By Simon Grey, on November 10th, 2011
Recently, I’ve been thinking a lot about how pointless economic models are. A good portion of this is probably due to being in a college microeconomics class (which uses Mankiw’s book, which Vox picks apart here.). What passes for mainstream economics is nothing more than gussied-up tautologies and pretty models that don’t actually prove anything. What irks me most is how the proponents of the mainstream view—in this case my professor and Mankiw’s textbook—act so certain about everything they assert.
For example, Mankiw devotes a chapter to the actions of a monopoly in the free market, complete with very precise mathematical models. Yet, this entire chapter is utter B.S. because all the models in the world don’t change the fact that said models are based entirely on assumptions and tautologies, and stupid tautologies at that. The models, then, only say what the programmer thereof tells them to say. This isn’t science; it’s an appeal to personal authority, writ large.
In the same vein, an economic prediction is only going to be as accurate as the assumptions upon which it is based. The reason why so many mainstream pundits missed the housing bubble is not because they lacked computational power with which to build their macroeconomic models; it’s because their assumptions about the housing market and banking industry were wrong. They thought housing growth could be sustained forever, or that the current housing growth was organic instead of government-subsidized, or that the government would be able to subsidize housing forever, etc.
Trying to build on assumptions of arbitrarily-defined variables being correlated to one another is a recipe for failure, as is evidenced by decades of repeatedly doing so. The time has come to put away the pretentious belief in the need for mathematically precise models; this isn’t physics, after all. Ultimately, the foundation of economics must be built on understanding Man as an economic actor. This paradigm shift cannot come soon enough.
By Ajay Shah, on October 27th, 2011
I was at a meeting in London recently, organised by the IGC, on the subject of the research agenda in macroeconomics for developing countries. This made me think about how to make progress.
The US as the shared dataset for mainstream macroeconomics
All existing knowledge on macroeconomics is rooted in data about the US economy. The US is seen as a canonical developed
country. Economists all over the world have treated it as a common object of study, when building macroeconomics. It is a shared
dataset. Researchers and Ph.D. students routinely pull out a paper from the literature, and replicate the results, as a first stage of
offering innovations: all this is rendered convenient by using the US as a shared dataset. New work is generally obliged to demonstrate value-add in the context of the US dataset.
The US works as a shared dataset because it has high quality data. Good quality data starts right after 1945, because there was no
destruction within the country, hence the early post-war years are not distorted by unusual reconstruction. There was a steady shift away from dirigisme from 1945 onwards, but for the rest there has been no regime change: events like the breakdown of communism or the rise of the European Union or the Euro have not taken place.
In the US, a high quality statistical system has produced good aggregative data. Organisations like NBER have processed this data
nicely to create datasets about the business cycle. High quality datasets are available about households, firms and financial markets. Household- and firm-level data has been nicely utilised to obtain numerical values for parameters in macroeconomic models: why
estimate something using macro data when you know it using gigantic and well trusted micro datasets? Finally, the major question
for macro today is the fusion with finance, and the US has nice data for the financial system.
As a consequence, facts about the US are the shared dataset used in all mainstream macro research across the world.
The insights developed in this literature, which has examined the US economy, have been transported with fair success, into other
developed countries. Thus, this emphasis on the US as a common dataset has delivered good results. As an example, the revolution in monetary policy which was thought through by Friedman, Lucas, etc. was created using US data. It has usefully reshaped central banks worldwide. US data was essential for inventing inflation targeting, but inflation targeting has worked well outside the US.
The major obstacle on building a macroeconomics for developing countries
The major obstacle that interferes with doing macroeconomics in developing countries is data.
India is a good example of what goes wrong. The standard GDP data is in bad shape. The annual GDP data is deplorable, and the quarterly GDP data that is so essential for doing macroeconomics is worse. The IIP is untrustworthy. Put these together, and we don’t have an output series, really.
The BOP data is measured fairly well. Some plausible inflation data is now starting to come together. The statistical system run by the government does not produce seasonally adjusted data [succor]. Given the absence of the Bond-Currency-Derivatives Nexus, the bulk of data about interest rates that is required is missing; policy makers are flying blind. The standard household survey (NSSO) is in bad shape: it does not produce panel data, surveys are only conducted once in a few years, and there are incentive issues about the front-line staff who interact with households.
