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Appendix B. Autocorrelation for u’_t, an estimator of u_t by the moving-average method. A pdf file of this appendix is also available.

Figure A2 (see Appendix A) shows the sample autocorrelation functions, r_uu(l) and r_u’u’(l), for the density-independent influence u_t and its estimator u’_t (Fig. 6b). The r_uu(l), as samples of r_uu(l) º 0, l  > 0, are insignificantly different from zero at all lags shown, whereas the r_u’u’(l) exhibit a significant negative spike at lag 2, which can be explained as follows.

Consider that the series {R_t} is made up of two components: a low-frequency cycle (g_t, say) due to the autocorrelated density-dependent process and a rapid fluctuation (about the cycle) due to the density-independent influence u_t with no autocorrelation. Thus, we may write

After the moving-average transformation on both sides of (A2), we have

in which the single and double primes indicate the moving-average transformations of the respective letters in Eq. A.2. Recalling the definition u’_t = R_t - R’_t in the main text, the difference (A.2) - (A.3) yields

Now that, by assumption, g_t is void of the rapidly fluctuating component, the difference

|g_t - g’_t| would be small compared with |u_t - u”_t (= z_t, say)| such that u’_t would be approximately equal to z_t. Hence, letting k (an odd positive integer) be the order of the moving-average transformation, the autocorrelation function r_u’u’(l) would be approximated by r_zz(l) whose theoretical function is readily shown to be

 For k = 5, r_zz(1) = - 0.3, r_zz(2) = - 0.35, r_zz(3) = 0.1, r_zz(4) = 0.05, and r_zz(l ³ 5) = 0, as in the r_u’u’(l) based on the length 50, although for length 4000 only the r_u’u’(2) is distinct. The tendency for r_u’u’(2) to be distinctly negative shows up in the cross-correlation function r_u’v’(l) (Fig. 7, middle column) that tends to exhibit significant spikes at lags other than 0.

The results (A.4) show that r_zz(l) tends to zero for all l as the order k increases, apparently suggesting that the higher the order k, the better r_u’v’(l) approximates r_uv(l). This is true when g_t is linear, or flat, across all t. Otherwise, g’_t would over-smooth g_t sooner or later as k increases, resulting in the difference (g_t  - g’_t) to increase in variance. Conversely, too low an order results in an under-smoothing that also results in an increase in variance. Thus, there must be an optimal order at which the variance of the difference (g_t  - g’_t) is minimized. Delving further into this subject is beyond the present scope. However, a graphical inspection (cf. Fig. 6a) would probably be adequate in practice.