2.1 Dutch household panel

We fielded a purpose-built module to measure ambiguity and risk attitudes in the CentERpanel, a representative household survey of about 2,000 respondents conducted by CentERdata at Tilburg University in the Netherlands. The survey is computer-based and subjects can participate from their homes. To limit selection bias, households lacking internet access at the recruiting stage were provided with a set-top box for their television sets (and with a TV if they had none). Each year, the DNB Household Survey (DHS) is fielded in the panel to obtain detailed information about the members’ income, assets, and liabilities.Footnote ⁶ We merged the DHS data with results from our survey module on ambiguity and risk attitudes. The CentERpanel is representative of the Dutch population and the DHS has previously been used to provide insight into household financial decisions (e.g., Guiso et al., Reference Guiso, Sapienza and Zingales2008; van Rooij et al., Reference van Rooij, Lusardi and Alessie2011; and von Gaudecker, Reference von Gaudecker2015).

As the panel is nationally representative, it includes a large number of people who do not invest in financial markets: about 85% of the panel members as of December 2016. Ambiguity about all investments is likely to be high in this group, without much meaningful variation between sources. To better utilize resources, our questionnaire was targeted at the 15% of DHS respondents who invested in financial assets, defined to include mutual funds (about 68% of the investors), individual company stocks (45%), bonds (11%), or options (2.5%).Footnote ⁷ Our survey module was fielded from 27 April-14 May 2018, yielding 295 complete and valid responses.Footnote ⁸ Our survey was also given to a random sample of non-investors from the general population, with 230 complete responses. The non-investor sample allows us to compare the ambiguity attitudes of investors and non-investors, which we do in Sect. 5.4. For our main results, we focus only on investors, as our goal is to assess ambiguity attitudes of investors in financial markets and to validate our measures by confirming that ambiguity attitudes are associated with investment decisions.

Summary statistics on the investor sample appear in Appendix Table 1. Education is an ordinal variable ranging from 1 to 6, where 1 indicates primary education and 6 indicates a university degree. Household Income averages €3193 per month. Household Financial Wealth consists of the sum of all current accounts, savings accounts, term deposits, cash value of insurance policies, bonds, mutual funds, stocks, options, and other financial assets such as loans to friends or family, all reported as of 31 December 2017. Mean (median) wealth was €142,357 (€84,489). We also have measures for Age, Female, Single, Number of Children living at home, Employed, and Retired. Appendix Table 1 shows that the average Dutch investor in financial markets is relatively old, male, and well educated. We note that this is the profile of a typical Dutch individual investor, as the DHS data is representative, and it is also in line with other studies of investors in the Netherlands (e.g., Cox et al., Reference Cox, Kamolsareeratana and Kouwenberg2020; von Gaudecker, Reference von Gaudecker2015).

2.2 Elicitation of ambiguity attitudes

We elicit ambiguity attitudes toward real-world investments following the method of Baillon et al. (Reference Baillon, Huang, Selim and Wakker2018b). The first source of ambiguity we evaluate is the return on the Amsterdam Exchange Index (AEX) over a 1-month period.Footnote ⁹ The method divides the possible outcomes of the AEX into three mutually exclusive and exhaustive events, denoted as $E_1,E_2,\text{and}{E}_3$ :

$E_1=(-\infty ,-4\%]$ : the AEX index decreases by 4% or more;

$E_2=(-4\%,+4\%)$ : the AEX index decreases or increases by less than 4%;

$E_3=[+4\%,\infty )$ : the AEX index increases by 4% or more.

For each event $E_i$ separately, we elicit the respondent’s matching probability with a choice list, shown in Fig. 1 for event $E_1$ as an example. The matching probability $m_i$ is the known probability of winning p = $m_i$ at which the respondent is indifferent between Option A (winning €15 if Event $E_1$ happens) and Option B (winning €15 with known chance $p$ ).Footnote ¹⁰ We approximate the matching probability by taking the average of the probabilities p in the two rows that define the respondent’s switching point from Option A to B. For example, in Fig. 1 the matching probability is: $m_1=\frac{20\%+30\%}{2}=25\%$ .

Fig. 1 Example of a Choice List for Eliciting Ambiguity Attitudes

We also elicit a matching probability for the complement of each event:

$E_{23}=(-4\%,\infty )$ : the AEX index does not decrease by 4% or more;

$E_{13}=\left(-,\infty ,,,-,4,\%\right)\cup [+4\%,\infty )$ : the AEX index decreases or increases by 4% or more;

$E_{12}=(-\infty ,+4\%)$ : the AEX index does not increase by 4% or more.

The matching probability for the composite event $E_{\mathrm{ij}}=E_i\cup E_j$ is denoted by $m_{\mathrm{ij}}$ , with $i\ne j$ . For example, Fig. 2 shows the choice list for the composite event $E_{23}$ , with $m_{23}=55\%$ .

Fig. 2 Second Choice List for Eliciting Ambiguity Attitudes about the AEX Index

A key insight of the method is that, for an ambiguity neutral decision maker, the matching probabilities of an event and its complement add up to 1 ( $m_1+m_{23}=1)$ , but under ambiguity aversion, the sum is less than 1 ( $m_1+m_{23}<1$ ). For example, the choices in Figs. 1 and 2 imply that ${1-m}_1-m_{23}=1-0.25-0.55=0.2$ , indicating ambiguity aversion. Baillon et al. (Reference Baillon, Huang, Selim and Wakker2018b) define their ambiguity aversion index b, averaging over the three events, as follows:

(1) $b=1-{\overline{m}}_c-{\overline{m}}_s,$

with $-1\le b\le 1.$ Here ${\overline{m}}_s=(m_1+m_2+m_3)/3$ denotes the average single-event matching probability, and ${\overline{m}}_c=(m_{12}+m_{13}+m_{23})/3$ is the average composite-event matching probability. The decision-maker is ambiguity averse for $b>0$ , ambiguity seeking for $b<0$ , and ambiguity neutrality implies $b=0$ .

