Updated: Nov 2, 2022
Estimating a trend in a time series and using that trend to predict future values of the time series is the simplest method of forecasting. We can examine two types of trends, linear trends, and log-linear trends, and discuss how to choose between them.
Linear Trend Models
The simplest type of trend is a linear trend, one in which the dependent variable changes at a constant rate with time. If a time series has a linear trend, then we can model the series using the following regression equation:
Log-Linear Trend Models
Sometimes a linear trend does not correctly model the growth of a time series. In those cases, we often find that fitting a linear trend to a time series leads to persistent rather than uncorrelated errors. If the residuals from a linear trend model are persistent, we then need to employ an alternative model satisfying the conditions of linear regression.
For financial time series, an important alternative to a linear trend is a log-linear trend. Log-linear trends work well in fitting time series that have exponential growth.
Exponential growth means constant growth at a particular rate. So, annual growth at a constant rate of 5 percent is exponential because the series continues to increase without an upper bound.
How does exponential growth work?
Suppose we describe a time series by the following equation:
Exponential growth is growth at a constant rate with continuous compounding.
If we take the natural log the equation becomes:
Therefore, if a time series grows at an exponential rate, we can model the natural log of that series using a linear trend.
Consequently, if we want to use a log-linear model, we must estimate the following equation:
Formulas and knowledge are taken from:
DeFusco, Richard A. Quantitative Investment Analysis. Wiley, 2007.