Asymptotics for the Generalized Autoregressive Conditional Duration Model

ArXiv ID: 2307.01779 “View on arXiv”

Authors: Unknown

Abstract

Engle and Russell (1998, Econometrica, 66:1127–1162) apply results from the GARCH literature to prove consistency and asymptotic normality of the (exponential) QMLE for the generalized autoregressive conditional duration (ACD) model, the so-called ACD(1,1), under the assumption of strict stationarity and ergodicity. The GARCH results, however, do not account for the fact that the number of durations over a given observation period is random. Thus, in contrast with Engle and Russell (1998), we show that strict stationarity and ergodicity alone are not sufficient for consistency and asymptotic normality, and provide additional sufficient conditions to account for the random number of durations. In particular, we argue that the durations need to satisfy the stronger requirement that they have finite mean.

Keywords: Autoregressive conditional duration (ACD) model, Exponential QMLE, Random number of durations, Time series analysis, General Financial Time Series

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is densely packed with advanced asymptotic theory, stochastic processes, and proofs (e.g., central limit theorems for random sample sizes, functional convergence to Brownian motion), earning a high math score. It lacks any empirical implementation, backtesting, or data analysis, focusing solely on theoretical asymptotic results, resulting in low empirical rigor.

  flowchart TD
    A["Research Goal:<br/>Asymptotic Analysis of ACD Model"] --> B{"Key Assumption<br/>Strict Stationarity & Ergodicity?"}
    B -->|Engle & Russell 1998| C["Apply GARCH Results<br/>Assume Finite Mean"]
    B -->|This Paper| D["Identify Flaw:<br/>Random N of Durations"]
    D --> E["Derive Sufficient<br/>Conditions: Finite Mean Required"]
    E --> F["Prove Asymptotic<br/>Properties<br/>Consistency & Normality"]
    C --> G["Outcome:<br/>Validates QMLE in ACD Model"]
    F --> G
    G --> H["Key Contribution:<br/>Generalized Theory for ACD"]