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In statistics, the term linear model is used in different ways according to the context. 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. Given a (random) sample the relation between the observations Y_{i} and the independent variables X_{ij} is formulated as
where may be nonlinear functions. In the above, the quantities ε_{i} are random variables representing errors in the relationship. The "linear" part of the designation relates to the appearance of the regression coefficients, β_{j} in a linear way in the above relationship. Alternatively, one may say that the predicted values corresponding to the above model, namely
are linear functions of the β_{j}.
Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β_{j} 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. Here the model for values {X_{t}} in a time series can be written in the form
where again the quantities ε_{t} are random variables representing innovations which are new random effects that appear at a certain time but also affect values of X at later times. In this instance the use of the term "linear model" refers to the structure of the above relationship in representing X_{t} 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 relative simple to derive relations for the mean and covariance properties of the time series. Note that here the "linear" part of the term "linear model" is not referring to the coefficients φ_{i} and θ_{i}, as it would be in the case of a regression model, which looks structurally similar.
There are some other instances where "nonlinear model" is used to contrast with a linearly structured model, although the term "linear model" is not usually applied. One example of this is nonlinear dimensionality reduction.
