Retirement Expectations under Ambiguity
Retirement Expectations under Ambiguity Lars Kleinhuis – June 5, 2020
A thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Economics from the University of Groningen. Master’s Thesis Economics (EBM877A20)
JEL codes: D81, D14, G11, J26, H55
Keywords: ambiguity, ambiguity aversion, uncertainty, retirement expectations, retirement planning
“The quest for certainty blocks the search for meaning.
Uncertainty is the very condition to impel man to unfold his powers.” – Erich Fromm (1947)
Abstract
This thesis empirically estimates the role of attitudes towards ambiguity in the formation of retirement expectations in the Netherlands. Ambiguity aversion (a general distaste for uncertainty in cases of unknown probabilities, relative to cases with known probabilities) and a-insensitivity (the overweighing of extreme events by treating them as fifty-fifty chances) are shown to capture distinct preferences unexplained by demographics. We find that a-insensitive women are less likely to know their expected retirement age. For full
a-insensitivity, this is about ten percentage points. Ambiguity averse women are more
likely to expect a longer working life than desired when faced with uncertainty surrounding
pension system reforms. a-insensitive women do not adjust this expectation. Mens’
ambiguity attitudes show no relation with their willingness or capability to form an expectation regarding their expected retirement age or with their expectation to work longer than desired. In response to increases in the state pension age, extremely ambiguity averse men do not adjust their expected retirement age, but this finding is not robust. a-insensitive men adjust their expected retirement age more than non-a-insensitive men, in
response to the same increase in state pension age. In all cases, the effects of
a-insensitivity is driven by an aversion to ambiguity for high a-neutral probabilities of winning. These effects can thus be seen as due to an overweighing of extreme events with bad outcomes (becoming disabled, etc.).
Author Supervisor
Lars B. Kleinhuis Prof. Dr. Rob J.M. Alessie
L.B.Kleinhuis@student.rug.nl (student) Department of Economics,
LBKleinhuis@gmail.com (personal) Econometrics, and Finance
S3109771 University of Groningen
1
Introduction
Economic models predominantly stylise uncertainty in the form of a game of roulette. The probability of the ball landing on any number or colour can be computed mathematically. The exact probabilities are assumed to be known but this is not true for most actual decision making. Real life situations are more like horse race lotteries; no one truly knows the exact probability
that a certain horse will win the race. More knowledge about the horses will increase the
precision of this subjective probability, but all factors can never be taken into account fully. Cases of such horse race lotteries are called Knightian uncertainty or ambiguity (we will stick with the latter). Different from the roulette case, i.e. risk, subjectivity plays a central role in ambiguity. Given the prominence of uncertainty in our lives, it is vital to know if, how, and to what extent ambiguity impacts our capabilities to plan, decide, and form expectations.
Adequately planning for retirement can be a tedious and cumbersome task which is made more difficult when faced with uncertainty about e.g. pension income, retirement benefits, or the state pension age, to name but a few. This thesis delves into the implications of ambiguity for the formation of expectations about retirement and is one of the first to use ambiguity attitudes in a panel setting. It is, to our knowledge, also one of the first to consider the relationship of ambiguity and retirement (expectations) in an empirical setting.
There is substantial theoretical literature surrounding the topic of ambiguity but surprisingly few empirical works. Relatively recent research efforts (Abdellaoui, Baillon, Placido, & Wakker,
2011;Abdellaoui, Bleichrodt, l’Haridon, & Van Dolder, 2016) resulted in tractable methods to empirically determine attitudes towards ambiguity providing a plethora of potentially fruitful empirical applications. This thesis is one such as we employ measures from a survey by Dimmock,
Kouwenberg, and Wakker (2016) which elicits ambiguity attitudes for a large sample that is
representative of the Dutch population.
The measures for ambiguity aversion and a-insensitivity of Dimmock et al. (2016) can
characterise the full scope of (local) ambiguity attitudes. Ambiguity aversion is relate to a
general dislike of uncertainty, relative to cases of risk. In other words, ambiguity aversion
captures a preference of roulette over horse race lotteries. a-insensitivity is the general tendency to overweigh extreme events, i.e. with a low probability. a-insensitivity is related to optimism surrounding long-shot good outcomes, such as winning a lottery, and very rare bad outcomes, such as a plane crash.
Ambiguity aversion and a-insensitivity are found to capture distinct preferences that cannot
explained by demographics. Risk aversion and financial literacy are significantly related to
a-insensitivity but have limited predictive power. Risk aversion is also related to ambiguity aversion but, again, does not predict it well. Education, income, and age differences all fail in predicting these attitudes. Most peculiarly is the lack of an observational relationship between dispositional optimism and ambiguity aversion, which are shown to be related in theory and experiments.
expectations. We find that, overall, a-insensitive women are more likely to not know their expected retirement age. They forgo planning for retirement out of a fear of rare bad outcomes (with a low a-neutral probability), such as becoming disabled, dying, etc., which would have
serious implications for one’s retirement. Men’s inability or unwillingness to for retirement
expectations is unaffected by their attitude towards ambiguity.
Next, we use a period of heavy pension reform in the Netherlands (2011-2015) as a source of uncertainty surrounding retirement. Adverse responses to this period are estimated by ambiguity aversion and a-insensitivity using a difference-in-differences approach. For both men and women, the period increased the probability of not knowing the expected retirement age but this effect was not differential for a-insensitive or ambiguity averse individuals. We do find that ambiguity averse women were more fearful of having to work longer than desired during the period of reform, compared to their ambiguity neutral counterparts (who did also became more likely to have this expectation). Simultaneously, a-insensitive women did not adjust this expectation, or even saw a decrease in the expectation to work longer than desired. For men, there were no differential effects across ambiguity attitudes.
If we zoom in on the effects of one policy reform in particular, the increasing of the state pension age, we do find a small differential effect for ambiguity averse and a-insensitive men. Men that exhibit extreme ambiguity aversion do not appear to adjust their expected retirement age in response to raises in the state pension age whereas their ambiguity neutral counterparts do adjust. This differential effect is not very robust. In response to a one year increase in the state pension age, men with low a-insensitivity raise their expected working life by about seven months versus a full year for a-insensitive men. This too is caused by an overweighing of rare bad outcomes (with a low a-neutral probability), similar as before.
Our findings are robust to different specifications of the ambiguity attitudes, controlling for clarity and difficulty of the questions in the ambiguity, and allowing for endogenous selection. They do appear to be driven mostly by individuals without a tertiary education which hints at the omission of an important confounding factor: probabilistic numeracy which may both affect interpretations of the ambiguity survey and have implications for the formation of retirement expectations. Other confounding factors or simultaneity between the ability to form a retirement expectation and attitude towards uncertainty cannot be ruled out.
This thesis is certainly not without its flaws. Next to the omission of probabilistic numeracy,
we make several assumptions that are not without scrutiny. We assume time-invariance of
ambiguity attitudes; we assume ambiguity attitudes are identical for gains and losses (and sizes); we assume no correlation between perceived ambiguity and attitudes towards it. All of these are subject to debate, both from theoretical evidence and experimental, but nevertheless necessary for our purposes.
appear to have adverse effects on the retirement age expectation so men with non-neutral
attitudes towards ambiguity. The costs of wrongly or failingly planning for retirement are
substantial and real. Adequate retirement planning can prevent drops in consumption around retirement, having to work during retirement, and successful retirement in general. Successful planning is more likely when policy uncertainty is reduced.
We still have much to learn about ambiguity and its role in retirement planning. Future research would benefit greatly from obtaining ambiguity attitudes of the same individuals over time to check for time-variance, the relationship to perceived ambiguity, and source dependancy of these ambiguity attitudes. Distinguishing ambiguity driven saving motives is also of interest. This thesis is organised as follows. Section 2 will provide an introduction to the theoretical literature concerning ambiguity. This will be accompanied by a discussion of notable approaches of incorporating ambiguity into utility functions. The notion of ambiguity prudence will be introduced in section 3 based on a model by Berger (2014). These two sections together provide a thorough treatise of decision-theoretic modelling under ambiguity.1 Findings of the limited amount of empirical studies will be discussed in section 4.
