Analyzing and Forecasting Time Series Data Chapter 13 Analyzing and Forecasting Time Series Data

Chapter 13 - Chapter Outcomes After studying the material in this chapter, you should be able to: Apply the basic steps in developing and implementing forecasting models. Identify the components present in a time series. Use smoothing-based forecasting models including, single and double exponential smoothing. Apply trend-based forecasting models, including linear trend, nonlinear trend, and seasonally adjusted trend.

Forecasting Model specification refers to the process of selecting the forecasting technique to be used in a particular situation.

Forecasting Model fitting refers to the process of determining how well a specified model fits its past data.

Forecasting Model diagnosis refers to the process of determining how well the model fits the past data and how well the model’s assumptions appear to be satisfied.

Forecasting The forecasting horizon refers to the number of future periods covered by the forecast, sometimes referred to as forecast lead time.

Forecasting The forecasting period refers to the unit of time for which the forecasts are to be made.

Forecasting The forecasting interval refers to the frequency with which the new forecasts are prepared.

Forecasting Time-Series data are data which are measured over time. In most applications the period between measurements is uniform.

Components of Time Series Data Trend Component Seasonal Component Cyclical Component Random Component

Time Series Forecasting A time-series plot is a two-dimensional plot of the time series. The vertical axis measures the variable of interest and the horizontal axis corresponds to the time periods.

Time-Series Plot (Figure 13-1)

Time Series Forecasting A linear trend is any long-term increase or decrease in a time series in which the rate of change is relatively constant.

Time Series Forecasting A seasonal component is a pattern that is repeated throughout a time series and has a recurrence period of at most one year.

Time Series Forecasting A cyclical component is a pattern within the time series that repeats itself throughout the time series and has a recurrence period of more than one year.

Time Series Forecasting The random component refers to changes in the time-series data that are unpredictable and cannot be associated with the trend, seasonal, or cyclical components.

Trend-Based Forecasting Techniques LINEAR TREND MODEL where: yi = Value of trend at time t 0 = Intercept of the trend line 1 = Slope of the trend line t = Time (t = 1, 2, . . . )

Linear Trend Model (Example 13-2) Taft Ice Cream Sales

Linear Trend Model (Example 13-2) Taft Sales

Linear Trend Model (Example 13-2) LEAST SQUARES EQUATIONS where: n = Number of periods in the time series t = Time period independent variable yt = Dependent variable at time t

Linear Trend Model (Example 13-2)

Linear Trend Model (Example 13-2)

Linear Trend Model - Forecasting - Trend Projection: Forecasting Period t = 11:

Linear Trend Model - Forecasting - MEAN SQUARE ERROR where: yt = Actual value at time t Ft = Predicted value at time t n = Number of time periods

Linear Trend Model - Forecasting - MEAN ABSOLUTE DEVIATION where: yt = Actual value at time t Ft = Predicted value at time t n = Number of time periods

Linear Trend Model - Forecasting - MEAN ABSOLUTE DEVIATION or:

Nonlinear Trend Models (Example)

Trend-Based Forecasting A seasonal index is a number used to quantify the effect of seasonality for a given time period.

Trend-Based Forecasting MUTIPLICATIVE TIME SERIES MODELS where: yt = Value of the time series at time t Tt = Trend value at time t St = Seasonal value at time t Ct = Cyclical value at time t It = Residual or random value at time t

Trend-Based Forecasting A moving average is the average of n consecutive values in a time series.

Trend-Based Forecasting RATIO-TO-MOVING-AVERAGE

Trend-Based Forecasting DESEASONALIZATION

Trend-Based Forecasting A seasonally unadjusted forecast is a forecast made for seasonal data that does not include an adjustment for the seasonal component in the time series.

Steps in the Seasonal Adjustment Process Compute each moving average from the k appropriate consecutive data values. Compute the centered moving averages. Isolate the seasonal component by computing the ratio-to-moving-average values. Compute the seasonal indexes by averaging the ratio-to-moving-averages for comparable periods.

Steps in the Seasonal Adjustment Process (continued) Normalize the seasonal indexes. Deseasonalize the time series. Use least-squares regression to develop the trend line using the deseasonalized data. Develop the unadjusted forecasts using trend projection. Seasonally adjust the forecasts by multiplying the unadjusted forecasts by the appropriate seasonal index.

Forecasting Using Smoothing Techniques Exponential smoothing is a time-series smoothing and forecasting technique that produces an exponentially weighted moving average in which each smoothing calculation or forecast is dependent upon all previously observed values.

Forecasting Using Smoothing Techniques EXPONENTIAL SMOOTHING MODEL or:: where: Ft+1= Forecast value for period t + 1 yt = Actual value for period t Ft = Forecast value for period t  = Alpha (smoothing constant)

Forecasting Using Smoothing Techniques DOUBLE EXPONENTIAL SMOOTHING MODEL where: yt = Actual value in time t  = Constant-process smoothing constant  = Trend-smoothing constant Ct = Smoothed constant-process value for period t Tt = Smoothed trend value for period t forecast value for period t Ft+1= Forecast value for period t + 1 t = Current time period

Key Terms Alpha () Beta () Cyclical Component Deseasonalizing Double Exponential Smoothing Exponential Smoothing Forecast Bias Forecast Error Forecasting Forecasting Horizon Forecasting Interval Forecasting Period Linear Trend Mean Absolute Deviation (MAD) Mean Squared Error (MSE)

Key Terms (continued) Model Diagnosis Model Fitting Model Specification Moving Average Nonlinear Trend Qualitative Forecasting Quantitative Forecasting Random Component Ratio-To-Moving-Average Method Residual Seasonal Component Seasonal Index Seasonally Unadjusted Forecast Smoothing Constant Splitting Samples

Key Terms (continued) Time-Series Data Time-Series Plot Trend