Skip to content

Autoregressive Conditional Duration

Model positive intervals between irregular events as a unit-mean innovation times a conditional expected duration that evolves from past durations and past conditional means.

Version
v2 · 2026-09-06 · History
Domain-specific #
1324
Origin domain
economics
Subdomain
financial econometrics
Aliases
ACD model, Autoregressive conditional duration model

Core Idea

Let event times satisfy \(t_i>t_{i-1}\) and define duration \(x_i=t_i-t_{i-1}>0\). An autoregressive conditional duration (ACD) model writes

\[ x_i=\psi_i\varepsilon_i,\qquad E(\varepsilon_i\mid\mathcal F_{i-1})=1, \]

where \(\psi_i=E(x_i\mid\mathcal F_{i-1})\) evolves from prior durations and conditional means. In a linear ACD\((p,q)\),

\[ \psi_i=\omega+\sum_{j=1}^{q}\alpha_jx_{i-j}+ \sum_{k=1}^{p}\beta_k\psi_{i-k}, \]

with positivity restrictions. Engle and Russell introduced the model for clustered, irregularly spaced financial transaction data.

Scope of Application

ACD models handle trade, quote, price-change, volume, and order-arrival durations in ultra-high-frequency finance. They also provide a template for irregular event sequences in other domains. Engle embeds duration analysis in marked point-process views of complete transaction records and links surprising duration to market-microstructure quantities.

Clarity

Define the event, timestamp resolution, zero-duration treatment, overnight gaps, censoring, time-zone/session boundaries, and deterministic seasonality adjustment. State lag orders, recursion form, innovation distribution, covariates, positivity constraints, likelihood, and residual tests. Do not interpret a shorter predicted duration as higher activity without declaring the reciprocal intensity convention.

Manages Complexity

ACD models move analysis from arbitrary calendar bins to event time. A compact recursion summarizes duration clustering, while the innovation distribution separates predictable pace from unpredictable waiting. The same framework supports likelihood, hazard, forecast, and marked-event extensions.

Abstract Reasoning

  1. Define and clean the event sequence.
  2. Compute positive durations and handle ties.
  3. Estimate or model intraday seasonality.
  4. Select ACD lag orders and recursion family.
  5. Choose a positive unit-mean innovation distribution.
  6. Estimate parameters under admissibility constraints.
  7. Standardize durations by fitted \(\psi_i\).
  8. Test residual dependence and distributional fit.
  9. Compare forecasts and alternative duration models.

Knowledge Transfer

The portable pattern is factor a positive waiting time into predictable local pace and unit-scale surprise, then let pace inherit memory from earlier waits. It transfers to irregular event-stream forecasting. The proposed immediate parent is Stochastic Process.

Relationships to Other Abstractions

Local relationship map for Autoregressive Conditional DurationParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.AutoregressiveConditional DurationDOMAINPrime abstraction: Stochastic Process — is a kind ofStochasticProcessPRIME

Current abstraction Autoregressive Conditional Duration Domain-specific

Parents (1) — more general patterns this builds on

  • Autoregressive Conditional Duration is a kind of Stochastic Process Prime

    Stochastic Process is the proposed immediate parent.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Autoregressive Conditional Duration sits in a sparse region of the domain-specific corpus (97th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08