BEHAVIORAL INSIGHTS • APR 2019

Thoughts on the Hot-Hand Debate in Basketball

AUTHORS “In science, convictions have no rights of citizenship.” – Nietzsche

The debate on the “hot hand” in basketball has gone on for decades. On one side, most players, coaches and fans are adamant that hot hands exist, with players experiencing winning streaks that cannot be explained by chance. On the other side, since the seminal paper of Gilovich, Vallone and Tversky (henceforth GVT) in 1985, most academic research has been unable to find statistical evidence to support the idea of the hot hand. The stark contrast between the two sides has made Emmanuel Roman Chief Executive Officer the belief in the hot hand in basketball one of the most massive and widespread cognitive illusions Managing Director cited in the literature on heuristics and biases (see, for example, Kahneman 2011).

Although for decades the dust seemed to have settled on the issue, at least for the academic community, a recent paper by Miller and Sanjurjo (2018) revived the debate by pointing out a finite sample bias previously overlooked by GVT and other researchers.1 After correcting for the bias, the authors claim that the data used by GVT actually provided evidence for the hot hand in basketball.

Jamil Baz Managing Director We find this paper fascinating in many ways. It identifies a subtle but substantial bias underlying a Client Solutions and Analytics standard measure of conditional dependence of random outcomes on past outcomes in sequential data, and therefore has wide implications for future research in finance. In addition, the overlooked bias itself can be interpreted as a result of the researchers’ belief in the law of small numbers. And the fact that it withstood three decades of scrutiny by fellow researchers (with more than 1,200 citations as of 2015) is startling, reminding us that even sophisticated researchers with solid statistical backgrounds can be fooled by their intuitions. In this article, we share our key takeaways from this recent development in the debate, as well as a simple Helen Guo empirical exercise in applying the proposed bias adjustment methodology to examine the short- Senior Vice President run persistence of mutual fund performance. Quantitative Research Analyst THE MATHEMATICS OF STREAKS AND HOT HAND IN BASKETBALL

In GVT, the main test for hot hand in basketball is the analysis of the conditional probabilities in the authors’ controlled study of basketball shooting by 26 NCAA players. Consider the basketball shooting as independent and identically distributed coin flips under the null hypothesis of “no hot hand.” This implies that the probability of a head (hit) immediately following k consecutive heads or k consecutive tails (misses) should be the same as the unconditional probability of heads, p(H). The hot-hand hypothesis predicts that the conditional probability of heads immediately after k consecutive heads, P(H|kH), would be higher than p(H)

1 We t hank Professor Richard H. Thaler, Charles R. Walgreen Distinguished Service P rofessor of Behavioral Science a nd Economics at the U niversity of Chicago Booth School of Business, for bringing the r ecent development in the h ot-hand debate t o our attention. 2 APR 2019 • BEHAVIORAL INSIGHTS or the conditional probability of heads immediately after k a head immediately following a head, , is not an consecutive tails, P(H|kT). To test this hypothesis, GVT unbiased estimator of the true conditional probability. P H H constructed a test statistic that is the difference of the two Although the variables Z and Y in Exhibit 1 are unbiased empirical conditional probabilities – estimators of the expected number of flips after a head and the – for each player in the study and tested whether the mean of D = P H kH − P H kT expected number of heads among those flips, the ratio between the difference is zero with a paired t-test2 for k=1,2,3.3 them is a biased estimator of the true conditional probability One problem with this seemingly intuitive test, however, is that – i.e., while 2 . in finite samples the empirical conditional probabilities are Y 5 E[Y] 3 1 4 biased estimators of the true conditional probabilities under One wayE Zto =understand12 E [theZ] = bias3 = is2 to recognize that the unit of the null hypothesis of i.i.d. coin flips. Miller and Sanjurjo measure here is a sequence instead of an event (a head or a identified this subtle bias and quantified it, showing it is tail immediately following a head). The sequences have the substantial enough for the sample size in GVT (in their same probabilities, but there are different numbers of events controlled study of 100 shots by each of the 26 players) to in individual sequences. In the HHH sequence, there are two reverse the conclusion of the test. positive events (two heads following a head), but both events collapse into one because our unit of measurement is the Exhibit 1 illustrates this bias with a simple example of coin sequence. So the HHH sequence (with two positive events) is flips. If one flips a fair coin three times and records the given the same weight as the HTH sequence (with one proportion of heads immediately following a head, the negative event). expected proportion is, surprisingly, only about 41.7%, much lower than the true conditional probability of 50%. Appendix 1 Another way to understand the bias is to think about how the shows the mathematical derivation of the bias in this case empirical conditional probability, , is constructed. If the (i.e., k = 1). first flip is a head (hit), the outcome of the second flip will be P H H recorded. If the second flip is a tail (miss), is zero Exhibit 1: The biased empirical conditional probability regardless of the outcome of the third flip because it will not be P H H Sequence # of flips # of H Proportion of H recorded. However, if the second flip is a head (hit), is (n=3) following H (Z) following H (Y) following H (X=Y/Z) not always equal to one because the head for the second flip TTT P H H triggers the recording of the outcome for the third flip, which TTH could be either a head or a tail. Because it is equally possible to THT 1 0 0 have a head or a tail in the second flip, this asymmetry means THH 1 1 1 HTT 1 0 0 the expectation of conditional on the first flip being a head will be less than . This selection bias penalizes the head HTH 1 0 0 P H H following a head by forcing1 the recording to continue to the next HHT 2 1 1/2 2 HHH 2 2 1 flip. This is why the bias disappears whenk = n – 1; i.e., there is Expectation 4/3 2/3 5/12 only one flip afterk consecutive heads.

