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Time Series and Their Components

Time Series and Their Components. Chapter 5. Time Series and Their Components. 2. Time series are often recorded at fixed time intervals. ¾. ¾ For example, Y might represent sales, and the associated time series could be a sequence of annual sales figures.

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Time Series and Their Components

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  1. Time Series and Their Components Chapter 5

  2. Time Series and Their Components 2 Time series are often recorded at fixed time intervals. ¾ ¾ For example, Y might represent sales, and the associated time series could be a sequence of annual sales figures. ¾ Other examples of time series include quarterly earnings, monthly inventory levels, and weekly exchange rates. ¾ In general, time series do not behave like a random sample and require special methods for their analysis. ¾ Observations of a time series are typically related to one another (autocorrelated). ¾ This dependence produces patterns of variability that can be used to forecast future values and assist in the management of business operations. ¾ Consider these situations. ¾ It is important that managers understand the past and use historical data and sound judgment to make intelligent plans to meet the demands of the future.

  3. Time Series and Their Components 2 Time series are often recorded at fixed time intervals. ¾ ¾ For example, Y might represent sales, and the associated time series could be a sequence of annual sales figures. ¾ Other examples of time series include quarterly earnings, monthly inventory levels, and weekly exchange rates. ¾ In general, time series do not behave like a random sample and require special methods for their analysis. ¾ Observations of a time series are typically related to one another (autocorrelated). ¾ This dependence produces patterns of variability that can be used to forecast future values and assist in the management of business operations. ¾ Consider these situations. ¾ It is important that managers understand the past and use historical data and sound judgment to make intelligent plans to meet the demands of the future.

  4. Time Series and Their Components 3 ¾ Properly constructed time series forecasts help eliminate some of the uncertainty associated with the future and can assist management in determining alternative strategies. ¾ Forecasting is done by a set of procedures followed by judgments.

  5. Time Series and Their Components 4 Decomposition ¾ It is an approach to the analysis of time series data involves an attempt to identify the component factors that influence each of the values in a series. ¾ The components of time series are: Time-Series Trend Seasonal Cyclical Random Component Component Component Component

  6. Time Series and Their Components 5 1. Trend Component ¾ It represents the growth and the decline in a time series, denoted by T. ¾ Long-run increase or decrease over time (overall upward or downward movement) and they could linear or nonlinear ¾ Data taken over a long period of time

  7. Time Series and Their Components 6 2. Cyclical Component ¾ It represents a long-term wavelike fluctuations or cycles of more than one yearʹs duration in a time series, denoted by C. ¾ Practically it is difficult to identify and frequently regarded as part of trend.¾ Regularly occur but may vary in length ¾ Often measured peak to peak

  8. Time Series and Their Components 7 3. Seasonal Component ¾ It represents the seasonal variation in a time series which refers to a more or less stable pattern of change that appears short-term regular wave-like patternsand repeats itself season after season, denoted by S. ¾ Observed within 1 year. ¾ Often monthly or quarterly.

  9. Time Series and Their Components 8 4. Irregular Component ¾ It represents the unpredictable or random fluctuations in a time series, denoted by I. ¾ Unpredictable, random, “residual” fluctuations¾ Due to random variations of ¾ Nature ¾ Accidents or unusual events “Noise” in the time series ¾ ¾ To study the components of a time series, the analyst must consider how the components relate to the original series.

  10. Time Series and Their Components 9 Time Series Components Models Additive Components Model ¾ It is suggested to use when the variability are the same throughout the length of the series. Multiplicative Components Model ¾ It is suggested to use when the variability are increasing throughout the length of the series. ¾ Note that it is possible to convert the multiplicative model to the additive model using logarithms. i.e.

  11. Time Series and Their Components 10 Time series with constant variability

  12. Time Series and Their Components 11 Time series with increasing variability

  13. Time Series and Their Components 12 Estimation of Time Series Components Estimation of Trend Component ¾ Trends are long term movements in a time series that can be sometimes be described by a straight line or a smooth curve. Remark ¾ Fitting a trend curve helps us in providing some indication of the general direction of the observed series, and in getting a clear picture of the seasonality after removing the trend from the original series. The Linear Trend The Quadratic Trend

  14. Time Series and Their Components 13 The Exponential Trend Where T is the predicted value of the trend at time t , b0 , b1 and are called t the model parameters. We can forecast the trend using the above models as and so on.Note that the Error Sum of Squares (SSE) is measured by

  15. Time Series and Their Components 14 Example 5.1 Data on annual registrations of new passenger cars in the United States from 1960 to 1992 are shown in the following table and plotted in the laterfigure.

