Testing the Elasticity of Volatility with Respect to the Level of An Integrated Process Masahito Kobayashi a , Michael McAleer b a Faculty of Economics, Yokohama National University (mkoba@ynu.ac.jp) b Department of Economics, University of Western Australia Abstract: The volatility, or conditional variance, of some variables in economics and finance can be proportional to a power function of the levels. It is shown that this process, which is known as the constant elasticity of volatility process, can be generated by the inverse Box-Cox transformation of an integrated series with small innovations. A test is proposed for the hypothesis that the power parameter, or volatility elasticity, has a specific value, when the innovation follows a GARCH(p,q) process. The test statistic detects the correlation between the conditional variance and the level of the integrated process, is shown to be a function of Brownian motions under the null hypothesis, and has a nonstandard asymptotic distribution. Keywords: Box-Cox transformation, GARCH, Integrated processes, Constant elasticity of volatility. 1. INTRODUCTION It is widely accepted in economics and finance that the volatility of the short-term interest rate, namely the conditional variance of interest rate changes, is sensitive to its level. For example, the Cox, Ingersoll, and Ross (1985) model assumes that the conditional volatility of changes in the interest rate is proportional to the level of the interest rate. Although the Cox, Ingersoll, and Ross model was developed to analyze a single-factor general equilibrium term structure, it has been used extensively in the analysis of valuation models for contingent claims that are sensitive to interest rates. Marsh and Rosenfeld (1983) assume that volatility is proportional to the square of the interest rate, so that the interest rate follows a geometric Brownian motion process. Courtadon (1982) uses a similar process to develop a model of discount bond option prices. Assuming that volatility is proportional to the cube of the interest rate, Constantinides and Ingersoll (1984) value bonds in the presence of taxes. A useful comparison of alternative economic models of the dynamics of short-term interest rate volatility is given in Chan et al. (1992). For recent developments of the constant elasticity of volatility (CEV) process, see Conley et al. (1997) and Smith (2002). The magnitude of the elasticity of volatility with respect to the level of interest rates has been a key empirical issue, but no consensus seems to have yet been reached. Chan et al. (1992) reported that the actual elasticity is higher than those typically assumed in theory. However, Brenner et al. (1996) obtained a lower estimate of the elasticity under the assumption of serially correlated volatility. In particular, they suggested that the reported high elasticity could be explained by neglected time-varying (conditional) heteroscedasticity. In addition to conditional heteroscedasticity, it is necessary to deal with nonstationarity appropriately in order to estimate the dynamics of volatility. It is natural to propose that short-term interest rates follow a random walk process, as mean reversion is rarely supported in empirical analysis. Owing to the presence of an integrated process, the estimators and test statistics are likely to have nonstandard distributions, so that conventional inferences might be invalid. However, to date there does not seem to have been any development of statistical tools to accommodate such nonstationary data, in which volatility depends upon the level of an integrated process. In this paper we propose a test for the hypothesis that volatility is proportional to a power transformation of nonstationary interest rates, and derive the asymptotic distribution of the test statistic when the innovation follows a GARCH(p,q) process. The test statistic is expressed as a function of Brownian motions under the null, and has a nonstandard asymptotic distribution. Although the motivation of the test procedure is based on analyzing short-term interest rates, the method developed in the paper is applicable to a variety of other data. For example, as argued in