Francisco Blasques

Stationarity and Ergodicity of Univariate Generalized Autoregressive Score Processes

This discussion paper led to a publication in the Electronic Journal of Statistics , 2014, 8, 1088-1112. We characterize the dynamic properties of Generalized Autoregressive Score (GAS) processes by identifying regions of the parameter space that imply stationarity and ergodicity. We show how these regions are affected by the choice of parameterization and scaling, which are key features of GAS models compared to other observation driven models. The Dudley entropy integral is used to ensure the non-degeneracy of such regions. Furthermore, we show how to obtain bounds for these regions in models for time-varying means, variances, or higher-order moments.

Maximum Likelihood Estimation for Generalized Autoregressive Score Models

We establish the strong consistency and asymptotic normality of the maximum likelihood estimator for time-varying parameter models driven by the score of the predictive likelihood function. We formulate primitive conditions for global identification, invertibility, strong consistency, and asymptotic normality under both correct specification and mis-specification of the model. A detailed illustration is provided for a conditional volatility model with disturbances from the Student’s t distribution. (Inclusive Supplemental Appendix)

Time Varying Transition Probabilities for Markov Regime Switching Models

We propose a new Markov switching model with time varying probabilities for the transitions. The novelty of our model is that the transition probabilities evolve over time by means of an observation driven model. The innovation of the time varying probability is generated by the score of the predictive likelihood function. We show how the model dynamics can be readily interpreted. We investigate the performance of the model in a Monte Carlo study and show that the model is successful in estimating a range of different dynamic patterns for unobserved regime switching probabilities. We also illustrate the new methodology in an empirical setting by studying the dynamic mean and variance behavior of U.S. Industrial Production growth. We find empirical evidence of changes in the regime switching probabilities, with more persistence for high volatility regimes in the earlier part of the sample, and more persistence for low volatility regimes in the later part of the sample.

Semiparametric score driven volatility models

A new semiparametric observation-driven volatility model is proposed. In contrast to the standard semiparametric generalized autoregressive conditional heteroskedasticity (GARCH) model, the form of the error density has a direct influence on both the semiparametric likelihood and the volatility dynamics. The estimator is shown to consistently estimate the conditional pseudo true parameters of the model. Simulation-based evidence and an empirical application to stock return data confirm that the new statistical model realizes substantial improvements compared to GARCH type models and quasi-maximum likelihood estimation if errors are fat-tailed and possibly skewed.

A Dynamic Stochastic Network Model of the Unsecured Interbank Lending Market

This paper introduces a structural micro-founded dynamic stochastic network model for the unsecured interbank lending market. Banks are profit optimizing agents subject to random liquidity shocks and can engage in costly counterparty search to find suitable trading partners and peer monitoring to reduce counterparty risk uncertainty. The structural parameters are estimated by indirect inference using appropriate network statistics of the Dutch interbank market. The estimated model is shown to explain accurately important dynamic features of the interbank market network. In particular, monitoring of counterparty risk and directed search are shown to be key factors in the formation of interbank trading relationships that are associated with improved credit conditions. Finally, the model is used to filter the optimal search and monitoring expenditures in the network and to analyze optimal network responses to changes in the policy of the central bank.

Stationarity and Ergodicity Regions for Score Driven Dynamic Correlation Models

We describe stationarity and ergodicity (SE) regions for a recently proposed class of score driven dynamic correlation models. These models have important applications in empirical work. The regions are derived from sufficiency conditions in Bougerol (1993) and take a non-standard form. We show that the non-standard shape of the sufficiency regions cannot be avoided by reparameterizing the model or by rescaling the score steps in the transition equation for the correlation parameter. This makes the result markedly different from the volatility case. Observationally equivalent decompositions of the stochastic recurrence equation yield regions with different sizes and shapes. We illustrate our results with an analysis of time-varying correlations between UK and Greek equity indices. We find that also in empirical applications different decompositions can give rise to different conclusions regarding the stability of the estimated model.

