Measurement Error and the Hot Hand

Abstract

This article shows that the first autocorrelation of basketball shot results is a highly biased and inconsistent estimator of the first autocorrelation of the ex ante probabilities with which the shots are made. Shot result autocorrelation is close to zero even when shot probability autocorrelation is close to one. The bias is caused by what is equivalent to a severe measurement error problem. The results imply that the widespread belief among players and fans in the hot hand is not necessarily a cognitive fallacy.

KEY WORDS::

• Autocorrelation
• Basketball
• Errors-in-variables
• Overinference
• Psychology
• Semi-streakiness

Notes

In addition to the psychology literature, false belief in the hot hand is also regularly cited in the behavioral economics literature (e.g., Rabin 1998). See Lehrer (2009) for another example from the popular press, in addition to Thaler and Sunstein (2008).

Arkes’ results differed from those of previous literature primarily because, first, he used a relatively large dataset (data on all free throws from the 2005–2006 NBA season), and, second, he pooled the analysis across players, using a fixed effects approach to account for heterogeneity in average ability. Dorsey-Palmateer and Smith (2004) is one of numerous articles that found evidence of a hot hand existing in a context other than basketball. See Reifman (2011) for a general discussion of the hot hand in sports; other literature is discussed further later.

As a side note, if the probabilities were not serially correlated, but still stochastic, a player would have higher probabilities for some shots than others and thus, in a sense, be hotter at some times than others. For example, the probability could be 70% on one shot, and so the player would be hot for that shot, but still be expected to be the player’s overall mean of 50% on the next shot. But this is not what people mean when they refer to the hot hand: they are referring to above average ability that is at least somewhat persistent, that is, that likely lasts for at least two consecutive shots. Otherwise, it would be completely irrelevant for behavior: there would be no point in passing to someone hot if the hot state immediately fully dissipates. The original definition of Gilovich, Vallone, and Tversky was based on persistence as well. I also note that a hot player will not actually stay hot with certainty if the probabilities are stochastic, but think it is clear that fans, players, etc., understand this.

It is also worth noting that if this alternative definition were to be satisfied, players would sometimes experience the cold hand as well: if their shot probability was, for example, 30% for one shot, it would be expected to be less than 50% on subsequent shots. This is not at all problematic; most fans and players seem to believe in the cold hand’s existence as well, and this phenomenon would also occur if the definition of Gilovich, Vallone, and Tversky was satisfied.

This assumes that the probabilities are equal to their expected values (70%, 66%, 63%, and 60%).

[See, for example, Stock and Watson (2007, pp. 319–321) for a discussion of how errors-in-variables (measurement error affecting the independent variables) causes regression coefficients to be inconsistent.]

Runs tests yield extremely similar results, consistent with the finding of Wardop (1999) discussed earlier in the article; they are not reported but available upon request.

This is determined using homoscedasticity-only standard errors to be consistent with some of the work done in previous literature; power would likely be even lower if appropriate heteroscedasticity-robust standard errors were used.

Oskarsson et al. (2009), for example, discussed how evidence of the hot hand has been found more often in sports that are more “controllable,” such as bowling and archery, as opposed to more “chaotic” sports such as basketball.