UTILIZATION OF THE ALGORITHMS

In document The estimation of parametric change in time-series models (Page 45-48)

This was due to lim ita tio n s on the analog equipment used in the hybrid (analog-digital) mechanisation of the corresponding

CHAPTER 5 UTILIZATION OF THE ALGORITHMS

5 . 1 I n t r o d u c t i o n

Thus f a r we have c o n s i d e r e d means by which one might model p a r a m e t r i c v a r i a t i o n in t h e models I and I I , and e s t i m a t e p a r ame te r s i n t h e s e models. There remain, however, some d i f f i c u l t i e s t o be overcome in t h e p r a c t i c a l impl e me n ta t io n o f t h e a l g o r i t h m s we have o b t a i n e d . In S e c t i o n 5 . 2 , t h e p r o c e s s which may l e a d t o the

a d o p t i o n o f a t i m e - v a r y i n g p a r a m e t e r model i s d i s c u s s e d . This can be t h o u g h t of as ' i d e n t i f i c a t i o n o f s t r u c t u r e ' , in the s e n s e of Box and J e n k i n s (1970). S e c t i o n 5 . 3 i s c once rne d w i th ways o f o b t a i n i n g va l ue s o f the

program parameters

. These a r e t h e

v a r i a n c e s , Q and a 2 , and the i n i t i a l c o n d i t i o n s 0q>Po|O’ (as d e f i n e d in Ch apt ers 2, 3 and 4) which have been so f a r assumed known. F i n a l l y in S e c t i o n 5 . 4 , some a s y m p t o t i c p r o p e r t i e s o f t h e e s t i m a t i o n p r oc e du re s a r e c o n s i d e r e d . 5 . 2 I d e n t i f i c a t i o n of Time-Varying S t r u c t u r e We can r e c o g n i z e t h r e e p o s s i b l e s t a g e s i n the p r o c e s s o f a d o p t i n g a t i m e - v a r y i n g p a r a m e t e r model. F i r s t l y , e xa mi n at io n o f c o n s t a n t p a r a m e t e r r e s u l t s ; s e c o n d l y , h y p o t h e s i s t e s t i n g c o nc e r n i n g t h e p o s s i b i l i t y o f p a r a m e t r i c change, and t h i r d l y , t h e e s t i m a t i o n o f a time v a r yi ng p a r a m e t e r model. The t h i r d s t a g e has been c o n s i d e r e d in some d e t a i l a l r e a d y , so we w i l l here b r i e f l y c o n s i d e r some a s p e c t s o f t h e f i r s t two s t a g e s .

5.2.1 Examination of constant parameter re sul ts

The use of recursive estimation methods in constant

parameter time series and regression models lias come into favour recently (Young, 1974; Söderström et a l . , 1974). Not only have they been found to provide computationally a t t r a c t i v e means of obtaining consistent, e f f i c i e n t , parameter estimates (Young,

1976), but also covergence c h a r a c te r i s ti c s can be conveniently examined by reference to graphical outputs o f the recursive parameter estimates. In th i s way, i t is possible to ascertain whether the estimates are slow in converging, or i f , indeed

they f a i l to converge.

Slow convergence or f a i l u r e to converge can occur f o r a

4. number of reasons. F i r s t l y , there could be an i d e n t i f i a b i 1i t y 1 problem associated with the model. In the case of model I , th i s could arise through mu lticol 1i n e a r i t y of the inputs (regressors in th i s case) u ^ 1*^, = 1 , 2 , . . . , M . Tests to detect t h i s , such as the m u lt ip le c o r re l a t io n t e s t , are well known (Kendall and S tu a rt, 1961). Mu lticol 1i n e a r i t y is manifested in near­ s i n g u l a r i t y o f the information matrix U^U, where

/u o>

u <mh

i • • • i

lU (1) u M

' l l • ■ • N /

See Hannan (1971) f o r a general discussion of i d e n t i f i a b i 1i t y .

42

which leads to a high (normalized) estimation e r r o r covariance matrix S^. In model I I , an i d e n t i f i a b i 1i t y problem could arise

ß through pole-zero can cell atio n in the t r a n s f e r function

i n d ic a ti n g tha t a model of too high an order is being f i t t e d to the data. Once again, th i s is manifested in a large estimation e r r o r covariance matrix, and a number of procedures can be

used to t e s t whether th is is the case (Young e t a l . , 1978). Again i d e n t i f i a b i l i t y problems can arise because the inp ut signal u^ is not ' s u f f i c i e n t l y e x c i t i n g ' (Aström and Bohlin, 1966). For example, a second order system is not i d e n t i f i a b l e when perturbed only by a single sinusoidal input : at le a s t two d i f f e r e n t

frequency components are required to avoid i d e n t i f i a b i 1i t y problems (see Young et a l ., 1971).

I f the p o s s i b i l i t y o f n o n - i d e n t i f i abi 1i t y has been

eliminated, then the reason f o r slow convergence o f the parameters is th a t a single model is not appropriate a t a l l time po in ts , and th a t there appears to be some v a r ia t i o n in the parameters. An examination of p l o tte d residuals (Draper and Smith, 1967;

f o r model I) or innovations (Harvey and P h i l l i p s , 1976, f o r model I I ) in a constant parameter model, may also corroborate evidence

of t h i s kind, since certai n types of parametric v a r ia t i o n may appear as a systematic component in residuals or innovations. I f there is such evidence of parametric v a r i a t i o n , then we may proceed to the second stage ou tl in ed above, provided the

5.2.2 Testing the hypothesis of parametric change

This stage in the procedure outlined at the s t a r t of this

In document The estimation of parametric change in time-series models (Page 45-48)