[Paper Review] Local Polynomial Estimation of Time-Varying Parameters in Nonlinear Models

Dennis Kristensen, Young Jun Lee|arXiv (Cornell University)|Apr 10, 2019

Financial Risk and Volatility ModelingEconomics, Econometrics and Finance15 references3 citations

TL;DR

This paper develops a general asymptotic theory for local polynomial quasi-maximum likelihood estimators of time-varying parameters in nonlinear Markov models, enabling consistent and normally distributed inference under weaker regularity and moment conditions than existing methods. The approach allows for broader application—particularly to non-concave likelihoods like Poisson autoregressions—and demonstrates improved performance in estimating time-varying default risk in U.S. corporate defaults.

ABSTRACT

We develop a novel asymptotic theory for local polynomial extremum estimators of time-varying parameters in a broad class of nonlinear time series models. We show the proposed estimators are consistent and follow normal distributions in large samples under weak conditions. We also provide a precise characterisation of the leading bias term due to smoothing, which has not been done before. We demonstrate the usefulness of our general results by establishing primitive conditions for local (quasi-)maximum-likelihood estimators of time-varying models threshold autoregressions, ARCH models and Poisson autogressions with exogenous co--variates, to be normally distributed in large samples and characterise their leading biases. An empirical study of US corporate default counts demonstrates the applicability of the proposed local linear estimator for Poisson autoregression, shedding new light on the dynamic properties of US corporate defaults.

Motivation & Objective

• To develop a general asymptotic theory for local polynomial (quasi-)maximum-likelihood estimators of time-varying parameters in nonlinear Markov models.
• To relax the smoothness and moment conditions required by existing theories, especially important in financial applications with fat-tailed data.
• To extend applicability to non-concave likelihoods—such as in Poisson autoregressions—where prior theories fail.
• To improve bandwidth selection flexibility by weakening restrictions on the bandwidth sequence.
• To demonstrate the method’s utility through empirical analysis of U.S. default counts, revealing time-varying dynamics missed by time-invariant models.

Proposed method

• Proposes local polynomial (quasi-)maximum-likelihood estimators for time-varying parameters in nonlinear models, including local constant and local linear variants.
• Derives asymptotic normality and consistency under weak regularity conditions on the data-generating process and its likelihood function.
• Uses a kernel-weighted likelihood approach with bandwidth sequences that allow standard bandwidth selection procedures.
• Applies the theory to three model classes: time-varying VARs, ARCH models, and Poisson autoregressions with exogenous covariates.
• Employs probability integral transform (PIT) and residual autocorrelation checks to assess model fit in empirical analysis.
• Uses randomized PITs and Kolmogorov-Smirnov tests to evaluate goodness-of-fit for time-varying versus time-invariant models.

Experimental results

Research questions

• RQ1 Can local polynomial estimation be extended to general nonlinear Markov models with time-varying parameters under weaker regularity conditions?
• RQ2 How do the proposed estimators perform in models with non-concave likelihoods, such as Poisson autoregressions?
• RQ3 To what extent does allowing time-varying parameters improve in-sample fit and predictive power in default intensity modeling?
• RQ4 How do macroeconomic and financial variables influence default risk over time, and does this relationship change across business cycles?
• RQ5 What is the impact of including additional covariates—such as industrial production and S&P 500 returns—on the time-varying parameter estimates?

Key findings

• The proposed local polynomial estimators are consistent and asymptotically normal under weaker smoothness and moment conditions than existing theories.
• The bias terms in the estimators take a simpler form, and the local linear estimator benefits from automatic boundary adjustment, outperforming the local constant estimator at sample edges.
• For Poisson autoregressions with time-varying parameters, the proposed method is the first to provide a valid asymptotic theory, as prior methods do not apply.
• Empirical analysis of U.S. default counts shows that time-varying parameters significantly improve model fit: log-likelihood increases from 700.6 (time-invariant) to 777.7 (time-varying), with a PIT p-value of 0.5432, indicating good fit.
• The Leading Index remains highly significant over time, while the impact of short-term interest rates on default risk shifts over time—positive during recessions, consistent with monetary policy transmission.
• During the Great Recession (2007–2011), financial, credit market, and macroeconomic variables become significant predictors of default intensity, a dynamic not captured by time-invariant models.