Specification Testing in Nonlinear Time Series with Long-Range Dependence

This paper proposes a model specification testing procedure for parametric specification of the conditional mean function in a nonlinear time series model with long–range dependence. An asymptotically normal test is established even when long–range dependence is involved. In order to implement the proposed test in practice using a simulated example, a bootstrap simulation procedure is established to find a simulated critical value to compute both the size and power values of the proposed test.Asymptotic theory, Gaussian process, nonlinear time series, long-range dependence, parametric specification

Estimation and model specification testing in nonparametric and semiparametric econometric models

This paper considers two classes of semiparametric nonlinear regression models, in which nonlinear components are introduced to reflect the nonlinear fluctuation in the mean. A general estimation and testing procedure for nonparametric time series regression under the strong-mixing condition is introduced. Several test statistics for testing nonparametric significance, linearity and additivity in nonparametric and semi-parametric time series econometric models are then constructed. The proposed test statistics are shown to have asymptotic normal distributions under their respective null hypotheses. Moreover, the proposed testing procedures are illustrated by several simulated examples. In addition, one of the proposed testing procedures is applied to a continuous-time model and implemented through a set of the US Federal interest rate data. Our research suggests that it is unreasonable to assume the linearity in the drift for the given data as required by some existing studies.Estimation; model specification; semi-parametric error correction model; stochastic process

A New Test in Parametric Linear Models against Nonparametric Autoregressive Errors

This paper considers a class of parametric models with nonparametric autoregressive errors. A new test is proposed and studied to deal with the parametric specification of the nonparametric autoregressive errors with either stationarity or nonstationarity. Such a test procedure can initially avoid misspecification through the need to parametrically specify the form of the errors. In other words, we propose estimating the form of the errors and testing for stationarity or nonstationarity simultaneously. We establish asymptotic distributions of the proposed test. Both the setting and the results differ from earlier work on testing for unit roots in parametric time series regression. We provide both simulated and real-data examples to show that the proposed nonparametric unit-root test works in practice.Autoregressive process; nonlinear time series; nonparametric method; random walk; semiparametric model; unit root test.

Price dynamics and trading volume: A semiparametric approach

In this paper we investigate the relation between price impact and trading volume for a sample of stocks listed on the New York Stock Exchange. The parametric VAR-models that have been used in the literature impose strong proportionality and symmetry restrictions on the price impact of trades, although market microstructure theory provides many reasons why these restrictions would not hold. We analyze a more flexible semiparametric partially linear specification and establish significant evidence for a nonlinear, asymmetric, increasing, and concave relation between trading volume and both immediate and persistent price impact. Moreover, we compare the price-impact functions obtained in the partially linear model to the ones generated by the parametric models and show that there are considerable differences. We test the parametric specifications against the partially linear model and show that the parametric models are rejected in favor of the semiparametric model. We also test the partially linear model against a more flexible fully nonparametric specification and show that this test does not reject the partially linear model

Further empirical evidence of nonlinearity in the us monetary policy rule

Given conflicting results on whether the US monetary policy rule exhibited nonlinearity in the post-war period we employ a new Granger non-causality nonlinearity test and non-parametric procedures to re-examine the issue. Both procedures suggest that the Fed followed a nonlinear Taylor rule with respect to expected inflation and expected output gap prior to 1979 but not post 1982.Taylor rule, nonlinearity, Granger non-causality nonlinearity, non-parametric

Generalized spectral tests for the martingale difference hypothesis

^aThis article proposes a test for the Martingale Difference Hypothesis (MDH) using dependence measures related to the characteristic function. The MDH typically has been tested using the sample autocorrelations or in the spectral domain using the periodogram. Tests based on these statistics are inconsistent against uncorrelated non-martingales processes. Here, we generalize the spectral test of Durlauf (1991) for testing the MDH taking into account linear and nonlinear dependence. Our test considers dependence at all lags and is consistent against general pairwise nonparametric Pitman's local alternatives converging at the parametric rate n^(-1/2), with n the sample size. Furthermore, with our methodology there is no need to choose a lag order, to smooth the data or to formulate a parametric alternative. Our approach can be easily extended to specification testing of the conditional mean of possibly nonlinear models. The asymptotic null distribution of our test depends on the data generating process, so a bootstrap procedure is proposed and theoretically justified. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. A Monte Carlo study examines the finite sample performance of our test and shows that it is more powerful than some competing tests. Finally, an application to the S and P 500 stock index and exchange rates highlights the merits of our approach

Identification Techniques Applied to a Passive Elasto-magnetic Suspension

The paper presents an experimental passive elasto-magnetic suspension based on rare-earth permanent magnets, characterized by negligible dependence on mass of its natural frequency. The nonlinear behaviour of this system, equipped with a traditional linear elastic spring coupled to a magnetic spring, is analysed in time domain, for non-zero initial conditions, and in frequency domain, by applying sweep excitations to the test rig base. The dynamics of the system is very complex in dependence of the magnetic contribution, showing both hardening behaviour in the elasto-magnetic setup, and softening motion amplitude dependent behaviour in the purely magnetic case. Hence it is necessary to adopt nonlinear identification techniques, such as non-parametric restoring force mapping method and direct parametric estimation technique, in order to identify the system parameters in the different configurations. Finally, it is discussed the ability of identified versus analytical models in reproducing the nonlinear dependency of frequency on motion amplitude and the presence of jump phenomen

A loss function approach to model specification testing and its relative efficiency

The generalized likelihood ratio (GLR) test proposed by Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193] and Fan and Yao [Nonlinear Time Series: Nonparametric and Parametric Methods (2003) Springer] is a generally applicable nonparametric inference procedure. In this paper, we show that although it inherits many advantages of the parametric maximum likelihood ratio (LR) test, the GLR test does not have the optimal power property. We propose a generally applicable test based on loss functions, which measure discrepancies between the null and nonparametric alternative models and are more relevant to decision-making under uncertainty. The new test is asymptotically more powerful than the GLR test in terms of Pitman's efficiency criterion. This efficiency gain holds no matter what smoothing parameter and kernel function are used and even when the true likelihood function is available for the GLR test.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1099 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org