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Thoughts on the Hot-Hand Debate in Basketball 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 Sequence # of flips # of H Proportion of H P H 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.
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