In practice, ambiguity attitudes have a second component apart from ambiguity aversion, namely a tendency to treat all uncertain events as though they had a 50–50% chance, which is called ambiguity-generated insensitivity or a-insensitivity (Abdellaoui et al., Reference Abdellaoui, Baillon, Placido and Wakker2011; Tversky & Fox, Reference Tversky and Fox1995). For unlikely events, a-insensitivity leads to overweighting and more ambiguity-seeking choices. Empirical studies have shown that a-insensitivity is a typical feature of decision-making under ambiguity (Trautmann and van de Kuilen Reference Trautmann, van de Kuilen, Keren and Wu2015; Dimmock et al., Reference Dimmock, Kouwenberg and Wakker2016a). Baillon et al. (Reference Baillon, Huang, Selim and Wakker2018b) define the following index to measure a-insensitivity:

(2) $a=3\times \left(\frac{1}{3},-,\left({\overline{m}}_c,-,{\overline{m}}_s\right)\right),$

with $-2\le a\le 4.$ For ambiguity neutral decision-makers, $a=0$ , while $a>0$ denotes a-insensitivity. Negative values, $a<0$ , indicate that the decision-maker is overly sensitive to changes in likelihood, implying underweighting of unlikely events.

In the neo-additive ambiguity model of Chateauneuf et al. (Reference Chateauneuf, Eichberger and Grant2007), index a is also a measure of the decision maker’s perceived level of ambiguity about a source, as long as $0\le \alpha \le$ 1 (see Dimmock et al., Reference Dimmock, Kouwenberg, Mitchell and Peijnenburg2015; Baillon et al., Reference Baillon, Bleichrodt, Keskin, I’Haridon and Li2018a; and Online Appendix A). When $a>1$ the respondent has violated monotonicity, as then the average matching probability of the single events exceeds the average for the composite events $({\overline{m}}_s>{\overline{m}}_c)$ . We will later analyze how frequently such violations occur and how often index a falls within the boundaries $0\le \alpha \le$ 1 where it can be interpreted as the level of perceived ambiguity.

The method of Baillon et al. (Reference Baillon, Huang, Selim and Wakker2018b) has two major advantages: first, risk preferences (both the utility and probability weighting function) cancel out in the comparison between Options A and B, so they do not need to be estimated to identify ambiguity attitudes (Dimmock et al., Reference Dimmock, Kouwenberg and Wakker2016a). Second, using events and their complements in the calculation of index b and a ensures that the unknown subjective probabilities drop out of the equation (Baillon et al., Reference Baillon, Bleichrodt, Li and Wakker2021). Accordingly, we can measure ambiguity aversion without knowing respondents’ subjective beliefs. This solves the important issue that, when observing a dislike of ambiguity, it is difficult to disentangle whether this is due to ambiguity aversion or pessimistic beliefs.

2.3 Elicitation of risk attitudes

The module also included four separate choice lists to measure risk attitudes (a screenshot is provided in Online Appendix B). The first risk attitude choice list elicited a certainty equivalent for a known 50% chance of winning €15 or €0 otherwise, based on a fair coin toss. The other three choice lists elicited a certainty equivalent for winning chances of €15 of 33%, 17%, and 83%, respectively, using a die throw. Respondents could win real money for the risk questions, and the order was randomized of the risk and ambiguity question sets in the survey.

Following Abdellaoui et al. (Reference Abdellaoui, Baillon, Placido and Wakker2011), we use index b _r for risk as a measure of Risk Aversion, and index a _r for risk as a measure of Likelihood Insensitivity, which is the tendency to treat all known probabilities as 50–50% and thus overweight small-probability events. To estimate these measures, we assume a rank-dependent utility model with a neo-additive probability weighting function and a linear utility function. We do this for two reasons. First, these two risk attitude measures are conceptually related to index b for ambiguity aversion and index a for a-insensitivity (Abdellaoui et al., Reference Abdellaoui, Baillon, Placido and Wakker2011). Second, utility curvature is often close to linear for small payoffs. We refer to Online Appendix C for more details about these measures, and for a robustness check using two non-parametric risk measures that do not rely on any model assumptions.

The Appendix Table 1 shows that on average investors were risk averse (mean > 0) but with strong heterogeneity, and about one third of the investors were risk seeking. Further, the Likelihood Insensitivity measure is positive for 85% of the investors, indicating a tendency to overweight small probabilities, in line with the findings of previous studies (see, e.g., Fehr-Duda & Epper, Reference Fehr-Duda and Epper2011 and Dimmock et al., Reference Dimmock, Kouwenberg, Mitchell and Peijnenburg2021).

2.4 Real incentives

At the outset of the survey, each subject was told that one of his or her choices in the ambiguity and risk questions would be randomly selected and played for real money. Hence all respondents who completed the survey had a chance to win a prize based on their choices, and a total of €2,758 in real incentives was paid out. The incentives were determined and paid by CentERdata one month after the end of the survey, when the changes in the asset values were known. As panel members regularly receive payments for their participation, the involvement of CentERdata minimizes subjects’ potential concerns about the credibility of the incentives.

2.5 Financial literacy and asset ownership

Our survey module also collected data on financial literacy and asset ownership. Financial literacy is one of our key independent variables, as we aim to assess whether financial knowledge relates to ambiguity attitudes. To measure this, we use 12 questions from Lusardi and Mitchell (Reference Lusardi and Mitchell2007) and van Rooij et al. (Reference van Rooij, Lusardi and Alessie2011), shown in Online Appendix C. Financial Literacy is the number of correct responses to the 12 questions (the average is 10.6; see Appendix Table 1).Footnote ¹²

We validate our ambiguity measures by examining whether they relate to the financial assets owned by the investors. Our survey module asked the panel members whether they currently invested in the familiar company stock they mentioned, in mutual funds tracking the MSCI World index, or any crypto-currencies such as Bitcoin. Invests in Familiar Stock is an indicator variable equal to one if the investor currently held the familiar company stock, which 30.2% did (see Appendix Table 1). Invests in Crypto-Currencies and Invests in MSCI World are equal to one if the investor held any crypto-currencies or funds tracking the MSCI World stock index, which was true for 2.4% and 1.4%, respectively. Finally, none of the investors in the sample owned funds tracking the domestic AEX stock index.

We note that even in the group of financial assets owners we label as “investors,” ownership of the four assets presented in our ambiguity survey is rather low, as most only own some mutual funds or bonds. Further, only half (55%) in the investor group could name a listed company stock with which they were familiar. All of this indicates that financial sophistication is not that high on average among Dutch retail investors, as is typically found in other household finance studies as well (see Gomes et al., Reference Gomes, Haliassos and Ramadorai2021, for a review).

