Measuring Loss Aversion under Ambiguity: A Method to Make Prospect Theory Completely Observable

We propose a simple, parameter‐free method that, for the first time, makes it possible to completely observe Tversky and Kahneman’s (1992) prospect theory. While methods existed to measure event weighting and the utility for gains and losses separately, there was no method to measure loss aversion under ambiguity. Our method allows this and thereby it can measure prospect theory’s entire utility function. Consequently, we can properly identify properties of utility and perform new tests of prospect theory. We implemented our method in an experiment and obtained support for prospect theory. Utility was concave for gains and convex for losses and there was substantial loss aversion. Both utility and loss aversion were the same for risk and ambiguity, as assumed by prospect theory, and sign‐comonotonic trade‐off consistency, the central condition of prospect theory, held. 


Introduction
Loss aversion, the assumption that people are more sensitive to losses than to commensurate gains, is a central element of prospect theory Tversky 1979, Tversky andKahneman 1992) and key to explaining deviations from expected utility (Rabin 2000(Rabin , pp. 1288(Rabin -1289. There is abundant qualitative evidence for loss aversion, from both the lab and the field (Barberis 2013, Fox and Poldrack 2014, Wakker 2010. However, measuring loss aversion is difficult. It requires the simultaneous measurement of utility for gains and utility for losses, which is complicated by prospect theory's assumption that decision weighting for gains and losses may differ. As a result, existing measurements of loss aversion impose simplifying assumptions, typically linear utility for gains and losses and no probability weighting. Abdellaoui et al. (2007) was the first to propose a method for measuring loss aversion that did not have to impose simplifying assumptions about utility or probability weighting. Their method is design ed for decision under risk, where objective probabilities are known. In most real-world decisions (e. g., the success of new medicines, the dangers of climate change, returns on investments in R&D), objective probabilities do not exist or are unknown and such decisions under ambiguity are now widely studied in both the empirical and the theoretical literature. It is difficult to extend the method of Abdellaoui et al. (2007) to decisions under ambiguity. 1 This paper introduces a method to measure loss aversion under ambiguity without making simplifying assumptions about prospect theory's parameters. It makes it possible, for the first time, to completely observe Tversky and Kahneman's (1992) prospect theory. 2 Parameter-free methods to measure prospect theory's other parameters had been introduced before. Wakker and Deneffe (1996) showed how utility for gains and losses can be measured separately. Abdellaoui (2000) and Bleichrodt and Pinto (2000) showed how probability weights can be measured in decision under risk. Abdellaoui et al. (2005) showed how event weights can be measured in decision under ambiguity. Abdellaoui et al. (2007) showed how loss aversion can be measured in decision under risk. There did not yet exist a method to measure loss aversion under ambiguity and this is what our paper achieves.
Hence, this paper completes a program to make prospect theory observable. 1 This extension requires finding events with decision weight ½, which can be complex. 2 Throughout this paper we use the term prospect theory for the 1992 version of the theory and the term original prospect theory (OPT) for the 1979 version. Because we only consider two-outcome prospects, OPT is the special case of prospect theory for decision under risk in which probability weighting for gains and losses are the same.
Our method is simple and uses only one response mode, which reduces the cognitive burden on subjects. It can quantify loss aversion through three preference elicitations and it does not require the complete measurement of utility. Our method is based on the trade-off method of Wakker and D eneffe (1996). In its original form the trade-off method can only measure the utility for gains and the utility for losses separately and, consequently, it cannot measure loss aversion. We extend the tradeoff method so that it can measure the utility for gains and losses simultaneously, and thus loss aversi on. This extension is not only useful from an empirical perspective, but also has theoretical merits. T here is a close connection between measurements using the trade-off method and axiomatizations o f decision theories (Köbberling and Wakker 2003). Our method may help to simplify existing preferen ce characterization and to develop new ones.
Because our method can completely measure prospect theory's utility function, it also permits new t ests of prospect theory. We implemented our method in an experiment and show that our measure ments can easily be used to test the central condition of prospect theory, sign-comonotonic trade-off consistency. We also test whether both utility and loss aversion are the same under risk and ambiguit y, as assumed by prospect theory. Our data is consistent with prospect theory. We could neither reject sign-comonotonic trade-off consistency nor in most cases the null hypotheses that utility and loss aversion were the same under risk and ambiguity. Utility had prospect theory's hypothesized shape, concave for gains and convex for losses, and there was substantial loss aversion.

