Time Series Forecasting

Outline:

1. Measuring forecast error

2. The multiplicative time series model

3. Naïve extrapolation

4. The mean forecast model

5. Moving average models

6. Weighted moving average models

7. Constructing a seasonal index using a centered moving average

8. Exponential smoothing

Forecast error

Month/Year

(1)Forecasted

Value

(2)Actual Value

(3) = (2) – (1)

Error

July 2000 $390 $423 $33

Aug 2000 450 429 -21

Sept 2000 289 301 12

Forecasting Convenience Store Ice Sales

3 measures of forecast error

• Mean absolute deviation

• Mean square error

• Root mean square error.

Actual

Predicted

Time

Average Absolute Error (AAE) is given by:

m

t

tt YYm

AAE1

ˆ1

Where Yt is the actual value of variable that we seek to forecast and is the fitted or forecasted value of the variable.

tY

Actual

Predicted

Time

Mean Square Error (MSE) is given by:

2

1

)ˆ(1

i

m

ti YY

mMSE

Where Yt is the actual value of variable that we seek to forecast and is the fitted or forecasted value of the variable.

tY

You can think of MSE as the average forecast error.

If we have a perfect forecast, then MSE = 0.

Actual

Predicted

Time

Root Mean Square Error (root MSE) is given by:

2

1

)ˆ(1

t

m

t

t YYm

rootMSE

Root MSE is a statistic that is typically is reported

by forecasting software applications

The time path of a variable (such as monthly sales of building materials by supply stores) is produced by the interaction of 4 factors or components. These components are:

1. The trend component (T)

2. The seasonal component (S)

3. The cyclical component (C); and

4. The irregular component (I)

The trend component (T)

Trend is the gradual, long-run (or secular) evolution

of the variables that we are seeking to forecast.

Factors affecting the trend component of a time series

•Population changes

•Demographic changes. For example, spending for healthcare services is likely to rise due to the aging of the population. Sales of fast food are up due to the secular increase in the female labor force participation rate.

•Technological change. Sales of music on DVD have slumped due to Ipods. Typewriter sales have plumetted.

•Changes in consumer tastes and preferences.

-60

-40

-20

0

20

40

10 20 30 40 50 60 70 80 90 100

Linear trends

Trend = 10 – 25t

Trend = -50 + .8t

0

1000

2000

3000

4000

10 20 30 40 50 60 70 80 90 100

Non-linear, increasing trend

Trend = 10 + .3t + .3t2

-5000

-4000

-3000

-2000

-1000

0

1000

10 20 30 40 50 60 70 80 90 100

Non-linear, decreasing trend

Trend = 10 - .4t - .4t2

The seasonal component (S)

•Many series display a regular pattern of variability depending on the time of year.

•For example, sales of toys and scotch whiskey peak in December each year.

•Ice cream sales are higher in summer months than in winter months.

•Car sales tend typically to be strong in May and June and weaker in November and December.

The cyclical component (C)

•The time path of a series can be influenced by business cycle fluctuations.

•For example, we expect housing starts to decline in the contractionary phase of the business cycle.

•The same holds true for federal or state tax receipts

•The time path of spending for consumer durable goods is also shaped by cyclical forces.

•Spending for capital goods is likewise cyclical.

•The movie industry has the reputation for being “counter-cyclical”—for example, it flourished during the Depression.

The irregular component (I)

•The irregular component of the series, sometimes called white noise, is the remaining variability (relative to trend) that cannot be explained by seasonal or cyclical factors. The irregular component is an unexpected, non-recurring factor that affects the series.

•For example, hamburger sales plunge due to panic about E-Coli bacteria.

•Production of trucks slumps because of a strike at a GM parts plant in Ohio.

•Airline slump after 9/11.

•A cold snap affects July ice cream sales in upstate NY.