3.1 Descriptive statistics

Figure 3 shows the fraction of respondents who were ambiguity averse, neutral, and seeking, for the four sources of ambiguity: the familiar stock, the domestic stock market index (AEX), a foreign stock market index (MSCI World), and Bitcoin. To account for possible measurement error, we classify small values of index b that are not significantly different from zero as ambiguity neutral.Footnote ¹³ About 58% of the respondents were ambiguity averse, while 30% were ambiguity seeking, a pattern that is similar across the sources of financial ambiguity. Furthermore, ambiguity neutrality was less common (12%), implying that only few investors’ choices were consistent with the expected utility model. Our results confirm for real-world sources of uncertainty that ambiguity aversion is common, but not universal. These findings are comparable to earlier large-scale studies that used artificial sources, such as Dimmock et al. (Reference Dimmock, Kouwenberg, Mitchell and Peijnenburg2015), Dimmock et al., (Reference Dimmock, Kouwenberg and Wakker2016a), and Kocher et al. (Reference Kocher, Lahno and Trautmann2018), showing that ambiguity seeking choices are not limited to Ellsberg urns.

Fig. 3 Ambiguity Attitudes toward Financial Sources (Averse, Neutral and Seeking). This Figure shows the percent of investors who are ambiguity averse (b-index > 0, significant at 5%), ambiguity neutral (cannot reject b-index = 0), and ambiguity seeking (b-index < 0, significant at 5%) for the local stock market index (b_aex), a familiar company stock (b_stock), the MSCI World stock index (b_msci), and Bitcoin (b_bitcoin). The sample consists of n = 295 investors

Table 1 shows descriptive statistics for the b-indexes. Investors on average display higher ambiguity aversion toward the foreign stock index (0.21), compared to the domestic AEX index (0.17), the familiar individual stock (0.16), and Bitcoin (0.17), in line with the well-document home bias (French & Poterba, Reference French and Poterba1991). There was strong heterogeneity in ambiguity aversion between investors, as indicated by the high standard deviation of the b-indexes (about 0.5 on average). We use Hotelling’s T-squared statisticFootnote ¹⁴ to test the hypothesis that the mean b-index is equal for the four investments, which cannot be rejected at the 5% level (T ² = 7.65; p = 0.057). This implies that the mean level of ambiguity aversion does not depend strongly on the source of financial uncertainty. This is in line with Baillon and Bleichrodt (Reference Baillon and Bleichrodt2015), who found that Dutch financial-economics students did not display source preference for the AEX over the Indian SENSEX index. Still, in our sample, ambiguity aversion for the familiar stock (0.16) is 34% lower compared to the MSCI World index (0.21; p = 0.009 for the mean difference), so investors on aggregate did display some source preference for the familiar stock over a foreign stock market index.

Table 1 Descriptive Statistics for Ambiguity Attitudes

	Mean	Median	St dev	Min	Max	n (obs.)
Panel A: Ambiguity aversion
b_aex	0.17	0.10	0.48	− 1.00	1.00	295
b_stock	0.16	0.10	0.48	− 1.00	1.00	295
b_msci	0.21	0.16	0.48	− 1.00	1.00	295
b_bitcoin	0.17	0.13	0.52	− 1.00	1.00	295
b_avg	0.18	0.15	0.43	− 1.00	1.00	295
Panel B: A-Insensitivity
a_aex	0.83	1.00	0.53	− 0.70	2.99	295
a_stock	0.69	0.85	0.64	− 1.81	2.90	295
a_msci	0.78	0.90	0.52	− 1.51	2.80	295
a_bitcoin	0.84	1.00	0.50	− 1.02	2.61	295
a_avg	0.79	0.88	0.33	− 0.29	1.73	295

Panel A shows summary statistics for ambiguity aversion regarding the local stock market index (b_aex), a familiar company stock (b_stock), the MSCI World stock index (b_msci) and Bitcoin (b_bitcoin), as well as the average of the four b-indexes (b_avg). Positive values of the b-index denote ambiguity aversion, and negative values indicate ambiguity seeking. Panel B shows summary statistics of index a (a-insensitivity) for the local stock market index (a_aex), a familiar company stock (a_stock), the MSCI World stock index (a_msci) and Bitcoin (a_bitcoin), as well as the average of the four a-indexes (a_avg). The sample consists of n = 295 investors

Figure 4 illustrates the relation between the ambiguity aversion measures for the four different investment sources at the subject level, shown with scatter plots. The correlations are all relatively strong, ranging between 0.62 and 0.74. This implies that if an investor had relatively high ambiguity aversion toward one specific financial source (e.g., the AEX index), he also tended to display high ambiguity aversion toward the other three investments. A factor analysis shows that the first factor explains 77% of the cross-sectional variation in the four ambiguity aversion measures, indicating that a single underlying variable is driving most of the variation.

Fig. 4 Scatter Plots of Ambiguity Attitudes toward Different Financial Sources. This Figure shows scatter plots of the relationships between ambiguity aversion (the b-indexes) for different investments: the local stock market index (b_aex), a familiar company stock (b_stock), the MSCI World stock index (b_msci), and Bitcoin (b_bitcoin). The correlation (r) is shown above each scatter plot, with *, **, *** denoting significance at the 10%, 5% and 1%, respectively. The sample consists of n = 295 investors

3.2 Econometric model

Previous empirical studies by Borghans et al. (Reference Borghans, Heckman, Golsteyn and Meijers2009), Stahl (Reference Stahl2014), and l’Haridon et al. (Reference I’Haridon, Vieider, Aycinena, Bandur, Belianin, Cingl, Kothiyal and Martinsson2018) found high levels of unexplained heterogeneity in ambiguity attitudes when measured with Ellsberg urns, which l’Haridon et al. (Reference I’Haridon, Vieider, Aycinena, Bandur, Belianin, Cingl, Kothiyal and Martinsson2018) interpreted as noise (e.g., errors or random responses). An open question is: to what extent does using real-world investments help to improve measurement reliability? In this section, we analyze the variation in ambiguity attitudes using econometric models, following the approach of Dimmock et al. (Reference Dimmock, Kouwenberg, Mitchell and Peijnenburg2015) and l’Haridon et al. (Reference I’Haridon, Vieider, Aycinena, Bandur, Belianin, Cingl, Kothiyal and Martinsson2018). We estimate a panel regression model, where the cross-sectional unit i is the individual respondent, and the “time dimension” s (or repeated measurement) comes from the four investments:

(3) $b_{i,s}={\beta }_1+{\sum }_{s=2}^4({\beta }_s+v_{i,s}^b)d_s+{\sum }_{k=1}^K{\gamma }_k^b{X}_{i,k}+u_i^b+{\epsilon }_{i,s}^{b}i=1,2,\cdots ,I,\mathrm{and}s=1,2,3,4,$

where $b_{i,s}$ is index b (ambiguity aversion) of respondent i toward source s, for the AEX index (s = 1), the familiar stock (s = 2), the MSCI World index (s = 3), and Bitcoin (s = 4).

