Deterministic Trends
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
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.
Note: we do not difference the data first. The data is assumed to be non-stationary.
