Chapter 16: Times Series Forecasting and Index Numbers

Times Series Forecasting and Index NumbersChapter 16Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinTime Series Forecasting16.1 Time Series Components and Models16.2 Time Series Regression16.3 Multiplicative Decomposition16.4 Simple Exponential Smoothing16.5 Holt-Winter’s Models16.6 The Box Jenkins Methodology (Optional Advanced Section)16.7 Forecast Error Comparisons16.8 Index Numbers16-*16.1 Time Series Components and ModelsTrend Long-run growth or declineCycle Long-run up and down fluctuation around the trend levelSeasonal Regular periodic up and down movements that repeat within the calendar yearIrregular Erratic very short-run movements that follow no regular patternLO16-1: Identify the components of a times series.16-*Time Series Exhibiting Trend, Seasonal, and Cyclical ComponentsLO16-1Figure 16.116-*SeasonalitySome products have demand that varies a great deal by periodCoats, bathing suits, bicyclesThis periodic variation is called seasonalityConstant seasonality: the magnitude of the swing does not depend on the level of the time seriesIncreasing seasonality: the magnitude of the swing increases as the level of the time series increasesSeasonality alters the linear relationship between time and demandLO16-116-*16.2 Time Series RegressionWithin regression, seasonality can be modeled using dummy variablesConsider the model:yt = b0 + b1t + bQ2Q2 + bQ3Q3 + bQ4Q4 + etFor Quarter 1, Q2 = 0, Q3 = 0 and Q4 = 0For Quarter 2, Q2 = 1, Q3 = 0 and Q4 = 0For Quarter 3, Q2 = 0, Q3 = 1 and Q4 = 0For Quarter 4, Q2 = 0, Q3 = 0 and Q4 = 1The b coefficient will then give us the seasonal impact of that quarter relative to Quarter 1Negative means lower sales, positive lower salesLO16-2: Use time series regression to forecast time series having linear, quadratic, and certain types of seasonal patterns.16-*TransformationsSometimes, transforming the sales data makes it easier to forecastSquare rootQuartic rootsNatural logarithmsWhile these transformations can make the forecasting easier, they make it harder to understand the resulting modelLO16-3: Use data transformations to forecast time series having increasing seasonal variation.16-*16.3 Multiplicative DecompositionWe can use the multiplicative decomposition method to decompose a time series into its components:TrendSeasonalCyclicalIrregularLO 4: Use multiplicative decomposition and moving averages to forecast time series having increasing seasonal variation.16-*16.4 Simple Exponential SmoothingEarlier, we saw that when there is no trend, the least squares point estimate b0 of β0 is just the average y valueyt = β0 + tThat gave us a horizontal line that crosses the y axis at its average valueSince we estimate b0 using regression, each period is weighted the sameIf β0 is slowly changing over time, we want to weight more recent periods heavierExponential smoothing does just thisLO 16-5: Use simple exponential smoothing to forecast a time series.16-*16.5 Holt–Winters’ ModelsSimple exponential smoothing cannot handle trend or seasonalityHolt–Winters’ double exponential smoothing can handle trended data of the formyt = β0 + β1t + tAssumes β0 and β1 changing slowly over timeWe first find initial estimates of β0 and β1Then use updating equations to track changes over timeRequires smoothing constants called alpha and gammaLO16-6: Use double exponential smoothing to forecast a time series.16-*Multiplicative Winters’ MethodDouble exponential smoothing cannot handle seasonalityMultiplicative Winters’ method can handle trended data of the formyt = (β0 + β1t) · SNt + tAssumes β0, β1, and SNt changing slowly over timeWe first find initial estimates of β0 and β1 and seasonal factorsThen use updating equations to track over timeRequires smoothing constants alpha, gamma and deltaLO16-7: Use multiplicative Winters’ method to forecast a time series.16-*16.6 The Box–Jenkins Methodology (Optional Advanced Section)Uses a quite different approachBegins by determining if the time series is stationaryThe statistical properties of the time series are constant through timePlots can helpIf non-stationary, will transform seriesLO16-8: Use theBox–Jenkinsmethodology toforecast a timeseries.16-*16.7 Forecast Error ComparisonForecast errors: et = yt - ŷtError comparison criteriaMean absolute deviation (MAD)Mean squared deviation (MSD)LO16-9: Compare time series models by usingforecast errors.16-*16.8 Index NumbersIndex numbers allow us to compare changes in time series over timeWe begin by selecting a base periodEvery period is converted to an index by dividing its value by the base period and then multiplying the results by 100Simple (quantity) indexLO16-10: Use index numbers to compare economic data over time.16-*