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Unformatted text preview: Res Ec 797A: Forecasting Order Of Topics And Readings For Lecture 8 (September 28, 2011) Source Pagegs) Top'c Material Not Covered Last Class BM 32 A Summary Of The Smoothing Equations Involved With
The Three Basic Exponential Smoothing Methods: One
Parameter; Two Parameters; Three Parameters Material Distributed Today
BM 33 Overview of multiparameter exponential smoothing methods Holt’s TwoParameter Method BM 34 Equation development for Holt’s method BM 35 The Dynamics of Holt’s Method BM Unnumbered The Dynamics of Holt’s Method (the “picture” directly above)
Single with calculations obtained from BM pp. 3637 or NB pages
Page 193199. BM 3637 Presenting the dynamics of Holt’s method. This represents numerical calculations that go with the graph on BM page 35.
This also parallels NB pages 193199. Winters’ ThreeParameter Method
NB 199~210 Winter’s Method (Triple Exponential Smoothing) BM 3839 Winter’s Additive Seasonal Method.
This goes with NB pages 199205. BM 3940 Presenting the dynamics of Winters’ method. These pages are
nowhere as welldeveloped as BM pages 3537 that illustrate
the dynamics of Holt’s method. You are not responsible for the
dynamics involved with Winters’ method. BM 41 Winter’s Multiplicative Seasonal Method
This goes with NB pages 205210. MULTI—PARAMETER EXPONENTIAL SMOOTHING In the late 19503 and early 19603, at the same time that Brown was developing the single parameter
exponential smoothing methods, a different and ultimately more useful approach was being
developed by Holt and by Winters. This approach is descn'bed in Newbold and B03, Sections 6.3
and 6.4 of their text. We have: 1. Holt's twoparameter method (for level and trend). This method is optimal for an ARIMA
(0,2,2) process.
2. Winters‘ additiveseasonal method. lts three parameters allow for level, trend, and seasonal smoothing. There is an equivalent ARIMA representation. 3. Winters' multiplicativeseasonal method. It has three parameters and is the widest method
used. It has no equivalent ARIMA representation. The threeparameter method is generally referred to as the HoltWinters method. Assume that the
multiplicative version is meant unless there is an explicit statement to the contrary. One attraction of the HoltWinters method is that it is quite ﬂexible due to the two additional
parameters (relative to simple exponential smoothing) that accommodate trend and seasonality.
However, the increased ﬂexibility afforded by three parameters comes with some risk.
Specifically, estimates of parameters are always subject to error. With HoltWinters, the risk appears worthwhile. In forecasting competitions, these methods
outperform Brown's oneparameter methods. Their forecasting performance relative to other
approaches is mixed. In terms of complexity, the HoltWinters method is relatively simple compared to BoxJenkins
ARIMA models and econometric models. As an example, if a data series is known to be
nonseasonal (as with annual data), the seasonal smoothing parameter in the Holt—Winters method
simply can be set to zero. With Holt and HoltWinters, we are estimating two and three smoothing parameters, respectively.