  16. Time Series and Their Components 15 We definitelyhave a trend here!

  17. Time Series and Their Components 16 The values from 1960 to 1992 are used to develop the trend equation. Registrations is the dependent variable, and the independent variable istime t coded as 1960 = 1, 1961 = 2, and so on. The fitted trend line has theequation The slope of the trend equation indicates that registrations are estimated toincrease an average of 68,700 each year. The figure shows a straight-linetrend fitted to the actual data. Italso shows forecasts of new carregistrations for the years 1993and 1994 (t = 34 and t = 35)obtained by extrapolating thetrend line.

  18. Time Series and Their Components 17 ¾ The estimated trend values for passenger car registrations from 1960 to1992 are shown in the table. For example, the trend equation estimatesregistrations in 1992 (t =33) to be or 10,255,000 registrations. ¾ Registrations of new passenger cars were actually 8,054,000 in 1992. ¾ For 1992, the trend equation overestimates registrations by approximately 2.2 million. ¾ This error and the remaining estimation errors were listed in the table. ¾ The estimation errors were used to compute the measures of fit, MAD,MSD, and MAPE also were shown in the figure.

  19. Time Series and Their Components 18 Forecasting a Trend ¾ Which trend model is appropriate? ¾ Linear, quadratic or exponential ¾ Linear models assume that a variable is increasing (or decreasing) by a constant amount each time period. ¾ A quadratic curve is needed to model the trend. ¾ Based on the accuracy measures, a quadratic trend appears to be a better representation of the general direction of the data.

  20. Time Series and Their Components 19 ¾ When a time series starts slowly and then appears to be increasing at an increasing rate such that the percentage difference from observation to observation is constant, an exponential trend can he fitted. ¾ The coefficient b1 is related to the growth rate. ¾ If the exponential trend is fit to annual data, the annual growth rate is estimated to be 100(b1 — 1)%. ¾ The figure next contains the number of mutual fund salespeoplefor several consecutive years. ¾ The increase in the number of salespeople is not constant. ¾ It appears as if increasingly larger numbers of people are being added in the later years.

  21. Time Series and Their Components 20 ¾ A linear trend fit to the salespeople data would indicate a constant average crease of about nine salespeople per year. ¾ This trend overestimates the actual increase in the earlier years and underestimates the increase in the last year. ¾ It does not model the apparent trend in the data as well as the exponential curve. ¾ It is clear that extrapolating an exponential trend with a 31 % growth rate will quickly result in some very big numbers. ¾ This is a potential problem with an exponential trend model. ¾ What happens when the economy cools off and stock prices begin to retreat? ¾ The demand for mutual fund salespeople will decrease and the number of salespeople could even decline. ¾ The trend forecast by the exponential curve will be much too high.

  22. Time Series and Their Components 21 ¾ Growth curves of the Gompertz and logistic types reflect a situation in which sales begin low, then increase as the product catches on, and finally ease off as saturation is reached. ¾ Judgment and common sense are very important in selecting the right approach. ¾ As we will discuss later, the line or curve that best fits a set of data points might not make sense when projected as the trend of the future.

  23. Time Series and Their Components 22 ¾ Suppose we are presently at time t = n (end of series) and we want to use a trend model to forecast the value of Y, p steps ahead. ¾ The time period at which we make the forecast, n in this case, is called the forecast origin. ¾ The value p is called the lead time. ¾ For the linear trend model, we can produce a forecast by evaluating ¾ Using the trend line fitted to the car registration data in Example 5.1 , a forecast of the trend for 1993 (t = 34) made in 1992 (t = n = 33) would be the p = 1 step ahead forecast ¾ Similarly, the p = 2 step ahead forecast (1994) is given by

  24. Time Series and Their Components 23 ¾ Using the quadratic trend curve for the car registration data, we can calculate forecasts of the trend for 1993 and 1994 by setting t = 33 + 1 = 34and t = 33 + 2 = 35. ¾ The forecasts are = 8.690 and = 8.470 (respectively) ¾ Recalling that car registrations are measured in millions, the two forecasts of trend produced from the quadratic curve are quite differentfrom the forecasts produced by the linear trend equation. ¾ Moreover, they are headed in the opposite direction. ¾ If we were to extrapolate the linear and quadratic trends for additional time periods, their differences would be magnified. ¾ This example illustrates why great care must be exercised in using fitted trend curves for the purpose of forecasting future trends. ¾ The differences can be substantial for large lead times (long-run forecasting).

