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Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
How Do Behavioral Assumptions Affect
Structural Inference? Evidence From a Laboratory
Experiment
Daniel Houser & Joachim Winter
To cite this article: Daniel Houser & Joachim Winter (2004) How Do Behavioral Assumptions
Affect Structural Inference? Evidence From a Laboratory Experiment, Journal of Business &
Economic Statistics, 22:1, 64-79, DOI: 10.1198/073500103288619386
To link to this article: http://dx.doi.org/10.1198/073500103288619386
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Date: 13 January 2016, At: 00:20
How Do Behavioral Assumptions
Affect Structural Inference?
Evidence From a Laboratory Experiment
Daniel H OUSER
Interdisciplinary Center for Economic Science & Department of Economics, George Mason University, Fairfax,
VA 22030 (dhouser@gmu.edu)
Joachim W INTER
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Mannheim Research Institute for the Economics of Aging, University of Mannheim, D-68131 Mannheim, Germany
(winter@uni-mannheim.de)
We use laboratory experiments to investigate the effect that assuming rational expectations has on structural inference in a dynamic discrete decision problem. Our design induces preferences up to the subjective
rate of time preference, leaving unrestricted both this parameter and subjects’ decision rules. We estimate
subjects’ discount rates under the assumption that all subjects use the rational expectations decision rule,
and under weaker behavioral assumptions that allow decision rule heterogeneity. We nd that certain
sophisticated heuristics t subjects’ decisions statistically signicantly better than rational expectations.
However, the rational expectations assumption does not distort inferences about the cross-sectional discount rate distribution.
KEY WORDS: Discounting; Heterogeneity; Heuristics; Rational expectations; Search; Statistical classication algorithm.
1.
INTRODUCTION
from human behavior. Our study, which examines the effects
of different expectations specications, takes advantage of this
feature of laboratory experiments.
Rational expectations is the standard assumption in virtually
all applied structural econometric work, and this has generated
some criticism. As Blundell (1994, p. 4) pointed out, “structural models are often accused (in many cases quite rightly) of
imposing untenable and unbelievable restrictions on behavior.”
Nevertheless, there seems to be very little evidence suggesting
that the rational expectations assumption has had detrimental
effects in any positive sense. Research using eld data to estimate life-cycle labor supply models with expectations-robust
procedures (see, e.g., Hotz and Miller 1988; Houser 2003) typically obtains results that line up well with results based on rational expectations specications.
Unfortunately, there is a fundamental limitation to the
amount that one can learn about the effect of expectation specication errors from the analysis of eld data. In particular, Rust
(1994) showed that under very general conditions, individuals’
preferences and their expectations processes are jointly nonparametrically unidentied. Because persons’ true period return
functions are not generally known, and because the identication problem implies that multiple period return function and
expectation specication combinations will t the data equally
well in-sample, examining the consequences of expectation
misspecication with eld data presents a substantial challenge.
In contrast, the experimental laboratory provides an environment in which period return functions can be controlled,and the
identication problem consequentlycircumvented.The result is
The latter half of the twentieth century saw the emergence
of a large literature documenting apparently irrational behavior by subjects in economics and psychology experiments. This
has led theorists, particularly over the last 10 years, to begin to assess the equilibrium implications of certain types of
rule-of-thumb behavior (see, e.g., Ellison and Fudenberg 1993;
Krusell and Smith 1995; Lettau and Uhlig 1999; Anderlini and
Canning 2001). Interestingly, however, relatively little effort
has been made by empirical economists to understand whether,
as a practical matter, assuming rational expectations leads to
misspecied models that substantially distort inferences about
structural parameters. A reason for this is that both the analysis of eld data and simulation procedures are limited in the
extent to which they can address this issue. This article uses
a laboratory experiment to investigate the sensitivity of inferences about one structural parameter—the subjective rate of
time preference—to the rational expectations assumption.
The experimental laboratory has proved a useful tool for investigating statistical questions not easily answered through the
analysis of eld data or Monte Carlo simulations. For example,
Cox and Oaxaca (1999) recently used lab experiments to assess whether supply-and-demand parameters can be recovered
from data generated by various common market institutions.
They pointed out that standard procedures to recover these parameters require assumptions about the data-generating process
(DGP), and, to the extent that these assumptions generate specication errors, inferences can be erroneous. Whereas Monte
Carlo simulations can be used to study the effects of any particular misspecication, the DGPs considered in such exercises
might not correspond to any naturally occurring DGP. Cox and
Oaxaca (1999) pointed out that one advantage of the experimental laboratory is that it can be used to study the effects of the
specic misspecication that arises when the true DGP stems
© 2004 American Statistical Association
Journal of Business & Economic Statistics
January 2004, Vol. 22, No. 1
DOI 10.1198/073500103288619386
64
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Houser and Winter: Behavioral Assumptions and Structural Inference
that inferences about the effect of expectation misspecication
on structural inference become possible.
This article analyzes data from a price search experiment that
induces preferences (but not the way in which people form expectations) up to a single structural parameter: the rate of subjective time preference (or the “subjective discount rate”). Our
experimental design, which builds on work of Rapoport and
Tversky (1970) and Seale and Rapoport (1997), provides information on discount rates because it incorporates the possibility that a subject’s experimental earnings might be paid to
them 2 weeks after they complete their laboratory session. Inferences about time preference have been drawn in many other
laboratory experimental studies; Frederick, Loewenstein, and
O’Donoghue (2002) have provided a comprehensive review of
this literature. A recent study by Warner and Pleeter (2001)
took advantage of a natural experiment, providing data on real
choices with large stakes, to estimate time discount rates. Finally, some econometric studies have estimated discount rates
from eld data using indirect inference methods (e.g., Lang
and Ruud 1986; Samwick 1998; Gourinchas and Parker 2001;
Cagetti 2003). In contrast to these studies, our focus is not so
much on estimating the discount rate per se, but rather on understanding the interaction of assumptions about individuals’
behavior and such estimates.
We compare three approaches to inference about subjects’
discount rates. One assumption common to each approach is
that subjects are risk neutral. This assumption is standard in the
price search literature, and using it here allows our results to
be more easily compared to that literature. Moreover, although
preferences regarding risk and intertemporal substitution can
be separately conceptualized in abstract models, say, using the
framework of Kreps and Porteus (1978), they are typically difcult to distinguish in practice. Still the assumption of risk neutrality might fail. Because risk-averse subjects should stop the
search sooner on average, and because it turns out that more impatient subjects should also stop the search sooner on average,
the failure of risk neutrality would tend to bias our discount rate
estimates downward.
Our rst results are derived from a simple revealed preference analysis based on subjects’ accepted payoffs only. This
procedure enables us to establish discount rate bounds under the
weak assumption that subjects prefer higher payoffs to lower
payoffs (and that they are risk neutral). Although this procedure
is robust, it ignores all continuation decisions that, in general,
also contain information about discount rates. In particular, a
subject’s stopping rule within the sequential search task will
vary with his on her rate of time preference, both under rational
expectations and under many alternative decision rules (which
we call “heuristics” in the sequel). Incorporating stronger behavioral assumptions, therefore, typically allows more precise
inference about rates of time preference, albeit at the possible
cost of bias due to model misspecication.
Our second set of results is derived using a very strong behavioral assumption, namely that subjects have rational expectations. Although evidence shows that people do not behave
according to rational expectations in many dynamic decision
tasks (see, e.g., Carbone and Hey 1999), other studies (e.g., Hey
1982) have found that behavior in price-search tasks is often
65
quite close to optimal. This is an important advantage of our design, because in our view, people likely behave nearly optimally
in many actual intertemporal (life-cycle) decision problems.
The nal set of results is based on a more robust approach
that is relatively common in experimental game theory (e.g.,
Weber and Tietz 1975; Selten 1991). We specify a set of candidate decision rules from which the decision rule that best ts observed behavior is selected for each subject. Similar ideas have
been explored by Hey (1981, 1982, 1987) and Moon and Martin (1990) in the experimental analysis of price-search problems, and by Müller (2001) in an experiment on behavior in
life-cycle choice situations. In our case, the set of candidate decision rules includes the optimal, rational expectations decision
rule as well as various more- or less-sophisticated heuristics.
The main difference between earlier studies and our approach
is that for each subject, we jointly determine both the decision
rule and a structural parameter (the subjective discount rate).
Moreover, in contrast to the literature cited earlier, we use a
formal statistical estimation approach rather than informal classication procedures. Our estimation procedure, related to that
of El-Gamal and Grether (1995), relaxes both the assumption
of rational expectations and the assumption that all subjects use
the same decision rule. At the same time, as we explain in Section 5, our procedure avoids the problem of overtting.
Relative to the revealed preference analysis, we nd that imposing behavioral restrictions substantially sharpens inferences
about discount rates. We also show that the data are described
less well by rational expectations behavior than by simpler
heuristics. The reason is that subjects tend to stop searching
sooner on average than is consistent with rational expectations
behavior and low time discounting (i.e., high willingness to accept delayed payoffs). At the same time, rational expectations
behavior under high time discounting is inconsistent with the
frequency of delayed payoffs that are accepted. It turns out that
some of the heuristics that we consider accommodate both relaxed stopping criteria and little time discounting,and it is these
heuristics that best explain our data. Nevertheless, we do not
nd statistically signicant differences between the estimates
of the cross-sectional distribution of discount rates that emerge
under rational expectations and weaker behavioral assumptions
(heuristics). The mean and median of the estimates are .65 and
.68 under the rational expectations analysis and both .70 under
weaker behavioral assumptions.
Although population-level inference seems robust to behavioral specications, inferences about any given individual’s
discount rate can vary substantially across behavioral specications. We cannot know any subject’s true rate of time preference, but nevertheless we can investigate the relationship
between subjects’ estimated discount rates and an external measure of the extent to which they are forward-looking. The external measure that we use is based on a personality survey,
described in detail below, that we administered as part of the
price-search experiment. We nd that neither revealed preference nor rational expectations estimates are statistically signicantly correlated with this external measure, whereas our
heuristic-based discount rate estimates and the external measure are statistically signicantly positively correlated.
The remainder of the article is structured as follows. In Section 2 we introduce our experimental design. We then present
66
Journal of Business & Economic Statistics, January 2004
evidence of discounting and derive bounds on subjective discount rates in Section 3. In Sections 4 and 5 we discuss structural inference about discount rates under rational expectations
and under weaker behavioral assumptions, respectively. In Section 6 we discuss differences between the estimates at both the
aggregate and individual levels. We provide some concluding
remarks in Section 7.
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2.
EXPERIMENTAL DESIGN
In our experiment, subjects make decisions in a standard
price-search problem (Rapoport and Tversky 1970; Seale and
Rapoport 1997) modied to incorporate delayed payments.
Each subject’s goal is to purchase a “widget” that they value
at 100 tokens. The widget is sold at many different locations,
and visiting a new location costs one token. Each location posts
both a price and availability (today or in 14 days) for the widget;
this location-specic information is revealed when the subject
visits that location. Subjects are informed that the price at each
location is drawn independentlyfrom a normal distributionwith
a mean of 100 and standard deviation of 10. Subjects are also
informed that the availability at each location is drawn independently (both across locations and with respect to the price
draw); availabilityis immediate with probability.5 or in 14 days
with probability .5. Subjects may stop their search at any time
and purchase the widget from any of the locations that they visited. (Hence this it is a search problem “with recall”.) An exact
transcript of the instructions for this experiment, along with all
data used in this article, are provided on the JBES website.
This design allows inference about time preference within
an environment where past research has shown behavior to be
nearly rational. Experiments conducted by Hey (1982, 1987)
have suggested that search behavior is closer to optimal when
there is no recall and when subjects have full information about
the price distribution. Our design combines the possibility of
recall and the availability of information about the price distribution. We do this because (1) the possibility of recall together
with the time-differentiated availability of the product (and possibility of a delayed payment of the search surplus) allows us
to observe within-game trade-offs between different points in
time, and (2) giving subjects full knowledge about the price
distribution, along with the opportunity to become experienced
with the task, avoids problems arising from learning about the
parameters of the price distribution that might distort inference
about time preference.
The perception that delayed payment involves risks or transaction costs can bias inference about a subject’s rate of time
preference. We designed the experiment with these concerns in
mind. Specically, subjects were told that at the end of the experiment, their earnings would be put into an envelope. If they
chose an immediate payment, then the envelope would be given
to them immediately. If they chose a delayed payment, their envelope would be stored in a safe until it was delivered or picked
up in 14 days. The instructions read: “We absolutely guarantee
that 14 days from today you will be able to obtain your envelope with your earnings. Moreover, we will pay any costs (such
as mailing costs) required to get your earnings to you after 14
days.”
The experiment was conducted at the University of Arizona’s
Economic Science Laboratory in the Spring and Summer of
2000. Subjects were recruited from the general student population. All experiments were run entirely on computers using
Visual Basic software written by the authors. (This software is
available from either author on request.)
Upon arrival, each subject was given an instruction sheet and
seated privately at a computer terminal. To ensure that subjects
were experienced with the task and comfortable with the computer interface, they were allowed to play an unlimited number
of practice games before playing 10 games that were relevant
to their nal payment. Subjects earned $5 for participating in
the experiment (paid to everybody at the end of the session regardless of their choices) and an additional amount, possibly
time-delayed, determined by selecting randomly the outcome
of 1 of the 10 payment-relevant games. Subjects were paid privately at the end of the experiment. On average, subjects were
in the laboratory for about an hour and earned about $20.
Our results are based on the play of 68 subjects. This yielded
a total of 680 payoff-relevant games and a total of 4,527 rounds
(continuation decisions). On average, subjects played 10 practice games before starting their 10 payoff-relevant games. Subjects on average played more rounds per game, accepted higher
prices, obtained lower payoffs, and also obtained zero payoffs
more often during practice games than in the payment-relevant
games.
3.
EVIDENCE OF DISCOUNTING AND BOUNDS ON
SUBJECTIVE DISCOUNT RATES
Table 1 contains an overview of the intertemporal choices
made by subjects. The top part contains data for all 680 games.
In 371 games subjects chose a product with instant delivery
(corresponding to an instant payoff), and in 309 games subjects chose a product with delayed delivery (corresponding to
a payoff in 14 days). On average, accepted prices are higher,
and the resulting payoffs lower, for chosen products with instant delivery. This is consistent with the notion that at least
some of our subjects discount future payments. Discounting is
also suggested by Figure 1, which shows the distributionsof accepted prices for both delivery dates. Figure 1(a), which shows
the distribution of prices for widgets with immediate delivery,
has more mass at higher prices than Figure 1(b), which shows
the distribution of accepted prices for widgets with delayed delivery.
Our rst approach to drawing inferences about subjects’ discount rates, which can be viewed as revealed intertemporal
Table 1. Accepted Prices and Corresponding Payoffs
Delivery date of the chosen product
Now
Later
Including games with zero payoff
Number of games (680 total)
371
Mean accepted price
86.33
Mean resulting payoff
9.20
Excluding games with zero payoff
Number of games (604 total)
329
Mean accepted price
85.23
Mean resulting payoff
10.37
NOTE:
Standard deviations are in parentheses.
(6.51)
(6.52)
309
85.35
9.73
(5.64)
(6.25)
(5.46)
(5.97)
275
84.71
10.94
(5.18)
(5.55)
Houser and Winter: Behavioral Assumptions and Structural Inference
67
Table 2. Combinations of Payoffs for Accepted and
Best Alternative Prices
(a)
The payoff associated with
the best price available for a
product with the alternative
delivery date is
Higher than the payoff for
the chosen product
Case 1
109 games
Case 4
92 games
Equal to the payoff for the
chosen product
Case 2
3 games
Case 5
4 games
Less than the payoff for
the chosen product
Case 3
217 games
Case 6
179 games
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NOTE:
(b)
Figure 1. Distribution of Accepted Prices. (a) Prices accepted by subjects for immediate-deliverywidgets. (b) Prices accepted by subjects for
delayed-delivery widgets. Accepted prices higher than 100 imply a loss,
because the widget is valued at 100; prices lower than 100 might also
imply a loss, depending on accumulated search cost. In the case of a
loss, actual payoffs were restricted to 0.
preference analysis, requires only the assumption that subjects
prefer higher payoffs to lower payoffs (and the assumption of
risk neutrality that we maintain throughout). With this assumption, we can derive bounds on each subject’s discount rate by
observing their location choices. For example, suppose that a
subject stops searching, and his or her best payoffs are 88 for
immediate-delivery and 90 for delayed-delivery widgets. If the
subject chooses the location that offers immediate delivery, then
it follows that his or her discount rate can be no larger than
88/90, whereas if he or she chooses the location offering delayed delivery, then his or her discount rate can be no smaller
than 88/90. This fact allows us to construct bounds on a sub-
Delivery date of the chosen product
Now
Later
Game counts exclude cases where the payoff is zero.
ject’s discount rate, given that the subject has a meaningful
choice between delivery dates. In particular, only those games
in which the subject has prices that would result in positive payoffs at both delivery dates contain information about his or her
discount rate. This is the case for 604 of the total 680 games.
The bottom part of Table 1 provides descriptive statistics for
the sample that excludes the 76 zero-payoff games from the revealed preference analysis.
Table 2 lists the six situations that a subject can face when he
or she decides to stop searching. These situations are dened by
comparing payoffs for the chosen delivery date with the payoffs
corresponding to the best price available at the alternative delivery date (of which many will be zero as well). Observations
for cases 1, 2, and 3 (delivery now) are shown in Figure 2(a);
observations for cases 4, 5, and 6 (delivery later) are shown
in Figure 2(b). Observations that correspond to cases 1 and 4
are above the 45-degree line, observations that correspond to
cases 2 and 5 are on the 45-degree line, and observations that
correspond to cases 3 and 6 are below the 45-degree line. Table 2 also contains the counts of these observations.
From these observed decisions on intertemporal tradeoffs,
we can compute bounds on subjects’ discount rates. We exclude those cases in which subjects chose combinations of price
and date that were strictly dominated in terms of the associated
payoffs (e.g., all case 4 observations) and all observations with
zero payoff for the best available price at the alternative delivery
date.
We end up with observed choices that allow us to obtain
lower and/or upper bounds that lie in the [0; 1] interval for
many of our subjects. Because all subjects play 10 games, we
have more than one lower or upper bound for some subjects.
In these cases, we compute the intrasubject maximum of the
lower bound for the former, or the intrasubject minimum of the
upper bound for the latter. There were several cases in which a
subject’s decisions did not reveal an upper or a lower bound. In
these cases we set the lower bound to 0 and/or the upper bound
to 1. Once these bounds were determined, we computed the “revealed preference” estimate of the subjective discount rate for
all subjects as the mean of the lower and upper bounds.
Our revealed preference estimation procedure can be interpreted as follows. We rst impose a uniform distribution over
the [0; 1] interval as a prior for the subjective discount rate.
In particular, we rule out the possibility that the discount rates
can be above 1 or below 0. Observed lower bounds above zero
and upper bounds below 1 allow us to narrow the prior. For
those subjects for whom we cannot construct bounds away from
68
Journal of Business & Economic Statistics, January 2004
Table 3. Estimates of Subjective Discount Rates Under Various
Behavioral Assumptions
(a)
Behavioral assumption
No restriction (revealed preference)
Lower bound
Upper bound
Mean of bounds
Optimal behavior
Assigned heuristic
Mean
.22
.97
.60
.64
.70
Minimum
Maximum
Standard
deviation
0
1.00
1.00
1.00
1.00
1.00
.31
.15
.18
.33
.22
:25
:13
0
0
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NOTE: The fth column reports the standard deviation of the cross-sectional distribution of
discount rate point estimates. The total number of subjects is 68.
(b)
Table 3 summarizes the distribution of the bounds on subjective discount rates and the estimates of subjective discount
rates that we obtain. In general, the bounds on discount rates are
relatively wide, and they contain a fair amount of heterogeneity; this corresponds to the scattered price observations shown
in Figure 2. It is evident that the revealed preference approach
without stronger behavioral assumptions does not narrow the
at prior substantially.
In the following sections we discuss how imposing additional
assumptions on subjects’ behaviors allow us to use the information generated in our experiment to obtain more precise estimates of their subjective discount rates. These approaches offer a major improvement. Relative to the revealed preference
analysis presented in this section, more informative estimates
can be obtained when observed behavior in all games is used,
instead of behavior only in those games that have nonzero payoffs and a valid intertemporal trade-off (a subset that is empty
for some subjects).
4.
Figure 2. Payoffs Resulting From the Accepted Price (horizontal
axis) Versus Payoffs That Would Result From the Best Price at the
Alternative Delivery Date (vertical axis). (a) Prices accepted by subjects for immediate-delivery widgets. (b) Prices accepted by subjects
for delayed-delivery widgets. In both panels the diamonds represent the
trade-off between prices for widgets delivered at two different points in
time; therefore, they provide relative prices across time and can be used
to bound discount rates. In (b), diamonds above the 45-degree line represent inconsistent choices, because the payoff associated with the accepted price for delayed-deliverywidgets is lower than the potential payoff that would result from the best available price for immediate-delivery
widgets.
0 and 1, we obtain an estimate of .5. (This is the case for 31 subjects.) Note that using the mean of the set of possible discount
rates as our point estimate is arbitrary, and this choice is made
mainly for convenience.Moreover, this is not a central aspect of
our analysis. Our primary interest is in whether differences appear between approaches, and we use this estimate within each
inferential approach presented later.
INFERENCE UNDER RATIONAL EXPECTATIONS
Figure 3 suggests that the revealed intertemporal preference
approach provides quite imprecise inference about our subjects’
discount rates. Although it is attractive in that it requires very
weak behavioral assumptions, this also means that only the nal
decision in each game—the location decision—informs the discount rate. Under stronger behavioral assumptions, it is usually
the case that continuation decisions inform the discount rate as
well. We adopt this approach in this and subsequent sections.
A natural starting point is to assume that subjects follow the
rational expectations stopping rule. Under this assumption, the
subjective discount rate is identied and can be estimated as the
only free parameter of the corresponding decision process. To
see this, it is useful to derive the optimal solution to the search
task. For the remainder of this article, we label a subject whose
behavior is consistent with this rule as a “rational expectations”
agent.
Because the value of the widget in the experimental task is
100 tokens, and because each search costs 1 token, it is reasonable to derive the solution to the problem under the restriction that the number of searches cannot exceed 100. Let
t 2 f1; 2; : : : ; 100g denote the number of searches that the subject has made. (For notational convenience, we suppress a subject index as long as no confusion can arise.) After making the
tth search the agents’ state vector is St D ft; pb0 ; pb1 g, where pb0 is
the lowest price that the subject has encountered for immediate
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Houser and Winter: Behavioral Assumptions and Structural Inference
69
Figure 3. Discount Rate Estimates Obtained With a Revealed Preference Procedure. The revealed preference estimation procedure provides
only bounds on individuals’ discount rates. Each vertical line indicates an estimated range of possible discount rates for an individual in our sample.
Ranges reect the prior restriction that discount rates must be between 0 and 1. We choose to report the mean of each subject’s estimated range as
the point estimate of his or her discount rate (represented as a solid circle). Although behaviorally robust, taking our data to the revealed preference
procedure yields wide bounds on the discount rate for most subjects, and does not inform the discount rate at all for 31 of our 68 subjects (their
lower bound remains at 0 and their upper bound at 1).
delivery and pb1 is the lowest price encountered for delayed delivery. After the tth search, the agent may either stop searching
and choose a location from which to buy the item, or continue
to search. After the agent has stopped searching, he or she will
buy the item from the location that provides the highest payoff.
The available payoffs are
50 .St / D maxf0; 100 ¡ pb0 ¡ tg;
51 .St / D maxf0; ±.100 ¡ pb1 ¡ t/g;
(1)
where ± is the agent’s rate of time preference. It is convenient to dene the chosen payoff, conditional on stopping, as
5.St / D max.50 .St /; 51 .St //. The agent stops searching only
if the stopping value is higher than the continuation value. The
recursive formulation of this decision problem is, therefore,
Vt .St / D maxf5.St /; E[VtC1 .StC1 /jSt ]g;
(2)
where E represents the mathematical expectations operator and
the expectation is taken with respect to the distribution of
StC1 jSt .
It is immediate from (1) and (2) that this problem has, at
every t, the reservation price property. That is, there is some
best available price for immediate-delivery items such that at
this and any lower price, it is optimal to stop searching, whereas
at every higher best available price, it is optimal to continue
searching. This reservation price depends both on the number of
completed searches and on the best available price for delayeddelivery items. To see this, note that both arguments that enter the maximization operator in (2) are weakly decreasing (in
both prices). When prices are sufciently low, the continuation
value is equal to the stopping payoff less 1. At sufciently high
prices, the stopping payoff is 0, whereas the continuationpayoff
is necessarily bounded above 0 (because all price draws are independent,and the measure of prices that generate positive payoffs is strictly positive). The intermediate value theorem then
implies that the stopping and continuation value cross at least
once. That this crossing is unique follows from the fact that the
slope of the stopping payoff at the crossing point must be ¡1
(because the crossing occurs at a positive value for the stopping
and continuation payoff), and the slope of the continuationpayoff function is evidently bounded from below by ¡1 (because a
one-unit improvement generated by the current price draw can
never do better than improve the continuation payoff by one
unit).
From (1) and (2), it is also easily seen that both reservation
prices depend on the rate of time preference, ±. One way to
see this is simply to observe that the continuation value incorporates both immediate-delivery and delayed-delivery events,
and that the value of delayed delivery clearly depends on the
rate of time preference. Figure 4 plots the path of reservation
prices, calculated by solving the dynamic discrete choice problem implied by (1) and (2) numerically, for the cases ± D 1:0
and ± D :1. Note that by “reservation price,” we mean that the
subject should stop searching if the price draw is less than or
equal to the reservation value. Note also that we report only
integer-valued prices, consistent with the design of our experiment.
When the discount rate is unity, the problem reduces to standard search with recall with a single reservation price. This
reservation price begins at 90, decays slowly to about 87 by
round 10, reaches 80 by the 19th search, and decays at a rate
of about 1 per round from that point forward. When the discount rate is .1, the reservation price for the current payoff follows roughly the same pattern as the earlier case, but is slightly
higher. The reason is that the continuation value falls with the
discount rate, so the person is willing to accept smaller stopping payoffs. Reservation prices for delayed delivery follow an
inverted U-shaped pattern. The very low initial values arise because the continuation value incorporates the chance of nding an item for immediate delivery, and offsetting this expected
value under heavy time discounting requires very low prices for
delayed delivery.
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70
Figure 4. Rational Expectations Reservation Prices for Alternative
Discount Rates and Delivery Dates. The three curves describe the highest integer price that a subject with rational expectations would pay for a
widget, by cumulative number of search attempts. With a discount rate
(delta) of 1, the subject is indifferent between delivery now and delivery
later, so there is only one reservation price curve. When the discount
rate is less than 1, subjects prefer delivery now to delivery later, all other
factors being equal. Because the task is search with recall, to calculate
the reservation prices, we assume that the best available price for delivery at the alternative date is 100. Subjects with discount rates less
than 1 are willing to pay slightly more for delivery now than are subjects with unit discount rates. Subjects incur a 1-unit cost per search,
and their reservation prot level declines with the number of searches.
When delta is 1, the reservation prot begins at 9, but declines to 1 by
the 15th search. ( ¥ , delta D 1:0;
, delta D 0 :1, now; N ,
delta D 0 :1, later.)
Journal of Business & Economic Statistics, January 2004
With reservation prices in hand, it is a simple matter to estimate the discount rate for each individual. The decision task
requires the subject to decide when to stop searching and, once
stopped, the location from which to purchase the widget. Given
any discount rate, in each period, each state-contingentcontinuation decision is either consistent or inconsistent with the reservation price associated with that discount rate. When the subject
stops searching and chooses a location, that choice is either consistent or inconsistent with the given discount rate. Hence if Tj
represents the number of locations searched before the subject
stopped in game j, then the subject makes T1 C ¢ ¢ ¢ C T10 C 10
payoff-relevant decisions during the course of the experiment.
We associate each subject with the discount rate that is consistent with the greatest number of his or her actual decisions.
(This procedure is formalized in the next section.)
In this research we consider only 11 discount rate values,
f0; :1; : : : ; 1:0g. Such a coarse grid reduces the computational
burden, and prior checks conrmed that choices for most subjects in our sample are not sensitive to smaller changes in the
discount rate. Figure 5 illustrates our discount rate inferences
for each subject. The solid circle in the gure represents the
point estimate of the discount rate. For 36 of our subjects, the
best discount rate is not unique; several discount rates explain
the subject’s choices equally well. In these cases the point estimate is assumed to be the mean of the candidate values, which
is described in the gure by a solid circle in the middle of a
line segment. These line segments, therefore, have the same interpretation as those reported in the revealed preference analysis results. In particular, there is a sense in which a longer line
segment implies more uncertainty about a subject’s discount
rate, but the line segments do not represent standard errors.
(Although we do not report them here, it is straightforward to
bootstrap standard errors for the lengths, positions, and, consequently, midpoints of the line segments.) The mean of the discount rate point estimates is .65, and the median is .68. Overall,
the model predicts about 91% of our subjects’ choices.
Figure 5. Estimates of Discount Rates Under Rational Expectations. The procedure for generating these estimates uses the assumption that
individuals behave as if they have rational expectations. Each vertical line indicates an estimated range of possible discount rates for an individual
in our sample. Ranges reect the prior restriction that discount rates must be between 0 and 1. We choose to report the mean of each subject’s
estimated range as the point estimate of his or her discount rate (represented as a solid circle). With the rational expectations procedure, we obtain
relatively tight bounds on the discount rate for most subjects, and every subject’s discount rate is informed by his or her experimental data.
Houser and Winter: Behavioral Assumptions and Structural Inference
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5.
INFERENCE UNDER HEURISTICS
An alternative to assuming that all subjects use the rational
expectations decision rule is to impose the weaker assumption
that each subject’s decision rule is a member of a prespecied,
nite candidate set of decision rules. This approach, therefore,
relaxes the rational expectations assumption and allows different subjects to use different decision rules. The candidate set
might (and in our case does) include the optimal rational expectations rule.
The intuition for our approach is as follows. If the rules in
the candidate set generate different state-contingent decisions
for at least one state that occurs with positive probability, then
the decision rule used by each subject is asymptotically (in the
number of decision problems played by the subject) identied.
Moreover, if the choices predicted by the decision rules also
depend on an intertemporal trade-off, then it is possible to estimate jointly the subject’s subjective discount rate and the decision rule he or she uses. The primary advantage of this procedure is that inferences about discount rates are less likely to be
biased by misspecication of the decision rule.
5.1
Specication of the Set of Admissible Heuristics
Following the approach taken in earlier experimental work
on behavior in search tasks by Hey (1982) and Moon and
71
Martin (1990), and in a slightly different context by Müller
(2001), we base our analysis on a prespecied set of candidate
decision rules. We assume in our statistical analysis that subjects use one of these rules to determine their stopping decision
in every round of the search problem, conditionalon observable
state variables and unobservable preference parameters. (In our
case, these are the discount rate and, possibly, parameters of the
decision rules themselves.)
The decision rules we consider fall into three broad classes.
The rst class is the solution to the search problem derived under rational expectations (rule 1). The second class comprises
six sophisticated heuristics that reect some properties of the
optimal solution but are still simple to apply (rules 2–7). The
third class consists of 19 naive heuristics that are not directly
related to the optimal solution but are extremely simple to apply (rules 8–26). Table 4 contains an overview of all 26 rules.
Decision rule 1 is the optimal search rule. Subjects who follow this rule stop searching if the best available price for delivery now is smaller than the reservation price for delivery now,
or if the best available price for delivery later is smaller than
the reservation price for delivery later. The reservation prices
are given by the solution to the search problem under rational
expectations, as discussed earlier in Section 4.
The next class of decision rules is sophisticated heuristics,
which have the reservation price property. We designed these
Table 4. Candidate Decision Rules for the Sequential Search Problem
Number
Description
Decision rule
varies with the
discount rate?
Parameter
values
1
Optimal search
Stop searching when the best available price for delivery now
is lower than the reservation price for delivery now, or when the best
available price for delivery later is lower than the reservation price
for delivery later.
Yes
2, 3, 4
Reservation price heuristic (version 1)
Stop searching when the best available payoff exceeds ° times the
standard deviation of the price distribution.
Yes
° D :5; 1:0; 1:5
5, 6, 7
Reservation price heuristic (version 2)
Stop searching when the best available payoff exceeds ° times the
standard deviation of the price distribution less the accumulated
search cost.
Yes
° D :5; 1:0; 1:5
8, 9, 10
One-bounce heuristic
Search at least ° rounds and then stop as soon as a price
draw would result in a present-value payment that is lower than the
best payment available from earlier rounds.
Yes
° D 2; 3; 4
11, 12
Fixed number of searches heuristic
Stop searching after ° rounds.
No
° D 2; 3
13, 14, 15, 16
Winning streak heuristic (price version)
Stop if you receive more than °1 consecutive price draws below E .p / ¡ °2 ¾ .p /.
No
° 1 D 2; 3
° 2 D :5 ; 1 :0
17, 18, 19, 20
Winning streak heuristic (payoff version)
Stop if you receive more than °1 consecutive price draws that
would result in a payment over °2 ¾ .p /.
Yes
° 1 D 2; 3
° 2 D :5 ; 1 :0
21, 22, 23, 24
Losing streak heuristic (price version)
Stop if you receive more than °1 consecutive price draws above
E .p/ C °2 ¾ .p /.
No
° 1 D 2; 3
° 2 D :5 ; 1 :0
25, 26
Losing streak heuristic (payoff version)
Stop if you receive more than ° consecutive price draws that
would result in a zero payment.
Yes
° D 2; 3
NOTE:
We assume that after stopping according to one of these decision rules, subjects accept the price that results in the greatest present-value payment.
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72
rules to take into account two important determinants of the
optimal reservation price: the known standard deviation of the
price distribution and the accumulated search cost. Subjects
who use one of these heuristics stop searching when the best
available present-value payment exceeds a certain threshold
level. In the rst version (rules 2, 3, and 4), this threshold is
a multiple (.5, 1.0, or 1.5) of the standard deviation of the price
distribution; note that this threshold is constant over time. In
contrast, in the second version we assume that the threshold
falls linearly in the accumulated search cost (rules 5, 6, and 7).
In that version, subjects always stop after a maximum number
of searches. (Because the standard deviation of the price distribution is 10, this happens in round 5, 10, or 15, depending on
the threshold parameter.)
An intuitive interpretation of these heuristics is that subjects
understand that a good realization of the random price process
is a price that is well below the mean of the known distribution, and natural focal points for the distance between a price
draw and the mean of the distribution are multiples of the standard deviation. Following these sophisticated heuristics does
not require solving the underlying dynamic decision problem.
In particular, subjects do not form expectationsover future price
draws.
Figures 6 and 7 compare the reservation prices for various decision rules, discount rates, and delivery dates. Figure 6 shows
reservation prices for items with immediate availability implied
by rules 2 and 5 (i.e., ° D :5) and a discount rate of 1.0 to the
correspondingoptimal reservation price path. The path of rule 2
Figure 6. Reservation Prices for Immediate Delivery Under Rational Expectations and Heuristic Decision Rules. The curves describe the
highest integer price that a subject would pay for an immediately delivered widget for three different decision rules and by cumulative number
of search attempts. The rational expectations reservation prices assume
a discount rate of unity, and were calculated as described in the legend
of Figure 4. The three proles are different. A necessary and sufcient
condition for separate (asymptotic) identication of decision rules is that
they have different reservation price proles. Table 5 shows a sense in
which heuristics 2 and 5 t the data better than the rational expectations rule. A reason is that these heuristics imply relatively high reservation prices, and this is consistent with most subjects’ decisions to stop
searching earlier than rational expectations predicts. ( ¥ , rational expectations,
, heuristic No. 2; N , heuristic No. 5.)
Journal of Business & Economic Statistics, January 2004
begins above the optimal path, is about the same at round 5 and
then falls and stays below. Hence, relative to the rational expectations, following this rule would see subjects either stopping
too easily in early rounds or continuing their search when they
should stop in later rounds. Rule 5, on the other hand, raises the
reservation price at all search durations. Under rule 5, searchers
stop with any positive payoff beginning in round 5, whereas under the optimal rule this does not occur until about round 20.
Figure 7 provides a similar comparison for delayed delivery
reservation prices under a discount rate of .5. The reservation
price rules are qualitatively similar.
Decision rules 8–12 were suggested by Moon and Martin
(1990), following earlier work by Hey (1982). Subjects who
follow rule 8, 9, or 10, the one-bounce rule, search at least two,
three, or four rounds, respectively, and then stop as soon as a
price draw results in a present-value payment that is lower than
the best payment available from earlier rounds. Rules 11 and
12 are even simpler; subjects just do a xed number of searches
(two or three, respectively) and then take the best available
price. These rules were designed for an environment with unknown price distributions.
The remaining decision rules are based on winning streaks
(rules 13–20) and losing streaks (rules 21–26). Subjects who
follow these rules stop searching if they receive two or three
consecutive price draws that are above or below some xed
threshold level. Both winning and losing streak heuristics are
formulated in two versions that differ in how these threshold
levels are constructed. The rst version is based on the price
draws themselves, whereas the second version is based on the
present-value payments implied by the current price draw. The
threshold levels are, once again, formulated as multiples (.5
or 1.0) of the standard deviation, which we interpret as focal
points; details are given in Table 4. All rules based on winning or losing streaks take into account knowledge of the price
Figure 7. Reservation Prices for Delayed Delivery Under Rational
Expectations and Heuristic Decision Rules. The discount factor used to
derive these reservation price proles is .5, and the rational expectations
prole was calculated as detailed in the legend of Figure 4. The shape
of the reservation price prole is inuenced by the discount rate. This is
the source of discount rate identication. ( ¥ , rational expectations;
, heuristic No. 2; N , heuristic No. 5.)
Houser and Winter: Behavioral Assumptions and Structural Inference
distribution—the idea is that subjects use the standard deviation
to determine whether a draw is good or bad. However, subjects
who follow these rules falsely assume, at least implicitly, that
there is some dependence in the price process. That subjects
might use these streak-based rules in search situations such as
that of our experiment can be motivated by results from psychological research on behavior in uncertain environments (see
Rabin 2002).
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5.2
Statistical Classication Procedure
Our approach to drawing joint inferences about preferences
and heuristics is to determine for each subject the number of
choices consistent with each possible combination of decision
rule and discount rate, and then choose that which best ts the
subject’s behavior. The drawback of this procedure is that it can
lead to overtting. The number of decision rules assigned to
subjects is unrestricted, and, at l
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
How Do Behavioral Assumptions Affect
Structural Inference? Evidence From a Laboratory
Experiment
Daniel Houser & Joachim Winter
To cite this article: Daniel Houser & Joachim Winter (2004) How Do Behavioral Assumptions
Affect Structural Inference? Evidence From a Laboratory Experiment, Journal of Business &
Economic Statistics, 22:1, 64-79, DOI: 10.1198/073500103288619386
To link to this article: http://dx.doi.org/10.1198/073500103288619386
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Date: 13 January 2016, At: 00:20
How Do Behavioral Assumptions
Affect Structural Inference?
Evidence From a Laboratory Experiment
Daniel H OUSER
Interdisciplinary Center for Economic Science & Department of Economics, George Mason University, Fairfax,
VA 22030 (dhouser@gmu.edu)
Joachim W INTER
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Mannheim Research Institute for the Economics of Aging, University of Mannheim, D-68131 Mannheim, Germany
(winter@uni-mannheim.de)
We use laboratory experiments to investigate the effect that assuming rational expectations has on structural inference in a dynamic discrete decision problem. Our design induces preferences up to the subjective
rate of time preference, leaving unrestricted both this parameter and subjects’ decision rules. We estimate
subjects’ discount rates under the assumption that all subjects use the rational expectations decision rule,
and under weaker behavioral assumptions that allow decision rule heterogeneity. We nd that certain
sophisticated heuristics t subjects’ decisions statistically signicantly better than rational expectations.
However, the rational expectations assumption does not distort inferences about the cross-sectional discount rate distribution.
KEY WORDS: Discounting; Heterogeneity; Heuristics; Rational expectations; Search; Statistical classication algorithm.
1.
INTRODUCTION
from human behavior. Our study, which examines the effects
of different expectations specications, takes advantage of this
feature of laboratory experiments.
Rational expectations is the standard assumption in virtually
all applied structural econometric work, and this has generated
some criticism. As Blundell (1994, p. 4) pointed out, “structural models are often accused (in many cases quite rightly) of
imposing untenable and unbelievable restrictions on behavior.”
Nevertheless, there seems to be very little evidence suggesting
that the rational expectations assumption has had detrimental
effects in any positive sense. Research using eld data to estimate life-cycle labor supply models with expectations-robust
procedures (see, e.g., Hotz and Miller 1988; Houser 2003) typically obtains results that line up well with results based on rational expectations specications.
Unfortunately, there is a fundamental limitation to the
amount that one can learn about the effect of expectation specication errors from the analysis of eld data. In particular, Rust
(1994) showed that under very general conditions, individuals’
preferences and their expectations processes are jointly nonparametrically unidentied. Because persons’ true period return
functions are not generally known, and because the identication problem implies that multiple period return function and
expectation specication combinations will t the data equally
well in-sample, examining the consequences of expectation
misspecication with eld data presents a substantial challenge.
In contrast, the experimental laboratory provides an environment in which period return functions can be controlled,and the
identication problem consequentlycircumvented.The result is
The latter half of the twentieth century saw the emergence
of a large literature documenting apparently irrational behavior by subjects in economics and psychology experiments. This
has led theorists, particularly over the last 10 years, to begin to assess the equilibrium implications of certain types of
rule-of-thumb behavior (see, e.g., Ellison and Fudenberg 1993;
Krusell and Smith 1995; Lettau and Uhlig 1999; Anderlini and
Canning 2001). Interestingly, however, relatively little effort
has been made by empirical economists to understand whether,
as a practical matter, assuming rational expectations leads to
misspecied models that substantially distort inferences about
structural parameters. A reason for this is that both the analysis of eld data and simulation procedures are limited in the
extent to which they can address this issue. This article uses
a laboratory experiment to investigate the sensitivity of inferences about one structural parameter—the subjective rate of
time preference—to the rational expectations assumption.
The experimental laboratory has proved a useful tool for investigating statistical questions not easily answered through the
analysis of eld data or Monte Carlo simulations. For example,
Cox and Oaxaca (1999) recently used lab experiments to assess whether supply-and-demand parameters can be recovered
from data generated by various common market institutions.
They pointed out that standard procedures to recover these parameters require assumptions about the data-generating process
(DGP), and, to the extent that these assumptions generate specication errors, inferences can be erroneous. Whereas Monte
Carlo simulations can be used to study the effects of any particular misspecication, the DGPs considered in such exercises
might not correspond to any naturally occurring DGP. Cox and
Oaxaca (1999) pointed out that one advantage of the experimental laboratory is that it can be used to study the effects of the
specic misspecication that arises when the true DGP stems
© 2004 American Statistical Association
Journal of Business & Economic Statistics
January 2004, Vol. 22, No. 1
DOI 10.1198/073500103288619386
64
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Houser and Winter: Behavioral Assumptions and Structural Inference
that inferences about the effect of expectation misspecication
on structural inference become possible.
This article analyzes data from a price search experiment that
induces preferences (but not the way in which people form expectations) up to a single structural parameter: the rate of subjective time preference (or the “subjective discount rate”). Our
experimental design, which builds on work of Rapoport and
Tversky (1970) and Seale and Rapoport (1997), provides information on discount rates because it incorporates the possibility that a subject’s experimental earnings might be paid to
them 2 weeks after they complete their laboratory session. Inferences about time preference have been drawn in many other
laboratory experimental studies; Frederick, Loewenstein, and
O’Donoghue (2002) have provided a comprehensive review of
this literature. A recent study by Warner and Pleeter (2001)
took advantage of a natural experiment, providing data on real
choices with large stakes, to estimate time discount rates. Finally, some econometric studies have estimated discount rates
from eld data using indirect inference methods (e.g., Lang
and Ruud 1986; Samwick 1998; Gourinchas and Parker 2001;
Cagetti 2003). In contrast to these studies, our focus is not so
much on estimating the discount rate per se, but rather on understanding the interaction of assumptions about individuals’
behavior and such estimates.
We compare three approaches to inference about subjects’
discount rates. One assumption common to each approach is
that subjects are risk neutral. This assumption is standard in the
price search literature, and using it here allows our results to
be more easily compared to that literature. Moreover, although
preferences regarding risk and intertemporal substitution can
be separately conceptualized in abstract models, say, using the
framework of Kreps and Porteus (1978), they are typically difcult to distinguish in practice. Still the assumption of risk neutrality might fail. Because risk-averse subjects should stop the
search sooner on average, and because it turns out that more impatient subjects should also stop the search sooner on average,
the failure of risk neutrality would tend to bias our discount rate
estimates downward.
Our rst results are derived from a simple revealed preference analysis based on subjects’ accepted payoffs only. This
procedure enables us to establish discount rate bounds under the
weak assumption that subjects prefer higher payoffs to lower
payoffs (and that they are risk neutral). Although this procedure
is robust, it ignores all continuation decisions that, in general,
also contain information about discount rates. In particular, a
subject’s stopping rule within the sequential search task will
vary with his on her rate of time preference, both under rational
expectations and under many alternative decision rules (which
we call “heuristics” in the sequel). Incorporating stronger behavioral assumptions, therefore, typically allows more precise
inference about rates of time preference, albeit at the possible
cost of bias due to model misspecication.
Our second set of results is derived using a very strong behavioral assumption, namely that subjects have rational expectations. Although evidence shows that people do not behave
according to rational expectations in many dynamic decision
tasks (see, e.g., Carbone and Hey 1999), other studies (e.g., Hey
1982) have found that behavior in price-search tasks is often
65
quite close to optimal. This is an important advantage of our design, because in our view, people likely behave nearly optimally
in many actual intertemporal (life-cycle) decision problems.
The nal set of results is based on a more robust approach
that is relatively common in experimental game theory (e.g.,
Weber and Tietz 1975; Selten 1991). We specify a set of candidate decision rules from which the decision rule that best ts observed behavior is selected for each subject. Similar ideas have
been explored by Hey (1981, 1982, 1987) and Moon and Martin (1990) in the experimental analysis of price-search problems, and by Müller (2001) in an experiment on behavior in
life-cycle choice situations. In our case, the set of candidate decision rules includes the optimal, rational expectations decision
rule as well as various more- or less-sophisticated heuristics.
The main difference between earlier studies and our approach
is that for each subject, we jointly determine both the decision
rule and a structural parameter (the subjective discount rate).
Moreover, in contrast to the literature cited earlier, we use a
formal statistical estimation approach rather than informal classication procedures. Our estimation procedure, related to that
of El-Gamal and Grether (1995), relaxes both the assumption
of rational expectations and the assumption that all subjects use
the same decision rule. At the same time, as we explain in Section 5, our procedure avoids the problem of overtting.
Relative to the revealed preference analysis, we nd that imposing behavioral restrictions substantially sharpens inferences
about discount rates. We also show that the data are described
less well by rational expectations behavior than by simpler
heuristics. The reason is that subjects tend to stop searching
sooner on average than is consistent with rational expectations
behavior and low time discounting (i.e., high willingness to accept delayed payoffs). At the same time, rational expectations
behavior under high time discounting is inconsistent with the
frequency of delayed payoffs that are accepted. It turns out that
some of the heuristics that we consider accommodate both relaxed stopping criteria and little time discounting,and it is these
heuristics that best explain our data. Nevertheless, we do not
nd statistically signicant differences between the estimates
of the cross-sectional distribution of discount rates that emerge
under rational expectations and weaker behavioral assumptions
(heuristics). The mean and median of the estimates are .65 and
.68 under the rational expectations analysis and both .70 under
weaker behavioral assumptions.
Although population-level inference seems robust to behavioral specications, inferences about any given individual’s
discount rate can vary substantially across behavioral specications. We cannot know any subject’s true rate of time preference, but nevertheless we can investigate the relationship
between subjects’ estimated discount rates and an external measure of the extent to which they are forward-looking. The external measure that we use is based on a personality survey,
described in detail below, that we administered as part of the
price-search experiment. We nd that neither revealed preference nor rational expectations estimates are statistically signicantly correlated with this external measure, whereas our
heuristic-based discount rate estimates and the external measure are statistically signicantly positively correlated.
