Re: Forecasting Seasonal Indices Question (Long)
You definitely should drop the idea that the seasonla factors are neutral (sum or product of them is equal to unity) when working with MOVING seasonality (thsi can be simply demonstrated). The only case where it is true is the regression approach (with dummy variables). I advise you to try the Census bureau website, querying for Census X12 regarima or the website of the bank of Spain www.bde.es (free powerfull soft tramo/seats). Also, the European commission site (== look for DEMETRA) offer a common platform for running X12 and Tramo/seats. [EMAIL PROTECTED] (after my holidays back on 20/8) Ronny Richardson [EMAIL PROTECTED] wrote in message [EMAIL PROTECTED]">news:[EMAIL PROTECTED]... The seasonal indices represent the amount by which the seasons vary from after. If there is no seasonality, then you would expect all of the indices to be 1.00 so the total (for quarterly data) should be 4.00. With seasonality, some are above 1.00 and others are below 1.00 but the total should still be 4.00. However, I am getting results that are significantly different. I want to lay out the problem here to see if anyone has any suggestions. (Everything below is has quarterly seasonality but the comments would apply to seasonality.) The approach I have been teaching for finding seasonal indices is the same one that is covered in Forecasting Principles and Applications by Stephen DeLurgio (McGraw-Hill). That approach is to: 1. Take period data and produce a 4-period moving average 2. Take that average and produce a 2-period moving average (This is required because (1+2+3+4)/4=2.5 so these averages are not whole periods. (2.5+3.5)/2=3 so this second moving average gives us whole periods. 3. Compute a percentage as Sales/Moving Average 4. Average all these percentage for period 1 to get the index for period 1. Do the same for the other periods. The first question that comes up for me is should the first moving average in step 1 be centered or forward-ended? DeLurgio shows it centered and that makes a little more sense to me since for the first four periods, (1+2+3+4)/4=2.5 which would be centered. However, I have seen other books where this was treated as a forward-ended average. Since every average will contain one sales figure from each period, you could justify writing down the average either centered or forward-ended. I've tried it both ways and the results are sometimes significantly different. The second question concerns the resulting seasonal indices. Since the indices represent deviations from average, you would expect the average indices to be 1.00 and so quarterly indices should total to about 4.00. Most of the examples I have seen in textbooks total to something near 4.00 and they either scale it to 4.00 as DeLurgio does or they ignore the small difference. However, the series I have been working with produces indices with a total significantly different from 4.00. That series follows: 147, 119, 153, 267, 225, 201, 1,011, 895, 865, 372, 305, 309, 178, 147, 144, 262, 222, 208, 1,297, 1,122, 1,091, 484, 412, 407, 191, 175, 149, 288, 243, 223, 1,468, 1,252, 1,246, 496, 459, 447 Using a centered moving average in step 1, the indices I obtain are: 0.87351 0.54884 1.14164 1.21995 === 3.78393 Using a forward-ended moving average in step 1, the indices I obtain are: 1.13681 0.74235 2.92445 1.44557 === 6.24918 It bothers me that these numbers are so different and that the total for the forward-ended moving average is so different from 4.00. I thought that the difference might be due to scaling so I scaled both series to force them to total to 4.00 and I got: 0.923389 0.58018 1.20683 1.289612 and 0.727654 0.475166 1.871894 0.925286 respectively. These are very different from one another. Several more approaches to seasonal indices are given in Production and Operations Management by Chase, Aquilano, Jacobs (McGraw-Hill.) The first one calls for simply dividing each period by the overall period average and then averaging these factors for each period. This approach forces the total to be 4.00 and the indices were: 1.008696 0.55492 1.23524 1.201144 Another approach is to fit a regression line to the data, find a ratio of actual to trend and then average the indices for each period. That approach yields the indices: 1.01504 0.559943 1.232433 1.184109 3.991525 Scaling everything to total to 4.00 and comparing the results, we have: Forward Average Centerd MA Ended MA DifferenceRegression -- ---- 0.9234 0.7277 1.00871.0172 0.5802 0.4752 0.55490.5611 1.2068 1.8719 1.23521.2350 1.2896 0.9253 1.20111.1866 Now, I understand why the results might be slightly different but it seems to me that they should be closer than they are. Any
