Generalized Autoregressive Method of Moments

Abstract

We introduce a new estimation framework which extends the Generalized Method of Moments (GMM) to settings where a subset of the parameters vary over time with unknown dynamics. To filter out the dynamic path of the time-varying parameter, we approximate the dynamics by an autoregressive process driven by the score of the local GMM criterion function. Our approach is completely observation driven, rendering estimation and inference straightforward. It provides a unified framework for modeling parameter instability in a context where the model and its parameters are only specified through (conditional) moment conditions, thus generalizing approaches built on fully specified parametric models. We provide examples of increasing complexity to highlight the advantages of our method.

The Importance of Heterogeneity in Dynamic Network Models Applied to European Systemic Risk

“…2 Additional unreported simulations show that the new model can also adequately track the true model parameters even in cases where the statistical model is misspecified. This is in line with the theoretical results in Blasques et al (2015) and Creal et al (2020). take some kind of average of these, both in terms of the overall level and in terms of the specific dynamics over time.…”

“…The model then remains easy to estimate via standard maximum likelihood methods. It is also known note score-driven dynamics possess information theoretic optimality properties and yield updates of the parameters that (in expectation) improve the fit of the model to the data (as measured in the so-called Kullback-Leibler divergence); see Blasques et al (2015) and Creal et al (2020).…”

Nonlinear autoregressive models with optimality properties

“…We now proceed by showing that an update function /ð Á ; hÞ in (4) is only optimal in an information theoretic sense if it is based on the score of the predictive log-density for y t , that is on @ logp t =@f t : Such an update locally results in an expected decrease in the Kullback-Leibler (KL) divergence between the true conditional density p t and the conditional model densityp t : KL divergence is an important and widely applied measure of statistical divergence in various fields; see, for example, Ullah (1996Ullah ( , 2002. The results we derive extend the results of Blasques et al (2015) and Creal et al (2018) to the context of autoregressive models with time-varying dependence parameters.…”

“…Here we adopt the time-varying parameter representation in (2) to find a nonlinear specification for the nonlinear AR model in (1) that possesses particular optimality properties. We do so by studying how to select the function f ðy tÀ1 Þ: Specifically, we extend the results in Blasques et al (2015) and Creal et al (2018) to dynamic autoregressive models. This allows us to find a parametric functional form for w that at each time point t is guaranteed to improve the local Kullback-Leibler divergence between the true unknown conditional density of y t and the conditional density implied by the fitted parametric model.…”

Closed-Form Multi-Factor Copula Models With Observation-Driven Dynamic Factor Loadings

“…Intuitively, this adjust the loadings in a steepest ascent direction using the local log-likelihood fit at time t as the criterion function. The approach has information theoretic optimality properties as argued in Blasques, Koopman, and Lucas (2015) and its generalizations in Creal et al (2018). As an example in our context, consider a 1-Factor equicorrelation copula, such thatL t =λ t ι N for a scalar parameterλ t = λ t / 1 + λ 2 t and f t = λ t , such thatλ t ∈ [−1, 1] by design, where ι N denotes an N × 1 vector filled with ones.…”