If you have a well-designed forecasting model, then forecasting errors should be mainly accounted

for by irregular factors

The modelttttt ICSTY

Where:

•Yt is the value of the time series variable in period t (month t, quarter t, etc.)

•Tt trend component of the series in period t

•St is the seasonal component of the series in period t

•Ct is the cylical component of the series at period t; and

•It is the irregular component of the series in period t.

The trend component (T) is measured in the units in which the time series itself is

measured. So, for example, the trend component for state revenues would be measured in dollars; whereas the trend

component for steel production might be measured in tons.

The Problem: Forecast Sales of Home Furnishing Stores, October-December, 2007

The data:

•We have monthly data of sales of home furniture stores January 1992 to July 2007(187 monthly observations).

•The data are expressed in millions of current dollars, not seasonally adjusted

t Yr Mo $1 1992 1 14602 1992 2 14533 1992 3 15564 1992 4 16225 1992 5 16756 1992 6 17597 1992 7 17898 1992 8 18149 1992 9 172110 1992 10 183911 1992 11 192512 1992 12 2246" " " "" " " "

187 2007 7 4803

The Data

1000

2000

3000

4000

5000

6000

7000

92 94 96 98 00 02 04 06

Sales of Home Furnishing Stores, 1992-2007(millions of dollars, NSA)

Year/Month

Source: Economagic.com

•OLS is a method of finding the line, or curve, of “best fit.”

•The trend function of best fit is the one that minimizes the squared sum of the vertical distances of the sample points (the actual monthly values of home furnishing sales) from the trend line (fitted values of monthly building materials sales).

Our first step is to estimate thetrend component of our series.This is accomplished using a

ordinary least squares, or OLS for short.

Let:

•Yt be the actual value of furniture store sales in month t;

•Let Ŷt be the trend value of furniture store sales in month t. The trend function we are seeking satisfies the following condition:

2187

1

)ˆ(. tt

t

YYMIN

We estimate a lineartrend function with Excel.It is displayed on the next slide.

t

R2 = 0.83

1000

2000

3000

4000

5000

6000

7000

92 94 96 98 00 02 04 06

Actual TREND

Year/Month

Actual and Trend Values of Hom Furniture Sales (in millions)

Month IndexJan 0.8799Feb 0.8475Mar 0.9823Apr 0.9004May 0.9939Jun 1.0197Jul 0.9729Aug 1.0487Sep 1.0042Oct 0.9962Nov 1.123Dec 1.2969

•If you sum the monthly values and divide by 12, you get 1.00.•Later we show a simple technique for computing a seasonal index.

Seasonal Index

Performing an in-sample forecast of home furnishing sales

•An in-sample forecast means we are forecasting home furshing sales for those months for which we already have data that have been used to estimate the trend, seasonal, and other components. Comparing forecasted, or fitted values of home furnishing sales with actual time series data gives us an idea of how well this performs.

•We will assume that the cyclical index is equal to 1 (Ct = 1). This is a poor assumption since our period contains two business cycle contractions.

tttApr CSTF 98ˆ

71.532,2$1900.0]1475)7662.17[(ˆ98 AprF

Let’s give an example how we use this model to Home

furnishing sales for a particular month, say, April 1998 . t = 76 for this month

1000

2000

3000

4000

5000

6000

7000

94 96 98 00 02 04 06

Multiplcative model Home furnishing sales (millions)

In-Sample Forecast of Home Furnishing Sales Using Multiplicative Model

Year/Month

-300

-200

-100

0

100

200

300

94 96 98 00 02 04 06

Residuals from In-sample Forecast of Home Furnishing Sales (in millions)

Recession is shaded

Year/Month275.103$MSE

t Yr/Mo Trend Seasonal Cyclical Forecast

190 2007/Oct 4822.8 0.9962 0.999 4799.669

191 2007/Nov 4840.42 1.123 0.979 5321.64

192 2007/Dec 4858.04 1.2969 0.975 6142.882

Forecasting Using the Multiplicative Model