The dummy variable $d_s$ is 1 for source s, and 0 otherwise. The constant ${\beta }_1$ represents ambiguity aversion for the AEX index, whereas the coefficients ${\beta }_2,{\beta }_3$ and ${\beta }_4$ for the familiar stock, MSCI World and Bitcoin represent differences in mean ambiguity aversion relative to the AEX index. A set of K observable individual characteristics $X_{i,k}$ , such as age and gender, can also impact ambiguity aversion, with regression slope coefficients ${\gamma }_k^b$ . The error term ${\epsilon }_{i,s}^b$ is identically and independently distributed, with $Var\left[{\epsilon }_{i,s}^b\right]={\left({\sigma }_{\epsilon }^b\right)}^2$ .

Random effect $u_i^b$ represents unobserved heterogeneity in ambiguity aversion, which is independent of the error term and uncorrelated between individuals, with $Var\left[u_i^b\right]={\left({\sigma }_u^b\right)}^2$ . Further, random effect $v_{i,s}^b$ is known as a “random slope” (with $Var\left[v_{i,s}^b\right]={\left({\sigma }_{v,s}^b\right)}^2$ ), as it changes the beta of the source dummy $d_s$ . For example, $v_{i,2}^b$ captures individual heterogeneity in ambiguity aversion toward the familiar stock (s = 2), in addition to the heterogeneity in ambiguity aversion that affects all sources captured by the “random constant” $u_i^b$ . Correlations between random effects ( $u_i^b$ , $v_{i,s}^b$ ) are also estimated as part of the model.

The total variance of ambiguity attitudes can now be decomposed as follows:Footnote ¹⁵

(4) $Var\left(b_{i,s}\right)=Var\left({\beta }^{\prime },D,+,{\gamma }^{b^{\prime }},X\right)+Var\left(u_i^b\right)+Var\left[{\sum }_{s=2}^4{\beta }_s{v}_{i,s}^b{d}_s\right]+Var\left[{\epsilon }_{i,s}^b\right],$

with the four right-hand-side components representing variance explained by observed variables, unobserved individual heterogeneity in ambiguity attitudes that is common to all sources ( $u_i^b$ ), unobserved source-specific individual heterogeneity ( $v_{i,s}^b)$ , and random errors ( ${\epsilon }_{i,s}^b)$ .

Our estimation approach is as follows: first, we estimate a model with only a random constant, and then random slopes are added to the model one at a time, followed by a test for their significance (a likelihood-ratio test).Footnote ¹⁶ Suppose $v_{i,2}^b$ (familiar stock) and $v_{i,4}^b$ (Bitcoin) are significant individually: then a model with both random slopes is estimated and tested as well. Finally, if an estimated random slope model turns out to have insignificant variance ( ${\sigma }_{v,s}^b=0$ ), or perfect correlation with the random constant ( $Cor(u_i^b,v_{i,s}^b)$ = 1 or -1), then it is considered invalid and not used.

3.3 Analysis of heterogeneity in ambiguity aversion

The estimation results for index b, ambiguity aversion, appear in Table 2. The sample consists of all 295 investors. All values of index $b_{i,s}$ are included, even if the respondent violated monotonicity or made other errors, to show the impact of noise in the data. Model 1 in Table 2 includes only a random effect capturing individual heterogeneity in ambiguity aversion that is common to the four investments, which explains 69% of the variation. The constant in the model is 0.177 (p < 0.001), implying that investors on average are ambiguity averse toward the investments. Table 2 also reports the interclass correlation coefficient (ICC), which in our dataset captures the correlation of ambiguity aversion toward the four sources.Footnote ¹⁷ The ICC is 0.69, indicating that the correlation in ambiguity aversion for the four investments is high.

Table 2 Analysis of Heterogeneity in Ambiguity Aversion

	Dependent variable: ambiguity aversion (Index b)
	Model 1	Model 2	Model 3	Model 4	Model 5
Constant	0.177***	0.168***	0.168***	0.153	0.212
	(0.025)	(0.028)	(0.028)	(0.219)	(0.240)
Dummy Familiar stock		− 0.012	− 0.012	− 0.012	− 0.012
		(0.020)	(0.020)	(0.020)	(0.020)
Dummy MSCI World		0.042**	0.042**	0.042**	0.042**
		(0.021)	(0.021)	(0.021)	(0.021)
Dummy Bitcoin		0.007	0.007	0.007	0.007
		(0.024)	(0.024)	(0.024)	(0.024)
Education				− 0.010	− 0.018
				(0.017)	(0.016)
Age				0.006***	0.003*
				(0.002)	(0.002)
Female				0.072	0.059
				(0.062)	(0.053)
Single				− 0.116**	− 0.090*
				(0.058)	(0.049)
Employed				− 0.040	− 0.042
				(0.070)	(0.058)
Number of Children (log)				0.059	0.048
				(0.062)	(0.057)
Family income (log)				− 0.011	0.016
				(0.014)	(0.016)
HH Fin. Wealth (log)				− 0.016*	− 0.011*
				(0.009)	(0.007)
HH wealth imputed				− 0.130	− 0.050
				(0.115)	(0.092)
Financial literacy					− 0.015
					(0.017)
Risk aversion					0.466***
					(0.065)
Likelihood insensitivity					− 0.084*
					(0.048)
Random slope: bitcoin	No	No	Yes	Yes	Yes
Observations n	1180	1180	1180	1180	1180
ICC of Random effect $u_i^b$	0.69	0.69	0.74	0.72	0.65
$Var\left[{\epsilon }_{i,s}^b\right]$ , error	0.075 (31%)	0.075 (31%)	0.061 (25%)	0.061 (25%)	0.061 (25%)
$Var\left[u_i^b\right]$ , Random constant	0.165 (69%)	0.165 (69%)	0.167 (70%)	0.152 (63%)	0.112 (47%)
$Var\left[v_{i,4}^b\right]$ , Slope Bitcoin	–	–	0.011 (5%)	0.012 (5%)	0.012 (5%)
$Var[\beta \prime D+\gamma \prime X]$ , Observed	–	0.0004 (0%)	0.0004 (0%)	0.015 (6%)	0.056 (23%)

