# #StackBounty: #time-series #estimation #stationarity #garch #estimators Moving estimators for nonstationary time series, like loglikeli…

### Bounty: 50

While in standard (“static”) e.g. ML estimation we assume that all values are from a distribution of the same parameters, in practice we often have nonstationary time series: in which these parameters can evolve in time.

It is usually considered by using sophisticated models like GARCH conditioning sigma with recent errors and sigmas, or Kalman filters – they assume some arbitrary hidden mechanism.

I have recently worked on a simpler and more agnostic way: use moving estimator, like loglikelihood with exponentially weakening weights of recent values:
$$theta_T=argmax_theta l_Tqquad textrm{for}qquad l_T= sum_{t
intended to estimate local parameters, separately on each position. We don’t assume any hidden mechanism, only shift the estimator.

For example it turns out that EPD (exponential power distribution) family $$rho(x) propto exp(-|x|^kappa)$$, which covers Gaussian ($$kappa=2$$) and Laplace ($$kappa=1$$) distributions, can have cheaply made such moving estimator (plots below), getting much better loglikelihood for daily log-returns of Dow Jones companies (100 years DJIA, 10 years individual), even exceeding GARCH: https://arxiv.org/pdf/2003.02149 – just using the $$(sigma_{T+1})^kappa=eta (sigma_{T})^kappa+(1-eta)|x-mu|^kappa$$ formula: replacing estimator as average with moving estimator as exponential moving average:

I have also MSE moving estimator for adaptive least squares linear regression: page 4 of https://arxiv.org/pdf/1906.03238 – can be used to get adaptive AR without Kalman filter, also analogous approach for adaptive estimation of joint distribution with polynomials: https://arxiv.org/pdf/1807.04119

Are such moving estimators considered in literature?

What applications they might be useful for?

Get this bounty!!!

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