The large firms are observed using the CMIE database; the small firms are not observed using the ASI dataset. The CMIE household
survey is starting to generate knowledge about households, but this only got started a few years ago. While the CMIE datasets (on firms and households) can be aggregated up to create many interesting macro series, so far this process has only begun in a small way.
Faced with these problems, it is not surprising that little is known, at present, about macroeconomics in India. We know numerous
important questions, and we know that we don’t know the answers. The roadmap to progress is often, though not always, blockaded by data constraints.
Many such problems bedevil the statistical system in other developing countries also.
Economists have complained about bad data in developing countries for decades, and that hasn’t changed things. And there is a uniquely perverse problem. Incremental progress with a gradually improving statistical system does not get the job done for us: By the time a country gets to good institutions and thus a good statistical system (e.g. Taiwan, South Korea, Israel, Chile), the country is not a developing country anymore and is thus not a useful dataset for studying the macroeconomics of developing countries. Chile has world class databases on households and firms, but you can’t extract microeconomic facts using these datasets and use them in
calibration if your object of inquiry is the canonical developing country.
A proposal
How can we make progress? I feel the first idea that we need to agree on is that we do not need many developing countries to build a
great literature. We need a shared dataset, a lingua franca, a replication platform, using which we will build a literature. We need
a country that will play the role, for the macroeconomics of developing countries, that has been played by the United States in
conventional macroeconomics.
The second idea is that we should be a little more ambitious. We should not merely sit around hand-wringing, complaining about a
problem that isn’t going to solve itself. When scientists in other disciplines identify questions that call for evidence, they write
funding proposals (sometimes running to billions of dollars) and organise themselves to create those datasets. Could we do similarly?
Specifically, imagine that we pick one canonical developing country. It’s got to be a typical developing country in most respects. And, it should not be a conflict zone, it should have the basics of law and order and physical safety so that operations can be mounted in it. Christopher Adam of Oxford suggests that Tanzania is a good choice.
Imagine that, the system of interest (a developing country) keeps running, but it gets instrumented up to world class. In essence, we
try to place first world instrumentation into a third world country. (To the extent that this data improves decision making in the
country, we would suffer from `Heisenberg’ effects).
This will call for financial resources and, more importantly, organisational capability. The physicists know how to organise themselves to build the Large Hadron Collider. Most of the time, economists do not organise themselves as laboratories or teams doing complex projects. This will be a bridge that we will have to cross.
As with the Large Hadron Collider, this is not a short-term project. It is a project that needs to run for 25 years, in order to
generate a strong dataset.
At first, the project will generate useful facts for calibration, drawing on household survey and firm databases. Gradually, as the span
of the time-series builds up, the full picture will start becoming clear.
If this works, it can ignite a literature where researchers from all across the world do replicable work off a common dataset. Perhaps
Tanzania could then play a role, for the macroeconomics of developing countries, that is comparable with the role played by the United States in mainstream macroeconomics.
By Christopher Briem, on September 29th, 2011
I read this headline and really couldn’t believe it. ComputerWorld of all places has this story today: Pirates tap BI tools to forecast, boost attendance. Notice there is no mention of improving on field performance as a means to improve attendance. They just want to narrow in on what poor folks are still willing to sit and watch another losing season.
Hey, I’m all for applying wonkery in all the weird corners of the world, but there is something perverse about that story and the Pirates endless losing streak. Moneyball is about how Billy Beane used some fundamentally econometric techniques to actually improve Oakland’s performance on the field. The Pirates’ version of that skips the box score and focuses entirely on squeezing more efficiency from the box office. I think that article says it all.
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By Ajay Shah, on September 2nd, 2009
For over 20 years, CMIE has computed `market shares’ of the companies who compete in a certain product. They have also computed the Herfindahl index of concentration. This data is valuable in obtaining a snapshot of what is happening in an industry. But more interesting, this data goes back to 1990-91, and thus constitutes a valuable historical series using which we can obtain insights into individual industries and the economy.
I often get asked how this information base can be accessed, other than by going through print documents over the years. Here are the steps through which you can get data on markets shares, and the time-series of the Herfindahl index of concentration from 1990-91 onwards, for all industries:
- Use the CMIE website http://www.business-beacon.com. This is a pay-per-use site. You can create an account to use this over the net.
- Start at the tree-structured industry classification.
- Pick an industry of interest: e.g. Paper.
- Click on Market shares.
