## Document Type

Discussion Paper

## Publication Date

8-11-2020

## CFDP Number

2250

## CFDP Pages

56

## Journal of Economic Literature (JEL) Code(s)

C14, C22

## Abstract

Limit distribution theory in the econometric literature for functional coefficient cointegrating (FCC) regression is shown to be incorrect in important ways, influencing rates of convergence, distributional properties, and practical work. In FCC regression the cointegrating coefficient vector \beta(.) is a function of a covariate z_t. The true limit distribution of the local level kernel estimator of \beta(.) is shown to have multiple forms, each form depending on the bandwidth rate in relation to the sample size n and with an optimal convergence rate of n^{3/4} which is achieved by letting the bandwidth have order 1/n^{1/2}.when z_t is scalar. Unlike stationary regression and contrary to the existing literature on FCC regression, the correct limit theory reveals that component elements from the bias and variance terms in the kernel regression can both contribute to variability in the asymptotics depending on the bandwidth behavior in relation to the sample size. The trade-oﬀ between bias and variance that is a common feature of kernel regression consequently takes a different and more complex form in FCC regression whereby balance is achieved via the dual-source of variation in the limit with an associated common convergence rate. The error in the literature arises because the random variability of the bias term has been neglected in earlier research. In stationary regression this random variability is of smaller order and can correctly be neglected in asymptotic analysis but with consequences for ﬁnite sample performance. In nonstationary regression, variability typically has larger order due to the nonstationary regressor and its omission leads to deficiencies and partial failure in the asymptotics reported in the literature. Existing results are shown to hold only in scalar covariate FCC regression and only when the bandwidth has order larger than 1/n and smaller than 1/n^{1/2}. The correct results in cases of a multivariate covariate z_t are substantially more complex and are not covered by any existing theory. Implications of the findings for inference, confidence interval construction, bandwidth selection, and stability testing for the functional coefficient are discussed. A novel self-normalized t-ratio statistic is developed which is robust with respect to bandwidth order and persistence in the regressor, enabling improved testing and confidence interval construction. Simulations show superior performance of this robust statistic that corroborate the ﬁnite sample relevance of the new limit theory in both stationary and nonstationary regressions.

## Recommended Citation

Phillips, Peter C. B. and Wang, Ying, "When Bias Contributes to Variance: True Limit Theory in Functional Coefficient Cointegrating Regression" (2020). *Cowles Foundation Discussion Papers*. 2558.

https://elischolar.library.yale.edu/cowles-discussion-paper-series/2558

*Supplemental material*