We then turn to the data in section 5 and see to what extent we can predict ambiguity attitudes in section 6. Our empirical strategy and some hypotheses are laid out in section 7 followed by the presentation of our results in section 8. Subsequently, we take a closer look at retirement age expectations, ambiguity attitudes, and state pension age increases in section 9.
What remains is to conclude by means of a discussion of the results and their policy implications in section 10 and our final words in section 11.
2
A Brief Review of Decision Making under Ambiguity
Savage’s (1954) theory of subjective expected utility relies on several axioms. One of these is the Sure-Thing Principle (P2). Savage explains this axiom in the following way:
A businessman contemplates buying a certain piece of property. He considers the outcome of the next presidential election relevant. So, to clarify the matter to himself, he asks whether he would buy if he knew that the Democratic candidate were going to win, and decides that he would. Similarly, he considers whether he would buy if he knew that the Republican candidate were going to win, and again finds that he would. Seeing that he would buy in either event, he decides that he should buy, even though he does not know which event obtains, or will obtain, as we would ordinarily say. (Savage,1954, p. 21)
Put in other words, if an act f is preferred over another act g both when an event E occurs and when it does not, the Sure-Thing Principle states that the preference should be the same when the probability of E is unknown.
1Section 2 and 3 can be skipped by readers not particularly interested in the decision-theoretic predictions
Table 1: The Ellsberg Paradox A B C f 10 0 0 g 0 10 0 f0 10 0 10 g0 0 10 10
Source: Klibanoff, Marinacci, and Mukerji (2005, Table 1)
Seven years later, Ellsberg (1961) constructed an example, known as the Ellsberg Paradox, in which the Sure-Thing Principle is violated. Table 1 displays such a case. It shows the pay-offs of four acts (f , g, f0, and g0) contingent on three events (A, B, and C). These events are mutually exclusive and exhaustive. Savage’s Sure-Thing Principle implies that if f is preferred over g, then f0 should be preferred over g0 because the payoffs in case of event C are equal and therefore should not be considered.
Now, suppose the probability of event A (πA) is known to be 13. πB and πC are unknown
but the decision maker deductively knows that πB+ πC = 23. This gives rise to the following
expected utilities for each act:
E {u(f )} = 13u(10) 13u(10) ≤ Eu(f0) ≤ u(10)
0 ≤ E {u(g)} ≤ 23u(10) Eu(g0) = 23u(10).
(2.1)
The decision maker now assigns preferences based on how she assigns subjective probabilities. She prefers f over g out of fear of wrongly assigning a probability πB exceeding 13. Similarly,
she argues prefers g0 over f0 out of fear of wrongly assigning a too high of event C. This is a direct contradiction of the Sure-Thing Principle but a behaviourally justifiable set of preferences. Also note that risk aversion alone cannot explain this as the choice is between two equally risky options.
The above indicates the preferences of an ambiguity (or uncertainty) averse individual. A decision maker who is ambiguity averse prefers acts whose pay-offs depend less on the assignment of subjective probabilities (Klibanoff et al.,2005, p. 1852). She would rather face known than unknown probabilities.
A decision maker who has the preferences f ∼ g and f0 ∼ g0, where ∼ denotes indifference, is
extremely a-insensitive (ambiguity-generated likelihood insensitivity).2 “a-insensitivity captures the limited ability of the decision-maker to discriminate between likelihood levels.” (Baillon,
2017, p. 1742) In our case, the decision maker acts as if πB = πC = 13, i.e. as if all unknown
probabilities are equal.
These preferences imply an overweighing of small probabilities and an underweighing of large probabilities. In other words, this decision maker exhibits ambiguity seeking at low probabilities
2
and ambiguity aversion at high probabilities.
2.1 Alternatives to Expected Utility Theory
The previous subsection shows that decision-making is differs based on whether probabilities are known or unknown. The remainder of this section discusses theories that model the above behaviour by explicitly incorporating ambiguity attitudes.
First, a brief recap of the standard expected utility with subjective probabilities. Assume preferences over outcomes are state-independent and can be represented by a Von Neumann-Morgenstern utility function u : X → R where X denotes a set of all possible consequences (with x ∈ X). Beliefs are formed about the likelihood of an event E ⊆ S where S denotes the state space (with s ∈ S). Preferences over actions f (with f ∈ F and F = {f |f : S → X}) can be represented by the expected utility function
U (f ) = Z
u(f (s)) dπ(s). (2.2)
Here, utility is maximised based on subjective probabilities π(s) which are functions of the state of the world.
The Ellsberg paradox in the previous section shows that one of the axioms of underlying expected utility theory with subjective probabilites (the Sure-Thing Principle) fails when faced
with ambiguity aversion. The Choquet expected utility from Schmeidler (1989) weakens this
axiom. It takes the form
U (f ) = Z
s∈S
u(f (s)) dν(s). (2.3)
Two important adjustments are made. First, the use of capacities ν(s) allows confidence in subjective probabilities to be taken into account. To illustrate, a capacity based on a probability distribution π(s) can be given by
ν(E) = (1 − γ)π(E) + γν0(E) (2.4)
where ν0(E) = 0 (∀ E " S) implies complete ignorance regarding events E. 1 − γ captures
the degree of confidence in π.3 Second, a standard Riemann integral is inappropriate because
capacities are not continuous. Hence, the use of a Choquet integral and the name of this type of expected utility.
Another approach is the Max-Min expected utility from Gilboa and Schmeidler (1989). It
essentially allows for multiple possible probability distributions.
Actions characterised as horse-race lotteries h ∈ H each of which contains a mutually
exclusive and exhaustive set of outcomes with unknown probabilities. Much like an actual
horse race, the decision maker assigns subjective probabilities based on available information
(previous performances of the horse, quality of the rider, etc.) which is not sufficient to
construct objective probabilities (a tug of wind cannot be accounted for). Anscombe and
3An extension of this includes ν1
Aumann (1963) contrast this with a roulette lottery from which the probability of each outcome can be fully calculated (if the roulette wheel is well made).4
Let A denote a unique, compact, and convex set of probability distributions on S. Preferences over horse-race lotteries h are represented by the utility function
U (h) = min π∈A X s∈S π(s) X x∈supp(hs) hs(x)u(x) . (2.5)
In words, actions are determined by the maximisation of the utility of the probability distribution π ∈ A that provides the lowest utility. Reversely, the Max-Max expected utility is represented by the utility function
U (h) = max π∈A X s∈S π(s) X x∈supp(hs) hs(x)u(x) . (2.6)
In words, actions are determined by the maximisation of the utility based on the probability distribution π ∈ A that provides the highest utility. Notice that under Max-Min, the agent is extremely pessimistic as she expects the worst. Under Max-Max, the agent is extremely optimistic.
A linear combination of the Max-Min and Max-Max utility function can be constructed which allows a degree of pessimism (or optimism) that is not in the extremes. This function is
U (h) =α min π∈A X s∈S π(s) X x∈supp(hs) hs(x)u(x) + (1 − α) max π∈A X s∈S π(s) X x∈supp(hs) hs(x)u(x) . (2.7)
In this utility function, α captures the degree of pessimism and 1 − α captures the degree of optimism. It is called α-Max-Min expected utility due to this incorporation of the degree of pessimism/optimism. In this specification, pessimism is analogous to ambiguity aversion. When faced with uncertainty, the decision maker feels it is unlikely to produce a beneficial outcome. A typically measures the degree of ambiguity.
A third way of incorporating ambiguity attitudes is constructed by Klibanoff, Marinacci, and
Mukerji (2005) (Henceforth KMM). Compared to the Choquet and Max-Min expected utility
functions, the utility function of KMM (2005) has the huge advantage of being differentiable (because it has no kinks). The KMM utility function is defined as
V (f ) = Z ∆ φ Z S u(f ) dπ dµ ≡ Eµφ (E {u(c)}) (2.8)
which can also be written in a double expectational form (right-most term). ∆ denotes the set of possible probabilities π over S and µ is the decision makers subjective prior over ∆ (π ∈ ∆).
