This paper focuses on recursive estimation of time varying autoregressive processes in a nonparametric setting. The stability of the model is revisited and uniform results are provided when the time-varying autoregressive parameters belong to appropriate smoothness classes. An adequate normalization for the correction term used in the recursive estimation procedure allows for very mild assumptions on the innovations distributions. The rate of convergence of the pointwise estimates is shown to be minimax in $β$-Lipschitz classes for $0<β\leq1$. For $1<β\leq 2$, this property no longer holds. This can be seen by using an asymptotic expansion of the estimation error. A bias reduction method is then proposed for recovering the minimax rate.
In this paper we propose a recursive online algorithm for estimating the parameters of a time-varying ARCH process. The estimation is done by updating the estimator at time point $t-1$ with observations about the time point $t$ to yield an estimator of the parameter at time point $t$. The sampling properties of this estimator are studied in a non-stationary context -- in particular, asymptotic normality and an expression for the bias due to non-stationarity are established. By running two recursive online algorithms in parallel with different step sizes and taking a linear combination of the e…
This paper considers the quantile regression approach for partially linear spatial autoregressive models with possibly varying coefficients. B-spline is employed for the approximation of varying coefficients. The instrumental variable quantile regression approach is employed for parameter estimation. The rank score tests are developed for hypotheses on the coefficients, including the hypotheses on the non-varying coefficients and the constancy of the varying coefficients. The asymptotic properties of the proposed estimators and test statistics are both established. Monte Carlo simulations are …
In this work, we study the problem of aggregating a finite number of predictors for nonstationary sub-linear processes. We provide oracle inequalities relying essentially on three ingredients: (1) a uniform bound of the $\ell^1$ norm of the time varying sub-linear coefficients, (2) a Lipschitz assumption on the predictors and (3) moment conditions on the noise appearing in the linear representation. Two kinds of aggregations are considered giving rise to different moment conditions on the noise and more or less sharp oracle inequalities. We apply this approach for deriving an adaptive predicto…
We study some regularity properties in locally stationary Markov models which are fundamental for controlling the bias of nonparametric kernel estimators. In particular, we provide an alternative to the standard notion of derivative process developed in the literature and that can be used for studying a wide class of Markov processes. To this end, for some families of V-geometrically ergodic Markov kernels indexed by a real parameter u, we give conditions under which the invariant probability distribution is differentiable with respect to u, in the sense of signed measures. Our results also co…
The paper proposes a time-varying parameter global vector autoregressive (TVP-GVAR) framework for predicting and analysing developed region economic variables. We want to provide an easily accessible approach for the economy application settings, where a variety of machine learning models can be incorporated for out-of-sample prediction. The LASSO-type technique for numerically efficient model selection of mean squared errors (MSEs) is selected. We show the convincing in-sample performance of our proposed model in all economic variables and relatively high precision out-of-sample predictions w…
We develop a new statistical model to analyse time-varying ranking data. The model can be used with a large number of ranked items, accommodates exogenous time-varying covariates and partial rankings, and is estimated via the maximum likelihood in a straightforward manner. Rankings are modelled using the Plackett-Luce distribution with time-varying worth parameters that follow a mean-reverting time series process. To capture the dependence of the worth parameters on past rankings, we utilise the conditional score in the fashion of the generalised autoregressive score (GAS) models. Simulation e…
In this paper, we study a notion of local stationarity for discrete time Markov chains which is useful for applications in statistics. In the spirit of some locally stationary processes introduced in the literature, we consider triangular arrays of time-inhomogeneous Markov chains, defined by some families of contracting Markov kernels. Using the Dobrushin's contraction coefficients for various metrics, we show that the distribution of such Markov chains can be approximated locally with the distribution of ergodic Markov chains and we also study some mixing properties. From our approximation r…
We consider parameter estimation, hypothesis testing and variable selection for partially time-varying coefficient models. Our asymptotic theory has the useful feature that it can allow dependent, nonstationary error and covariate processes. With a two-stage method, the parametric component can be estimated with a $n^{1/2}$-convergence rate. A simulation-assisted hypothesis testing procedure is proposed for testing significance and parameter constancy. We further propose an information criterion that can consistently select the true set of significant predictors. Our method is applied to autor…
