1.3 What “linear” means

A regression model is called linear when it is linear in the parameters \(\beta _0, \beta _1, \dots \), not necessarily in \(x\). For example, \begin {align*} y &= \beta _0 + \beta _1 x + \varepsilon && \text {(a straight line: linear)}, \\ y &= \beta _0 + \beta _1 x + \beta _2 x^2 + \varepsilon && \text {(a parabola: still linear)}, \\ y &= \beta _0 + \beta _1 \log x + \varepsilon && \text {(linear)}, \\ y &= \beta _0 e^{\beta _1 x} + \varepsilon && \text {(nonlinear in $\beta _1$)} . \end {align*}

In the first three, each parameter multiplies a known quantity and the terms are added, so the same estimation method works for all of them. The last is a nonlinear regression model, treated in a later section.

A model with one regressor is a simple regression model; with two or more it is a multiple regression model.

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