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. We propose a new estimation approach that considers the bounds of the interval as realizations of the max/min order statistics coming from a sample of n_t random draws from the conditional density of an underlying stochastic process {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. We are interested in the characterization of the latent process as well as in the modeling of the bounds themselves. We estimate a dynamic model 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 {n_t}, which follows a negative binomial density function. Under these assumptions, together with the densities of order statistics, we obtain maximum likelihood estimates of the parameters of the model, which are needed to estimate the expected value of the bounds of the interval. We implement this approach with the time series of livestock prices, of which only low/high prices are recorded making the price process itself a latent process. We find that the proposed model provides an excellent fit of the intervals of low/high returns with an average coverage rate of 83%. We also offer a comparison with current models for interval-valued data.