Estimate event-clock (information) time from log-odds variation

The information accumulated about a scheduled event over a window $[t, T]$ — event-clock time $A_{t,T}$ — equals the quadratic variation of the log-odds of the traded event probability, $A_{t,T} = [\mathrm{logit}(q)]_{t,T}$. event_clock() estimates it by the realized variation $$\widehat A_{t,T} = \sum_i \left[\mathrm{logit}(q_{t_{i+1}}) - \mathrm{logit}(q_{t_i})\right]^2$$ together with standard robustness variants.

Usage

event_clock(
  x,
  from = NULL,
  to = NULL,
  methods = ec_default_params()$methods,
  sample_every = ec_default_params()$sample_every,
  trunc_sd = ec_default_params()$trunc_sd,
  scale_fn = stats::mad,
  clip = NULL,
  se = FALSE,
  conf = 0.95,
  se_method = c("quarticity", "bootstrap"),
  boot_reps = 999
)

Arguments

x: An event_prices object (see as_event_prices()), or a data.frame coercible to one.
from: Valuation date/time (default: first observation).
to: One or more horizon dates/times (default: last observation = full sample). Names are used as horizon labels.
methods: Character vector of estimator variants; see Details.
sample_every: Integer, use every k-th observation (sparse-sampling robustness; default 1).
trunc_sd: Numeric, truncation threshold in robust standard deviations (default 3).
scale_fn: Function computing the robust scale of the increments for truncated (default stats::mad()).
clip: Numeric length-2 clipping bounds for q before the log-odds transform; defaults to the bounds stored in x.
se: Logical; if TRUE, add a standard error and confidence interval for the rv estimate (columns are NA for the other methods). The asymptotic variance is estimated by the quarticity analogue $\widehat{Var}(\widehat A) = \tfrac{2}{3}\sum (\Delta L_i)^4$; the interval is log-based, $\exp\{\log \widehat A \pm z\, se/\widehat A\}$. With se_method = "bootstrap", a wild bootstrap with two-point multipliers (moment-matched to the same asymptotic variance) is used and the interval is the percentile interval. Conditional drift contributes at order $(\Delta t)^2$ and is ignored.
conf: Confidence level (default 0.95).
se_method: "quarticity" (default) or "bootstrap".
boot_reps: Bootstrap replications (default 999).

Value

A tibble with one row per horizon and method:

market_id: market label (from x).
from, to: window bounds as supplied.
horizon: horizon label (names of to, or the date).
n_obs: number of non-missing observations in the window.
n_incr: number of increments used (after subsampling).
n_gaps: number of increments spanning more than 1.5 times the median observation spacing (see Details).
max_gap_days: largest spacing (in days) between consecutive observations used.
method: estimator variant.
A: the estimate $\widehat A_{t,T}$.
se, ci_lo, ci_hi: (only with se = TRUE) standard error and confidence bounds for the rv rows.

Details

Because quadratic variation ignores finite-variation drift and is invariant under equivalent measure changes, $\widehat A$ is measure-robust: it does not require a martingale assumption under the physical measure, and any (approximately) constant level distortion of q — discounting, a constant state-price tilt, a constant cross-market wedge — drops out entirely.

Available methods:

rv: plain realized variation (the baseline).
truncated: drops increments larger than trunc_sd robust standard deviations, where the robust scale is scale_fn(dL) (stats::mad() by default). Note that with daily data and short windows this criterion is coarse; reference results reported as "truncated" often coincide with dropping the single largest increment (largest1). Compare both. When the robust scale is degenerate (e.g. more than half of the increments are identical), truncation is disabled with a message and plain realized variation is returned.
bipower: bipower variation $BV = \frac{\pi}{2}\sum_{i\ge 2} |\Delta L_i||\Delta L_{i-1}|$, a jump-robust companion; the gap $RV - BV$ is a descriptive jumpiness index. No finite-sample correction is applied, so BV is biased downward in very short windows (a 1-week window has only six neighbor products); read it as descriptive, not as an unbiased estimate.
largest1, largest2: realized variation after removing the one or two largest absolute increments.

Windows. Windows are defined in calendar time from the valuation date from to each horizon date in to (they are anchored windows, not trailing ones). All observations in the window are used, including weekends if the series has them. Date-typed bounds combined with a POSIXct-typed series are interpreted in the series' timezone, with to covering the full horizon day.

Missing observations and gaps. Missing q values inside the window are skipped with a warning; the increment then bridges the gap and aggregates more elapsed time than a regular one-period increment. Plain realized variation remains a valid (sparser) estimate of the window's total variation, but the robustness variants treat all increments as homogeneous: a gap-spanning increment is mechanically larger and can be misclassified as a jump by truncated/largest1/largest2, and it distorts the neighbor products of bipower. The output columns n_gaps (increments spanning more than 1.5 times the median observation spacing) and max_gap_days flag affected windows — interpret the robustness variants cautiously whenever n_gaps > 0.

Examples

data(brexit2016)
ep <- as_event_prices(brexit2016,
  time = "date", price = "q_leave",
  market_id = "Brexit: Leave", event_date = as.Date("2016-06-23")
)
# the working paper's 1M valuation date and 1W/2W/1M horizons
event_clock(ep,
  from = as.Date("2016-05-24"),
  to = c(`1W` = as.Date("2016-05-31"), `2W` = as.Date("2016-06-07"),
         `1M` = as.Date("2016-06-23"))
)
#> # A tibble: 15 × 10
#>    market_id     from       to         horizon n_obs n_incr n_gaps max_gap_days
#>    <chr>         <date>     <date>     <chr>   <int>  <int>  <int>        <dbl>
#>  1 Brexit: Leave 2016-05-24 2016-05-31 1W          8      7      0            1
#>  2 Brexit: Leave 2016-05-24 2016-05-31 1W          8      7      0            1
#>  3 Brexit: Leave 2016-05-24 2016-05-31 1W          8      7      0            1
#>  4 Brexit: Leave 2016-05-24 2016-05-31 1W          8      7      0            1
#>  5 Brexit: Leave 2016-05-24 2016-05-31 1W          8      7      0            1
#>  6 Brexit: Leave 2016-05-24 2016-06-07 2W         15     14      0            1
#>  7 Brexit: Leave 2016-05-24 2016-06-07 2W         15     14      0            1
#>  8 Brexit: Leave 2016-05-24 2016-06-07 2W         15     14      0            1
#>  9 Brexit: Leave 2016-05-24 2016-06-07 2W         15     14      0            1
#> 10 Brexit: Leave 2016-05-24 2016-06-07 2W         15     14      0            1
#> 11 Brexit: Leave 2016-05-24 2016-06-23 1M         31     30      0            1
#> 12 Brexit: Leave 2016-05-24 2016-06-23 1M         31     30      0            1
#> 13 Brexit: Leave 2016-05-24 2016-06-23 1M         31     30      0            1
#> 14 Brexit: Leave 2016-05-24 2016-06-23 1M         31     30      0            1
#> 15 Brexit: Leave 2016-05-24 2016-06-23 1M         31     30      0            1
#> # ℹ 2 more variables: method <chr>, A <dbl>