Testing for the Box–Cox parameter for an integrated process

This paper analyses the constant elasticity of volatility (CEV) model suggested by Chan et al. [K.C. Chan, G.A. Karolyi, F.A. Longstaff, A.B. Sanders, An empirical comparison of alternative models of the short-term interest rate, Journal of Finance 47 (1992) 1209–1227]. The CEV model without mean reversion is shown to be the inverse Box–Cox transformation of integrated processes asymptotically. It is demonstrated that the maximum likelihood estimator of the power parameter has a nonstandard asymptotic distribution, which is expressed as an integral of Brownian motions, when the data generating process is not mean reverting. However, it is shown that the t-ratio follows a standard normal distribution asymptotically, so that the use of the conventional t-test in analyzing the power parameter of the CEV model is justified even if there is no mean reversion, as is often the case in empirical research. The model may applied to ultra high frequency data.