This paper proposes a procedure to test for the correct specification of the functional form of the volatility process, within the class of eigenfunction stochastic volatility models (Meddahi, 2001). The procedure is based on the comparison of the moments of realized volatility measures with the corresponding ones of integrated volatility implied by the model under the null hypothesis. We first provide primitive conditions on the measurement error associated with the realized measure, which allow to construct asymptotically valid specification tests. Then we establish regularity conditions under which realized volatility, bipower variation (Barndorff-Nielsen & Shephard, 2004d), and modified subsampled realized volatility (Zhang, Mykland & Aït Sahalia, 2003), satisfy the given primitive assumptions.\ud Finally, we provide an empirical illustration based on three stock from the Dow Jones Industrial Average
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