  25. Time Series and Their Components 24 ¾ Trend curve models are based on the following assumptions: ¾ The correct trend curve has been selected. ¾ The curve that fits the past is indicative of the future. ¾ We must be able to argue that the correct trend has been selected the future will be like the past. ¾ There are objective criteria for selecting a trend curve. We will discuss two of these criteria, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), in later chapters. ¾ However, although these and other criteria help to determine an appropriate model, they do not replace good judgment.

  26. Time Series and Their Components 25 Estimation of Seasonal Component ¾ A seasonal pattern is one that repeats itself year after year. ¾ For annual data, seasonality is not an issue because there is no chance to model a within year pattern with data recorded once per year. ¾ Time series consisting of weekly, monthly, or quarterly observations often exhibit seasonality. ¾ The analysis of the seasonal component of a time series has direct short- term implications and is of greatest importance to mid- and lower-level management. ¾ Marketing plans have to take into consideration expected seasonal patterns in consumer purchases. ¾ Several methods for measuring seasonal variation have been developed.¾ The basic idea in all of these methods is to first estimate and remove the trend from the original series and then smooth out the irregular component.

  27. Time Series and Their Components 26 ¾ The seasonal values are collected and summarized to produce a number (generally an index number) for each observed interval of the year (week, month, quarter, and so on). ¾ The identification of the seasonal component in a time series differs from trend analysis in at least two ways: 1 . The trend is determined directly from the original data, but the seasonal component is determined indirectly after eliminating the other componentsfrom the data so that only the seasonality remains. 2. The trend is represented by one best-fitting curve, or equation, but a separate seasonal value has to be computed for each observed interval (week, month, quarter) of the year and is often in the form of an index number. ¾ Always we estimate the seasonality in form of index numbers, percentages that show changes over time, are called seasonal index. ¾ If an additive decomposition is used, estimates of the trend, seasonal, and irregular components are added together to produce the original series.

  28. Time Series and Their Components 27 ¾ If a multiplicative decomposition is used, the individual components must be multiplied together to reconstruct the original series, and in thisformulation, the seasonal component is represented by a collection ofindex numbers. ¾ These numbers show which periods within the year are relatively low and which periods are relatively high. ¾ The seasonal indices trace out the seasonal pattern. ¾ Index numbers are percentages that show changes over time.Remark ¾ In this chapter we study the multiplicative model and leave the additive model to chapter 8 if we have time. ¾ In multiplicative decomposition model, the ratio to moving average is a popular method for measuring seasonal variation.

  29. Time Series and Their Components 28 Finding Seasonal Indexes Ratio-to-moving average method: ¾ Begin by removing the seasonal and irregular components (St and It), leaving the trend and cyclical components (Tt and Ct) Example: Four-quarter moving average ¾ First average: Q1+Q2+Q3+Q4 Moving average = 1 4 ¾ Second average: Moving average¾ etc… Q2+Q3+Q4+Q5 4 = 2

  30. Time Series and Their Components 29 Quarter Sales Quarterly Sales 1 23 2 40 60 3 25 50 4 27 40 5 32 30 20 6 48 10 7 33 0 8 37 1 2 3 4 5 6 7 8 9 10 11 9 37 Quarter 10 50 11 40 etc… etc…

  31. Time Series and Their Components 30 Centered Seasonal Index 4-Quarter Average Moving Quarter Sales Period Average 1+2+3+4 1 23 2.5= 2.5 28.75 4 2 40 3.5 31.00 3 25 23+40+25+27 4.5 33.00 28.75= 4 4 27 5.5 35.00 5 32 6.5 37.50 6 48 7.5 38.75 7 33 etc… 8.5 39.25 8 37 9.5 41.00 9 37 Each moving average is for a consecutive block of 4quarters 10 50 11 40

  32. Time Series and Their Components 31 ¾ Average periods of 2.5 or 3.5 don’t match the original quarters, so we average two consecutive moving averages to get centered moving averages 4-Quarter Centered Average Moving Centered Moving Period Average Period Average 2.5 28.75 3 29.88 3.5 31.00 4 32.00 4.5 33.00 5 34.00 5.5 35.00 6 36.25 etc… 6.5 37.50 7 38.13 7.5 38.75 8 39.00 8.5 39.25 9 40.13 9.5 41.00 ¾ Now estimate the St x It value by dividing the actual sales value by the centered moving average for that quarter. QMIS 320, CH 5 by M. Zainal

  33. Time Series and Their Components 32 Yt ¾ Ratio-to-Moving Average formula: S×I = t t Tt×Ct Centered Ratio-to- Moving Moving Quarter Sales Average Average 1 23 Example 2 40 25 0.837 = 3 25 29.88 0.837 29.88 4 27 32.00 0.844 5 32 34.00 0.941 6 48 36.25 1.324 7 33 38.13 0.865 8 37 39.00 0.949 9 37 40.13 0.922 10 50 etc… etc… 11 40 … … … … … …

