Time is knowledge: What response times reveal

Benkert, Jean-Michel; Liu, Shuo; Netzer, Nick Working Paper Time is knowledge: What response times reveal Discussion Papers, No. 24-07 Provided in Cooperation with: Department of Economics, University of Bern Suggested Citation: Benkert, Jean-Michel; Liu, Shuo; Netzer, Nick (2024) : Time is knowledge: What response times reveal, Discussion Papers, No. 24-07, University of Bern, Department of Economics, Bern This Version is available at: https://hdl.handle.net/10419/302144 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Faculty of Business, Economics and Social Sciences Department of Economics Time is Knowledge: What Response Times Reveal Jean-Michel Benkert, Shuo Liu, Nick Netzer 24-07 August, 2024 Schanzeneckstrasse 1 CH-3012 Bern, Switzerland http://www.vwi.unibe.ch DISCUSSION PAPERS Time is Knowledge: What Response Times Reveal Jean-Michel Benkert, Shuo Liu and Nick Netzer∗ August 2024 Abstract Response times contain information about economically relevant but unobserved variables like willingness to pay, preference intensity, quality, or happiness. Here, we provide a general characterization of the properties of latent variables that can be detected using response time data. Our characterization generalizes various results in the literature, helps to solve identification problems of binary response models, and paves the way for many new applications. We apply the result to test the hypothesis that marginal happiness is decreasing in income, a principle that is commonly accepted but so far not established empirically. Keywords: response times, chronometric effect, binary response model, non-parametric identification, decreasing marginal happiness JEL Classification: C14, D60, D91, I31 ∗Benkert: Department of Economics, University of Bern, [email protected]. Liu: Guanghua School of Management, Peking University, sh[email protected]. Netzer: Department of Economics, University of Zurich, nic[email protected]. We are grateful for very helpful comments by Blaise Melly, Costanza Naguib, Martin Vaeth and seminar participants at University of Bern, Peking University and Xiamen University. Shuo Liu acknowledges financial support from the National Natural Science Foundation of China (Grant No. 72322006). 1 Introduction Traditionally, economists have ignored choice process data like response times. Only recently has the literature realized that response times can contain valuable information about economically relevant but unobserved variables like, among others, willingness to pay or accept (Krajbich et al., 2012; Cotet and Krajbich, 2021), preference intensity (Chabris et al., 2009; Alós-Ferrer, Fehr and Netzer, 2021; Alós-Ferrer and Garagnani, 2022), product quality (Card et al., 2024), or happiness (Liu and Netzer, 2023). In this paper, we take a systematic approach to study what information response times contain. We do this in the context of a canonical binary response model, which has been used extensively in the economics literature and is applicable to all the settings described above. The model posits that an unobservable latent variable generates binary choices. The common observation in the aforementioned strands of literature is that decisions are faster when the value of the latent variable is larger. The observable response times are therefore informative about the unobservable latent variable. We phrase our question as one about the identification of the binary response model. The existing econometrics literature has studied questions along this line using assumptions on the distribution of the latent variable and exogenous variation of observables (e.g., Manski, 1988; Matzkin, 1992). We take a complementary approach and study the identification of distributional properties using response time data. This allows us to circumvent controversial assumptions and to solve identification problems noted in the literature (e.g., Haile, Hortaçsu and Kosenok, 2008; Bond and Lang, 2019). We provide a full characterization of the distributional properties of latent variables that can be detected with the help of response time data, depending on the assumptions an analyst is willing to make about the relation between the latent variable and response time. The approach that we adopt is more general than the existing literature. In the context of stochastic choice, Alós-Ferrer, Fehr and Netzer (2021) have shown that response time data can be used to obtain revealed preferences without making distributional assumptions about the random utility component and to improve out-of-sample predictions. Several of their results follow as immediate corollaries from our characterization. We then generalize these results, for example allowing for additional individual heterogeneity and dispensing with unnecessary symmetry assumptions. In the context of happiness surveys, Liu and Netzer (2023) have shown that response time data can help to solve identification problems of ordered response models. Once more, our approach yields several of their results as corollaries and allows for further generalization. Our approach paves the way for a range of new applications. Here, we briefly highlight 1 three, each of which will be discussed in greater detail in the theoretical part of the paper. First, we show how to detect polarization of political attitudes (Lelkes, 2016) from simple ordinal survey questions. This task would be difficult, if not impossible, without response times (Vaeth, 2023). Second, we demonstrate how to infer properties of demand functions that are important for the optimal pricing of firms (Johnson and Myatt, 2006) from observed purchase decisions at a single price. Third, we show how to uncover correlational patterns that are not directly observable to an analyst because subjects’ responses may be distorted when they conflict with authoritarian governments or social norms (Coffman, Coffman and Ericson, 2017; Guriev and Treisman, 2020). Our theoretical results make it possible to tackle long-standing empirical challenges and debates in the economics literature. We showcase this potential by applying the results to test the hypothesis of decreasing marginal happiness of income, a principle that is central to redistributive policies. Oswald (2008) and Kaiser and Oswald (2022) question the empirical foundation of this principle by arguing that an observed concave relationship between income and self-reported happiness may result from a concave reporting function rather than from decreasing marginal happiness. Conventional approaches used in the happiness literature are insufficient to establish the principle (Bond and Lang, 2019). In the empirical part of the paper, we show that the principle becomes testable with our response time-based method. Our analysis of existing survey data reveals that the hypothesis of decreasing