The remainder of the article is structured as follows. In Section 2 we introduce our experimental design. We then present
66
Journal of Business & Economic Statistics, January 2004
evidence of discounting and derive bounds on subjective discount rates in Section 3. In Sections 4 and 5 we discuss structural inference about discount rates under rational expectations
and under weaker behavioral assumptions, respectively. In Section 6 we discuss differences between the estimates at both the
aggregate and individual levels. We provide some concluding
remarks in Section 7.
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2.
EXPERIMENTAL DESIGN
In our experiment, subjects make decisions in a standard
price-search problem (Rapoport and Tversky 1970; Seale and
Rapoport 1997) modied to incorporate delayed payments.
Each subject’s goal is to purchase a “widget” that they value
at 100 tokens. The widget is sold at many different locations,
and visiting a new location costs one token. Each location posts
both a price and availability (today or in 14 days) for the widget;
this location-specic information is revealed when the subject
visits that location. Subjects are informed that the price at each
location is drawn independentlyfrom a normal distributionwith
a mean of 100 and standard deviation of 10. Subjects are also
informed that the availability at each location is drawn independently (both across locations and with respect to the price
draw); availabilityis immediate with probability.5 or in 14 days
with probability .5. Subjects may stop their search at any time
and purchase the widget from any of the locations that they visited. (Hence this it is a search problem “with recall”.) An exact
transcript of the instructions for this experiment, along with all
data used in this article, are provided on the JBES website.
This design allows inference about time preference within
an environment where past research has shown behavior to be
nearly rational. Experiments conducted by Hey (1982, 1987)
have suggested that search behavior is closer to optimal when
there is no recall and when subjects have full information about
the price distribution. Our design combines the possibility of
recall and the availability of information about the price distribution. We do this because (1) the possibility of recall together
with the time-differentiated availability of the product (and possibility of a delayed payment of the search surplus) allows us
to observe within-game trade-offs between different points in
time, and (2) giving subjects full knowledge about the price
distribution, along with the opportunity to become experienced
with the task, avoids problems arising from learning about the
parameters of the price distribution that might distort inference
about time preference.
The perception that delayed payment involves risks or transaction costs can bias inference about a subject’s rate of time
preference. We designed the experiment with these concerns in
mind. Specically, subjects were told that at the end of the experiment, their earnings would be put into an envelope. If they
chose an immediate payment, then the envelope would be given
to them immediately. If they chose a delayed payment, their envelope would be stored in a safe until it was delivered or picked
up in 14 days. The instructions read: “We absolutely guarantee
that 14 days from today you will be able to obtain your envelope with your earnings. Moreover, we will pay any costs (such
as mailing costs) required to get your earnings to you after 14
days.”
The experiment was conducted at the University of Arizona’s
Economic Science Laboratory in the Spring and Summer of
2000. Subjects were recruited from the general student population. All experiments were run entirely on computers using
Visual Basic software written by the authors. (This software is
available from either author on request.)
Upon arrival, each subject was given an instruction sheet and
seated privately at a computer terminal. To ensure that subjects
were experienced with the task and comfortable with the computer interface, they were allowed to play an unlimited number
of practice games before playing 10 games that were relevant
to their nal payment. Subjects earned $5 for participating in
the experiment (paid to everybody at the end of the session regardless of their choices) and an additional amount, possibly
time-delayed, determined by selecting randomly the outcome
of 1 of the 10 payment-relevant games. Subjects were paid privately at the end of the experiment. On average, subjects were
in the laboratory for about an hour and earned about $20.
Our results are based on the play of 68 subjects. This yielded
a total of 680 payoff-relevant games and a total of 4,527 rounds
(continuation decisions). On average, subjects played 10 practice games before starting their 10 payoff-relevant games. Subjects on average played more rounds per game, accepted higher
prices, obtained lower payoffs, and also obtained zero payoffs
more often during practice games than in the payment-relevant
games.
3.
EVIDENCE OF DISCOUNTING AND BOUNDS ON
SUBJECTIVE DISCOUNT RATES
Table 1 contains an overview of the intertemporal choices
made by subjects. The top part contains data for all 680 games.
In 371 games subjects chose a product with instant delivery
(corresponding to an instant payoff), and in 309 games subjects chose a product with delayed delivery (corresponding to
a payoff in 14 days). On average, accepted prices are higher,
and the resulting payoffs lower, for chosen products with instant delivery. This is consistent with the notion that at least
some of our subjects discount future payments. Discounting is
also suggested by Figure 1, which shows the distributionsof accepted prices for both delivery dates. Figure 1(a), which shows
the distribution of prices for widgets with immediate delivery,
has more mass at higher prices than Figure 1(b), which shows
the distribution of accepted prices for widgets with delayed delivery.
Our rst approach to drawing inferences about subjects’ discount rates, which can be viewed as revealed intertemporal
Table 1. Accepted Prices and Corresponding Payoffs
Delivery date of the chosen product
Now
Later
Including games with zero payoff
Number of games (680 total)
371
Mean accepted price
86.33
Mean resulting payoff
9.20
Excluding games with zero payoff
Number of games (604 total)
329
Mean accepted price
85.23
Mean resulting payoff
10.37
NOTE:
Standard deviations are in parentheses.
(6.51)
(6.52)
309
85.35
9.73
(5.64)
(6.25)
(5.46)
(5.97)
275
84.71
10.94
(5.18)
(5.55)
Houser and Winter: Behavioral Assumptions and Structural Inference
67
Table 2. Combinations of Payoffs for Accepted and
Best Alternative Prices
(a)
The payoff associated with
the best price available for a
product with the alternative
delivery date is
Higher than the payoff for
the chosen product
Case 1
109 games
Case 4
92 games
Equal to the payoff for the
chosen product
Case 2
3 games
Case 5
4 games
Less than the payoff for
the chosen product
Case 3
217 games
Case 6
179 games
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NOTE:
(b)
Figure 1. Distribution of Accepted Prices. (a) Prices accepted by subjects for immediate-deliverywidgets. (b) Prices accepted by subjects for
delayed-delivery widgets. Accepted prices higher than 100 imply a loss,
because the widget is valued at 100; prices lower than 100 might also
imply a loss, depending on accumulated search cost. In the case of a
loss, actual payoffs were restricted to 0.
preference analysis, requires only the assumption that subjects
prefer higher payoffs to lower payoffs (and the assumption of
risk neutrality that we maintain throughout). With this assumption, we can derive bounds on each subject’s discount rate by
observing their location choices. For example, suppose that a
subject stops searching, and his or her best payoffs are 88 for
immediate-delivery and 90 for delayed-delivery widgets. If the
subject chooses the location that offers immediate delivery, then
it follows that his or her discount rate can be no larger than
88/90, whereas if he or she chooses the location offering delayed delivery, then his or her discount rate can be no smaller
than 88/90. This fact allows us to construct bounds on a sub-
Delivery date of the chosen product
Now
Later
Game counts exclude cases where the payoff is zero.
ject’s discount rate, given that the subject has a meaningful
choice between delivery dates. In particular, only those games
in which the subject has prices that would result in positive payoffs at both delivery dates contain information about his or her
discount rate. This is the case for 604 of the total 680 games.
The bottom part of Table 1 provides descriptive statistics for
the sample that excludes the 76 zero-payoff games from the revealed preference analysis.
Table 2 lists the six situations that a subject can face when he
or she decides to stop searching. These situations are dened by
comparing payoffs for the chosen delivery date with the payoffs
corresponding to the best price available at the alternative delivery date (of which many will be zero as well). Observations
for cases 1, 2, and 3 (delivery now) are shown in Figure 2(a);
observations for cases 4, 5, and 6 (delivery later) are shown
in Figure 2(b). Observations that correspond to cases 1 and 4
are above the 45-degree line, observations that correspond to
cases 2 and 5 are on the 45-degree line, and observations that
correspond to cases 3 and 6 are below the 45-degree line. Table 2 also contains the counts of these observations.
From these observed decisions on intertemporal tradeoffs,
we can compute bounds on subjects’ discount rates. We exclude those cases in which subjects chose combinations of price
and date that were strictly dominated in terms of the associated
payoffs (e.g., all case 4 observations) and all observations with
zero payoff for the best available price at the alternative delivery
date.
We end up with observed choices that allow us to obtain
lower and/or upper bounds that lie in the [0; 1] interval for
many of our subjects. Because all subjects play 10 games, we
have more than one lower or upper bound for some subjects.
In these cases, we compute the intrasubject maximum of the
lower bound for the former, or the intrasubject minimum of the
upper bound for the latter. There were several cases in which a
subject’s decisions did not reveal an upper or a lower bound. In
these cases we set the lower bound to 0 and/or the upper bound
to 1. Once these bounds were determined, we computed the “revealed preference” estimate of the subjective discount rate for
all subjects as the mean of the lower and upper bounds.
Our revealed preference estimation procedure can be interpreted as follows. We rst impose a uniform distribution over
the [0; 1] interval as a prior for the subjective discount rate.
In particular, we rule out the possibility that the discount rates
can be above 1 or below 0. Observed lower bounds above zero
and upper bounds below 1 allow us to narrow the prior. For
those subjects for whom we cannot construct bounds away from
68
Journal of Business & Economic Statistics, January 2004
Table 3. Estimates of Subjective Discount Rates Under Various
Behavioral Assumptions
(a)
Behavioral assumption
No restriction (revealed preference)
Lower bound
Upper bound
Mean of bounds
Optimal behavior
Assigned heuristic
Mean
.22
.97
.60
.64
.70
Minimum
Maximum
Standard
deviation
0
1.00
1.00
1.00
1.00
1.00
.31
.15
.18
.33
.22
:25
:13
0
0
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NOTE: The fth column reports the standard deviation of the cross-sectional distribution of
discount rate point estimates. The total number of subjects is 68.
