We investigate a model in which we connect slowly time varying unconditional long-run volatility with short-run conditional volatility whose representation is given as a semi-strong GARCH(1,1) process with heavy tailed errors. We focus on robust estimation of both long-run and short-run volatilities. Our estimation is semiparametric since the long-run volatility is totally unspecified whereas the short-run conditional volatility is a parametric semi-strong GARCH(1,1) process. We propose different robust estimation methods for nonstationary and strictly stationary GARCH parameters with nonparametric long-run volatility function. Our estimation is based on a two-step LAD procedure. We establish the relevant asymptotic theory of the proposed estimators. Numerical results lend support to our theoretical results.

Financial time series is often assumed to be stationary and has a normal distribution in the literature. Both assumptions are however unrealistic. This paper proposes a new methodology with a focus on volatility estimation that is able to account for nonstationarity and heavy tails simultaneously. In particular, a local exponential smoothing (LES) approach is developed, in which weak estimates with different memory parameters are aggregated in a locally adaptive way. The procedure is fully automatic and the parameters are tuned by a new propagation approach. The extensive and practically oriented numerical results confirm the desired properties of the constructed estimate: it performs stable in a nearly time homogeneous situation and is sensitive to structural shifts. Our main theoretical “oracle” result claims that the aggregated estimate performs as good as the best estimate in the considered family. The results are stated under realistic and unrestrictive assumptions on the model.

We propose a statistical model to assess whether individuals strategically use mixed strategies in repeated games. We formulate a hidden Markov model in which the latent state space contains both pure and mixed strategies. We apply the model to data from an experiment in which human subjects repeatedly play a normal form game against a computer that always follows its part of the unique mixed strategy Nash equilibrium profile. Estimated results show significant mixed strategy play and nonstationary dynamics. We also explore the ability of the model to forecast action choice.

This paper considers estimation and inference for varying-coefficient models with nonstationary regressors. We propose a nonparametric estimation method using penalized splines, which achieves the same optimal convergence rate as kernel-based methods, but enjoys computation advantages. Utilizing the mixed model representation of penalized splines, we develop a likelihood ratio test statistic for checking the stability of the regression coefficients. We derive both the exact and the asymptotic null distributions of this test statistic. We also demonstrate its optimality by examining its local power performance. These theoretical findings are well supported by simulation studies.

This paper studies the robust estimation and inference of threshold models with integrated regressors. We derive the asymptotic distribution of the profiled least squares (LS) estimator under the diminishing threshold effect assumption that the size of the threshold effect converges to zero. Depending on how rapidly this sequence converges, the model may be identified or only weakly identified and asymptotic theorems are developed for both cases. As the convergence rate is unknown in practice, a model-selection procedure is applied to determine the model identification strength and to construct robust confidence intervals, which have the correct asymptotic size irrespective of the magnitude of the threshold effect. The model is then generalized to incorporate endogeneity and serial correlation in error terms, under which, we design a Cochrane–Orcutt feasible generalized least squares (FGLS) estimator which enjoys efficiency gains and robustness against different error specifications, including both I(0) and I(1) errors. Based on this FGLS estimator, we further develop a sup-Wald statistic to test for the existence of the threshold effect. Monte Carlo simulations show that our estimators and test statistics perform well.

In this paper, we consider a nonparametric model with a time-varying regression function and locally stationary regressors. We are interested in the question whether the regression function has the same shape over a given time span. To tackle this testing problem, we propose a kernel-based L2-test statistic. We derive the asymptotic distribution of the statistic both under the null and under fixed and local alternatives. To improve the small sample behavior of the test, we set up a wild bootstrap procedure and derive the asymptotic properties thereof. The theoretical analysis of the paper is complemented by a simulation study and a real data example.

This paper is an extension of Magnus (2002, Econometrics Journal 5, 225–236) to multiple dimensions. We consider estimation of a multivariate normal mean under sum of squared error loss. We construct the efficiency bound (the lowest achievable risk) for minimax shrinkage estimation in the class of minimax orthogonally invariate estimators satisfying the sufficient conditions of Efron and Morris (1976, Annals of Statistics 4, 11–21). This allows us to compare the regret of existing orthogonally invariate shrinkage estimators. We also construct a new shrinkage estimator which achieves substantially lower maximum regret than existing estimators.

It is well known that the least squares estimator (LSE) of an AR(p) model with i.i.d. (independent and identically distributed) noises is n1/αL(n)-consistent when the tail index α of the noise is within (0,2) and is n1/2-consistent when α ≥ 2, where L(n) is a slowly varying function. When the noises are not i.i.d., however, the case is far from clear. This paper studies the LSE of AR(p) models with heavy-tailed G-GARCH(1,1) noises. When the tail index α of G-GARCH is within (0,2), it is shown that the LSE is not a consistent estimator of the parameters, but converges to a ratio of stable vectors. When α ε [2,4], it is shown that the LSE is n1–2/α-consistent if α ε (2,4), logn-consistent if α = 2, and n1/2 / logn-consistent if α = 4, and its limiting distribution is a functional of stable processes. Our results are significantly different from those with i.i.d. noises and should warn practitioners in economics and finance of the implications, including inconsistency, of heavy-tailed errors in the presence of conditional heterogeneity.

Despite their theoretical advantages, Integrated Conditional Moment (ICM) specification tests are not commonly employed in the econometrics practice. An important reason is that the employed test statistics are nonpivotal, and so critical values are not readily available. This article proposes an omnibus test in the spirit of the ICM tests of Bierens and Ploberger (1997, Econometrica 65, 1129–1151) where the test statistic is based on the minimized value of a quadratic function of the residuals of time series econometric models. The proposed test falls under the category of overidentification restriction tests started by Sargan (1958, Econometrica 26, 393–415). The corresponding projection interpretation leads us to propose a straightforward wild bootstrap procedure that requires only linear regressions to estimate the critical values irrespective of the model functional form. Hence, contrary to other existing ICM tests, the critical values are easily calculated while the test preserves the admissibility property of ICM tests.