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I was told (correctly) that the ability to forecast something is not the same as understanding it. I was also told that inflation is pretty well understood by macroeconomists nowadays.

How do macroeconomists assess how good/accurate their understanding/model of inflation is? (I am guessing not by forecasting inflation based on their models/understanding, and checking the results against reality as it unfolds.)

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How do macroeconomists assess how good/accurate their understanding/model of inflation is?

tl;dr

In short macroeconomists do that by checking model against historical data and checking if we can find historically that coefficients or directions of causality agree with our model(s) of inflation.

If one can find evidence of causality, or coefficient estimates that agree with our theoretical model we consider that as an evidence that our understanding is correct, and when we find evidence of causal relationship or lack of there-off or coefficient estimates inconsistent with the models of how inflation works we would consider that as an evidence that our understanding is not completely correct. Moreover, in modern Popperian understanding of science there is no such thing as absolutely true model, and hence no phenomenon can be considered to be understood absolutely, but the more the model implications match historical data the more correct our understanding becomes.

Of course, decisions are not made based on a single study or result, researchers will either formally or informally look at results from various studies, weighted by their quality.

Full Answer:

Why/how is this different to forecasting?

It is different because with forecasting you need to use presently available information to guess what future value of something is. As a result even if you know for sure x causes y you might simply not even be able to have data on x so even if you would want to use such 'structural' model for forecasting you would first have to forecast x. That might end up giving you worse forecast than just trying to predict future from some variables that might be superficially correlated even if there is no causal relationship (e.g. this could be used for forecasting and could even be accepted by professionals whereas nobody would accept it as a causal relationship).

However, in regular (non-forecasting) empirical analysis you have no such limitations. You don't care about forecasting what will happen in future from currently available limited noisy and often inaccurate data (GDP figures are routinely revised even 5 years after initial publication). You can take full advantage of revised/corrected 'vintage' macro data series where there is much less noise and inaccuracy and you can also easily include contemporaneous variables so if you for example believe that current GDP affects inflation, such knowledge is useless for forecasting (as GDP figures will typically be published long after current inflation figures) but such relationship can be tested or modeled in standard retrospective empirical analysis.

How is this done?

Now to be clear I will describe here the best practice or how it should be done (of course in real life you might see some research, especially one published in bad journals, deviate from this).

  1. You start with some model (ideally explicit rigorous micro-founded one) you want to test.

For example you could start with quantitative theory of money which is described by:

$$MV=PY$$

where $M$ is money supply, $V$ velocity of money, $P$ price level (change of which is inflation) and $Y$ real output. Or New Keynesian LM curve:

$$M/P = L(Y,i)$$

where $L$ is money demand $i$ nominal interest rate (other variables have the same meaning as in QTM). Or Philips curve (which is technically still part of IS-LM-PC model, and actually consistent with above LM curve):

$$\pi = \pi^e - b(u -u_n)$$

where $\pi$ is inflation, $\pi^e$ inflation expectation, $u$ current unemployment and $u_n$ natural rate of unemployment, and so on.

  1. You come up with appropriate identification (i.e. estimation) strategy (or alternatively numerical simulation which I will cover separately later).

Of course, you can't just use naive OLS to estimate every relationship, and you can't usually just take model exactly as it is as for example multiplicative model such as MV=PY would be nightmare to estimate in such form. Lets continue using QTM as an example. You would first think about the model (which variables are exogenous which endogenous). In canonical QTM $P$ is the only endogenous and $V$, $Y$ and $M$ exogenous (Mankiw Macroeconomics pp 87). Hence you would first solve the model for $P$:

$$P = \frac{MV}{Y}$$

Next typically linearize a model because non-linear models are nightmare to estimate, so you would take logs of both sides to to make the model linear:

$$\ln P = \ln M + \ln V - \ln Y$$

Next pick some model that could the the relationship above. Here one has to be careful as not every model is appropriate in every situation. For example, it is likely that in the relationship above $P$, $M$, $V$ and $Y$ are non-stationary so you should not use regular regression to estimate the relationship. In such case you could test for cointegration or use different transformation of the relationship (e.g. by taking time derivate you could show that relationship will hold between growth rates as well and even if levels are non-stationary growth rates can be). In this case there should be cointegration, so I will continue this example as if we would found it, in that case the model can be still estimated by OLS (although one should use more specifically Fully Modified OLS - see Verbeek Guide to Modern Econometrics 4th ed pp 346).

So you will set up model like:

$$\ln P_t = \alpha + \beta \ln M_t + \gamma \ln V_t + \omega Y_t + \epsilon_t$$

Next before you go to estimation itself you need to think about appropriate variable selection. For example, do I use CPI as a good measurement of $P_t$ or GDP deflator? Or should we use $M2$ as measure of $M_t$ or $M3$? Or velocity data might be hard to find so one could use interest rates which affect velocity, and allow for coefficients to not to be unity. People generally try to use variables that they believe are most relevant subject to data availability constraint. Sometimes, there can be disagreement about which measurement for a variable is appropriate (e.g. people often argue in Philips curve how inflation expectations are supposed to be measured) in such case one might run in parallel multiple models with different measurements of same variable and see if they agree or not.

In addition to thinking about variables people also need to think about model specification in sense that one could argue that in the test above we can directly impose restriction such as ($\alpha=0$) or it could be argued that $\alpha$ should unconstrained as there could be some level shifts in data.

