3.1 The true model

In practice, any economic phenomenon is affected by many factors (regressors), some of which are highly influential, defined as relevant variables, and some are less relevant. The best model thus is the one that includes all the relevant variables as regressors and combines all the irrelevant ones into one single variable as a disturbance term denoted \(\varepsilon_t\) as in the simple linear regression model.

Throughout this chapter, we will use the following notations:

• \(y_t\): The value of the dependent variable \(y\) at the time \(t\).
• \(x_i\): The regressor vector \(i\).
• \(x_{it}\): The value of the regressor \(x_i\) at the time (or of the individual) \(t\)

The true model then can be written:

\[\begin{equation*} y_t=\beta_1+\beta_2x_{1t}+\beta_3x_{2t}+...+\beta_{k+1}x_{kt}+\varepsilon_t \end{equation*}\]

We see that this writing is a bit confusing as the index of the regressor is different from the index of its parameter. So we can set the same index by removing back one step the index of the regressors as follows:

\[\begin{equation*} y_t=\beta_1x_{1t}+\beta_2x_{2t}+\beta_3x_{3t}+...+\beta_{k}x_{kt}+\varepsilon_t \end{equation*}\]

Note that this model now does not have a constant term, so to keep the constant term we can simply set \(x_{1t}=1\) and hence the model will be:

\[\begin{equation} y_t=\beta_1+\beta_2x_{2t}+\beta_3x_{3t}+...+\beta_{k}x_{kt}+\varepsilon_t \tag{3.1} \end{equation}\]

With this form, the model have \(k-1\) regressors and \(k\) parameters.

The last model expression can be detailed over the index \(t\) as follows:

\[\begin{align*} y_1&=\beta_1+\beta_2x_{21}+\beta_3x_{31}+...+\beta_{k}x_{k1}+\varepsilon_1\\ y_2&=\beta_1+\beta_2x_{22}+\beta_3x_{32}+...+\beta_{k}x_{k2}+\varepsilon_2\\ &.....................................\\ y_n&=\beta_1+\beta_2x_{2n}+\beta_3x_{3n}+...+\beta_{k}x_{kn}+\varepsilon_n \end{align*}\]

This system can be reformulated in matrix form:

\[\begin{equation*} \underset{(n\times 1)}{ \begin{pmatrix} y_1\\ y_2\\ ..\\ y_n \end{pmatrix}} =\underset{(n\times k)}{ \begin{pmatrix} 1&x_{21}&..&x_{k1}\\ 1&x_{22}&..&x_{k2}\\ ..&..&..&..\\ 1&x_{2n}&..&x_{kn} \end{pmatrix}}\times \underset{(k\times 1)}{ \begin{pmatrix} \beta_1\\ \beta_2\\ ..\\ \beta_k \end{pmatrix}}+ \underset{(n\times 1)}{ \begin{pmatrix} \varepsilon_1\\ \varepsilon_2\\ ..\\ \varepsilon_n \end{pmatrix}} \end{equation*}\]

Then the matix and the vectors will be denoted by:

\[\begin{equation*} y=\underset{(n\times 1)}{ \begin{pmatrix} y_1\\ y_2\\ ..\\ y_n \end{pmatrix}}\quad X=\underset{(n\times k)}{ \begin{pmatrix} 1&x_{21}&..&x_{k1}\\ 1&x_{22}&..&x_{k2}\\ ..&..&..&..\\ 1&x_{2n}&..&x_{kn} \end{pmatrix}}\quad \beta=\underset{(k\times 1)}{ \begin{pmatrix} \beta_1\\ \beta_2\\ ..\\ \beta_k \end{pmatrix}}\quad \varepsilon=\underset{(n\times 1)}{ \begin{pmatrix} \varepsilon_1\\ \varepsilon_2\\ ..\\ \varepsilon_n \end{pmatrix}} \end{equation*}\]

Conventionally, a matrix \(A\) is defined by \(\underset{n\times k}{A}=\left(a_{ij}\right)\) where the first subscript \(i=1,..,n\) indicates the row number and the second one \(j=1,..,k\) indicates the column number. However, in our formulation, the matrix \(X\) is defined by \(\underset{n\times k}{X}=\left(x_{ij}\right)\) the column subscript \(i=1,..,k\) comes first then followed by the row subscript \(j=1,..,n\)

Finally, the true multiple linear model with \((k-1)\) regressors will be written in matrix form as:

\[\begin{equation} y=X\beta+\varepsilon \tag{3.2} \end{equation}\]