The trend in a time series is a persistent rise or fall in the value of the series. The variable t is is often used to indicate the time period for the observation. For example, given monthly data that starts in January 1980, t=1 would represent January 1980, t=2 would represent February 1980 and so forth.
The trend in a series can take one of many different forms.
Linear trend
The model for a linear trend is Yt = β0 + β1 t + εt (additive model) or Yt = (β0 + β1 t) x εt (multiplicative model).
In this model, the coefficient β1 measures the change in Y over one time period. That is, Yt - Y(t-1) = ß1. The coefficients can be estimated by Ordinary Least Squares regression.
Quadratic trend
The quadratic model is 
. In this case, the rate of change in the variable Y is equal to β1 + β2 t at time period t. In contrast to the linear trend, the series does not increase by the same magnitude every period.
Exponential Trend
The model for an exponential trend is 
. We can rewrite this expression by taking the natural log of each side of the equation, so the model becomes 
. This model says that the log of the variable has a linear trend.
This model can only be used for series that are positive since the natural log of a negative number does not exist.
Stochastic Trend
The model for a stochastic trend is Yt = Yt-1 + ε~t. A series such as this is said to have a unit root.
Detrended data
Detrended data is data that has had the trend component removed. For an additive model, the detrended data is calculated by subtracting the estimated trend component from the observed data. For a linear trend:
Detrended Data = Yt - β0 + β1 t.
For a multiplicative model, the detrended data is calculated by dividing the observed series by the trend component. For a linear trend: