2.1 Introduction

A Multivariate linear regression model (MLRM) is a generalization of the univariate linear regression model which has been dealt with in Chapter 2. In this chapter we consider an endogenous or response variable denoted by $ y$, which depends on a set of k variables $ x_{j}$ ($ j=1,...,k$), called "regressors", "independent variables" or "explanatory variables", and an unobservable random term called "disturbance" or "error" term. The latter includes other factors (some of them non-observable, or even unknown) associated with the endogenous variable, together with possible measurement errors.

Given a sample of $ n$ observations of each variable, in such a way that the information can be referred to time periods (time-series data) or can be referred to individuals, firms, etc. (cross-section data), and assuming linearity, the model can be specified as follows:

$\displaystyle y_{i}=\beta_{1}x_{1i}+\beta_{2}x_{2i}+\ldots+\beta_{k}x_{ki}+u_{i}$ (2.1)

with (i=1,2,...,n), where usually $ x_{1i}=1$ $ \forall$ $ i$, and their coefficient $ \beta_{1}$ is called the intercept of the equation.

The right-hand side of (2.1) which includes the regressors ($ x_{j}$), is called the $ \textsl{systematic component}$ of the regression function, with $ \beta_{j}$ ( $ j=1,\ldots,k$) being the coefficients or parameters of the model, which are interpreted as marginal effects, that is to say, $ \beta_{j}$ measures the change of the endogenous variable when $ x_{j}$ varies a unit, maintaining the rest of regressors as fixed. The error term $ u_{i}$ constitutes what is called the $ \textsl{random component}$ of the model.

Expression (2.1) reflects $ n$ equations, which can be written in matrix form in the following terms:

$\displaystyle y=X\beta+u$ (2.2)

where $ y$ is the $ n\times1$ vector of the observations of the endogenous variable, $ \beta $ is the $ k\times1$ vector of coefficients, and $ X$ is an $ n \times k$ matrix of the form:

$\displaystyle \begin{pmatrix}1 & x_{21} & x_{31} & \ldots & x_{k1} \\ 1 & x_{22... ...\vdots & \ldots & \vdots \\ 1 & x_{2n} & x_{3n} & \ldots & x_{kn} \end{pmatrix}$ (2.3)

in such a way that every row represents an observation. In expression (2.1) $ X\beta$ represents the systematic component, while $ u$ is the random component.

Model (2.2) specifies a causality relationship among $ y$, $ X$ and $ u$, with $ X$ and $ u$ being considered the factors which affect $ y$.

In general terms, model (2.2)(or(2.1)) is considered as a model of economic behavior where the variable $ y$ represents the response of the economic agents to the set of variables which are contained in $ X$, and the error term $ u$ contains the deviation to the average behavior.