(b)
Table 3 summarizes the distribution of the bounds on subjective discount rates and the estimates of subjective discount
rates that we obtain. In general, the bounds on discount rates are
relatively wide, and they contain a fair amount of heterogeneity; this corresponds to the scattered price observations shown
in Figure 2. It is evident that the revealed preference approach
without stronger behavioral assumptions does not narrow the
at prior substantially.
In the following sections we discuss how imposing additional
assumptions on subjects’ behaviors allow us to use the information generated in our experiment to obtain more precise estimates of their subjective discount rates. These approaches offer a major improvement. Relative to the revealed preference
analysis presented in this section, more informative estimates
can be obtained when observed behavior in all games is used,
instead of behavior only in those games that have nonzero payoffs and a valid intertemporal trade-off (a subset that is empty
for some subjects).
4.
Figure 2. Payoffs Resulting From the Accepted Price (horizontal
axis) Versus Payoffs That Would Result From the Best Price at the
Alternative Delivery Date (vertical axis). (a) Prices accepted by subjects for immediate-delivery widgets. (b) Prices accepted by subjects
for delayed-delivery widgets. In both panels the diamonds represent the
trade-off between prices for widgets delivered at two different points in
time; therefore, they provide relative prices across time and can be used
to bound discount rates. In (b), diamonds above the 45-degree line represent inconsistent choices, because the payoff associated with the accepted price for delayed-deliverywidgets is lower than the potential payoff that would result from the best available price for immediate-delivery
widgets.
0 and 1, we obtain an estimate of .5. (This is the case for 31 subjects.) Note that using the mean of the set of possible discount
rates as our point estimate is arbitrary, and this choice is made
mainly for convenience.Moreover, this is not a central aspect of
our analysis. Our primary interest is in whether differences appear between approaches, and we use this estimate within each
inferential approach presented later.
INFERENCE UNDER RATIONAL EXPECTATIONS
Figure 3 suggests that the revealed intertemporal preference
approach provides quite imprecise inference about our subjects’
discount rates. Although it is attractive in that it requires very
weak behavioral assumptions, this also means that only the nal
decision in each game—the location decision—informs the discount rate. Under stronger behavioral assumptions, it is usually
the case that continuation decisions inform the discount rate as
well. We adopt this approach in this and subsequent sections.
A natural starting point is to assume that subjects follow the
rational expectations stopping rule. Under this assumption, the
subjective discount rate is identied and can be estimated as the
only free parameter of the corresponding decision process. To
see this, it is useful to derive the optimal solution to the search
task. For the remainder of this article, we label a subject whose
behavior is consistent with this rule as a “rational expectations”
agent.
Because the value of the widget in the experimental task is
100 tokens, and because each search costs 1 token, it is reasonable to derive the solution to the problem under the restriction that the number of searches cannot exceed 100. Let
t 2 f1; 2; : : : ; 100g denote the number of searches that the subject has made. (For notational convenience, we suppress a subject index as long as no confusion can arise.) After making the
tth search the agents’ state vector is St D ft; pb0 ; pb1 g, where pb0 is
the lowest price that the subject has encountered for immediate
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Houser and Winter: Behavioral Assumptions and Structural Inference
69
Figure 3. Discount Rate Estimates Obtained With a Revealed Preference Procedure. The revealed preference estimation procedure provides
only bounds on individuals’ discount rates. Each vertical line indicates an estimated range of possible discount rates for an individual in our sample.
Ranges reect the prior restriction that discount rates must be between 0 and 1. We choose to report the mean of each subject’s estimated range as
the point estimate of his or her discount rate (represented as a solid circle). Although behaviorally robust, taking our data to the revealed preference
procedure yields wide bounds on the discount rate for most subjects, and does not inform the discount rate at all for 31 of our 68 subjects (their
lower bound remains at 0 and their upper bound at 1).
delivery and pb1 is the lowest price encountered for delayed delivery. After the tth search, the agent may either stop searching
and choose a location from which to buy the item, or continue
to search. After the agent has stopped searching, he or she will
buy the item from the location that provides the highest payoff.
The available payoffs are
50 .St / D maxf0; 100 ¡ pb0 ¡ tg;
51 .St / D maxf0; ±.100 ¡ pb1 ¡ t/g;
(1)
where ± is the agent’s rate of time preference. It is convenient to dene the chosen payoff, conditional on stopping, as
5.St / D max.50 .St /; 51 .St //. The agent stops searching only
if the stopping value is higher than the continuation value. The
recursive formulation of this decision problem is, therefore,
Vt .St / D maxf5.St /; E[VtC1 .StC1 /jSt ]g;
(2)
where E represents the mathematical expectations operator and
the expectation is taken with respect to the distribution of
StC1 jSt .
It is immediate from (1) and (2) that this problem has, at
every t, the reservation price property. That is, there is some
best available price for immediate-delivery items such that at
this and any lower price, it is optimal to stop searching, whereas
at every higher best available price, it is optimal to continue
searching. This reservation price depends both on the number of
completed searches and on the best available price for delayeddelivery items. To see this, note that both arguments that enter the maximization operator in (2) are weakly decreasing (in
both prices). When prices are sufciently low, the continuation
value is equal to the stopping payoff less 1. At sufciently high
prices, the stopping payoff is 0, whereas the continuationpayoff
is necessarily bounded above 0 (because all price draws are independent,and the measure of prices that generate positive payoffs is strictly positive). The intermediate value theorem then
implies that the stopping and continuation value cross at least
once. That this crossing is unique follows from the fact that the
slope of the stopping payoff at the crossing point must be ¡1
(because the crossing occurs at a positive value for the stopping
and continuation payoff), and the slope of the continuationpayoff function is evidently bounded from below by ¡1 (because a
one-unit improvement generated by the current price draw can
never do better than improve the continuation payoff by one
unit).
From (1) and (2), it is also easily seen that both reservation
prices depend on the rate of time preference, ±. One way to
see this is simply to observe that the continuation value incorporates both immediate-delivery and delayed-delivery events,
and that the value of delayed delivery clearly depends on the
rate of time preference. Figure 4 plots the path of reservation
prices, calculated by solving the dynamic discrete choice problem implied by (1) and (2) numerically, for the cases ± D 1:0
and ± D :1. Note that by “reservation price,” we mean that the
subject should stop searching if the price draw is less than or
equal to the reservation value. Note also that we report only
integer-valued prices, consistent with the design of our experiment.
When the discount rate is unity, the problem reduces to standard search with recall with a single reservation price. This
reservation price begins at 90, decays slowly to about 87 by
round 10, reaches 80 by the 19th search, and decays at a rate
of about 1 per round from that point forward. When the discount rate is .1, the reservation price for the current payoff follows roughly the same pattern as the earlier case, but is slightly
higher. The reason is that the continuation value falls with the
discount rate, so the person is willing to accept smaller stopping payoffs. Reservation prices for delayed delivery follow an
inverted U-shaped pattern. The very low initial values arise because the continuation value incorporates the chance of nding an item for immediate delivery, and offsetting this expected
value under heavy time discounting requires very low prices for
delayed delivery.
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70
Figure 4. Rational Expectations Reservation Prices for Alternative
Discount Rates and Delivery Dates. The three curves describe the highest integer price that a subject with rational expectations would pay for a
widget, by cumulative number of search attempts. With a discount rate
(delta) of 1, the subject is indifferent between delivery now and delivery
later, so there is only one reservation price curve. When the discount
rate is less than 1, subjects prefer delivery now to delivery later, all other
factors being equal. Because the task is search with recall, to calculate
the reservation prices, we assume that the best available price for delivery at the alternative date is 100. Subjects with discount rates less
than 1 are willing to pay slightly more for delivery now than are subjects with unit discount rates. Subjects incur a 1-unit cost per search,
and their reservation prot level declines with the number of searches.
When delta is 1, the reservation prot begins at 9, but declines to 1 by
the 15th search. ( ¥ , delta D 1:0;
, delta D 0 :1, now; N ,
delta D 0 :1, later.)
Journal of Business & Economic Statistics, January 2004
With reservation prices in hand, it is a simple matter to estimate the discount rate for each individual. The decision task
requires the subject to decide when to stop searching and, once
stopped, the location from which to purchase the widget. Given
any discount rate, in each period, each state-contingentcontinuation decision is either consistent or inconsistent with the reservation price associated with that discount rate. When the subject
stops searching and chooses a location, that choice is either consistent or inconsistent with the given discount rate. Hence if Tj
represents the number of locations searched before the subject
stopped in game j, then the subject makes T1 C ¢ ¢ ¢ C T10 C 10
payoff-relevant decisions during the course of the experiment.
We associate each subject with the discount rate that is consistent with the greatest number of his or her actual decisions.
(This procedure is formalized in the next section.)
In this research we consider only 11 discount rate values,
f0; :1; : : : ; 1:0g. Such a coarse grid reduces the computational
burden, and prior checks conrmed that choices for most subjects in our sample are not sensitive to smaller changes in the
discount rate. Figure 5 illustrates our discount rate inferences
for each subject. The solid circle in the gure represents the
point estimate of the discount rate. For 36 of our subjects, the
best discount rate is not unique; several discount rates explain
the subject’s choices equally well. In these cases the point estimate is assumed to be the mean of the candidate values, which
is described in the gure by a solid circle in the middle of a
line segment. These line segments, therefore, have the same interpretation as those reported in the revealed preference analysis results. In particular, there is a sense in which a longer line
segment implies more uncertainty about a subject’s discount
rate, but the line segments do not represent standard errors.
(Although we do not report them here, it is straightforward to
bootstrap standard errors for the lengths, positions, and, consequently, midpoints of the line segments.) The mean of the discount rate point estimates is .65, and the median is .68. Overall,
the model predicts about 91% of our subjects’ choices.
