The first-order moving average model or MA(1) is given by $X_t=Z_t-\theta_0 Z_{t-1}$, with independent and identically distributed $\{Z_t\}$. This is arguably the simplest time series model that one can write down. The MA(1) with unit root ($\theta_0=1$) arises naturally in a variety of time series applications. For example, if an underlying time series consists of a linear trend plus white noise errors, then the differenced series is an MA(1) with unit root. In such cases, testing for a unit root of the differenced series is equivalent to testing the adequacy of the trend plus noise model. The unit root problem also arises naturally in a signal plus noise model in which the signal is modeled as a random walk. The differenced series follows a MA(1) model and has a unit root if and only if the random walk signal is in fact a constant.

The asymptotic theory of various estimators based on Gaussian likelihood has been developed for the unit root case and nearly unit root case ($\theta=1 +\beta/n, \beta\le0$). Unlike standard $1/\sqrt{n}$-asymptotics, these estimation procedures have $1/n$-asymptotics and a so-called pile-up effect, in which P$(\hat\theta= 1)$ converges to a positive value. One explanation for this pile-up phenomenon is the lack of identifiability of $\theta$ in the Gaussian case. That is, the Gaussian likelihood has the same value for the two sets of parameter values $(\theta,\sigma^2)$ and $(1/\theta,\theta^2\sigma^2$). It follows that $\theta=1$ is always a critical point of the likelihood function. In contrast, for non-Gaussian noise, $\theta$ is identifiable for all real values. Hence it is no longer clear whether or not the same pile-up phenomenon will persist in the non-Gaussian case. In this paper, we focus on limiting pile-up probabilities for estimates of $\theta_0$ based on a Laplace likelihood. In some cases, these estimates can be viewed as Least Absolute Deviation (LAD) estimates. Simulation results illustrate the limit theory.

We introduce a recursive algorithm of conveniently general form for estimating the coefficient of a moving average model of order one and obtain convergence results for both correct and misspecified MA(1) models. The algorithm encompasses Pseudolinear Regression (PLR--also referred to as AML and $\mbox{RML}_1$) and Recursive Maximum Likelihood ($\mbox{RML}_2$) without monitoring. Stimulated by the approach of Hannan (Hannan, E. J. (1980), Recursive estimation based on ARMA models, Ann. Statist.8 (4) 762–777, MR572620), our convergence results are obtained indirectly by showing that the recursive sequence can be approximated by a sequence satisfying a recursion of simpler (Robbins-Monro) form for which convergence results applicable to our situation have recently been obtained.

In this paper the stochastic complexity criterion is applied to estimation of the order in AR and ARMA models. The power of the criterion for short strings is illustrated by simulations. It requires an integral of the square root of Fisher information, which is done by Monte Carlo technique. The stochastic complexity, which is the negative logarithm of the Normalized Maximum Likelihood universal density function, is given. Also, exact asymptotic formulas for the Fisher information matrix are derived.

In this article asymptotic expressions for the final prediction error (FPE) and the accumulated prediction error (APE) of the least squares predictor are obtained in regression models with nonstationary regressors. It is shown that the term of order $1/n$ in FPE and the term of order $\log n$ in APE share the same constant, where $n$ is the sample size. Since the model includes the random walk model as a special case, these asymptotic expressions extend some of the results in Wei (1987) and Ing (2001). In addition, we also show that while the FPE of the least squares predictor is not affected by the contemporary correlation between the innovations in input and output variables, the mean squared error of the least squares estimate does vary with this correlation.

Previous analysis on forecasting theory either assume knowing the true parameters or assume the stationarity of the series. Not much are known on the forecasting theory for nonstationary process with estimated parameters. This paper investigates the recursive least square forecast for stationary and nonstationary processes with unit roots. We first prove that the accumulated forecast mean square error can be decomposed into two components, one of which arises from estimation uncertainty and the other from the disturbance term. The former, of the order of $\log(T)$, is of second order importance to the latter term, of the order T. However, since the latter is common for all predictors, it is the former that determines the property of each predictor. Our theorem implies that the improvement of forecasting precision is of the order of $\log(T)$ when existence of unit root is properly detected and taken into account. Also, our theorem leads to a new proof of strong consistency of predictive least squares in model selection and a new test of unit root where no regression is needed.