Binary prospect theory
Consider a decision maker who has to make a choice in the face of ambiguity. Ambiguity is modeled through a state space . Exactly one of the states will obtain, but the decision maker does not know which one. Subsets of are events and denotes the complement of .
Prospects map states to outcomes. Outcomes are money amounts and more money is preferred to less. In our measurements, we will only use two-outcome prospects , signifying that the decision maker obtains € if event occurs and € otherwise. If probabilities are known, we will write for the prospect that pays € with probability and € with probability 1 − . We will refer to as an ambiguous prospect (meaning that probabilities are unknown) and to as a risky prospect (meaning that probabilities are known).
The decision maker has preferences over prospects and we use the conventional notation ≻, ≽, and ∽ to denote strict preference, weak preference, and indifference. Preferences are defined relative to a reference point 0 . Gains are payoffs higher than 0 and losses are payoffs lower than 0 . A prospect is mixed if it involves both a gain and a loss. For mixed prospects, the notation signifies that is a gain and is a loss. A gain prospect involves no losses (i.e., both and are at least as great as 0 ) and a loss prospect involves no gains. For gain and loss prospects the notation signifies that the absolute value of exceeds the absolute value of (i.e., for gains ≥ and for losses ≤ ).
Under binary prospect theory (PT) the decision maker's preferences over mixed prospects are evaluated by: and preferences over gain or loss prospects by: where = + for gains and = − for losses. is a strictly increasing, real-valued utility function that satisfies ( 0 ) = 0. The utility function is a ratio scale and we are free to choose the utility of one outcome other than the reference point. is an overall utility function that includes loss aversion. In empirical applications is often decomposed in a basic utility function, which captures the decision maker's attitudes towards final outcomes and which can be interpreted as the rational part of utility, and a loss aversion coefficient capturing attitudes towards gains and losses (Köbberling and Wakker 2005, Köszegi and Rabin 2006, Sugden 2003. Our method does not require this decomposition. However, it does allow to decompose into , the basic utility function, and loss aversion if this is considered desirable. The event weighting functions , = +, −, assign a number ( ) to each event such that The event weighting functions depend on the sign of the outcomes and may be different for gains and losses. They need not be additive. For gains, binary PT contains most ambiguity models as special cases, 3 as was pointed out by Luce (1991) and Ghirardato and Marinacci (2001). These ambiguity models only differ when the number of outcomes is at least three. Equations (1a) and (1b) represent the extension of these models to include sign-dependence.
Binary PT evaluates mixed risky prospects by and gain and loss risky prospects as is a strictly increasing probability weighting function that satisfies (0) = 0 and (1) = 1 and that, again, may differ between gains and losses. Hence, in the evaluation of risky prospects the event weighting functions are replaced by probability weighting functions . Equations (2a-b) include most theories of decision under risk as special cases. 4

