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Hypothesis Tests

IEF 217a: Lecture 2.b

Fall 2002

Hypothesis Testing

• Correct models?

• Data similar?– Use one series to predict another

• Has something changed in the data?– Quality control, portfolio strategies

Outline

• Introduction (Basketball)

• Proportion changes (Political polls)

• Difference in means (Airline arrivals, Firestone)

• Testing a distribution (die)

• Causality

• Multiple comparisons and data snooping

• Statistical power

Hypothesis Testing

• Null hypothesis– Assumption about how the world works– Assume this is true– Could data have come from this

machine/theory/conjecture???– Do you need more/other data?

Basketball and Larry Bird

• Facts– Bird normally makes 48 percent of his shots– Bird has just finished a series of games where

he made only 20 of 57 shots– Question: Is this the usual Larry Bird, or has

something changed?– Is he in a slump?– On to matlab (bird1.m)

Hypothesis Testing Terms

• Null hypothesis– Assumption about the world

• Test statistic – Observed statistic (Random variable)

• p-value (probability null is true)– Prob( shots <= 20 )

Outline





• Causality



Political Poll

• Gore/Bush 0/1

• Two polls (100 people)– First 50/50– Second 55/45

• What is the probability that something has changed in the population?

• Matlab: pollchange.m

Outline





• Causality



Differences in Means

• Two samples

• Different means

• Could they be drawn from the same population?

• Examples– Has something changed?

• Flights (time)

• Tires (Firestone)

Flight Delays

• Two series (minutes late)– Before mechanics threat of delays– After mechanics threat of delays

• More delays after threat

• Compare to pooled data

• Null = two series are the same

• Could the mean difference between the two come from the pooled series?

Flight Delays

• Matlab code: airline.m

• Note: Fancy histogram code

Firestone

• Overall tires have a failure rate of 5 in 1000

• You have observed in a sample of 10,000 tires a failure rate of 60

• Is something wrong with Firestone tires?

• Matlab: firestone.m

Outline





• Causality



Testing a Die

• Problem:– You’ve observed the following rolls of a die

out of 6000 rolls• 1: 1014, 2: 958, 3: 986, 4: 995, 5: 1055, 6:992

– Could this have come from a fair die with probs of 1/6 for each side?

Dietest.m

• Method:– Think up a test statistic– Roll 6000 dies with sample– Check how the value of the test statistic from

the original data compares with the distribution from the simulations

• dietest.m

Outline





• Causality



Causality

• Stock returns and weather

• Are returns higher when it is sunny?

• Given some data on weather and returns test this hypothesis

• on to matlab: sunny.m

Outline





• Causality



Multiple Tests and Data Snooping

• In the search for patterns you often look at many different things– Different trading rules– Different regression runs– Different drugs

• Each is often tested alone

• Then get excited when 1 is significant

Data Snooping and Trading Strategies

• Efficient markets world (no predictability)

• Someone claims to have a buy/sell (short/long) strategy which generates significantly large returns

• They pretested 10 strategies and chose the best out of the 10

• Return sample is independent and normal

Questions

• What is the likelihood that some “best” strategy beats a buy and hold benchmark?

• What if this strategy were tested to see if it was “significant” using traditional statistical tests, ignoring that it had been snooped?

• Matlab: snooptest.m

Other Applications

• Many other trading strategies– More later

• Multiple regressions– Run 20 regressions of y = a + bx for different x– Report only those with significant b– Common economist sin

Outline





• Causality



Hypothesis Tests Again

• P-value or significance level– Probability of rejecting null hypothesis given

that it is true

P-Value, Size, and Type I error

Observe 2Prob(x>2)Null: Normal(0,1)


• Type II error– Probability of accepting null hypothesis given

that it is false


• Power– Probability of rejecting null hypothesis when it

is false– Probability of catching a deviation

Type I and Type II errorsWhich do you prefer?

• Mushroom/Toadstool(poison) test– Null = Mushroom– Type I: Reject mushroom given mushroom– Type II: Accept mushroom given toadstool

• Makes a difference

Hypothesis Tests: Final Word

• Traditional Goals– Correct Size– Maximum Power

• Specific situations– Costs of Type II error (mushrooms)– Finance:

• Using incorrect model

• Missing risks (LTCM)

Problems for Monte-Carlo Tests of Power

• Test a null hypothesis under some alternative

• Need to commit to which alternative

• Power(alternative)

Outline

• Introduction (Basketball)• Proportion changes (Political polls)• Difference in means (Airline arrivals, Firestone)• Testing a distribution (die)• Causality (stocks and weather)• Multiple comparisons and data snooping• Statistical power