An additional attraction of these methods is that the extra parameters permit trend and the seasonal
effect to change or update for each new time period. This possibility is the precursor to treating
trend and seasonality as stochastic in more complicated models later on. It is also possible for the trend or seasonal parameter to have a value of zero. This does not mean
that there is no trend or seasonal pattern. A trend or seasonal parameter with a value of zero
means that the pattern does not update. (Shortly, we will see this graphically). If the smoothed
trend and seasonal series (i.e., the T and F series) are nonzero, the data have trend and seasonal
patterns, but they are deterministic only. 1. HOLT'S METHOD
(Newbold and Bos, pages 193199) Deﬁnitions L1 = the new smoothed estimate for the level at time period t;
a = the smoothing constant (parameter) for the level; LH = the old smoothed estimate of level found at time period t — 1; TH = the old smoothed estimate of the slope found at time period t — 1;
T, = the new smoothed estimate of the slope at time period t;
B = the smoothing constant (parameter) for the slope; Ll  LH = the difference between the new estimate and previous estimate for the level. This is an estimate of the slope based primarily on Y1 . Here, the
change in the level of the series is due to the change in the actual data. {{t(h) = the forecast made at time t for time t + 11. Level:
LI=aYl+(l—0t) [L4_1+T1_1] 0<0.<l = Lt—l + Ttl + 0[Yt ' {Lll + Tll = LH + TH + aet , from the definition of Yt(h) below. That is, update level by:
previous level + previous slope estimate + some fraction (a) of current error. Also note, for use below, that L[  LH  TH = (tel Trend:
Tt=BlLt141]+(1I3)TtI 0<B<1 The change in level [L  LH] is an estimate of recent slope [3. = Tll + ML: ' Ltl ' Ttl] = Tll +6016: Forecast:
Yt(1) = L + T [ onestep ahead
Yt(2) = L + 2T: two—steps ahead
{(411) = L + th hsteps ahead Notice that the 1 through hstepsahead forecasts lie on a straight line. 34 The Dynamics Of Holt’s Method The way that the algorithm adjusts forecasts over time is shown in the ﬁgure below. Using the
notation presented on the previous page, Yt_1(1) is deﬁned as the forecast made at time period tl for one period ahead; i.e., it is the forecast for Y1. In the ﬁgure, notice that Yt_1 (1) an underestimate of Yt. Therefore, the level L will be revised upward because of existing trend and underestimation of error. The trend Tl likewise is raised because it failed to increase enough to
predict Yt. ‘ X1   e Lt: Lt_1+ Tt_1+ otet 35 240 i The Dynamics Of ____________ 1' L10: 153001
Holt’s Method 230 2' T10: 4015
3. Y10(1) = 1530.01+40.15 = 1570.16
4. Y11= 2162
220 ___________ 13 e12: 24199 5. e11: 2162—1570.16 = 591.84
6. one”: (.05)(591.84) = 295.92
210 11' V11(1) 199501 7. L11: 1570.16+295.92 = 1866.08
9, T10: 40,15 8. Eden: (.03)(.05)(591..84) = 88.776
10' T“ 1289221 81361911: 88.7 9 10. T11: 40.15+88.776 = 128.926
190 5. e11: 591.84 I K 11. \7 0(1) = 1866.08+129.926 = 1995.01 12. Y12= 2337 7. L11: 1866.08 180 €12: 71
6. (1811: 295.92
170 N
1606 R 2. T10: 40.15 9 3. Y10(1) = 1570.16 1. L10: 1530.01 / 1O 11 12 To get an idea of the dynamics of Holt’s algonthm, we provide an example. We start with the
annual sales data X[ found in Table 6.2 on page 196 of Newbold and Bos. They use (1 = 0.5 and
I3 ' 0.3 as smoothing constants to generate their results. For time periods t=1, 2, ..., 30, they
repoﬁ calculations for Ll and for TL that coincide with each value of X We change the notation for the annual sales data from XI to Y1. We expand their table to include
columns that result from calculations for the one—stepahead forecasts 311(1) and the forecast errors e1. We provide a portion of our revised table below. Notice that several of the
calculations for time periods 10 through 12 have numbers (1 through 10 in bold) with matching
arrows that point to the calculations. This sequence represents the dynamics of how the updating
occurs in the algorithm. After the table, the details of each calculation are presented that match
the numbers reported in the table. Expansion ofNewbold and Bos' Table 6.2 to Include {(41) and et __ . _ . __ (GO5,13%” 0.32. _ . . _
t Y1 LI T1 YtU) et
1 1813
2 1650 1650.00 163.00
3 1822 1654.50 112.75 1487.00 335.00
4 1778 1659.88 —77.31 1541.75 236.25
9 1431 1323.96 30.95 1216.92 214.08
10 1767 1—>1530.01 2—> 40.15 1293.02 473.98
11 4—>2162 6—>1866.08 7—>128.93 3—>1570.16 5—>591.84
12 9—237 2166.00 180.23 841995.01 104341.99
13 2608 2477.11 219.49 2346.23 261.77
30 1297 1312.23 72.45 1327.46 30.46 To appreciate the dynamics, go to the line in the table that corresponds with t = 10. Notice that
L10 = 1530.01 and T10 = 40.15. These two items get things going for our example. The numbers
in bold below correspond to the numbers in bold in the table. 1 L10=1530.01 2 T10=40.15 3 A onestepahead forecast is made at the end of time period 10. This is 910(1).
