Overview
A determistic trend is a trend that is not random. That is, the time series can be modeled as a determistic function of time plus a random component.
{% Y_t = f(t) + X_t %}
where {% X_t %} is a stationary process.
Deciding whether a trend in a time series is determistic can be a bit of an art. The trend must not be random, even though there will be random fluctuations around the trend line.
An example may be analyzing goverment spending. The baseline of the government spending is largely determined by law, that is, the legislature determines the budget. However, the government may spend more or less based on certain random components. That is, one could assert that the trend of goverment spending has already been specified by law, but the exact amount may have a random component.
Fitting a Deterministic Trend and De-trending
While a deterministic trend is not random, its exact form may not be known ahead of time. That is, one may need to fit a function to the data in order to determine what the likely trend actually is.
The typical way to fit a deterministic trend is to fit a polynomial to the raw un-differenced data using OLS regression.
That is, the functional trend that we are looking for is assumed to be of the form
{% f(t) = \alpha +\beta_0 t + \beta_1 t^2 + ... + \beta_n t ^n %}
The time series can then be de-trended using subtraction
{% X_t = Y_t - f(t) %}