marginal happiness cannot be rejected. The central assumption underpinning our analysis is the so-called chronometric function that associates each choice with a response time. This function is monotone, in the sense that a larger absolute value of the latent variable generate a faster decision, possibly after controlling for individual heterogeneity. From a theoretical perspective, such a monotone relationship emerges naturally in evidence-accumulation models (see, e.g., Chabris et al., 2009; Fudenberg, Strack and Strzalecki, 2018; Card et al., 2024), where a stronger stimulus generates faster decisions. The empirical evidence for a monotone chronometric function in the laboratory is vast. For example, and among many others, Kellogg (1931), Moyer and Bayer (1976), and Palmer, Huk and Shadlen (2005) document the effect for choice situations with an objective stimulus, and Moffatt (2005), Chabris et al. (2009), Konovalov and Krajbich (2019), and Alós-Ferrer and Garagnani (2022) for value-based environments. Field evidence is also emerging. Card et al. (2024) document that editorial decisions take longer when the submitted paper’s quality implies a closer decision. Using eBay data on bargaining behavior, Cotet and Krajbich (2021) show that sellers’ response times to an offer systematically depend on its perceived value. In the context of an online survey, Liu and Netzer (2023) demonstrate that faster responses are associated with a stronger sense of 2 approval for the selected answer. The consistent support across diverse settings and studies underscores the validity and robustness of the chronometric effect as modeled in our work. To gain a more concrete idea of our main insight, consider a standard decision-making environment where one or multiple agents choose between two options. Choice is determined by the realization xof an unobservable latent variable, with x≤0generating choice of option aand x > 0generating choice of option b. An analyst observing the choices naturally wonders what can be learned about the underlying binary response model, and in particular about the cumulative distribution function Gof the latent variable, from those data. Unfortunately, the only property of Gthat is identified without additional assumptions or data is its value at zero, G(0), which is given by the observed probability or frequency of choosing a. This information is extremely limited and does, for example, not imply anything about the mean of Gwithout additional distributional assumptions. Now suppose that the speed of the decision is given by c(|x|)for a strictly decreasing chronometric function c, assumed here to be identical for both choice options and all agents just for easy of exposition. Since a choice of aarises at time tor earlier if xis sufficiently far below 0, where “sufficiently far” is determined by the chronometric function, the observed probability or frequency of choosing aat time tor earlier pins down the value of G(−c−1(t)), and analogous for choices of b. Observing the joint distribution of responses and response times therefore allows the analyst to identify a composition of the distribution Gand the chronometric function c. Consequently, if the analyst had perfect knowledge of the chronometric function linking values to response times, she could recover the entire latent distribution from the data. However, such detailed knowledge is not necessary for inferring only specific distributional properties. Our main result fully characterizes which properties (or their violation) can be detected under which assumptions on the chronometric function. For example, detecting properties that are preserved under monotone transformations requires only knowledge of monotonicity of the chronometric function. This includes properties such as full support or, because our approach allows for settings with multiple latent variables, first-order stochastic dominance between distributions (as in Liu and Netzer, 2023). Knowing more about the chronometric function beyond its monotonicity enables the detection of a broader class of properties. For example, if we restrict attention to chronometric functions that are identical for both choice options, as in the above illustration, then we can detect properties that are preserved under symmetric monotone transformations. This includes sufficient conditions for the mean of a distribution to be positive (as in Alós-Ferrer, Fehr and Netzer, 2021) and for the ranking of the means of multiple distributions (as in Liu and Netzer, 2020). Our main result also provides a simple recipe how to detect or reject any property of interest. It involves constructing a candidate distribution based on the empirical data using 3 a representative chronometric function that the analyst deems possible. Then, if the property of interest holds for this candidate distribution, it must hold for all chronometric functions that can be obtained from the representative one using any transformation under which the property is preserved. An analogous statement applies when the property is violated. We also discuss an extension that combines this approach with direct distributional assumptions and we provide necessary and sufficient conditions for rationalizability of response time data in this scenario. The generality of our approach lends itself to a wide range of applications. When applied to a single distribution, it enables the detection of properties such as the sign of the mean, inequality, and unimodality. These properties play important roles in the context of revealed preference theory, optimal pricing, and political analysis. For multiple distributions, we can detect first-order stochastic dominance, the ranking of means, likelihood-ratio dominance, and correlations with observable variables. These properties matter for the analysis of survey data, out-of-sample prediction of behavior, and monotone comparative statics. The paper is organized as follows. Section 2 introduces the formal framework and presents the main result, along with two extensions. Section 3 applies the main result and derives theoretical conditions for detecting the various distributional properties of interest discussed above. Section 4 uses several of these results to empirically test the hypothesis that marginal happiness is decreasing in income. Section 5 concludes. Additional material can be found in the Appendix. 2 General Theory In this section, we develop our general theoretical framework and present our main result, which shows how and under which conditions distributional properties can be detected using response time data. 2.1 Binary Response Model We first introduce the binary response model (e.g. Manski, 1988). There is a random variable ˜xwith values x∈Rthat induce binary responses by comparison with a decision threshold. We normalize the threshold to zero without loss of generality. Thus, the response is i= 0 if ˜xtakes a value x≤0and i= 1 if ˜xtakes a value x > 0. We describe the distribution of the latent variable ˜xby a cumulative distribution function (cdf) G, which we assume to be continuous. It follows that the probabilities of the two responses are p0=G(0) and p1= 1 −G(0). 