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The definition of term "Structural Equation" in Economics as defined in Introductory Econometrics: A Modern. Approach, Fifth Edition. Jeffrey M. Wooldridge. (page 859)

Structural Equation: An equation derived from economic theory or from less formal economic reasoning

However upon reading earlier works in Studies in Econometric Method I find that structural equations are far from informal.

In this passage of chapter 3 (page 86 in the PDF) by Herbert Simon, he makes a distinction between structural and non-structural equations which I don't fully understand.

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How does identifiablity define whether an equation is structural or not? does this mean structural equations assume causality?

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First, structural equations assume causality. The following quotation is from Cameron and Trivedi's textbook Microeconometrics: Methods and Applications:

At one end of the spectrum can be found very highly structured models, derived from detailed specification of the underlying economic behavior, that analyze causal (behavioral) or structural relationships for interdependent microeconomic variables. At the other end are reduced form studies that aim to uncover correlations and associations among variables, without necessarily relying on a detailed specification of all relevant interdependencies.

Second, identifiability does not exactly define whether an equation is structural or not. I would rather say that identifiability is the reason why we are forced to work with nonstructural equations. Parameters in structural equations cannot be identified most of the time. What can be estimated instead are reduced-form (or nonstructural) parameters. For instance, take the following simulatenous equation model \begin{align*} x_i =\alpha_0 y_i +\alpha_1 z_i +\epsilon_i\\ y_i = \beta_0 x_i+ \beta_1 z_i +\mu_i \end{align*} with two endogenous variables $x$ and $y$, an exogenous variable $z$, and $\epsilon$ and $\mu$ as error terms (suppose also $\alpha_0\beta_0\neq 1$). These two equations are structural. They may correspond to demand and supply equations of a given good, with $x$ the quantity and $y$ the price for instance. Without more assumptions, the coefficients are not identified and cannot be estimated. We can however estimate the parameters $\zeta$ and $\gamma$ of the reduced-form equation: \begin{align*} x_i = \zeta z_i + e_i\\ y_i = \gamma z_i +u_i \end{align*} with $\zeta=\frac{\alpha_1+\alpha_0\beta_1}{1-\alpha_0\beta_0}$ and $\gamma=\frac{\beta_1+\alpha_1\beta_0}{1-\alpha_0\beta_0}$.

In the second paragraph of the extract, the point is that identifiability (of the structural equations) is not necessary "as long as the structure remains unaltered". For instance, you can well predict the values of $x$ given the knowledge of $z$ and a good estimate of the nonstructural parameters $\zeta$. The last sentence illustrates the fact that, in our case, i) we do not need to know the causal structure to make a good prediction, ii) it does not help to distinguish between structural and nonstructural equations, and iii) we do not care about identifiability of the structural model.

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