John C. Nankervis

TESTING FOR ZERO AUTOCORRELATION IN THE PRESENCE OF STATISTICAL DEPENDENCE

The problem addressed in this paper is to test the null hypothesis that a time series process is uncorrelated up to lag K in the presence of statistical dependence. We propose an extension of the Box–Pierce Q-test that is asymptotically distributed as chi-square when the null is true for a very general class of dependent processes that includes non-martingale difference sequences. The test is based on a consistent estimator of the asymptotic covariance matrix of the sample autocorrelations under the null. The finite sample performance of this extension is investigated in a Monte Carlo study.

Testing for Autocorrelation Using a Modified Box-Pierce Q Test

This article investigates the finite-sample performance of a modified Box-Pierce Q statistic (Q*) for testing that financial time series are uncorrelated without assuming statistical independence. The finite-sample rejection probabilities of the Q* test under the null and its power are examined in experiments using time series generated by an MA (1) process where the errors are generated by a GARCH (1, 1) model and by a long memory stochastic volatility model. The tests are applied to daily currency returns.

The Level and Power of the Bootstrap t Test in the AR(1) Model With Trend

This article considers a first-order autoregressive (AR) model that may include an intercept and trend in which the innovations are independently and identically distributed. The innovation distribution is assumed unknown. The AR parameter is tested using the conventional t statistic. The article presents Monte Carlo estimates of the rejection probability of the test with bootstrap-based critical values. The results show that the test with the bootstrap-based critical value has essentially the right rejection probability for sample sizes comparable to or smaller than those that occur in practice and essentially the same power as the test with level-corrected critical values.

Multiple optima and asymptotic approximations in the partial adjustment model

This paper examines statistical problems which arise in empirical applications of the partial adjustment model with autoregressive errors when the model is nearly nonidentified. The results of Monte Carlo experiments show that the NLS estimation criterion function is multipeaked with high probability when the model is nearly nonidentified. In the cases examined the finite-sample distributions of the NLS estimators and the Wald test statistics are poorly approximated by their asymptotic distributions. The asymptotic approximation works better for the likelihood ratio (LR) test statistics, but still can be unsatisfactory. When the Wald and LR tests are based on bootstrap critical values the size distortions are effectively eliminated.

Mirror-Image and Invariant Distributions in ARMA Models

The finite sample distributions of estimators and test statistics in ARMA time series models are generally unknown. For typical sample sizes, the approximations provided by asymptotic distributions are often unsatisfactory. Hence simulation or numerical integration methods are used to investigate the distributions. In practice only a limited part of the parameter space is examined using these methods. Thus any results which allow us to infer properties from one portion of the parameter space to another or to establish symmetry are most welcome.

Disaggregated Letter Traffic Demand in the UK

All national post offices offer an extensive range of mail products to customers. It is likely that the various demand elasticities of these products differ across services (for example, products grow at different rates in response to economic growth or are more or less responsive to a given price change) and that there is some degree of substitutability between at least some products. It is also the case that the impact on volumes from developments such as electronic substitution and competitive entry may vary across services. Consequently, it is important to have an understanding of the demand for mail products at a disaggregated level. There are various approaches that, in principle, could be adopted to gain such understanding but, where data are available, one which is well suited at a relatively high level of aggregation is econometric modelling using time series data.

Aggregate Letter Traffic Demand in the United Kingdom and the Economy

The prospect of a dramatic fall in letter traffic levels has been a constant theme in the discussion of the mail market for several years. The development of electronic data interchange, fax, and a host of other electronically based means of communication have been seen as posing a fundamental and, ultimately, decisive threat to communication by letter. So far, however, the trend in letter volumes has remained upwards. As table 1 shows, average annual growth rates in domestic postal traffic through the 1980s were quite rapid in most developed countries. The highest rates of increase were found in France, the United Kingdom and the United States at over 4% per annum, and in none of these major economies did total volumes come close to remaining static, let alone showing signs of decline.

How Important have Price and Quality of Service Been to Mail Volume Growth

Mail volumes everywhere appear to be under threat from technological substitutes such as fax, EDI, electronic mail, the Internet, and the ever expanding range of telephony services. Yet in many countries, not least in the United States where these factors are perhaps at their most developed and mail volumes per head are high, total letter mail volumes continue to rise, albeit rather more slowly than for much of the 1980s. How can this, to many, surprising conjuncture be accounted for and what factors appear to be encouraging growth in traffic volumes? Further, and perhaps more pertinently for postal operators, to what extent are these positive factors directly under the control of postal administrations themselves?

Forecast Error Symmetry in ARIMA Models

Abstract Forecasts and residuals from estimated autoregressive integrated moving average (ARIMA) models are investigated with respect to the symmetry of their distributions. For models estimated with and without intercept terms and, more generally, for regression models with correlated errors, it is shown that both the set of residuals and the set of forecast errors at leads 1, 2, …, l have joint distributions that are symmetric about 0 under mild symmetry conditions on the true process generating the observations. We consider most common estimation methods including maximum likelihood, conditional least squares, and unconditional least squares. We also show that the forecast error t statistics are symmetrically distributed about 0.