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 Subject: quantitative methods/stats Category: Business and Money > Economics Asked by: k9queen-ga List Price: \$50.00 Posted: 09 Nov 2003 09:07 PST Expires: 09 Dec 2003 09:07 PST Question ID: 274108
 ```1) Let's invent data where there is no seasonality but there is a trend. Suupose these are order for a certain brand of computers. You have been asked to give a forecast for 2004 yr.2001 yr. 2002 yr. 2003 ------------------------------------------- 1900 2100 2300 1950 2150 2350 2000 2200 2400 2050 2250 2450 2)Now invent data where we have no trend but pronounced seasonality yr. 2001 yr. 2002 yr. 2003 ---------------------------------------------- 1900 1900 1900 2100 2100 2100 2000 2000 2000 3000 3000 3000 3)Now invent data where we have a trend and seasonality, but the trend is always a constant and the seasonality is always the same. yr. 2001 yr. 2002 yr. 2003 -------------------------------------------- 1800 2000 2200 2000 2200 2400 2100 2300 2500 3500 3700 3900 4)Now invent data where we have a trend and seasonality but there is some variation in both. yr. 2001 yr.2002 yr. 2003 --------------------------------------- 1900 2250 2325 1950 2060 2100 2000 2340 2560 2050 2325 2765```
 ```Hi again k9queen-ga, I have embedded my responses with the questions you have provided by adding the 2004 column (where each row in the tables represents quarters in the given year): 1) Let's invent data where there is no seasonality but there is a trend. Suupose these are order for a certain brand of computers. You have been asked to give a forecast for 2004 yr.2001 yr. 2002 yr. 2003 yr. 2004 (forecasted) ------------------------------------------------------------------- 1900 2100 2300 2500 1950 2150 2350 2550 2000 2200 2400 2600 2050 2250 2450 2650 Notes: The trend here is fairly clear..the orders are increasing by 50 for each subsequent period. So we can carry that pattern through for 2004. 2)Now invent data where we have no trend but pronounced seasonality yr. 2001 yr. 2002 yr. 2003 yr. 2004 (forecasted) --------------------------------------------------------------------- 1900 1900 1900 1900 2100 2100 2100 2100 2000 2000 2000 2000 3000 3000 3000 3000 Notes: This time we are only considering the same period in previous years to forecast. In all cases, the 2001/2/3 values for a season are the same, so we can carry them over to 2004. 3)Now invent data where we have a trend and seasonality, but the trend is always a constant and the seasonality is always the same. yr. 2001 yr. 2002 yr. 2003 yr. 2004 (forecasted) ---------------------------------------------------------------------- 1800 2000 2200 2400 2000 2200 2400 2600 2100 2300 2500 2700 3500 3700 3900 4100 Notes: This problem can be solved by considering the seasonality alone - notice that the values for a given quarter in 2001/2/3 are increasing by a constant 200. We can continue this constant trend to 2004. Another way of looking at this is the difference between quarters in the same period. For example, the difference between Q1 and Q2 is always the same, etc. 4)Now invent data where we have a trend and seasonality but there is some variation in both. yr. 2001 yr.2002 yr. 2003 yr. 2004 (forecasted) ---------------------------------------------------------------------- 1900 2250 2325 2426.36 1950 2060 2100 2319.30 2000 2340 2560 2542.78 2050 2325 2765 2648.53 Notes: This is the only question that doesn't have a clear pattern and contains elements of both trend and seasonality. This is a perfect application of Winter's model: "Winters' Model for Seasonality" http://www.cba.uh.edu/~ekao/D6360S02LEC11A.pdf (slides 26-28) We need to first define three smoothing constants that relate to the forecast, trend, and seasonality (you can play around with these to weight the result of one component of the forecast more heavily compared to the others). For this forecast, I will use all three (alpha, beta, and sigma) equal to 0.2 and the number of periods is 4. Same-period next year forecast = F(t+n) = (F(t) + T(t+ 1))*S(n) Where n is the period number (in your case this is the quarter) and t is the current period. Note that this only works after we have a full year of actual data (since the forecast needs to go back to the same period last year). Continuing this through years 2002 and 2003 we can forecast 2004 figures incorporating trend (which includes previous forecasts) and seasonality. Hopefully this has helped you understand the different methods of forecasting - keep in mind that in real life the results are not a perfect fits as in questions 1, 2, and 3 :) Cheers! answerguru-ga``` Request for Answer Clarification by k9queen-ga on 09 Nov 2003 10:50 PST ```I need to see how you arrived at these answers. For instance: #2 You do the center moving average for one quarter and you have done it for all the same quarters. The value is always 2250. Why?