Previous evidence
Because we concentrate on utility and loss aversion in this paper, we will only discuss the empirical literature on these two elements of prospect theory. For an extensive review of the literature on probability weighting and event weighting see Wakker (2010) and Fox and Poldrack (2014). Tversky and Kahneman (1992) assume that utility differs between gains and losses and is S-shaped: concave for gains and convex for losses. In addition, they assume that utility is steeper for losses than for gains, reflecting loss aversion. Nearly all the empirical evidence on utility comes from decision under risk. There is much evidence that utility for gains is indeed concave (Wakker 2010), but for losses the evidence is somewhat mixed. Although most studies found convex utility, some studies also found linear or concave utility (for example, Bruhin et al. 2010). For losses, utility usually was closer to linearity than for gains.
Empirical evidence on utility under ambiguity is scarce. Abdellaoui et al. (2005) confirmed that utility under ambiguity was concave for gains and slightly convex for losses. Their parametric estimates were close to those previously obtained under risk, but they did not directly measure utility under risk. Abdellaoui et al. (2011) and Vieider et al. (2013) measured utility under risk andunder 1989), -maxmin expected utility (Ghirardato et al. 2004), and contraction expected utility (Gajdos et al. 2008). ambiguity for small stakes and under parametric assumptions about utility. Abdellaoui et al. (2011) found that utility was moderately concave for both risk and ambiguity, while Vieider et al. (2013) found linear utility.
Nearly all empirical measurements of loss aversion made simplifying assumptions, typically assuming linear utility and either ignoring probability weighting ( Most studies found loss aversion coefficients around 2, meaning that losses weight approximately twice as much as absolutely commensurate gains (Booij et al. 2010, Fox andPoldrack 2014). A difficulty in comparing the results of these studies is that they not only made different parametric assumptions, but also used different definitions of loss aversion.
Finally, even though binary PT is consistent with much of the empirical data on decision under risk and ambiguity and includes many models as special cases, there is some evidence challenging it. For example, Birnbaum and Bahra (2007) and Wu and Markle (2008) obtained violations of binary PT for mixed prospects. Because of this negative evidence, we included a test of sign-comonotonic tradeoff consistency, the main condition underlying binary PT, in our experiment. This test is explained below.

Measurement method
Our method for measuring utility and loss aversion consists of three stages and is summarized in Table 1. In the first stage, a gain and a loss are elicited that connect utility for gains (measured in the second stage) with utility for losses (measured in the third stage). The measurements in the second and in the third stage employ the trade-off method of Wakker and Deneffe (1996). Within each domain, we determine a standard sequence of outcomes such that the utility difference between successive elements of the sequence is constant. The trade-off method is commonly used in decision theory (Wakker 2010), but thus far it could only be used to measure utility for gains and utility for losses separately. It could not be used to measure loss aversion, which requires that the utility for gains and the utility for losses can be compared. Our method measures utility for gains and utility for losses jointly by eliciting a standard sequence of outcomes that goes through the reference point, and, consequently, it can measure loss aversion. In all the derivations presented below we impose no parametric assumptions on utility and the weighting functions and , = +, −. Hence, our method is parameter-free. Our method only asks subjects to respond in terms of money and uses no other response scale. This reduces the cognitive demands on subjects. The third column shows the quantity that is assessed in each of the three stages of the procedure. The fourth column show s the indifference that is elicited. The fifth column shows the stimuli used in our experiment. ℓ and were used to tes t binary PT (see Section 4 for explanation).

Assessed quantity
Indifference Choice variables Stage 1 ~0 = €2000 = color of a ball drawn from an unknown Ellsberg urn (for the case of risk we replace by = ½)

First stage: connecting utility for gains and utility for losses
We start by selecting an event that will be kept constant throughout the first stage and a gain .
(3) (5) gives Equation (6) defines the first elements 1 + and 1 − of the standard sequences of gains and losses that we will elicit in the second and third stages.
For choice under risk, the elicitation of 1 + and 1 − is similar except that the event is replaced by a known probability , and that the weights + ( ) and − ( ) are replaced by + ( ) and − (1 − ), respectively.

Second stage: measurement of utility for gains
In the second stage, we elicit a standard sequence of gains. Let ℓ be a prespecified loss. We first elicit the loss ℒ < ℓ such that the decision maker is indifferent between the prospects 1 + ℒ and ℓ 0 , where 1 + is the gain that was elicited in the first stage. We may select an event ′ different from the event used in the first stage, but, for notational convenience, we will continue using the symbol for the selected event. In our experiment, we used the same event in all stages to simplify the tasks for the subjects. The indifference 1 + ℒ~ℓ 0 implies that Rearranging Eq. (7) and using ( 0 ) = 0 gives, Next, we elicit the gain 2 + such that 2 + ℒ~1 + ℓ. From this indifference we obtain after rearranging Combining Eqs. (8) and (9) gives : We proceed by eliciting a series of indifferences + ℒ~− 1 + ℓ, = 2, … , , to obtain the sequence { 0 , 1 + , 2 + , … , + }. It is easy to see that for all , ( + ) − ( −1 + ) = ( 1 + ) − ( 0 ). For decision under risk, we apply the above procedure with the event replaced by a probability (which can be different from the probability used in the first stage).