- To go further, you need to have a prefunded account.
- Click on Market share of companies. At this point, you are charged Rs.100. This shows the market share of each company in the paper business. The last row (at the bottom) shows the Herfindahl index. This has been around 0.02 in recent years — suggesting very little market power.
- At the right hand top, there is access to a spreadsheet where there is a full time-series for the market share of 425 paper companies, going back to 1990-91. At this point, you are charged Rs.250. This spreadsheet has the bottom row (row 430) with the time-series of the Herfindahl index.
By Ajay Shah, on July 23rd, 2009
Risk management failures have clearly taken place. It has become fashionable to criticise risk models.
A fair amount of the naive criticism is not well thought out. Too many people today read Nassim Taleb and pour scorn upon hapless economists who inappropriately use normal distributions. That’s just not a fair depiction of how risk analysis gets done either in the real world or in the academic literature.
Another useful perspective is to see that a 99% value at risk estimate should fail 1% of the time. If a VaR implementation that seeks to find that 99% threshold does not have actual losses exceeding the VaR on 2-3 trading days each year, then it is actually faulty. Civil engineers do not design homes for once-in-a-century floods or earthquakes. When the TED Spread did unbelievable things:
the loss of a short position on the TED Spread should have been bigger than the Value at Risk reported by a proper model on many days.
The really important questions lie elsewhere. Risk management was a new engineering discipline which was pervasively used by traders and their regulators. Does the field contain fundamental problems at the core? And, are there some consequences of the use of risk management which, in itself, create or encourage crises?
Implementation problems
There are a host of practical problems in building and testing risk models. Model selection of VaR models is genuinely hard. Regulators and boards of directors sometimes push into Value at Risk at a 99.99% level of significance. This VaR estimate should be exceeded in one trading day out of ten thousand. Millions of trading days would be required to get statistical precision in testing the model. In most standard situations, there is a semblence of meaningful testing for VaR at a 99% level of significance [example], and anything beyond that is essentially untested for all practical purposes.
Similar concerns afflict extrapolation into longer time horizons. Regulators and boards of directors sometimes push for VaR estimates with horizons like a month or a quarter. The models actually know little about those kinds of time scales. When modellers go along with simple approximations, even though the underlying testing is weak, model risk is acute.
In the last decade, I often saw a problem that I used to call `the Riskmetrics illusion’: the feeling that one only needed a short time-series to get a VaR going. What was really going on was that Riskmetrics assumptions were driving the risk measure. Adrian and Brunnermeier (2009) emphasise that the use of short windows was actually inducing procyclicality: When times were good, the VaR would go down and leverage would go up, and vice versa. Today, we would all be much more cautious in (a) Using long time-series when doing estimation and (b) Not trusting models estimated off short series when long series are unavailable.
The other area where the practical constraints are onerous is that of going from individual securities to portfolios. In practical settings, financial firms and their regulators always require estimates of VaR for portfolios and not individual instruments.
Even in the simplest case with only linear positions and multivariate normal returns, this requires an estimate of the covariance matrix of returns. Ever since atleast Jobson and Korkie (JASA, 1980), we have known that the historical covariance matrix is a noisy estimator. The state of the art in asset pricing theory has not solved this problem. So while risk measures at a portfolio level are essential, this is a setting where our capabilities are weak. Realworld VaR systems that try to make do using poor estimators of the covariance matrix of returns are fraught with model risk.
As an example, when we look at the literature on portfolio optimisation, there is a lot of caution about the complexity of jumping into portfolio optimisation using estimated covariance matrices. As an example, see this paper by DeMiguel, Garlappi, Nogales and Uppal, which is one of the first papers to gain some traction in trying to actually make progress on estimating a covariance matrix that’s useful in portfolio optimisation. This paper is very recent – it appeared in May 2009 – and highlights the fact that these are not solved problems. It seems easy to talk about covariance matrices but obtaining useful estimates is genuinely hard.
Similar problems afflict Value at Risk in multivariate settings. Sharp estimates seem to require datasets which do not exist in most practical settings. And all this is when discussing only the simplest case, with linear products and multivariance normality. The real world is not such a benign environment.
With all these implementation problems, VaR models actually fared rather well in most areas
There is immense criticism of risk models, and certainly we are all amazed at the events which took place on (say) the money market, which were incredible in the eyes of all modellers. But at the same time, it is not true that all risk models failed.
