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Unformatted text preview: Res Ec 797A: Forecasting Order Of Topics And Readings For Lecture 6 (September 21, 2011) Source Page} s) Topic Cover The Following Material Distributed Last Class Assignment 1 Simple Moving Averages And Seasonality
3 Due Monday, September 26, 2011
BM 2429 Simple Exponential Smoothing Material Distributed Today NB 183193 Simple Exponential Smoothing
(Single Exponential Smoothing) BM 3031 Prediction intervals when doing simple exponential smoothing
Assignment 4 Single (or Simple) Exponential Smoothing:
4 (Due Wednesday, September 28, 2011 or Tuesday, October 4, 201 1) Please note: We will meet on Tuesday, October 4 at 4:00 pm.
rather than on Monday, October 3 at 10:10 am. Notes to the table The forecast for year I, done in year 0, is unknown, so we assume that L0 = 103. This appears as the
forecast for year 1. It is the onestep—ahead forecast. Because of this assumption, there is no forecast error
for year 1. Consequently, there is no error to correct, and L1 = 103 also. Proceeding with calculating L2
using both the recurrence form and the error correction form, we have: Recurrence form: L; = 0.3(86) + (1  0.3) 103 = 97.9
Error correction form: L2 = 103 + 0.3 (17) = 97.9 . This is the forecast for year 3. W also could have referred to this as ‘AYJ . (Recall that L2 5% ). In the table on the previous page, notice that each onestepahead forecast lines up with the actual value for which it
is the forecast and not with the year in which the forecast is calculated. (Table 6.1 on page 187 of
Newbold and Bos is different from the table above for this reason.) The standard deviation for the prediction interval is the square root of SSE/(n—1). Since 21 errors are used
in the calculation of SSE, n  1 = 20. The 90% prediction interval (see next section) is:
the forecast :t 1.721 (standard deviation)
= forecast i 1.721 [square root of (2248.41/20)]
= forecast i 1.721 (10.6028)
= forecast i 18.24 . As expected from the search for optimum or, the naive no—change forecast, which corresponds to
a = 1, has a larger within—sample SSE and, hence, a larger RMSE. It also gives less accurate forecasts in the outofsample period. Standing at the end of the withinsample data set, the onestep through hsteps ahead forecast is a
horizontal straight line. Within sample period 1 thro g 8 steps
ahead fo ecast Outofsample period 15 20
Year __ Actual _ Forecast _ Upper p i _ Lower p i 30 Prediction intervals This topic is addressed in Newbold and Bos on pages 229233. As they point out, because single
exponential smoothing is an algorithm, the concept of a prediction interval is a bit elusive. However, the
algorithm does make forecasts and does so with error. Thus, the idea of a prediction interval is certainly
applicable. There are two basic approaches. 1. From the hstepsahead withinsample prediction errors. Once the smoothing parameter has been ﬁtted, then the sample used for ﬁtting can also be used for
calculating hstepsahead errors. The errors are e, (h) = Y, 4?, (h) +11 The measure of forecast error variance is nh
sz(h)= Z ef(h)/(n—h—m), forh=1,2,... t=m+ where n is the number of withinsample predictions, and m+1 is the ﬁrst observation used in the variance
calculation (to avoid distortions caused by choice of initial smoothed value). Assuming that the errors are
normally distributed, the usual critical t or 2 value can be used to compute prediction intervals. For
example, a 95% prediction interval about a point forecast at time t will be r, (h) i196 *s(h). When plotting such forecasts and intervals, it is important that they be located at the correct date. If h = 1,
the above forecast would be compared with the actual value at time t+1, and the actual value would lie
inside or outside the prediction interval plotted at time t+1. As h increases, the forecast is more uncertain and the prediction interval widens. 2. Using the exact variance derived from a corresponding ARIMA model Recall equation (3): Yt — Yt_1 = 8t — (1 — Ot)8t_1 . The variance for the hstepsahead forecast error is var(el (h)) = G:[l + (h —1)0L2] . Ifor = 1, this is the random walk model, and var(et (h)) = ho: . In both cases, the variance increases without limit as the forecasting horizon goes to inﬁnity. (This topic is also
addressed in Enders on page160). 31 University of Massachusetts
Department of Resource Economics RESEC 797A Fall 2011
Assignment 4 SINGLE (or SIMPLE) EXPONENTIAL SMOOTHING 1. For Variable 1 and Variable 2 use single exponential smoothing. See the instructions below
on how to ﬁnd the optimal smoothing parameter. Calculate the RMSE for onestepahead
forecasts both withinsample and postsample. Calculate the RMSE for fourstepsahead
forecasts for the postsample only. 2. Compare the single exponential smoothing method with the naive nochange method for one
stepahead forecasts both withinsample and postsample. 3. Repeat Number 2 above for four—stepahead forecasts for the postsample only. Excel Spreadsheet Instructions Look at the third and fourth pages. The third page is a program called ESstarter. This is
designed to show you how to ﬁnd the optimal smoothing parameter in a spreadsheet framework.