4
µ captures the measures the subjective relevance of a particular π as the “right” probability. It can be seen as a second-order probability over first-order probabilities π.
KMM (2005) represent ambiguity attitudes by the function φ. A concave (convex) φ, implies ambiguity aversion (loving). The Von Neumann-Morgenstern utility function u characterises standard risk preferences in the absence of ambiguity. A distinct feature of this representation is that for an ambiguity neutral decision maker returns to the subjective expected utility form of Savage (1954). This is because the function φ is linear for an ambiguity neutral decision maker. The same holds if the decision maker is completely certain.
The KMM utility function allows for a clear distinction between acts. First-order acts involve ambiguity and order (or Savage) acts do not involve ambiguity. Also note that for second-order acts, Equation 2.8 reduces to the standard expected utility function. Because the KMM representation contains a smooth ambiguity function (unlike Choquet and Max-Min) it is often called the smooth expected utility function (under ambiguity). This smooth nature also allows for differentiation whereas the others do not.5
3
The Notion of Ambiguity Prudence
KMM have extended the utility function of Equation 2.8 in a subsequent paper (KMM, 2009)
to facilitate intertemporal decision making. This recursive utility representation allows for the notion of ambiguity prudence to be explored. It allows for dynamically consistent preferences while maintaining the benefits of the original KMM utility function (differentiability, distinction between ambiguity and ambiguity attitudes). The recursive representation takes the form
Vst(f ) = u f (st) + βφ−1 " Z Θ φ Z χt+1 Vst,x t+1(f ) dπθ(xt+1|s t) ! dµ(θ|st) # (3.1)
where Vst(f ) is the recursively defined value function. As before, u denotes a Von Neumann-Morgenstern utility function that is additively time-separable. β is the discount factor. Θ is a finite parameter space capturing the domain of uncertainty. χt+1consists of possible observations
just before time t + 1. xt+1 denotes a realisation of the random variable χt+1 (xt+1 ∈ χt+1).
θ is an element of Θ and vector of parameters exhaustively describing a particular stochastic process πθ. πθ(xt+1|st) denotes the probability distribution that the next observation will be xt+1
conditional on reaching the decision tree node st.6 µ described the decision makers subjective prior belief about the stochastic process πθ. µ(·|st) denotes the decision maker’s subjective
5
To my knowledge, there are no theories in which subjective probabilities are themselves set to maximise utility. All the utility functions mentioned here presume the beliefs about subjective probabilities are formed with the goal of forming a representation of reality. The representation is then used as a given in utility maximisation. It could be that belief formation itself is part of the control space used to maximise utility. An individual may form beliefs about subjective probabilities that best fit with her worldview. Models explicitly exploring this possibility could be a source of fruitful future research.
6
Bayesian posterior distribution about which stochastic process applies at st.7
The representation of KMM (2009) in Equation 3.1 has multiple advantageous characteristics. First, it is differentiable. Second, it contains a distinction between attitudes toward ambiguity and (perceived) levels of ambiguity. It also maintains the typical representations of u under uncertainty with known probabilities allowing risk attitudes to have a clear distinction from ambiguity attitudes whilst conserving standard theories regarding risk.
Berger (2014) used the representation of KMM (2009) to show that ambiguity prudence, like its risk counterpart, can increase saving when future income is ambiguous. To illustrate, Berger (2014) considers a stylised two-period model. First period income is given by y1 and second
period income follows a distribution ˜y2(θ) depending on a parameter θ for which the agent has
prior beliefs. The agent sets consumption c1 and implicitly saving s = y1 − c1 to maximise
lifetime utility. In the second period all wealth (˜y2(˜θ) + (1 + r)s) is consumed (i.e. there is no
bequest motive). The recursive representation of Equation 3.1 thus takes the form (in double expectations) max s u (y1− s) + βφ −1h Eθφ E n u(˜y2(˜θ) + (1 + r)s) oi . (3.2)
r denotes the interest rate. Eθ is the expectational operator over the distribution of θ. As before,
the function φ captures ambiguity attitudes. A concave (convex) φ implies ambiguity aversion (seeking).
For an ambiguity averse individual, the certainty equivalent is lower than an ambiguity neutral agent (with a linear φ). More formally,
φ−1 h Eθφ E n u(˜y2(˜θ) + (1 + r)s) oi ≤ EθE n u(˜y2(˜θ) + (1 + r)s) o . (3.3)
An ambiguity neutral agent (with linear φ) produces expectations over the distribution of θ so that each possible second period income ys,2 is assigned a probability ¯ps = Eθps(˜θ). This
probability, though subjective, is then treated as given just as in Savage’s (1954) expected utility framework.8
In the case of a non-linear φ, the FOC of Equation 3.2 yields (Berger,2014, Eq. 4)
−u0(y1− s) + (1 + r)β Eθφ0 E n u(y2+ ˜z(˜θ) + (1 + r)s) o E n u0(y2+ ˜z(˜θ) + (1 + r)s) o φ0nφ−1h Eθφ0 E n u(y2+ ˜z(˜θ) + (1 + r)s) oio (3.4)
where Berger defines ˜y2(θ) = y2+ ˜z(θ) to separate second period income into an expected part
EθE˜y2(˜θ) = y2 and an ambiguous part ˜z(˜θ). Ambiguity aversion increases optimal saving if the
7
In the non-dynamic setting of KMM (2005), this subjective prior can be taken as a given but in a dynamic setting subjective priors are updated as new information becomes available. A structural model needs to account for this and include a method by which beliefs are updated (often Bayesian updating is used) which is why structural modelling is not as straightforward as in the case of uncertainty with known probabilities. For an example with a Choquet expected utility function, see Groneck, Ludwig, and Zimper (2016).
8Optimal saving is then given by u0
(y1− s∗) = (1 + r)βEθE n
u0(˜y2(˜θ) + (1 + r)s∗)
o
. The right-hand term is equal to Enu0(˜y2+ (1 + r)s∗)
o
left-hand side of Equation 3.4 is positive when computed with the ambiguity neutral optimal saving s∗. (The reverse holds for ambiguity loving agents.) This is the case iff (Berger, 2014, Eq. 5) Eθφ0 E n u(y2+ ˜z(˜θ) + (1 + r)s∗) o E n u0(y2+ ˜z(˜θ) + (1 + r)s∗) o φ0 n φ−1 h Eθφ0 E n u(y2+ ˜z(˜θ) + (1 + r)s∗) oio ≥ EθE n y2+ ˜z(˜θ) + (1 + r)s∗ o . (3.5) Do note that s∗ denotes the optimal saving under a linear φ or ambiguity neutral agent. Berger dubs the individual ambiguity prudent if Equation 3.5 holds. The formal definition of ambiguity prudence is given by Berger:
An agent is ambiguity prudent if the introduction of ambiguity through a mean-preserving spread in the space of conditional second period expected utility raises his optimal level of saving. (Berger,2014, p. 249)
This definition is the logical conclusion of the condition in Equation 3.5.
Berger’s (2014) definition is problematic as it is a derivation of this specific model. In different specifications or cases, Equation 3.5 does not appear, at least not in the same form. Berger’s definition also describes a finding (ambiguity prudent if introducing ambiguity raises saving) which is not useful from an empirical or causal viewpoint. Is the agent ambiguity prudent because she saves more when faced with ambiguity or does the agent save more when faced with ambiguity because she is ambiguity prudent? Berger’s definition does not claim either. We would argue it should be the latter. Berger’s definition does not allow for ex ante hypotheses regarding the saving behaviour of ambiguity prudent individuals as the characterisation of prudence itself hinges on an increase in saving when introducing ambiguity.
