Fisher information matrix of binary time series

TL;DR

This paper derives an exact Fisher information matrix for binary time series models, improving inference accuracy and confidence interval precision over traditional empirical methods, especially for short series.

Contribution

It introduces a novel exact conditional Fisher information matrix for logistic autoregressive models with endogenous covariates, applicable to any series length.

Findings

Exact Fisher information yields narrower confidence intervals.

Simulation studies confirm improved inference accuracy.

Application to real data shows increased statistical power.

Abstract

A common approach to analyzing categorical correlated time series data is to fit a generalized linear model (GLM) with past data as covariate inputs. There remain challenges to conducting inference for short time series length. By treating the historical data as covariate inputs, standard errors of estimates of GLM parameters computed using the empirical Fisher information do not fully account the auto-correlation in the data. To overcome this serious limitation, we derive the exact conditional Fisher information matrix of a general logistic autoregressive model with endogenous covariates for any series length $T$. Moreover, we also develop an iterative computational formula that allows for relatively easy implementation of the proposed estimator. Our simulation studies show that confidence intervals derived using the exact Fisher information matrix tend to be narrower than those utilizing the empirical Fisher information matrix while maintaining type I error rates at or below nominal levels. Further, we establish that the exact Fisher information matrix approaches, as T tends to infinity, the asymptotic Fisher information matrix previously derived for binary time series data. The developed exact conditional Fisher information matrix is applied to time-series data on respiratory rate among a cohort of expectant mothers where it is found to provide narrower confidence intervals for functionals of scientific interest and lead to greater statistical power when compared to the empirical Fisher information matrix.