The simulation results confirm our theoretical findings. In addition, we find that while mis-specification of AR order and under-specification of the number of unit root have marginal impact on forecasting precision, over-specification of the number of unit root strongly deteriorates the quality of long term forecast. As for the empirical study using Taiwanese data, the results are mixed. Adaptive forecast and imposing unit root improve forecast precision for some cases but deteriorate forecasting precision for other cases.

In the application of autoregressive models the order of the model is often estimated using either a sequence of likelihood ratio tests, a likelihood based information criterion, or a residual based test. The properties of such procedures has been discussed extensively under the assumption that the characteristic roots of the autoregression are stationary. While non-stationary situations have also been considered the results in the literature depend on conditions to the characteristic roots. It is here shown that these methods for lag length determination can be used regardless of the assumption to the characteristic roots and also in the presence of deterministic terms. The proofs are based on methods developed by C. Z. Wei in his joint work with T. L. Lai.

The finite-sample as well as the asymptotic distribution of Leung and Barron’s (2006) model averaging estimator are derived in the context of a linear regression model. An impossibility result regarding the estimation of the finite-sample distribution of the model averaging estimator is obtained.

Employing recent results of Robinson (2005) we consider the asymptotic properties of conditional-sum-of-squares (CSS) estimates of parametric models for stationary time series with long memory. CSS estimation has been considered as a rival to Gaussian maximum likelihood and Whittle estimation of time series models. The latter kinds of estimate have been rigorously shown to be asymptotically normally distributed in case of long memory. However, CSS estimates, which should have the same asymptotic distributional properties under similar conditions, have not received comparable treatment: the truncation of the infinite autoregressive representation inherent in CSS estimation has been essentially ignored in proofs of asymptotic normality. Unlike in short memory models it is not straightforward to show the truncation has negligible effect.

It has been shown that some macroeconomic time series, especially those where outliers could be present, can be well modelled using heavy tailed distributions for the noise components. Methods for deciding when and where heavy-tailed models should be preferred are investigated. These investigations primarily focus on automatic methods for model identification and selection. Current methods are extended to incorporate a non-Gaussian selection element, and various different criteria for deciding on which overall model should be used are examined.

This paper develops a European option pricing formula for fractional market models. Although there exist option pricing results for a fractional Black-Scholes model, they are established without accounting for stochastic volatility. In this paper, a fractional version of the Constant Elasticity of Variance (CEV) model is developed. European option pricing formula similar to that of the classical CEV model is obtained and a volatility skew pattern is revealed.

The Sharpe ratio, which is defined as the ratio of the excess expected return of an investment to its standard deviation, has been widely cited in the financial literature by researchers and practitioners. However, very little attention has been paid to the statistical properties of the estimation of the ratio. Lo (2002) derived the $\sqrt{n}$-normality of the ratio's estimation errors for returns which are iid or stationary with serial correlations, and pointed out that to make inference on the accuracy of the estimation, the serial correlation among the returns needs to be taken into account. In the present paper a class of time series models for returns is introduced to demonstrate that there exists a factor other than the serial correlation of the returns that dominates the asymptotic behavior of the Sharpe ratio statistics. The model under consideration is a linear process whose innovation sequence has summable coefficients and contains a latent volatility component which is long-memory. It is proved that the estimation errors of the ratio are asymptotically normal with a convergence rate slower than $\sqrt{n}$ and that the estimation deviation of the expected return makes no contribution to the limiting distribution.

We review the advancement of nonstationary time series analysis from the perspective of Cowles Commission structural equation approach. We argue that despite the rich repertoire nonstationary time series analysis provides to analyze how do variables respond dynamically to shocks through the decomposition of a dynamic system into long-run and short-run relations, nonstationarity does not invalid the classical concerns of structural equation modeling — identification and simultaneity bias. The same rank condition for identification holds for stationary and nonstationary data and some sort of instrumental variable estimators will have to be employed to yield consistency. However, nonstationarity does raise issues of inference if the rank of cointegration or direction of nonstationarity is not known a priori. The usual test statistics may not be chi-square distributed because of the presence of unit roots distributions. Classical instrumental variable estimators have to be modified to ensure valid inference.

