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Time Series

Time Series. A time series is a sequence of measurements of a variable that is a function of time. Components of a time series. Trend Seasonal Cyclical Random or irregular. trend. the long term movement in a time series. seasonal variation.

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Time Series

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  1. Time Series

  2. A time series is a sequence of measurements of a variable that is a function of time.

  3. Components of a time series • Trend • Seasonal • Cyclical • Random or irregular

  4. trend the long term movement in a time series

  5. seasonal variation fluctuations that repeat themselves within a fixed period of a year.

  6. cyclical component pattern repeated over time periods of differing lengths, usually longer than one year example: business cycles

  7. The essential difference between the seasonal and cyclical components is that seasonal effects occur at regular, predictable intervals, whereas the timing of cyclical effects is less predictable.

  8. random variation the random movement up and down of a time series after adjustment for the trend, seasonal, and cyclical components examples: natural disasters, fads

  9. In a multiplicative time series model, a times series Y is decomposed as where T, C, S, and I are the trend, cyclical, seasonal, and random or irregular components respectively.

  10. Example: Suppose at a particular point in time, the value of the trend component is 340 dollars, the cyclical component is 120%, the seasonal component is 85%, and the irregular component is 110%. Then the observed value will be

  11. The linear trend is usually predicted using ordinary least squares (OLS) regression. Example: SALESi = a + b (YEARi) + ei Nonlinear trends are often estimated as well. Examples: SALESi = a + b1 (YEARi) + b2 (YEARi)2 + ei ln(SALESi) = a + b (YEARi) + ei

  12. Use OLS to estimate the trend line: SALES = a + b (YEAR) for the data below.

  13. Use OLS to estimate the trend line: SALES = a + b (YEAR) for the data below. So the estimated trend line is

  14. SALES(thousands of dollars) slope = 1.3indicates that sales tend to increase 1.3 thousand dollars or $1300 per year. 5.1 0 1 2 3 4 5 6 7 8 YEAR

  15. If the estimated trend continues, what would expected sales be in year 10?

  16. Seasonal variations may be studied because of • specific interest in their movements, or • attempts to eliminate them from time series measures, so that business cycle fluctuations can be more clearly seen.

  17. One way of measuring seasonal variation is called the ratio to moving average method.

  18. Calculating seasonal indices and seasonally adjusting a time series We specify the procedure for quarterly data, but the method can be modified for other periods of time. • Calculate a four-quarter moving average. • Average two moving averages to get a centered moving average. • Divide the series by the centered moving average to get S x I . • Average S x I to get the (unadjusted) seasonal index.(If there are enough years, drop the highest and lowest values before averaging). • Calculate the adjustment factor by dividing 400 by the sum of the unadjusted seasonal indices. • Multiply the adjustment factor by the unadjusted seasonal indices to get the adjusted seasonal indices. • Divide the series by the adjusted seasonal indices to get the seasonally adjusted series.

  19. Example: For the following time series Y, calculate the seasonal indices and seasonally adjust the series.

  20. Calculate a four-quarter moving average. Each number is entered in the table in the middle of the 4 numbers of which it is the average.

  21. 2. Average two moving averages to get a centered moving average.

  22. 3. Divide the series (Y = T x C x S x I ) by the centered moving average (T x C) to get S x I . [Divide column 3 by column 5.]

  23. Average S x I to get the (unadjusted) seasonal indices or the quarter averages. • To do this, we create a small table from columns 1, 2, and 6 of the big table.

  24. Average S x I to get the (unadjusted) seasonal indices (the quarter averages). • To do this, we create a small table from columns 1, 2, and 6 of the big table. • Average the numbers in each column. (If there are enough years, drop the highest and lowest values before averaging. We don’t have enough years to drop any values in this example.)

  25. Calculate the adjustment factor by dividing 400 by the sum of the quarter averages or unadjusted seasonal indices. • The sum of the quarter averages is 83.0369 + 95.0588 + 111.4286 + 113.2376 = 402.7619. • So the adjustment factor is 400/402.7619= 0.99314.

  26. 6. Multiplying each of the quarter averages by the adjustment factor, we get the adjusted quarter averages or adjusted seasonal indices.

  27. 7. To get the seasonally adjusted series, put the adjusted seasonal indices in the original table, ….

  28. 7. …. and divide the series (column 3) by the adjusted seasonal indices (column 7). But, …

  29. Be aware that the seasonal indices are often expressed in percentage terms, as we did here. So the first quarter index is 82.4675% or 0.824675 in decimal terms. When you compute the seasonally adjusted series in column 8, you need to divide by the adjusted seasonal indices in column 7 in decimal form. Otherwise, your final numbers will be 100 times too small. For example, the original sales numbers are between 3 and 18. So the numbers in your seasonally adjusted series should be on the same order of magnitude. However, if you divide your first number, 3, by the adjusted seasonal index for the first quarter, in percentage instead of decimal form, you get 3 ÷82.4675 = 0.0364, which is much too small. The correct seasonally adjusted number is 3 ÷ 0.824675 = 3.64.

  30. 7. So when we divide the series (column 3) by the adjusted seasonal indices (column 7) in decimal form, we get this:

  31. Forecasting through Exponential Smoothing This technique uses a weighted average of the actual value for a time period and the forecasted value for that period to make a forecast for the next period. Ft is the forecast for period t. At-1 is the actual value for period t-1. Ft-1 is the forecast for period t-1 w is a weight between 0 and 1. If w = 0, then we forecast the same value for each period. If w = 1, then we forecast for the next period whatever we observed in the previous period.

  32. Example: Each week we forecast the next week’s demand using our observation of the actual demand and our most recent forecast. In this particular case, demand is constant, but there is a peculiarity in period 5.

  33. In one case, we use a weight of 0.1 on our actual value and 0.9 on our most recent forecast. So, for example, for F6 we get

  34. In one case, we use a weight of 0.1 on our actual value and 0.9 on our most recent forecast. So, for example, for F6 we get

  35. In a second case, we use a weight of 0.8 on our actual value and 0.2 on our most recent forecast. So, for F6 we get

  36. In a second case, we use a weight of 0.8 on our actual value and 0.2 on our most recent forecast. So, for F6 we get

  37. In a situation like this, having a large weight on the most recent actual value enables us to get our future forecasts back in line more quickly. However, our forecast immediately following the peculiarity is way off.

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