Source: PIMCO Because means 5

Under the assumption of i.i.d. fair coin flips, E P H H = E P T T =for12 a sequence of three fair for each flip, regardless of the outcome of the previous flip, if1 7 1 P H = P T = 2 coinE P flips,T H a= naturalE P H questionT = 12 that> 2 may arise is whether we can there was one. In other words, the true conditional probability of make a profit by betting on a head following a tail, and vice a head immediately following a head is equal to the versa, for many sequences of three fair coin flips. The answer is unconditional probability of a head – i.e., .. no. As seen in Exhibit 1, although the expected proportion of However, the empirical (or observed) conditional probability of1 P H H = P H = 2 heads following a head is lower than , there tend to be more recorded flips in the realized sequences1 with more heads. If we 2

2 The p aired t-test assumes the difference between each player’s two empirical conditional probabilities is drawn from a normal distribution with zero mean under H 0. 3 Miller and Sanjurjo (2018) focus on the case w here k=3 or more because 1 ) shorter streak lengths, as weaker signals of entering a “hot state,” exacerbate a ttenuation bias due to measurement errors and 2) common perception of streaks starts with three c onsecutive e vents. APR 2019 • BEHAVIORAL INSIGHTS 3 weight the proportion in each sequence by the number of bets Miller and Sanjurjo then perform bias-adjusted tests using the (flips following a head), we will get back without any bias, as original data in GVT for the controlled study of basketball shown below: 1 shooting. They find that, after correcting for the bias, the 2 original conclusion of no evidence for hot hand is reversed.

8 1 1 1 1 A simple empirical exercise: hot hand in mutual 8 zixi = 1×0 + 1×1 + 1×0 + 1×0 + 2× + 2×1 = i=3 zi i=3 8 2 2 fund performance where is the number of bets (flips following a head) along path i andi x is the proportion of heads immediately following a To better understand the bias-correction methodology in Miller z i head along path i.4 and Sanjurjo (2018), we apply it to real financial data instead of describing the example in the paper. However, unlike sports, For k > 1, the bias can be even more especially controlled basketball shooting, the richness of negative. The bias also depends on the number of flips and the financial market settings makes it difficult to find potentially i.i.d. b = E P H kH − P(H) unconditional probability of heads, P(H). Exhibit 2 plots Bernoulli random variables. Binarizing nonbinary data also risks