This paper describes a new approach to time series modeling that combines subject-matter knowledge of the system dynamics with statistical techniques in time series analysis and regression. Applications to American option pricing and the Canadian lynx data are given to illustrate this approach.

Correlations between asset returns are important in many financial applications. In recent years, multivariate volatility models have been used to describe the time-varying feature of the correlations. However, the curse of dimensionality quickly becomes an issue as the number of correlations is $k(k-1)/2$ for $k$ assets. In this paper, we review some of the commonly used models for multivariate volatility and propose a simple approach that is parsimonious and satisfies the positive definite constraints of the time-varying correlation matrix. Real examples are used to demonstrate the proposed model.

Consider a multi-phase project management problem where the decision maker needs to deal with two issues: (a) how to allocate resources to projects within each phase, and (b) when to enter the next phase, so that the total expected reward is as large as possible. We formulate the problem as a multi-armed bandit problem with precedence relations. In Chan, Fuh and Hu (2005), a class of asymptotically optimal arm-pulling strategies is constructed to minimize the shortfall from perfect information payoff. Here we further explore optimality properties of the proposed strategies. First, we show that the efficiency benchmark, which is given by the regret lower bound, reduces to those in Lai and Robbins (1985), Hu and Wei (1989), and Fuh and Hu (2000). This implies that the proposed strategy is also optimal under the settings of aforementioned papers. Secondly, we establish the super-efficiency of proposed strategies when the bad set is empty. Thirdly, we show that they are still optimal with constant switching cost between arms. In addition, we prove that the Wald’s equation holds for Markov chains under Harris recurrent condition, which is an important tool in studying the efficiency of the proposed strategies.

This exposition explains the basic ideas of Stein’s method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigration-death process and Palm theory. The latter approach also enables us to define local dependence of point processes and use it to study Poisson process approximation for locally dependent point processes and for dependent superposition of point processes.

Design of experiment with related factors can be implemented by using the technique of sliding levels. Taguchi (1987) proposed an analysis strategy by re-centering and re-scaling the slid factors. Hamada and Wu (1995) showed via counter examples that in many cases the interactions cannot be completely eliminated by Taguchi’s strategy. They proposed an alternative method in which the slid factors are modeled by nested effects. In this work we show the inadequacy of both methods when the objective is response prediction. We propose an analysis method based on a response surface model, and demonstrate its superiority for prediction. We also study the relationships between these three modeling strategies.

In a market with transaction costs, the price of a derivative can be expressed in terms of (preconsistent) price systems (after Kusuoka (1995)). In this paper, we consider a market with binomial model for stock price and discuss how to generate the price systems. From this, the price formula of a derivative can be reformulated as a stochastic control problem. Then the dynamic programming approach can be used to calculate the price. We also discuss optimization of expected utility using price systems.

The maximum product of spacings (MPS) is employed in the estimation of the Generalized Extreme Value Distribution (GEV) and the Generalized Pareto Distribution (GPD). Efficient estimators are obtained by the MPS for all $\gamma$. This outperforms the maximum likelihood method which is only valid for $\gamma< 1$. It is then shown that the MPS gives estimators closer to the true parameters compared to the maximum likelihood estimates (MLE) in a simulation study. In cases where sample sizes are small, the MPS performs stably while the MLE does not. The performance of MPS estimators is also more stable than those of the probability-weighted moment (PWM) estimators. Finally, as a by-product of the MPS, a goodness of fit statistic, Moran's statistic, is available for the extreme value distributions. Empirical significance levels of Moran's statistic calculated are found to be satisfactory with the desired level.

We consider the Gittins index for a normal distribution with unknown mean $\theta$ and known variance where $\theta$ has a normal prior. In addition to presenting some monotonicity properties of the Gittins index, we derive an approximation to the Gittins index by embedding the (discrete-time) normal setting into the continuous-time Wiener process setting in which the Gittins index is determined by the stopping boundary for an optimal stopping problem. By an application of Chernoff's continuity correction in optimal stopping, the approximation includes a correction term which accounts for the difference between the discrete and continuous-time stopping boundaries. Numerical results are also given to assess the performance of this simple approximation.