This paper considers a semiparametric generalized autoregressive conditional heteroskedasticity (S-GARCH) model. For this model, we first estimate the time-varying long run component for unconditional variance by the kernel estimator, and then estimate the non-time-varying parameters in GARCH-type short run component by the quasi maximum likelihood estimator (QMLE). We show that the QMLE is asymptotically normal with the parametric convergence rate. Next, we construct a Lagrange multiplier test for linear parameter constraint and a portmanteau test for model checking, and obtain their asymptot…
Department of Economics, University of Bologna（博洛尼亚大学经济系）；Department of Economics, University of Exeter（埃克塞特大学经济系）；Department of Economics, University of Copenhagen（哥本哈根大学经济系）
We consider a time-varying first-order autoregressive model with irregular innovations, where we assume that the coefficient function is Hölder continuous. To estimate this function, we use a quasi-maximum likelihood based approach. A precise control of this method demands a delicate analysis of extremes of certain weakly dependent processes, our main result being a concentration inequality for such quantities. Based on our analysis, upper and matching minimax lower bounds are derived, showing the optimality of our estimators. Unlike the regular case, the information theoretic complexity depen…
In this paper we introduce the Kumaraswamy autoregressive moving average models (KARMA), which is a dynamic class of models for time series taking values in the double bounded interval $(a,b)$ following the Kumaraswamy distribution. The Kumaraswamy family of distribution is widely applied in many areas, especially hydrology and related fields. Classical examples are time series representing rates and proportions observed over time. In the proposed KARMA model, the median is modeled by a dynamic structure containing autoregressive and moving average terms, time-varying regressors, unknown param…
Quasi-Maximum Likelihood (QML) procedures are theoretically appealing and widely used for statistical inference. While there are extensive references on QML estimation in batch settings, it has attracted little attention in streaming settings until recently. An investigation of the convergence properties of the QML procedure in a general conditionally heteroscedastic time series model is conducted, and the classical batch optimization routines extended to the framework of streaming and large-scale problems. An adaptive recursive estimation routine for GARCH models named AdaVol is presented. Th…
The identification of the lag length for vector autoregressive models by mean of Akaike Information Criterion (AIC), Partial Autoregressive and Correlation Matrices (PAM and PCM hereafter) is studied in the framework of processes with time varying variance. It is highlighted that the use of the standard tools are not justified in such a case. As a consequence we propose an adaptive AIC which is robust to the presence of unconditional heteroscedasticity. Corrected confidence bounds are proposed for the usual PAM and PCM obtained from the Ordinary Least Squares (OLS) estimation. The volatility s…
In this paper we propose a new time-varying econometric model, called Time-Varying Poisson AutoRegressive with eXogenous covariates (TV-PARX), suited to model and forecast time series of counts. {We show that the score-driven framework is particularly suitable to recover the evolution of time-varying parameters and provides the required flexibility to model and forecast time series of counts characterized by convoluted nonlinear dynamics and structural breaks.} We study the asymptotic properties of the TV-PARX model and prove that, under mild conditions, maximum likelihood estimation (MLE) yie…
The problem of test of fit for Vector AutoRegressive (VAR) processes with unconditionally heteroscedastic errors is studied. The volatility structure is deterministic but time-varying and allows for changes that are commonly observed in economic or financial multivariate series. Our analysis is based on the residual autocovariances and autocorrelations obtained from Ordinary Least Squares (OLS), Generalized Least Squares (GLS)and Adaptive Least Squares (ALS) estimation of the autoregressive parameters. The ALS approach is the GLS approach adapted to the unknown time-varying volatility that is …
We consider the estimation of the transition matrix in the high-dimensional time-varying vector autoregression (TV-VAR) models. Our model builds on a general class of locally stationary VAR processes that evolve smoothly in time. We propose a hybridized kernel smoothing and $\ell^1$-regularized method to directly estimate the sequence of time-varying transition matrices. Under the sparsity assumption on the transition matrix, we establish the rate of convergence of the proposed estimator and show that the convergence rate depends on the smoothness of the locally stationary VAR processes only t…
The autoregressive (AR) models are used to represent the time-varying random process in which output depends linearly on previous terms and a stochastic term (the innovation). In the classical version, the AR models are based on normal distribution. However, this distribution does not allow describing data with outliers and asymmetric behavior. In this paper, we study the AR models with normal inverse Gaussian (NIG) innovations. The NIG distribution belongs to the class of semi heavy-tailed distributions with wide range of shapes and thus allows for describing real-life data with possible jump…
We consider the problem of tracking an unknown time varying parameter that characterizes the probabilistic evolution of a sequence of independent observations. To this aim, we propose a stochastic gradient descent-based recursive scheme in which the log-likelihood of the observations acts as time varying gain function. We prove convergence in mean-square error in a suitable neighbourhood of the unknown time varying parameter and illustrate the details of our findings in the case where data are generated from distributions belonging to the exponential family.