as a function of the number of flips n, the losing helpful information in the process. Therefore, this exercise true probability of heads p=P(H) and the streak length k. For a is done only to illustrate how the bias correction methodology E P1k : = E P H kH sequence of 100 fair coin flips, the bias for the proportion of developed by Miller and Sanjurjo can potentially reverse the heads following a streak of three heads is still pretty substantial conclusion of a hypothesis test. It is not meant to replace more at about -4%. The bias for the difference of the two empirical rigorous or comprehensive analysis of this topic. conditional probabilities is about -8%. The bias can be even bigger in magnitude for lower values Our exercise uses quarterly excess returns over prospective D = P H 3H − P H 3T primary benchmarks (after fees) of active open-ended U.S. of p. To put the -8% bias into perspective, the difference between the median three-point shooter and the top three- mutual funds (the oldest institutional share classes) in the point shooter in the 2015–2016 NBA season was -12% (Miller Morningstar Intermediate-Term Bond category (the largest bond and Sanjurjo 2018). category). Our sample includes 21 funds with excess returns data from first-quarter 1993 to fourth-quarter 2017, with a total of Exhibit 2: Expected proportion of heads immediately 100 quarters. We are trying to see whether there is evidence for following k consecutive heads short-run serial dependence in the relative performance of these k=1 k=2 k=3 0.70 funds with a long history of survival, beyond what can be explained by chance.5 To apply the bias correction method p=0.7 directly, we convert the quarterly excess returns to binary outcomes of H (outperform) and T (underperform). 0.50

] The null hypothesis is that each fund’s quarterly performance 1k 1k P [ i E p=0.5 (H/T) follows an i.i.d. Bernoulli process with p (H), the unconditional probability of (outperform) for fund . The test 0.30 H i is a joint test of both the independence and the identical distribution assumptions in the null hypothesis. If we assume p=0.3 further that the probability of outperformance is constant over 0.10 time for each fund, we can interpret the rejection of the null 0 10 20 30 40 50 60 70 80 90 100 n hypothesis as evidence in favor of the existence of hot hand (or Source: PIMCO cold hand) in the relative performance of these funds. For n ≤ 20, we c alculate t he e xpected proportion of heads immediately following k consecutive h eads directly by complete enumeration of all possible sequences. We also verify our estimates are the s ame a s those b ased on the f ormula in Miller and Sanjurjo (2018). For n > 20, we a pply the f ormula that is more numerically tractable. The formula can be f ound in Appendix 2.

4 i starts from three because is not defined for the fi rst two paths.

5 By construction, this test cannot address the survivorship bias, so the t est result should be interpreted with this caveat. zi 4 APR 2019 • BEHAVIORAL INSIGHTS