A more general definition of ambiguity prudence is given by Baillon:
Ambiguity prudent decision-makers do not like bearing two harms simultaneously (exactly like risk prudent decision-makers) but these two harms are a decrease in the probability of a good outcome and additional ambiguity (instead of a loss and additional risk as in the risk prudence definition). (Baillon,2017, p. 1732)
Baillon likens ambiguity prudence to its risk counterpart. A risk prudent decision maker prefers a riskless act with an additional cost over a risky act that has the same expected outcome as the act without additional cost. Baillon’s definition of ambiguity prudence applies the same logic but to probabilities instead of outcomes. An ambiguity prudent decision maker prefers an unambiguous act with a lower probability of a winning outcome (i.e., a probability cost) over the same act with an ambiguous winning probability. This definition, unlike Berger’s (2014), allows ex ante predictions and applications to other settings.
Proposition 1. Non-increasing absolute ambiguity aversion is a sufficient condition for ambiguity prudence in any of the following situations:
(i) the decision maker is risk neutral,
(ii) the decision maker has a Von Neumann-Morgenstern utility function of the constant absolute risk aversion (CARA) form,
(iii) the decision maker is risk averse and there are only two states of the world, (iv) the decision maker is risk prudent and ambiguity is concentrated on a particular
state i.
In the last part (iv) of Proposition 1, concentration of ambiguity on a particular state of the world i refers to a situation in which: state i is ambiguous, and conditional to the information that the state i does not occur, the distribution of final wealth is unambiguous. (Berger,2014, p. 250)
Proofs can be found in the original paper.
Berger’s (2014) theoretical findings regarding ambiguity prudence have a several implications. First, ambiguity and risk prudence represent two distinct preferences which are non-excludable; ambiguity prudence does not imply risk prudence and vice versa. Second, ambiguity prudence is enough to facilitate a precautionary saving motive by itself without requiring risk prudence (u000 = 0). Risk neutral agents may therefore still save more when faced with ambiguous future income if they are ambiguity prudent. Third, precautionary saving can simultaneously be driven by risk and ambiguity prudence. The size of each and effects on one another are unclear. Also note that Berger’s analysis relies on ambiguity attitudes and makes no statements regarding levels of perceived ambiguity. His findings therefore only require the presence of ambiguity but make no claims regarding its extent.9
Osaki and Schlesinger (2014) add to the findings of Berger (2014). They find that u000> 0 and φ000> 0 together are not enough to induce precautionary saving. Only under CAAA, this feature disappears. Whereas in the time-separable models of Berger (2014) and Osaki and Schlesinger (2014) find that φ000> 0 by itself is does not necessarily imply ambiguity prudence, Peter (2019) argues that it is not even necessary in a non-separable version of the model.
Gierlinger and Gollier (2015) apply the Max-Min expected utility and find that more
perceived ambiguity does not always induces additional saving. Theoretical intertemporal
decision making models using the Choquet or (α-)Max-Min representations are quite sparse.
Peter and Toquebeuf (2019) use the Choquet representation and find that ambiguity aversion
alone is enough increase the demand for saving. Moreover, saving demand increases when more ambiguity is perceived.10
9This is a rather common feature in models using the KMM expected utility representation.
10For a more comprehensive review of the precautionary saving motive, we recommend Trautmann and Van
The previous two paragraphs indicate that intertemporal models with ambiguity are not set in stone. The interplay between risk and ambiguity prudence is not theoretically obvious. There is no clarity on their relative importance or to what extent either is responsible for precautionary saving. The possibilities of risk loving and ambiguity aversion (or the reverse) is also not entertained. Comparisons of different expected utility representations in a life-cycle setting are also scarce. As Peter (2019, p. 250) ironically concludes: “This calls for a horse race between different intertemporal models of decision-making under ambiguity.”
4
Empirical Findings
Though theory provides some testable predictions, empirical studies are unbelievable rare. This is partly because of a lack of straightforward measures of ambiguity attitudes11 and the source dependency of ambiguity. Empirically estimating the relevance of ambiguity is therefore more demanding from the data and, ideally, requires information specific to the situation of interest. Regardless, some important findings have to be noted. Baillon, Schlesinger and Van de Kuilen
(2018) show in an experiment among students that ambiguity prudence12 is a common finding
and that ambiguity prudence has significant correlation with risk prudence. They also find more ambiguity seeking for low probability events and ambiguity aversion for high probability events. This is what is referred to as ambiguity generated likelihood insensitivity (a-insensitivity) where small and large probabilities are over- and underweighed, respectively (Abdellaoui et al.,2011;
Dimmock, Kouwenberg, & Wakker,2016).
Baillon and Placido (2019) test characterisations of ambiguity preferences using an
experiment on the individual level. The assumption of CAAA is valid for about 40 percent of individuals in their sample. A similar proportion can be characterised as DAAA and half the subjects exhibit DRAA. Important for our purposes, assuming CAAA for the whole population can lead to errors in the estimation of preferences of up to 10 percent.
Baillon, Bleichrodt, Keskin, l’Haridon, and Lin (2018) find that behaviour moves towards subjective expected utility as more information becomes available, i.e. ambiguity attitudes
become less important. They also find that at full information, there are still significant
deviations from expected utility. Even at full information, ambiguity attitudes can help
explain behaviour that subjective expected utility cannot by itself. Baillon, Huang, Selim, and Wakker (2018) add by finding that ambiguity aversion is time invariant but a-insensitivity not; time pressure increases a-insensitivity. The effects of ambiguity attitudes do thus depend on information and time pressures.
The above are all experimental studies into attitudes and preferences regarding ambiguity but none empirically investigates the implications of these attitudes for behaviours. There is a limited amount of empirical studies that do this. a-insensitivity is found to be negatively 11Wakker (2010, p. 327) argues that because uncertainty is a far more complex domain than risk and trying
to capture attitudes in a single measure is “crude.”
12
related to stock market participation and private business ownership in a representative Dutch sample whereas ambiguity aversion had no effects (Dimmock, Kouwenberg, & Wakker, 2016). For a sample of Dutch investors, ambiguity aversion can be described by one measure across
different types of investments but perceived ambiguity cannot (Anantanasuwong, Kouwenberg,
Mitchell, & Peijnenberg,2019). The same study finds that investments which are perceived to be surrounded by a lot of ambiguity are less likely to be owned. In the US, ambiguity averse individuals are less likely to participate in the stock market (Dimmock, Kouwenberg, Mitchell, & Peijnenburg, 2016) and a measure similar to a-insensitivity is positively related to portfolio underdiversification (Dimmock, Kouwenberg, Mitchell, & Peijnenburg,2018;Bianchi & Tallon,
2019).
The above shows that the importance of ambiguity attitudes in financial behaviour is being documented in recent years. Meanwhile, there are virtually no empirical studies into the role of ambiguity and ambiguity attitudes for (precautionary) saving and retirement.
5
Data
In our empirical analysis, we make use of data of the LISS (Longitudinal Internet Studies for the Social sciences) panel administered by CentERdata (Tilburg University, The Netherlands). The LISS panel is a representative sample of Dutch individuals who participate in monthly Internet surveys. The panel is based on a true probability sample of households drawn from the population register. Households that could not otherwise participate are provided with a computer and Internet connection.
The following subsection elaborates on the measurement of ambiguity attitudes. Afterwards some descriptive statistics and the sample are discussed. Lastly, the remaining control variables are explained.
5.1 Measuring Ambiguity Attitudes
We use the indices of ambiguity attitudes of Dimmock et al. (2016) constructed using the
approach introduced by Abdellaoui, Baillon, Placido, and Wakker (2011). Dimmock et al. (2016) constructed a questionnaire regarding Ellsberg type behaviour and confronted participants of the LISS panel with it. The questions considered preferences between lotteries with know and unknown probabilities.
probability of winning in the urn K is increased. These iterations are competed up to six iterations or until the subject is indifferent between the two.
