Dag Tjøstheim

Nonparametric Estimation and Identification of Nonlinear ARCH Time Series Strong Convergence and Asymptotic Normality: Strong Convergence and Asymptotic Normality

We consider the estimation and identification of the functional structures of nonlinear econometric systems of the ARCH type. We employ nonparametric kernel estimates for the nonlinear functions characterizing the systems, and we establish strong consistency along with sharp rates of convergence under mild regularity conditions. We also prove the asymptotic normality of the estimates.

Nonparametric Identification of Nonlinear Time Series: Projections

Abstract We study the possibility of identifying general linear and nonlinear time series models using nonparametric methods. The kernel estimators of the conditional mean and variance are used as a basis, and the properties of these quantities as model indicators are briefly discussed. Some drawbacks are pointed out, and motivated by these we introduce projections as tools of identification. The projections are especially useful for additive modeling. Expressions for the asymptotic bias and variance are obtained. The projection of the conditional variance is suggested as a tool for identifying heteroscedastic time series. The results are illustrated by simulations for both the estimators of the projections and the estimators of the conditional mean and variance.

Nonparametric estimation in a nonlinear cointegration type model

We derive an asymptotic theory of nonparametric estimation for an nonlinear transfer function model Z(t) = f (Xt) + Wt where {Xt} and {Zt} are observed nonstationary processes and {Wt} is a stationary process. IN econometrics this can be interpreted as a nonlinear cointegration type relationship, but we believe that our results have wider interest. The class of nonstationary processes allowed for {Xt} is a subclass of the class of null recurrent.. Markov chains. This subclass contains the random walk model and the unit root processes. WE derive the asymptotics of an nonparametric estimate of f(z) under two alternative sets of assumptions on {Wt}: i) {Wt} is a linear process ii) {Wt} is a Markov chain satisfying some mixing conditions. The latter requires considerably more work but also holds larger promise for further developments. The finite sample properties f(x) are studied via a set of simulation experiments.

The Dynamics of Coordinated Group Hunting and Collective Information Transfer among Schooling Prey

Predator-prey interactions are vital to the stability of many ecosystems. Yet, few studies have considered how they are mediated due to substantial challenges in quantifying behavior over appropriate temporal and spatial scales. Here, we employ high-resolution sonar imaging to track the motion and interactions among predatory fish and their schooling prey in a natural environment. In particular, we address the relationship between predator attack behavior and the capacity for prey to respond both directly and through collective propagation of changes in velocity by group members. To do so, we investigated a large number of attacks and estimated per capita risk during attack and its relation to the size, shape, and internal structure of prey groups. Predators were found to frequently form coordinated hunting groups, with up to five individuals attacking in line formation. Attacks were associated with increased fragmentation and irregularities in the spatial structure of prey groups, features that inhibit collective information transfer among prey. Prey group fragmentation, likely facilitated by predator line formation, increased (estimated) per capita risk of prey, provided prey schools were maintained below a threshold size of approximately 2 m(2). Our results highlight the importance of collective behavior to the strategies employed by both predators and prey under conditions of considerable informational constraints.

Nonparametric identification of nonlinear time series: selecting significant lags

Abstract In this article we suggest a nonparametric procedure for selecting significant lags in the model description of a general nonlinear stationary time series. The procedure can be applied to both the conditional mean and the conditional variance and is valid for heteroscedastic series. The procedure is illustrated by simulations and sunspot data, lynx data, and blowfly data are analyzed. It is indicated that projectors can be used in conjunction with the procedure for selecting significant lags to check the adequacy of an additive time series model.

Linearity testing using local polynomial approximation

We use local polynomial approximation to estimate the conditional mean and conditional variance, and test linearity by using a functional measuring the deviation between the nonparametric estimates and the parametric estimates based on a linear model. We also employ first-and second-order derivatives for this purpose, and we point out some advantages of using local polynomial approximation as opposed to kernel estimation in the context of linearity testing. The asymptotic theory of the test functionals is developed in some detail for a special case. It is used to draw qualitative conclusions concerning the bandwidth, but in order to apply the asymptotic distribution to specific testing problems very large sample sizes are needed. For moderate sample sizes we have examined a bootstrap alternative in a large variety of situations. We have tried bandwidths suggested by asymptotic results as well as bandwidths obtained by cross-validation.

Aspects of modelling nonlinear time series

This paper surveys some of the recent developments in nonlinear analysis of economic time series. The emphasis lies on stochastic models. Various classes of nonlinear models appearing in the economics and time series literature are presented and discussed. Linearity testing and estimation of nonlinear models, both parametric and nonparametric, are considered as well as post-estimation model evaluation. Data-based nonlinear model building is illustrated with an empirical example.

Additive nonlinear ARX time series and projection estimates

We propose projections as means of identifying and estimating the components (endogenous and exogenous) of an additive nonlinear ARX model. The estimates are nonparametric in nature and involve averaging of kernel-type estimates. Such estimates have recently been treated informally in a univariate time series situation. Here we extend the scope to nonlinear ARX models and present a rigorous theory, including the derivation of asymptotic normality for the projection estimates under a precise set of regularity conditions.

Avoidance behaviour in cod (Gadus morhua) to a bottom-trawling vessel

The reaction of fish induced by a trawling vessel was measured using the Bergen Acoustic Buoy. It is a free-floating buoy with a split beam echo sounder system. Individual fish trajectories were obtained by target tracking methods, and average swimming velocities as a function of depth and time before and after passage of the vessel was calculated. A measure for the change in behaviour was applied, showing a significant response during and after propeller passage. The change in horizontal displacement speed is significant at all depths, while the change in vertical displacement velocity is significant at all but one layer of depth. The horizontal reaction seems to occur a bit later than the diving reaction. After the main response, a slightly higher mean horizontal displacement speed was observed for the deepest layers. This indicates a change in the fish state after being exposed to the vessel/gear.

A Cautionary Note on the Use of the Kolmogorov–Smirnov Test for Normality

Abstract The Kolmogorov–Smirnov goodness-of-fit test is used in many applications for testing normality in climate research. This note shows that the test usually leads to systematic and drastic errors. When the mean and the standard deviation are estimated, it is much too conservative in the sense that its p values are strongly biased upward. One may think that this is a small sample problem, but it is not. There is a correction of the Kolmogorov–Smirnov test by Lilliefors, which is in fact sometimes confused with the original Kolmogorov–Smirnov test. Both the Jarque–Bera and the Shapiro–Wilk tests for normality are good alternatives to the Kolmogorov–Smirnov test. A power comparison of eight different tests has been undertaken, favoring the Jarque–Bera and the Shapiro–Wilk tests. The Jarque–Bera and the Kolmogorov–Smirnov tests are also applied to a monthly mean dataset of geopotential height at 500 hPa. The two tests give very different results and illustrate the danger of using the Kolmogorov–Smirnov ...