This question is addressed to expert in econometrics. I generally fit econometric models and statistical learning models to financial time series and some discretionary traders usually asked me if I try to "catch momentum or mean-reversion" and I always reply to them "both". Am I right?

Using the best fit, for instance using AIC to select a vector ARMA, I should be able to capture the dynamic of the financial time series both auto correlation and causality/cross-correlation. Depending on the input I could have a forecast telling me the prices are trending or mean-reverting.

I realised though that the vector ARMA can rarely catch both effect at the same time. For instance an AR(1) the simplest model is always a mean-reverting model (if stationary). An integrated AR(1) (ARIMA(1,1,0)) is always a momentum model, as the change of prices will converge to the mean change of price (if the mean is positive and the mean is large compare to the standard deviation, this is a momentum model). Can more complex models Vector ARMA achieve to catch both dynamic? Or is it always one or the other?

  • $\begingroup$ I think its one or the other. $\endgroup$ May 2 '17 at 18:17

It's one or the other, depending on how the parameter estimates come out. Here's an example. Say we're modelling log-returns. If we have an ARIMA(1,1,0) model: $$ x_t = \mu + \phi(x_{t-1} - \mu) + \epsilon_t, $$ then the conditional mean/1-step ahead prediction is $\mu(1 - \phi) + \phi x_{t}$. Let's always assume that $\mu > 0$, and that we don't have to worry about trading costs.

  1. If we assume that $0 < \phi < 1$, then we are supposed to sell when $$ x_t < \frac{\mu(\phi-1)}{\phi} < 0. $$ Notice that the right hand side is negative, so we are supposed to sell when we experienced a significantly negative return. On the other hand, we buy when that inequality is reversed. So we buy when it's positive, or not-so-negative.

  2. Now if $-1 < \phi < 0$ then we buy when $$ x_t < \frac{\mu(\phi-1)}{\phi}. $$ The right hand side here is positive this time. This means we only don't buy when we experienced a significantly positive return.

So, the first model induces a kind of "momentum" strategy, and the second induces a "reversion strategy." Mode (1) can be described as "never take profits but bail out of positions when they start going really wrong", and mode (2) can be seen as "take profits early and never abandon your positions when they move against you."

  • $\begingroup$ @Tayor: This is interesting but one question: Are you saying that the first case is kind of "momentum" and the second case is kind of "reversion". If so, then it's a nice example but I would still argue that you can't do both because the sign of the AR(1) coefficient has to be positive or negative so you can only do one thing at a time. Also, if the sign is frequently changing over time so that you can do both, I'd be worried about the appropriateness of the model. Still, a neat example and thanks. $\endgroup$
    – mlofton
    Oct 31 '18 at 14:36
  • $\begingroup$ @mlofton no problemo. Yeah i guess you're right: the first is a momentum strategy, and the second is a reversion one. Let me edit this a little bit. $\endgroup$
    – Taylor
    Oct 31 '18 at 15:43
  • $\begingroup$ Thanks for edit. But it is doing both if the AR coefficient sign changes often enough. I'm just saying that, if it changes, I'd be less comfortable with whatever model is being estimated because, generally speaking, sign changes of model coefficients are not a good thing. Still, very interesting regardless. $\endgroup$
    – mlofton
    Oct 31 '18 at 18:24

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