In this paper, we give a general time-varying parameter model, where the multidimensional parameter follows a continuous local martingale. As such, we call it the locally parametric model  . The quantity of interest is defined as the integrated value over time of the parameter process \Theta := T^{-1} \int_0^T \theta_t^* dt. We provide a local parametric estimator  of \Theta based on the original (non time-varying) parametric model estimator and conditions under which we can show consistency and the corresponding limit distribution. We show that the LPM class contains some models that come from popular problems in the high-frequency financial econometrics literature (estimating volatility, high-frequency covariance, integrated betas, leverage effect, volatility of volatility), as well as a new general asset-price diffusion model which allows for endogenous observations and time-varying noise which can be auto-correlated and correlated with the efficient price and the sampling times. Finally, as an example of how to apply the limit theory provided in this paper, we build a time-varying friction parameter extension of the (semiparametric) model with uncertainty zones (Robert and Rosenbaum (2012)) , which is noisy and endogenous, and we show that we can verify the conditions for the estimation of integrated volatility  .
In this paper, we give a general time-varying parameter model, where the multidimensional parameter follows a continuous local martingale. As such, we call it the locally parametric model (LPM) . The quantity of interest is defined as the integrated value over time of the parameter process \Theta := T^{-1} \int_0^T \theta_t^* dt. We provide a local parametric estimator (LPE) of \Theta based on the original (non time-varying) parametric model estimator and conditions under which we can show the central limit theorem. As an example of how to apply the limit theory provided in this paper, we build a time-varying friction parameter extension of the (semiparametric) model with uncertainty zones (Robert and Rosenbaum (2012)) and we show that we can verify the conditions for the estimation of integrated volatility . Moreover, practical applications in time series, such as the optimal block length and local bias-correction, are discussed and numerical simulations are carried on the local MLE of a time-varying parameter MA(1) model to illustrate them .
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(LPM)
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(LPE)
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consistency and the corresponding limit distribution. We show that the LPM class contains some models that come from popular problems in the high-frequency financial econometrics literature (estimating volatility, high-frequency covariance, integrated betas, leverage effect, volatility of volatility), as well as a new general asset-price diffusion model which allows for endogenous observations and time-varying noise which can be auto-correlated and correlated with the efficient price and the sampling times. Finally, as
the central limit theorem. As
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, which is noisy and endogenous, and
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. Moreover, practical applications in time series, such as the optimal block length and local bias-correction, are discussed and numerical simulations are carried on the local MLE of a time-varying parameter MA(1) model to illustrate them