Exhibit 3 displays the results of the Wald-Wolfowitz runs test (p-value > 0.20). Simply adjusting the differences by the (Siegel 1956) for each fund. Most of the funds have a negative Z estimated biases for the paired t-test would lead to rejection of score for the number of runs, and three of them have significantly the null hypothesis (p-value < 0.05). An approximate z-test negative Z scores. This indicates fewer runs and more clustering using the number of trials in each category (3H versus 3T) to or streakiness in the data than explained by chance. inform the standard errors under the i.i.d. and normality assumptions, as in Miller and Sanjurjo (2018),7 confirms the Exhibit 3: The Wald-Wolfowitz runs test previous conclusion (p-value < 0.01). Therefore, we found an Fund ID # Quarters Heads Tails # of runs E (# of runs) Z empirical example where the bias correction can potentially 1 100 49 51 49 51.0 -0.4 change the conclusion of the test. 2 100 46 54 31 50.7 -4.0 Exhibit 4: Raw and bias-adjusted differences in the 3 100 59 41 44 49.4 -1.1 4 100 62 38 46 48.1 -0.5 empirical conditional probabilities 5 100 54 46 42 50.7 -1.8 D=P(H|3H)-P(H|3T) 6 100 51 49 46 51.0 -1.0 Fund ID # Quarters P(H) P(H|3H) P(H|3T) Raw Bias-adjusted 7 100 55 45 47 50.5 -0.7 1  0.49 0.57 0.46 0.11 0.19 8 100 41 59 44 49.4 -1.1 2 100 0.46 0.55 0.24 0.31 0.39 9 100 62 38 39 48.1 -1.9 3 100 0.59 0.50 0.67 -0.17 -0.08 10 100 43 57 35 50.0 -3.1 4 100 0.62 0.54 1.00 -0.46 -0.37 11 100 63 37 49 47.6 0.3 5 100 0.54 0.65 0.54 0.11 0.19 12 100 56 44 50 50.3 -0.1 6 100 0.51 0.56 0.38 0.19 0.27 13 100 69 31 48 43.8 1.0 7 100 0.55 0.59 0.56 0.03 0.11 14 100 35 65 44 46.5 -0.6 8 100 0.41 0.56 0.27 0.29 0.37 15 100 32 68 42 44.5 -0.6 9 100 0.62 0.76 0.33 0.43 0.52 16 100 55 45 43 50.5 -1.5 10 100 0.43 0.69 0.23 0.46 0.55 17 100 45 55 47 50.5 -0.7 11 100 0.63 0.58 1.00 -0.42 -0.33 18 100 48 52 41 50.9 -2.0 12 100 0.56 0.63 0.63 0.01 0.09 19 100 38 62 48 48.1 0.0 13 100 0.69 0.60 — — — 20 100 58 42 52 49.7 0.5 14 100 0.35 0.25 0.42 -0.17 -0.08 21 100 67 33 43 45.2 -0.5 15 100 0.32 0.50 0.32 0.18 0.28 16 100 0.55 0.65 0.29 0.37 0.45 Source: PIMCO calculations based on Morningstar data as of 31 December 2017 17 100 0.45 0.38 0.53 -0.15 -0.07 Appendix 3 explains how the r uns test is performed. 18 100 0.48 0.64 0.35 0.29 0.37 19 100 0.38 0.29 0.41 -0.12 -0.03 Exhibit 4 shows the empirical conditional probabilities and the 20 100 0.58 0.56 0.80 -0.24 -0.16 differences between them before and after the bias adjustment. 21 100 0.67 0.62 0.67 -0.05 0.05 We compute the difference between the empirical conditional Average 0.52 0.56 0.50 0.05 0.14 probability of outperformance immediately following three Source: PIMCO calculations based on Morningstar data as of 31 December 2017 consecutive quarters of outperformance and the See Appendix 4 for the f ormula for bias in the difference in the empirical conditional probabilities. empirical conditional probability of outperformance p H|3H immediately following three consecutive quarters of underperformance .6 We then estimate the expected differences (biases) between the two empirical conditional p H|3T probabilities, assuming the true probability of heads for player , . i i Ifi wep performH = p aH paired t-test on the raw statistics, as in GVT, we cannot reject the null hypothesis at the usual significance level

6 We f ocus on the c ase where k =3 for similar reasons as outlined in footnote 3 . 7 In addition to the bias-adjusted paired t-test and approximate z -test, Miller and Sanjurjo (2018) also perform a nonparametric permutation test based on the e xact distribution of the difference conditional on the total number of heads for each player as a robustness check. Our permutation test confirms the c onclusion of the other two tests (p-value < 0 .01). For simplicity and illustration, we s how only the fi rst two tests for our simple e xercise in this article. APR 2019 • BEHAVIORAL INSIGHTS 5

