Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Offering a unifying theoretical perspective not readily available in any other text, this innovative guide to econometrics uses simple geometrical arguments to develop students' intuitive understanding of basic and advanced topics, emphasizing throughout the practical applications of modern theory and nonlinear techniques of estimation. One theme of the text is the use of artificial regressions for estimation, reference, and specification testing of nonlinear models, including diagnostic tests for parameter constancy, serial correlation, heteroscedasticity, and other types of mis-specification. Explaining how estimates can be obtained and tests can be carried out, the authors go beyond a mere algebraic description to one that can be easily translated into the commands of a standard econometric software package. Covering an unprecedented range of problems with a consistent emphasis on those that arise in applied work, this accessible and coherent guide to the most vital topics in econometrics today is indispensable for advanced students of econometrics and students of statistics interested in regression and related topics. It will also suit practising econometricians who want to update their skills. Flexibly designed to accommodate a variety of course levels, it offers both complete coverage of the basic material and separate chapters on areas of specialized interest.

Estimation and Inference in Econometrics

Features of Financial Returns Modelling Price Volatility Forecasting Standard Deviations The Accuracy of Autocorrelation Estimates Testing the Random Walk Hypothesis Forecasting Trends in Prices Evidence Against the Efficiency of Futures Markets Valuing Options Appendix: A Computer Program for Modelling Financial Time Series.

Modelling Financial Time Series

Artificial neural networks (ANNs) are flexible computing frameworks and universal approximators that can be applied to a wide range of time series forecasting problems with a high degree of accuracy. However, despite all advantages cited for artificial neural networks, their performance for some real time series is not satisfactory. Improving forecasting especially time series forecasting accuracy is an important yet often difficult task facing forecasters. Both theoretical and empirical findings have indicated that integration of different models can be an effective way of improving upon their predictive performance, especially when the models in the ensemble are quite different. In this paper, a novel hybrid model of artificial neural networks is proposed using auto-regressive integrated moving average (ARIMA) models in order to yield a more accurate forecasting model than artificial neural networks. The empirical results with three well-known real data sets indicate that the proposed model can be an effective way to improve forecasting accuracy achieved by artificial neural networks. Therefore, it can be used as an appropriate alternative model for forecasting task, especially when higher forecasting accuracy is needed.

An artificial neural network (p,d,q) model for timeseries forecasting

Improving forecasting especially time series forecasting accuracy is an important yet often difficult task facing decision makers in many areas. Both theoretical and empirical findings have indicated that integration of different models can be an effective way of improving upon their predictive performance, especially when the models in combination are quite different. Artificial neural networks (ANNs) are flexible computing frameworks and universal approximators that can be applied to a wide range of forecasting problems with a high degree of accuracy. However, using ANNs to model linear problems have yielded mixed results, and hence; it is not wise to apply ANNs blindly to any type of data. Autoregressive integrated moving average (ARIMA) models are one of the most popular linear models in time series forecasting, which have been widely applied in order to construct more accurate hybrid models during the past decade. Although, hybrid techniques, which decompose a time series into its linear and nonlinear components, have recently been shown to be successful for single models, these models have some disadvantages. In this paper, a novel hybridization of artificial neural networks and ARIMA model is proposed in order to overcome mentioned limitation of ANNs and yield more general and more accurate forecasting model than traditional hybrid ARIMA-ANNs models. In our proposed model, the unique advantages of ARIMA models in linear modeling are used in order to identify and magnify the existing linear structure in data, and then a neural network is used in order to determine a model to capture the underlying data generating process and predict, using preprocessed data. Empirical results with three well-known real data sets indicate that the proposed model can be an effective way to improve forecasting accuracy achieved by traditional hybrid models and also either of the components models used separately.

A novel hybridization of artificial neural networks and ARIMA models for time series forecasting

We generalize the Gaussian mixture transition distribution (GMTD) model introduced by Le and co-workers to the mixture autoregressive (MAR) model for the modelling of non-linear time series. The models consist of a mixture of K stationary or non-stationary AR components. The advantages of the MAR model over the GMTD model include a more full range of shape changing predictive distributions and the ability to handle cycles and conditional heteroscedasticity in the time series. The stationarity conditions and autocorrelation function are derived. The estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The shape changing feature of the conditional distributions makes these models capable of modelling time series with multimodal conditional distributions and with heteroscedasticity. The models are applied to two real data sets and compared with other competing models. The MAR models appear to capture features of the data better than other competing models do.

https://www.stat.fi/isi99/proceedings/arkisto/varasto/wong0064.pdf

On a mixture autoregressive model

We propose a mixture autoregressive conditional heteroscedastic (MAR-ARCH) model for modeling nonlinear time series. The models consist of a mixture of K autoregressive components with autoregressive conditional heteroscedasticity; that is, the conditional mean of the process variable follows a mixture AR (MAR) process, whereas the conditional variance of the process variable follows a mixture ARCH process. In addition to the advantage of better description of the conditional distributions from the MAR model, the MARARCH model allows a more flexible squared autocorrelation structure. The stationarity conditions, autocorrelation function, and squared autocorrelation function are derived. Construction of multiple step predictive distributions is discussed. The estimation can be easily done through a simple EM algorithm, and the model selection problem is addressed. The shape-changing feature of the conditional distributions makes these models capable of modeling time series with multimodal conditional distr...

On a mixture autoregressive conditional heteroscedastic model

SUMMARY We generalise the mixture autoregressive, MAR, model to the logistic mixture autoregressive with exogenous variables, LMARX, model for the modelling of nonlinear time series. The models consist of a mixture of two Gaussian transfer function models with the mixing proportions changing over time. The model can also be considered as a generalisation of the self-exciting threshold autoregressive, SETAR, model and the open-loop threshold autoregressive, TARSO, model. The advantages of the LMARX model over other nonlinear time series models include a wider range of shape-changing predictive distributions, the ability to handle cycles and conditional heteroscedasticity in the time series and better point prediction. Estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The models are applied to two real datasets and compared with other competing models.

https://www.jstor.org/stable/pdfplus/2673450.pdf

On a logistic mixture autoregressive model

This paper addresses the null distribution of the Lagrange-multiplier statistic for the threshold autoregression with conditional heteroscedasticity. The problem is nonstandard because the threshold parameter is a nuisance parameter which is absent under the null hypothesis. We generalise the results of Chan (1990) and Chan & Tong (1990) to show that the asymptotic null distribution of the Lagrange-multiplier statistic is a functional of a zero-mean Gaussian process. The generalisation is not direct as the conditional variance is changing and the unconditional distribution of the process variable is no longer normal. In some special cases, we can reduce the problem to the asymptotic distribution of certain functions of Brownian bridges and the upper percentage points can be tabulated as in Chan (1991). Monte Carlo experiments show that the approximation and the power of the test are quite good. We also demonstrate the importance of using our test if the true model has conditional heteroscedasticity.

Testing for threshold autoregression with conditional heteroscedasticity

The authors show how to extend univariate mixture autoregressive models to a multivariate time series context. Similar to the univariate case, the multivariate model consists of a mixture of stationary or nonstationary autoregressive components. The authors give the first and second order stationarity conditions for a multivariate case up to order 2. They also derive the second order stationarity condition for the univariate mixture model up to arbitrary order. They describe an EM algorithm for estimation, as well as a diagnostic checking procedure. They study the performance of their method via simulations and include a real application.

C. S. Wong

Papers

On a mixture autoregressive model

On a mixture autoregressive conditional heteroscedastic model

On a logistic mixture autoregressive model

Testing for threshold autoregression with conditional heteroscedasticity

On a mixture vector autoregressive model