Rolf Tschernig

Nonlinear Interest Rate Dynamics and Implications for the Term Structure

This paper explores nonlinear dynamics in the time series of the short-term interest rate in the United States. The proposed model is an autoregressive threshold model augmented by conditional heteroskedasticity. The performance of the model is evaluated by considering its implications for the term structure of interest rates. The nonlinear dynamics imply a form of nonlinearity in the levels relation between the long and the short rate.

Multivariate bandwidth selection for local linear regression

The existence and properties of optimal bandwidths for multivariate local linear regression are established, using either a scalar bandwidth for all regressors or a diagonal bandwidth vector that has a different bandwidth for each regressor. Both involve functionals of the derivatives of the unknown multivariate regression function. Estimating these functionals is difficult primarily because they contain multivariate derivatives. In this paper, an estimator of the multivariate second derivative is obtained via local cubic regression with most cross-terms left out. This estimator has the optimal rate of convergence but is simpler and uses much less computing time than the full local estimator. Using this as a pilot estimator, we obtain plug-in formulae for the optimal bandwidth, both scalar and diagonal, for multivariate local linear regression. As a simpler alternative, we also provide rule-of-thumb bandwidth selectors. All these bandwidths have satisfactory performance in our simulation study.

Nonparametric lag selection for time series

A nonparametric version of the Final Prediction Error (FPE) is proposed for lag selection in nonlinear autoregressive time series. We derive its consistency for both local constant and local linear estimators using a derived optimal bandwidth. Further asymptotic analysis suggests a greater probability of overfitting (too many lags) than underfitting (missing important lags). Thus a correction factor is proposed to increase correct fitting by reducing overfitting. Our Monte-Carlo study also corroborates that the correction factor generally improves the probability of correct lag selection for both linear and nonlinear processes. The proposed methods are successfully applied to the Canadian lynx data and daily returns of DM/US-Dollar exchange rates.

A SIMPLE VARIABLE SELECTION TECHNIQUE FOR NONLINEAR MODELS

Applying nonparametric variable selection criteria in nonlinear regression models generally requires a substantial computational effort if the data set is large. In this paper we present a selection technique that is computationally much less demanding and performs well in comparison with methods currently available. It is based on a polynomial approximation of the nonlinear model. Performing the selection only requires repeated least squares estimation of models that are linear in parameters. The main limitation of the method is that the number of variables among which to select cannot be very large if the sample is small and the order of an adequate polynomial at the same time is high. Large samples can be handled without problems.

Non- and Semiparametric Identification of Seasonal Nonlinear Autoregression Models

Non- or semiparametric estimation and lag selection methods are proposed for three seasonal nonlinear autoregressive models of varying seasonal flexibility. All procedures are based on either local constant or local linear estimation. For the semiparametric models, after preliminary estimation of the seasonal parameters, the function estimation and lag selection are the same as nonparametric estimation and lag selection for standard models. A Monte Carlo study demonstrates good performance of all three methods. The semiparametric methods are applied to German real gross national product and UK public investment data. For these series our procedures provide evidence of nonlinear dynamics.

Web quantlets for time series analysis

New and advanced methods for nonlinear time series analysis are in general not available in standard software packages. Their implementation requires substantial time, computing power as well as programming skills. In time series analysis such a scenario is given by a recently suggested nonparametric lag selection procedure for univariate nonlinear autoregressive models which is based on the Corrected Asymptotic Final Prediction Error. In this paper we suggest a worldwide Web based specific client/server architecture that provides empirical researchers with fast access to new methods and powerful computing environments without knowing the statistical computing language and the server location. This architecture is implemented using the XploRe Quantlet technology and illustrated for nonparametric lag selection. Access to the Quantlet computing service can be obtained via standard WWW browsers or a Java client. The XploRe Quantlet service can be helpful in constructing research books and interactive teaching environments as the electronic version of this paper, available from http://ise.wiwi.hu-berlin.de/∼rolf/webquant.pdf, demonstrates.

Illusive Persistence in German Unemployment

The non-stationarity of many macroeconomic time-series has lead to an increased demand for economic models that are able to generate fragile equilibria. For instance, the natural unemployment rate is al- lowed to shift over time depending on past unemployment. Actually, many European unemployment series seem to exhibit a unit root or persistence. This view is questioned in the paper using German data on unemployment. A new class of time-series models, the fractionally integrated ARMA model, that allows the difference parameter to take real values, enables the researcher to separate long memory and short memory in the data. It is shown that using this approach the unit root hypothesis is rejected but unemployment exhibits long memory.

Long-Run Identification in a Fractionally Integrated System

We propose an extension of structural fractionally integrated vector autoregressive models that avoids certain undesirable effects on the impulse responses that occur if long-run identification restrictions are imposed. We derive the model’s Granger representation and investigate the effects of long-run restrictions. Simulations illustrate that enforcing integer integration orders can have severe consequences for impulse responses. In a system of U.S. real output and aggregate prices, the effects of structural shocks strongly depend on the specification of the integration orders. In the statistically preferred fractional model, shocks that are typically interpreted as demand disturbances have a very brief influence on GDP. Supplementary materials for this article are available online.

Nonparametric Time Series Modelling

Introduction As the previous chapters have shown, parametric time series modeling offers a great wealth of modeling tools. Linear time series models are generally the starting point for modeling both univariate and multivariate time series data. Such data may also exhibit nonstationary behavior caused by the presence of unit roots, structural breaks, seasonal influences, and the like. If one is interested in nonlinear dynamics, it is no longer sufficient to consider linear models. This situation arises if, for example, the strength of economic relationships depends on the state of the business cycle or if the adjustment speed toward long-run equilibrium relationships is not proportional to the deviation from the long-run equilibrium. Chapter 6 discusses how to build nonlinear parametric models for various kinds of nonlinear dynamics, including those of business cycles. There it is also explained how the parameter estimates can be used to understand the underlying nonlinear dynamic behavior. However, nonlinear parametric modeling also has its drawbacks. Most importantly, it requires an a priori choice of parametric function classes for the function of interest. The framework of smooth transition regression models discussed in Chapter 6, for example, is widely used. Although it is an appropriate modeling framework for many empirical problems, it may not always capture features that are relevant to the investigator. In the latter case, one has to choose alternative nonlinear parametric models like neural networks or Markov-switching models to name a few. Thus, nonlinear parametric modeling implies the difficult choice of a model class. In contrast,when using the nonparametric modeling approach, one can avoid this choice.

Nonparametric estimation of generalized impulse response function

A local linear estimator of generalized impulse response (GIR) functions for nonlinear conditional heteroskedastic autoregressive processes is derived and shown to be asymptotically normal. A plug-in bandwidth is obtained that minimizes the asymptotical mean squared error of the GIR estimator. A local linear estimator for the conditional variance function is proposed which has simpler bias than the standard estimator. This is achieved by appropriately eliminating the conditional mean. Alternatively to the direct local linear estimators of the k-step prediction functions which enter the GIR estimator the use of multi-stage prediction techniques is suggested. Simulation experiments show the latter estimator to perform best. For quarterly data of the West German real GNP it is found that the size of generalized impulse response functions varies across different histories , a feature which cannot be captured by linear models.