Given a data set $\{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n$ of n statistical units, a linear regression model assumes that the relationship between the dependent variable yi and the p-vector of regressors xi is linear. This relationship is modeled through a disturbance term or error variable εi — an unobserved random variable that adds noise to the linear relationship between the dependent and regressors.

The MLR is described as follows:

$$ y_i = \beta_1 x_{i1} + \cdots + \beta_p x_{ip} + \varepsilon_i = \mathbf{x}^{\rm T}_i\boldsymbol\beta + \varepsilon_i, \qquad i = 1, \ldots, n, $$

Where:

- $\mathbf{x}^{\rm T}_i$ is the transpose matrix

Often these n equations are stacked together and written in vector form as

$$\mathbf{y} = \mathbf{X}\boldsymbol\beta + \boldsymbol\varepsilon, \, $$

$$\mathbf{y} = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, \quad \mathbf{X} = \begin{pmatrix} \mathbf{x}^{\rm T}_1 \\ \mathbf{x}^{\rm T}_2 \\ \vdots \\ \mathbf{x}^{\rm T}_n \end{pmatrix} = \begin{pmatrix} x_{11} & \cdots & x_{1p} \\ x_{21} & \cdots & x_{2p} \\ \vdots & \ddots & \vdots \\ x_{n1} & \cdots & x_{np} \end{pmatrix}, \quad \boldsymbol\beta = \begin{pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_p \end{pmatrix}, \quad \boldsymbol\varepsilon = \begin{pmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \vdots \\ \varepsilon_n \end{pmatrix}. $$

**Remarks**

- $y_i\,$ is called the regressand, response variable, measured variable, or dependent variable (see dependent and independent variables.)
- $\mathbf{x}_i\,$ are called regressors, exogenous variables, explanatory variables, covariates, input variables, predictor variables, or independent variables (see dependent and independent variables, but not to be confused with independent random variables).
- Usually, a constant is included as one of the regressors called the intercept.

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## References

- Hamilton, J .D.; Time Series Analysis, Princeton University Press (1994), ISBN 0-691-04289-6
- Kenney, J. F. and Keeping, E. S. (1962) "Linear Regression and Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252-285

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