THE LAW OF SMALL NUMBERS AND THE The bias identified by Miller and Sanjurjo provides a possible “GAMBLER’S FALLACY” theoretical explanation for how the gambler’s fallacy can persist, even for people with extensive experience. Due to The law of large numbers generally states that the average limited working memory capacity or attention span, people result of a large number of random draws from a population will may observe many sequences of finite length of random converge to the population mean. The law of small numbers, outcomes, such as heads and tails. Without knowing the bias in however, describes the erroneous tendency to regard a small the observed conditional probabilities, or because of sample sample randomly drawn from a population as having similar size neglect well documented in the literature, they may update essential characteristics as the population. The term was their prior beliefs with the biased empirical proportions without coined by Tversky and Kahneman (1971), who illustrated the adjusting for the bias. The bias can persist without correction prevalence of this belief with a survey of 84 participants at after extensive experience if the experience consists of many meetings of the Mathematical Psychology Group and of the such finite sequences. American Psychological Association. Most of these sophisticated researchers, when asked to give quick and The prevalence of the gambler’s fallacy has important intuitive answers to realistic research questions, believed implications for investment. For example, the disposition effect, erroneously that the law of large numbers applied to small which refers to investors’ tendency to sell winning assets too samples as well. As a result, they systematically overestimated early and losing assets too late, can be linked to the gambler’s the power of their statistical tests and underestimated the fallacy (see, for example, Shefrin and Statman 1985, and Odean sample size needed for a meaningful test. 1998). The implicit mindset is that if an asset keeps generating good returns, a negative return will be more and more likely, The survey results indicate that statistical intuitions can be even if it is not supported by the data. Similarly, if an asset has substantially biased and that mathematically trained been losing value, the chance it will generate good returns will researchers can be equally susceptible to these biases even be higher. A similar behavior pattern has been documented though they have the ability to derive the right answers once among mutual fund investors. Barber et al. (2000) find they analyze the problem rigorously. Therefore, Tversky and investors are twice as likely to sell a winning mutual fund than a Kahneman propose that researchers always maintain healthy losing mutual fund; thus, nearly 40% of fund sales occur in suspicion about their statistical intuitions and replace intuitions funds ranked in the top quintile of past annual returns. Hsu et al. with computation whenever possible. (2016) find that investors in value mutual funds have produced The mistaken belief in the law of small numbers can be an average internal rate of return that is meaningfully lower than manifested in various ways. One is the gambler’s fallacy, or the the average returns reported by the corresponding funds, negative recency effect, a misconception that any recent because of their poor timing. deviation from the population distribution in one direction will soon be cancelled out by deviation in the other direction. For RELATED FINITE SAMPLE BIASES example, a person influenced by the gambler’s fallacy may tend The bias identified and quantified by Miller and Sanjurjo is to believe a head is more likely than a tail to follow a streak of closely related to some well-known finite sample biases for the tails, and vice versa, in a sequence of fair coin flips. After standard estimates in time-series models. These biases are Tversky and Kahneman (1971), more evidence on the cognitive often ignored by practitioners, either because the magnitudes bias related to the belief in the law of small numbers, especially are assumed to be negligible or because practitioners are not the gambler’s fallacy, started to emerge in the labs and the field aware of the biases. For example, consider the following simple (see, for example, Kahneman and Tversky 1972, Gold and AR(1) model: Hester 2008, and Asparouhova et al. 2009).

yt = α + βyt−1 + εt. 6 APR 2019 • BEHAVIORAL INSIGHTS

The Gauss-Markov theorem – which states that the ordinary FINAL REMARKS least-squares (OLS) estimator for a linear regression model with Heuristics (mental shortcuts) are helpful for quick decision- uncorrelated, homoscedastic and mean zero errors is the best making when there is a large amount of data, but they are prone linear unbiased estimator (BLUE) – no longer holds. The least- to biases. Statisticians tend to make better decisions not squares estimator will be consistent (converging in probability to necessarily because they have better intuitions but probably the true value when the number of observations goes to infinity) because they know better when to replace their intuitions with but biased. rigorous analyses. If α is unknown, the least-squares estimator for the serial The recent development in the hot-hand debate in basketball correlation parameter β has an approximate bias of reminds us that the belief in the law of small numbers is 1+3β where T is the sample size (Kendall 1954). This means that for − T−1 prevalent even among sophisticated researchers with relatively short and persistent time series, which are very quantitative training when they rely on intuitions rather than common for macroeconomic data, the bias can be quite computations. The associated cognitive biases, such as the substantial. For example, if the true β is less than but close to gambler’s fallacy and sample size neglect, could lead to one and the time series consist of 21 observations, the bias systematic deviations from optimal decisions. Finite sample will be close to -20%, which can have a considerable impact biases are often ignored without proper justification in practice. on multiperiod forecasting (such as in an impulse There is also a common tendency to underestimate the sample response function). size needed for valid statistical inference.

Even if β = 0, there will still be a negative bias for in the For individual investors and professional portfolio managers magnitude of .. This implies that even if yt is i.i.d., the alike, these and other cognitive biases can lead to poor 1 β estimated AR(1) autoregression parameter will still tend to be − T−1 investment decisions and performance. Being aware of these negative, on average (-5% for a sample size of 21). This is closely biases is the necessary first step to mitigating their negative related to the bias identified by Miller and Sanjurjo whenk = 1. To impact on investment outcomes. Although it is not possible to see this, consider an AR(1) process under the coin-flipping setup eliminate the biases completely, a natural way to help overcome them is to develop carefully vetted investment strategies and

stick to them with discipline. yt = α + βyt−1 + εt where and REFERENCES .. It can be shown that the OLS yt~ i. i. d. Bernoulli p , α = p, β = 0 estimators are related to the empirical conditional probabilities Asparouhova, Elena, Michael Hertzel, and Michael Lemmon. εt~ i. i. d. Bernoulli p − p as follows: “Inference from Streaks in Random Outcomes: Experimental Evidence on Beliefs in Regime Shifting and the Law of Small Numbers.” Management Science, November 2009 and

