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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.

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chapter 13

Chapter 13

Analyzing and Forecasting Time Series Data

chapter 13 chapter outcomes

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

Forecasting

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

forecasting4

Forecasting

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

forecasting5

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.

forecasting6

Forecasting

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

forecasting7

Forecasting

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

forecasting8

Forecasting

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

forecasting9

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
Components of Time Series Data
  • Trend Component
  • Seasonal Component
  • Cyclical Component
  • Random Component
time series forecasting

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 forecasting13

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 forecasting14

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 forecasting15

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 forecasting16

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

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 13 220

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 forecasting
Linear Trend Model- Forecasting -

Trend Projection:

Forecasting Period t = 11:

linear trend model forecasting24

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 forecasting25

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 forecasting26

Linear Trend Model- Forecasting -

MEAN ABSOLUTE DEVIATION

or:

trend based forecasting

Trend-Based Forecasting

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

trend based forecasting29

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 forecasting30

Trend-Based Forecasting

A moving average is the average of n consecutive values in a time series.

trend based forecasting31

Trend-Based Forecasting

RATIO-TO-MOVING-AVERAGE

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Trend-Based Forecasting

DESEASONALIZATION

trend based forecasting33

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
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
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

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 techniques37

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 techniques38

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
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)