My first point is the one emphasised above, it was not wrong to have VaR models being surprised at once-in-a-century events.
By and large, the models worked pretty well with equities, currencies and commodities. By and large, the models used by clearing corporations worked pretty well; derivatives exchanges did not get into trouble even when we think of the eurodollar futures contract at CME which was explicitly about the London money market.
Fairly simple risk models worked well in the determination of collateral that is held by futures clearing corporations. See this paper by Jayanth Varma. If the field of risk modelling was as flawed as some make it out to be, clearing corporations worldwide would not have handled the unexpected events of 2007 and 2008 as well as they did. These events could be interpreted as suggesting that, as an engineering approximation, the VaR computations that were done here were good enough. Jayanth Varma argues that the key elements that are required are the use of coherent risk measures (like expected shortfall), fat tailed distributions and nonlinear dependence structures.
As boring as civil engineering?
In his article Blame the models, Jon Danielsson shows a very nice example of the simplest possible VaR problem: the estimation of VaR for a $1000 position on IBM common stock. He points out that across a reasonable range of methodologies and estimation periods, the VaR estimates range over a factor of two (from 1.77% to 3.26%).
This large range is disconcerting. But look back at how civil engineers work. A vast amount of sophisticated analysis is done, and then a safety factor of 2x or 2.5x is layered on. The highest aspiration of the field of risk modeling should be to become as humdrum and useful as civil engineering. My optimistic reading of what Danielsson is saying is that a 2x safety factor adequately represents model risk in that problem.
This suggests a pragmatic approach. All models are wrong; some models are useful. Risk modeling would then go forward as civil engineering has, with an attempt at improving the scientific foundations, and with a final coup de grace of a safety factor thrown in at the end. Civil engineering evolved over the centuries, learning from the cathedrals that collapsed and the bridges that were swept away, continually improving the underlying science and improving the horse sense on what safety factors are a reasonable tradeoff between cost and safety.
Fundamental criticism: the `Lucas critique of risk management’
When an econometric model finds a reduced form relationship between y and x, this is not a useful guide for policy formulation. Hiding inside the slope parameter of x is the optimisation of economic agents, which reflect a certain policy environment. When policy changes are made, these optimisations change, giving structural change in the slope parameter. When policy changes take place, the old model will break down; the modeller will be surprised at what large deviations from the model have popped up. The Lucas critique is an integral part of the intellectual toolkit of every macroeconomist.
It should be much more prominent in the thinking of financial economists also. The most fundamental criticism of risk models is that they also suffer from the Lucas critique. As Avinash Persaud, Jon Danielsson and others have argued, risk modeling should not only be seen in a microeconomic sense of one economic agent using the model. When many agents use the same model, or when policy makers or clearing corporations start using the model, then the behaviour of the system changes.
As a consequence of this fundamental problem, an ARCH model estimated using historical data is vulnerable to getting surprised by what comes in the future. The coefficients of the ARCH model are not deep parameters; they are reduced form parameters. They suffer from structural breaks when enough traders start estimating that model and using it. The reduced-form parameters are time varying and endogenous to decisions of traders about what models they use, and the kinds of model-based prudential risk systems that regulators or clearing corporations use.
In the field of macroeconomics, the Lucas critique was a revolutionary idea, which pretty much decimated the old craft of macro modelling. Today, we walk on two very distinct tracks in macroeconomics. Forecasters do things like Bayesian VAR models where there are no deep parameters, but these models are not used for policy analysis. Policy analysis is done using DSGE models, which try to explicitly incorporate optimisations of the economic agents.
In addressing the problem of endogeneity of risk, or the Lucas critique, we in finance could do as the macroeconomists did. We could retreat into writing models with optimising agents, which is what took macroeconomists to DSGE models (though it took thirty years to get there). One example of this is found in Risk appetite and endogenous risk by Jon Danielsson, Hyun Song Shin and Jean-Pierre Zigrand, 2009.
In the field of macro, the Lucas critique decimated traditional work. But we should be careful to worry about the empirical significance of the problem. While people do optimise, the extent to which the reduced form parameters change (when policy changes take place) might not be large enough for reduced form models to be rendered useless.
