Practical Time Series Analysis

Trend models

This type of model aims to capture the long run trend in the time series that can be fitted as linear regression of the time index. When the time series does not exhibit any periodic or seasonal fluctuations, it can be expressed just as the sum of the trend and the zero mean model as x_t = μ(t) + y_t, where μ(t) is the time-dependent long run trend of the series.

The choice of the trend model μ(t) is critical to correctly capturing the behavior of the time series. Exploratory data analysis often provides hints for hypothesizing whether the model should be linear or non-linear in t. A linear model is simply μ(t) = wt + b, whereas quadratic model is μ(t) = w₁t + w₂t² + b. Sometimes, the trend can be hypothesized by a more complex ...