  34. Time Series and Their Components 33 Centered Ratio-to- Moving Moving Quarter Sales Average Average 1 23 2 40 Fall 3 25 29.88 0.837 Average all of the Fallvalues to get Fall’s seasonalindex 4 27 32.00 0.844 5 32 34.00 0.941 6 48 36.25 1.324 Fall 7 33 38.13 0.865 Do the same for the otherthree seasons to get theother seasonal indexes 8 37 39.00 0.949 9 37 40.13 0.922 10 50 etc… etc… Fall 11 40 … … … … … …

  35. Time Series and Their Components 34 ¾ Suppose we get these seasonal indices: Seasonal Season Index Interpretation: Spring sales average 82.5% of the annualaverage sales Spring 0.825 Summer sales are 31.0% higher than theannual average sales Summer 1.310 etc… Fall 0.920 Winter 0.945 Σ = 4.000 -- four seasons, so must sum to 4 QMIS 320, CH 5 by M. Zainal

  36. Time Series and Their Components 35 ¾ The data is deseasonalized by dividing the observed value by its seasonal index Yt Tt×Ct×It= St ¾ This smoothes the data by removing seasonal variation Quarter Sales Seasonal Index Deseasonalized Sales 23 27.88 27.88 = 1 23 0.825 0.825 2 40 1.310 30.53 3 25 0.920 27.17 4 27 0.945 28.57 5 32 0.825 38.79 6 48 1.310 36.64 7 33 0.920 35.87 8 37 0.945 39.15 9 37 0.825 44.85 10 50 1.310 38.17 11 40 0.920 43.48 … … …

  37. Time Series and Their Components 36

  38. Time Series and Their Components 37 Example 5.3 In Example 3.5 the analyst for the Outboard Marine Corporation, used autocorrelation analysis to determine that sales were seasonal on a quarterlybasis. Now, he uses decomposition to understand the quarterly salesvariable. Minitab was used to produce the following table and figure. Tokeep the seasonal pattern current, only the last seven years (1990 to 1996) ofsales data (Y), were analyzed. The trend is computed using the linear model:

  39. Time Series and Their Components 38 I =CI /C =1.187 /1.146=1.036 SCI YT= 255.026= =232.7 .912 Tˆ = 253.742+1.284(1) = 255.026 S=(.912+.788+...+.812) 7=0.780 1 TCI Y S= .7796= =232.7 298.486 CI =Y TS = 232.7 (255.026×.7796) C =(1.170+1.187+1.080) 3=1.146

  40. Time Series and Their Components 39 ¾ The cyclical indices can be used to answer the following questions: ¾ The series cycle? ¾ How extreme is the cycle? ¾ The series follow the general state of the economy (business cycle)? ¾ One way to investigate cyclical patterns is through the study of business indicators. ¾ A business indicator is a business-related time series that is used to help assess the general state of the economy. ¾ The most important list of statistical indicators originated during the sharp business setback of 1937 to 1938. ¾ Leading indicators. ¾ Coincident indicators.¾ Lagging indicators.

  41. Time Series and Their Components 40 Forecasting A Seasonal Time Series ¾ In forecasting a seasonal time series, the decomposition process is reversed. ¾ Instead of separating the series into individual components for examination, the components are recombined to develop the forecasts forfuture periods. Example 5.4 Forecasts of Outboard Marine Corporation sales for the four quarters of 1997can he developed using the previous table. 1. Trend. The quarterly trend equation is: T = 253.742 + 1.284t. The forecast origin is the fourth quarter of 1996, or time period t = n = 28.Sales for the first quarter of 1997 occurred in time period t = 28 + 1 = 29. 9 9 This notation shows we are forecasting p = 1 period ahead from the end of the time series.Setting t = 29, the trend projection is then T29 = 253.742 + 1.284(29) = 290.978 9 9

  42. Time Series and Their Components 41 2. Seasonal. The seasonal index for the first quarter is .7796. 3. Cyclical. The cyclical projection must be determined from the estimated cyclicalpattern (if any) and any other information generated by indicators of the generaleconomy for 1997. Projecting the cyclical pattern for future time periods is fraught with uncertainty, and aswe indicated earlier, is generally assumed for forecasting purposes to be included in thetrend. 9 To demonstrate the completion of this example, we set the cyclical index to 1.0. 9 4. Irregular. Irregular fluctuations represent random variation that can’t beexplained by the other components. For forecasting, the irregular component is set to the average value 1.0 9 ¾ The forecast for the first quarter of 1997 is ¾ The forecasts for the rest of 1997 are

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