The history of the development of statistical hypothesis testing in time series analysis is reviewed briefly and it is pointed out that the hypothesis testing procedure is not adequately defined as the procedure for statistical model identification. The classical maximum likelihood estimation procedure is reviewed and a new estimate minimum information theoretical criterion (AIC) estimate (MAICE) which is designed for the purpose of statistical identification is introduced. When there are several competing models the MAICE is defined by the model and the maximum likelihood estimates of the parameters which give the minimum of AIC defined by AIC = (-2)log-(maximum likelihood) + 2(number of independently adjusted parameters within the model). MAICE provides a versatile procedure for statistical model identification which is free from the ambiguities inherent in the application of conventional hypothesis testing procedure. The practical utility of MAICE in time series analysis is demonstrated with some numerical examples.

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A new look at the statistical model identification

SUMMARY A generalization of the sampling method introduced by Metropolis et al. (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties of assessing the error in Monte Carlo estimates. Examples of the methods, including the generation of random orthogonal matrices and potential applications of the methods to numerical problems arising in statistics, are discussed. For numerical problems in a large number of dimensions, Monte Carlo methods are often more efficient than conventional numerical methods. However, implementation of the Monte Carlo methods requires sampling from high dimensional probability distributions and this may be very difficult and expensive in analysis and computer time. General methods for sampling from, or estimating expectations with respect to, such distributions are as follows. (i) If possible, factorize the distribution into the product of one-dimensional conditional distributions from which samples may be obtained. (ii) Use importance sampling, which may also be used for variance reduction. That is, in order to evaluate the integral J = X) p(x)dx = Ev(f), where p(x) is a probability density function, instead of obtaining independent samples XI, ..., Xv from p(x) and using the estimate J, = Zf(xi)/N, we instead obtain the sample from a distribution with density q(x) and use the estimate J2 = Y{f(xj)p(x1)}/{q(xj)N}. This may be advantageous if it is easier to sample from q(x) thanp(x), but it is a difficult method to use in a large number of dimensions, since the values of the weights w(xi) = p(x1)/q(xj) for reasonable values of N may all be extremely small, or a few may be extremely large. In estimating the probability of an event A, however, these difficulties may not be as serious since the only values of w(x) which are important are those for which x -A. Since the methods proposed by Trotter & Tukey (1956) for the estimation of conditional expectations require the use of importance sampling, the same difficulties may be encountered in their use. (iii) Use a simulation technique; that is, if it is difficult to sample directly from p(x) or if p(x) is unknown, sample from some distribution q(y) and obtain the sample x values as some function of the corresponding y values. If we want samples from the conditional dis

https://www.jstor.org/stable/pdfplus/2334940.pdf

Monte Carlo Sampling Methods Using Markov Chains and Their Applications

Large sample properties of generalized method of moments estimators

This paper introduces an ARCH model (exponential ARCH) that (1) allows correlation between returns and volatility innovations (an important feature of stock market volatility changes), (2) eliminates the need for inequality constraints on parameters, and (3) allows for a straightforward interpretation of the "persistence" of shocks to volatility. In the above respects, it is an improvement over the widely-used GARCH model. The model is applied to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987. Copyright 1991 by The Econometric Society.

Conditional heteroskedasticity in asset returns: a new approach

Any series of observations ordered along a single dimension, such as time, may be thought of as a time series. The emphasis in time series analysis is on studying the dependence among observations at different points in time. What distinguishes time series analysis from general multivariate analysis is precisely the temporal order imposed on the observations. Many economic variables, such as GNP and its components, price indices, sales, and stock returns are observed over time. In addition to being interested in the contemporaneous relationships among such variables, we are often concerned with relationships between their current and past values, that is, relationships over time.

Time Series Analysis

SUMMARY It is shown that a strongly consistent estimation procedure for the order of an autoregression can be based on the law of the iterated logarithm for the partial autocorrelations. As compared to other strongly consistent procedures this procedure will underestimate the order to a lesser degree.

The determination of the order of an autoregression

Multiple time series

Publisher Summary The chapter discusses the development of a rather complete inferential theory for ARMAX models. The first problem in the development is the coordinatization of spaces of such structures. Coordinates are needed both for computations and because a central limit theorem must be expressed in terms of them. One such coordinatization would follow from the use of the coefficient matrices in g, h, j in the scalar m.f.d., but there are many others. The chapter highlights the algebraic and topological description of ARMAX systems. In this connection, control engineers have played a premier part. The chapter focuses on the asymptotic properties of maximum likelihood (ML) estimators. The ML estimator is obtained by optimizing this likelihood. The chapter explains the asymptotic properties of these estimators without assuming the data to be Gaussian and also discusses the basis of the assumptions that appear to be minimal.

The statistical theory of linear systems

Under general conditions strong consistency of certain estimates of the maximum lags of an autoregressive moving average process is established. A theorem on weak consistency is also proved and in certain cases where consistency does not hold the probability of over-estimation of a maximum lag is evaluated.

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The Estimation of the Order of an ARMA Process

SUMMARY The order, (p, q), of an autoregressive-moving average sequence, y(t), may be estimated- by minimizing a criterion, log &2 + (p + q) log T/T, with respect to p and q, where ^2 is the maximum likelihood estimate of the variance of the innovations, ?(t). It is suggested that, instead, a2 be estimated from a series of regressions of y(t) on y(t-1),y(t-2),...,y(t-p), 5(t-1),...,?(t-q), where the ?(t) are obtained by fitting a long autoregression to the data. It is shown how the sequence of regressions may, for p = q, be economically recursively calculated by embedding them in a sequence of bivariate autoregressions. Asymptotic properties of the procedure are established under very general conditions.

E. J. Hannan

Papers

The determination of the order of an autoregression

Multiple time series

The statistical theory of linear systems

The Estimation of the Order of an ARMA Process

Recursive estimation of mixed autoregressive-moving average order