The questionnaire is set up so that it finds the probability X of drawing a winning ball in the known urn that makes the participant indifferent between the known and unknown urn. If the maximum number of iterations is exceeded, X is set as the midpoint between the upper and lower bound of the subjects choices. These upper and lower bounds are based on the choices made by the subject. If the known urn is preferred in Figure 1, the upper bound is updated to be 50. If the subject then prefers the unknown urn in Figure 2, the lower bound is updated to be 25.
Using this estimated probability X (in percentages), the matching probability attached to the questions depicted in Figure 1 is constructed as follows (Dimmock, Kouwenberg, & Wakker,
2016, Eq. 5)
m(0.5) = X/100. (5.1)
Because one out of two colours is a winning colour, an ambiguity neutral individual has a matching probability of 0.5. This is because she assigns the subjective probability of drawing the winning ball from the unknown urn at 0.5 and sets the matching probability equal to this subjective probability because she has no distaste for ambiguity. She is therefore indifferent between a lottery from the known urn with 50% chance of winning and the lottery from the unknown urn. Matching probability’s are often called probability equivalents to resemble their risk counter parts (certainty equivalents).
The same procedure is done to construct two other probabilities. One questions considers an urn with ten different colours of balls, one of which is a winning colours. This produces the matching probability m(0.1) for which an ambiguity neutral individual has a value of 0.1. Another questions considers a case where nine out of ten colours are winning. The resulting matching probability m(0.9) equals 0.9 for an ambiguity neutral individual. The exact questions for these can be found in Appendix A.1.
Dimmock et al. (2016) use the matching probabilities to compute event-specific ambiguity
indices ESAp. Each shows how much “local” ambiguity aversion the individual has when
choosing between the known and unknown urn with ambiguity neutral probability p. For the three derived matching probabilities, this implies (Dimmock, Kouwenberg, & Wakker, 2016, Eq. 6, 7, and 8)
ESA0.1= 0.1 − m(0.1); ESA0.5= 0.5 − m(0.5); ESA0.9= 0.9 − m(0.9). (5.2)
Under ambiguity aversion (seeking), these indices are positive (negative).
Ambiguity-insensitivity implies a negative ESA0.1 and a positive ESA0.9. A positive ESA0.1
implies an overweighing of low a-neutral probabilities with a good outcome and an
underweighing of high a-neutral probabilities with a bad outcome. The reverse holds for
ESA0.9; a positive ESA0.9 implies an underweighing of high a-neutral probabilities with a
Figure 1: Screenshot of Question 1 regarding a-Neutral Probability of 0.5
Source: Dimmock, Kouwenberg, and Wakker (2016, Fig. 1).
Figure 2: Screenshot of Question 1 regarding a-Neutral Probability of 0.5 after Choice K
The matching probabilities are also used to calculate global ambiguity attitudes. To do this Dimmock et al. (2016) fit a linear line through the matching probabilities using OLS for each subject separately. The model fitted is (Dimmock, Kouwenberg, & Wakker,2016, Eq. 9)
m(p) 7→ c + s × p (5.3)
which is fitted individually using the three matching probabilities. The slope s is used to
define ambiguity generated likelihood insensitivity (a-insensitivity) (Dimmock, Kouwenberg, & Wakker,2016, Eq. 10)
a = 1 − s. (5.4)
a-insensitivity captures to what extent the agent discriminates between different levels of
ambiguity.13 A positive a implies an overweighing of small and underweighing of large
probabilities. Instead of considering the how many different colours of balls are in the urn, the agent tends toward a fifty-fifty chance; she assigns equal probabilities to winning and losing. Given the empirical similarity of risk insensitivity and prudence, this measure could seen as similar to ambiguity prudence (Baillon,2017).
a-insensitivity is related to a negative ESA0.1 and a positive ESA0.9. This is because
at low a-neutral probabilities of the good outcome, an a-insensitive individual overweighs the probability. At a high a-neutral probability of a good outcome, she underweighs the probability of a good outcome.
Next, the index of ambiguity aversion is computed as follows (Dimmock, Kouwenberg, &
Wakker,2016, Eq. 11)
b = 1 − s − 2c. (5.5)
Alternatively, if we set d = 1 − c − s as the distance of the fitted regression line from point p = 1, we have b = d − c. A positive (negative) b implies ambiguity aversion (seeking). In this setting, ambiguity aversion is analogous to pessimism; the subject tends to assign a (relatively) lower subjective probability to good outcomes. a and b are used as our main measures of ambiguity attitudes.
The two derived indices are sufficient to capture the characteristics of the subjects’ source
functions. These functions are used in the analysis of non-expected utility and capture
interactions between tastes and beliefs (Abdellaoui et al.,2011). As the name suggests, source functions typically depend on the source of uncertainty which in our case is the ambiguous urn.14 Dimmock et al. (2016, §3.2) provide a formal decision-theoretic foundation for the use of matching functions (which we estimated). The essence of their argument is that matching functions capture differences in a subject’s attitude towards known and unknown probabilities. For our purposes, it is important to clarify that applying these measures to different setting entails the vital assumption that the source function is similar, if not identical, across different settings. At the very least, we need to assume that the attitudes are applicable to our setting.
13See Appendix A.2 exact derivations.
14When probabilities are known, the source function reduces to a standard probability weighting function
Figure 3: Matching Functions and the Ambiguity Aversion (b) and a-Insensitivity (a) Indices
Source: Dimmock, Kouwenberg, and Wakker (2016, Fig. 3). (Figure is also found in Abdellaoui et al.,2011, Fig. 2)
To get a grasp of the intuition behind the ambiguity measures, it is useful to plot some possibilities and the implied matching functions. Figure 3 contains these. It shows the fitted line of Equation 5.3 across different a and b and their implied matching functions. In panel (a), the individual is ambiguity neutral and does not exhibit ambiguity-insensitivity. Here, ambiguity is approached in the same way as risk where the unknown probabilities presumed to be equal. Figure 3b shows the matching function of an ambiguity averse agent. The bold straight line is the line fitted from Equation 5.3. Because at m(0) = 0 and m(1) = 1 the implied matching function
is curved. An ambiguity averse (seeking) agent has a convex (concave) matching function.
Figure 3c and d show what the implied matching functions are for an ambiguity neutral or averse, respectively, who is also ambiguity insensitive.
5.2 Sample & Descriptive Statistics
We follow Dimmock et al. (2016) and exclude subjects that answered too quickly on a subset
of questions (three seconds or less), gave wrong answers to two check questions, or answered indifferent to the first iterations of all questions. This leaves us with a sample of 1, 647 for the year 2010. Of these, 774 received real incentives incentives and 873 did not. Who was given real incentives was randomly decided. At the end of the survey, the real incentives group played for a real reward based on one of the questions and their answers (which question was used was also randomly decided).
Summary statistics for the ambiguity measures can be found in Table 2. Let us first consider the incentivised group. As can be expected, the summary statistics are close to identical to the
statistics in Dimmock et al. (2016, Table 3). Reason for some minor differences is that our
sample is slightly larger as we do not cut observations due to missing data.
Table 2: Summary Statistics of Ambiguity Attitudes
Incentivised Group (774 obs.) Non-Incentivised Group (873 obs.)
mean med. SD min. max. mean med. SD min. max.
m(0.1) 0.218 0.100 0.246 0.013 0.986 0.256 0.142 0.271 0.013 0.986 m(0.5) 0.393 0.391 0.238 0.016 0.984 0.428 0.438 0.247 0.016 0.984 m(0.9) 0.691 0.872 0.325 0.014 0.988 0.724 0.886 0.306 0.014 0.988 ESA0.1 -0.118 0.000 0.246 -0.886 0.088 -0.156 -0.042 0.271 -0.886 0.088 ESA0.5 0.107 0.109 0.238 -0.484 0.484 0.072 0.063 0.247 -0.484 0.484 ESA0.9 0.209 0.028 0.325 -0.088 0.886 0.176 0.014 0.306 -.0875 0.886 a-insensitivity (a) 0.408 0.295 0.434 -0.219 2.210 0.415 0.352 0.424 -0.219 2.210 Ambiguity aversion (b) 0.132 0.100 0.419 -0.972 0.972 0.061 0.053 0.430 -0.972 0.972
this does not hold. The median is ambiguity neutral, but there is a lot of variation above that level implying ambiguity seeking. These findings indicate a source function similar to Figure 3d, a common finding in the literature (Abdellaoui et al., 2011). The summary statistics of the event specific indices concur.
