Non-standard Confidence Sets for Ratios and Tipping Points with Applications to Dynamic Panel Data

We study estimation uncertainty when the object of interest contains one or more ratios of parameters. The ratio of parameters is a discontinuous parameter transformation; it has been shown that traditional confidence intervals often fail to cover this true ratio with very high probability. Constructing confidence sets for ratios using Fieller’s method is a viable solution as the method can avoid the discontinuity problem. This paper proposes an extension of the multivariate Fieller method beyond standard estimators, focusing on asymptotically mixed normal estimators that commonly arise in dynamic panel polynomial regression with persistent covariates. We discuss the cases where the underlying estimators converge to various distri- butions, depending on the persistence level of the covariates. We show that the asymptotic distribution of the pivotal statistic used for constructing a Fieller’s confidence set remains a standard Chi-squared distribution regardless of rates of convergence, thus the rates are being ‘self-normalized’ and can be unknown. A simulation study illustrates the finite sample properties of the proposed method in a dynamic polynomial panel. Our method is demonstrated to work well in small samples, even when the persistence coefficient is unity.