Prediction markets function best when liquidity is high, but they break completely if the liquidity exceeds the price of influencing the outcome. Prediction markets function only in situations where outcomes are expensive to influence.
There are a ton of fun examples of this failing:
Option expiries (I don’t have a good single link for this)
I don’t know enough about how equities trade during earnings, but I do know a little about how some other products trade during data releases and while people are speaking.
In general, the vast, vast, vast majority of liquidity is withdrawn from the market before the release. There will be a few stale orders people have left by accident + a few orders left in at levels deemed ridiculously unlikely. As soon as the data is released, the fastest players will general send quotes making a (fairly wide market) around their estimate of the fair price. Over time (and here we’re still talking very fast) more players will come in, firming up that new market.
The absolute level of money which is being made during this period is relatively small. It’s not like the first person to see the report gets to trade at the old price, they get to trade with any stale orders—the market just reprices with very little trading volume.
All of the money-making value was redeemed before people like you and me even had a chance to trade. Right?
Correct, you absolutely did not have the chance to be involved in this trade unless you work at one of a handful of firms which have spent 9 figure sums on doing this really, really well.
I agree identifying model failure is something people can be good at (although I find people often forget to consider it). Pricing it they are usually pretty bad at.
I’d personally be more interested in asking someone for their 95% CI than their 68% CI, if I had to ask them for exactly one of the two. (Although it might again depend on what exactly I plain to do with this estimate.)
I’m usually much more interested in a 68% CI (or a 50% CI) than a 95% CI because:
People in general arent super calibrated, especially at the tails
You won’t find out for a while how good their intervals are anyway
What happens most often is usually the main interest. (Although in some scenarios the tails are all that matters, so again, depends on context—emphasis usually). I would like people to normalise narrower confidence intervals more.
(as you note) the tails are often dominated by model failure, so you’re asking a question less about their forecast, and more about their estimate of model failure. I want information about their model of the world rather than their beliefs about where their beliefs breakdown.
Under what assumption?
1/ You aren’t “[assuming] the errors are normally distributed”. (Since a mixture of two normals isn’t normal) in what you’ve written above.
2/ If your assumption is then yes, I agree the median of is ~0.45 (although
from scipy import stats
stats.chi2.ppf(.5, df=1)
>>> 0.454936would have been an easier way to illustrate your point). I think this is actually the assumption you’re making. [Which is a horrible assumption, because if it were true, you would already be perfectly calibrated].
3/ I guess you’re new claim is “[assuming] the errors are a mixture of normal distributions, centered at 0”, which okay, fine that’s probably true, I don’t care enough to check because it seems a bad assumption to make.
More importantly, there’s a more fundamental problem with your post. You can’t just take some numbers from my post and then put them in a different model and think that’s in some sense equivalent. It’s quite frankly bizarre. The equivalent model would be something like:
I think the controversy is mostly irrelevant at this point. Leela performed comparably to Stockfish in the latest TCEC season and is based on Alpha Zero. It has most of the “romantic” properties mentioned in the post.
That isn’t a “simple” observation.
Consider an error which is 0.5 22% of the time, 1.1 78% of the time. The squared errors are 0.25 and 1.21. The median error is 1.1 > 1. (The mean squared error is 1)
Metaculus uses the cdf of the predicted distribution which is better If you have lots of predictions, my scheme gives an actionable number faster
You keep claiming this, but I don’t understand why you think this
If you suck like me and get a prediction very close then I would probably say: that sometimes happen :) note I assume the average squared error should be 1, which means most errors are less than 1, because 02+22=2>1
I assume you’re making some unspoken assumptions here, because is not enough to say that. A naive application of Chebyshev’s inequality would just say that .
To be more concrete, if you were very weird, and either end up forecasting 0.5 s.d. or 1.1 s.d. away, (still with mean 0 and average squared error 1) then you’d find “most” errors are more than 1.
Go to your profile page. (Will be something like https://www.metaculus.com/accounts/profile/{some number}/). Then in the track record section, switch from Brier Score to “Log Score (continuous)”
I’d be happy to.