Re: Forecasting Seasonal Indices Question (Long)
A time series plot of the data shows three things going on: 1. there is seasonality of period 12 2. there is trend 3. there is increasing spread. When you use moving averages, the number of periods averaged should match your seasonality, which means you would have to use a 12 period MA. Suggestion is to use a stat package, (I'm most familiar with Minitab, but most others should have comparable capabilities) and try a decomposition approach. The advantage is that you get indices automatically in a decomposition. Most simple models such as Moving Average and Exponential Smoothing work best for a stationary series. This series is not one of them. Ronny Richardson wrote: The seasonal indices represent the amount by which the seasons vary from after. If there is no seasonality, then you would expect all of the indices to be 1.00 so the total (for quarterly data) should be 4.00. With seasonality, some are above 1.00 and others are below 1.00 but the total should still be 4.00. However, I am getting results that are significantly different. I want to lay out the problem here to see if anyone has any suggestions. (Everything below is has quarterly seasonality but the comments would apply to seasonality.) The approach I have been teaching for finding seasonal indices is the same one that is covered in Forecasting Principles and Applications by Stephen DeLurgio (McGraw-Hill). That approach is to: 1. Take period data and produce a 4-period moving average 2. Take that average and produce a 2-period moving average (This is required because (1+2+3+4)/4=2.5 so these averages are not whole periods. (2.5+3.5)/2=3 so this second moving average gives us whole periods. 3. Compute a percentage as Sales/Moving Average 4. Average all these percentage for period 1 to get the index for period 1. Do the same for the other periods. The first question that comes up for me is should the first moving average in step 1 be centered or forward-ended? DeLurgio shows it centered and that makes a little more sense to me since for the first four periods, (1+2+3+4)/4=2.5 which would be centered. However, I have seen other books where this was treated as a forward-ended average. Since every average will contain one sales figure from each period, you could justify writing down the average either centered or forward-ended. I've tried it both ways and the results are sometimes significantly different. The second question concerns the resulting seasonal indices. Since the indices represent deviations from average, you would expect the average indices to be 1.00 and so quarterly indices should total to about 4.00. Most of the examples I have seen in textbooks total to something near 4.00 and they either scale it to 4.00 as DeLurgio does or they ignore the small difference. However, the series I have been working with produces indices with a total significantly different from 4.00. That series follows: 147, 119, 153, 267, 225, 201, 1,011, 895, 865, 372, 305, 309, 178, 147, 144, 262, 222, 208, 1,297, 1,122, 1,091, 484, 412, 407, 191, 175, 149, 288, 243, 223, 1,468, 1,252, 1,246, 496, 459, 447 Using a centered moving average in step 1, the indices I obtain are: 0.87351 0.54884 1.14164 1.21995 === 3.78393 Using a forward-ended moving average in step 1, the indices I obtain are: 1.13681 0.74235 2.92445 1.44557 === 6.24918 It bothers me that these numbers are so different and that the total for the forward-ended moving average is so different from 4.00. I thought that the difference might be due to scaling so I scaled both series to force them to total to 4.00 and I got: 0.923389 0.58018 1.20683 1.289612 and 0.727654 0.475166 1.871894 0.925286 respectively. These are very different from one another. Several more approaches to seasonal indices are given in Production and Operations Management by Chase, Aquilano, Jacobs (McGraw-Hill.) The first one calls for simply dividing each period by the overall period average and then averaging these factors for each period. This approach forces the total to be 4.00 and the indices were: 1.008696 0.55492 1.23524 1.201144 Another approach is to fit a regression line to the data, find a ratio of actual to trend and then average the indices for each period. That approach yields the indices: 1.01504 0.559943 1.232433 1.184109 3.991525 Scaling everything to total to 4.00 and comparing the results, we have: Forward Average Centerd MA Ended MA DifferenceRegression -- ---- 0.9234 0.7277 1.00871.0172 0.5802 0.4752 0.55490.5611 1.2068 1.8719 1.23521.2350 1.2896 0.9253 1.20111.1866 Now, I understand why the results might be slightly different but it seems to me that they should be closer than they are. Any
