Moments of the terminal probability and log-odds

Closed-form moments of the resolution-date log-odds \(L_T\) and event probability \(q_T\) implied by the exact logistic-normal transition law (see ec_transition_density()), given current probability q and event-clock time A until resolution.

Usage

ec_moments(q, A)

Arguments

q

Numeric vector of current event probabilities in \((0,1)\).

A

Numeric vector of event-clock time (non-negative).

Value

A tibble with columns q, A, L, E_qT, E_LT, var_LT, sd_LT, m3_LT, var_qT (approximation), and var_qT_bound.

Details

Exact results: \(E[q_T] = q\) (the martingale property — free of A), \(E[L_T] = L + (q - 1/2)A\), \(Var(L_T) = A + q(1-q)A^2\), and the third central moment \(m_3(L_T) = q(1-q)(1-2q)A^3\). The variance of \(q_T\) is reported in its small-A approximation \(Var(q_T) \approx (q(1-q))^2 A\) together with the exact upper bound \(\min\{A/16,\; q(1-q)\}\).

Examples

# Brexit, two weeks before the referendum: q = 0.195, A = 0.166
ec_moments(0.195, 0.166)
#> # A tibble: 1 × 10
#>       q     A     L  E_qT  E_LT var_LT sd_LT    m3_LT  var_qT var_qT_bound
#>   <dbl> <dbl> <dbl> <dbl> <dbl>  <dbl> <dbl>    <dbl>   <dbl>        <dbl>
#> 1 0.195 0.166 -1.42 0.195 -1.47  0.170 0.413 0.000438 0.00409       0.0104