  1. You estimate the model using historical data and test how well it fits the theory

Continuing QTM example, once you estimate the model you will get some result such as (I took the numbers from a random study Omanukwue 2010 - its not necessary best study, I wanted to just find some quick numbers so I literally took first google scholar result, also the study uses interest rate to proxy velocity and has some extra controls but I just include numbers for this example):

$$\ln P = 11.15 + 0.55 \ln M -0.01 V - 1.34 \ln Y$$

Then you first test whether the coefficients statistically significantly different from zero. If not then likely the variable did not had influence on dependent variable. Next you check whether the signs correspond to the theoretical model or not (here M and Y has correct sign V not - although it was proxied by interest rate which can explain that).

Then in addition you run some extra tests to see whether the values are statistically different from values that are predicted by theory. For example, you can run Wald test that will test whether $\beta = \gamma = \omega =1 $.

In addition, it is not always possible to run such a neat tests. So sometimes tests are done indirectly, for example you could take advantage of quasi-experimental design such as DiD or synthetic control or other method to try to see if there indeed is causality from money supply to inflation as predicted by QTM or not.

  1. Compare fit of alternative models

In the current, instrumentalist-Popperian-Kuhnian, approach to philosophy of science people accept that no theory is perfect, so theories are not simply rejected if they do not perfectly fit every data point, but you comparatively select theories that best fit the data among competing theories.

For example, a competing theory to QTM is New Keynesian LM relation given by:

$$M/P = L(Y,i)$$

This theory is less restrictive as it allows for $Y$ and $i$ to have effect different from unity like QTM. This model could be rewriten for estimation purposes as:

$$\ln P_t = \alpha + \beta \ln M_t - \omega\ln Y_t - \gamma i_t + \epsilon_t$$

So if we consistently find evidence that lets say $\omega \neq 1$ we can take that as an evidence that New Keynesian LM provides better understanding of price level (and thus inflation) than QTM. In addition for example LM relation allows for endogeneity so if you find also simultaneity to be present you can view that as a score for New Keynesian LM vis-a-vis QTM.

This is being done iteratively in such a way that as time progresses we are getting closer and closer to true model (complete understanding of the phenomenon) although of course that goal is never reached we just constantly approach it.

However, in the instrumentalist spirit that is fine, as long as we accept nothing can be understood 100%. For example, even the flawed QTM above would correctly allow us to understand that increase in money supply leads to higher price level (inflation) even if QTM prediction that this is perfectly proportional relationship might not be supported.

  1. Numerical modeling (alternative to the empirical estimation above)

Sometimes its impossible to find enough data to run appropriate empirical model (e.g. for models that allow for endogeneity you need to usually find instrument that might be extremely difficult), so people also try to test theories via numerical modelling.

In a numerical modelling you would start from theoretical model or some well known empirical estimates that you take as a given parameters (i.e. you don't re-estimate them), and then feed the model historical data and see if output of the model matches real observed values. For example, continuing with our QTM example, you can could run very simplistic simulation by just taking log of $M$, $Y$ and $V$ and adding them together to see if add up values follow real life observed values for log of $P$ (of course this is extremely simplistic example nowadays simulations are much more complex and this wouldn't pass even as a bachelor thesis but I am trying to keep things as simple as possible).

This might sound as forecasting, and indeed numerical models can also be used for forecasting, but here I am specifically referring to numerical models using retrospective data, since again forecasting has its own challenges (needing forecasts for x for any structural model etc).

You can see lot of examples of numerical modeling being used to support theory of inflation in the recently published Fiscal Theory of Price Level by Cochrane if you want see some examples.

  1. Researchers aggregate results

Finally, as one of my professors used to quip, you do not pee your pants every time new study is published with interesting results. Due to stochastic nature of empirical work it is always possible to find negative results even if there actually was true relationship in the data. Hence, researchers will aggregate research results both formally (meta studies) and informally (literature review).

This is finally where you can start assessing also how good our understanding is. If lets say 100 studies show that there is negative relationship between output and price level and single study shows there isn't that single study can be dismissed as a statistical fluke. If 100 studies show there is such relationship but 20 studies show there isn't and those 20 studies are not just random but all focus on some specific time period that would be indication that although we understand general nature of the relationship there are some specific cases which we can't explain with our current models. Of course, this gets little bit tricky in real life as one has to always keep in mind possibility of publication bias, and it is important to weight studies by quality (i.e. Q1 journal study with excellent identification strategy outweighs even thousands of some Q4 predatory journal studies with terrible identification strategy that is well known to produce biased results).

Currently, the most generally accepted model of inflation is one based on $IS-LM-PC$ model, which is why this model is included in most contemporary textbooks (e.g. Blanchard et al Macroeconomics or Mankiw Macroeconomics etc). In this model you can think about inflation in long run through the LM relation $M/P = L(Y,i)$ and in short run through Philips curve $\pi = \pi^e - \beta (u - u_n)$. These models are widely accepted because, generally speaking they can explain quite a lot. Philips curve is quite good in explaining month-to-month variation, in fact Philips curve in hindsight predicts current high inflation (e.g. see Gorodnichenko et al. 2019), even though the timing was off. The model can also explain why interest rates worked well when central banks tried to adjust inflation in period before 2009 and not after, when nominal interest was close to zero, as in the model liquidity trap can occur (contrast it with QTM that cannot explain liquidity traps well).

Of course, the model cannot explain everything. For example one thing that IS-LM-PC model cannot explain well is for example why QE1-2 and partially 3 had not much effect on inflation but QE4 had large effect. There are actually already newer theories that can explain that such as fiscal theory of price level that builds upon Woodford's seminal work, but since this is quite new theory it does not yet have mainstream acceptance. Yet even with IS-LM-PC model by my guesstimate you can explain about 70% of all variation in inflation rate over time.

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