The table shows estimation results for the panel regression model in Eq. (3), with index b (ambiguity aversion) toward the four investments as the dependent variable. Model 1 includes a constant and a random effect for individual-level heterogeneity in ambiguity aversion common to all sources. Model 2 adds dummies for differences in mean b between the four investments. Model 3 includes a random slope to capture heterogeneity in ambiguity aversion toward Bitcoin, shown to be significant by a likelihood ratio test (see Online App. E.1). Model 4 includes education, age, gender, single, employment, log number of children, family income, and household financial wealth, plus a missing wealth dummy. Model 5 adds financial literacy, risk aversion and likelihood insensitivity. Sample: n = 1180 observations of index b, for 295 investors. Standard errors clustered by investor shown in parentheses

Model 2 adds dummies to allow for differences in the mean level of ambiguity aversion toward the four investments. The dummy for the MSCI World index is positive (p = 0.042), implying investors are more ambiguity averse toward foreign stocks. Random slopes for source-specific ambiguity aversion are added next, and a chi-square test (reported in Online Appendix E.1) shows that only adding a random slope for Bitcoin leads to an improvement of model fit (p < 0.001). Heterogeneity in ambiguity aversion toward Bitcoin (the random slope) explains 5% of the variation in Model 3, on top of the 70% captured by ambiguity aversion toward all four sources (random constant). Overall, the results imply that ambiguity aversion toward investments is driven mainly by one underlying factor, with high correlation between measurements for different sources.

3.4 Variation in ambiguity attitudes explained by individual characteristics

Model 4 in Table 2 adds observed individual socio-demographic variables such as age, gender, education, employment, income, and financial assets, which together account for about 6% of the variation. Ambiguity aversion toward investments is lower for younger investors (p = 0.008) and singles (p = 0.044). In Model 5, proxies for financial literacy and risk attitudes are added, which account for an additional 17% of the variation. Specifically, ambiguity aversion toward investments and risk aversion are positively related (p < 0.001). Ambiguity aversion is not related to financial literacy and also not lower for the familiar stock, suggesting limited competence effects (Heath & Tversky, Reference Heath and Tversky1991) in ambiguity aversion for investments.

All observed variables together explain 23% of the variation in ambiguity aversion in Model 4, while 52% is driven by unobserved heterogeneity (of which 5% specific to the Bitcoin source), and the remaining 25% is error. The relatively low percentage attributed to error suggests that measurement reliability is relatively high for ambiguity aversion toward investments.

4.1 Descriptive statistics

Panel B of Table 1 summarizes the a-index values. On average, a-insensitivity is high (0.79), implying that investors do not much discriminate between single and composite events, suggesting they perceive high ambiguity about their probabilities.Footnote ¹⁸ Further tests show that the mean of index a is significantly lower for the familiar stock (0.69).

Overall, the large majority of investors is insensitive to the likelihood of ambiguous events (a > 0) for the four investment sources, as shown in Panel A of Table 3. Hence, a-insensitivity is common, in line with earlier results for stocks in Fox et al. (Reference Fox, Rogers and Tversky1996) and Kilka and Weber (Reference Kilka and Weber2001), but we can now confirm this in a large field study of investors, using measurements that control for beliefs and risk preferences. These findings are also in line with results for Ellsberg urns in Dimmock et al. (Reference Dimmock, Kouwenberg, Mitchell and Peijnenburg2015) and Dimmock et al., (Reference Dimmock, Kouwenberg and Wakker2016a).

Table 3 Descriptive Statistics for Perceived Ambiguity

	Within limits for perceived ambiguity	Over-sensitive to likelihoods	Violation of monotonicity
	% with 0 ≤ a ≤ 1	% with a < 0	% with a > 1
Panel A: Negative values of index a and violations of monotonicity
a_aex	65.1	8.8	26.1
a_stock	65.1	12.5	22.4
a_msci	69.5	7.8	22.7
a_bitcoin	69.5	5.4	25.1
a_avg	77.6	2.0	20.3

	Mean	Median	St. dev.	Min	Max	n (obs.)
Panel B: Summary statistics of perceived ambiguity
a_aex	0.74	0.89	0.30	0.00	1.00	192
a_stock	0.64	0.74	0.35	0.01	1.00	192
a_msci	0.72	0.80	0.30	0.00	1.00	205
a_bitcoin	0.75	0.91	0.30	0.01	1.00	205
a_avg	0.71	0.76	0.26	0.02	1.00	229

The table shows summary statistics for index a, for the local stock market index (a_aex), a familiar company stock (a_stock), the MSCI World stock index (a_msci) and Bitcoin (a_bitcoin), as well as the average of the four a-indexes (a_avg). Panel A of the table shows the percentage of a-index values that are negative (over-sensitive to likelihoods), between 0 and 1 (in line with the interpretation of index a as perceived ambiguity), and larger than 1 (violations of monotonicity). The sample consists of n = 295 investors. In Panel B, the sample has been restricted to only those observations of index a that are between 0 and 1, after pairwise deletion, so that the a-indexes can be interpreted as measures of perceived ambiguity. For this reason, in Panel B the sample size varies, as indicated in the last column

Panel A of Table 3 shows about two thirds of the a-index values fell in the range 0 to 1, such that they can be interpreted as the perceived level of ambiguity. About 25% of the respondents had a > 1 and thus violated monotonicity when looking at each investment separately, and 20% after averaging over the four investments (a_avg > 1). Earlier studies such as Li et al. (Reference Li, Müller, Wakker and Wang2018) and Dimmock et al., (Reference Dimmock, Kouwenberg and Wakker2016a) found similar rates of monotonicity violations, as summarized in Sect. 5.4.2.

As we aim to interpret index a as a proxy for perceived ambiguity, from now on we exclude monotonicity violations (a > 1) and negative values of a, using pairwise deletion.Footnote ¹⁹ Panel B of Table 3 shows descriptive statistics for perceived ambiguity. On average, investors perceived less ambiguity about the familiar individual stock (0.64) than toward the foreign index (0.72), the domestic stock index (0.74), and Bitcoin (0.75).Footnote ²⁰ We note that perceived ambiguity about investments is relatively high, at 0.71 on average. For comparison, in Dimmock et al., (Reference Dimmock, Kouwenberg and Wakker2016a), perceived ambiguity toward Ellsberg urns was 0.35 on average.