I provide instructions directly below. The fourth page bﬁngs ESstarter to a conclusion. The top
half of page 4 shows you the different SSEs corresponding to different smoothing parameter
values. The bottom half of page 4 provides formulas for all of the cells so that you can apply this
framework to your own vaﬁables. Let’s begin with ESstarter. But note that column references are provided with the top table on
page 4. My actual data are in Column C. I am reserving the last eight observations (in bold) for
expost analysis. I am averaging the ﬁrst three actual values to start the process and get my ﬁrst
forecast. I place my forecasts in Column D. I set up an error column (Column E) and an error
squared column (Column F). Notice that their formulas are shown in the bottom half of the
fourth page. I use the error con‘ection fonn to get all forecasts after the ﬁrst forecast which was
based on an average. Column D in the second table contains the error correction forecast
formula. Cell G4 contains a label. The label is “Input cell.” It acts as a visual marker for you to place a
value (i.e., an alpha value) in the adjacent cell (i.e., in Cell H4) which at this point is blank. To
create the alpha series (Column I) enter the ﬁrst two values. Rather than typing in the remaining
values, let’s see if we can get them automatically. First, highlight the two cells in which values
do appear. Second, stick the cursor in the lower righthand corner of the second cell. This is the
ﬁll handle’s position. (The ﬁll handle does what its name suggests. It ﬁlls cells with pattemed
values according to what was previously highlighted). Third, drag the ﬁll handle down as far as
you need (i.e., up to an alpha value of one). Fourth, take your ﬁnger off the mouse Bingoll Enter the formula for summing the squared errors [=SUM (F3:F15) in this particular example]. I
place this formula in cell F25. For withinsample calculations, notice that I use only the
italicized values in Column F and that I omit the ﬁrst error (affected by my startup value). For
postsample work, I use the obsewations in bold. Here comes the programming stuff done automatically by Excel. It’s pretty neat! Go to Column J. Click on the cell in Column J above where you entered 0.05 in Column I. In this
example, this is cell J2. Enter the cell address for the error sums of squares, prefaced with an “=”
sign. My example uses “ =F25”. So, enter :F25 in cell J 2. Now, highlight the rectangle containing the alpha values and the “=F25” formula cell. (This rectangle is about to become a very informative table). Go to the Tool Bar and click the Data tab. In the rectangular boxes that are presented
horizontally, search for the one that says “Data Tools” at the bottom. In the right part of this
rectangle, click on “WhatIf Analysis.” Then, click “Data Table.” Enter H4 to the right of
“Column input cellz”. Click OK. My results to this point are presented in the top half of the
fourth page. On this fourth page, examine the SSE column values and ﬁnd the smallest. Enter
the corresponding alpha value in cell H4. Your table of forecasts and errors should now compute. Date Time Actual Forecast
1969 l 92.04 86.06
1970 2 81.44 86.06
1971 3 84.71 86.06
1972 4 88.61 86.06
1973 5 96.92 86.06
1974 6 91.12 86.06
1975 7 78.75 86.06
1976 8 87.82 86.06
1977 9 95.39 86.06
1978 10 100.88 86.06
1979 11 100.09 86.06
1980 12 79.89 86.06
1981 13 79.59 86.06
1982 14 70.68 86.06
1983 15 75.19 86.06
1984 16 85.59 86.06
1985 17 87.98 86.06
1986 18 87.28 86.06
1987 19 85.37 86.06
1988 20 85.26 86.06
1989 21 84.97 86.06
1990 22 82.47 86.06 Program Name: ESstarter It stands for: How the Excel program
ought to look before I specify possible
values for the smoothing parameter which,
in turn, lead to matching error sums of
squares (SSEs) and onestepahead
forecasts. Error 5.977
4.623
—1.353
2.547
10.857
5.057
7.313
1.757
9.327
14.817
14.027
6.173
6.473
15.383
10.873
0.473
1.917
1.217
0.693
0.803
—1.093
3.593 Errorsq
35.721
21.375 1.832
6.486
11 7.867
25.570
53.485
3.086
86.987
219.534
196. 747
38.110
41.904
236.647
118.229
0.224
3.674
1.480
0.481
0.645
1.195
12.912 812.982 Input cell Alpha SSE 1 ime I SE
2 1 I 812.98
3 2 I 79941
4 3nputcell 0.1807983
s 4 I 826.438
5 I 02 848.634
7 I 871412
7 I 0.3 893083
I
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WWWl
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20 1987 85.37 86.06 0.481 972.426
Emma—l
mmm—l
ram — Errorsq —
__
=+csDs —
=TABLE<H4>
—
0.2 =TABLE<.H4>
m_
=+caDe 0.3 =TABLE<.H4>
_
0.4 =TABLE<H4>
—
0.5 =TABLE<H4>
=TABLE(,H4)
=TABLE<H4>
=TABLe<,H4>
=TABLE(,H4)
Elm—
—uu
ill—
im—Iw
0.95
EII
1990 El—
—————
=SUM F32F15 _ ...
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 Fall '11
 BernardMorzuch

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