4 The model has different applications and interpretations. For example, the latent variable could be a random utility difference ˜x=u(1) −u(0) + ˜ϵ(1) −˜ϵ(0) between two options, inducing stochastic choices of a single agent (as in Alós-Ferrer, Fehr and Netzer, 2021). In a different application, ˜xcould describe the distribution of happiness in a population of agents, inducing frequencies of responses to a binary survey question about life happiness (as in Liu and Netzer, 2023). The same logic applies to other survey questions, where the responses could be driven by a distribution of political attitudes or other preference parameters in the population. In yet another application, ˜xcould capture the random quality of papers that are submitted to a journal, inducing the editor’s decision to accept or reject (as in Card et al., 2024). The same logic applies to other settings where quality determines a binary decision, such as whether to invest in an innovation project. Finally, the latent variable ˜x= ˜v−p could describe the difference between willingness to pay for a product among consumers and the product price, inducing the demand for the product at price p(in the spirit of Cotet and Krajbich, 2021). We now follow Alós-Ferrer, Fehr and Netzer (2021) and Liu and Netzer (2023) and assume that the realized value xof ˜xnot only determines the response but also the response time, with larger absolute values implying faster responses, in line with the well-established chronometric effect. Formally, we denote by c:R→[t, t]the chronometric function, where 0≤t < t < ∞. The function cmaps each realized value xinto a response time c(x). It is assumed to be continuous, strictly increasing on R−and strictly decreasing on R+whenever c(x)> t, and to satisfy c(0) = tand limx→−∞ c(x) = limx→+∞c(x) = t. Figure 1 illustrates two examples of chronometric functions that adhere to all these conditions. The restriction of cto x∈R−is denoted c0. This function c0has a well-defined inverse (c0)−1: (t, t]→R−that is continuous and strictly increasing. We extend it to tby setting (c0)−1(t) = −∞ if c(x)> t for all x∈R−and (c0)−1(t) = max{x∈R−|c(x) = t}otherwise. Analogously, the restriction of cto x∈R+is denoted c1, with continuous and strictly decreasing inverse (c1)−1: (t, t]→R+, which we extend to tby setting (c1)−1(t)=+∞or (c1)−1(t) = min{x∈R+|c(x) = t}, as appropriate. In addition to the response probabilities, the model (G, c)also induces distributions of response times. We denote by Fithe cdf of response times conditional on a response of i= 0,1. Since a response i= 0 at time tor earlier arises if x≤(c0)−1(t), we obtain that p0F0(t) = G((c0)−1(t)) (1) for all t∈[t, t], where we use the convention G(−∞) = 0. Analogously, a response i= 1 at 5 • • t ¯ t 0x response time c0(x)c1(x) (a) c0(x)=0.1 + 1 1−2x, c1(x)=0.1 + 1 1+x. • • t ¯ t 0x response time c0(x)c1(x) (b) c0(x) = c1(−x) = max{0.1x+ 1.1,0.1}. Figure 1: Examples of chronometric functions. Notes: The left panel in the figure depicts an asymmetric chronometric function cthat asympotically approaches the fastest response time t, while the right panel shows a symmetric one that attains tat finite absolute values of the latent variable. In both panels, the red and blue curves correspond to the restrictions of cto R−and R+, respectively. time tor earlier arises if x≥(c1)−1(t), so that p1F1(t) = 1 −G((c1)−1(t)) (2) for all t∈[t, t], where we use G(+∞)=1. The induced response-time cdfs Fiare continuous on [t, t]and satisfy Fi(t)=1. In summary, the binary response model (G, c)induces the data (p, F) = (p0, p1, F0, F1)according to (1) and (2). 2.2 Detecting Properties We now ask what we can learn from observed data about the underlying binary response model and, in particular, about the distribution Gof the latent variable. Taking observed data as given, different binary response models could have generated those data, so that inference about the model is not straightforward. This is true especially if the analyst is not willing to make potentially strong assumptions about the form of the chronometric function or the latent distribution. We ask whether there are some properties that all models which are consistent with the data satisfy. These respective properties are then detected from the data rather than assumed by the modeller. Consider a profile of data (pj, Fj)j= (p0 j, p1 j, F0 j, F1 j)jindexed by j∈J. The set Jcould be a singleton, e.g. when studying the choices of a single agent between two options, or the 6 (still assumed to be also bijective and hence continuous). We denote the set of all these transformations by Ψall . Suppose we want to allow all chronometric functions that approach tasymptotically in the limit but never reach t. We impose no other assumptions on their shape, such as symmetry across the two different responses. Denote the set of all these functions by C∗ a.all (where astands for asymptotic) and observe that it is generated by the representative member (3) together with the transformations Ψall . We now construct an empirical function Haccording to (5) using c∗from (3), which yields H(x) =    1−p1F1t+1 x+1/(t−t)if x > 0, p0F0t+1 −x+1/(t−t)if x≤0. (6) Theorem 1 tells us that full support is detected if His strictly increasing in x, and a violation of full support is detected if His not strictly increasing in x. Expressed directly in terms of the observed data, full support is detected if pi>0and Fihas full support on [t, t], for both i= 0,1. A violation of full support is detected otherwise. Suppose instead that we had reasons to believe that all chronometric functions reach c(x) = tfor finite absolute values of x, again without making any other assumptions on their shape. Denote this set by C∗ f.all (where fstands for finite) and observe that it is generated by (4) together with Ψall . Theorem 1 now tells us that we need to check whether H(x) =                1if (t−t)< x, 1−p1F1(t−x)if 0< x ≤(t−t), p0F0(t+x)if −(t−t)≤x≤0, 0if x < −(t−t), (7) is strictly increasing in x. This is not the case, and we therefore detect that the distribution violates full support. Intuitively, since the chronometric functions reach tbut the response time distributions have no atoms at t, the latent distributions cannot have full support. If we allow the union C∗ a.all ∪C∗ f.all of chronometric functions, we can apply the extension discussed in Subsection 2.5.1. If (6) is not strictly increasing, we detect that all distributions that are compatible with the data do not have full support. If (6) is strictly increasing, then