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Concept
Linear model
Summary
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.
Linear regression
For the regression case, the statistical model is as follows. Given a (random) sample the relation between the observations and the independent variables is formulated as
where may be nonlinear functions. In the above, the quantities are random variables representing errors in the relationship. The "linear" part of the designation relates to the appearance of the regression coefficients, 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 .
Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters 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:
the function to be minimised is a quadratic function of the for which minimisation is a relatively simple problem;
the derivatives of the function are linear functions of the making it easy to find the minimising values;
the minimising values are linear functions of the observations ;
the minimising values are linear functions of the random errors which makes it relatively easy to determine the statistical properties of the estimated values of .
An example of a linear time series model is an autoregressive moving average model. Here the model for values {} 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 at later times.
Official source
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