In statistics, the term linear model refers to any model which assumes linearity in the system.
The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model.
However, the term is also used in time series analysis with a different meaning.
In each case, the designation "linear" is used to identify a subclass of models for which substantial reduction in the complexity of the related statistical theory is possible.
For the regression case, the statistical model is as follows.
are random variables representing errors in the relationship.
The "linear" part of the designation relates to the appearance of the regression coefficients,
Alternatively, one may say that the predicted values corresponding to the above model, namely are linear functions of the
are determined by minimising a sum of squares function From this, it can readily be seen that the "linear" aspect of the model means the following: An example of a linear time series model is an autoregressive moving average model.
} in a time series can be written in the form where again the quantities
are random variables representing innovations which are new random effects that appear at a certain time but also affect values of
In this instance the use of the term "linear model" refers to the structure of the above relationship in representing
as a linear function of past values of the same time series and of current and past values of the innovations.
[1] This particular aspect of the structure means that it is relatively simple to derive relations for the mean and covariance properties of the time series.
, as it would be in the case of a regression model, which looks structurally similar.