we detect neither full support nor a violation of full support, because the data are compatible with some distributions that have full support and others that have not. 13 3.1.2 Sign of Mean Our first economically relevant application concerns the sign of the mean. In a random utility application where ˜x=u(1) −u(0) + ˜ϵ(1) −˜ϵ(0) and the errors have mean zero, the mean equals u(1) −u(0). Detecting the sign of the mean is therefore the same as deducing the agent’s true (non-distorted) ordinal preference between the two options. The sign of the mean is invariant to linear transformations of the form ψ(x) = bx for b > 0. Denote the set of these transformations by Ψlin. Unfortunately, the sets of chronometric functions which can be generated using Ψlin are rather restrictive. For example, when starting from the symmetric linear function (4), we can generate the set of all symmetric linear functions. To detect the sign of the mean u(1) −u(0) assuming this class, by Theorem 1 we just need to calculate the sign of the mean of H, defined in (7), which can easily be done empirically.6 We can achieve more robust detection by working with a property that is sufficient for a positive mean (the argument for a negative mean is analogous). Consider the asymmetry property that G(−x)≤1−G(x)for all x∈R+. This property implies that the mean of Gis positive (see Alós-Ferrer, Fehr and Netzer, 2021). It is invariant to all transformations that are symmetric around zero but not necessarily linear. Denote this set by Ψsym. Suppose that we once more allow chronometric functions that approach tasymptotically, but further restrict attention to those which are symmetric across responses. The set of all these functions, denoted C∗ a.sym, is generated by (3) together with Ψsym. Intuitively, this set captures the assumption that the chronometric effect is identical for the two choice options. We obtain that the desired asymmetry of Gis detected if H, defined in (6), exhibits the desired asymmetry. Taken together and expressed directly in terms of the observed data, it follows that p0F0(t)≤p1F1(t)for all t∈[t, t](8) is a sufficient condition for a revealed preference u(0) ≤u(1). We remark here that the same condition obtains when considering the set C∗ f.sym of all symmetric chronometric functions that reach tand that analogous statements hold for revealed strict preferences. Condition (8) is the same as in Theorem 1of Alós-Ferrer, Fehr and Netzer (2021). They discuss in detail that observing choice frequencies p0≤p1is not sufficient for a preference u(0) ≤u(1) to be revealed without the additional assumptions on error distributions (be6Some distributions do not have a mean. This can be dealt with either by using the procedure described in Subsection 2.5.2 to restrict the set of distributions to those which have a mean, or by refining the desired property, for example to “the mean exists and is positive.” Analogous arguments apply whenever a property is not well-defined for all possible distributions. 14 yond mean zero) that are made in conventional logit or probit models. However, if the inequality holds for all response times, as stated in (8), then a preference is robustly revealed without distributional assumptions. Alós-Ferrer, Fehr and Netzer (2021) report that slightly more than 60% of all stochastic choices in the data of Clithero (2018) do robustly reveal a preference.7Alós-Ferrer, Garagnani and Fehr (2023) use the same condition to show that a sizable fraction of choices that violate stochastic transitivity in different experiments reveal non-transitive preferences and can thus not be explained by transitive preferences together with noise. Our approach suggests possible generalizations of Alós-Ferrer, Fehr and Netzer (2021). For example, if we have reasons to believe that the chronometric effect is different for the two choice options, we can construct Hbased on a representative chronometric function c∗that is asymmetric across the options, and we obtain a modified version of condition (8) which reflects our prior knowledge of the asymmetry. Assume, for example, that the representative function satisfies c∗1(x) = m(c∗0(−x)) for some m: [t, t]→[t, t]and all x∈R+. Then, p0F0(t)≤p1F1(m(t)) for all t∈[t, t] is sufficient for detecting u(0) ≤u(1). If responses i= 1 are a priori known to be faster than responses i= 0, formalized by m(t)≤tfor all t, then the right hand side is smaller than in (8) and the inequality is harder to satisfy. The converse is true if responses i= 0 are faster. If we want to allow for some degree of asymmetry without knowing details, we can construct multiple functions Hkusing different representative functions ck∗with varying degrees of asymmetry. For a revealed preference, all modified versions of condition (8) have to hold simultaneously, ultimately resulting in a requirement that the difference between the left and the right hand side of (8) must be large enough. This would give rise to a more demanding but even more robust test than in Alós-Ferrer, Fehr and Netzer (2021). As a side remark—and to illustrate the logic of additional distributional assumptions discussed in Subsection 2.5.2—let us finally assume that any admissible Gis symmetric around its mean. In that case, we can try to detect the sign of the median instead of the mean because the two are identical. Whatever representative chronometric function c∗we use in our construction of H—and therefore irrespective of which C∗we want to generate— the sign of the median of Hequals the sign of p1−p0. This mirrors a well-known result (stated, for example, as Proposition 2in Alós-Ferrer, Fehr and Netzer, 2021): under the assumption of symmetric noise distributions, choice frequencies reveal preferences without the need to rely on response time data. 7Alós-Ferrer, Fehr and Netzer (2021) allow all chronometric functions that are symmetric. One minor difference is that they assume the chronometric function to be unbounded while we assume that tis finite. 