RDP 8906: A Random Walk Around the $A: Expectations, Risk, Interest Rates and Consequences for External Imbalance Appendix D
October 1989
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We describe here a non-parametric statistical test for the skewness of the distribution D_{i} [a/b]. Assume we have a sample with an odd number (2n + 1) of independent^{[46]} observations from D_{i} [a/b].^{[47]} Order the sample from the most negative to the most positive and define d_{j} as the j^{th} observation (so d_{j−1} ≤ d_{j} ≤ d_{j + 1} for 1 < j < 2n + 1). d_{n + 1} is the median of the sample. Define y_{j} = d_{j} − d_{n + 1}, j = 1,…, 2n+1, and form the random variables Y_{k}, k = 1,…,2n, defined by Y_{k} = − 1 when y_{j} is the k^{th} largest of the y_{j} 's in absolute value and y_{j} is negative; Y_{k} = 1 when y_{j} is the k^{th} largest of the y_{j} 's in absolute value and y_{j} is non-negative. Finally, define the random walk Z_{k}, by
Provided that d_{j} ≠ d_{n + 1} for all j ≠ n + 1,^{[48]} there are exactly n ‘−1’ values and n ‘+1’ values taken by the random variables Y_{k} , k = 1,…,2n, and hence the random walk, Z_{k}, walks from Z_{0} = 0 to Z_{2n} = 0. Crucially, under the null hypothesis that the distribution D_{i} [a/b] is symmetric, all distributions of the n ‘−1’ values and n ‘+1’ values among the random variables Y_{k} , k = 1,…,2n, are equally likely and all walks Z_{k} from Z_{0} = 0 to Z_{2n} = 0 are also equally likely. By contrast, if D_{i} [a/b] is negatively (positively) skewed, Z_{k}, 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 D_{i} [a/b] is necessary – the null hypothesis is simply that D_{i} [a/b] is symmetric.
Define the random variable W_{M} as the number of the random variables Y_{k} , k = 1,…,M, which take the value ‘−1’. Under H_{0}, Pr(W_{M} = w), is
Our one-sided test for negative [positive] skewness involves evaluating the probability, Pr(W_{M} ≥ w) [ Pr(W_{M} ≤ w)]. For the results in Table 9, n = 84, and M = 10 was chosen. Evaluation of (D.1) gives: Pr(W_{10} = 0) = Pr(W_{10} = 10) = 0.0007, Pr(W_{10} ≤ 1) = Pr(W_{10} ≥ 9) = 0.0090, Pr(W_{10} ≤ 2) = Pr(W_{l0} ≥ 8) = 0.0494. Thus, sample values of W_{10} of 9 or 10 (0 or 1) imply rejection of H_{0} 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 ≤ W_{10} ≤ 7 are insignificant.^{[49]}
As discussed in Appendix B, in general the distribution of D_{i} [a/b] at time t (D^{t}_{i} [a/b]) depends on observations of s_{τ + i} − s_{τ}, 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 D^{t}_{i} [a/b] distributions is symmetric with a common mean, μ. Under this null, the distribution of W_{10} depends on how different are the distributions D^{t}_{i} [a/b], t = 1,…, 2n+1. At one extreme is the case already examined when all the distributions are identical, and each Y_{k}, k = 1,…,2n has an equal chance of coming from any of the D^{t}_{i} [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 D^{t(j)}_{i} [a/b] ≡ D^{+}, and at all other times, τ, τ ≠ t(j), j = 1,…,10, D^{τ}_{i} [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(W_{10} = w) is simply
Equation (D.2) gives: Pr(W_{10} = 0) = Pr(W_{10} = 10) = 0.00098, Pr(W_{10} ≤ 1) = Pr(W_{10} ≥ 9) = 0.011, Pr(W_{10} ≤ 2) = Pr(W_{10} ≥ 8) = 0.055. Thus, even in this extreme case, the critical values of W_{10} are only changed slightly.
Footnotes
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 d_{j} as described, but now define y_{j} = dj − (d_{n + 1} + d_{n + 2})/2, j = 1,…, 2n+2. The random variables Y_{k} , k = 1,…,2n, are defined as described and equation (D.1) is again the basis of our non-parametric test for the skewness of D_{i} [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 d_{j} would be the same. [48]
An alternative test based on the distribution of the maximum (or minimum) value taken by the walk, Z_{k}, 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 D_{5} [a/b] has a distribution of the stable Paretian form, The results are similar to those reported here. [49]