Forecasting Techniques: Naïve Methods

Su, Chapter 10, sections I-II

Forecasting Exercises: Data

• Table 10.2 in Su

• Annual New Car Sales (in thousands) and a New Automobile Price Index (1982-1984=100) for 1971-1991

New Car Sales

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1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991

Forecasting Exercises

• First, read the file table10-2.dat into excel– This file contains three columns, containing

dates (Col. A), New Car Sales (Col. B) and the New Car Price Index (Col. C)

• Extend the date column through 1999

• Label Columns D-H: No Change, Same Change, Same Ratio, MA, Partial Adjustment

No Change Model

• Simplest Naïve Model

• Often used without even realizing it

• Requires only one period of historical data

• Anticipated level of the variable this period is the same as last periodX*t = Xt-1

X*t: Forecast value

Same Change Model

• No change model in first differencesX*t = Xt-1

X*t - Xt-1 = Xt-1 - Xt-2

• Requires only 2 periods of past data

Same Ratio Model

• Same change model in multiplicative form(X*t / Xt-1) = (Xt-1 / Xt-2)

X*t = Xt-1 (Xt-1/Xt-2 )

Three Naïve Forecasts

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1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999

Evaluating these Forecasts

• What are the underlying assumptions?

• How much historical data were used by each

• How accurate are they?

• Over how long a period should these forecasts be evaluated?

Defining and Measuring Accuracy

• Reading: Su, Chapter 16, section I-II

• The criteria that should be used to measure forecast accuracy are open to debate; we’ll look at the main competing methods

• Assessment of forecast accuracy is a very important component of forecast evaluation

Definitions: Forecast Errors

• Forecast Error in LevelsFEt in level = Ft - At

FEt: Forecasting Error in period t

Ft: Forecast in period t

At: Actual (or Realized) value in period t

• Forecast error measured in same units as variable

• FEt > 0 Overestimate FEt < 0 Underestimate

Summary Statistics

• Must avoid problems associated with signs of forecast errors - can’t simply add them up!

• Two ways to correct for this:– Absolute Value– Squaring

Three Summary Statistics

• Mean Absolute Error (MAE)MAE = |FEt| / n = |Ft - At| / n

• Mean Square error (MSE)MSE = (FEt)2 / n = (Ft - At)2 / n

• Root Mean Square Error (RMSE)RMSE = SQRT[ (FEt)2 / n = (Ft - At)2 / n]

Naïve Forecasts: In-Sample Measurement of Accuracy

• Use these definitions to evaluate the accuracy of these three naïve methods

• We’ll use “In-Sample” evaluation, as we have a lot of historical data but require very little to make these forecasts

• Step 1: Copy Table to a new sheet• Step 2: Calculate in-sample forecasts• Step 3: Calculate forecast error• Step 4: Calculate Summary Statistics

Summary Statistics

• No ChangeMAE = 812.3 MSE = 1077729.1 RMSE = 1038.1

• Same Change

MAE = 1561.4 MSE = 6080819.2 RMSE = 2465.9

• Same RatioMAE = 1561.9 MSE = 6015850.2 RMSE = 2452.7

Conclusions From Summary Statistics

• Which is the “best” at one-period ahead forecasts?

Moving Average Methods• Provides more efficient mechanical projections of

short-term movements• Has advantage of flexibility and presents a more

realistic picture of long-run movements• Data are not forced into any particular patterns

MA: X*t = (1/n)niXt-i

=(1/n)[Xt-1 +Xt-2 + Xt-3 + ...+ Xt-n]

• Note this is not a centered moving average• Must only decide on n• Can be applied to first differences or % changes

Moving Average Example

• Start with an MA(4) forecast– For ease of coding, copy the car sales values to

the MA column, then the out of sample MA forecast can be easily written and copied

• Compute the within sample, one period ahead MAE, MSE, RMSE

Naïve Forecasts and MA(4)

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1971 1974 1977 1980 1983 1986 1989 1992 1995 1998

Sales

NoChange

SameChange

SameRatio

MA(4)

Changing the Order of an MA Forecast

• Economists refer to MA forecasts by the number of periods they use, which is called the “order” of the moving average– MA(2): Two period moving average– MA(3): Three period moving average– etc.

• The forecast depends on the MA order

Effect of Changing Order

5000

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1991 1992 1993 1994 1995 1996 1997 1998 1999

MA(4)MA(5)MA(6)MA(7)MA(3)