910(1) = Llo +(1)T10 = 1530.01 + 40.15 = 1570.16 —> forecast for period 11 36 10 Actual value of Y in period 11: Y” = 2162 Total error between Y1. and 810(1) = 2162  1570.16 = 591.84 : en
1 ErrorCon'ection Updating Equation for Lt: L‘ = Ll_I +T(_I + onel some fraction of . ‘ + r. i ’1‘
Update level by p1 ev1ous level preuous slope estimate current error %,—1_.V—_J Ek—J
L11 L10 T10 0“’“1
= . 1 + _ + (0.50)(591.84)
LA + 295.92 71866.08 1570.16
ErrorCorrection Updating Equation for TI: T1 = Tl_l + OLBei U p d at e trend by previous trend + a proportion of the fraction of current en‘or ¥_W—/ *4
T11 T10 l3 0‘ 611
= 40.15 + 0.3 .05 591.84
= 40.15 + 88.776 = 128.926 A one—stepahead forecast is made at the end of time period 1 1. This is 'AYHU) .
811(1) 2 LH + (1)Tll = 1866.08 + 128.93 = 1995.01 —> forecast for period 12 Actual value on in period 12: Y12 = 2337 Total errorbetweenYlg and 1%,,(1) = 2337 l995.01=341.99:e12 Finally, and just for comparison, suppose that a forecast were made at the end of period 10 for two periods into the ﬁiture. Notation for this forecast is 810(2) , and it is calculated as 560(2) = LIO +2Tlo = 1530.01 + 2(40.15) = 1610.31 Because this forecast is twosteps ahead from the forecast at period 10, there is no update to the
level and no update to trend. Trend for the second step continues at the same slope as the ﬁrst step. 37 2. WINTERS' ADDITIVE SEASONAL METHOD
(Newbold and Bos, pages 199205) Additional deﬁnitions Y1  Ft . S = actual seasonally adjusted data for current time period t, quarter or month s;
Fl = the new smoothed estimate for the seasonal component at time period t; y = the smoothing constant (parameter) for the seasonal components; :<
I
F
II a measure of the actual seasonal variation in the data, obtained by subtracting
the new estimate of level from the actual data point; Fl  s = the old smoothed estimate of the seasonal component found at time period t  s
(i.e., same period one year earlier); Ft + h _ s = the smoothed estimate for the t + h season made at time period t+hs (i.e., a
year earlier).
Level:
L4=a[Yt Fl_s]+(1a)[L1_1+Tl_1] 0<a<1 Yt  F1 5 term takes out the seasonal component from Yt using the seasonal factor from last year = Ltl + Ttl + a[Yl ' {Ll1+ Ttl + Fts}]
= LH + TH + oret , from the deﬁnition of Sigh) below. Trend:
TI=B[LI'Lll]+(1'B)Ttl 0<l3<1
= Tl1+[3[Ll LH  Tl.1]= TH + Bore;
This is identical to the trend equation in Holt’s twoparameter algorithm shown earlier.
