Inference on stochastic time-varying coefficient models

Recently, there has been considerable work on stochastic time-varying coefficient models as vehicles for modelling structural change in the macroeconomy with a focus on the estimation of the unobserved paths of random coefficient processes. The dominant estimation methods, in this context, are based on various filters, such as the Kalman filter, that are applicable when the models are cast in state space representations. This paper introduces a new class of autoregressive bounded processes that decompose a time series into a persistent random attractor, a time varying autoregressive component, and martingale difference errors. The paper examines, rigorously, alternative kernel based, nonparametric estimation approaches for such models and derives their basic properties. These estimators have long been studied in the context of deterministic structural change, but their use in the presence of stochastic time variation is novel. The proposed inference methods have desirable properties such as consistency and asymptotic normality and allow a tractable studentization. In extensive Monte Carlo and empirical studies, we find that the methods exhibit very good small sample properties and can shed light on important empirical issues such as the evolution of inflation persistence and the purchasing power parity (PPP) hypothesis.     

Online adaptive lasso estimation in vector autoregressive models for high dimensional wind power forecasting

Wind power forecasts with lead times of up to a few hours are essential to the optimal and economical operation of power systems and markets. Vector autoregression (VAR) is a framework that has been shown to be well suited to predicting for several wind farms simultaneously by considering the spatio-temporal dependencies in their time series. Lasso penalisation yields sparse models and can avoid overfitting the large numbers of coefficients in higher dimensional settings. However, estimation in VAR models usually does not account for changes in the spatio-temporal wind power dynamics that are related to factors such as seasons or wind farm setup changes, for example. This paper tackles this problem by proposing a time-adaptive lasso estimator and an efficient coordinate descent algorithm for updating the VAR model parameters recursively online. The approach shows good abilities to track changes in the multivariate time series dynamics on simulated data. Furthermore, in two case studies it shows clearly better predictive performances than the non-adaptive lasso VAR and univariate autoregression.     

Space-varying Coefficient Simultaneous Autoregressive Models for the Structural Analysis of Residential Water Demand

Abstract

This study develops two space-varying coefficient simultaneous autoregressive (SVC-SAR) models for areal data and applies them to the discrete/continuous choice model, which is an econometric model based on the consumer's utility maximization problem. The space-varying coefficient model is a statistical model in which the coefficients vary depending on their location. This study introduces the simultaneous autoregressive model for the underlying spatial dependence across coefficients, where the coefficients for one observation are affected by the sum of those for the other observations. This model is named the SVC-SAR model. Because of its flexibility, we use the Bayesian approach and construct its estimation method based on the Markov chain Monte Carlo simulation. The proposed models are applied to estimate the Japanese residential water demand function, which is an example of the discrete/continuous choice model.     

Dynamic stochastic copula models: estimation,inference and applications

We propose a new dynamic copula model in which the parameter characterizing dependence follows an autoregressive process. As this model class includes the Gaussian copula with stochastic correlation process, it can be viewed as a generalization of multivariate stochastic volatility models. Despite the complexity of the model, the decoupling of marginals and dependence parameters facilitates estimation. We propose estimation in two steps, where first the parameters of the marginal distributions are estimated, and then those of the copula. Parameters of the latent processes (volatilities and dependence) are estimated using efficient importance sampling. We discuss goodness‐of‐fit tests and ways to forecast the dependence parameter. For two bivariate stock index series, we show that the proposed model outperforms standard competing models. Copyright © 2010 John Wiley & Sons, Ltd.     

The method of elimination and substitution in the GMM estimation of mixed regressive,spatial autoregressive models

This paper proposes a computationally simple GMM for the estimation of mixed regressive spatial autoregressive models. The proposed method explores the advantage of the method of elimination and substitution in linear algebra. The modified GMM approach reduces the joint (nonlinear) estimation of a complete vector of parameters into estimation of separate components. For the mixed regressive spatial autoregressive model, the nonlinear estimation is reduced to the estimation of the (single) spatial effect parameter. We identify situations under which the resulting estimator can be efficient relative to the joint GMM estimator where all the parameters are jointly estimated.     

Testing for Parameter Stability in Dynamic Models across Frequencies*

This paper contributes to the econometric literature on structural breaks by proposing a test for parameter stability in vector autoregressive (VAR) models at a particular frequency ω, where ω ∈ [0, π]. When a dynamic model is affected by a structural break, the new tests allow for detecting which frequencies of the data are responsible for parameter instability. If the model is locally stable at the frequencies of interest, the whole sample size can then be exploited despite the presence of a break. The methodology is applied to analyse the productivity slowdown in the US, and the outcome is that local stability concerns only the higher frequencies of data on consumption, investment and output.     

Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

Combination schemes for turning point predictions

We propose new forecast combination schemes for predicting turning points of business cycles. The proposed combination schemes are based on the forecasting performances of a given set of models with the aim to provide better turning point predictions. In particular, we consider predictions generated by autoregressive (AR) and Markov-switching AR models, which are commonly used for business cycle analysis. In order to account for parameter uncertainty we consider a Bayesian approach for both estimation and prediction and compare, in terms of statistical accuracy, the individual models and the combined turning point predictions for the United States and the Euro area business cycles.     

A note on Model Reference Adaptive System (MRAS) estimate with infinite variance

This paper discusses the estimation of a parameter in an autoregressive model with infinite variance. A recursive estimation procedure based on minimizing the prediction errors is provided. It is also shown that the model reference adaptive system estimate for an AR (1) model given in Aase (1983) is a special case.     

Probabilistic forecasting of industrial electricity load with regime switching behavior

This paper suggests a novel inhomogeneous Markov switching approach for the probabilistic forecasting of industrial companies’ electricity loads, for which the load switches at random times between production and standby regimes. The model that we propose describes the transitions between the regimes using a hidden Markov chain with time-varying transition probabilities that depend on calendar variables. We model the demand during the production regime using an autoregressive moving-average (ARMA) process with seasonal patterns, whereas we use a much simpler model for the standby regime in order to reduce the complexity. The maximum likelihood estimation of the parameters is implemented using a differential evolution algorithm. Using the continuous ranked probability score (CRPS) to evaluate the goodness-of-fit of our model for probabilistic forecasting, it is shown that this model often outperforms classical additive time series models, as well as homogeneous Markov switching models. We also propose a simple procedure for classifying load profiles into those with and without regime-switching behaviors.     

Averaging estimators for autoregressions with a near unit root

This paper uses local-to-unity theory to evaluate the asymptotic mean-squared error (AMSE) and forecast expected squared error from least-squares estimation of an autoregressive model with a root close to unity. We investigate unconstrained estimation, estimation imposing the unit root constraint, pre-test estimation, model selection estimation, and model average estimation. We find that the asymptotic risk depends only on the local-to-unity parameter, facilitating simple graphical comparisons. Our results strongly caution against pre-testing. Strong evidence supports averaging based on Mallows weights. In particular, our Mallows averaging method has uniformly and substantially smaller risk than the conventional unconstrained estimator, and this holds for autoregressive roots far from unity. Our averaging estimator is a new approach to forecast combination.     

Forecasting risk measures using intraday data in a generalized autoregressive score framework

A new framework for the joint estimation and forecasting of dynamic value at risk (VaR) and expected shortfall (ES) is proposed by our incorporating intraday information into a generalized autoregressive score (GAS) model introduced by Patton et al., 2019 to estimate risk measures in a quantile regression set-up. We consider four intraday measures: the realized volatility at 5-min and 10-min sampling frequencies, and the overnight return incorporated into these two realized volatilities. In a forecasting study, the set of newly proposed semiparametric models are applied to four international stock market indices (S&P 500, Dow Jones Industrial Average, Nikkei 225 and FTSE 100) and are compared with a range of parametric, nonparametric and semiparametric models, including historical simulations, generalized autoregressive conditional heteroscedasticity (GARCH) models and the original GAS models. VaR and ES forecasts are backtested individually, and the joint loss function is used for comparisons. Our results show that GAS models, enhanced with the realized volatility measures, outperform the benchmark models consistently across all indices and various probability levels.     

Bayesian median autoregression for robust time series forecasting

We develop a Bayesian median autoregressive (BayesMAR) model for time series forecasting. The proposed method utilizes time-varying quantile regression at the median, favorably inheriting the robustness of median regression in contrast to the widely used mean-based methods. Motivated by a working Laplace likelihood approach in Bayesian quantile regression, BayesMAR adopts a parametric model bearing the same structure as autoregressive models by altering the Gaussian error to Laplace, leading to a simple, robust, and interpretable modeling strategy for time series forecasting. We estimate model parameters by Markov chain Monte Carlo. Bayesian model averaging is used to account for model uncertainty, including the uncertainty in the autoregressive order, in addition to a Bayesian model selection approach. The proposed methods are illustrated using simulations and real data applications. An application to U.S. macroeconomic data forecasting shows that BayesMAR leads to favorable and often superior predictive performance compared to the selected mean-based alternatives under various loss functions that encompass both point and probabilistic forecasts. The proposed methods are generic and can be used to complement a rich class of methods that build on autoregressive models.     

GENERALIZED AUTOREGRESSIVE SCORE MODELS WITH APPLICATIONS

We propose a class of observation‐driven time series models referred to as generalized autoregressive score (GAS) models. The mechanism to update the parameters over time is the scaled score of the likelihood function. This new approach provides a unified and consistent framework for introducing time‐varying parameters in a wide class of nonlinear models. The GAS model encompasses other well‐known models such as the generalized autoregressive conditional heteroskedasticity, autoregressive conditional duration, autoregressive conditional intensity, and Poisson count models with time‐varying mean. In addition, our approach can lead to new formulations of observation‐driven models. We illustrate our framework by introducing new model specifications for time‐varying copula functions and for multivariate point processes with time‐varying parameters. We study the models in detail and provide simulation and empirical evidence. Copyright © 2012 John Wiley & Sons, Ltd.     