15 3.1.3 Inequality Our method can also be used to detect inequality or dispersion of a distribution, which has applications across multiple fields. For instance, within the literature on subjective well-being, there is a substantial interest in understanding the inequality of happiness (e.g. Stevenson and Wolfers, 2008). Similarly, researchers have studied societal polarization by measuring the dispersion of individual attitudes towards social and political issues (DiMaggio, Evans and Bryson, 1996; Evans, 2003). In the context of market competition, the spread of consumer preferences has direct implications for the optimal pricing and advertising strategies of firms (Johnson and Myatt, 2006; Hefti, Liu and Schmutzler, 2022). Unfortunately, standard measures of inequality like the Gini index require cardinal information, which makes their application to ordered response data questionable (see the discussion in Dutta and Foster, 2013). Response times may serve as the source of cardinal information, even when the decisions are binary such as consumers’ decisions to buy or not buy a product (Cotet and Krajbich, 2021). The Lorenz curve is a convenient graphical representation of a distribution’s inequality, and interesting measures of inequality like the Gini index are based on the Lorenz curve (Atkinson, 1970; Cowell, 2011). For any distribution G, the associated Lorenz curve is defined by L(q, G) = Rq 0G−1(x)dx R1 0G−1(x)dx for all q∈[0,1], where G−1(x) := inf{x|G(x)≥q}denotes the left inverse of G. In the context of subjective well-being, L(q, G)could be understood as the proportion of total happiness allocated to the least happy 100qpercent of the population. How far the Lorenz curve falls below the 45degree line is an indication of how unequal the distribution is. The curve is invariant to all linear transformations Ψlin of the distribution G. Therefore, if we plot the Lorenz curve for an empirical function Hlike in (6) or (7), or based on any other representative function c∗, exactly this Lorenz curve (and any measure based on it like the Gini index) is detected for the class of chronometric functions that are linearly generated from c∗. As before, we can repeat this procedure for various different representative functions ck∗to obtain bounds on the true Lorenz curve for larger sets of chronometric functions. We just remark that analogous arguments apply to other distributional properties that are invariant to linear transformations, like skewness or kurtosis. 16 3.1.4 Unimodality We close the section on single distributions with the property that the distribution is unimodal with mode at zero, i.e., Gis convex below zero and concave above zero, and strictly so except when G(x)∈ {0,1}. Unimodality is of interest once more in political applications where Gdescribes the distribution of political attitudes in a population. Unimodality reflects a centered population where more extreme positions receive less support. An ongoing debate in political science disagrees whether political attitudes follow unimodal distributions or are polarized and better described by bimodal distributions (see Lelkes, 2016; Vaeth, 2023). Analysts often want to learn about these properties from survey responses, but as Vaeth (2023) points out, ordinal responses are inadequate to test for properties like unior bimodality of the underlying distribution. The property of unimodality is invariant to (sigmoid) transformations that satisfy ψ(0) = 0and are weakly convex below zero and weakly concave above zero, the set of which is denoted Ψsig. Starting from a representative chronometric function c∗, the transformations Ψsig allow us to generate all chronometric functions which are weakly “more convex” than c∗on R−and on R+separately (because c∗is increasing on R−but decreasing on R+). We will use this insight in a slightly different way than before. Assume that the observed cdfs Fiare strictly increasing on [t, t]. Then consider the representative chronometric function constructed from the data by c∗(x) =                tif 1< x, (F1)−1(1 −x)if 0< x ≤1, (F0)−1(1 + x)if −1≤x≤0, tif x < −1, (9) which is a member of C∗ f.all. With this function, the empirical distribution Hbecomes H(x) =                1if 1< x, 1−p1+p1xif 0< x ≤1, p0+p0xif −1≤x≤0, 0if x < −1, which is piece-wise linear. Applying any strictly sigmoid transformation to (9) generates a chronometric function that is strictly more convex (for each response separately) and results in a unimodal H. By the same logic, applying any strictly inverse-sigmoid transformation 17 generates a strictly more concave chronometric function and a distribution Hthat is not unimodal. Function (9) therefore delimits sets of chronometric functions for which we can detect or reject unimodality. It follows from Theorem 1 that unimodality is detected for all those functions that are more convex than (9) and rejected for those that are more concave. This approach does not cover all possible chronometric functions but may yield expressive results. For example, if (9) plotted from the data is already strongly convex, then it appears unlikely that the true chronometric function is even more convex, and we may be able to reject the assumption of unimodality of the distribution. 3.2 Multiple Distributions 3.2.1 First-Order Stochastic Dominance As a first application involving more than one distribution, consider the property of (G1, G2) that G1first-order stochastically dominates G2, i.e., G1(x)≤G2(x)for all x∈R. In the context of happiness surveys, Bond and Lang (2019) have pointed out that conventional probit or logit models make that assumption when comparing two (or more) groups of survey participants. The assumption is crucial for the results of these models, as it yields a ranking of the groups’ average happiness for any choice of the happiness scale. Without the assumption, the sign of estimated parameters can often be flipped by using a different scale. For example, rich survey participants may be less happy on average than poor participants despite responding to be happy more frequently. It is difficult to test FOSD using only response data. For our approach here, observe that FOSD (and also its violation) is invariant to all profiles (ψ1, ψ2)of increasing transformations that satisfy ψ1=ψ2. Denote the set of all these profiles by Ψall.i (where istands for identical across the index j). Let C∗ a.all.ibe the set of all profiles of chronometric functions that approach tasymptotically and are identical across indices. This set is generated by the representative chronometric function (3) for all j∈Jtogether with Ψall.i. Intuitively, it embodies the assumption that the chronometric effect is identical for all groups but otherwise unrestricted. By Theorem 1, we now need to check whether H1first-order stochastically dominates H2, where Hjis defined as in (6) using the observed data (p0 j, p1 j, F0 j, F1 j)of group j= 1,2. Expressed directly in terms of the data, we detect FOSD if p0 1F0 1(t)−p0 2F0 2(t)≤0≤p1 1F1 1(t)−p1 2F1 2(t)(10) for all t∈[t, t], and a violation of FOSD otherwise. We remark that the same condition 18 obtains when considering the set C∗ f.all.iof identical chronometric functions that reach tand that an analogous statement holds when a strict inequality for some xis required in the definition of FOSD. Condition (10) equals conditions (i)and (ii)of Proposition 2in Liu and Netzer (2023).8 As they point out, for t=tcondition (10) reduces to p0 1≤p0 2, which is the condition under which conventional probit or logit models conclude that group j= 1 is happier than group j= 2. To arrive at this conclusion without making distributional assumptions, the inequalities in (10) must hold for all t. Liu and Netzer (2023) test these inequalities using data from an online survey. They show that the null hypothesis of FOSD often cannot be rejected, in particular in cases where the probit model yields significant parameter estimates, indicating that the results of conventional models often seem to be robust at least qualitatively. Our approach here suggests possible generalizations. One worry when comparing response times across individuals as in a survey is that they may differ in their decision speed. Liu and Netzer (2023) address this problem by normalizing individual response times using a baseline question and by showing that i.i.d. heterogeneity does not affect the necessary detection conditions used for testing. A different approach that accounts for group-specific decision speed would be to construct the functions Hjbased on group-specific representative chronometric functions c∗ j, utilizing prior knowledge about group differences. The result would be an asymmetric version of (10). Imprecise knowledge of group differences could once more be captured by working with multiple functions Hk jand checking multiple corresponding conditions, giving rise to a more demanding but more robust test. 3.2.2 Ranking of Means Suppose we want to detect whether the mean of G1is larger than the mean of G2. A first application where the comparison of means matters is once more the case of surveys, where we want to learn whether one group is happier than another on average. We will discuss a second application at the end of this subsection. The ranking of the means is invariant to identical positive affine transformations of the form ψj(x) = a+bx for b > 0. Since we can only use the subset of those transformations which satisfy a= 0 when generating chronometric functions, we restrict attention to the set Ψlin.i of identical linear transformations right away. The set of chronometric functions that can be generated by linear transformations is rather restrictive, as we discussed before. However, if the assumption holds, the analysis can be remarkably simple. As an example, 8Proposition 2in Liu and Netzer (2023) characterizes detection of first-order stochastic dominance using general ordered response models for surveys with more than two response categories. Their condition (iii) applies to the intermediate response categories and coincides with the demanding condition by Bond and Lang (2019), as response times are not monotone and hence not informative in the intermediate categories. 19 if we have reasons to believe that the chronometric functions are symmetric, linear, and identical for both j= 1,2, then we can simply compare the means of H1and H2, where Hj is defined as in (7) using the observed data (p0 j, p1 j, F0 j, F1 j)of group j= 1,2. It is again possible to achieve more robust detection using conditions that are sufficient for a ranking of the means. One sufficient condition is of course first-order stochastic dominance of G1over G2, which we discussed before. Hence, (10) is a sufficient condition for detecting that the mean of G1is larger than the mean of G2, using only the assumption that the chronometric function is the same in both groups j= 1,2. Another sufficient condition would be the detection that the mean of G1is positive while the mean of G2is negative, which we discussed in Subsection 3.1.2. It thus follows immediately that p1 2F1 2(t)−p0 2F0 2(t)≤0≤p1 1F1 1(t)−p0 1F0 1(t)(11) for all t∈[t, t]is also sufficient for detecting a ranking of the means. A comparison of conditions (11) and (10) is instructive. In inequality (10), we compare response times between the groups j(by calculating a difference of the distributions) but not between the response categories i. The result therefore generates detection under the assumption that the chronometric function is identical for the two groups but not necessarily symmetric across the two responses. In inequality (11), we compare response times between response categories ibut not between groups j. It therefore requires the assumption that the chronometric function is symmetric across responses but not necessarily identical between the two groups. Formally, the condition that we are aiming to detect with (11) is invariant to transformations that are symmetric around zero but possibly different between the groups, the set of which is denoted Ψsym.d (where dstands for different across the index j). These transformations can, for example, be used to generate the sets C∗ a.sym.d or C∗ f.sym.d of profiles of chronometric functions with the just-discussed properties.9 We now discuss a third sufficient condition for a ranking of the means. Consider the property that G1(x)+G1(−x)≤G2(x)+G2(−x)for all x∈R+. This condition implies that the mean of G1is larger than the mean of G2(see Appendix B). Furthermore, the condition is invariant to transformations that are identical for both j= 1,2and symmetric around zero. Denote this set by Ψsym.i. Starting from the representative functions (3) for all j∈J, we can use Ψsym.i to generate the set C∗ a.sym.i of all chronometric functions that approach tasymptotically and are symmetric across responses and identical across indices. Using this set, Theorem 1 implies that 9Condition (11) and the results in the next two paragraphs were first derived in our earlier working paper Liu and Netzer (2020) and are unpublished as yet. Other results from Liu and Netzer (2020) were published as Liu and Netzer (2023). 20 our desired inequality condition is detected if it holds for the functions H1and H2that are constructed based on (6). Taken together and expressed directly in terms of the observed data, it follows that p0 1F0 1(t)−p0 2F0 2(t)≤p1 1F1 1(t)−p1 2F1 2(t)(12) for all t∈[t, t]is another sufficient condition for detecting that the mean of G1is larger than the mean of G2. We remark that the same condition obtains when allowing the respective set C∗ f.sym.i of chronometric functions that reach tand that an analogous statement holds for detecting a strict inequality of means. Inequality (12) is a weaker requirement than the directly comparable conditions (10) or (11). However, as it implements a comparison of response times across the groups and across the responses, it generates detection only under the stronger combination of assumptions required in (10) and (11), namely that the chronometric functions are symmetric across responses and identical between groups. We now discuss another application that involves the comparison of two means. Suppose we observe the choices of a single agent between the two options xand zand between the two options yand z. Can we infer the agent’s preference between xand yfrom these choices? Sometimes this is possible based on transitivity of preferences, for example if xis chosen over zand zis chosen over y. If, by contrast, both xand yare chosen over z, then we cannot rank xand