It would be very nice if we could now get an research literature on this. I can think of three examples of avenues for progress. Simulations from the Danielsson/Shin/Zigrand paper could be conducted under different policy regimes, and reduced form parameters compared. Researchers could look back at natural experiments where policy changes took place (e.g. a fundamental change in rules for initial margin calculations at a futures clearing corporation) and ask whether this induced structural change in the reduced form parameters of the data generating process. Experimental economics could contribute something useful: it would be neat to setup a simulated market with 100 people trading in it, watch what reduced form parameters come out, then introduce a policy change (e.g. an initial margin requirement based on an ARCH model), and watch whether and how much the reduced form parameters change.
In the field of macro, there is a clear distinction between problems of policy analysis versus problems of forecasting. Even if the `Lucas critique’ problem of risk modelling is economically significant (i.e. the parameters of the data generating process of IBM significantly change once traders and regulators start using risk modeling), one could sometimes argue that there is a problem of risk modelling which is not systemic. I suppose Avinash Persaud and Jon Danielsson would say that in finance, there is no such comparable situation. If a new time series model is useful to you in forecasting, it’s useful to a million other traders, and the publication of the model generates drift in the reduced form parameters.
Regulators have focused on the risk of individual financial firms and on making individual firms safe. Today there is an increased sense that regulators need to run a capability which looks at the risk of the system and not just one firm at a time. A lot of work is now underway on these questions and it will yield improved insights and regulatory strategies in the days to come.
Why did risk models break down in some situations but not in others?
I find it useful to ask: Why did risk models work pretty well in some fields (e.g. the derivatives exchanges) but not in others (e.g. the OTC credit markets)? I think the endogenous risk perspective has something valuable to contribute in understanding this.
There are valuable insights in the ECB working paper by Lagana, Perina, von Koppen-Mertes and Persaud in 2006. They think of liquidity as made up of two stories: `search liquidity’ as opposed to `systemic liquidity’. Search liquidity is about setting up a nice computer-driven market which can be accessed by as many people as possible. `Systemic liquidity’ is about the consequences of endogenous risk. If a market is dominated by the big 20 financial firms, all of whom run the same models and have the same regulatory compulsions, this market will exhibit inferior systemic liquidity.
This gives us some insight into what went right with exchange-traded derivatives: the diversity of players on the exchanges (i.e. many different forecasting models, many different regulatory compulsions) helped to contain the difficulties.
The lesson then, is perhaps this one. If a market is populated with a diverse array of participants, then risk modelling as we know it works relatively well, as an engineering approximation. The big public exchange-traded derivatives fit this bill. We will all, of course, refine the practice of risk modeling, drawing on the events of 2007 and 2008 much as the civil engineers of old learned from spectacular disasters. But by and large, the approach is not broken.
Where the approach gets into trouble is in markets with just a few participants, i.e. `club markets’. A typical example would be an OTC derivative with just a handful of banks as players. In these settings, there is much more inter-dependence. When a market is populated by just a small set of players, all of whom think alike and all of whom are regulated alike, this is a much more dangerous world for the use of risk modeling. The application of standard techniques is going to run afoul of economically significant parameter instability and acute liquidity risk.
Implications for harmonisation of regulation
Harmonisation of regulation is a popular solution in regulatory circles these days. But if all large financial firms are regulated alike, the likelihood of the failure of risk management could go up. Until we get the tools to do risk modeling under conditions of economically significant risk endogeneity, all we can say is that we do not know how to compute VaR under those conditions. Harmonisation of regulation will give us more of those situations.
In the best of times, there seem to be limits of arbitrage; there is not enough rational arbitrage capital going around to fix all market inefficiencies. With non-harmonised regulation, if a certain firm is constrained by regulation to not take a rational trade, some other firm will be able to do so. The monoculture induced by harmonised regulation will likely make the world more unsafe.
Acknowledgement
Tarun Ramadorai, Avinash Persaud, and Viral Acharya gave me valuable feedback on this.
By Ajay Shah, on July 14th, 2009
In order to make progress on doing macroeconomics in India, one weak link has been business cycle measurement. This, in turn, requires access to a wide range of seasonally adjusted time-series. In most countries, the infrastructure of seasonally adjusted data is produced by the statistical system, but in India this has not come about.
Seasonally adjusted series are particularly important in tracking current developments in the economy. The familiar year-on-year change is the moving average of the latest twelve monthly changes. In order to know what is happening in the economy, it is better to look at recent months, rather than looking back 12 months. The familiar y-o-y changes are a sluggish indicator of what is happening. Month-on-month changes are more informative: but this requires seasonal adjustment.
