Interval-valued time series models: Estimation based on order statistics exploring the Agriculture Marketing Service data
The current regression models for interval-valued data ignore the extreme nature of the lower and upper bounds of intervals. A new estimation approach is proposed; it considers the bounds of the interval as realizations of the max/min order statistics coming from a sample of nt n t random draws from the conditional density of an underlying stochastic process {Yt} { Y t } . This approach is important for data sets for which the relevant information is only available in interval format, e.g., low/high prices. The interest is on the characterization of the latent process as well as in the modeling of the bounds themselves. A dynamic model is estimated for the conditional mean and conditional variance of the latent process, which is assumed to be normally distributed, and for the conditional intensity of the discrete process {nt} { n t } , which follows a negative binomial density function. Under these assumptions, together with the densities of order statistics, maximum likelihood estimates of the parameters of the model are obtained. They are needed to estimate the expected value of the bounds of the interval. This approach is implemented with the time series of livestock prices, of which only low/high prices are recorded making the price process itself a latent process. It is found that the proposed model provides an excellent fit of the intervals of low/high returns with an average coverage rate of 83%. In addition, a comparison with current models for interval-valued data is offered.
order statistics interval-valued data intensity maximum likelihood estimation