``` Clarification of Answer by answerguru-ga on 09 Nov 2003 11:43 PST ```Hi again, The reason as to how I arrived to answers #1-3 is based purely on the "hint" given in the questions: 1. "there is no seasonality but there is a trend" You already have all the actual values for the first 12 periods, but the forecast is still done to factor in trends throughout that period. Since we already have the actual values we can verify how close the forecast was and adjust our value accordingly for the next period if it is above or below. This is known as exponential smoothing. Since the trend is perfectly linear (each value is exactly 50 higher than the previous period), we just continue the linear trend. Formula: Forecast of next period = (Forecast current period) + (smoothing factor)*(Acutal value current period - Forecasted value current period) Where the smoothing factor is between 0 and 1 2. "no trend but pronounced seasonality" This means that seasons are the only factor that matters in this case - we forecast in a manner similar to #1 but only consider the previous period in the same season. Again, there is a linear relationship (no growth by season) so our forecast would be an exact reflection of the previous season. Formula: Forecast of next period = (Forecast same season last year) + (smoothing factor)*(Acutal same season last year - Forecasted value same season last year) 3. "we have a trend and seasonality, but the trend is always a constant and the seasonality is always the same" This time there is a linear trend for each season (across the rows of the table) - there is an increase of 200 between each given period and the same period in the previous year. The difference between the periods within a year are always the same as well. That means there are two ways of looking at the problem; either one will yield the correct forecast. Formula: Forecast of next period = (Forecast same season last year) + (constant factor) Where the constant factors are (incidentally all the same here): Q1: Y2002 - Y2001 = 200 Q2: Y2002 - Y2001 = 200 Q3: Y2002 - Y2001 = 200 Q4: Y2002 - Y2001 = 200 4. "doesn't have a clear pattern and contains elements of both trend and seasonality" I already gave a description of Winter's model, which is the appropriate way to approach this type of problem - I did the calculations in excel and the result was as follows: Period Actual Forecast Trend Seasonality Winters 1 1900 1900 1900 1 2 1950 1900 1900 1 3 2000 1960 1912 1 4 2050 2008 1921.6 1 5 2250 2058.4 1931.68 1.018616401 2055.600493 6 2060 2288.32 1977.664 0.980044749 2063.575951 7 2340 2014.336 1922.8672 1.032334625 2072.598902 8 2325 2405.1328 2001.02656 0.993336518 2178.847637 9 2325 2308.97344 1981.794688 1.016281319 2182.268245 10 2100 2328.205312 1985.641062 0.964432261 2053.795678 11 2560 2054.358938 1930.871788 1.075093866 2207.481945 12 2765 2661.128212 2052.225642 1.002475812 2375.007091 13 2661.128212 2785.774358 2077.154872 1.004076294 2426.35747 14 2785.774358 2636.198983 2047.239797 0.982893615 2319.303037 15 2636.198983 2815.689432 2083.137886 1.047325785 2542.775887 16 2815.689432 2600.300894 2040.060179 1.018547081 2648.528886 (the formatting for the table may get skewed...sorry) The formulas for these are taken from the slides referenced in the original answer - I haven't posted them here because the special characters don't transfer well. The reason I didn't include the formulas in the original question for #1-3 is because this question is using special (easy) cases to teach you that you don't always need a formula. It was more of a conceptual question that was trying to get you to "see" the patterns rather than just plugging them into a formula. I hope that the formulas above help you understand how I arrived at the answers. answerguru-ga```
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