There are several reasons why perceived ambiguity about investment can be expected to be high in our field study. First, the household finance literature shows that typical retail investors are not sophisticated, lack financial knowledge, and suffer from a host of biases (see Gomes et al., Reference Gomes, Haliassos and Ramadorai2021 for a review). In line with this, even in the group of financial asset owners that we labelled as investors, only 55% could mention a familiar stock name, and less than half owned individual stocks. Second, the probabilities of the one-month investment return events are hard to estimate, as both the mean and standard deviation are time-varying (Lettau & Ludvigson, Reference Lettau, Ludvigson, Ait-Sahalia and Hansen2010), requiring quite a high level of financial sophistication. Finally, our choice lists do not reveal any information about the event probabilities; a respondent who always selects the middle row due to lack of knowledge or attention will end up with an a-index equal to one (100%).Footnote ²¹

Figure 5 shows scatter plots of the relations between perceived ambiguity toward the four financial sources. The correlations between the a-indexes are positive, ranging from 0.35 to 0.55, but lower than correlations between the b-indexes. A factor analysis indicates that the first component accounts for about 60% of the cross-sectional variation in the four measures. This implies that, for a given respondent, the perceived ambiguity toward different investments is related, but not strongly. Hence, the same investor may perceive relatively low ambiguity about a familiar stock, while concurrently perceiving high ambiguity about another investment.Footnote ²²

Fig. 5 Scatter Plots of Perceived Ambiguity about Different Financial Sources. This Figure shows scatter plots of the relation between perceived ambiguity (the a-indexes) for different investments: the local AEX stock market index (a_aex), a familiar company stock (a_stock), the MSCI World stock index (a_msci), and Bitcoin (a_bitcoin). The correlation (r) is shown above each scatter plot, with *, **, *^** denoting significance at the 10%, 5% and 1%, respectively. The original sample consists of n = 295 investors, but values of index a that are negative or larger than 1 are excluded pairwise

4.2 Analysis of heterogeneity in perceived ambiguity

We now analyze the variance in perceived ambiguity, using a similar panel model as above:

(5) $a_{i,s}={\lambda }_1+{\sum }_{s=2}^4({\lambda }_s+v_{i,s}^a)d_s+{\sum }_{k=1}^K{\gamma }_k^a{X}_{i,k}+u_i^a+{\epsilon }_{i,s}^a,i=\text{1, 2,}\cdots \text{,}I,s=1,2,3,4,$

where $a_{i,s}$ is index a (perceived ambiguity) of respondent i toward source s, with $0{\le a}_{i,s}\le 1$ . The constant ${\lambda }_1$ represents perceived ambiguity for the AEX index, whereas the coefficients ${\lambda }_2,{\lambda }_3$ and ${\lambda }_4$ for the familiar stock, MSCI World and Bitcoin represent differences in the mean relative to the AEX. The random effect and the error term are denoted by $u_i^a$ and ${\epsilon }_{i,s}^a$ , respectively, whereas $v_{i,s}^a$ represents random slopes (added if significant based on a likelihood ratio test).

Table 4 shows the estimation results. Model 1 includes only a random effect, capturing individual heterogeneity in perceived ambiguity that is common to the four sources, which explains 44% of the variation. Model 2 shows that investors perceive less ambiguity about the familiar stock, ${\lambda }_2$ = -0.091 (p < 0.001), relative to ${\lambda }_1$ = 0.718 for the other investments on average. The ICC is 0.45, implying that levels of perceived ambiguity toward the four different investments have a moderate positive correlation.

Table 4 Analysis of Heterogeneity in Perceived Ambiguity

	Dependent variable: perceived ambiguity (Index a, between 0 and 1)
	Model 1	Model 2	Model 3	Model 4	Model 5
Constant	0.696***	0.718***	0.721***	0.796***	0.915***
	(0.015)	(0.021)	(0.021)	(0.117)	(0.143)
Dummy familiar stock		− 0.091***	− 0.099***	− 0.102***	− 0.103***
		(0.026)	(0.025)	(0.025)	(0.025)
Dummy MSCI world		− 0.011	− 0.013	− 0.014	− 0.016
		(0.023)	(0.023)	(0.023)	(0.023)
Dummy Bitcoin		0.013	0.014	0.013	0.011
		(0.025)	(0.025)	(0.025)	(0.025)
Education				− 0.041***	− 0.034***
				(0.010)	(0.009)
Age				0.003***	0.002*
				(0.001)	(0.001)
Female				0.019	0.005
				(0.030)	(0.030)
Single				− 0.059*	− 0.045
				(0.032)	(0.030)
Employed				0.027	0.028
				(0.036)	(0.034)
Number of children (log)				− 0.029	− 0.032
				(0.038)	(0.037)
Family income (log)				− 0.019**	− 0.010
				(0.008)	(0.010)
HH Fin. Wealth (log)				0.005	0.007
				(0.006)	(0.006)
HH wealth imputed				0.068	0.069
				(0.049)	(0.055)
Financial literacy					− 0.022**
					(0.009)
Risk aversion					0.041
					(0.030)
Likelihood Insensitivity					0.087***
					(0.031)
Random slope: stock/bitcoin	No	No	Yes	Yes	Yes
Observations n	794	794	794	794	794
ICC of Random effect $u_i^b$	0.44	0.45	0.49	0.44	0.41
$Var\left[{\epsilon }_{i,s}^a\right]$ , Error	0.057 (56%)	0.055 (54%)	0.046 (46%)	0.046 (46%)	0.047 (47%)
$Var\left[u_i^a\right]$ , Random Constant	0.044 (44%)	0.044 (44%)	0.044 (43%)	0.035 (35%)	0.030 (30%)
$Var\left[v_{i,4}^a\right]$ , Slope bitcoin	–	–	0.004 (4%)	0.004 (4%)	0.004 (4%)
$Var\left[v_{i,2}^a\right]$ , Slope stock	–	–	0.005 (5%)	0.005 (5%)	0.004 (4%)
$Var[{\alpha }^{\prime }D+{\gamma }^{\prime }X]$ , Observed	–	0.002 (2%)	0.002 (2%)	0.010 (10%)	0.014 (14%)

The table shows estimation results for the panel regression model in Eq. (5), with index a toward the four investments as dependent variable. Only values of index a between 0 and 1 are included, so index a can be interpreted as perceived ambiguity. Model 1 includes a constant and a random effect for individual-level heterogeneity in perceived ambiguity common to all sources. Model 2 adds dummies for differences in the mean of perceived ambiguity between investments. Model 3 includes a random slope to capture heterogeneity in perceived ambiguity toward the familiar stock and Bitcoin (see Online App. E.2). Model 4 includes observed socio-demographic variables. Model 5 adds financial literacy, risk aversion and likelihood insensitivity. Sample: n = 794 observations of perceived ambiguity (a-index values between 0 and 1), for 295 investors. Standard errors clustered by investor in parentheses.