α = P 1 0 β = D = P 1 1 − P 1 0 . Barber, Brad M., Terrance Odean, Lu Zheng. "The Behavior of Mutual Fund Investors." Working paper, UC Davis Graduate Therefore, 1 School of Management 2000. One easy wayE β to =correctE P 1for1 this− Pfinite1 0 sample= − n −bias1. is to replace Gilovich, Thomas, Robert Vallone, and Amos Tversky. “The Hot with in the approximate bias equation, Hand in Basketball: On the Misperception of Random , and solve for β (Orcutt and Winokur 1969): E[β] β 1+3β Sequences.” Cognitive Psychology, July 1985 E β − β = − T−1 Gold, E. and G. Hester. “The Gamblerʼs Fallacy and the

(T − 1)β + 1 Coinʼs Memory.” βBC = . This error correction method canT −be4 generalized to higher-order autoregression models (Patterson 2000). APR 2019 • BEHAVIORAL INSIGHTS 7

Rationality and Social Responsibility: Essays in Honor of Appendix 1: The math: closed form for the expected Robyn Mason Dawes (Joachim I. Krueger, ed.). New York: proportion for k = 1 Psychology Press This proof is adapted from Rinott and Bar-Hillel (2015). Hsu, Jason, Brett W. Myers, and Ryan Whitby. “Timing Poorly: A is the event defined by a head among the first (n-1) flips in a Guide to Generating Poor Returns While Investing in Successful A sequence being followed by a head. is the event defined by Strategies.” Journal of Portfolio Management, January 2016 B the first (n-1) flips not being all tails. If q is the probability of a Kahneman, Daniel. Thinking, Fast and Slow. New York: Farrar, tail, then Straus and Giroux, 2011

m Kahneman, Daniel and Amos Tversky. “Subjective Probability: P B = 1 − q a Judgment of Representativeness.” Cognitive Psychology, where .. We now calculate . In the firstm July 1972 tosses, the probability of getting i Hs and (m–i) T’s is m ≡ n − 1 P A ⋂ B Kendall, M.G. “Note on the Bias in the Estimation of

Autocorrelation.” Biometrika, December 1954 m i m−i If you place one of the i Hs randomlyi p qin one. of the first m slots of Miller, Joshua B. and Adam Sanjurjo. “Surprised by the the sequence, then the head will be last with probability 1/m and Gambler’s and Hot Hand Fallacies? A Truth in the Law of Small will be followed by a head with probability p. Or it will not be last Numbers.” Econometrica, November 2018. An earlier version with probability (m – 1)/m and will be followed by another head with was published as IGIER Working Paper 552, Bocconi University, probability (i – 1)/(m – 1). Summing over all the possible i’s, Milan, 2016. this gives:

Odean, Terrance. “Are Investors Reluctant to Realize their

Losses?” Journal of Finance, October 1998 m m i m−i p m−1 i−1 P A ⋂ B = i=1 i p q m + m m−1 Orcutt, Guy H. and Herbert S. Winokur Jr. “First Order

Autoregression: Inference, Estimation, and Prediction.” m m i m−i p−1+i = i=1 i p q m Econometrica, January 1969 and

Patterson, Kerry D. “Bias Reduction in Autoregressive Models.”

pv1+i Economics Letters, August 2000 m m i m− i P A⋂B i=1 i p q m m P A B = P(B) = 1−q . Rinott, Yosef and Maya Bar-Hillel. “Comments on a ‘Hot Hand’ To complete this calculation, first note that Paper by Miller and Sanjurjo.” 2015.

Shefrin, Hersh and Meir Statman. “The Disposition to Sell m m i m−i m m i m−i m 0 m m i=1 i p q = i=0 i p q − 0 p q = 1 − q . Winners Too Early and Ride Losers Too Long: Theory and Also, because this corresponds to the expectation of a binomial Evidence.” Journal of Finance, July 1985 distribution:

Siegel, Sidney. Nonparametric Statistics for the Behavioral m m i m−i Sciences. New York: McGraw-Hill, 1956 i=1 i i p q = mp.

