Overview
A trend in a time series is a pattern in the data, such as a general upward slope or wavelike behaviour that persists over time.
Deterministic vs Stochastic Trends
The difference between deterministic versus stochastic trends is subtle.
- Deterministic Trends - A deterministic trend is a trend that is not random. That is, the time series can be modelled as a deterministic function of time plus a random component.
- Stochastic Trends - Stochastic trends are trends that may appear in any particular realization of a time series, but is not present in all points in the sample space. Stochastic trends appear out of the randomness, as their name suggests, and are not trends that can be relied on to make forecasts.
Deterministic vs Stochastic Trends
Deterministic Trend A deterministic trend is a trend that is not random. That is, the time series can be modeled as a deterministic function of time plus a random component.
{% Y_t = f(t) + X_t %}
where {% X_t %} is a stationary process.
A deterministic trend shows a great degree of stability to the trend.
The random walk often displays the presence of stochastic trends. Trends within a random walk tend not to be stable, and can reverse course or vary length.
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