When you ask people to trade an asset with an unknown terminal payout in a lab experiment, it’s really common to observe a boom in the asset’s price followed by a sudden crash right before the trading session ends. In other words, it’s very common to observe price paths that resemble a bubble. Bubbles are a regular occurrence in the lab even when participants are given no additional info about the assets they are trading and are not allowed to talk with one another.
By contrast, the recent literature on real-world asset bubbles tends to emphasize the role played by specific asset characteristics and social interactions. For example, Greenwood, Shleifer, and You (2019) document that price booms in industries with lots of young firms are more likely to followed by a crash. In my recent paper, The Ex Ante Likelihood of Bubbles, I show that speculative bubbles are more likely in assets where small price increases make excited speculators much more persuasive to their friends.
What are we to make of these two contradictory sets of results? I have a hunch that the boom/bust price paths observed in lab experiments are not the same phenomenon as the asset bubbles observed out in the wild. But this is just a hunch. And, as highlighted above, I’m definitely not an impartial judge on this topic.
So how might you check if the trading phenomenon that you’re studying in the lab is the same as the one you observe in real-world financial markets? It’s common to hear asset-pricing researchers say something like: “There’s *ALWAYS* a chance that a lab experiment has missed some important real-world consideration.” It’s certainly true that researchers should worry about omitted variables. But is it really the case that this problem is totally unsolvable? The *ALWAYS* part of the claim strikes me as too strong.
In this post, I describe an analogous problem from the physical sciences and show how you might solve it using dimensional analysis. My point is not that asset-pricing researchers should use dimensional analysis (but that ain’t a terrible idea). My point is that you can theoretically analyze the relationship between the lab and real-world asset markets. Asset-pricing researchers should be doing more of that. Is trading in the lab governed by the same economic forces as trading in real life? This need not be a philosophical question.
An analogous problem
Imagine you’re an engineer who works at a company that makes personal submarines, and your team’s most recent prototype turned out to be much slower in field tests than expected. Something about the craft must be generating excess drag, and you want to figure out what.
You have the data on the prototype submarine’s performance during the recent field tests done out in the ocean. But this craft is 18ft long and weighs several tons. It’s entirely impractical to tinker with. So you create a 1ft mock-up that you can quickly and cheaply experiment on in a wave pool.
But how can you be sure that these results will apply to the full-sized submarine? How can you be sure that your lab experiment isn’t omitting an important real-world consideration? And what is the correct way to scale up the results in the wave pool to results in the ocean?
This an analogous problem to the one outlined above in the introduction. Instead of worrying about the drag force experienced by a submarine, asset-pricing researchers are interested in some property of speculative bubbles (i.e., timing/severity of the crash, likelihood of future occurrence, etc). And, instead of using a 1ft model in a wave pool, they run market simulations in university labs. But the resulting question are identical: are the findings in the lab informative about the real world?
Here are the relevant variables for determining the submarine’s drag, : the length of the ship, , the speed at which the ship is traveling, , the density of the liquid it is traveling through, , and the viscosity of this liquid, . The functional form of the relationship between these variables
(1)
will be determined by the nitty-gritty details of the submarine design. You don’t know this function. But, for your wave-pool experiments to be informative about the drag experienced by a full-sized submarine, this function needs to be the same for both. Any differences between the drag experienced by an 18ft submarine in the ocean and the drag experienced by a 1ft in your lab should be determined by the inputs , , , and .
Consider the alternative. Suppose there are other important factors in determining drag in the open ocean besides , , , and . Or suppose your wave pool is introducing a new variable that is irrelevant to drag force in the open ocean. In either case, your analysis of a toy 1ft submarine will not generalize to the real-world 18ft object. If it turns out to be the case that
(2)
then your lab experiments will not be informative.
Dimensional analysis
You want to know when/how/whether there is any way to address this concern. Dimensional analysis offers one way to do this. “We are allowed to add, or subtract, various quantities together only if they are expressed in the same units. Thus, the left hand and right hand sides of an equation… must have the same physical dimensions.” And it turns out that it’s possible to leverage this seemingly banal observation to test whether . Here’s how.
