ISEN 315Spring 2011

Dr. Gary Gaukler

Forecasting for Stationary Series

A stationary time series has the form:Dt = m + e t where m is a constant and e

t is a random variable with mean 0 and var s2 .

Two common methods for forecasting stationary series are moving averages and exponential smoothing.

Moving Averages

In words: the arithmetic average of the n most recent observations. For a one-step-ahead forecast:

Ft = (1/n) (Dt - 1 + Dt - 2 + . . . + Dt - n )

(Go to Example.)

Exponential Smoothing Method

A type of weighted moving average that applies declining weights to past data.

1. New Forecast = a (most recent observation)+ (1 - a) (last forecast)

or2. New Forecast = last forecast -

a (last forecast error)

where 0 < a < 1 and generally is small for stability of forecasts ( around .1 to .2)

Comparison of ES and MA

• Similarities– Both methods are appropriate for stationary series– Both methods depend on a single parameter– Both methods lag behind a trend

• Differences– –

Two-equation Smoothing ModelAdd linear trend:

Assume Dt = m + t G + et

St = a Dt + (1-a ) [St-1 + 1 Gt-1],

where Gt -1 = 1-period trend estimate

Two-equation Smoothing Model:

Update G by exponential smoothing:

Gt = b (St - St-1) + (1 - b) Gt-1

Then forecast is:

Ft, t+t = St + t Gt

Example

Demand: 200, 250, 175

Estimates: S0=200, G0=10

Parameters: a= b=0.1

Estimate demand in weeks 4 - 6

Using Regression for Forecasting(Linear) regression methods can be used when trend is present

– Model: Dt = a + bt, or y = a + bx

How do we find the a and b?

Deriving the Regression Parameters

Deriving the Regression Parameters

Deriving the Regression Parameters

Deriving the Regression Parameters

Deriving the Regression Parameters

Using Regression for Forecasting

Least squares estimates for a and b are computed as follows:

1) Set Sxx = n2 (n+1)(2n+1)/6 - [n(n+1)/2]2

2) Set Sxy = n Σ (i Di)- [n(n + 1)/2] Σ Di

3) Let b = Sxy / Sxx and a = Davg - b (n+1)/2

Example

Assume demand for periods 1 through 5 is as follows:

200, 250, 175, 186, 235

What is the regression forecast for period 7?

The Difficulty with Long-Term Forecasts