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I have to estimate a price time series for my thesis. We found that the price is reasonably well described by a process on this form, which looks like an ARMA(p, q) process but has an extra term:

$$x_t = \mu + \sum_{i=1}^p \alpha_i x_{t-i} + \sum_{i=1}^q \beta_i \varepsilon_{t-i} + \varepsilon_t + \underbrace{\sum_{i=1}^r \gamma_i y_{t-i}}_\text{this is new}$$

The $y_t$'s are exogenous input. Does this kind of process have a name, and if so, could anyone provide pointers to relevant literature about it?

Also, statistics and time series analysis is really not my forte, so if anyone could provide input on (1) what properties this series has and (2) whether this model is a sensible at all, that would be great. Thanks in advance.

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You're probably looking for ARMAX, which is ARMA with exogenous inputs.

Wikipedia covers it here

And Rob J Hyndman has a write up on his blog which gives is a slightly more extended treatment than the wikipedia article

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The name you're looking for is ARIMA - Auto Regressive Integrated Moving Average. There is also such a thing as VARMA, where one of your regressors is an additional time series.

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  • $\begingroup$ Habu, I think @user4853516 has has the correct answer while yours does not really hit the point. I suggest you to consider removing your answer. $\endgroup$
    – Richard Hardy
    May 1, 2015 at 9:40
  • $\begingroup$ @RichardHardy - ARMAX models use nondifferenced covariates, while the equation provided by the OP appears to include a differenced one (note the t-i subscript for the exogenous variable). That would, IMO, place the model closer to a classical ARIMA than ARMAX; should the exogenous variable be another timeseries, then the model would more aptly be described as VARMA. The RJH blog post that also discusses the impact of using a nondifferenced covariate on the interpretation of model results. I believe my answer is valid and would prefer to leave it here unless the OP specifically states otherwise $\endgroup$
    – habu
    May 1, 2015 at 10:09
  • $\begingroup$ OK, I accept your point about differencing. $\endgroup$
    – Richard Hardy
    May 1, 2015 at 12:21
  • $\begingroup$ Hi: What you have is an autoregressive distributed lag with p terms in the AR component, r terms in the exogenous variable ( technically, the non-autoregressive term ) and an ma(q) term. So an ADL(p,r) with an MA(q) error term .There is tons of literature on ADL models. You may be able to eliminate the ma(q) error term if the x_t already have an error structure because the autoregressive component may already take care of the lagged e_t's. Assuming you can ( if you can't it's still possible but tricker ), I would use numerical optimization to minimize sum (residuals^2). $\endgroup$
    – mlofton
    May 1, 2015 at 23:54
  • $\begingroup$ @mlofton I think you should post that as an answer to the original question, if that was your intention, rather than as a comment to habu's answer. $\endgroup$
    – Silverfish
    Sep 9, 2015 at 13:49
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Hi: What you have is an autoregressive distributed lag with p terms in the AR component, r terms in the exogenous variable ( technically, the non-autoregressive term ) and an ma(q) term. So an ADL(p,r) with an MA(q) error term .There is tons of literature on ADL models. You may be able to eliminate the ma(q) error term if the x_t already have an error structure because the autoregressive component may already take care of the lagged e_t's. Assuming you can ( if you can't it's still possible but tricker ), I would use numerical optimization to minimize sum (residuals^2). – mlofton May 1 at 23:54

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