Re: Forecasting Seasonal Indices Question (Long)
Hi Ronny, First, both the centred and non-centred moving averages are used in smoothing time series. The latter is used in short term forecasting - you are assuming a model where the mean is locally constant, so the forecasting process consists of estimating the current value of that mean and projecting it ahead; so it is appropriate to use the last M observations to estimate it. For long term forecasting the aim is to smooth both the random fluctuations and the seasonal fluctuations, leaving only trend and cyclic variation. This can be done by using a moving average whose length equals the length of the season, so for example a 4 quarter moving average is used for quarterly data. Here we are clearly obtaining an average over a full year, so it clearly should be located at the middle of the year - hence the centred moving average is used. Second, if you graph your time series you will see why you are in this case getting peculiar results - the series shows cyclic behaviour but not seasonal - the peaks are at times 8, 19, 31. The absolutely crucial characteristic of 'seasonal' variation is not that it relates to spring, summer, etc, but that the length of the cycle is fixed and known - so public transport shows a daily cycle (for hourly data) or a weekly cycle (for daily data) - no relationship to the seasons, but this is a seasonal variation! (A seasonal variation is usually also relatively short cycle.) Regards, Alan Ronny Richardson wrote: The seasonal indices represent the amount by which the seasons vary from after. If there is no seasonality, then you would expect all of the indices to be 1.00 so the total (for quarterly data) should be 4.00. With seasonality, some are above 1.00 and others are below 1.00 but the total should still be 4.00. However, I am getting results that are significantly different. I want to lay out the problem here to see if anyone has any suggestions. (Everything below is has quarterly seasonality but the comments would apply to seasonality.) The approach I have been teaching for finding seasonal indices is the same one that is covered in Forecasting Principles and Applications by Stephen DeLurgio (McGraw-Hill). That approach is to: 1. Take period data and produce a 4-period moving average 2. Take that average and produce a 2-period moving average (This is required because (1+2+3+4)/4=2.5 so these averages are not whole periods. (2.5+3.5)/2=3 so this second moving average gives us whole periods. 3. Compute a percentage as Sales/Moving Average 4. Average all these percentage for period 1 to get the index for period 1. Do the same for the other periods. The first question that comes up for me is should the first moving average in step 1 be centered or forward-ended? DeLurgio shows it centered and that makes a little more sense to me since for the first four periods, (1+2+3+4)/4=2.5 which would be centered. However, I have seen other books where this was treated as a forward-ended average. Since every average will contain one sales figure from each period, you could justify writing down the average either centered or forward-ended. I've tried it both ways and the results are sometimes significantly different. The second question concerns the resulting seasonal indices. Since the indices represent deviations from average, you would expect the average indices to be 1.00 and so quarterly indices should total to about 4.00. Most of the examples I have seen in textbooks total to something near 4.00 and they either scale it to 4.00 as DeLurgio does or they ignore the small difference. However, the series I have been working with produces indices with a total significantly different from 4.00. That series follows: 147, 119, 153, 267, 225, 201, 1,011, 895, 865, 372, 305, 309, 178, 147, 144, 262, 222, 208, 1,297, 1,122, 1,091, 484, 412, 407, 191, 175, 149, 288, 243, 223, 1,468, 1,252, 1,246, 496, 459, 447 Using a centered moving average in step 1, the indices I obtain are: 0.87351 0.54884 1.14164 1.21995 === 3.78393 Using a forward-ended moving average in step 1, the indices I obtain are: 1.13681 0.74235 2.92445 1.44557 === 6.24918 It bothers me that these numbers are so different and that the total for the forward-ended moving average is so different from 4.00. I thought that the difference might be due to scaling so I scaled both series to force them to total to 4.00 and I got: 0.923389 0.58018 1.20683 1.289612 and 0.727654 0.475166 1.871894 0.925286 respectively. These are very different from one another. Several more approaches to seasonal indices are given in Production and Operations Management by Chase, Aquilano, Jacobs (McGraw-Hill.) The first one calls for simply dividing each period by the overall period average and then averaging these factors for each period. This approach forces the total to be 4.00 and the indices were: 1.008696
