On the Autocorrelation Structure and Identification of Some Bilinear Time Series (original) (raw)

ON THE IDENTIFICATION OF SOME BILINEAR TIME SERIES MODELS

Journal of Time Series Analysis, 1986

Abstract. In their book on bilinear time series models Granger and Andersen (1978, p. 43) dismiss the use of third order moments for identifying models on the grounds that for some bilinear models they will all be zero and hence are of no use in discriminating between true white noise and some bilinear models. However, in this paper it is shown that some of the third order moments do not vanish for some superdiagonal and diagonal bilinear models and the pattern of non zero moments can be used to discriminate between true white noise and these bilinear models and also between different bilinear models. Simulation experiments are used to study the applicability of theoretical results.

A note on recognizing autocorrelation and using autoregression

Agricultural and Forest Meteorology, 1999

The presence of autocorrelation in the analysis of a variable sampled sequentially at regular time intervals appears to be unknown to many agricultural meteorologists despite abundant documentation found in the traditional meteorological and statistical literature. It follows that the statistical consequences as well as methodological alternatives are also unknown. Through an example using paired radiometer observations, this note discusses recognition of autocorrelation as well as the importance of testing ordinary least squares regression parameters in the presence of autocorrelated residuals. An autoregression example is presented as one alternative way to analyze the given dataset. # 0168-1923/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 -1 9 2 3 ( 9 9 ) 0 0 0 2 5 -8

Analysis of the correlation structure of square time series

Journal of Time Series Analysis, 2004

This paper analyses the asymptotic behaviour of the autocorrelation structure exhibited by squares of time series with a Wold expansion where the input error is a sequence of random variables with mean zero and finite kurtosis. Two important cases are discussed: (i) when the errors are independent and, (ii) when the errors are uncorrelated but their squares are correlated. Both situations are addressed when the process exhibits short or long memory. Consequences of these results on certain models widely used in many disciplines are also discussed.

On the Structure of Third Order Moment and Identification of Bilinear Time Series Model

Calcutta Statistical Association Bulletin, 1988

Kumar(l986) has proposed the criterion of third order moment for the identification of bilinear time series model and discussed its properties with respect to a simple bilinear model, In this paper, we have derived the explicit expressions of third order moment with respect to two terms diagonal model and discussed their properties in identifying the structure of a diagonal model.

ON THE ESTIMATION OF THE AUTOCORRELATION FUNCTION

The autocorrelation function has a very important role in several application areas involving stochastic processes. In fact, it assumes the theoretical base for Spectral analysis, ARMA (and generalizations) modeling, detection, etc. However and as it is well known, the results obtained with the more current estimates of the autocorrelation function (biased or not) are frequently bad, even when we have access to a large number of points. On the other hand, in some applications, we need to perform fast correlations. The usual estimators do not allow a fast computation, even with the FFT. These facts motivated the search for alternative ways of computing the autocorrelation function. 9 estimators will be presented and a comparison in face to the exact theoretical autocorrelation is done. As we will see, the best is the AR modified Burg estimate.

New types of nonlinear auto-correlations of bivariate data and their applications

Applied Mathematics and Computation, 2012

The paper introduces new types of nonlinear correlations between bivariate data sets and derives nonlinear auto-correlations on the same data set. These auto-correlations are of different types to match signals with different types of nonlinearities. Examples are cited in all cases to make the definitions meaningful. Next correlogram diagrams are drawn separately in all cases; from these diagrams proper time lags/delays are determined. These give rise to independent coordinates of the attractors. Finally three dimensional attractors are reconstructed in each case separately with the help of these independent coordinates. Moreover for the purpose of making proper distinction between the signals, the attractors so reconstructed are quantified by a new technique called 'ellipsoid fit'.

Covariance analysis of the squares of the purely diagonal bilinear time series models

Brazilian Journal of Probability and Statistics, 2011

The covariance structure among other properties of the square of the purely diagonal bilinear time series model is obtained. The time series properties of these squares are compared with those of the linear moving average time series model. It was discovered that the square of a linear moving average process is also identified as a moving average process whereas, while

Estimation and Prediction for Subset Bilinear Time Series Models with Applications

An Introduction to Bispectral Analysis and Bilinear Time Series Models, 1984

It has been pointed out earlier that some of the coefficients for the full bilinear models of the form (5.8.1) when fitted to a realisation may be "small" when compared to other coefficients. Therefore, it is useful to see whether it is possible to fit a subset bilinear model to the data which leads to a parsimonious representation. In this chapter, we consider the estimation of a subset bilinear model and we give an algorithm for the estimation of its parameters (see also Gabr and Subba Rao, 1981). The method is illustrated with real data. A comparison is then made between the forecats obtained from the subset bilinear models and other time series models. Some comments about the transformation of the series are included. (See Subba Rao and Gabr, 1981). 6.2 AN ALGORITHM FOR FITTING SUBSET BILINEAR MODELS In general, bi 1 inear model s are models whi ch are 1 inear in the's tate' and linear in the errors, but not jOintly. In the description of the model (5.8.1), the 'linear part' of the model as described by the autoregressive model, explains the 'linear variation' in the series, and the rest of the variation is explained by the non-linear terms {b ij X ti e t _ j }. Of course, this interpretation is only heuristic, since the linear terms and the non

On the Third-Order Moment Structure and Bispectral Analysis of Some Bilinear Time Series

Journal of Time Series Analysis, 1988

For the bilinear time series model X, = p X ,-k et-, + e , , k 1, k = 1 and k i 1 formulae for the third-order theoretical moments and an expression for the bispectral density function are obtained. These results can be used to distinguish between bilinear models and white noise and, in general, linear models. Furthermore, they give an indication of the type combination (k, Q in the above model. The modulus of the bispectral density function of the above bilinear time series model for different combinations of (k, Q and values of /? are computed and the properties are studied.

Lag-one autocorrelation in short series: Estimation and hypotheses testing

Psicológica, 2010

In the first part of the study, nine estimators of the first-order autoregressive parameter are reviewed and a new estimator is proposed. The relationships and discrepancies between the estimators are discussed in order to achieve a clear differentiation. In the second part of the study, the precision in the estimation of autocorrelation is studied. The performance of the ten lag-one autocorrelation estimators is compared in terms of Mean Square Error (combining bias and variance) using data series generated by Monte Carlo simulation. The results show that there is not a single optimal estimator for all conditions, suggesting that the estimator ought to be chosen according to sample size and to the information available on the possible direction of the serial dependence. Additionally, the probability of labelling an actually existing autocorrelation as statistically significant is explored using Monte Carlo sampling. The power estimates obtained are quite similar among the tests associated with the different estimators. These estimates evidence the small probability of detecting autocorrelation in series with less than 20 measurement times.