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w up article in bloomberg noah says much of the blame comes from abusing mathematical theory by failing to draw a tight link between mathematical elements and the real world hopefully romer is talking about something more than just mathematical errors in papers by prominent researchers if this is his main gripe let me break the bad news to you every paper has errors sometimes the errors are innocuous sometimes they are fatal errors are not confined to mathematical papers either there are plenty of mistakes in purely empirical papers the famous mistake in the reinhart rogoff debt paper the results from levitt s abortion paper and so on mistakes and mess ups are part of research criticizing someone for making lots of mistakes is almost like criticizing them for doing lots of research if you aren t making mistakes you aren t working on stuff that is sufficiently hard this isn t to say that i encourage careless or intellectually dishonest work all i am saying is that mistakes are an inevitable byproduct of research particularly research on cutting edge stuff moreover mistakes will often live on mistakes are most likely to be exposed and corrected if the paper leads to follow up work unfortunately most research doesn t lead to such subsequent work and thus any mistakes in the original contributions simply linger this isn t a big problem of course since no one is building on the work focusing on mistakes is also not what we should be spending our time on we should not be discussing or critiquing papers that we don t value very much we should be focused on papers that we do value this is how we judge academics in general i don t care about whether people occasionally or frequently write bad papers i care whether they occasionally write good ones we don t care about the average paper we care about the best papers the orderstatistics if romer is indeed talking about something other than mistakes then i suspect that his point is closer to what noah describes in his recent columns mathiness is a kind of mathematical theory that lacks a sufficiently tight link to reality certainly having the tight link that noah talks about is advantageous such a connection allows researchers to make direct make comparisons between theory and data in a way that is made much more difficult if the mapping between the model and the data is not explicit on the other hand valuable insights can certainly be obtained even if the theorist appeals to mathematical sloppiness hand waving mathinesss whatever you want to call it in fact i worry that the pressure on many researchers is often in the opposite direction instead of being given the freedom to leave some of their theories somewhat loose reduced form partial equilibrium researchers are implored to solve things out in general equilibrium to micro found everything to be as precise and explicit as possible often at the expense of realism i would welcome a bit of tolerance for some hand waviness in economics outside of economics the famous story of the proof of fermat s last theorem includes several important instances of at least what can be described as incompleteness if not outright hand waving the initial taniyama shimura conjecture was a guess the initial statement of gerhard frey s epsilon conjecture was described by ken ribet who ultimately proved the conjecture as a plausibility argument even though they were incomplete these conjectures were leading the researchers in important directions indeed these guesses and sketches ultimately led to the modern proof of the theorem by andrew wiles wiles himself famously described his work experience like stumbling around in a dark room if the room is very dark and very cluttered then you will certainly knock things over and stub your toes searching for a lightswitch fn1 in economics some of my favorite papers have a bit of mathiness that serves the papers brilliantly a good example occurs in mankiw romer and weil s famous paper a contribution to the empirics of economic growth qje 1992 as the title suggests the paper is an analysis of the sources of differences in economic growth experiences the paper includes a simple theory section and a simple data section the theory essentially studies simple variations of the standard solow growth model augmented to include human capital skills know how the model is essentially a one good economy with exogenous savings rates in some corners of the profession using a model with an exogenous savings rate might be viewed as a stoning offense but it is perfectly fine in this context the paper is about human capital not about the determinants of the saving rate but that s not the end of the mathiness their analysis proceeds by constructing a linear approximation to the growth paths of the model and then using standard linear regression methods together with aggregate data naturally such regressions are typically not identified but mankiw romer and weil don t let that interfere with the paper they simply assume that the error terms in their regressions are uncorrelated with the savings rates and proceed with ols there is a ton of mathiness in this work and the consequence mankiw romer and weil s 1992 paper is one of the most cited and influential papers in the field of economic growth think about how this paper would be changed if some idiot referee decided that it needed optimizing savings decisions after all we can t allow hand waving about exogenous savings rates multiple goods and a separate human capital production function no hand waving about an aggregate production function or one good economies micro data no hand waving about aggregate data and instruments for savings rates population growth rates and human capital savings rate no hand waving about identification the first three modifications suggested by the referee would simply be a form of hazing combined with obfuscation the modifications make the authors jump through hoops for no good reason and the end product has an analysis that is less clear the last one insistence on a valid instrument would probably be the end of the paper since such instruments probably don t exist thank god this paper didn t run into a referee like this my own opinion is that mathematical sloppiness can be perfectly fine if it deals with a feature that is not a focus of the paper hand waving of this sort likely comes at very little cost and may have benefits by eliminating a lengthy discussion of issues only tangentially related to the paper on the other hand if the hand waving occurs when analyzing or discussing central features of the paper then i am much more inclined to ask the researcher to do the analysis right this type of hand waving happens sometimes but it is not clear that it happens more often in macroeconomics or in freshwater macro at that ironically the freshwater guys that romer criticizes are much more likely to demand a tight specification and precise analysis of the model whether it is called for or not fn1 if you are interested in the modern proof of fermat s last theorem i highly recommend this documentary 13 comments may 10 2015 by chris house 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 14 comments may 10 2015 by chris house back to blogging i haven t written a post in a long long time research referee reports teaching and administrative work have all been preventing me from contributing to the blog this summer i m going to try to get back into writing somewhat regularly famous last words leave a comment search recent posts homo economicus vs homo economicus straw icus is big data the answer to all our ...