Forecasting Seasonal Indices Question (Long)
The seasonal indices represent the amount by which the seasons vary from after. If there is no seasonality, then you would expect all of the indices to be 1.00 so the total (for quarterly data) should be 4.00. With seasonality, some are above 1.00 and others are below 1.00 but the total should still be 4.00. However, I am getting results that are significantly different. I want to lay out the problem here to see if anyone has any suggestions. (Everything below is has quarterly seasonality but the comments would apply to seasonality.) The approach I have been teaching for finding seasonal indices is the same one that is covered in Forecasting Principles and Applications by Stephen DeLurgio (McGraw-Hill). That approach is to: 1. Take period data and produce a 4-period moving average 2. Take that average and produce a 2-period moving average (This is required because (1+2+3+4)/4=2.5 so these averages are not whole periods. (2.5+3.5)/2=3 so this second moving average gives us whole periods. 3. Compute a percentage as Sales/Moving Average 4. Average all these percentage for period 1 to get the index for period 1. Do the same for the other periods. The first question that comes up for me is should the first moving average in step 1 be centered or forward-ended? DeLurgio shows it centered and that makes a little more sense to me since for the first four periods, (1+2+3+4)/4=2.5 which would be centered. However, I have seen other books where this was treated as a forward-ended average. Since every average will contain one sales figure from each period, you could justify writing down the average either centered or forward-ended. I've tried it both ways and the results are sometimes significantly different. The second question concerns the resulting seasonal indices. Since the indices represent deviations from average, you would expect the average indices to be 1.00 and so quarterly indices should total to about 4.00. Most of the examples I have seen in textbooks total to something near 4.00 and they either scale it to 4.00 as DeLurgio does or they ignore the small difference. However, the series I have been working with produces indices with a total significantly different from 4.00. That series follows: 147, 119, 153, 267, 225, 201, 1,011, 895, 865, 372, 305, 309, 178, 147, 144, 262, 222, 208, 1,297, 1,122, 1,091, 484, 412, 407, 191, 175, 149, 288, 243, 223, 1,468, 1,252, 1,246, 496, 459, 447 Using a centered moving average in step 1, the indices I obtain are: 0.87351 0.54884 1.14164 1.21995 === 3.78393 Using a forward-ended moving average in step 1, the indices I obtain are: 1.13681 0.74235 2.92445 1.44557 === 6.24918 It bothers me that these numbers are so different and that the total for the forward-ended moving average is so different from 4.00. I thought that the difference might be due to scaling so I scaled both series to force them to total to 4.00 and I got: 0.923389 0.58018 1.20683 1.289612 and 0.727654 0.475166 1.871894 0.925286 respectively. These are very different from one another. Several more approaches to seasonal indices are given in Production and Operations Management by Chase, Aquilano, Jacobs (McGraw-Hill.) The first one calls for simply dividing each period by the overall period average and then averaging these factors for each period. This approach forces the total to be 4.00 and the indices were: 1.008696 0.55492 1.23524 1.201144 Another approach is to fit a regression line to the data, find a ratio of actual to trend and then average the indices for each period. That approach yields the indices: 1.01504 0.559943 1.232433 1.184109 3.991525 Scaling everything to total to 4.00 and comparing the results, we have: Forward Average Centerd MA Ended MA DifferenceRegression -- ---- 0.9234 0.7277 1.00871.0172 0.5802 0.4752 0.55490.5611 1.2068 1.8719 1.23521.2350 1.2896 0.9253 1.20111.1866 Now, I understand why the results might be slightly different but it seems to me that they should be closer than they are. Any comments? Dr. Ronny Richardson Associate Professor of Management Southern Polytechnic State University School of Management 1100 South Marietta Parkway Marietta, GA 30060-2896 Phone: (770) 528-5542 Fax:(770) 528-4967 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