Next, including random slopes for the familiar stock and Bitcoin in Model 3 leads to a significant improvement of model fit (see Online Appendix E.2). Individual variation in perceived ambiguity toward the familiar stock explains 5% of the total variation, versus 4% for Bitcoin, on top of the 43% captured by general perceived ambiguity about all investments (random constant). Hence, whereas ambiguity aversion is mostly driven by one underlying preference variable, perceived levels of ambiguity tend to differ more depending on the specific source considered.

4.3 Variation in perceived ambiguity explained by individual characteristics

Model 4 adds observed individual socio-demographic variables to the model, which account for 8% of the variation (= 10–2%) in perceived ambiguity. Older investors perceive more ambiguity about investments (p = 0.005), whereas investors with higher education (p < 0.001) and more income (p = 0.026) perceive less ambiguity. Model 5 adds proxies for financial literacy and risk attitudes, which explain an additional 4%. Specifically, investors with better financial literacy perceive less ambiguity (p = 0.011). Further, perceived ambiguity is positively related to index a _r for risk (p = 0.005), a proxy for likelihood insensitivity. All variables together explain 14% of the variation, whereas 38% is driven by unobserved heterogeneity, and 47% is error. Together, the results indicate that measurement reliability for perceived ambiguity is reasonable, although clearly lower than for ambiguity aversion. The likely reason is that index a is measured using differences in matching probabilities for composite events and single events (see Eq. (2)), and therefore more sensitive to errors and violations of monotonicity.

In Online Appendix F we repeat the analyses above using all values of index a, without screening out monotonicity violations and negative values: in that case the ICC is just 0.16 and measurement error accounts for 75% of the variation. The high level of noise implies that unfiltered a-insensitivity is strongly influenced by errors such as violations of monotonicity (a > 1). Further, screening out such violations leads to substantially better reliability for index a.

5.1 Relation with risk preferences, education and financial literacy

We assess the validity of the ambiguity measures by testing whether they relate to other variables in expected ways. For example, a priori we expect that ambiguity aversion is positively related to risk aversion, as that is the most common finding in previous studies summarized by Trautmann and van de Kuilen (Reference Trautmann, van de Kuilen, Keren and Wu2015) and Baillon et al. (Reference Baillon, Schlesinger and van de Kuilen2018c). Similarly, we expect that likelihood insensitivity (overweighting of small probabilities) is positively related to a-insensitivity (overweighting of unlikely events), and thus to perceived ambiguity. The results in Tables 2 and 4 confirm these relations (p < 0.01).Footnote ²³

A priori, we also expect that investors with better financial knowledge and higher education perceive less ambiguity about the distribution of investment returns. Table 4 confirms both of these relations (p < 0.05), suggesting that more knowledge reduces the level of perceived ambiguity.Footnote ²⁴ As an additional test, we also included a dummy for investors who named a specific stock they are familiar with (see Online Appendix G). As expected, investors naming a familiar stock perceive lower ambiguity about it (− 0.182; p < 0.001), compared to those answering questions about Philips.

The competence hypothesis of Heath and Tversky (Reference Heath and Tversky1991) predicts that ambiguity aversion is stronger when the decision maker feels less competent or less knowledgeable about the source, suggesting a negative relation between financial literacy and ambiguity aversion. The coefficient in Table 2 is negative as expected, but the effect is too small to be statistically significant.Footnote ²⁵ Further, investors who named a specific familiar stock also did not display significantly lower ambiguity aversion to it (see Online Appendix G). Still, investors were significantly more ambiguity averse toward the foreign MSCI stock index in Table 2 compared to the domestic AEX index, displaying home bias (French & Poterba, Reference French and Poterba1991). In addition, the difference in ambiguity aversion for the familiar stock (b = 0.156) and the MSCI index (b = 0.210) is 34% in relative terms, a sizeable difference (p = 0.009). All in all, we find that there are strong competence effects in perceived ambiguity (a-insensitivity), but less so in ambiguity aversion.

5.2 The relation to investments

Next, we evaluate whether ambiguity attitudes correlate with actual investment choices. Based on theory, we expect that investments in risky assets are negatively affected by ambiguity aversion (Dow & Werlang Reference Dow and da Costa Werlang1992) and by the level of perceived ambiguity (Boyle et al., Reference Boyle, Garlappi, Uppal and Wang2012; Uppal & Wang, Reference Uppal and Wang2003). One caveat is that these relations also depend on how much ambiguity the investor perceives about all other available investment opportunities considered, for which we lack complete information. Further, many other unknown parameters also affect these relations, such as the investor’s subjective expectations about the return distribution of all available assets (mean, risk, skewness, and cross-correlations) and risk preferences. Although the exact signs are hard to predict, negative relations of index b and a with investments are what we expect to find on aggregate, based on earlier work by Dimmock et al., (Reference Dimmock, Kouwenberg and Wakker2016a).

We estimate a pooled probit model for asset ownership, ${\mathrm{DI}}_{i,s}$ , a dummy variable indicating ownership of the familiar stock (s = 2), the MSCI World index (s = 3), and Bitcoin (s = 4):

(6) ${P[DI}_{i,s}=1]={\mu }_s{d}_s+{{\theta }_{1}b}_{i,s}+{\theta }_2{a}_{i,s}+{\sum }_{k=1}^K{\theta }_{k+2}{X}_{i,k}+{ϵ}_{i,s},i=\text{1, 2,}\cdots ,I;s=2,3,4,$

where indexes $b_{i,s}$ and $a_{i,s}$ have slope coefficients ${\theta }_1$ and ${\theta }_2$ . The constant ${\mu }_2$ represents average ownership of the familiar stock, whereas ${\mu }_3$ and ${\mu }_4$ indicate differences in ownership rates for MSCI World and Bitcoin. Investment in the AEX index (s = 1) is excluded, as noone in our sample invested in a fund tracking the AEX. The model includes K observable individual characteristics $X_{i,k}$ as control variables, with slope coefficients ${\theta }_{k+2}$ , for $k=\text{1, 2,}\cdots ,K$ .

