CONDITIONS FOR THE OPTIMALITY OF EXPONENTIAL ...

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of Exponential Smoothing Forecast Procedures. Johannes Ledolter* and George E.P. Box**. PREFACE. The use at time t of available observations from a time.
CONDITIONS FOR THE OPTIMALITY OF EXPONENTIAL SMOOTHING FORECAST PROCEDURES

Johannes Ledolter

March 1976

Professional Papers are not official publications of the International Institute for Applied Systems Analysis, but are reproduced and distributed by the Institute as an aid t o staff members in furthering their professional activities. Views or opinions expressed herein are those of the author and should not be interpreted as representing the views of either the Institute or the National Member Organizations supporting the Institute.

Conditions for the Optimality of Exponential Smoothing Forecast Procedures Johannes Ledolter* and George E.P. Box**

PREFACE

The use at time t of available observations from a time series to forecast its value at some future time t+ provides an important basis for planning and control. The obtaining of good forecasts is an important part of model building at IIASA.

* currently at IIASA, Laxenburg, Austria ** University of Wisconsin-Madison, USA.

C o n d i t i o n s. --y of -

E x p o n e n t i a l Smoothing F o r e c a s t P r o c e d u r e s J o h a n n e s L e d o l t e r and G e o r g e E . P .

Box

A bstract E x p o n e n t i a l smoothing p r o c e d u r e s , i n p a r t i c u l a r t h o s e recommended by 3rown [ 3 ] a r e u s e d e x t e n s i v e l y i n many a r e a s I t i s shown i n o f e c o n o m i c s , b u s i n e s s and e n g i n e e r i n g . t h i s paper t h a t :

i ) S r o w n ' s f o r e c a s t i n g p r o c e d u r e s a r e o p t i m a l i n terms o f a c h i e v i n g minimum mean s q u a r e e r r o r f o r e c a s t s o n l y i f t h e underlying s t o c h a s t i c p r o c e s s is inc l u d e d i n a l i m i t e d s u b c l a s s o f ARIPJA ( p I d I q ) processes. Hence, it i s shown w h a t a s s u m p t i o n s a r e made when u s i n g t h e s e p r o c e d u r e s . i i ) The i m p l i c a t i o n o f p o i n t ( i ) i s t h a t t h e u s e r s o f B r o w n ' s p r o c e d u r e s t a c i t l y assume t h a t t h e s t o c h a s t i c p r o c e s s e s which o c c u r i n t h e r e a l w o r l d a r e f r o m t h e p a r t i c u l a r r e s t r i c t e d s u l ~ c l n s so f A R I N A ( p , d , q ) processes. No r e a s o n c a n b e f o u n d why t h e s e p a r t i c u l a r m o d e l s s h o u l d o c c u r more f r e q u e n t l y t h a n o t h e r s . i i i ) I t i s f u r t h e r shown t h a t e v e n i f a s t o c h a s t i c p r o c e s s w h i c h would l e a d t o B r o w n ' s model o c c u r r e d , t h e a c t u a l m e t h o d s u s e d f o r making t h e f o r e c a s t s a r e clunlsy and much s i m p l e r p r o c e d u r e s c a n b e employed.

1.

-The -

-.-------

c l a s s o f a u t o r e g r e s s i v e i n t e g r a t e d moving a v e r a g e p r o -

cesses a n d t h e i r minimum mean s a u a r e e r r o r f o r e c a s t s An a p p r o a c h t o t h e m o d e l l i n g and f o r e c a s t i n g o f s t a t i o n a r y a n d n o n s t a t i o n a r y p r o c e s s e s , s u c h a s commonly o c c u r i n b u s i n e s s , e c o n o m i c s and e n g i n e e r i n g , i s d i s c u s s e d by Box a n d J e n k i n s [ 2 ] . U t i l i z i n g e a r l i e r work by Kolmogorov [ 7 , 8 ] , Wold [ 1 2 ] , Yaglom [ 1 3 ] , Yule [ 1 4 ] , i t u s e s a t h r e e s t a g e i t e r a t i v e model b u i l d i n g di a g n o s t i c checking. p r o c e d u r e o f --.i d e n t i f i c a t i o n , e s t i m a t i o n a n d The c l a s s o f a u t o r e g r e s s i v e i n t e g r a t e d moving a v e r a g e (ARIVA) m o d e l s o f o r d e r ( p , d , q ) w h i c h i s d i s c u s s f X l i n [ 2 ] c a n be written

where i) ii)

zt is a discrete stochastic process Op(B) = ~-$I~B-...-@~B P

r

d Op(B) (1-B)

rlB-.. .-

p+d r-,+dB m and B is the backshift operator: B zt = zt-m p+d

iii)

(B)

=

= l-

{at) is a white noise sequence

The roots of J$ (B) = P outside the unit circle.