Figure 5. Estimates of Discount Rates Under Rational Expectations. The procedure for generating these estimates uses the assumption that
individuals behave as if they have rational expectations. Each vertical line indicates an estimated range of possible discount rates for an individual
in our sample. Ranges reect the prior restriction that discount rates must be between 0 and 1. We choose to report the mean of each subject’s
estimated range as the point estimate of his or her discount rate (represented as a solid circle). With the rational expectations procedure, we obtain
relatively tight bounds on the discount rate for most subjects, and every subject’s discount rate is informed by his or her experimental data.
Houser and Winter: Behavioral Assumptions and Structural Inference
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5.
INFERENCE UNDER HEURISTICS
An alternative to assuming that all subjects use the rational
expectations decision rule is to impose the weaker assumption
that each subject’s decision rule is a member of a prespecied,
nite candidate set of decision rules. This approach, therefore,
relaxes the rational expectations assumption and allows different subjects to use different decision rules. The candidate set
might (and in our case does) include the optimal rational expectations rule.
The intuition for our approach is as follows. If the rules in
the candidate set generate different state-contingent decisions
for at least one state that occurs with positive probability, then
the decision rule used by each subject is asymptotically (in the
number of decision problems played by the subject) identied.
Moreover, if the choices predicted by the decision rules also
depend on an intertemporal trade-off, then it is possible to estimate jointly the subject’s subjective discount rate and the decision rule he or she uses. The primary advantage of this procedure is that inferences about discount rates are less likely to be
biased by misspecication of the decision rule.
5.1
Specication of the Set of Admissible Heuristics
Following the approach taken in earlier experimental work
on behavior in search tasks by Hey (1982) and Moon and
71
Martin (1990), and in a slightly different context by Müller
(2001), we base our analysis on a prespecied set of candidate
decision rules. We assume in our statistical analysis that subjects use one of these rules to determine their stopping decision
in every round of the search problem, conditionalon observable
state variables and unobservable preference parameters. (In our
case, these are the discount rate and, possibly, parameters of the
decision rules themselves.)
The decision rules we consider fall into three broad classes.
The rst class is the solution to the search problem derived under rational expectations (rule 1). The second class comprises
six sophisticated heuristics that reect some properties of the
optimal solution but are still simple to apply (rules 2–7). The
third class consists of 19 naive heuristics that are not directly
related to the optimal solution but are extremely simple to apply (rules 8–26). Table 4 contains an overview of all 26 rules.
Decision rule 1 is the optimal search rule. Subjects who follow this rule stop searching if the best available price for delivery now is smaller than the reservation price for delivery now,
or if the best available price for delivery later is smaller than
the reservation price for delivery later. The reservation prices
are given by the solution to the search problem under rational
expectations, as discussed earlier in Section 4.
The next class of decision rules is sophisticated heuristics,
which have the reservation price property. We designed these
Table 4. Candidate Decision Rules for the Sequential Search Problem
Number
Description
Decision rule
varies with the
discount rate?
Parameter
values
1
Optimal search
Stop searching when the best available price for delivery now
is lower than the reservation price for delivery now, or when the best
available price for delivery later is lower than the reservation price
for delivery later.
Yes
2, 3, 4
Reservation price heuristic (version 1)
Stop searching when the best available payoff exceeds ° times the
standard deviation of the price distribution.
Yes
° D :5; 1:0; 1:5
5, 6, 7
Reservation price heuristic (version 2)
Stop searching when the best available payoff exceeds ° times the
standard deviation of the price distribution less the accumulated
search cost.
Yes
° D :5; 1:0; 1:5
8, 9, 10
One-bounce heuristic
Search at least ° rounds and then stop as soon as a price
draw would result in a present-value payment that is lower than the
best payment available from earlier rounds.
Yes
° D 2; 3; 4
11, 12
Fixed number of searches heuristic
Stop searching after ° rounds.
No
° D 2; 3
13, 14, 15, 16
Winning streak heuristic (price version)
Stop if you receive more than °1 consecutive price draws below E .p / ¡ °2 ¾ .p /.
No
° 1 D 2; 3
° 2 D :5 ; 1 :0
17, 18, 19, 20
Winning streak heuristic (payoff version)
Stop if you receive more than °1 consecutive price draws that
would result in a payment over °2 ¾ .p /.
Yes
° 1 D 2; 3
° 2 D :5 ; 1 :0
21, 22, 23, 24
Losing streak heuristic (price version)
Stop if you receive more than °1 consecutive price draws above
E .p/ C °2 ¾ .p /.
No
° 1 D 2; 3
° 2 D :5 ; 1 :0
25, 26
Losing streak heuristic (payoff version)
Stop if you receive more than ° consecutive price draws that
would result in a zero payment.
Yes
° D 2; 3
NOTE:
We assume that after stopping according to one of these decision rules, subjects accept the price that results in the greatest present-value payment.
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72
rules to take into account two important determinants of the
optimal reservation price: the known standard deviation of the
price distribution and the accumulated search cost. Subjects
who use one of these heuristics stop searching when the best
available present-value payment exceeds a certain threshold
level. In the rst version (rules 2, 3, and 4), this threshold is
a multiple (.5, 1.0, or 1.5) of the standard deviation of the price
distribution; note that this threshold is constant over time. In
contrast, in the second version we assume that the threshold
falls linearly in the accumulated search cost (rules 5, 6, and 7).
In that version, subjects always stop after a maximum number
of searches. (Because the standard deviation of the price distribution is 10, this happens in round 5, 10, or 15, depending on
the threshold parameter.)
An intuitive interpretation of these heuristics is that subjects
understand that a good realization of the random price process
is a price that is well below the mean of the known distribution, and natural focal points for the distance between a price
draw and the mean of the distribution are multiples of the standard deviation. Following these sophisticated heuristics does
not require solving the underlying dynamic decision problem.
In particular, subjects do not form expectationsover future price
draws.
Figures 6 and 7 compare the reservation prices for various decision rules, discount rates, and delivery dates. Figure 6 shows
reservation prices for items with immediate availability implied
by rules 2 and 5 (i.e., ° D :5) and a discount rate of 1.0 to the
correspondingoptimal reservation price path. The path of rule 2
Figure 6. Reservation Prices for Immediate Delivery Under Rational Expectations and Heuristic Decision Rules. The curves describe the
highest integer price that a subject would pay for an immediately delivered widget for three different decision rules and by cumulative number
of search attempts. The rational expectations reservation prices assume
a discount rate of unity, and were calculated as described in the legend
of Figure 4. The three proles are different. A necessary and sufcient
condition for separate (asymptotic) identication of decision rules is that
they have different reservation price proles. Table 5 shows a sense in
which heuristics 2 and 5 t the data better than the rational expectations rule. A reason is that these heuristics imply relatively high reservation prices, and this is consistent with most subjects’ decisions to stop
searching earlier than rational expectations predicts. ( ¥ , rational expectations,
, heuristic No. 2; N , heuristic No. 5.)
Journal of Business & Economic Statistics, January 2004
begins above the optimal path, is about the same at round 5 and
then falls and stays below. Hence, relative to the rational expectations, following this rule would see subjects either stopping
too easily in early rounds or continuing their search when they
should stop in later rounds. Rule 5, on the other hand, raises the
reservation price at all search durations. Under rule 5, searchers
stop with any positive payoff beginning in round 5, whereas under the optimal rule this does not occur until about round 20.
Figure 7 provides a similar comparison for delayed delivery
reservation prices under a discount rate of .5. The reservation
price rules are qualitatively similar.
Decision rules 8–12 were suggested by Moon and Martin
(1990), following earlier work by Hey (1982). Subjects who
follow rule 8, 9, or 10, the one-bounce rule, search at least two,
three, or four rounds, respectively, and then stop as soon as a
price draw results in a present-value payment that is lower than
the best payment available from earlier rounds. Rules 11 and
12 are even simpler; subjects just do a xed number of searches
(two or three, respectively) and then take the best available
price. These rules were designed for an environment with unknown price distributions.
The remaining decision rules are based on winning streaks
(rules 13–20) and losing streaks (rules 21–26). Subjects who
follow these rules stop searching if they receive two or three
consecutive price draws that are above or below some xed
threshold level. Both winning and losing streak heuristics are
formulated in two versions that differ in how these threshold
levels are constructed. The rst version is based on the price
draws themselves, whereas the second version is based on the
present-value payments implied by the current price draw. The
threshold levels are, once again, formulated as multiples (.5
or 1.0) of the standard deviation, which we interpret as focal
points; details are given in Table 4. All rules based on winning or losing streaks take into account knowledge of the price
Figure 7. Reservation Prices for Delayed Delivery Under Rational
Expectations and Heuristic Decision Rules. The discount factor used to
derive these reservation price proles is .5, and the rational expectations
prole was calculated as detailed in the legend of Figure 4. The shape
of the reservation price prole is inuenced by the discount rate. This is
the source of discount rate identication. ( ¥ , rational expectations;
, heuristic No. 2; N , heuristic No. 5.)
Houser and Winter: Behavioral Assumptions and Structural Inference
distribution—the idea is that subjects use the standard deviation
to determine whether a draw is good or bad. However, subjects
who follow these rules falsely assume, at least implicitly, that
there is some dependence in the price process. That subjects
might use these streak-based rules in search situations such as
that of our experiment can be motivated by results from psychological research on behavior in uncertain environments (see
Rabin 2002).
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5.2
Statistical Classication Procedure
Our approach to drawing joint inferences about preferences
and heuristics is to determine for each subject the number of
choices consistent with each possible combination of decision
rule and discount rate, and then choose that which best ts the
subject’s behavior. The drawback of this procedure is that it can
lead to overtting. The number of decision rules assigned to
subjects is unrestricted, and, at l