ResEc 797A Lecture 6 - Fall 2011

ResEc 797A Lecture 6 - Fall 2011 - Res Ec 797A: Forecasting...

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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 24-29 Simple Exponential Smoothing Material Distributed Today NB 183-193 Simple Exponential Smoothing (Single Exponential Smoothing) BM 30-31 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 one-step—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 one-step-ahead 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 out-of-sample period. Standing at the end of the within-sample data set, the one-step through h-steps ahead forecast is a horizontal straight line. Within sample period 1 thro g 8 steps- ahead fo ecast Out-of-sample 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 229-233. 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 h-steps-ahead within-sample prediction errors. Once the smoothing parameter has been fitted, then the sample used for fitting can also be used for calculating h-steps-ahead errors. The errors are e, (h) = Y, 4?, (h) +11 The measure of forecast error variance is n-h sz(h)= Z ef(h)/(n—h—m), forh=1,2,... t=m+| where n is the number of within-sample 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 h-steps-ahead 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 one-step-ahead forecasts both within-sample and post-sample. Calculate the RMSE for four-steps-ahead forecasts for the post-sample only. 2. Compare the single exponential smoothing method with the naive no-change method for one- step-ahead forecasts both within-sample and post-sample. 3. Repeat Number 2 above for four—step-ahead forecasts for the post-sample 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 ex-post 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 right-hand 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 within-sample calculations, notice that I use only the italicized values in Column F and that I omit the first error (affected by my start-up value). For post-sample 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 “What-If 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 one-step-ahead 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 3|nputcell 0.1807983 s 4 I 826.438 5 I 02 848.634 7 I 871-412 7 I 0.3 893-083 I m—l _I -l _l —l-ma -. mam—l WWW-l —l-za Ila—l 20 1987 85.37 86.06 0.481 972.426 Emma—l mmm—l ram — Errorsq —- _--_ =+cs-Ds —- =TABLE<H4> —- 0.2 =TABLE<.H4> m_- =+ca-De 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 EI-I 1990 El—-- --————-— =SUM F32F15 _-- ...
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This note was uploaded on 12/08/2011 for the course ECON 797A taught by Professor Bernardmorzuch during the Fall '11 term at UMass (Amherst).

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ResEc 797A Lecture 6 - Fall 2011 - Res Ec 797A: Forecasting...

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