The 2000-2021 VIX has averaged 19.7, sp500 annualized vol 18.1.
I think you’re trying to say something here like 18.1 ⇐ 19.7, therefore VIX (and by extension) options are expensive. This is an error. I explain more in detail here, but in short you’re comparing expected variance and expected volatility which aren’t the same thing.
From a 2ndary source: “The mean of the realistic volatility risk premium since 2000 has been 11% of implied volatility, with a standard deviation of roughly 15%-points” from https://www.sr-sv.com/realistic-volatility-risk-premia/ . So 1/3 of the time the premia is outside [-4%,26%], which swamps a lot of vix info about true expect vol.
I’m not going to look too closely at that, but anything which tries to say the VRP was solidly positive post 2015 just doesn’t gel with my understanding of that market. (For example). (Also, fwiw anyone who quotes changes in volatility in percentages should be treated with suspicion at best)
-60% would the worst draw down ever, the prior should be <<1%. However, 8 years have been above 30% since 1928 (9%), seems you’re using a non-symetric CI.
Yeah, it’s not symmetric, but I wasn’t the person who suggested it. All I’m saying is “OP says [interval] has probability 90%” “market says [interval] has probability 90%”.
The reasoning for why there’d be such a drawdown is backwards in OP: because real rates are so low the returns for owning stocks has declined accordingly. If you expect 0% rates and no growth stocks are priced reasonably, yielding 4%/year more than bonds. Thinking in the level of rates not changes to rates makes more sense, since investments are based on current projected rates. A discounted cash flow analysis works regardless of how rates change year to year. Currently the 30yr is trading at 2.11% so real rates around the 0 bound is the consensus view.
OP being my post of arunto’s?
There’s several things unclear with this paragraph though:
Stocks are currently ‘yielding’ 1.3% (dividend yield) or 3.9% (‘earnings’ yield). Not sure exactly what yield you think is 4% over bonds. (Or which maturity bond you’re considering).
“Thinking in the level of rates not changes to rates makes more sense, since investments are based on current projected rates.”. The forward curve is upward sloping, yes, but if arunto thinks rates are going to change higher than what the market forecasts that will definitely change the price of equities. “A discounted cash flow analysis works regardless of how rates change year to year.” Yes, but if you change the rates in your DCF you will change your price
“Currently the 30yr is trading at 2.11% so real rates around the 0 bound is the consensus view.”. Currently 30y real rates are −15bps after a steep sell-off after the start of the year. 30y real rates were as low as −60bps in December.
10y real rates are more like −75bps (up from −110bps in December).
“the 0 bound” is something people talk about in nominal space because the yield on cash is somewhere in that ballpark. (These days people generally think that figure should be around −50 to −100bps depending on which euro rates trader you speak to). For real rates there’s no particular reason to think there is any significant bound − 10y real rates in the US have been negative since the start of 2020; in the UK they’ve been negative since the early 2010s.
I still think you’re missing my point.
If you’re making ~20 predictions a year, you shouldn’t be doing any funky math to analyse your forecasts. Just go through each one after the fact and decide whether or not the forecast was sensible with the benefit of hindsight.
I am even explaining what an normal distribution is because I do not expect my audience to know...
I think this is exactly my point, if someone doesn’t know what a normal distribution is, maybe they should be looking at their forecasts in a fuzzier way than trying to back fit some model to them.
All I propose that people sometimes make continuous predictions, and if they want to start doing that and track how much they suck, then I give them instructions to quickly getting a number for how well it is going.
I disagree that’s all you propose. As I said in an earlier comment, I’m broadly in favour of people making continuous forecasts as they convey more information. You paired your article with what I believe is broadly bad advise around analysing those forecasts. (Especially if we’re talking about a sample of ~20 forecasts)
I disagree with that characterisation of our disagreement, I think it’s far more fundamental than that.