ydirectly. Krajbich, Oud and Fehr (2014) note, however, that the preference can be deduced from response times under the assumption of a monotone chronometric effect. When the choice of xover zis faster than the choice of yover z, then u(x)−u(z)must be larger than u(y)−u(z)and we can conclude that u(y)≤u(x)(see also Echenique and Saito, 2017). Alós-Ferrer, Fehr and Netzer (2021) provide a generalization of this argument for stochastic choice under the assumption of symmetric utility distributions. Following their setting, suppose that the random utility difference between xand zis described by a cdf Gxz with mean u(x)−u(z), and the random utility difference between yand zis described by Gyz with mean u(y)−u(z). Deducing a revealed preference for xover ycan now be rephrased as detecting that the mean of Gxz is larger than the mean of Gyz. We can put several of our above results to work. First, one sufficient condition is that Gxz first-order stochastically dominates Gyz, which we detect (under the above-described assumptions on the chronometric functions) when pz xzFz xz(t)−pz yzFz yz(t)≤0≤px xzFx xz(t)−py yzFy yz(t) for all t∈[t, t], where lower indices describe the binary choice problem and upper indices describe the chosen option. Another sufficient condition is that the mean of Gxz is positive 21 while the mean of Gyz is negative, which we detect (under different assumptions on the chronometric functions) when py yzFy yz(t)−pz yzFz yz(t)≤0≤px xzFx xz(t)−pz xzFz xz(t) for all t∈[t, t]. Finally, a weaker sufficient condition for detecting the out-of-sample preference (but under stricter assumptions on the chronometric functions) is pz xzFz xz(t)−pz yzFz yz(t)≤px xzFx xz(t)−py yzFy yz(t) for all t∈[t, t]. To our knowledge, none of these conditions has been studied in the individual choice context. We emphasize that we obtain the revealed preference between xand y without making any assumptions on the shape of the utility distributions, but remark that a revealed preference translates into an out-of-sample prediction of choice probabilities only with additional distributional assumptions such as symmetry. We can also follow Alós-Ferrer, Fehr and Netzer (2021) and assume right away that Gxz and Gyz are symmetric around their means. Since mean and median coincide in this case, we can instead try to detect whether the median of Gxz is larger than that of Gyz. This property is invariant to the set Ψall.i of all transformations that are identical for the two distributions. We can therefore detect the property assuming either C∗ a.all.i or C∗ f.all.i, which means that we only have to assume that the chronometric effect is the same in the two binary decision problems. Consider then the case where px xz >1/2and py yz >1/2, which under symmetry implies that both u(x)−u(z)are u(y)−u(z)are strictly positive, so that a preference between xand ydoes not follow from transitivity. Define θxz and θyz as percentiles of the response time distributions when xor ywere chosen over z, respectively, as follows: Fx xz(θxz) = 1 2px xz and Fy yz(θyz) = 1 2py yz . It is now an easy exercise to show that the median of Hxz is larger than the median of Hyz, where both functions are constructed either as in (6) or as in (7), if and only if θxz ≤θyz.(13) Analogous statements hold for strict inequalities and for the case px xz <1/2and py yz <1/2. Inequality (13) is the condition stated in Theorem 2of Alós-Ferrer, Fehr and Netzer (2021) for a revealed preference u(y)≤u(x)under the assumption of symmetric distributions. As these authors discuss in detail, (13) formalizes that the choice of xover zis faster than the 22 0 20000 40000 60000 80000 100000 120000 140000 0.70 0.75 0.80 0.85 income average response Figure 2: The relationship between income and reported happiness. Notes: The eight colored curves correspond to the different methods used to determine the average income within each bin. where αis such that αwL+(1−α)wH=wM. Following the logic of Bond and Lang (2019), we can easily explain the data in Figure 2 with happiness distributions (GL, GM, GH)for which (16) is violated, for example distributions that become more right-skewed as income grows and which therefore go along with high average happiness among the rich. It is impossible to reject or verify such distributional assumptions based on response data alone. According to (16), the problem of detecting or rejecting decreasing marginal happiness is a problem of ranking the means of distributions, for which we have developed conditions in Subsection 3.2.2. The left-hand side of (16) is the average happiness in a mixed population composed of a fraction αof subjects with low income and a fraction 1−αof subjects with high income. Under the assumption that all income groups have the same chronometric function, which most of the criteria from Subsection 3.2.2 require anyway, we can therefore pool the two extreme income groups with appropriate weights and detect or reject (16) based on response time data. We follow Liu and Netzer (2023) and normalize individual response 29 Method αMean Happiness p-Value Pooled Middle (12) (10) (11) 10.500 98’787 129’335 0.9548 0.3380 0.0000 20.610 91’395 129’335 0.9501 0.4105 0.0000 30.660 88’036 129’335 0.9457 0.4397 0.0000 40.623 90’504 129’335 0.9481 0.4215 0.0000 50.660 88’003 129’335 0.9417 0.4405 0.0000 60.649 88’793 129’335 0.9427 0.4364 0.0000 70.646 88’985 129’335 0.9445 0.4350 0.0000 80.656 88’310 129’335 0.9482 0.4391 0.0000 Table 1: Summary of the tests for concavity. times by subtracting in logs a subject’s response time to the marital status question, which accounts for individual-specific speed and further corroborates the assumption of identical chronometric functions for the different income groups. If we are willing to assume that the identical chronometric functions are also linear and symmetric, we can use (7) to compute one distribution HMfor the middle income group and one distribution HPfor the pooled group of low and high incomes and simply compare their means. Since HP=αHL+ (1 −α)HH, we can compute the distributions HLand HHin the low and high income groups separately and then form a convex combination, rather than actually pooling the data to compute HP. Table 1 contains the results of this approach for our eight different methods of assigning income levels, which give weights αbetween 0.50 and 0.66. The mean happiness of the middle income group clearly exceeds that of the pooled group across all methods.11 This serves as a first indication of concavity of the relationship between income and happiness but is far from conclusive. In particular, the assumption of a linear chronometric function is probably not less controversial than the assumption of a linear reporting function. To obtain results under less stringent assumptions, we can try to detect if mean happiness is larger in the middle income group than in the pooled group based on inequality (12). This inequality is sufficient to detect a ranking of the means and requires only that the chronometric function is symmetric across responses and identical between groups, not that 11We use response times that are normalized by the marital status question but not by taking the logarithm here, as taking the logarithm would correspond to a non-linear transformation of the chronometric function and this matters for the approach. It is irrelevant for the approaches used later in this subsection, so there we report and depict all results using log normalized response times. 