Linear models

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 ${\displaystyle (Y_{i},X_{i1},\ldots ,X_{ip}),\,i=1,\ldots ,n}$ the relation between the observations Y_i and the independent variables X_ij is formulated as

where ${\displaystyle \phi _{1},\ldots ,\phi _{p}}$ 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

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