Question

Suppose that annual earnings and alcohol consumption are determined by the SEM $$\begin{array}{rl}\log (\text{earnings})& ={\beta }_0+{\beta }_1\text{ alcohol }+{\beta }_{2}educ+u_1\\ \text{ alcohol}& ={\gamma }_0+{\gamma }_1\log (\text{earnings})+{\gamma }_{2}educ+{\gamma }_3\log (\text{price})+u_2\end{array}$$ where price is a local price index for alcohol, which includes state and local taxes. Assume that $educ$ and price are exogenous. If ${\beta }_1,{\beta }_2,{\gamma }_1,{\gamma }_2,$ and ${\gamma }_3$ are all different from zero, which equation is identified? How would you estimate that equation?

Short answer

The equation for $\log (\text{earnings})$ is identified and can be estimated using Two-Stage Least Squares (2SLS).

Step-by-step solution

Understand the Model

The given model is a Simultaneous Equations Model (SEM) with two equations. The first equation models the log of earnings as a function of alcohol consumption and education. The second equation models alcohol consumption as a function of log earnings, education, log price, and includes exogenous variables (educ and price).

Assess Exogenous Variables

Identify the exogenous variables in the system. In the equation for $\log (\text{earnings})$, $\text{educ}$ is exogenous. In the equation for $\text{alcohol}$, both $\text{educ}$ and $\log (\text{price})$ are exogenous.

Determine Identification of Equations

To check for identification, we apply the order condition. For the first equation $\log (\text{earnings})$, we see that the number of excluded exogenous variables from the second equation $(\text{price})$ is one, matching the number of endogenous variables (alcohol) in the system, satisfying the order condition. Thus, it is exactly identified. For the second equation (alcohol), no exogenous variables are excluded from the first equation, hence it is under-identified.

Choose Estimation Method

Since the first equation $\log (\text{earnings})$ is exactly identified, we can estimate it using methods suitable for identified structural equations, like Two-Stage Least Squares (2SLS). This involves first using ordinary least squares (OLS) to predict values of the endogenous variables using the exogenous ones, and then using these predicted values to estimate the equation.

Key concepts

Endogenous Variables

In the context of a Simultaneous Equations Model (SEM), endogenous variables are variables whose values are determined by other variables in the system. They are essentially the output or the dependent variables in one or more equations of the SEM. In the given model, both $\log (\text{earnings})$ and alcohol consumption are endogenous. This is because each is determined within the model through their respective equations:

$$\log (\text{earnings})={\beta }_0+{\beta }_1\text{alcohol}+{\beta }_2\text{educ}+u_1$$ $$\text{alcohol}={\gamma }_0+{\gamma }_1\log (\text{earnings})+{\gamma }_2\text{educ}+{\gamma }_3\log (\text{price})+u_2$$
In these equations, there is a causal feedback loop: $\log (\text{earnings})$ depends on alcohol, and alcohol, in turn, depends on $\log (\text{earnings})$. This interdependence is a hallmark of endogenous variables. To properly estimate the parameters in the SEM, we need to account for this simultaneous relationship, which we achieve through specific estimation techniques like the Two-Stage Least Squares.

Exogenous Variables

Exogenous variables are those that are determined outside the model and are not affected by other variables within the system. They act as the independent variables, providing an external input to the model.

In the given simultaneous equations model, the exogenous variables are education ($\text{educ}$) and the logarithm of alcohol price ($\log (\text{price})$). These variables influence the outcomes but are not influenced by the variables determined by the model itself. Their main purpose is to help identify and estimate the parameters associated with the endogenous variables.

For example, in SEMs, exogenous variables provide the necessary instrumental variables that help in the correct estimation of the endogenous variables, ensuring that we can derive meaningful and unbiased estimates of the coefficients within the system. By excluding certain exogenous variables from specific equations, we achieve identification of each equation, determining which parameters can be accurately estimated. However, care must be taken to ensure the exclusion restrictions maintain proper identification as explained in the original step-by-step solution.

Two-Stage Least Squares

Two-Stage Least Squares (2SLS) is a robust estimation technique used in the context of simultaneous equations models, especially when endogenous variables are present. The main challenge with SEMs is dealing with the bias that arises from simultaneity, where the endogenous variables are correlated with the error terms.

The 2SLS method addresses this by breaking the estimation process into two key steps:

• **First Stage**: It uses ordinary least squares (OLS) to regress each endogenous variable on all the exogenous variables present in the system. This process obtains predicted values for the endogenous variables that are purified from the endogenous relationship part.
• **Second Stage**: These predicted values are then used in place of the actual endogenous variables to estimate the structural equations using OLS. Since the predicted values are not correlated with the error terms, they provide unbiased parameter estimates.

The use of 2SLS ensures that we correctly account for the simultaneity in the model. By separating the direct effect from the feedback loops present among endogenous variables, 2SLS helps achieve consistent and reliable estimates, making it an indispensable tool in simultaneous equation analysis.