I have a LOT of time series observations and I would like to estimate a simple AR(1) model $$y_t =c+ \phi y_{t-1}+ \varepsilon_t \qquad \varepsilon_t \sim \text{N}(0, \sigma^{2})$$ with parameters $\theta =\left\{ c, \phi,\sigma^{2} \right\}$. Because of the amount of observations I would like to use stochastic gradient descent (or "deterministic" Robbins and Monro algorithm) to estimate the model $$\theta^{i}_{t+1} = \theta^{i}_t + \gamma_t s^{i}_t , \quad \sum \gamma_t =\infty \quad \text{and}\quad \sum \gamma^{2}_t <\infty$$ where $\theta^{i}$ is an element of $\theta$ and $s^{i}_t$ is the derivative of log likelihood contribution at time t with respect to $\theta^{i}$.

I have the following questions:

• First of all is this supposed to work? What alternative online estimation procedures are out there? I have the feeling that this is really not robust and needs a lot of case by case tuning.

• How to set $\gamma_t$? I know the standard choice of $\gamma_t$ is $t^{-2/3}$ which satisfy the requirements. My experience so far is that I should set $\gamma_t$ differently for different parameters as the data is more informative about the constant $c$ than about $\sigma^2$ and the gradients have totally different magnitude. If i use the same gamma some of the parameters would diverge (too large step size) while others would converge slowly. Is there a "stochastic Newton-Raphson" type algorithm which would scale the gradients properly? I guess trying different combinations would work but there are too many combinations if I would set different gammas for different parameters.

Any comment is appreciated!

One assumption of stochastic gradient descent is that you should have independent identical distributed gradients, e.g. $s^i_t$ is independent over $t$, so that the law of large numbers ensures the stochastic gradient is a good approximation of the real gradient.
For $AR(1)$ and for most time series model, it is not.