### The Nobel Prize in Economics: A Note on Chris Sims' Contributions

Let me talk a bit about Sims contributions to economics, and if I have time I'll try to cover Sargent later.

Prior to Sims work, in particular his paper "Macroeconomics and Reality," the state of the art in macroeconometrics was to use large-scale structural models. These models often involved scores or even hundreds of equations, essentially a S=D equation for every important market, identities to make sure things add up correctly, etc. But in order to estimate the parameters of these models, the structural parameters as they are known, you had to overcome the identification problem.

Without getting into the details, the identification problem essentially asks if its possible to estimate the structural parameters at all. The answer, in general, is no. For example, if every variable in the model appears in every equation, then it won't be possible to estimate the structural model. Let me give an example to illustrate. Suppose that X and and Y are the endogenous variables, e.g. price and quantity for some market, and that the structural model is:

Y_{t} = a_{0} + a_{1}X_{t} + a_{2}Y_{t-1} + a_{3}X_{t-1} + u_{t}

X_{t} = b_{0} + b_{1}Y_{t} + b_{2}Y_{t-1} + b_{3}X_{t-1} + v_{t}

The a's and the b's are the parameters that economists are generally interested in, but in this form it is not possible to estimate them. There must be what are known as exclusion restrictions before estimation is possible. In this case, for example, identification can be achieved by making either a_{1} or b_{1} equal to zero (more on this below), i.e. excluding one of the variables from one of the equations. If there is a reason for this, then excluding the variable is okay, but a variable can't be left out simply to achieve identification -- there must be good reason for excluding X_{t} from the first equation, or Y_{t} from the second (or both). Omitting a variable that ought to be in a model in order to satisfy the identification restrictions results in a misspecified model and biased estimates.

In large models, these exclusions are numerous, and many researchers simply assumed whatever exclusion restrictions were needed to achieve identification, and then went on to estimate the model. In Macroeconomics and Reality, Sims pointed out the problem with this approach. The assumptions that researchers were imposing to achieve identification had no theoretical basis. They were ad hoc and difficult to defend (especially when expectations are in the model -- expectations tend to depend upon all the variables in a model making it difficult to exclude anything from an equation involving expectations).

What Sims suggested as an alternative was to drop structural modeling altogether, and to use generalized reduced forms as the basis for estimation. There would be no hope of recovering structural parameters in most cases, but there was still much that could be learned by using reduced forms instead of structural models.

For example, the reduced form for the model above is (you can find the reduced form by expressing the endogenous variables X_{t} and Y_{t} in terms of exogenous and predetermined variables):

X_{t} = [1/(1-a_{1}b_{1})]{(a_{0} + a_{1}b_{0}) + (a_{1}b_{2} + a2)Y_{t-1} + (a_{1}b_{3} + a_{3})X_{t-1 }+ a_{1}v_{t} + u_{t}}

Y_{t} = [1/(1-a_{1}b_{1})]{(b_{0} + b_{1}a_{0}) + (b_{1}a_{2} + b_{2})Y_{t-1} + (b_{1}a_{3} + b_{3})X_{t-1 }+ v_{t} + b_{1}u_{t}}

To estimate this, write it as:

X_{t} = c_{0} + c_{1}Y_{t-1} + c_{2}X_{t-1 }+ a_{1}v_{t} + u_{t}

Y_{t} = d_{0} + d_{1}Y_{t-1} + d_{2}X_{t-1 }+ v_{t} + b_{1}u_{t}

This is a VAR model. At first, Sims thought we could draw important conclusions from this model, e.g. suppose that X is money and Y is output. Then this model could tell us how a shock to money would change output over time (these are called impulse response functions -- you hit the system with a shock, and then use the estimated model to trace out the path of the endogenous variables over time). We could use this model to answer important questions such as whether money causes output (Sims' technique for testing causality was essentially the same as Granger causality, but Sims' made an important contribution in extending the causality techniques to systems with three or more variables when he introduced impulse response functions and variance decompositions).

But, as Cooley and LeRoy pointed out in an important paper, these models don't avoid structural assumptions after all, at least not if you want to say anything about how variables in the model respond to structural shocks. To see this, note first that the shock we are interested in is the shock to money, v_{t}. Now look at the errors in the two reduced form equations. We can estimate each equation by OLS, and when we do the error terms will be estimates of a_{1}v_{t} + u_{t} for the first equation and v_{t} + b_{1}u_{t} for the second. Thus, we get estimates of linear combinations of the v_{t} and u_{t} shocks we are interested in, but we don't get the shocks in isolation like we need. And there's no way to isolate the shocks, i.e. to determine their individual values. That's a problem because we need to find the money shock alone if we want to estimate its effect on output.

How can we do this? One way is to make either a_{1} or b_{1} equal to zero. Let's set b_{1}=0 because that's the easiest to discuss. In this case, when we estimate the second equation by OLS (the equation with the d parameters), the error will now be an estimate of v_{t}, which is just what we need. However, notice that this is nothing more than an exclusion restriction -- by assuming that b_{1}=0, we are excluding Y_{t} from the second equation (see the structural model). Thus, we have come full circle.

This is where Sims Structural VARS come into play. The reduced form above is known as a VAR model (in its estimable form, i.e. the second set of reduced for equations above involving the c and d parameters). It turns out that if we can often defend particular restrictions theoretically, e.g. if money can only respond to output with a lag, perhaps due to information problems, then there is no reason to have the contemporaneous value of output on the right-hand side of the structural equation for money, i.e. this implies that b_{1}=0.

Thus, while this still amounts to an exclusion restriction, the restriction is no longer ad hoc -- simply imposed as necessary to achieve identification as back in the old, large-scale structural model days -- it is grounded in theory. And the fact that we insist these restrictions be grounded in theory marks an important difference from the work that came before Sims.

And even better, this technique also allows the model to be identified without using exclusion restrictions at all. For example, if we think that some variables in the model have short-run but not long-run effects, e.g. that money can affect output in the short-run, but only produces price effects in the long-run -- a standard assumption in most macro models -- then the zero impact in the long-run can be imposed as an identifying restriction. Exclusion restrictions won't be needed (this is the Blanchard-Quah and Shapiro-Watson techniques).

This just scratches the surface of Sims' work -- I wish I had time to do more -- but *hopefully* this provides a window into one part of Sims' contributions.

Posted by Mark Thoma on Monday, October 10, 2011 at 11:43 AM in Economics, Methodology |
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