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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~tSqt/Sq0\tilde M_t \equiv S_{qt}/S^0_q, and there exists a hypothetical distribution sDs^*_{D} from which these values are drawn. If we are interested in the var[log(Sqt/Sq0)]var[\log(S_{qt}/S^0_q)]—which is equivalent to var[log(Sqt)]var[\log(S_{qt})] since Sq0S_q^0 is a fixed level—as opposed to var[Sqt/Sq0]var[S_{qt}/S_q^0] on the linear levels, we can use the relation:

var[log(M~t)]var[M~t]ln2(10)E2[M~t]var[\log(\tilde M_t)] \approx \frac{var[\tilde M_t]}{\ln^2(10) E^2[\tilde M_t]}

The variance of log levels is identified with σq2\sigma^2_q 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)]var[Sqt/Sq0]ln2(10)E2[Sqt/Sq0]=nqα102μ+σ^2ln(10)(10σ^2ln(10)1)ln2(10)102μ+σ^2ln(10)=nqα10σ^2ln(10)1ln2(10)var[\log(\tilde M_t)] \approx \frac{var[S_{qt}/S^0_q]}{\ln^2(10) E^2[{S_{qt}}/{S^0_q}]} = n_q^{-\alpha} \frac{10^{2 \mu + \hat \sigma^2 \ln(10)} (10^{\hat \sigma^2 \ln(10)} - 1)}{\ln^2(10) 10^{2 \mu + \hat \sigma^2 \ln(10)}} = n_q^{-\alpha} \frac{10^{\hat \sigma^2 \ln(10)} - 1}{\ln^2(10)}

In the limit of small fluctuations, this simplifies to:

var[log(M~t)]nqα(σ^2+12σ^4ln2(10)+o(σ^6))var[\log(\tilde M_t)] \approx n_q^{-\alpha} \left(\hat \sigma^2 + \frac{1}{2} \hat \sigma^4 \ln^2(10) + o(\hat \sigma^6)\right)

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)]var[Sqt/Sq0]ln2(10)E2[Sqt/Sq0]=nqαln2(10)((112σ^2ln2(10))212σ^2ln2(10)1)var[\log(\tilde M_t)] \approx \frac{var[S_{qt}/S^0_q]}{\ln^2(10) E^2[{S_{qt}}/{S^0_q}]} = \frac{n_q^{-\alpha}}{\ln^2(10)} \left(\frac{(1 - \frac{1}{2} \hat \sigma^2 \ln^2(10))^2}{1 - 2 \hat \sigma^2 \ln^2(10)} - 1\right)

In the limit of small fluctuations, this becomes:

var[log(M~t)]nqα(σ^2+94σ^4ln2(10)+92σ^6ln4(10)+o(σ^8))var[\log(\tilde M_t)] \approx n_q^{-\alpha} \left(\hat \sigma^2 + \frac{9}{4} \hat \sigma^4 \ln^2(10) + \frac{9}{2} \hat \sigma^6 \ln^4(10) + o(\hat \sigma^8)\right)

In both cases, the dependence for small fluctuations is primarily on σ^2\hat \sigma^2, but the nonlinear terms in the log-Laplace case are more than four times larger than those in the log-normal case.