I will use to denote the dimension of a variable. The length of the ship, , has dimension . It doesn’t matter whether this length has units of feet, meters, or parsecs. The speed at which the submarine is traveling, , has dimensions of length per unit of time, . Drag, , is a force, and Newton taught us that force equals mass times acceleration:
(3)
Density, , measures how much stuff there is per unit of volume, . Finally, viscosity, , measures how much force is needed to deform a given area per unit of time, .
Suppose it’s possible to input , , , and . When you do this, the output is . If the functional form in Equation (1) holds across a wide-range of values, then it should also hold in the special case where , , and . The original values of , , , and were just arbitrary choices from the domain of .
But notice the problem that this creates. By setting and , we’ve just stripped the dimension of from the inputs to : , , and . So there is no way for us to balance the dimensions on the right-hand and left-hand sides of Equation (1) without making an additional change.
The simplest possible change would be to multiply by . If we do that, we get:
(4)
That way, we would have times . This yields a combined dimension of for the right-hand side, which is exactly what we’re after.
Of course, there’s nothing special about the density input to . We could do a similar trick for length, , and speed, . If we set , , and , then we would need to multiply the right-hand side by an additional factor of to preserve dimensional consistency:
(5)
Again, and are chosen to balance the dimensions on either side of the equal sign in Equation (1).
What’s more, there was nothing special about our original choice of inputs: , , , and . These could have been anything in domain so long as we used the same values throughout. So, if we drop the ones as inputs to and rearrange things a bit, we get:
(6)
The left-hand side is dimensionless, , and commonly called the “drag coefficient”. The input to is called the Reynolds number, , and is also dimensionless, .
Why this helps
We didn’t know anything about the functional form of when we started our analysis. And we still don’t know anything about the functional form of . We have to empirically estimate both function based on observed data. So it might not be immediately obvious how dimensional analysis has improved things.
What’s more, every textbook I’ve read has talked about the benefits of dimensional analysis in terms of variable elimination. was a function of four variables. is a function of a single variable. These textbooks argue that it’s easier to estimate than because , , , and have been combined into a single variable, . This is true. But it’s not why dimensional analysis is helpful here.
Dimensional analysis is helpful for assessing the external validity of your wave-pool experiments because you know the way that the single variable, , combines the four other inputs , , , and . As a result, you can compare predictions made using Reynolds numbers that were arrived at in very different empirical settings.
For example, suppose that you observe a particular drag coefficient in the data from the recent open-ocean field tests involving the full-sized 18ft submarine:
(7)
If you fill your wave pool with a liquid that is -times as viscous as sea water, like honey , do you observe a similar drag coefficient on your 1ft model submarine?
(8)
The first calculation comes from real-world data. The second comes from your lab.
Do they match? If yes: great!! Your lab experiment captures the main forces at work in your open-ocean field test, . If no: bummer. Either your lab experiment is introducing unwanted variables or it is missing an important consideration. Either way, you should not assume your lab results can be applied to the next full-sized prototype submarine your company turns out.
The original problem
Asset-pricing researchers regularly see prices that boom and bust in trading experiments. These episodes look a whole helluva lot like speculative bubbles. But are they? Are the forces that explain these episodes the same ones that produced the Dotcom bubble? Or are these two different phenomenon? I really want to know the answer to this question.
The point of this post is not to argue that dimensional analysis is the obvious way forward. I wrote about an “analogous problem” rather than the problem I’m actually interested in because it’s not immediately obvious how to apply dimensional analysis to study speculative bubbles. It might be possible. But I haven’t managed to select the right variables. If I knew how to do this, this would be a paper and not a blog post.
Instead, my goal is to argue that it’s possible to assess whether the phenomenon observed in the lab is the same as the one observed out in the real world. This is true when the lab experiment involves a 1ft submarine and a wave pool. It’s also true when the lab experiment involves undergrads buying and selling a fictitious stock. Lab experiments are a major source of knowledge in the social sciences. Rather than complaining that a given experiment might be missing something important about the real world, asset-pricing researchers should work towards developing methods for verifying external validity. The application of dimensional analysis to the submarine example above is a proof of concept. It can be done. It is done in other fields.
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