The results in Model 1 of Table 5 show that index a has a negative relation with investing in an asset (p = 0.005). The coefficient of index b is also negative, as expected, but only marginally significant (p = 0.060). As more controls are added in Models 2 and 3, we note that the effect of index $a$ becomes smaller, probably because it is related to financial literacy and education.

Table 5 Investment in the Familiar Stock, MSCI World, and Crypto-Currencies

	Dependent variable: Invests in the Asset (1 = yes, 0 = no)
	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6
Perceived Ambiguity (index a)	− 0.043***	− 0.033**	− 0.029*
	(0.015)	(0.015)	(0.015)
Ambiguity Aversion (index b)	− 0.035*	− 0.020	− 0.028
	(0.018)	(0.019)	(0.022)
Perceived Ambiguity (fitted $\hat{a}$ )				− 0.134***	− 0.096**	− 0.088*
				(0.046)	(0.046)	(0.051)
Ambiguity Aversion (fitted $\hat{b}$ )				− 0.018	− 0.004	− 0.018
				(0.024)	(0.026)	(0.032)
Dummy MSCI World	− 0.241***	− 0.242***	− 0.242***	− 0.212***	− 0.218***	− 0.218***
	(0.029)	(0.028)	(0.028)	(0.035)	(0.033)	(0.033)
Dummy Bitcoin	− 0.204***	− 0.207***	− 0.207***	− 0.167***	− 0.175***	− 0.177***
	(0.024)	(0.023)	(0.022)	(0.028)	(0.026)	(0.026)
Education		0.011	0.008		0.008	0.005
		(0.007)	(0.007)		(0.008)	(0.008)
Age		− 0.000	− 0.000		− 0.000	− 0.000
		(0.001)	(0.001)		(0.001)	(0.001)
Female		− 0.058**	− 0.048**		− 0.062**	− 0.050*
		(0.024)	(0.024)		(0.028)	(0.027)
Single		0.008	0.002		− 0.015	− 0.019
		(0.022)	(0.021)		(0.025)	(0.024)
Employed		0.051**	0.053**		0.059**	0.059**
		(0.025)	(0.024)		(0.027)	(0.026)
Number of Children (log)		0.001	0.003		− 0.025	− 0.020
		(0.025)	(0.025)		(0.031)	(0.031)
Family Income (log)		0.002	− 0.000		− 0.003	− 0.005
		(0.009)	(0.009)		(0.009)	(0.009)
HH Fin. Wealth (log)		0.001	0.000		− 0.001	− 0.002
		(0.004)	(0.003)		(0.004)	(0.004)
HH Wealth Imputed		− 0.063	− 0.060		− 0.054	− 0.042
		(0.047)	(0.047)		(0.052)	(0.050)
Financial Literacy			0.016**			0.015*
			(0.007)			(0.008)
Risk Aversion			0.018			0.019
			(0.024)			(0.029)
Likelihood Insensitivity			− 0.007			0.015
			(0.018)			(0.022)
Observations n	885	885	885	602	602	602
Pseudo R-square	0.267	0.304	0.315	0.245	0.281	0.293

This table reports estimation results for a panel probit model explaining asset ownership with index a and b, see Eq. (6). The dependent variable is 1 if the respondent invests in the asset (familiar individual stock, MSCI World, or Bitcoin), and 0 otherwise. Investment in the AEX index is excluded, as no respondents hold an AEX fund. The data for ownership of the three assets is treated as a panel dataset similar to Tables 2 and 4, see model Eq. (6) in the text for details. The coefficients displayed are estimated marginal effects, with index a and b evaluated at the mean. Standard errors clustered by investor shown in parentheses. In Model 4, 5, and 6 index a and b are replaced by fitted values from the panel regression models in Tables 2 and 4, using specification Model 3 with source dummies and random slopes. Further, only observations with $0\le a\le 1$ are included in Model 4–6, so that fitted a can be interpreted as perceived ambiguity. The set of control variables is the same as in Tables 2 and 4

To reduce the impact of measurement error, Models 4 to 6 of Table 5 use as independent variables the predicted values ${\hat{b}}_{i,s}$ and ${\hat{a}}_{i,s}$ of ambiguity aversion and perceived ambiguity from the estimated panel models in Tables 2 and 4 (Model 3).Footnote ²⁶ Using the predicted values, we effectively remove the error terms ${\hat{\epsilon }}_{i,s}^b$ and ${\hat{\epsilon }}_{i,s}^a$ from index b and a. The sample in Models 4 to 6 is smaller, as it includes only observations with $0\le a_{i,s}\le 1$ , similar to Table 4. The results in Model 4 confirm that investors who perceive more ambiguity about an asset are less likely to invest in it, while ambiguity aversion is not significant.

Online Appendix E.3 shows results for several model specification tests. First, adding a random effect to the panel probit model (6) does not add value, because ownership of different investments is not very correlated. Second, allowing ambiguity aversion and perceived ambiguity to have a different effect on each investment does not improve the model fit either. Online Appendix H.5 also reports estimates using a probit model for each investment separately, as a robustness check. The results show that higher perceived ambiguity is negatively related to investing in MSCI World and Bitcoin, but it is not significant for the familiar stock. Further, investors with higher ambiguity aversion (index b) are less likely to invest in Bitcoin. Overall, these results show that the ambiguity attitude measures have significant correlations with actual investment choices.

5.3 Robustness tests

We performed several additional robustness checks for our main results, reported in Online Appendix H. First, we repeated the main analysis after screening out investors who make mistakes on the ambiguity choice lists, by preferring Option A or B on every row. The main effect is that the mean level of index b drops, as the most common error is selecting the unambiguous Option B on every row of the choice list; this results in high values of index b.Footnote ²⁷ Apart from that, the measurement reliability (ICC), the percentage of variance explained by observable variables, and the correlates of ambiguity attitudes are similar to the full-sample results.

Second, we repeated the main analysis after excluding values of b and a from subjects with many violations of set-inclusion monotonicity. Third, we excluded 30 respondents who spent less than 10 minutes on the ambiguity survey module, to screen out subjects who devoted insufficient attention to the questions. The results in Online Appendix H are similar to the main findings in Tables 2 and 4. Overall, the three robustness checks show that the main results are not driven by respondents who made many mistakes, or who spent little time on the ambiguity questions.

5.4 Cross-sample comparisons