Frederik Vilandt Rasmussen defends his PhD thesis at the Department of Economics

Candidate:

Frederik Vilandt Rasmussen, Department of Economics, University of Copenhagen

Title:

Econometric Analysis of Events, Counts, and Duration Models

Supervisor:

• Anders Rahbek, Professor, Department of Economics, University of Copenhagen

Assessment Committee:

• Rasmus Søndergaard Pedersen, Associate Professor, Department of Economics, University of Copenhagen
• Alessandra Luati, Professor, Chair in Statistics, Imperial College London
• Leopoldo Catania, Associate Professor, Department of Economics, Aarhus University

Summary:

This dissertation consists of four chapters. The first three chapters, co-authored with Giuseppe Cavaliere (University of Bologna, University of Exeter), Thomas Mikosch, and Anders Rahbek (University of Copenhagen), develop new asymptotic theory for likelihood-based estimators in autoregressive conditional duration models for (financial) durations. The fourth chapter, co-authored with Jonas Fiedler Frederiksen (University of Copenhagen), introduces a new asymmetric autoregressive conditional heteroscedasticity (ARCH) model for modeling conditional variances of financial returns: the Smooth Threshold ARCH (STARCH). This chapter establishes the stochastic properties of the STARCH model and provides asymptotic theory for the (quasi) maximum likelihood estimator.

Chapter 1: Tail Behavior of ACD Models and Consequences for Likelihood-Based Estimation

We establish new results for estimation and inference in financial durations models, where events are observed over a given time span, such as a trading day, or a week. For the classical autoregressive conditional duration (ACD) models by Engle and Russell (1998, Econometrica 66, 1127-1162), we show that the large sample behavior of likelihood estimators is highly sensitive to the tail behavior of the financial durations. In particular, even under stationarity, asymptotic normality breaks down for tail indices smaller than one or, equivalently, when the clustering behavior of the observed events is such that the unconditional distribution of the durations has no finite mean. Instead, we find that estimators are mixed Gaussian and have non-standard rates of convergence. The results are based on exploiting the crucial fact that for duration data the number of observations within any given time span is random. Our results apply to general econometric models where the number of observed events is random.

Chapter 2: A Comment on: “Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data”

Based on the GARCH literature, Engle and Russell (1998) established consistency and asymptotic normality of the QMLE for the autoregressive conditional duration (ACD) model, assuming strict stationarity and ergodicity of the durations. Using novel arguments based on renewal process theory, we show that their results hold under the stronger requirement that durations have finite expectation. However, we demonstrate that this is not always the case under the assumption of stationary and ergodic durations. Specifically, we provide a counterexample where the MLE is asymptotically mixed normal and converges at a rate significantly slower than usual. The main difference between ACD and GARCH asymptotics is that the former must account for the number of durations in a given time span being random. As a by-product, we present a new lemma which can be applied to analyze asymptotic properties of extremum estimators when the number of observations is random.

Chapter 3: Beyond the Mean: Limit Theory and Tests for Infinite-Mean Autoregressive Conditional Durations

Integrated autoregressive conditional duration (ACD) models serve as natural counterparts to the well-known integrated GARCH models used for financial returns. However, despite their resemblance, asymptotic theory for ACD is challenging and also not complete, in particular for integrated ACD. Central challenges arise from the facts that (i) integrated ACD processes imply durations with infinite expectation, and (ii) even in the non-integrated case, conventional asymptotic approaches break down due to the randomness in the number of durations within a fixed observation period. Addressing these challenges, we provide here unified asymptotic theory for the (quasi-) maximum likelihood estimator for ACD models; a unified theory which includes integrated ACD models. Based on the new results, we also provide a novel framework for hypothesis testing in duration models, enabling inference on a key empirical question: whether durations possess a finite or infinite expectation. We apply our results to high-frequency cryptocurrency ETF trading data. Motivated by parameter estimates near the integrated ACD boundary, we assess whether durations between trades in these markets have finite expectation, an assumption often made implicitly in the literature on point process models. Our empirical findings indicate infinite-mean durations for all the five cryptocurrencies examined, with the integrated ACD hypothesis rejected - against alternatives with tail index less than one – for four out of the five cryptocurrencies considered.

Chapter 4: Smooth Threshold ARCH: Properties and Estimation

Since the seminal work of Engle (1982), the ARCH model and its many extensions have become central tools in modeling financial volatility. Among these, the GJR model (Glosten, Jagannathan, and Runkle, 1993) is widely used due to its ability to capture leverage effects through its asymmetric specification of the news impact curve (NIC). Although asymmetric, the GJR imposes that volatility is minimized when returns are zero, i.e. that the NIC takes its minimum value at zero. We propose a new asymmetric ARCH model, the Smooth Threshold ARCH (STARCH), that nests the GJR, which (i) by introducing a threshold parameter allows a non-zero minimum point of the NIC, (ii) has a smooth NIC despite the threshold, and, as a biproduct, for which (iii) the quasi maximum likelihood estimator, which can be obtained without grid-search, is asymptotically normal at the standard root-n rate. Specifically, while the (Gaussian) log-likelihood is smooth enough to make estimation and inference standard, the second derivative is non-smooth. This requires non-standard arguments in the asymptotic analysis, for which we rely on conditions verified using empirical process theory. Applying our framework to large stock indices, we show that minimum point of the NIC is positive (and not zero)--providing empirical evidence for our new model.

An electronic copy of the dissertation can be requested here: lema@econ.ku.dk