# What is simply meant by reduced form?

In econometrics, what is meant by reduced form? Also, what are people looking for when they say "I would like to see the reduced form estimates." This has been thrown around at work and individual explanations and Google searches are overly technical. Hoping someone where would be able to give a simple example.

• What broad area of economics do you work in? Perhaps that information would allow for a more tailored intuitive example. Feb 10 '14 at 1:37
• @Dimitriy V. Masterov Work with sales data for a large corporation
– CJ12
Feb 10 '14 at 1:59
• Have you ever seen any attempts at demand estimation? Feb 10 '14 at 2:34
• For a more pragmatic explanation see econ.lse.ac.uk/staff/spischke/ec533/IV.pdf.
– o_v
Dec 25 '20 at 9:02

Take a look at this simple example showing how the Keynesian consumption function and equilibrium condition can be re-written in a reduced form.

The reduced form of a model is the one in which the endogenous variables are expressed as functions of the exogenous variables (and perhaps lagged values of the endogenous variables). Very roughly, reduced form estimates do not give you the structural, primitive policy-invariant behavioral parameters that you (sometimes) care about, such as parameters of an agent's utility function or the slopes of the demand and supply curves.

With RFEs, you only get functions of those parameters (and often not even that). For some purposes, that can be enough, which is why some people want to see them. For example, you can frequently get the sign of the relationship from RF estimates, but not the magnitude. Once is a blue moon, you can use algebra to solve for structural parameters from the RFEs.

Finally, it is also the case that some people will not believe the assumptions needed to estimate the structural parameters.

• This is great but still more on the technical side. I will look at this example. Is there an even plainer English version to start with?
– CJ12
Feb 10 '14 at 1:26
• That's the simplest one I know of. Feb 10 '14 at 1:29
• The other common example is supply and demand with an equilibrium condition. It very similar to the above example. See these lecture notes, especially pp. 19-27. Feb 10 '14 at 6:51
• Would it be fair to say that the reduced form of a model describes the data but not necessarily the underlying phenomenon? Jul 21 '16 at 2:37
• @BenOgorek Yes, that would be correct. Jul 21 '16 at 3:45

To complement Dimitriy's answer (+1), the structural form and the reduced form are two ways of thinking about your system of equations.

The structural form is what your economic theory says the economic relations between the variables are (like consumption and income in the linked Keynesian example). However, getting the estimates of the model coefficients requires jumping through multiple hoops to make sure these estimates are not biased because of endogeneity problems when one endogenous variable is regressed on another. So structural form is good for intuitive explanation, and terrible to work with when the numbers come in.

The reduced form complements the structural form in functionality. As Dimitriy said, and as shown in the consumption example, the reduced form solves for the endogenous variables (if at all possible) -- this is American Algebra II material, to my knowledge. In the end, in each equation, one and only one endogenous variable appears in the left hand side, and the right hand side only contains exogenous variables and error terms. If at all possible is an important qualifier: sometimes it will not be possible to arrive at such a transformation of the structural form, and it means that the model is not identified, and no amount of data will help you get estimates of your parameters. The reduced form is easily estimable though, as you can run something as basic as OLS on each equation to get some estimates (although these won't be the best possible estimates), and they will be unbiased for the reduced form parameters. However, there may or may not be a nice cross-walk back to the structural form, which had interpretable parameters. Thus the reduced form is good for estimation, but terrible for interpretation. Reduced form can also be used for prediction, including impulse response functions -- this may have been the reason somebody wanted to see these estimates.

When you do a regression involving two steps (two-step least squares, or 2sls) you have two equations. The first equations, named the structural equation, looks like any other regression equation. The second equation is the reduced form equation (and looks a lot like any other regression equation). The reason for doing a 2sls is that some variable in the first equation correlated with the error term, which violates the basic assumptions of regression analysis. To fix this problem you make the second equation (the reduced form equation) using the correlated variable as the dependent variable and a set of independent variables (which in this case get the fancy name of instrumental variables) that you think will correct the correlation problem along with all the independent variables from the first equation. Then you have the computer run it. The output will provide you the estimates you need.

So in short, I think the person asking for your reduced form estimates, wants to see your work. Particularly they want to see the second equations and the associated betas --- show them the regression output and they should be happy.

Hope this helps!

Agree with @user107905, if you use the 2SLS the reduced format equation is used to construct the IV, while the original structural equation can still be fitted through OLS by plugging in the fitted endogenous value. In that way, you can still get INTERPRETABLE parameters for the original/1st structural equation.

see chapter 15 Instrumental variables estimation and two stage least squares in 'Introductory Econometrics A Modern Approach' Wooldridge.

Jörn-Steffen Pischke provides a very pragmatic explanation of the reduced form in the context of instrumental variables analysis (IV) in his lecture notes.

He essentially distinguishes 3 causal effects of interest:

1. the causal effect of the instrument (Z) on the endogenous variable (X) - obtained in the first stage;
2. the causal effect of the instrument (Z) on the outcome of interest (Y) - obtained in the reduced form;
3. the causal effect of the endogenous variable (X) on the outcome of interest (Y) - obtained in the second stage.

Although what we ultimately care about (after all it's the whole reason why we do this IV exercise) is causal effect number 3, and we can obtain this also without estimating causal effect number 2, Pischke argues that this parameter might be interesting in it's own right:

"For example, the instrument might be a policy variable in which case it is the policy effect."

Another source of interest might be the lecture notes by Kurt Schmidheiny.

He mentions the reduced form in the context of weak instruments (F-stat of 1st stage < 10). In such case hypothesis testing based on the IV estimates is no longer valid. A 'reduced form test' might provide an alternative. In his words:

"Reduced form estimation offers a simple approach to test the null hypothesis H0 that all K coefficients βk related to the endogenous explanatory variables [...] are simultaneously equal to zero."

This essentially means that one can use the reduced form to assess whether the instruments have a direct effect on the outcome of interest. Under H0 they do not have an effect (are not significantly different from zero).