Definitions, Terminology and Notations

A repository of potentially useful definitions, terminology and notations

Terminology and notation

For $n$ individuals indexed by $i$ ( $i = 1,\dots,n$ ), in the linear regression model $y_i = \beta_0 + \beta_1 x_{1i} + ... + \beta_k x_{ki} + e_i$ ,

$y$ is called the dependent variable, the regressand, the explained variable, or the left-hand side variable,
$x_{1}, ..., x_{k}$ are the $k$ independent variables, the regressors, the explanatory variables, or the right-hand side variables,
$x \mapsto \beta_0 + \beta_1 x$ is the population regression function,
$\beta_0$ is the intercept,
$\beta_1$ is the slope parameter associated with $x_1$ ,
Together $\beta_0$ and $\beta_1$ are the coefficients or parameters of the regression model,
$e_i$ is the error or disturbance term.

In vector form, this is equivalent to $y = \beta_0 + \beta_1 x_1 + ... + \beta_k x_{k} + e$ where:

$y = \begin{bmatrix} y_1\\ \vdots\\ y_n \end{bmatrix}_{n\times 1}, \quad \forall j \in \{1, .., k\} \quad x_j = \begin{bmatrix} x_{j,1}\\ \vdots\\ x_{j,n} \end{bmatrix}_{n\times 1} \quad and \quad e = \begin{bmatrix} e_1\\ \vdots\\ e_n \end{bmatrix}_{n\times 1}$

In matrix form, this equivalent to $y = X\beta + e$ where:

$X = \begin{bmatrix} 1 & x_{1,1} & ... & x_{k,1}\\ 1 & x_{1,2} & ... & x_{k,2}\\ \vdots & \vdots & & \vdots\\ 1 & x_{1,n} & ... & x_{k,n}\\ \end{bmatrix}_{n\times (k + 1)} \quad and \quad \beta = \begin{bmatrix} \beta_{0}\\ \vdots\\ \beta_{k} \end{bmatrix}_{(k+1)\times 1}$

Note

In this class, vectors are in lower case (eg $y$ ) and matrices in caps (eg $X$ ). Most of the time, individual observations will be indexed by $i$ (eg $y_i$ ); these are scalars.