0

and 8 (B) = q

0

are assumed to lie

ARIMA (p,d,q) processes provide a class of models capable of representing time series which, although not necessarily stationary, are homogeneous and in statistical equilibrium.

The stochastic process in (1.1) can equivalently be written in terms of current and previous shocks at

where

or in terms of a weighted sum of previous values of the stochastic process and the current shock at.

where

Forecasts of ARIfilA ( p ,d,q) processes : Minimum mean square error forecasts for linear stochastic processes are given by the conditional expectation of future observations

Forecasts are calculated using the difference equation form of the model

where Izt+j

for j < O

=

for j > O for j cO =

for j > O

Forecasts can equivalently be expressed as a linear function of previous observations

In particular, for R = 1

Forecasts can be updated from one time origin to the other by

Although forecasts are calculated and updated most conveniently from the difference equation form (1.5), from the point of studying the nature of the forecasts it is profitable to consider the explicit form of the forecast function. The eventual forecast function is the solution of the difference equation Bt(e)

-

2 (2-p-d) c1 2 t (!~-i)-...-lg+~

=

bt (t)ft (el+. -+b;+d(t)fG+d (R) for R>q-p-d

= 0

for

R > q

and is given by 2,(e)

f~(e),...,f~+d(R) are functions of the lead time R and depend (B). In genonly on the autoregressive part of the model cP p+d eral, these functions can be polynomials, exponentials, sines, cosines or combinations of these functions. For a given forecast origin t, the coefficients

b* (t) =

[bt (t), . . . ,bG+d (t)] ' are constants and are the same

for all lead times R; however they change from one forecast origin to the next and as shown by Box and Jenkins [ 2 ] they can be updated by

where

and g = Fa*-1

-

?a

with V,

=

[@,I$,+

,....

1I'

for any R>q-p-d

2.

Exponential smoothins forecast procedures

Exponential smoothing techniques have received broad attention in the existing literature, especially in the area of management science. These procedures are fully automatic which means that once a computer program has been written, forecasts for any time series can be derived without manual intervention. The fact that they are automatic has been put forward as an advantage of the scheme. However it can equally well be argued that this is a great disadvantage since it discourages the use of the human mind in circumstances where this instrument could be used with profit. The basic exponential smoothing equation replaces an observed ser'ies zt by a smoothed series zt, an exponentially weighted average of current and past values of z .

The latest available smoothed value is used to forecast all future observations

This basic exponential smoothing procedure by Holt [6], Winters [11], Brown [3] was, and still is, used frequently to derive forecasts of economic and business data. Muth [9] investigated the conditions under which this procedure provides minimum mean square error forecasts. He showed that the underlying process has to be given by the ARIMA (0,1,1) process

Generalizations of exponential smoothing procedures have been considered by Brown [3], Brown and Meyer [41 . They select fitting functions f (R) = [f ( R ) , . . ,fm(R) 1 ' from the class of functions

-

.

L is a ( m x m ) non singular transition matrix and .f(0) is .,

specified. The coefficients b(t) .., the forecast function

=

[bl(t), . . . ,bm(t) ]' of

are fitted by discounted least squares minimizing

The fitting functions are chosen by visual inspection and the smoothing constant B(O 0

which coincides with the forecast function of model A given in (A.1). ad (ii). The updating algorithm for the coefficients of the for model A is given by forecast function bl(t)f(R) -

-

Dobbie [5] showed that for the case of exponential fitting functions h = (hlrh2,.-.,hn)' is given by

-

The updating algorithm for the coefficients of the eventual forecast function of the ARIFlA model in (A.2) is given by: b*(t) = L'b* - (t-1)

+

-

(1)]

Choosing R = 1 in (1.9), it is seen that

= -1

where

a

and

qk(1 < k < n) are the $-weights in

i

ui

l< i -< n

.

In order to prove theorem 1 we have to show that g = h .-.--

or equivalently that -1

,. .. ,dnl ':

di - ail$l + ai2$2 + ain$n and aij are the elements of the inverse of matrix A. di is the 1 1 coefficient of xo in -P. (-) $ (x), where Pi (x) is given in (A.3) .

where d = [dl .--

X

1 X

Using (A.3) and (A.9)

1 -P. X

1

(-) $ (x) = ~ X

1

xn(1-six)

k= 1

n

ak (ai-ak)

k#i It therefore remains to show that the coefficient of xo in n

rI (1 --XIB

vi(x)

f

k=l xn( 1

-

ak aix)

equals

aihi

II (ai - ak)

k+i

(A.10)

However,

where the c 's are the coefficients in the expansion of j

n. (X1 k= 1

B ak

-)

given by

(A.13) C3

=

-B

3

1

1 a a a k