Seasonal: Ft = 7(Yl  Ll ) + (1 y)Fl_ S is the seasonal factor to be used in calculating level in the same
season next year, not used in forecasting. =FlS+Y[Yl ‘ (L1+Ft'5)] The term (L + Ft _ S) directly above is a “predicted” value for Y1 based on an updated level
component. Now, substitute in the error correction form of the level equation from above. Ft = Fts+Y[Yt' +Tll + (161 +Fts)] = Fts+Y[Yt' +Ttl +Fls) ' Get] 38 Look at the form of the ﬁrst equation under the Forecasting section below. Compare it to the
form of the term (LN + TM + Ft _ S) on the right of the previous equation. Recognize that this
term is the forecast of Y1 , giving Fl = Fts +Y(et ' Ole! ) = Fts +Y[(1' (1)31]
Forecasting Yt(1)= Lt +Tt + Ft+1—s
m): Lt +hTt +Ft+h_s,h= 1, 2, . . . , 8 Once you forecast beyond one year you must reuse the same seasonal factors. So, the formula
changes to Ft + h _ 25 , etc.‘ The Dynamics Of Winters’ Method Adjustment of level, trend, and seasonal is illustrated below. {{t_1(1) is an underestimate of Y1 . Therefore, the level L is raised because of existing trend and underestimation of the error. The
trend is raised because it failed to increase enough to predict Y1 . The seasonal is raised (for one
year from now) because even after allowing for insufﬁcient trend, the seasonal factor was still
not enough. (Y1  Ft 2 was less than actual.) (Note: This is an old and unrevised graph.) Also, the pattern corresponds to the start of Table 6.3 on page 201 of Newbold and Bos. The smoothing
parameters are a = 0.3, B: 0.4, and y = 0.5. Time Actual Level Trend Seasonal Forecast Error
1 897 332.50 2 476 88.50 3 376 —188.50 4 509 564.50 0.00 ~55.50 5 967 585.50 8.40 357.00 897.00 70.00 Please note. I have yet to develop a graph demonstrating the dynamics of Winters’ method that is similar
to the one that I developed for Holt’s method on BM page 35. (I have a cryptic version of Winters’
dynamics in the graph on BM page 39). Likewise, I have yet to show how Winters’ smoothing equations
on BM pages 3839 can be applied to Newbold and Bos’s data in their Table 6.3 to illustrate the dynamics
of the algorithm as I did with Holt’s method on BM pages 3637. You are not responsible for the
dynamics involved with Winters’ method. 40 3. WINTERS' MULTIPLICATIVE SEASONAL METHOD
(Newbold and B03, pages 205210) In general terms, Whenever you performed addition in the previous section, here you multiply.
When you subtracted, here you divide. Level: Additive Winters’, for comparison L, = (1[Yl /Fl.5]+ (la) [LH + TH] L, = (l[Yl  Ft_s]+ (la) [LH + TH]
= Lu + TH + ((1 /Fts)[Yt  (Lll + TlI)Fls] = Ltl + Ttl + alYt  (Lll + Ttl + FlS)]
= LH + TH + (ti/13196; = 144+ TH + (let For level, the errorcorrection form for multiplicative is similar to additive but not identical. Trend:
T1 = Blh  Lu] +(1'B)Tll T: = l3[Lt  Lu] + (113) Ttl
= TH + (Ba/Ft_s)et = TH + Bore,
Trend similarities hold between error correction forms. From above Lt  LH — TH = (at /Ft _ S)et
Seasonal:
Fl=y(Yl/L,)+(1y)Ft_S Ft=y(YtLl)+(1—y)FI_S = FF, + y[(Yt /Ll) — FM]
= Fbs + (y/Ll) [Y1  Ll Fl_s]
Now, as before, substitute in the error correction form of the level equation from above:
Ft = Fts + (Y/Lt) [Y1 ' {1—41 + Ttl + (a/Fts) et} Fts]
= Fm + (y/Lt) [Y1  L(,_l + TH) F,_s  ael] From the Forecasting section below, recognize that L(t_1 + TH) Fl _ s in the brackets on the right
hand side of the equation above is the forecast of Yt, giving Fl = F[S + ('Y/Lt)(e[ ' (161)
= Fts+(Y/Lt)[(1' 00 er] Ft: Fts+Y[(1' (1)61] Forecasting m = (Lt +Tt>Ft+1_s Yt(h)=(Lt+hTt)Ft+h_s ,h= 1, 2, . . . , s Once you forecast beyond a year you must reuse the same seasonal factors so the formula changes to Ft + h  25 , etc., as before.
41 ...
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