Short and long-run trend I have a time series, let's call it $y(t)$. Augmented Dickey-Fuller and Phillips-Perron tests indicate no unit root but a deterministic trend. The series is strongly autocorrelated. I estimated the following short-run dynamic model: 
$$y(t) = \alpha + \beta\cdot y(t-1) + \gamma\cdot t + \varepsilon_t$$
where $\alpha$, $\beta$ and $\gamma$ are estimated coefficients. The estimated model has no statistical anomalies and the coefficients are significantly different from zero. The estimated value of $\gamma$ measures a trend. However, a long-run trend can be estimated by calculating $\gamma/(1-\beta)$. 
We can estimate directly the following static, long-run model: 
$$y(t) = \alpha + \gamma\cdot t + \varepsilon_t.$$
Question: What is the best, consistent model to apply in an analysis of a deterministic trend?
 A: There is no quick solution to the problem, since the deterministic trend is just a function of $t$. We may denote this trend by $f(t)$ and it is not evident that the trend is linear.
So some quick tips what to do in this case:


*

*Plot the original data, though the noise ratio could be huge therefore...

*Make technical decomposition of your time series into Trend + Cycle + Seasonal + Residual components and try to plot them (implementation of this step depends on the software you do use), usual approach is to apply Hodrick-Prescott filter for example. 

*After you decide on the form of trend you may conclude that some polynomial $\sum_{i=0}^n \alpha_i t^i$ is suitable, to check this you may apply $n$ times difference operator, the reminder term should be close to white noise then...

*If it is not search in addition for the best fit from the $(S)ARMA(p, q)$ class (since there is no unit root in your case, in general you will work with SARIMA, or some advanced users would go for SARFIMA-GARCH may be :)).

*Decide on the efficient estimate of the final model on the basis of any information criteria you think is relevant ($BIC$ for example).


Anyway, the time series models are needed for short term forecasting needs (without clear structure and theoretical restrictions long-term forecasts are useless), short and long term betas therefore are just a matter of curiosity. Since you do not have a structural model here you do not need to care about consistency. For forecasting needs any parsimonious (the simpler the better) model with lowest information criteria (the best) will do nicely in the short run, even if the true data generating process is more complicated.
