Indirect Inference with Time Series Observed with Error

We propose the indirect inference estimator as a consistent method to estimate the parameters of a structural model when the observed series are contaminated by measurement error by considering the noise as a structural feature. We show that the indirect inference estimates are asymptotically biased if the error is neglected. Instead, if the condition for identification is satisfied, the measurement error parameters can be estimated jointly with the structural ones leading to a consistent and asymptotically Gaussian estimator. The issues of identification and misspecification of ME are discussed in detail. Based on the encompassing principle, we show that II can still be consistent for the parameters of interest when the conditional distribution of ME is misspecified, as long as the structural model for the observed series encompasses the auxiliary. We illustrate the reliability of this procedure in the estimation of stochastic volatility models based on realized volatility measures contaminated by microstructure noise. The empirical application stresses the importance of a realistic specification of the microstructure noise distribution to match the features of the observed log-returns at high frequencies