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I have a stationary (constant mean, constant standard deviation) time series $y_t$, that contains some missing values and some incorrect values (values so far from the distribution they are obviously wrong).

I have a simple method to detect these incorrect values which works well. However, I would like to know if there are any standard or canonical ways of imputing these missing and wrong values that does not have look ahead bias i.e if we are at some point $t$ in the series, we can only use data and information from points at $k\leq t$.

What I have tried so far is simply to impute the wrong value with the median of the previous $n$ values. But I’m looking for a method that perhaps uses more descriptive statistics of the distribution of the time series.

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Fitting an state-space model and using smoothed values (rather than predicted of filtered values) would take care of your requirement of no look ahead bias: the smoother basically interpolates data using all past, present and future information available.

One step further would be to use the simulation smoother, which would give you not fitted values using all information, but rather simulated values. You could run the simulation smoother a number of times and have not one but several imputed values, giving you an idea of the uncertainty of the imputations. much in the spirit of multiple imputation (EXCEPT for the fact that the uncertainty in the estimated model parameters is not accounted for).

I find the book Durbin & Koopman and excellent introduction to the underlying theory.

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