048. 0.2209 -0.2750E-01 0.7502
9 WINTERS 0.1026E+07 9328. 7328. 0.2268 -0.3912E-01 0.7400
Summary Measures for in sample Forecast for Lag 1
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 N_N 0.3007E+06 4809. 2133. 0.1136 0.1577E-01 0.9236
2 N_A 0.2827E+06 4703. 1991. 0.1109 0.1064E-01 0.9286
3 N_M 0.2827E+06 4705. 1991. 0.1110 0.1077E-01 0.9286
4 A_N 0.3899E+06 5637. 2746. 0.1338 -0.1721E-01 0.9010
5 A_A 0.3526E+06 5267. 2483. 0.1252 -0.1536E-01 0.9111
6 A_M 0.3543E+06 5279. 2495. 0.1255 -0.1612E-01 0.9108
7 DA_N 0.3029E+06 4878. 2133. 0.1150 0.1060E-01 0.9233
8 DA_A 0.2827E+06 4724. 1991. 0.1117 0.7415E-02 0.9284
9 DA_M 0.2828E+06 4726. 1992. 0.1118 0.7309E-02 0.9283
10 M_N 0.4080E+06 5758. 2873. 0.1371 -0.2543E-01 0.8981
11 M_A 0.3669E+06 5360. 2584. 0.1277 -0.2244E-01 0.9089
12 M_M 0.3695E+06 5376. 2602. 0.1282 -0.2322E-01 0.9085
13 DM_N 0.3039E+06 4893. 2140. 0.1153 0.9431E-02 0.9229
14 DM_A 0.2836E+06 4731. 1997. 0.1119 0.6446E-02 0.9281
15 DM_M 0.2837E+06 4733. 1998. 0.1120 0.6432E-02 0.9281
Summary Measures for in sample Forecast for Lag 2
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 N_N 0.4495E+06 5852. 3211. 0.1393 0.2057E-01 0.8836
2 N_A 0.4183E+06 5674. 2967. 0.1345 0.1347E-01 0.8925
3 N_M 0.4190E+06 5675. 2972. 0.1345 0.1398E-01 0.8923
4 A_N 0.6362E+06 7155. 4512. 0.1710 -0.2388E-01 0.8387
5 A_A 0.5691E+06 6731. 4036. 0.1609 -0.2170E-01 0.8564
6 A_M 0.5709E+06 6742. 4049. 0.1612 -0.2223E-01 0.8561
7 DA_N 0.4692E+06 5998. 3328. 0.1423 0.1387E-01 0.8792
8 DA_A 0.4329E+06 5815. 3071. 0.1382 0.9351E-02 0.8886
9 DA_M 0.4336E+06 5818. 3075. 0.1383 0.9581E-02 0.8885
10 M_N 0.6725E+06 7342. 4769. 0.1760 -0.3511E-01 0.8338
11 M_A 0.5982E+06 6882. 4242. 0.1650 -0.3141E-01 0.8525
12 M_M 0.6011E+06 6900. 4263. 0.1654 -0.3193E-01 0.8520
13 DM_N 0.4714E+06 6016. 3343. 0.1428 0.1250E-01 0.8786
14 DM_A 0.4349E+06 5830. 3085. 0.1386 0.8244E-02 0.8881
15 DM_M 0.4354E+06 5832. 3088. 0.1386 0.8580E-02 0.8880
Summary Measures for in sample Forecast for Lag 3
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 N_N 0.5467E+06 6685. 3933. 0.1594 0.2805E-01 0.8569
2 N_A 0.5020E+06 6349. 3586. 0.1511 0.1853E-01 0.8693
3 N_M 0.5035E+06 6359. 3597. 0.1513 0.1963E-01 0.8690
4 A_N 0.8504E+06 8362. 6074. 0.2016 -0.2821E-01 0.7837
5 A_A 0.7504E+06 7756. 5360. 0.1880 -0.2607E-01 0.8090
6 A_M 0.7505E+06 7754. 5361. 0.1879 -0.2590E-01 0.8088
7 DA_N 0.5856E+06 6950. 4183. 0.1653 0.1992E-01 0.8475
8 DA_A 0.5329E+06 6543. 3806. 0.1564 0.1379E-01 0.8609
9 DA_M 0.5340E+06 6551. 3815. 0.1565 0.1463E-01 0.8606
10 M_N 0.9087E+06 8571. 6490. 0.2077 -0.4283E-01 0.7767
11 M_A 0.7974E+06 7930. 5696. 0.1931 -0.3875E-01 0.8029
12 M_M 0.7980E+06 7931. 5700. 0.1930 -0.3851E-01 0.8027
13 DM_N 0.5886E+06 6967. 4204. 0.1658 0.1844E-01 0.8466
14 DM_A 0.5355E+06 6555. 3825. 0.1568 0.1264E-01 0.8601
15 DM_M 0.5363E+06 6562. 3831. 0.1569 0.1359E-01 0.8600
Summary Measures for in sample Forecast for Lag 4
Obs METHOD RSS MAD MSE MAPE MPE CORR
1 N_N 0.5853E+06 6894. 4241. 0.1634 0.3711E-01 0.8466
2 N_A 0.5304E+06 6659. 3815. 0.1582 0.2619E-01 0.8608
3 N_M 0.5329E+06 6667. 3834. 0.1582 0.2796E-01 0.8605
4 A_N 0.9863E+06 9140. 7095. 0.2213 -0.2861E-01 0.7465
5 A_A 0.8595E+06 8592. 6183. 0.2087 -0.2685E-01 0.7771
6 A_M 0.8569E+06 8581. 6165. 0.2082 -0.2578E-01 0.7771
7 DA_N 0.6369E+06 7308. 4582. 0.1733 0.2919E-01 0.8327
8 DA_A 0.5718E+06 6991. 4114. 0.1666 0.2127E-01 0.8489
9 DA_M 0.5736E+06 6997. 4127. 0.1666 0.2279E-01 0.8486
10 M_N 0.1061E+07 9372. 7634. 0.2283 -0.4663E-01 0.7387
11 M_A 0.9195E+06 8801. 6615. 0.2151 -0.4255E-01 0.7700
12 M_M 0.9154E+06 8789. 6586. 0.2146 -0.4131E-01 0.7702
13 DM_N 0.6392E+06 7326. 4598. 0.1738 0.2770E-01 0.8319
14 DM_A 0.5737E+06 7000. 4128. 0.1669 0.2016E-01 0.8482
15 DM_M 0.5753E+06 7005. 4139. 0.1669 0.2180E-01 0.8480
______
Note that Forecasts for more than one period are not possible with the AVETD, MAVE, ES, methods. See Table 2.2 for definitions of the column headings.
Using the forecasting of gasout as an example, the no change extrapolation plus trend method (ncept) is far and away the best approach with values of 56.03, 323.0, 879.7 and 1583 for forecasts of from 1-4 periods ahead. The average to date method (avetd)had the worsterror sum of squares with and no capability for forecasts = 1. Using an automatically detected ARIMA model of gasout, , which is substantially better than the ncept method. The summary measures listed in Table 2.3, however, do not tell the whole story. For that we need to look at graphs of and for the nine methods discussed in the prior section which are shown in figures 2.1 to 2.9 below.
Figure 2.1 One Step ahead forecast using No Change Extrapolation Method
Figure 2.2 One Step ahead forecast using No Change Plus Trend Method
Figure 2.3 One Step ahead forecast using Average to Date Method
Figure 2.4 One Step ahead forecast using Moving Average Method