30 it is linear. In our context, the inequality becomes p0 MF0 M(t)−p0 PF0 P(t)≤p1 MF1 M(t)−p1 PF1 P(t) for all t∈[t, t], where pi PFi P(t) = αpi LFi L(t)+(1−α)pi HFi H(t). The condition intuitively rules out examples like the increasingly right-skewed distributions discussed above, as these would generate relatively fast (slow) happy (unhappy) responses in the pooled group. Figure 3 plots the left-hand side and the right-hand side of (12), again for all the eight methods used to impute incomes. Inequality (12) is not satisfied exactly in the data, because the right-hand side falls below the left-hand side for some small t. This indicates that there are indeed some fast happy responses in the pooled relative to the middle group. However, the crossing of the functions appears to be minor, so that the question of statistical significance arises. To obtain p-values for the null hypothesis that (12) holds, we employ a bootstrapbased method as in Liu and Netzer (2023) that rests on a test for conventional first-order stochastic dominance by Barrett and Donald (2003). Table 1 shows that the p-values are very large. We clearly cannot reject (12), which is a sufficient condition for a ranking of the means in line with decreasing marginal happiness. If we are even unwilling to accept the assumption of symmetry of the chronometric functions, we can still test the stronger condition that the happiness distribution of the middle income group first-order stochastically dominates that of the pooled group. This condition is formalized by (10) and requires that the functions in Figure 3 are separated by zero. While not satisfied exactly in the data, the p-values of the null hypothesis that (10) holds are smaller than for (12) but still large, as can be seen in Table 1.12 We cannot reject the hypothesis of first-order stochastic dominance and hence of a very strong sufficient condition for decreasing marginal happiness that applies under weak assumptions on the chronometric function. For completeness, Table 1 also reports p-values for the hypothesis that (11) holds, another sufficient condition for a ranking of the means. This condition is of less interest here. First, it is very strong and would detect a ranking only if one of the means happened to be smaller and the other larger than zero. Second, its advantage of allowing group-specific chronometric functions has no bite because our approach of pooling groups requires identical chronometric functions (after normalization) anyway. The hypothesis that (11) holds is clearly rejected. To summarize, our results are supportive of the idea that marginal happiness is decreasing in income, in a cross-sectional data set. While our tests avoid several of the pitfalls noted at 12The test again follows Liu and Netzer (2023) and implements the procedure of Barrett and Donald (2003) together with a joint hypothesis correction by Romano and Wolf (2016) to account for the fact that condition (10) contains two inequalities. We refer the reader to Liu and Netzer (2023) for more details. 31 −6 −4 −2 0 2 4 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 log normalized response time cumulative difference in response fractions Figure 3: Empirical conditions for detecting the income-happiness relation. Notes: The curves at the top represent the empirical functions p1 MF1 M(t)−p1 PF1 P(t), while those at the bottom represent p0 MF0 M(t)−p0 PF0 P(t). Different colors indicate the varying methods used to determine the average income within each bin. the beginning of this section, some limitations remain. Most importantly, we were not able to test concavity of µat any income level but only for the three income levels implied by the bins used in the survey, and the detected relation is bivariate without additional controls. 5 Conclusion The goal of this paper is to provide a systematic account of the information that response time data contain. We approach the problem by phrasing it as one of identification in the context of binary response models. Our main result relates the set of identifiable distributional properties to the set of admissible chronometric functions. The fundamental idea is that the joint distribution of responses and response times identifies a composition of the latent distribution and the chronometric function. Properties of the distribution that are preserved under a given set of transformations can therefore be identified if the chronometric function 32 is known up to these transformations. Several existing results in the literature follow as corollaries and can be generalized. Many new results emerge. To illustrate the applicability of our approach, we empirically test and cannot reject the hypothesis of decreasing marginal happiness of income. Our theoretical applications in Section 3 are merely examples of the scope of the method and not an exhaustive list. Additional properties that one could study include general linear relationships between observable variables and the latent variable like in regression models, as well as the extent to which responses are potentially distorted by the framing of a decision problem. Similarly, our empirical study in Section 4 is only one straightforward application showing how the method can be used to contribute to long-standing debates. Other applications that we have in mind include the study of polarization using surveys on political attitudes, as well as optimal product pricing using data on purchase decisions from online platforms. There are also several possible extensions of our framework that merit investigation. These include correlations between multiple latent variables, the case with more than two choice options, and the use of response times in settings where these times are affected by additional factors like player types or decision modes as in Rubinstein (2007, 2013, 2016). 33 References Alós-Ferrer, Carlos, and Michele Garagnani. 2022. “Strength of preference and decisions under risk.” Journal of Risk and Uncertainty, 64: 309–329. Alós-Ferrer, Carlos, and Michele Garagnani. 2024. “Improving Risky-Choice Predictions Using Response Times.” Journal of Political Economy: Microeconomics, 2(2): 335– 354. 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