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47 initial detrended A more detailed account of X11 and Hendersons weighted ~ ing average (step 9) can be found in Bo~ as Abraham and Johannes Ledolter. Statistical .Methods for Forecasting (Jolm \X ilc\ & Sons. New York. pp. 178191) Their book also includes a computer program for seasonal exponential smoothing FIGURE 73 Detrending com using a moving average with a yearly period. 2. Apply a weighted 5period MA to each month separately to get an estimate of the seasonal factors. 3. Compute a centered 12month MA of the seasonal factors (step 2) for the entire series. Fill in the six missing values at either end by repeating the first and last available MA values. Adjust the seasonal factors from stqa 2 by subtracting the centered 12 term MA. The adjusted seasonal factors will total approximately zero over any 12 month period. 4. Subtraa the seasonal feaor estimates (step 3) from the initial detrended seasonal series (step I). This is the irregular component series used for outlier adjustment. 5. Adjust the outliers in step 4 by the following procedure: a. Compute a 5year tnovirig standard deviation $ of the irregular component series (step 4). b. Assign weights tu the series components as follows. Oifc,>2.5s linearly scaled from 0 to 1 if 2.5s > c, > 1.5s 1 if c, < 1.5s Use this weighting function to adjust the detrended series in step 1. 6. Apply a weighted 7period MA to the acjusted series (step 5) to get the preliminary seasonal factor 7. Repeat step 3 to standardize the seasonal factors. 8. Subtract the senes resulting in step 7 from the onginal senes to find the preliminary seasonally adjusted series. 9. To get the frend estimate, apply a 9, 13, or 23period Hendersons weighted moving average to the seasonally acjusted series (step 8). Subtract this series from the original data to find a second estimate of the detrended series. can be f"Uiidin Bovas AtT.Jiam andjh.ames Ledolter, Satistical IJetb&t for
10. Apply a weighted 7period MA to eadi month separately to get a second estimate of the seasonal component. 11. Repeat step 3 to standardize the seasonal factors. 12. Subtract the final seasonal factors from the original series to get the final seasonally adjusted senes. Winters Method .A.nother technique for forecasting prices with a seasonal component is Winters method, a selfgenerating, heuristic approadi. it assumes that the only relevant characteristics of price movement are the frend and seasonal components, which are represented by the formula X,(a + bl)S. + e, wtiere X, is the esrimated value at time/ ( + bl) is a straight line that represents the trend S, is the seasonal weighting factor e, is the error in the estimate at each point If each season is represented by/V data points. S, repeats every/V enines, and The unique feature of Winters model is that eadi new observation is used to correct the previous components a, b, and S, Without that feature, it would have no applicability to commodity price forecasting. Starting with 2 or 3 years of price data, the yearly (seasonal) price average can be used to calculate both values a and b of the linear frend Each subsequent year can be used to correct the equation a + bt using any regression analjsis. Winters method actually uses a technique similar to exponential smoothing to estimate the next a and b components individually. The seasonal adjustment factors are assigned by calculating the average variance from the linear component, expressed as a ratio, al eadi point desired. As more observations are made, eadi component can be refined, and it will take on the form of the general longterm seasonal pattern. A Comparison of Seasonality Using Different Methods Heating oil and soybeans represent two fundamentally sound seasonal markets. Even before we analyze the pattems, we can expect heating oil to post higher prices during the winter months and soybeans to be high during the summer. Heating oil is accumulated beginning in midsummer in anticipation of the winter season, while soybeans are subject to speculative volatility during planting and growing seasons. For the pmposes of comparison, we will use four basic methods of calculating seasonality: average price, median price, percentage change from the previous month, and the moving average deseasonalizing. For our pmposes, more complicated methods are unnecessary Figure 74a shows that the peak price for heating oil is most likely to come in October, well below the coldest months of the year. According to these techniques, it is the anticipation of winter that drives prices up, while Februarj and March, tjpically colder, show lower average prices as most consumers use up the oil they had committed to buy during the previous September and October. Three of the methods for calculation gave similar results, while "oCbange seems to lead by one month. s " a.bancedm&lelE, ci FIGURE 74 (a) Heating oil seasonal patterns. Results are very uniform with "oChange leading the other methods by one month. Peak prices come in October, during the time of greatest accumulation, with lower prices in Februarj and March when consumers are normally depleting inventory, (b) Soybean seasonal pattems.The four basic calculations show soybean prices peaking in June and readiing lows in February .The use of the median gives results in July that are much lower than the average, and higher during the period November through January, indicating that there were unusually extreme moves during those months that distort the average
Heatmg oil seasonality 1980Mar 1994 All years from continuous data Soybean seasonality 19691993 All years from continuous data When applied to soybeans, all four basic seasonal calculations produce pattems that show the expected winter and summer cycles, with most lines rising and falling smoothly throughout the year (see Figure 74b). Again, it should be noted that the seasonal pattem of the percentage price changes leads the normal and detrended methods because it acts in the same way as momentum; a decline in the percentage change does not mean that current prices are lower than the previous month, but that prices did not increase by as much as the previous month. Seasonal Volatility Consistent seasonal pattems can be confirmed by a corresponding increase in volatility as the season readies its peak as seen in Figure 75. In this soybean chart, the peak volatility in June, represented by one standard deviation of price equal to about 12V shows that there is a 68° chance that sojlseans will vary by 24"/6 in June. For the purpose of confirming the validity of the seasonal results, the steadj increase and decrease in volatility surrounding the peak month of June indicates a building of the seasonal concems that cause wide fluctuations. If a single price shock had occurred, unrelated to seasonality, there would be a sha increase in volatility during one month, perhaps declining afterward, but with no steadj growth of volatility leading up to this event. WEATHER SENSITIVITY The effects of changes in weather, especially exfremes in weather, are an inseparable part of seasonal effects. Without weather, the price of an agricultural commodity would lack the surprises that cause them to jump around
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