# online estimation of autoregressive process

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You may be interested in reading the paper Efficient Online Estimation of Autoregressive Parameters by T. Sharia.

It is instructive to work out the details of the AR(1) case beforehand. In this case you have the model

$$x_{t+1} = \beta_t x_t + \varepsilon_t$$

with

$$\beta_t = \frac{\sum_{s=1}^tx_sx_{s-1}}{\sum_{s=1}^tx_{s-1}^2}$$

By considering $\beta_{t+1} - \beta_t$ and rearranging, you can derive

$$\beta_{t+1} = \beta_t + ( x_{t+1} - \beta_tx_t) \frac{x_t}{I_{t+1}}$$

$$I_{t+1} = I_t + x_t^2$$

where $I_t$ is the quantity $\sum_{s=1}^tx_{t-1}^2$.

This has an intuitive appeal - the autoregressive estimator at the next time step is the current autoregressive parameter, plus a correction proportional to the prediction error at the current step.

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