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Assume we have a time-series with a deterministic trend.

I'm wondering if the following model is well specified as an AR model:

$y_{t} = b_{0} + b_{1}t + b_{2}y_{t-1} + \epsilon_{t} \ \ \ \ \ \ $

Specifically, I'm interested in knowing if it's OK to put a time term and auto-regressive terms in the same model.

If it is not, what would be the better way of specifying the model?

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  • $\begingroup$ Well, it's not "an AR model", it's more complicated than that; it's a model with an AR component - but such a model can be written down - indeed you just did it. I imagine you mean to ask something else. $\endgroup$ – Glen_b Apr 11 '13 at 10:01
  • $\begingroup$ Thank you for your comment. I'm not asking whether it can be written down or not. I'm asking whether it is statistically well specified -i.e. if it has reliable statistical properties. Will the t and F-tests be biased? Will multicollinearity or serial correlation be introduced as a result of having both AR terms and the time term? That sort of thing. $\endgroup$ – J4y Apr 11 '13 at 10:39
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    $\begingroup$ "Will ... serial correlation be introduced as a result of having both AR terms and the time term?" -- clearly, since AR is already serially correlated; the time trend introduces an additional source of 'dependence'. Often this sort of thing can be identified, but there can be a number of issues. A couple of relevant papers - no.1 no.2 $\endgroup$ – Glen_b Apr 11 '13 at 11:24
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It's OK to put a time term and auto-regressive terms in the same model. But, the model may not be meaningful when $\epsilon_t$ is non stationary.

If the series is stationary after removing deterministic part, one can add deterministic part in the time series analysis.

In your case if $y_t$ is stationary after removing the time trend, you can proceed with your model.

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