RDP 8906: A Random Walk Around the $A: Expectations, Risk, Interest Rates and Consequences for External Imbalance Appendix D

We describe here a non-parametric statistical test for the skewness of the distribution Di [a/b]. Assume we have a sample with an odd number (2n + 1) of independent[46] observations from Di [a/b].[47] Order the sample from the most negative to the most positive and define dj as the jth observation (so dj−1 ≤ dj ≤ dj + 1 for 1 < j < 2n + 1). dn + 1 is the median of the sample. Define yj = dj − dn + 1, j = 1,…, 2n+1, and form the random variables Yk, k = 1,…,2n, defined by Yk = − 1 when yj is the kth largest of the yj 's in absolute value and yj is negative; Yk = 1 when yj is the kth largest of the yj 's in absolute value and yj is non-negative. Finally, define the random walk Zk, by

Provided that dj ≠ dn + 1 for all j ≠ n + 1,[48] there are exactly n ‘−1’ values and n ‘+1’ values taken by the random variables Yk , k = 1,…,2n, and hence the random walk, Zk, walks from Z0 = 0 to Z2n = 0. Crucially, under the null hypothesis that the distribution Di [a/b] is symmetric, all distributions of the n ‘−1’ values and n ‘+1’ values among the random variables Yk , k = 1,…,2n, are equally likely and all walks Zk from Z0 = 0 to Z2n = 0 are also equally likely. By contrast, if Di [a/b] is negatively (positively) skewed, Zk, k = 1,…,2n will be more likely to walk to large negative (positive) values before returning to zero when k = 2n. No specific assumption about the distribution of Di [a/b] is necessary – the null hypothesis is simply that Di [a/b] is symmetric.

Define the random variable WM as the number of the random variables Yk , k = 1,…,M, which take the value ‘−1’. Under H0, Pr(WM = w), is

Our one-sided test for negative [positive] skewness involves evaluating the probability, Pr(WM ≥ w) [ Pr(WM ≤ w)]. For the results in Table 9, n = 84, and M = 10 was chosen. Evaluation of (D.1) gives: Pr(W10 = 0) = Pr(W10 = 10) = 0.0007, Pr(W10 ≤ 1) = Pr(W10 ≥ 9) = 0.0090, Pr(W10 ≤ 2) = Pr(Wl0 ≥ 8) = 0.0494. Thus, sample values of W10 of 9 or 10 (0 or 1) imply rejection of H0 at the 1% level of significance against the alternative of negative (positive) skewness, while a value of 8 (2) implies rejection at the 5% level. Sample values 3 ≤ W10 ≤ 7 are insignificant.[49]

As discussed in Appendix B, in general the distribution of Di [a/b] at time t (Dti [a/b]) depends on observations of sτ + i − sτ, τ < t. Clearly, this invalidates our assumption of the independence of the observations, and our test of skewness must be modified. The null hypothesis is now that each of the Dti [a/b] distributions is symmetric with a common mean, μ. Under this null, the distribution of W10 depends on how different are the distributions Dti [a/b], t = 1,…, 2n+1. At one extreme is the case already examined when all the distributions are identical, and each Yk, k = 1,…,2n has an equal chance of coming from any of the Dti [a/b], t = 1,…,2n+1. At another extreme, assume that under the null there are only two distinct (symmetrical) distributions: for ten particular times, t(j), j = 1,…,10, the distributions Dt(j)i [a/b] ≡ D+, and at all other times, τ, τ ≠ t(j), j = 1,…,10, Dτi [a/b] ≡ D*. D* is assumed to have all its probability weight “near” μ while D+ is assumed to have all its probability weight in two tails “far from” μ, so that D* and D+ have no overlap. In this contrived case, we can be sure that for j = 1,…,10, Yj must come from D+ and hence from the ten particular times, t(j), j = 1,…,10. Then under the null hypothesis, Pr(W10 = w) is simply

Equation (D.2) gives: Pr(W10 = 0) = Pr(W10 = 10) = 0.00098, Pr(W10 ≤ 1) = Pr(W10 ≥ 9) = 0.011, Pr(W10 ≤ 2) = Pr(W10 ≥ 8) = 0.055. Thus, even in this extreme case, the critical values of W10 are only changed slightly.


The assumption of independence makes the analysis exact. We examine the removal of this assumption at the end of this appendix. [46]

If we have an even number (2n + 2) of independent observations, we define dj as described, but now define yj = dj − (dn + 1 + dn + 2)/2, j = 1,…, 2n+2. The random variables Yk , k = 1,…,2n, are defined as described and equation (D.1) is again the basis of our non-parametric test for the skewness of Di [a/b]. [47]

With the exception of the TWI data (which is quoted to three figures), all our exchange rate data is quoted to (at least) four significant figures, so it is unlikely that any two values of dj would be the same. [48]

An alternative test based on the distribution of the maximum (or minimum) value taken by the walk, Zk, k = 1,…,2n, was also examined but found to have little power. One of us (J. S.) examined the skewness of the data assuming that D5 [a/b] has a distribution of the stable Paretian form, The results are similar to those reported here. [49]