We have initiated some computation and release at cycle.in
At present, we have a dataset with seasonally adjusted levels for a few time-series. We will be updating this every Monday. At the above URL, you get a sense of what is happening with month-on-month changes of seasonally adjusted data in these series.
In the spirit of creating public goods, we make it easy for you to embed these graphs into your work products. We also have a .csv file with data for levels which can be the foundation of further work.
This will be useful in tracking current developments in the economy, and also make possible research in macroeconomics, which critically requires seasonally adjusted data.
We hope this is useful.
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By Cheryl Grey, on February 13th, 2009
Real gross domestic product (GDP) is the market value of all goods and services produced by a nation within a certain span of time, adjusted for inflation. It includes both the goods and services sold in the marketplace, such as a can of tuna at the grocery or server space leased from a website host, as well as those that are not, such as disaster relief provided by the Red Cross. Because this calculation is for what’s produced, it doesn’t include existing goods (re-sold homes, used cars) or those in transition (empty aluminum cans purchased by Coca-Cola to fill at a bottling facility). Nor does it include the value of stocks or bonds outstanding (although the sales commissions count), which is why the Dow Jones meltdown currently underway has not affected GDP estimates.
The adjustment for inflation is of primary importance. If an economy grew by 2.8% and inflation also rose by 2.8%, then the economy didn’t really grow. The same amount of goods and services were produced as before; only the prices increased. Economist Charles Wheelan calls it the equivalent of exchanging a $10 bill for ten $1 bills; your wallet feels fatter but there’s really no difference.
In the United States, the Bureau of Economic Analysis, a division of the Department of Commerce, keeps an index of inflation adjustments dating back to 1929, giving economists a stable means of comparison for U.S. economic performance across the years.
Nominal GDP has not been adjusted for inflation and is therefore merely raw data, which is why you don’t hear about it all that often.
Measuring GDP
GDP is measured in two ways: the raw figure, and the percent change from the previous time period. The U.S. real, inflation-adjusted GDP for the third quarter of 2008 reached $14,420,500,000,000.00, even if the economy is currently contracting rather than expanding. The sheer size of that number makes working with the raw figures rather cumbersome and also makes people’s eyes glaze over. It’s more easily understood if we simply say the U.S. economy contracted by 0.5% in the third quarter as compared to the second quarter of 2008, when it expanded by 2.8% over the first quarter.
Taking the real GDP figure and dividing it by that nation’s current population gives per capita GDP, another favorite economic scorecard, this one designed to compare economies by their average (not median) incomes and therefore standards of living. For example, Ireland’s 2007 GDP of $191,600,000,000.00, when divided by its population of 4,156,119, equals its per capita GDP of $46,600—higher than the $45,800 of the U.S.
If a nation’s population is growing, then its GDP must grow at least as quickly just to provide jobs for the new arrivals. An economic rule of thumb called Okun’s Law states that, in the U.S., GDP growth of 3% is required to prevent unemployment from rising. For every gain of 1% above that figure, the unemployment rate should fall by 0.5%. Although this pattern isn’t cast in stone, it’s been fairly consistent since the end of World War II, which unfortunately doesn’t bode well for job seekers through at least the end of this year, considering the current and expected fall in GDP worldwide.
Downsides to GDP
On average, GDP makes for a workable scorecard across economic borders; however, it does have its shortcomings. It doesn’t count values that aren’t easily transcribed into monetary figures, such as cultural, environmental, or historical values. An old-growth forest, in GDP terms, is worth no more than one planted for harvest by a forestry company (and it also doesn’t care what sort of owls call it home), while a building remains the total of its construction materials plus labor no matter who slept there.
GDP also doesn’t count work performed in the home unless it requires the purchase of cleaning materials or new siding. Nor does it count raising children as an investment for the future beyond braces and educational materials.
GDP also doesn’t include the “shadow economy,” constructed to avoid paying a tax or to bypass governmental regulations. The classic example is the waiter who doesn’t report his tips as income. Although it’s obviously difficult to calculate such things with any exactitude, a serious study performed by Friedrich Schneider of the Johannes Keppler Institute Linz claims that this shadow economy in the U.S. is approximately 7.9% the size of the official one, or around $1.14 trillion in 2007—larger than the official GDP of Australia.
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