Note on the Correlation of First Differences of Averages in a Random Chain Holbrook Working Econometrica, Volume 28, Issue(Oct., 1960), 916-918. Stable URL: http: //links.jstor.org/sici?sici=0012-9682%28196010%2928%3A4%3C916%3ANOTCOF3E2.0.%3B2-X Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http: //uk.jstor. org/about/terms. html. jstor's terms and Conditions of Use provides, in part, that unless you obtained prior permission, you may not download an entire issue of a joumal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Econometrica is published by The Econometric Society. Please contact the publisher for further permissions regarding the use of this work. Publisher contact information may be obtained at http: //uk. jstor.org/joumals/econosoc. html. Econometrica C1960 The Econometric Society d the ISTOR logo are trademarks of JSTOR, and are Registered in the U.S. Patent and Trademark Office. For more information on JSTOR contact jstor @mimas. ac.uk. C2002 JSTOR http://uk.jstor.org/ Mon Nov2513:44:022002
Econometrica, Vol 28, 4(October 1960) NOTE ON THE CORRELATION OF FIRST DIFFERENCES OF AVERAGES IN A RANDOM CHAIN BY HOLBROOK WORKING IN THE STUDY of serial correlations in prices series it is important to bear in mind that the use of averages can introduce correlations not present in in the original series. II consider here the effect of averaging successive groups of items in a random chain, which is the simplest sort of stochastic series to which stock prices and certain commodity prices have a close resemblance The equation for a random chain may be written (1)x4=X4-1+b(i=1,2,…;E(6)=0;cor(5,04+)=0 when 1≠0), where Xi-1, Xi are successive terms in the chain. In what follows I assume further, for convenience and without loss of generality, that var(8)=I Consider now a random chain that is treated as being composed of successive segments of m items each, corresponding to weeks, months or any other time intervals, in a price series, For purposes of illustration we may take m =3 and write the terms of three successive segments of a random chain by derivation from the 8s as follows 4 6;=+20-11-0.6+0.3+1.3-1.0+0.1+07-0.3 Xt=24443.327 3.0 It is obvious that if, from this segmented random chain, we take first 1 One example of the need is cited in the paper by Alfred Cowles elsewhere in this ssue of Econometrica. Another example is afforded by M. G. Kendall's conclusion that wheat prices and cotton prices have behaved differently, as evidenced by a first order serial correlation of first differences, n1 = +0.313, for cotton prices as against a corresponding coefficient, n1 =-0.071, for wheat prices( "The Analysis of Economic me-Series, "Journal of the Royal Statistical Society, CXVI(1953), pp. 15, 23). Because the cotton-price series that Kendall used consisted of monthly averages of, for the most part, daily prices, a serial correlation of about=+0.25 in the cotton price series was to have been expected simply as a result of the averaging process, as I show below.The wheat price series that Kendall used, on the other hand, was one that I had compiled without averaging, in order to avoid introducing the averaging effect Vhen this difference in constitution of the two series is taken into account there remains no clear evidence of difference in behaviour between wheat prices and cotton prices The example is from Holbrook Working, " A Random-Difference Series for Use in the Analysis of Time Series, " Journal of the American Statistical A ssociation, March, 934, taking the last figure in the second column of the table as Xo, but with al figures divided by 10 in order to have var(dx)=I 916
CORRELATION OF FIRST DIFFERENCES 917 differences between terms correspondingly positioned in each segment Aum)= Xi-Xi-m, these first differences will have a variance var(4 But suppose that we now average the m terms in each segment of the chain and take first differences between the averages Am=1(x+x+1..+x4+m)-1(xm+x E一+1 +Xt-1) sing the relationship in(1), as illustrated in the tabulation above, this may be written, with reversal of the order of terms in the second parenthesis above (m)=mX+(x4+4+)…+(x4+1+4+2…+b+m-1 [(X4-0)+(Xt-0-b4-1)..+(X-b-0 and then simplified to 彡们 A +b4+m-1+mbx+(m-1) +-m+1] Then, bearing in mind that the 8s are all mutually uncorrelated and have been assigned a variance of unity, we may derive var(di(m)) (m-1)2+(m-2)2.,+12+m2+(m-1 thich reduces to var Comparison of expression (4)with expression(2)shows that, with m only moderately large, the variance of first differences between averages over successive segments of a random chain approximates 2/3 of the variance of first differences between correspondingly positioned terms in the chain The covariance of successive first differences between averages may be written as, cov(4((m), 4(-m)(m)), which suggests that the covariance may be derived from the product of expression (3)multiplied by (5)4(-m)m[(m-1)b1-m+1+(m-2)5-m+2+ b-1+mb-m+(m-1)04-m-1+
918 HOLBROOK WORKING Inasmuch as the 8s are mutually uncorrelated, the only terms that will ppear in an expression for the covariance will be those resulting from multiplication of terms in(3)and (5) that have like subscripts for the 8s These are the terms involving &i-m+1, S-m+2,., 8i-1. Thus (6)cov((m),4(-m)(m)=m21(m-1)+2m-2)…+(m=1= Then from(4)we have, cor(4(m),4(-m)(m)= (2m2+1 From this expression it appears that even with m fairly small, the expected first-order serial correlation of first differences between averages of terms a random chain approximates e(1=+1/4. with m= 5, corresponding to weekly averages for a 5-day week, E(1)=+0. 235. With m= 2, as it would be if monthly averages were derived by averaging prices at the ends of the first and third full weeks of each month E(1) 0.167.I have no exact solution for the case of averages bases on the high and loze prices of each month(or of any other time interval), but I suspect that the correlation introduced by such averaging is close to that given above for Serial correlation coefficients of higher order than the first remain zero for first differences of averages of successive groups of terms in a random chain, as may readily be shown by proceeding in a manner similar to that by which expression(6)is derived ood Research Institute, Stanford University