The mean a-insensitivity is 0.408 with a median of 0.295 which implies a general a-insensitive attitude. The same goes for ambiguity aversion b. With a mean of 0.132 and a median of 0.1, most of the sample exhibits ambiguity aversion. What is noteworthy is that the maximum of a-insensitivity is 2.21. Such a value implies a negative slope of the matching function which is not a rationally justifiable attitude. It suggest that if this subject had to choose between two ambiguous urns both with ten balls, she prefers the urn with 1 out of 10 winning over the one
with 9 out of 10 winning. This goes undiscussed by Dimmock et al. (2016) but seems rather
strange. Regardless, we will leave these subjects in the sample just as Dimmock et al. (2016). On the first iteration of the question concerning an ambiguity neutral probability of 0.5, approximately 70 percent preferred the known urn, implying ambiguity aversion. 9 percent chose the ambiguity neutral option (indifference). The remaining 21 percent preferred the unknown urn implying ambiguity seeking. For the a neutral probability of 0.1, 32 percent chose the ambiguity averse option, 18 percent chose the ambiguity neutral option, and 50 percent chose the ambiguity seeking option (all on first iteration). For the ambiguity neutral probability of 0.9, the shares were 54, 12, and 34 percent for ambiguity averse, neutral, and seeking, respectively.
Table 3: Effects of Incentivising
Kolmogorov-Smirnov test (p-value) (1) Mean (2) Median (3) Full Sample (4) Low Edu. (5) High Edu.
m(0.1) -0.038*** (.0127) -0.042*** [.0123] 0.012 0.018 0.293 m(0.5) -0.034*** (.0120) -0.047*** [.0146] 0.055 0.036 0.573 m(0.9) -0.032* (.0156) -0.014 [.0279] 0.160 0.102 0.870 Constant of Eq. 5.3 (c) -0.039*** (.0138) -0.027* [.0114] 0.009 0.082 0.132 Slope of Eq. 5.3 (s) 0.007 (.0212) 0.057 [.0405] 0.482 0.619 0.601 a-insensitivity (a) -0.007 (.0212) -0.057 [.0417] 0.482 0.619 0.601 Ambiguity aversion (b) 0.070*** (.0210) 0.047* [.0220] 0.005 0.014 0.292
Notes: Robust standard errors in parenthesis. Bootstrapped standard errors in braces. *, **, and *** indicate significance at the 5, 1, and 0.5 percent level, respectively. Low education refers to the group which highest education is primary school, vmbo, havo, or vwo (the latter three are the Dutch equivalent of (junior) high school). High education refers to the group which highest education is mbo, hbo, or wo (US equivalents are junior college, college, and university, respectively).
Mean ambiguity aversion is significantly higher for the incentivised group (as this does depend on the constant).
Similar conclusions are reached when comparing medians. To compare medians, we run quantile regressions of which the results can be found in column 2 of Table 3. The medians of
m(0.1) and m(0.5) are lower for the incentivised group. The median of m(0.9) is not
significantly different across groups. Just as with the means, most of this difference goes
through the constants. The median a-insensitivity is does not significantly differ across groups but the median ambiguity aversion is significantly higher in the incentivised group.
The final formal test we run is the Kolmogorov-Smirnov test, the results of which can be found in column 3 of Table 3. The null-hypothesis of equal distributions is rejected at the five percent level for m(0.1), the constant, and ambiguity aversion. For m(0.5) it is rejected at the ten percent level. These results confirm the slight tendency towards ambiguity seeking when not incentivised.
As Dimmock et al. (2016) state, the disparity between the two incentive groups is driven by educational differences. Columns 4 and 5 show the result of the Kolmogorov-Smirnov tests for low and high education separately. The incentive and non-incentive samples are significantly different for the low education group when it comes to ambiguity aversion. This cannot be said of the high education group.15
Given that the distributions differ in a non-trivial way, we only consider individuals that received extra incentives.
There are no noticeable differences in the distributions of the ambiguity measures between genders (see Appendix A.6) or age groups (see Figure A.3). Younger people appear slightly more ambiguity averse. There is one difference between educational attainment groups that is
15
Table 4: Correlations between Ambiguity Attitudes a b ESA0.1 ESA0.5 b 0.2213 ESA0.1 -0.4396 0.7343 ESA0.5 0.0342 0.8003 0.5141 ESA0.9 0.7346 0.7908 0.2865 0.4248
worth mention. The distribution of both a-insensitivity and ambiguity aversion is much flatter for those with only a primary school education (see Figure A.4). These are comparatively more a-insensitive and ambiguity averse.
Some subjects also gave inconsistent answers when confronted with check questions. These check questions confronted subjects to the same set up as used for deriving the matching
probability m(0.5). The first increased the subject’s estimated m(0.5) by 20 percent. A
subject is be inconsistent if she now prefers the unknown urn, which about 20 percent did. The second questions decreased the subject’s estimated m(0.5) by 20 percent. Subsequently, 34 percent chose the known urn, again implying an inconsistency. Unfortunately, as Dimmock et al. (2016) explain, there was a coding error which resulted in one of the check questions not
taking the 20 percent change fully into account. We follow Dimmock et al. (2016) and keep
the inconsistent individuals in the sample as well as compare results of the full sample with that of the inconsistent and consistent subsamples. Results of these can be found in Appendix B.5 and C.6. A comparison of the distributions of the ambiguity measures across consistency groups can be found Appendix A.6.
The correlations between the local ambiguity attitudes and global indices can be found in Table 4. The correlation between ambiguity aversion and a-insensitivity is positive but rather low which illustrates that they do indeed capture different attitudes. As expected, ESA0.1 is
negatively correlated with a-insensitivity and ESA0.9 shows a positive correlation.
5.3 Control Variables
Next to the ambiguity measures described in the previous section, several other variables will be included. This section briefly explains a few notable control variables that will be used.
(negative) parameter implies risk aversion (loving).
Financial literacy is estimated in the same manner as Dimmock et al. (2016) by extracting the factor from two questions regarding finance. These can be found in Appendix A.3. The “I don’t know” answer to the second question (“Which of the investments below generally speaking offers the greatest return over an extended period (say 10 or 20 years): a savings account, bonds or shares?”) are used separately as a proxy of ambiguity.
Trust is taken form the Personality survey and is the answer the level of agreement with the statement “other people can be trusted” given on a zero to ten scale. Optimism is also taken from the personality survey and is the LOT-R optimism scale (Scheier, Carver, & Bridges,1994). The exact questioning can be found in Appendix A.4. The optimism questions are available from 2012 onward.
We are following Dimmock et al. (2016) and define financial assets as the sum of account balance, value of investments and value of insurances (single-premium insurance policy, life annuity insurance, and endowment insurance). All remaining covariates are obtained from the
LISS panel background variables. (Net) monthly income is taken at the personal level.
Descriptive statistics can be found in Table A.1.
6
Predicting Ambiguity Attitudes
To see what individual characteristics can predict ambiguity attitudes, we will first use the attitudes as dependent variables. Our set of characteristics is close to identical to Dimmock et al. (2016, Table 6). It is therefore no surprise our results yield similar conclusions.16 We do show more results and also include optimism as an independent variable.
First we discuss the predictors of a-insensitivity, found in columns 1 to 4 of Table 5. First and foremost, more risk averse individuals are less a-insensitive. Wealthier people are overall, less a-insensitivity. The average marginal effect of assets is only significantly negative for those with positive wealth of less than 300 thousand euros, which is most of the sample. In these estimations, those with over half a million worth of financial assets were excluded. Including these outliers makes the effect on assets highly significant. Most education dummies are individually significant but not jointly. The difference with the base group (primary education) is significant, but among the other there is no significant difference in a-insensitivity.
