In Praise of Linear Models …

Noah Smith and Francis Coppola have recent columns discussing the prevalence of linear methods in economics and in particular in macroeconomics.

Coppola acknowledges that adding financial components to DSGE models is a step in the right direction but that it “does not begin to address the essential non-linearity of a monetary economy. … Until macroeconomists understand this, their models will remain inadequate.” (This is ok but to me it sounds like she’s going into Deepak Chopra mode a bit.)

Noah gives a good description of the properties and benefits of linear models.

“[Linear models are] easy to work with. Lines can only intersect at one point, so […] there’s only one thing that can happen.” In non-linear models “the curves can bend back around and [could] meet in some faraway location. Then you have a second equilibrium – another possible future for the economy. […] if you go with the full, correct [non-linear] versions of your models, you stop being able to make predictions about what’s going to happen to the economy. […] Also, linearized models […] are a heck of a lot easier to work with, mathematically. […] As formal macroeconomic models have become more realistic, they’ve become nastier and less usable. Maybe their days are simply numbered.”

There are elements of truth in both columns but on the whole I don’t share their assessments. In fact, I am firmly of the opinion that linear solution methods are simply superior to virtually any other solution method in macroeconomics (or any other field of economics for that matter). Moreover, I suspect that many younger economists and many entering graduate students are gravitating towards non-linear methods for no good reason and I also suspect they will end up paying a price if they adopt the fancier modelling techniques.

Why am I so convinced that linear methods are a good way to proceed? Let me count the ways:

1. Most empirical work is done in a (log) linear context. In particular regressions are linear. This is important because when macroeconomists (or any economist) tries to compare predictions with the data, the empirical statements are usually stated in terms that are already linear. This comparison between theory and data is easier if the models are making predictions that are couched in a linear framework. In addition, there aren’t many empirical results that speak clearly on non-linearities in the data. There is a statement that I’ve been told by empirical economists several times that a well-known economist said that “either the world is linear, or it’s log-linear, or God is a son-of-a-bitch.”

2. Linear doesn’t mean simple. When I read Francis’ column it seems that her main complaint lies with the economic substance of the theories rather than the solution method or the approximate linearity of the solutions. She talks about lack of rationality, heterogeneity, financial market imperfections, and so forth. None of these things requires a fundamentally non-linear approach. On the contrary, the more mechanisms and features you shove into a theory, the more you will benefit from a linear approach. Linear systems can accommodate many features without much additional cost.

3. Linear DSGE models can be solved quickly and accurately. Noah mentioned this but it bears repeating. One of the main reasons to use linear methods is that they are extremely efficient and extremely powerful. They can calculate accurate (linear) solutions in milliseconds. In comparison, non-linear solutions often require hours or days (or weeks) to converge to a solution. [Fn 1]

4. The instances where non-linear results differ importantly from linear results are few and far between. The premise behind adopting a non-linear approach is that knowing about the slopes (or elasticities) of the demand and supply curves is not sufficient. We have to know about the curvatures of the demand and supply curves too. On top of the fact that we don’t really know much about these curvature terms, the presumption itself is highly suspect. If we are picking teams, I’ll pick the first order terms and let you pick all of the higher order terms you want any day of the week (and I’ll win). In cases in which we can calculate linear vs. non-linear solutions, the differences are typically embarrassingly small and even when there are noticeable differences, they often go away as we improve the non-linear solution. A common exercise is to calculate the growth convergence path for the neoclassical (Ramsey) growth model using discrete dynamic programming techniques and then compare the solution to the one you get from a linearized solution in the neighborhood of the balanced growth path. The dynamic programming approach is non-linear – it allows for an arbitrary reaction on a discrete grid space. When you plot out the two responses, it is clear that the two solutions aren’t the same. However, as we add more and more grid points, the two solutions start to look closer and closer. (The time required for computation of the dynamic programming solution grows of course.)

5. Unlike non-linear models, approximate analytical results can be recovered from the linear systems. This is an underappreciated side benefit of adopting a linear approach. The linear equations can be solved by hand to yield productive insights. There are many famous examples of log-linear relationships that have led to well-known empirical studies based on their predictions. Hall’s log-linear Euler equation, the New Keynesian Phillips Curve, log-linear labor supply curves, etc. Mankiw, Romer and Weil (1992) used a linear approximation to crack open the important relationship between human capital and economic growth.

Given the huge benefits to linear approaches in the field. The main question I have is not why researchers don’t adopt non-linear, global solution approaches but rather why linear methods aren’t used even more widely than they already are. One example in particular concerns the field of Industrial Organization (IO). IO researchers are famous for adopting complex non-linear modelling techniques that are intimidating and impressive. They often seem to brag about how it takes even the most powerful computers weeks to solve their models. I’ve never understood this aspect of IO. IO is also known to feature the longest publication lags and revision lags of any field. It’s possible that some of this is due to the techniques that they are using; I’m not sure. I have asked friends in IO why they don’t use linear solutions more often and the impression I am left with is that it is a combination of an assumption that a linear solution simply won’t work for the kinds of problems they analyze together with a lack of familiarity with linear methods.

There are of course cases in which there is important non-linear behavior that needs to be featured in the results. For these cases it does indeed seem like a linear approach is not appropriate. Brad DeLong argued that accommodating the zero lower bound on interest rates (i.e., the “flat part of the LM curve” ) is such a case. I agree. There are cases like the liquidity trap that clearly entail important aggregate non-linearities and in those instances you are forced to adopt a non-linear approach. For most other cases however, I’ve already chosen my team …

 

[Fn 1] This reminds me of a joke from the consulting business. I once asked an economic consultant why he thought he could give valuable advice to people who have been working in an industry their whole lives. He told me that I was overlooking one important fact – the consultants charge a lot of money.

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