Lecture 15: �Independence Assumptions

Stat 165

Jacob Steinhardt

Warm-up Exercise

• What is the probability that I (Jacob) eat cereal� for breakfast tomorrow?

25%, 55%, 5%, 40%, 20%�

• What is the probability that I eat cereal at least �once in the next week?��70%, 80%, 10%, 55%, 80%

Other Non-Independence Examples

• Covid lockdowns per state
• Rain in next week vs. on Wednesday
• Celtics losing next game to Nets vs. next 3 games��
• Brainstorming: other examples?
• Flights leaving airport in a week -> bad weather could make all days have few flights
• -> border closure
• Performance of overall stock market (and correlation across different stock markets)

Two Consequential Wrong Predictions

• 2008 financial crisis
• 2016 US presidential election������
• Both were (partly) failures to account for non-independence!

2016 Election – A Simple Model

• Each state s has N_s polls conducted in that state
• Poll i in state s has sample size n_i,s and k_i,s respondents who will vote for Clinton
• Aggregate polls together – total margin of error is approximately 1/sqrt(total sample size)
• Assume each state’s vote share has Gaussian error around the polling results
• Simulate draws of all 50 states, look at how often Clinton wins across many different draws
• This gives >99% probability to Clinton winning

2016 Election – A More Complicated Model

• Each pollster has some bias – tends to consistently favor either Republicans or Democrats
• Bias not fully known, but can be estimated from past elections
• Modify the Gaussian for each state:
• Change the mean based on estimate of polling bias
• Increase the variance based on unknown component of polling bias
• To be safe, replace Gaussian with Student-t�
• This still (often) gives >99% probability of Clinton winning

2016 Election – What’s the problem?

• Polls have systematic error (as already discussed)
• Some error affects all the polls (nonresponse bias)
• So, at the very least, should have additional global bias term that affects all polls (and hence all states) at once
• There was a systematic polling error of �~2% (plus states Clinton lost were� particularly bad in electoral college).�
• 2% was within range of �historical errors

2016 Election – Models That Work

• Correlated national error accounts for �election uncertainty�
• Fancier version: state-level correlations�(example)�
• Some models expressed uncertainty �with heavy tails, rather than correlations, �but this leads to weird effects: see this �article by Andrew Gelman�
• What other factors (beyond polls) �would you put into your predictive model?

2008 Financial Crisis

• Basic background: many people were offered “subprime” (i.e. risky) mortgages on their houses. Or in other words, they were offered loans with an (initially) low interest rate, despite not having strong finances.
• Rates increased over time, but housing values were also increasing.
• The loans were partly financed through retirement accounts. �
• But retirement accounts are �supposed to be low risk—so �how was this possible?

Collateralized Debt Obligations (CDOs)

• CDOs are a way to turn risky financial instruments (bets) into a less risky bet�
• Simplest way to reduce risk: take �N bets (mortgages) and average
• But can do better, with tranches, �ranked from senior to junior�
• If a mortgage defaults, most �junior tranches take losses first�
• Senior tranches should be very �unlikely to ever take heavy losses

CDOs illustrated (from Wikipedia)

Risk and Leverage

• Senior tranches were “AAA” rated, meaning considered very low risk (low enough for retirement accounts)�
• They also had pretty good returns.
• Low risk + high returns => want to be a large share of your portfolio�
• So companies “leveraged”: took out loans in order to buy more CDOs�
• This blew up in everyone’s faces. Why?

Non-Independence of Housing Market

• AAA rating + risk assessments assumed mortgage defaults were independent, or at least not too correlated�
• But if national housing prices dropped, many people would default at once. Even senior tranches might not pay out.�
• This happened, and highly leveraged investment banks collapsed (+many other bad things).

Non-Independence of Housing Market (contd.)

• Actual story is a bit more complicated: correlations were modeled, �using a Gaussian copula
• Basic idea: sample debt payouts from a multivariate Gaussian (with specified correlation structure)
• Issue: payouts don’t follow normal distribution. So apply 1-dimensional transformation to each payout to give it desired distribution shape.
• Big problem: despite correlation in bulk, no tail dependence (extremes in one variable don’t imply extremes in other)
• Colloquially: conditional on X₁ being in the 99.9^th percentile, the probability that X₂ is also in 99.9^th percentile is close to zero in the model, but not in reality.

Lessons for Forecasting

• Ignoring independence leads to underestimating tail events
• This is because tail events require many things to “go wrong” at once
• Correlated errors mean this happens more often than expected
• Models with bulk correlations might not have much correlation in tails!�
• How can we handle this in our own forecasts?

Strategies for Handling Correlation

• Add latent variables to simulations
• Ask “what could change all of these at once?”
• For “fancy” models, check/think about tail correlations
• Look at historical reference classes