Financial literacy is introduced in column 2. It enters significantly and its effect is negative. The education dummies have become mostly insignificant. Given that the estimates shrunk but not the standard errors, it suggests that some of the difference between primary school educated and the other levels of education is mediated through financial literacy (or some collinear omitted variable). The same holds for assets. The estimate for risk aversion has also decreased in 16The magnitudes of our results do differ somewhat from Dimmock et al. (2016, Table 6) which is likely
magnitude
Adding the trust variable to the regression has a minor impact on the results. The optimism scale enters insignificantly. This is not strange as it should theoretically be related to ambiguity aversion and not a-insensitivity. A significant finding implies that optimism has differential effect across different a-neutral probabilities. Do note that around a hundred observations had to be dropped. The effect of risk aversion declines and the effect of financial literacy increases by almost a full standard error. If we run a regression including an interaction of optimism with financial literacy, we find that only those with an above average score on the optimism scale and high financial literacy have significantly lower a-insensitivity.
Columns 5 to 8 of Table 5 contain the same regressions but with ambiguity aversion as the dependent variable. Risk averse individuals are less ambiguity averse. Financial assets only decreases ambiguity aversion if above 170 thousand, which is the case for only three percent of the sample. The education dummies are jointly significant throughout.
Adding financial literacy the model has negligible effect. The same hold when trust is
added. Adding optimism decreases the estimate for risk aversion by about a standard error but optimism itself is insignificant. This is peculiar because optimism and ambiguity aversion should theoretically be related. Even if we regress the answers to selected optimism questions separately none produce a significant coefficient.17 If optimism does not significantly predict ambiguity aversion, relating ambiguity aversion to pessimism or optimism, as in the α-Max-Min utility function, would be inappropriate. Our finding suggests that the two are distinct and capture different attitudes. This finding is contrary to earlier evidence which do find a significant relationship (Pulford,2009;Ahn, Choi, Gale, & Kariv,2014).
One explanation of this irrelevance of optimism lies in our use of it. The optimism questions were only added the personality survey of the LISS panel in 2012, two years after the ambiguity survey of Dimmock et al. (2016, Table 6). We have thus imputed these variables from 2012. If the computed optimism scale (or ambiguity aversion) is stable over time, i.e. people’s answers do not change much year over year, assuming it is constant over time would not be unreasonable. To test this, we run an autoregressive model on optimism. The coefficient of previous year’s optimism is 0.992 (p = .0000, R2 = .974, constant excluded). The optimism scale is extremely persistent over time which justifies our method. It also implies that optimism/pessimism and ambiguity aversion are indeed distinct.
For both a-insensitivity and ambiguity aversion, the regressions have little predictive power. The adjusted R2’s range from 5.9 to 7.7 percent for a-insensitivity and are around 3.3 percent for ambiguity aversion. Ambiguity attitudes are therefore empirically distinct measures that cannot be explained by the included variables. If estimations are done using logit regressions, similar pseudo-R2’s are obtained.
We also run Chow tests to see whether the predictors are different across incentive groups. 17
Table 5: Predicting a-Insensitivity and Ambiguity Aversion
a-Insensitivity (a) Ambiguity Aversion (b)
(1) (2) (3) (4) (5) (6) (7) (8) Risk Aversion -0.102*** -0.097*** -0.093*** -0.081** -0.100*** -0.099*** -0.097*** -0.068** (0.031) (0.031) (0.031) (0.032) (0.031) (0.031) (0.031) (0.034) Financial Literacy -0.060** -0.057* -0.086*** -0.009 -0.004 -0.028 (0.029) (0.030) (0.029) (0.027) (0.026) (0.031) Trust 0.004 0.008 (0.007) (0.008) Optimism 0.009 -0.006 (0.006) (0.005) Financial Assets -1.549** -1.331* -1.330* -1.687** 1.037 1.070 1.025 0.824 (0.747) (0.738) (0.739) (0.758) (0.735) (0.738) (0.735) (0.743) Financial Assets2 1.279 0.759 0.727 1.493 -4.355** -4.434** -4.333** -3.761* (2.039) (2.016) (2.018) (2.056) (2.072) (2.077) (2.064) (2.018) Income -0.507 -0.445 -0.395 -0.148 -0.072 -0.063 0.036 0.500 (0.370) (0.377) (0.381) (0.383) (0.409) (0.406) (0.407) (0.470) Income2 0.732 0.657 0.626 0.358 -0.163 -0.175 -0.269 -0.834 (0.491) (0.508) (0.515) (0.490) (0.751) (0.745) (0.753) (0.885) Age 0.042 0.045 0.038 -0.020 -0.023 -0.023 -0.033 -0.041 (0.057) (0.057) (0.057) (0.064) (0.051) (0.052) (0.051) (0.061) Age2 -0.001 -0.001 -0.001 0.006 0.001 0.001 0.002 0.003 (0.006) (0.006) (0.006) (0.007) (0.005) (0.005) (0.005) (0.006) Female -0.009 -0.029 -0.027 -0.009 0.025 0.022 0.031 0.038 (0.038) (0.038) (0.039) (0.042) (0.036) (0.037) (0.037) (0.040) HH size 0.003 0.005 0.004 0.004 0.023 0.023 0.022 0.025* (0.015) (0.015) (0.015) (0.016) (0.014) (0.014) (0.014) (0.015) Education Dummies (base is primary school)
Intermediate/Low -0.156* -0.145 -0.139 -0.126 -0.018 -0.016 -0.007 -0.050 (0.087) (0.088) (0.089) (0.094) (0.077) (0.077) (0.077) (0.095) Intermediate/High -0.195** -0.169* -0.177* -0.155 0.006 0.010 0.003 -0.018 (0.091) (0.093) (0.094) (0.098) (0.082) (0.083) (0.081) (0.099) Vocational 1 -0.178** -0.160* -0.163* -0.135 -0.032 -0.029 -0.028 -0.085 (0.088) (0.089) (0.091) (0.096) (0.079) (0.079) (0.078) (0.096) Vocational 2 -0.196** -0.171* -0.174* -0.169* -0.085 -0.082 -0.083 -0.101 (0.088) (0.090) (0.091) (0.098) (0.078) (0.078) (0.078) (0.094) University -0.137 -0.105 -0.106 -0.057 0.109 0.114 0.112 0.092 (0.094) (0.097) (0.099) (0.104) (0.086) (0.087) (0.088) (0.103) Cons. 0.469*** 0.444*** 0.438*** 0.388** 0.188 0.184 0.148 0.271* (0.138) (0.141) (0.144) (0.169) (0.123) (0.124) (0.129) (0.160)
Educ. joint p-value .292 .415 .383 .336 .019 .019 .021 .029
No. of Obs. 621 621 615 520 621 621 615 520
Adj. R2 .059 .071 .071 .077 .034 .035 .033 .030
We reject the null of equal coefficients across incentive groups for ambiguity aversion (p = .006) but not for a-insensitivity (p = .011). This is driven by differences in the local ambiguity attitude of low probabilities (ESA0.1, p = .0000). For the other two local indices, ESA0.5 (p = .244)
and ESA0.9 (p = .404), the coefficients are not significantly different. Earlier suspicions of the importance of properly incentivising are confirmed by these results.
Running the same Chow test but now by gender reveals that predictors of ambiguity attitudes are different be gender. Only for a-insensitivity and ESA0.1 are the predictors significantly
different (p = .005 and p = .003, respectively. For the other indices, there is no significant difference. This provides grounds for considering the ambiguity attitudes separately for men and women, which we shall do.
7
Empirical Strategy
Our second analysis concerns the effects of ambiguity and ambiguity attitudes on retirement expectations.