Tversky, Amos and Daniel Kahneman. “Belief in the Law of Small Numbers.” Psychological Bulletin, August 1971. 8 APR 2019 • BEHAVIORAL INSIGHTS

It follows directly that is the proportion of heads among trials

i∈Ik x xi k immediatelyP x := |I followingk x | k consecutive heads for sequence x. –

1−q q m is the set of sequences over which We now calculate the PbiasA B: = 1−q m. b n k theF := proportion{x ∈ {0,1 } : I ) xis) defined.≠ ∅}

k P x is the total number of heads for the sequence. 1−q q m n 1−q m b: = P A B − p = − − 1 − q 1 1 i=1 i N X = n := X is the number of tails for the sequence.

m 1−qm m 1−qm 1−qm m−1 −q 1−q − q + 1−q + q −q 1−q 1−q − mq N0 X = n0 := n _ n1 m m is the number of runs of heads of length k. = m 1−q = m 1−q .

Noting that , we have r1k m is the number of runs of heads of length k or more. 1−q m−1 1−q = 1 + q + ... + q s1k is the total number of runs of heads. k_1 1 j=1 1j 1k r : = j r + s m− 1 m−1 if and 0 otherwise except for the −q 1−q (1+q+...+q −mq ) m n! Because b = m (1−q ) . n special= k case! n_k ! n ≥ k. ≥ 0 + ( times),, k _1 we have b < 0. m−1 m−1 m_1 m_1 is the Kronecker= delta.1 1 + q + ... + q > q q + ... + q m _1 δ ij is the floor function. Appendix 2: Expected proportion of heads immediately . following k consecutive heads for k > 1

For n ≤ 20, we calculate the expected proportion of heads immediately following k consecutive heads directly by computing for each possible sequence with and taking the E[P1k(X)|I1k(X) ≠ ∅] P1k(x) weighted average of the valuesn with weights equal to the x ∈ {0,1} I1k(x) ≠ ∅ probabilities of the sequences. We also verify that our estimates are the same as those based on the formula.

For n > 20, we apply the formula that is more numerically tractable. The formula is based on the IGIER Working Paper version of Miller and Sanjurjo (2018).

Notations:

is a sequence of n independent and identically distributedn Bernoulli trials, each with a constant probability of X := {Xi}i=1 head (“1,” “win,” “hit”) (0,1).

p ∈ is the subset of trials immediately following k consecutivei_1 heads (k < n). Ik x := i ∈ {k + 1. ... , n} : j=i_k xj = 1 APR 2019 • BEHAVIORAL INSIGHTS 9

The formula for expected proportion:

n

E P1k X F = E P1k X N1 X = n1, F P N1 X = n1 F 1 where n =k

n1 n_n1 n 1k 1 1 _ U n, n p 1 _ p 1 1 n P N X = n F = n n i n_i i=k _ U1k n. i p 1 _ p i

E P1k X N1 X = n1, F

1 s1k f1k _ s1k n0 + 1 _ s1k f1k _ s1k 1k = C 0 1k + 0 1k n 1k 1 r11,r12,...,r1k_1,s1k n + 1 f _ 1 n + 1 f 1 _ U n, n k_1 n j=1 jr1j+k

n1_r1 1 0 k_1 min n ,n +1

0 l 1 1 1k 1 n + 1 r n _ 1 _ l k _ 1 U n, n = 1 _1 1 r1=1 r l=0 l r _ 1

n1_k_r1+1 1 0 k_1 min n _k=1,n +1

0 l 1 1 n1k n r _ 1 n _ k _ 1 _ l k _ 1 +δ + 1 _1 1 r1=2 r _ 1 l=0 l r _ 2

r1! 0 1k 1k k_1 n + 1 f _ 1 C = 1 1k s1k! j=1 r1j! r s _ 1

k_1 1k 1 1j 1k f = n _ j=1 jr _ k _ 1 s . 10 APR 2019 • BEHAVIORAL INSIGHTS

Appendix 3: The runs test is the proportion of tails among trials

iεjk x 1_xi A run is defined as a streak of consecutive heads or tails. The immediatelyQk x := |followingjk x | k consecutive tails for sequence x. runs test is a nonparametric test to check randomness for the elements of a sequence. Under the null hypothesis, the mean is the difference between the and the standard deviation of the number of runs conditional on proportions of heads immediately following a streak of k heads Dk x := Pk x _ (1 _ Qk x ) the number of heads n for a sequence of n trials are given by 1 and k tails.

is the set of sequences , and . n 2n1(n_N1) 2n1 n_n1 [2n1 n_n1 _n] over which the differencek is defined.k 2 H := {x ε {0,1} : I (x) ≠ ∅, j (x) ≠ ∅} μ = 1 + n σ = n (n_1) Appendix 4: Expected difference in the proportions for k > 1 is the number of runs of tails of length k.