Low Frequency and Weighted Likelihood Solutions for Mixed Frequency Dynamic Factor Models

The multivariate analysis of a panel of economic and financial time series with mixed frequencies is a challenging problem. The standard solution is to analyze the mix of monthly and quarterly time series jointly by means of a multivariate dynamic model with a monthly time index: artificial missing values are inserted for the intermediate months of the quarterly time series. In this paper we explore an alternative solution for a class of dynamic factor models that is specified by means of a low frequency quarterly time index. We show that there is no need to introduce artificial missing values while the high frequency (monthly) information is preserved and can still be analyzed. We also provide evidence that the analysis based on a low frequency specification can be carried out in a computationally more efficient way. A comparison study with existing mixed frequency procedures is presented and discussed. Furthermore, we modify the method of maximum likelihood in the context of a dynamic factor model. We introduce variable-specific weights in the likelihood function to let some variable equations be of more importance during the estimation process. We derive the asymptotic properties of the weighted maximum likelihood estimator and we show that the estimator is consistent and asymptotically normal. We also verify the weighted estimation method in a Monte Carlo study to investigate the effect of differen t choices for the weights in different scenarios. Finally, we empirically illustrate the new developments for the extraction of a coincident economic indicator from a small panel of mixed frequency economic time series.

A Stochastic Recurrence Equations Approach for Score Driven Correlation Models

ABSTRACT We describe stationarity and ergodicity (SE) regions for a recently proposed class of score driven dynamic correlation models. These models have important applications in empirical work. The regions are derived from sufficiency conditions in Bougerol (1993) and take a nonstandard form. We show that the nonstandard shape of the sufficiency regions cannot be avoided by reparameterizing the model or by rescaling the score steps in the transition equation for the correlation parameter. This makes the result markedly different from the volatility case. Observationally equivalent decompositions of the stochastic recurrence equation yield regions with different shapes and sizes. We use these results to establish the consistency and asymptotic normality of the maximum likelihood estimator. We illustrate our results with an analysis of time-varying correlations between U.K. and Greek equity indices. We find that also in empirical applications different decompositions can give rise to different conclusions regarding the stability of the estimated model.

Accelerating GARCH and Score-Driven Models: Optimality, Estimation and Forecasting

We first consider an extension of the generalized autoregressive conditional heteroskedasticity (GARCH) model that allows for a more flexible weighting of financial squared-returns for the filtering of volatility. The parameter for the squared-return in the GARCH model is time-varying with an updating function similar to GARCH but with the squared-return replaced by the product of the volatility innovation and its lagged value. This local estimate of the first order autocorrelation of volatility innovations acts as an indicator of the importance of the squared-return for volatility updating. When recent volatility innovations have the same sign (positive autocorrelation), the current volatility estimate needs to adjust more quickly than in a period where recent volatility innovations have mixed signs (negative autocorrelation). The empirical relevance of the accelerated GARCH updating is illustrated by forecasting daily volatility in return series of all individual stocks present in the Standard & Poor’s 500 index. Major improvements are reported for those stock return series that exhibit high kurtosis. The local adjustment in weighting new observational information is generalised to score-driven time-varying parameter models of which GARCH is a special case. It is within this general framework that we provide the theoretical foundations of accelerated updating. We show that acceleration in updating is more optimal in terms of reducing Kullback-Leibler divergence and in comparison to fixed updating. The robustness of our proposed extension is highlighted in a simulation study within a misspecified modelling framework. The score-driven acceleration is also empirically illustrated with the forecasting of US inflation using a model with time-varying mean and variance; we report significant improvements in the forecasting accuracy at a yearly horizon.

Smooth Transition Spatial Autoregressive Models

This paper introduces a new model for spatial time series in which cross-sectional dependence varies nonlinearly over space by means of smooth transitions. We refer to our model as the Smooth Transition Spatial Autoregressive (ST-SAR). We establish consistency and asymptotic Gaussianity for the MLE under misspecification and provide additional conditions for geometric ergodicity of the model. Simulation results justify the use of limit theory in empirically relevant settings. The model is applied to study spatio-temporal dynamics in two cases that differ in spatial and temporal extent. We study clustering in urban densities in a large number of neighborhoods in the Netherlands over a 10-year period. We pay particular focus to the advantages of the ST-SAR as an alternative to linear spatial models. In our second study, we apply the ST-SAR to monthly long term interest rates of 15 European sovereigns over 25-year period. We develop a strategy to assess financial stability across the Eurozone based on attraction of individual sovereigns toward the common stochastic trend. Our estimates reveal that stability attained a low during the Greek sovereign debt crisis, and that the Eurozone has remained to struggle in attaining stability since the onset of the financial crisis. The results suggest that the European Monetary System has not fully succeeded in aligning the economies of Ireland, Portugal, Italy, Spain, and Greece with the rest of the Eurozone, while attraction between other sovereigns has continued to increase. In our applications linearity of spatial dependence is overwhelmingly rejected in terms of model fit and forecast accuracy, estimates of control variables improve, and residual correlation is better neutralized.