To empirically estimate the importance of attitudes towards ambiguity for the formation of retirement expectations, we exploit an exogenous inducement of ambiguity regarding the State Pension Age (SPA) (among other factors). The Dutch pension system was subject to drastic reforms in the past decades. One of which was the increase of SPA from 65 to 67 in
2013.18 Before and after this reform, heavy debate ensued between the Dutch government and
the opposition, but also within the coalition parties themselves. The result was huge uncertainty for many people regarding when they were entitled to state retirement benefits (AOW).
Figure 4 shows that the public debate and uncertainty regarding SPA impacted people’s capabilities of forming retirement expectations. The first bill to increase the SPA was proposed in May 2011 which shows a sharp increase in the percentage of people not knowing their expected retirement age. Note that our data was collected in June and July for the year 2011, right after the initial proposal. In the following years, the bill was accepted and the Dutch government drew up another bill which proposed to move increase in SPA forward along with more austerity measures. The first bill went practice in 2013 and the final adjustment was accepted in 2015, again right before our data collection. The public debate faded out from there. Evident from Figure 4, this period of uncertainty between 2011 and 2015 had its effects.
Our first analysis sees to what extent not being able to form retirement expectations can be attributed to different attitudes towards ambiguity. To this end, we regress
yit= βbi+ δai+ γri+ x>itλ + ci+ uit (7.1)
where yit is the dependant variable of interest. bi and ai capture ambiguity aversion and a-insensitivity as discussed in subsection 5.1. ri is the risk aversion index. xit is a vector of
covariates including a female dummy, a quadratic net personal income polynomial, a quadratic 18
Figure 4: Share Not Knowing Expected Retirement Age (%)
Notes: Percentage shown is the share of subject that answered “I don’t know” to the question “At what age do you expect to retire or take early retirement or to stop working?” if age is below 65 or “At what age do you expect to stop working?” if age is 65 or above.
Source: Author’s calculations based on data from Longitudinal Internet Studies for the Social sciences (LISS).
financial assets polynomial, education dummies, a quadratic age polynomial, and number of household members. ci is the idiosyncratic error and uit is the error term.
We employ numerous dependent variables. The first is a dummy variable equaling one if the
individual answers “I don’t know” to questions regarding her expected retirement age.19 What
we aim to capture is an inability to form retirement expectations which is implied by a not knowing of ones retirement age. For those who are able to form an expectation, we consider another dependant variable which equals one if the subject reports a higher expected retirement age than desired. These individuals thus expect to work longer than they would like to.
We consider a simple linear probability model as opposed to different panel binary outcome models. Reason for this is, first, that we are not interested in producing predictions and, second, even if we were the linear probability model does not produce many predictions below zero or greater than one in any specification. Regressions are done using random effects and correlated random effects (seeMundlak,1978). Regarding outlier corrections, we exclude 41 observations
19
with a net personal income monthly income over 5.000 euros and 20 observations with financial assets exceeding half a million euros. Inclusion of these has disproportionate effect on the results. Some exante predictions and hypotheses can be formulated. Ambiguity averse individuals
may postpone or forgo planning for retirement when faced with a lot of uncertainty. This
reasoning comes from the assumption that retirement planning is costly in terms of time and effort and that this cost is greater for ambiguity averse individuals. This is supported by Snow (2010) who finds that information is more valuable to ambiguity averse individuals. Resolving ambiguity is more costly for ambiguity averse individuals. When uncertainty rises, so does the cost of retirement planning and ambiguity averse individuals may want to delay the formation of expectations, partially out of fear of disappointment and costly adjustment of expectations.20 We call this the greater cost hypothesis.
A second hypothesis regarding the effect of ambiguity aversion on the ability to form
retirement exceptions goes in the opposite direction. A distaste for ambiguity entails an
increased willingness to decrease ambiguity (Snow,2010). Ambiguity averse individuals should in theory be more eager to obtain information and reduce the perceived ambiguity. Ambiguity averse individuals have much more to gain from resolvement of ambiguity from a utility optimisation standpoint (Baillon, Bleichrodt, et al., 2018). Such individuals therefore invest more time and effort in planning than their a-neutral counterparts. We call this the planning hypothesis.
Notice that the greater cost hypothesis applies to situations in which ambiguity is high but individuals belief that it will be lower in the future. A postponement of planning would not necessarily be beneficial to ambiguity averse individuals if the ambiguity is not resolved in the future. The planning hypothesis is, a priori, more dominant in situations where uncertainty has a permanent character and where it cannot be resolved by time alone.
We thus have two hypotheses. First, ambiguity averse individuals forgo or postpone
planning as they are overcome by ambiguity and its resolvement is costly. This greater cost hypothesis operates through a “wait and see” attitude. Second, ambiguity averse individuals have an increased willingness to obtain information to reduce ambiguity and engage in more planning. This planning hypothesis operates when ambiguity is not resolved over time and the individual must herself take action te resolve it.
β captures whether the greater cost of planning hypothesis dominates. A negative β implies that for ambiguity averse individuals, the willingness to inform oneself exceeds the additional cost of forming wrong expectations, i.e. the greater cost outweighs the greater planning incentive. A positive β entails the reverse.
a-insensitive, and by extension ambiguity prudent (see section 3), individuals may have trouble forming retirement expectations. Remember that such individuals overweigh extreme events. They may, for instance, give excessive emphasis to a bad potential outcome (with low a-neutral probability) such as seeing a sharp drop in future retirement benefits. This overweighing 20For ambiguity seekers, the reverse goes. A resolvement of ambiguity is costly and they should, in principle,
may cause a postponement of forming retirement expectations because, as retirement approaches, such ambiguity decreases and the individual fears the bad outcome less (as she assigns a lower subjective probability). The reasoning here is also of a “wait and see” approach. As a bad outcome seems likely, extending the working life seems inevitable. Rather than adhering to this expectation, the ambiguity may cause an individual to wait and see what the outcome will be before forming the expectation, again out of fear of disappointment and having to adjust expectations.
The reverse can also be hold if the overweighed outcomes are good instead of bad.
a-insensitivity also entails an overweighing of good outcomes with a low a-neutral probability. This excess optimism for extreme events may create overly confident retirement predictions. The sign of δ is thus a priori unclear. Prospect theory suggests that bad outcomes are more pressing suggesting that δ is positive.
For the expectations to work longer than desired the predictions are slightly different. Ambiguity averse individuals can be characterised as more pessimistic (along the lines of the alpha-Max-Min expected utility function). This pessimism is reflected by a positive coefficient
for ambiguity aversion. Ambiguity averse individuals are expected to be more likely to be
pessimistic about their retirement age and are more likely expect to work longer than desired. The effect of a-insensitivity depends on a subject’s belief about the probabilities of uncertain outcomes. Suppose again there is an a-neutral low probability of seeing a drop in retirement benefits. An a-insensitive individual overweighs this negative outcome and should thus expect to work longer. The reverse can also be argued. The extreme case of winning a lottery, a good outcome with a low a-neutral probability, is also overweighed by an a-insensitive individual. An overweighing of such events should decrease the probability of expecting to work longer than desired. a-insensitivity can thus have either effect on this expectation depending on what outcomes the subject considers.
The above analysis is extended by interactions between the ambiguity attitudes, a and b,
and period dummies. The interactions with the period dummies essentially entail a
multi-period difference-in-differences with continuous treatment variables (the ambiguity attitudes). Important to note is that the survey data in 2011 was collected in the two months after the Government’s proposal to increase the SPA (along with other pension reforms). In 2015, data was collected right after the final major pension reform bill passed. We will add two interactions: one with a dummy equalling one if the year is 2011, 2012, 2013, or 2014, and one if the year is 2015 or later. The interactions with the dummy of the spell of public debate (2011, 2012, 2013, 2014) capture a time of increased uncertainty regarding state pensions. A substantial drop in uncertainty occurred from 2015 onwards as no substantial public debate or reforms occurred thereafter. These interactions should capture an increase and drop in uncertainty regarding retirement.