The formula is based on the IGIER Working Paper version of 0k r is the number of runs of tails of length k or more. Miller and Sanjurjo (2018). s0k is the total number of runs of tails. Additional notations to Appendix 2: k+1 r0: = j=1 jr0j + s0k

is the subset i_1 ojkf trialsx := immediatelyi ε {k + 1 ,following... , n} : kj =consecutivei_k(1 _ xj) heads= 1 (k < n).

The formula for expected difference in the proportions:

n

E Dk X H = E Dk X N1 X = n1, H P N1 X = n1 H where n1=k

n1 n_n1 n k 1 1 _ U n, n p 1 _ p 1 1 n P N X = n H = n n i n_i i=k _ Uk n, i p 1 _ p i

E Dk X N1 X = n1, H

1 s0k f1k _ s1k s0k _ 1 r0 _ s0k f1k _ s1k s0k = Ck _ + _ n 0 02 0k_1 0k 0 1k 0k 0 1k 0k k 1 r ,r ,...,r ,s r f f _ 1 r f f 1 _ U n
, n r11,r12,...,r1k_1,s1k n k_1 j=1 jr0j+k

1k 1k 1k 0k 1 1k 1k 1k 0k k s f _ s s r _ s f _ s s + 01 02 0k_1 0k C 1 1k _ 0k + 1 1k _ 0k r ,r ,...,r ,s r f _ 1 f r f f r11,r12,...,r1k_1,s1k k_1 j=1 jr0j+k≤n0,s0k≥1 k_1 j=1 jr1j+k

r0! R1! 0K 1k k k_1 k_1 F _ 1 f _ 1 C = 0k 1k s0k! j=1 r0j! s1k! j=1 r1j! s _ 1 s _ 1

k_1 1k 1 1j 1k f = n _ j=1 jr _ k _ 1 s

k_1 0k 0 0j 0k f = n _ j=1 jr _ k _ 1 s

n1_r1 1 0 k_1 min n ,n +1

0 l 1 1 k 1 n + 1 r n _ 1 _ l k _ 1 U n, n = 1 _1 1 r1=1 r l=0 l r _ 1

n _k_r +1 1 1 min n1_k+1,n0+1 k_1

0 n l r1 _ 1 n1 _ k _ 1 _ l k _ 1 + 1 _1 1 r1=2 r _ 1 l=0 l r _ 2

n0_r0

min n0,n1+1 k_1

n1 + 1 l r0 n0 _ 1 _ l k _ 1 + 0 _1 0 r0=1 r l=0 l r _ 1

n0_k_r0+1 0 1 k_1 min n _k+1,n +1 1 n l r0 _ 1 n0 _ k _ 1 _ l k _ 1 + 0 _1 0 r0=2 r _ 1 l=0 l r _ 2

n _r 0 0 min n1,n0+1 min r1+1,n0 k_1

l0 0 0 0 r1=r0 r1_r0 =1 r n _ 1 _ l k _ 1 _ 21 + 1 _1 0 0 r1=1 r0=max r1_1,1 l0=0 l r _ 1

n _r 1 1 k_1

l1 r1 n1 _ 1 _ l1 k _ 1 x _1 1 1 l1=0 l r _ 1

n0_k_r0+1

min n0_k+1,n1+1 min r0,n1 k_1

l0 r0 _ 1 n0 _ k _ 1 _ l0 k _ 1 _ _1 0 0 r0=2 r1=max r0_1,1 l0=0 l r _ 2

n _r 0 0 k_1

l0 r0 n0 _ 1 _ l0 k _ 1 x _1 0 0 . l1=0 l r _ 1 pimco.com blog.pimco.com

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