We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of $l$ consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero. Under appropriate conditions consistency is obtained if $l = l(n) \rightarrow \infty$ and $l(n)/n \rightarrow 0$. General statistics are approximated by an arithmetic mean. In regular cases this approximation determines the asymptotic behavior. Bootstrap replicates are constructed by selecting blocks of length $l$ randomly with replacement among the blocks of observations. The procedures are illustrated by using the sunspot numbers and some simulated data.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176347265

The Jackknife and the Bootstrap for General Stationary Observations

This article considers the application of two families of nonlinear autoregressive models, the logistic (LSTAR) and exponential (ESTAR) autoregressive models. This includes the specification of the model based on simple statistical tests: linearity testing against smooth transition autoregression, determining the delay parameter and choosing between LSTAR and ESTAR models are discussed. Estimation by nonlinear least squares is considered as well as evaluating the properties of the estimated model. The proposed techniques are illustrated by examples using both simulated and real time series.

Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models

We fit nonlinearly mean-reverting models to real dollar exchange rates over the post-Bretton Woods period, consistent with a theoretical literature on transactions costs in international arbitrage. The half lives of real exchange rate shocks, calculated through Monte Carlo integration, imply faster adjustment speeds than hitherto recorded. Monte Carlo simulations reconcile our results with the large empirical literature on unit roots in real exchange rates by showing that when the real exchange rate is nonlinearly mean reverting, standard univariate unit root tests have low power, while multivariate tests have much higher power to reject a false null hypothesis.

Nonlinear Mean‐Reversion in Real Exchange Rates: Toward a Solution To the Purchasing Power Parity Puzzles

We propose an exchange rate model that can explain both the observed volatility and the persistence of real and nominal exchange rate movements and thus in some measure resolves Rogoff's (1996) purchasing power parity puzzle. Our analysis reconciles the well-known difficulties in beating the random walk forecast model with the statistical evidence of nonlinear mean reversion in deviations from fundamentals. Our analysis also provides a compelling rationale for the long-horizon predictability of exchange rates. We find strong empirical support for long-horizon predictability, and we explain why it is difficult to exploit this predictability in real-time forecasts. Our results not only lend support to economists' beliefs that the exchange rate is inherently predictable, but they also help us to understand the reluctance of applied forecasters to abandon chartist methods in favor of models based on economic fundamentals.

https://papers.ssrn.com/sol3/Delivery.cfm?abstractid=356266

Why is it so Difficult to Beat the Random Walk Rorecast of Exchange Rates

https://fordschool.umich.edu/rsie/workingpapers/Papers451-475/r464.pdf

Why is it so difficult to beat the random walk forecast of exchange rates

Estimation in nonlinear time series models

. We consider time series models obtained by replacing the parameters of autoregressive models by stochastic processes. Special attention is given to the problem of finding conditions for stationarity and to the problem of forecasting. For the first problem we are only able to obtain solutions in special cases, and the emphasis is on techniques rather than obtaining the most general results in each case. For the second problem more complete results are obtained by exploiting similarities with discrete time (nonlinear) filtering theory. The methods introduced are illustrated on two standard examples, one of state space type and one where the parameter process is a Markov chain.

Some doubly stochastic time series models

. We study nonstationary autoregressive processes, where the variance of the generating white noise process is allowed to depend on time. It is shown that ordinary least squares estimates are strongly consistent and with a proper scaling factor asymptotically normal, but, as can be expected, they are not efficient. Furthermore, AIC type order determination criteria, used as if the underlying process is stationary, are consistent, whereas identification of order in terms of the partial autocorrelation function may lead one astray.

Dag Tjøstheim

Papers

Estimation in nonlinear time series models

Some doubly stochastic time series models

Least squares estimates and order determination procedures for autoregressive processes with a time dependent variance