2.1 Variable selection

Assuming unidirectional causality, where $x$ causes $y$ , there are several alternative terms that can be used for the variables $y$ and $x$

TABLE 2.2: Variables terminology
$~~~~~~$ Variable $y~~~~~~$	$~~~~~~~$ Variable $x~~~~~~$
dependent	independent
outcome	predictor
response	explanatory
regressand	regressor
endogenous	exogenous

Variables on the right-hand side can have different roles; some may serve as control variables, while others can be mutiplied to represent interaction term
When dealing with time-series data (data observed over time) it is common for the dependent variable $y$ to appear also as an independent variable, making it endogenous

Exercise 1. Which variable is endogenous and which one is exogenous in the following equation? What does subscript

$t$ represent? Which variable is lagged?

$y_t=\beta_0+\beta_1y_{t-1}+\beta_2x_t+u_t~~~~~~~~~t=1,~2, ...,T$

Solution

Variable

$x$ is exogenous because

$x$ causes

$y$ , but not the other way around. Variable

$y$ is endogenous as it appears on both sides of the equation, meaning that

$y$ is both the consequence and the cause simultaneously. This is common when dealing with time-series data, and thus subscript

$t$ represents time unit (such as week, day, hour, month or year). Variable

$y_{t-1}$ is agged because it is observed in previous time period (subscript

$t-1$ ), e.g. lagged consumption might be used as RHS variable to account for how past consumption impacts present consumption. Likewise, a variable lagged for two periods is noted as

$y_{t-2}$ , variable lagged for three periods is noted as

$y_{t-3}$ , etc.

Exercise 2. Which variable is endogenous and which one is exogenous in the system of equations? How many parameters we need to estimate? System of two equations write in a matrix form!

$y_t=\beta_{1,0}+\beta_{1,1}y_{t-1}+\beta_{1,2}x_{t-1}+u_{1,t}$

$x_t=\beta_{2,0}+\beta_{2,1}y_{t-1}+\beta_{2,2}x_{t-1}+u_{2,t}$

Solution

Considering the system of equations both variables are endogenous, meaning that

$x$ causes

$y$ and

$y$ causes

$x$ . From this point none of the variables is strictly exogenous. Matrix form of the system is:

$\begin{bmatrix}y_t \\ x_t \end{bmatrix}=\begin{bmatrix} \beta_{1,0} \\ \beta_{2,0} \end{bmatrix}+ \begin{bmatrix} \beta_{1,1} \quad \beta_{1,2} \\ \beta_{2,1} \quad \beta_{2,2} \end{bmatrix} \begin{bmatrix}y_{t-1} \\ x_{t-1} \end{bmatrix}+\begin{bmatrix} u_{1,t} \\ u_{2,t} \end{bmatrix}$

Exercise 3. Is the following model bivariate or multivariate? What does subscript

$i$ represent? Which terms are variables and which are parameters? Which variables are known (observed) and which are unknown?

$y_i=\alpha+\beta x_i+\gamma z_i+u_i~~~~~~~~~i=1,~2, ...,n$

Solution

It is multivariate model due to more than one observed RHS variable. Subscript

$i$ represent cross-sectional unit. Variables are

$y$ ,

$x$ ,

$z$ and

$u$ , while

$\alpha$ ,

$\beta$ and

$\gamma$ are parameters. Known variables are

$y$ ,

$x$ and

$z$ . Error term is uknown (unobserved) variable

$u$ .

Exercise 4. Which parameter represents the constant term, and which one represents the interaction term? Explain the interaction term, assuming that

$y=$ income,

$x=$ years of working experience and

$z=$ gender (

$1$ for males and

$0$ for females).

$y_i=\alpha+\beta x_i+\gamma z_i+ \lambda (x_i z_i) + u_i$

Solution

Parameter

$\alpha$ is the constant term, and parameter

$\lambda$ is the interaction term associated with the multiplication of the two variables

$x$ and

$z$ . In a given example, parameter

$\lambda$ represents the difference in the change of income with respect to

$1$ unit change in working experience between males and females. For example, if

$\lambda \gt 0$ , it indicates that the income increase due to additional

$1$ year of experience is greater for males than the females.