ON THE INSTABILITY INDEX OF TIME SERIES DATA: A GENERALIZATION*
| Published date | 01 August 1979 |
| DOI | http://doi.org/10.1111/j.1468-0084.1979.mp41003007.x |
| Date | 01 August 1979 |
ON THE INSTABILITY INDEX OF TIME SERIES DATA:
A GENERALIZATION*
By P. A. DELLA VALLE
In the paper Measuring the Instability of Time Series Data" Cuddy and Della
Valle defined a new measure of the instability of time series data I,, as
SEE
I=lOO (1)
Y
where SEE is the standard error of the regression estimates, and y is the average
value of the time series data. It was then shown that I, could be measured as
I=CVJ(l_R)2 (2)
where C V is the coefficient of variation, and 2 is the adjusted coefficient of multiple
determination.
The use of (2) in the calculation of the instability indexes was correct for any
linear or non-linear time series model, and the results reported in the paper are
valid. However, it must be realized that the definition of I as set forth in (1) is only
applicable to the linear time series model. In the log-linear, or indeed, in any other
alternative regression model, that definition must be altered as follows:
Ix=loo -ySEE /1R1R2 (3)
The corrective factor, which generalizes the definition of the instability measure
I, is given in (3) as /(l R)/(l R2)
The coefficient of multiple determination R2 is obtained from the linear model of y
on time, while R is the coefficient of multiple determination from any alternative
model used. In the example given for commodity prices in the Cuddy-Della Valle
paper, R would be calculated from the log-linear model. It should be noted that
if R = R2 1, the corrective factor = 1. In such an extreme case, I,, defined by (3)
or by (1), would be identical.
The generalized definition of the instability measure as defined in (3) is arrived
at as follows. Let y, be the transformed original variable y, where j and
represent their estimated values. Given:Ç'
R21 L\Ya Ya) 4
a(Yaia)2
(5)
SEEa = \/(Ya a) 2/(N - K) (6)
and the standard deviation of the non-transformed data
*John Cuddy has made a substantial contribution to this note.
'BULLETIN, Vol. 40, No. 1, February 1978.247
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