<p>I would like to complement the interesting, and clearly researched answer by jayk, by sharing the outlook more from the perspective of current econometricians.</p>
<p>From the very good book &quot;Econometrics&quot; by Hansen, in chapter 12, the way he talks about the structural and reduced form equations seems to imply the following:</p>
<ul>
<li>A <strong>structural equation</strong> is the initial scientifically-based model, which may contain endogenous variables. This model usually comes from the underlying theory, i.e. labour economics, education economics, macroeconomics, etc.</li>
<li>A <strong>reduced form equation</strong>, usually implied by a structural model plus some possible extra conditions, results in a model of endogenous variables as a function of exogenous variables, and potentially some unobserved error terms.</li>
</ul>
<p>We could think of the following example.</p>
<p><span class="math-container">$Y=X´\beta + u$</span> derived from some scientific theory, with some of the columns of <span class="math-container">$X$</span> including some endogenous variables. This would be a structural equation.
For this structural equation, a reduced form equation could deduced with the following form:
<span class="math-container">$(Y, X_\text{end}) = F(X_\text{exo}, Z) + e$</span>, where end for the endogenous variables in <span class="math-container">$X$</span>, and exo for the exogenous variables in <span class="math-container">$X$</span>, and <span class="math-container">$Z$</span> for some instruments (which are also exogenous).</p>

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2024-01-16T15:55:17.913Z
2024-01-16T16:01:20.813Z
2024-01-16T16:01:20.813Z
An old man in the sea.
<p>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 "<a href="http://www.jstor.org/stable/1905689" rel="noreferrer">Identification Problems in Economic Model Construction</a>":</p>

<blockquote>
  <p>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.</p>
</blockquote>

<p><strong>Structural equations are then equations that come from an underlying economic (or physical, or legal) model</strong>. 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 <em>invariant</em>, and therefore counter-factuals taken from their estimates will be completely "correct". Counter-factuals were the main unit of interest to the Cowles commission.</p>

<p>Koopmans also discusses reduced form estimation:</p>

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

<p>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 <em>usually</em> 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 <a href="https://ideas.repec.org/a/jae/japmet/v8y1993isps85-118.html" rel="noreferrer">indirect inference</a> 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.</p>

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

<p>For a comparison of this kind of Cowles commission structural modelling with Rubin causal modelling, check out <a href="http://jenni.uchicago.edu/papers/koop2006/koop1-cowles_ho_2006-09-25_jlt.pdf" rel="noreferrer">this awesome set of slides</a> by Heckman.</p>

<p>For other resources I'd check out more of what Koopmans wrote, the book <a href="http://rads.stackoverflow.com/amzn/click/069115287X" rel="noreferrer">Structural Macroeconomics</a> by DeJong and Dave, these <a href="http://toni.marginalq.com/FMA.pdf" rel="noreferrer">lecture notes by Whited</a>, <a href="http://cowles.econ.yale.edu/conferences/koopmans/tck10/wolpin-2.pdf" rel="noreferrer">this paper</a> by Wolpin (written for the Cowles Foundation, in honour of Koopmans) and <a href="http://pages.stern.nyu.edu/~dbackus/Identification/Rust_Wolpin_review_Dec_13.pdf" rel="noreferrer">a response</a> by Rust.</p>

<p><strong>Addendum:</strong> A simple example of reduced form and structural models.</p>

<p>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 $v_t$</p>

<p>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.</p>

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

<p>\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}</p>

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

<p>\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}</p>

<p>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:</p>

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

<p>Another case of identification of structural parameters from reduced forms is the logit model in the case of valuations with extreme value errors (see <a href="http://eml.berkeley.edu/reprints/mcfadden/zarembka.pdf" rel="noreferrer">McFadden (1974)</a>). In general it is unlikely a given reduced form model will have a structural interpretation.</p>

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2017-01-17T18:10:36.447Z
I wonder if there is such a thing as *structural estimation*. I understand *a structural model* vs. *a reduced-form model*, but not quite *structural estimation* vs. *reduced-form estimation*. E.g. we can have structural vs. reduced form vector autoregressive model, but only the latter actually gets estimated, and then the former is backed up from the estimates of the latter one. (This is a rough example but it should illustrate my point.)
Richard Hardy
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2017-02-26T20:01:35.637Z
Take a simple example. Some models, especially in asset pricing, have closed form representations that describe moments in terms of structural parameters. For these models, we are *estimating* the parameters directly, just as you'd estimate the mean of a normally distributed random variable from the first moment in the data. Structural parameters are estimated, just as reduced form parameters are - the difference is in the required identifying assumptions.
jayk
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2017-04-29T07:36:21.120Z
@RichardHardy check jayk comment above.
An old man in the sea.
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2017-04-29T08:11:21.280Z
@Anoldmaninthesea, thanks. That is an interesting perspective. But does it not imply that in such situations the reduced-form and the structural parameters coincide? The ones being estimated are by definition reduced form, are they not?
Richard Hardy
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2021-05-04T21:20:15.727Z
How is "consistent estimates $\hat a$ and $\hat b$ can be formed" related to "This is a case where a reduced form equation will have a meaningful structural interpretation"?
Aqqqq
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2014-11-21T00:09:47.640Z
2017-04-29T09:24:49.300Z
2017-04-29T09:24:49.300Z
jayk
<p>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?</p>

CC BY-SA 3.0
2014-11-20T21:26:32.380Z
Great question. I always thought structural meant "build a theoretical model and estimate the parameters for the resulting functional form". But then an econometrician told me that my understanding was wrong and tried to explain structural modelling to me. I still don't get it and am looking forward to the answers here.
Ubiquitous
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2014-11-21T00:21:49.897Z
I love it when a question is just right in your area of expertise.
jayk
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2014-11-21T04:39:17.133Z
Adding to the links. Here's some discussion about structural vs reduced form in IO. [Nevo, Aviv, and Michael D. Whinston. 2010. "Taking the Dogma out of Econometrics: Structural Modeling and Credible Inference."](http://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.24.2.69)
Pburg
<econometrics><structural-estimation><reduced-form-estimation>
What is structural estimation compared to reduced form estimation?
jmbejara
# What is structural estimation compared to reduced form estimation?

