1.1 Why a separate discipline?

• Economic theory is primarily descriptive (qualitative), e.g. according to the theory of Keynes, consumption increases as income increases, but less than the increase in income

• From this statement, we can establish a hypothesis: the marginal propensity of consumption for a unit change in income is grater than zero but less than one

• Mathematics, being quantitative in nature, expresses the dependence between variables analytically as a single equation or a system of equations, typically in deterministic form

• To validate the theory of Keynes, we assume:
  $~~~~~y=$ consumption,
  $~~~~~x=$ income,
  $f(x)=$ linear function

• The mathematical model of the consumption theory is defined as $$\begin{equation} y=f(x)=\beta_0+\beta_1x, \tag{1.1} \end{equation}$$

$~~~~~~~$ with restrictions $\beta_0>0$ and $0<\beta_1<1$.

An econometric model includes not only deterministic component $f(x)$ but also unobserved random component, i.e. error term $u$ $$\begin{equation} y=\beta_0+\beta_1x+u \tag{1.2} \end{equation}$$

FIGURE 1.1: Interdisciplinarity of econometrics

$~~~$

• Econometrics is interdisciplinary because it integrates economics, statistics, mathematics, and computer science to analyze economic data (FIGURE 1.1)

• Population parameters $\beta_0$ (constant term) and $\beta_1$ (slope coefficient) are unknown as well as the error term $u$, and they need to be estimated using observed variables $x$ and $y$

• Observed variables are typically associated with multiple cross-sectional units (such as households, individuals, cities, countries, $\dots$) at a given point in time, and therefore we use subscript $i$ in the notation of these variables

• There is a finite number of cross-sectional units $i=1,2,3,...,n$, and thus EQUATION (1.2) holds for every observation

$$\begin{equation}\begin{matrix} y_1=\beta_0+\beta_1x_1+u_1 \\ y_2=\beta_0+\beta_1x_2+u_2 \\ \vdots \\ y_n=\beta_0+\beta_1x_n+u_n\end{matrix} \tag{1.3} \end{equation}$$

• A system of $n$ equations (1.3) with two unknowns can be presented in matrix form

$$\begin{equation}\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix}=\begin{bmatrix} 1 & x_1 \\ 1 & x_2 \\ \vdots & \vdots \\ 1 & x_n\end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \end{bmatrix}+\begin{bmatrix} u_1 \\ u_2 \\ \vdots \\u_n\end{bmatrix} \tag{1.4} \end{equation}$$

The matrix equation $$\begin{equation} y=x\beta+u \tag{1.5} \end{equation}$$ describes population regression model

• Data are usually presented as matrices and/or vectors, which simplifies computational operations in the background of any applied method.