Sequential numerical integration in nonlinear state space models for microeconometric panel data

This paper discusses the estimation of a class of nonlinear state space models including nonlinear panel data models with autoregressive error components. A health economics example illustrates the usefulness of such models. For the approximation of the likelihood function, nonlinear filtering algorithms developed in the time‐series literature are considered. Because of the relatively simple structure of these models, a straightforward algorithm based on sequential Gaussian quadrature is suggested. It performs very well both in the empirical application and a Monte Carlo study for ordered logit and binary probit models with an AR(1) error component. Copyright © 2008 John Wiley & Sons, Ltd.     

Realized volatility forecast with the Bayesian random compressed multivariate HAR model

We develop a Bayesian random compressed multivariate heterogeneous autoregressive (BRC-MHAR) model to forecast the realized covariance matrices of stock returns. The proposed model randomly compresses the predictors and reduces the number of parameters. We also construct several competing multivariate volatility models with the alternative shrinkage methods to compress the parameter’s dimensions. We compare the forecast performances of the proposed models with the competing models based on both statistical and economic evaluations. The results of statistical evaluation suggest that the BRC-MHAR models have the better forecast precision than the competing models for the short-term horizon. The results of economic evaluation suggest that the BRC-MHAR models are superior to the competing models in terms of the average return, the Shape ratio and the economic value.     

Dynamic spatial autoregressive models with autoregressive and heteroskedastic disturbances

We propose a new class of models specifically tailored for spatiotemporal data analysis. To this end, we generalize the spatial autoregressive model with autoregressive and heteroskedastic disturbances, that is, SARAR(1, 1), by exploiting the recent advancements in score‐driven (SD) models typically used in time series econometrics. In particular, we allow for time‐varying spatial autoregressive coefficients as well as time‐varying regressor coefficients and cross‐sectional standard deviations. We report an extensive Monte Carlo simulation study in order to investigate the finite‐sample properties of the maximum likelihood estimator for the new class of models as well as its flexibility in explaining a misspecified dynamic spatial dependence process. The new proposed class of models is found to be economically preferred by rational investors through an application to portfolio optimization.     

Forecasting volatility with component conditional autoregressive range model

In this paper, we propose a component conditional autoregressive range (CCARR) model for forecasting volatility. The proposed CCARR model assumes that the price range comprises both a long-run (trend) component and a short-run (transitory) component, which has the capacity to capture the long memory property of volatility. The model is intuitive and convenient to implement by using the maximum likelihood estimation method. Empirical analysis using six stock market indices highlights the value of incorporating a second component into range (volatility) modelling and forecasting. In particular, we find that the proposed CCARR model fits the data better than the CARR model, and that it generates more accurate out-of-sample volatility forecasts and contains more information content about the true volatility than the popular GARCH, component GARCH and CARR models.     

Nonparametric expected shortfall forecasting incorporating weighted quantiles

A new semi-parametric expected shortfall (ES) estimation and forecasting framework is proposed. The proposed approach is based on a two-step estimation procedure. The first step involves the estimation of value at risk (VaR) at different quantile levels through a set of quantile time series regressions. Then, the ES is computed as a weighted average of the estimated quantiles. The quantile weighting structure is parsimoniously parameterized by means of a beta weight function whose coefficients are optimized by minimizing a joint VaR and ES loss function of the Fissler–Ziegel class. The properties of the proposed approach are first evaluated with an extensive simulation study using two data generating processes. Two forecasting studies with different out-of-sample sizes are then conducted, one of which focuses on the 2008 Global Financial Crisis period. The proposed models are applied to seven stock market indices, and their forecasting performances are compared to those of a range of parametric, non-parametric, and semi-parametric models, including GARCH, conditional autoregressive expectile (CARE), joint VaR and ES quantile regression models, and a simple average of quantiles. The results of the forecasting experiments provide clear evidence in support of the proposed models.     

Covariance functions that are stationary or nonstationary in space and stationary in time

There is a great demand for statistical modelling of phenomena that evolve in both space and time, and thus, there is a growing literature on covariance function models for spatio-temporal processes. Although several nonseparable space–time covariance models are available in the literature, very few of them can be used for spatially anisotropic data. In this paper, we propose a new class of stationary nonseparable covariance functions that can be used for both geometrically and zonally anistropic data. In addition, we show some desirable mathematical features of this class. Another important aspect, only partially covered by the literature, is that of spatial nonstationarity. We show a very simple criteria allowing for the construction of space–time covariance functions that are nonseparable, nonstationary in space and stationary in time. Part of the theoretical results proposed in the paper will then be used for the analysis of Irish wind speed data as in HASLETT and RAFTERY ( Applied Statistics , 38 , 1989, 1).