Moments of Log Quantile Levels
Understanding Moments of Log Quantile Levels
In this section, we delve into the moments of log quantile levels. We have been working with M~t≡Sqt/Sq0, and there exists a hypothetical distribution sD∗ from which these values are drawn. If we are interested in the var[log(Sqt/Sq0)]—which is equivalent to var[log(Sqt)] since Sq0 is a fixed level—as opposed to var[Sqt/Sq0] on the linear levels, we can use the relation:
var[log(M~t)]≈ln2(10)E2[M~t]var[M~t]
The variance of log levels is identified with σq2 in previous sections, such as in equations related to linear approximations and variance of shocks.
Log-Normal Fluctuations
For log-normal fluctuations, by substituting the expressions for expected value and variance into the equation above, we have:
var[log(M~t)]≈ln2(10)E2[Sqt/Sq0]var[Sqt/Sq0]=nq−αln2(10)102μ+σ^2ln(10)102μ+σ^2ln(10)(10σ^2ln(10)−1)=nq−αln2(10)10σ^2ln(10)−1
In the limit of small fluctuations, this simplifies to:
var[log(M~t)]≈nq−α(σ^2+21σ^4ln2(10)+o(σ^6))
Log-Laplace Fluctuations
Similarly, for log-Laplace fluctuations, using the variance-to-mean ratio and the expressions for the kth moment of a log-Laplace, we get:
var[log(M~t)]≈ln2(10)E2[Sqt/Sq0]var[Sqt/Sq0]=ln2(10)nq−α(1−2σ^2ln2(10)(1−21σ^2ln2(10))2−1)
In the limit of small fluctuations, this becomes:
var[log(M~t)]≈nq−α(σ^2+49σ^4ln2(10)+29σ^6ln4(10)+o(σ^8))
In both cases, the dependence for small fluctuations is primarily on σ^2, but the nonlinear terms in the log-Laplace case are more than four times larger than those in the log-normal case.