What is structural estimation compared to reduced form estimation? I've heard a lot of definitions given for structural estimation. But it's never seemed entirely clear to me. Some times I've heard that what one person might call "reduced form" estimation should actually be called structural estimation. Sorry I don't have an example to illustrate, but I was wondering if somebody could clarify, hopefully with a link to a paper or some other source. What is structural estimation compared to reduced form estimation? Does the potential outcomes framework count as a structural equation?
Structural estimation is a term coined by the Cowles commission which at the time seems to have been dominated by Haavelmo, Koopmans and a few others. The motto of the Cowles commission (after 1965) was: "Theory and Measurement". The phrase represents the underlying rationale of structural modelling, that measurement cannot be done without some kind of theory. To my knowledge, the phrase was first used by Koopmans in "Identification Problems in Economic Model Construction":


  Systems of structural equations may be composed entirely on the basis
  of economic "theory." By this term we shall understand the combination of (a) principles of economic of behavior derived from general observation--partly introspective, partly through interview or experience--of the motives of economic decisions,(b) knowledge of legal and institutional rules restricting individual behavior (tax schedules, price controls, reserve requirements, etc.), (c) technological knowledge, and (d) carefully constructed definitions of variables.


Structural equations are then equations that come from an underlying economic (or physical, or legal) model. Structural estimation is precisely estimation which uses these equations to identify parameters of interest, and inform counter-factuals. Importantly, these parameters are usually taken to be invariant, and therefore counter-factuals taken from their estimates will be completely "correct". Counter-factuals were the main unit of interest to the Cowles commission.

Koopmans also discusses reduced form estimation:


  By the reduced form of a complete set of linear structural equations... we mean the form obtained by solving for each of the dependent (i.e., nonlagged endogenous) variables, and in terms of transformed disturbances (which are linear functions of the disturbances in the original structural equations).


The linearity is an artifact of the times (this was published in 1949!) but the point is that reduced-form equations are equations written in terms of economic variables which do not have a structural interpretation as defined above. So, a linear regression will be a reduced-form of some true structural model, because linear regression usually does not have a true economic interpretation. This does not mean that reduced form equations cannot be used to identify parameters in structural equations - in fact this is precisely how indirect inference works - just that they do not represent a deeper model of the data generating process. Reduced forms can (in principle) be used to identify structural parameters, in which cased you are still performing structural estimation, just through using the reduced form.

Another way to look at this is that structural models are generally deductive, whereas reduced forms tend to be used as part of some greater inductive reasoning.

For a comparison of this kind of Cowles commission structural modelling with Rubin causal modelling, check out this awesome set of slides by Heckman.

For other resources I'd check out more of what Koopmans wrote, the book Structural Macroeconomics by DeJong and Dave, these lecture notes by Whited, this paper by Wolpin (written for the Cowles Foundation, in honour of Koopmans) and a response by Rust.

Addendum: A simple example of reduced form and structural models.

Suppose we were looking at data on the prices, $p_t$ and quantities, $q_t$ produced by a monopolist. The monopolist faces a series of unknown costs in the future, and a linear demand curve (this would really have to be justified). Let's say the $\hat q_t$ and $\hat p_t$ we observe are measured with some kinds of mean-zero error, $e_t$, and $v_t$

Noting that both price and quantity seem to be associated with changes in cost, a reduced form equation for this model might be:
\begin{align}
\hat q_t &amp;= \gamma - \lambda c_t + \epsilon_t\\
\hat p_t &amp;= \alpha + \beta c_t + \nu_t
\end{align}
Because this is a reduced form model, it needs no justification other than that it might work empirically.

On the other hand, a structural model would start by specifying the demand curve (again to be strict this should start at the level of individual utility), and the monopolist's problem:

\begin{align}
\text{Demand curve: }&amp;p_t=a-bq_t\\
\text{Producer's problem: }&amp;\max E\left[\sum_{t=0}^\infty\delta^t (p_t-c_t)q_t(p_t)\right]\\
\text{Measurement equations: }&amp;\hat q_t = q_t + e_t\\
&amp;\hat p_t = p_t + v_t
\end{align}

From this further structural equations could be derived (structural because they are still representative of principles of economic behavior):

\begin{align}
\hat q_t&amp;=\frac{a-c_t}{2b} +e_t\\
\hat p_t&amp;=\frac{a+c_t}{2} + v_t\\
\end{align}

This is a case where a reduced form equation will have a meaningful structural interpretation, as consistent estimates $\hat a$ and $\hat b$ can be formed:

\begin{align}
\hat a&amp;= 2\hat\alpha \\
\hat b&amp;= \frac{1}{2\hat \lambda}
\end{align}

Another case of identification of structural parameters from reduced forms is the logit model in the case of valuations with extreme value errors (see McFadden (1974)). In general it is unlikely a given reduced form model will have a structural interpretation.