(Asked by: username_0 on 2014-11-20)

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?

**Tags:** econometrics, structural-estimation, reduced-form-estimation

> **Comments:**
> **username_1**: Great question. I always thought structural meant "build a theoretical model and estimate the parameters for the resulting functional form". But then an econometrician told me that my understanding was wrong and tried to explain structural modelling to me. I still don't get it and am looking forward to the answers here.  
> **username_2**: I love it when a question is just right in your area of expertise.  
> **username_3**: Adding to the links. Here's some discussion about structural vs reduced form in IO. [Nevo, Aviv, and Michael D. Whinston. 2010. "Taking the Dogma out of Econometrics: Structural Modeling and Credible Inference."](http://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.24.2.69)  

---

## 2 answers

### Answer by username_2 on 2014-11-21. Score: 62 | Accepted answer

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](http://www.jstor.org/stable/1905689)":

> 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](https://ideas.repec.org/a/jae/japmet/v8y1993isps85-118.html) 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](http://jenni.uchicago.edu/papers/koop2006/koop1-cowles_ho_2006-09-25_jlt.pdf) by Heckman.

For other resources I'd check out more of what Koopmans wrote, the book [Structural Macroeconomics](http://rads.stackoverflow.com/amzn/click/069115287X) by DeJong and Dave, these [lecture notes by Whited](http://toni.marginalq.com/FMA.pdf), [this paper](http://cowles.econ.yale.edu/conferences/koopmans/tck10/wolpin-2.pdf) by Wolpin (written for the Cowles Foundation, in honour of Koopmans) and [a response](http://pages.stern.nyu.edu/~dbackus/Identification/Rust_Wolpin_review_Dec_13.pdf) 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 &= \gamma - \lambda c\_t + \epsilon\_t\\
\hat p\_t &= \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: }&p\_t=a-bq\_t\\
\text{Producer's problem: }&\max E\left[\sum\_{t=0}^\infty\delta^t (p\_t-c\_t)q\_t(p\_t)\right]\\
\text{Measurement equations: }&\hat q\_t = q\_t + e\_t\\
&\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&=\frac{a-c\_t}{2b} +e\_t\\
\hat p\_t&=\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&= 2\hat\alpha \\
\hat b&= \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)](http://eml.berkeley.edu/reprints/mcfadden/zarembka.pdf)). In general it is unlikely a given reduced form model will have a structural interpretation.

> **Comments:**
> **username_5**: I wonder if there is such a thing as *structural estimation*. I understand *a structural model* vs. *a reduced-form model*, but not quite *structural estimation* vs. *reduced-form estimation*. E.g. we can have structural vs. reduced form vector autoregressive model, but only the latter actually gets estimated, and then the former is backed up from the estimates of the latter one. (This is a rough example but it should illustrate my point.)  
> **username_2**: Take a simple example. Some models, especially in asset pricing, have closed form representations that describe moments in terms of structural parameters. For these models, we are *estimating* the parameters directly, just as you'd estimate the mean of a normally distributed random variable from the first moment in the data. Structural parameters are estimated, just as reduced form parameters are - the difference is in the required identifying assumptions.  
> **username_4**: @username_5 check username_2 comment above.  
> **username_5**: @Anoldmaninthesea, thanks. That is an interesting perspective. But does it not imply that in such situations the reduced-form and the structural parameters coincide? The ones being estimated are by definition reduced form, are they not?  
> **username_6**: How is "consistent estimates $\hat a$ and $\hat b$ can be formed" related to "This is a case where a reduced form equation will have a meaningful structural interpretation"?  

---

### Answer by username_4 on 2024-01-16. Score: 4

I would like to complement the interesting, and clearly researched answer by username_2, by sharing the outlook more from the perspective of current econometricians.

From the very good book "Econometrics" by Hansen, in chapter 12, the way he talks about the structural and reduced form equations seems to imply the following:

* A **structural equation** is the initial scientifically-based model, which may contain endogenous variables. This model usually comes from the underlying theory, i.e. labour economics, education economics, macroeconomics, etc.
* A **reduced form equation**, usually implied by a structural model plus some possible extra conditions, results in a model of endogenous variables as a function of exogenous variables, and potentially some unobserved error terms.

We could think of the following example.

$Y=X´\beta + u$ derived from some scientific theory, with some of the columns of $X$ including some endogenous variables. This would be a structural equation.
For this structural equation, a reduced form equation could deduced with the following form:
$(Y, X\_\text{end}) = F(X\_\text{exo}, Z) + e$, where end for the endogenous variables in $X$, and exo for the exogenous variables in $X$, and $Z$ for some instruments (which are also exogenous).


---