I think you misrepresent the nature of forecasting (in it’s generality) versus modelling in some specifics
I think your methodology is needlessly complicated
I propose what I think is a better methodology
To expand on 1. I think (although I’m not certain, because I find your writing somewhat convoluted and unclear) that you’re making an implicit assumption that the error distribution is consistent from forecast to forecast. Namely your errors when forecasting COVID deaths and Biden’s vote share come from some similar process. This doesn’t really mirror my experience in forecasting. I think this model makes much more sense when looking at a single model which produces lots of forecasts. For example, if I had a model for COVID deaths each week, and after 5-10 weeks I noticed that my model was under or over confident then this sort of approach might make sense to tweak my model.
To expand on 2. I’ve read your article a few times and I still don’t fully understand what you’re getting at. As far as I can tell, you’re proposing a model for how to adjust your forecasts based on looking at their historic performance. Having a specific model for doing this seems to miss the point of what forecasting in the real world is like. I’ve never created a forecast, and gone “hmm… usually when I forecast things with 20% they happen 15% of the time, so I’m adjusting my forecast down” (which is I think what you’re advocating) it’s more likely a notion of, “I am often over/under confident, when I create this model is there some source of variance I am missing / over-estimating?”. Setting some concrete rules for this doesn’t make much sense to me.
Yes, I do think it’s much simpler for people to look at a list of percentiles of things happening, to plot them, and then think “am I generally over-confident / under-confident”? I think it’s generally much easier for people to reason about percentiles than standard-deviations. (Yes, I know 68-95-99, but I don’t know without thinking quite hard what 1.4 sd or 0.5 sd means). I think leaning too heavily on the math tends to make people make some pretty obvious mistakes.
d/ is actually completely consistent with the vol market (I point this out here), so it’s not clear that’s their recommendation.
If you think 2 data points are sufficient to update your methodology to 3 s.f. of precision I don’t know what to tell you. I think if I have 2 data point and one of them is 0.99 then it’s pretty clear I should make my intervals wider, but how much wider is still very uncertain with very little data. (It’s also not clear if I should be making my intervals wider or changing my mean too)
you are missing the step where I am transforming arbitrary distribution to U(0, 1)
I am absolutely not missing that step. I am suggesting that should be the only step.
(I don’t agree with your intuitions in your “explanation” but I’ll let someone else deconstruct that if they want)
you need less data to check whether your squared errors are close to 1 than whether your inverse CDF look uniform
I don’t understand why you think that’s true. To rephrase what you’ve written:
“You need less data to check whether samples are approximately N(0,1) than if they are approximately U(0,1)”
It seems especially strange when you think that transforming your U(0,1) samples to N(0,1) makes the problem soluble.
(If this makes no sense, then ignore it): Using an arbitrary distribution for predictions, then use its CDF (Universality of the Uniform) to convert to , and then transform to z-score using the inverse CDF (percentile point function) of the Unit Normal. Finally use this as in when calculating your calibration.
Well, this makes some sense, but it would make even more sense to do only half of it.
Take your forecast, calculate it’s percentile. Then you can do all the traditional calibration stuff. All this stuff with z-scores is needlessly complicated. (This is how Metaculus does it’s calibration for continuous forecasts)
There are a bunch of different metrics which you could look at on a prediction market / prediction platform to gauge how “uncertain” the forecast is:
Volatility—if the forecast is moving around quite a bit, there are two reasons:
Lots of new information arriving and people updating efficiently
There is little conviction around “fair value” so traders can move the price with little capital
Liquidity—if the market is 49.9 / 50.1 in millions of dollars, then you can be fairly confident that 50% is the “right” price. If the market is 40 / 60 with $1 on the bid and $0.50 on the offer, I probably wouldn’t be confident the probability lies between 40 and 60, let along “50% is the right number”. (The equivalent on prediction platforms might be number of forecasters, although CharlesD has done some research on this which suggests there’s little addition value being added by large numbers of forecasters)
“Spread of forecasts”—on Metaculus (for example) you can see a distribution of people’s forecasts. If everyone is tightly clustered around 50% that (usually) gives me more confidence that 50% is the right number than if they are widely spread out