Third stage: measurement of utility for losses
The standard sequence of losses is constructed similarly. We select a gain ℊ and an event and elicit the gain such that

Experiment
We will next implement our method in an experiment. By exploring whether utility and loss aversion are the same for risk and ambiguity we test prospect theory. The experiment also contains a test of sign-comonotonic trade-off consistency, the central condition of prospect theory.

Experimental set-up
Subjects were 75 economics students of the Erasmus School of Economics, Rotterdam (29 female).
Each subject was paid a flat fee of €10 for participation in the experiment. Before conducting the actual experiment, the experimental protocol was tested in several pilot sessions.
The experiment was run on computers. Subjects answered the questions individually in sessions of at most two subjects. They first received instructions about the tasks and then completed five training questions. Subjects were told that there were no right or wrong answers and that they should go through the experiment at their own pace. They could approach the experimenter if they had any questions regarding the experiment. A session lasted 40 minutes on average.
The order in which utility under risk and ambiguity were measured was randomized between 6 Again, we may select an event " different from the events employed in the other two stages.
sessions. When a subject had completed the first part of the experiment, the experimenter would approach her to explain the next part. Within the risk and ambiguity elicitations, the order in which the gain sequence and the loss sequence were elicited was also randomized. The first stage, the elicitation of the amounts 1 + and 1 − , always came first because it served as an input for the other stages.
We did not immediately ask subjects for their indifference values, but, instead, first used three binary choice questions to zoom in at them and only then asked subjects for their indifference value.
Examples of this zooming-in procedure can be found in the Appendix. We applied a choice-based elicitation procedure as previous research suggests that it leads to more reliable results than directly asking for indifference values (Bostic et al. 1990).

Details
The method described in Section 3 requires the prior specification of some stimuli. The final column of Table 1 shows the stimuli we selected for the experiment. We made the common assumption that the reference point 0 was equal to 0. In the risk condition, the outcome of a prospect was determined by drawing a ball from an urn containing five red balls and five black balls. Subjects could state which color they preferred to bet on with the chance of winning always equal to 50 percent. In the ambiguity condition, the outcome of a prospect was determined by drawing a ball from an urn containing ten red and black balls in unknown proportions. Again, subjects could select the color they preferred to bet on to avoid suspicion (Pulford 2009, Viscusi andMagat 1992).
For both gains and losses, we elicited six points of the utility function under risk and six points of the utility function under ambiguity. Next to these elicitations, we performed a second smaller sequence in the domain of gains where we used a different gauge amount ℓ. In the main elicitation we set ℓ = −€300. In the second elicitation, where we only elicited 2 + and 3 + , we set ℓ = € 0. Under binary PT the elicitations of 2 + and 3 + should not depend on the selected value of ℓ. This second elicitation tested sign-comonotonic trade-off consistency (Köbberling and Wakker 2003). 7 Figures 1-3 show the displays used under ambiguity. The screens under risk were similar, except that the two branches would simply say 50% rather than "Red" or "Black". Figure 1 displays the typical decision that subjects had to make. Subjects faced a choice between two prospects denoted as alternatives A and B. They could not state indifference. By choosing between the two prospects, the subject narrowed down the interval in which her indifference value should fall.

Figure 2: Scrollbar screen under ambiguity
After narrowing down the interval thrice, we presented subjects with a scrollbar (Figure 2). The scrollbar allowed subjects to specify their indifference value up to €1 precision. The range of the scrollbar was wider than the interval, so that subjects could correct any mistakes they might have made. The way in which subjects used the scrollbar also gives an indication of the quality of the data.
If many subjects would provide answers that did not align with their previous choices, possibly even violating stochastic dominance, this signals poor understanding of the task. After specifying a value with the scrollbar, subjects were asked to confirm their choice (Figure 3). If they cancelled their choice, the process started over. If subjects confirmed their choice, they moved on to the next elicitation.  We also analyzed the utility function by parametric estimation. We employed the power/constant relative risk aversion (CRRA) family, , the most commonly used parametric family. For gains [losses] > 1 corresponds to convex [concave] utility, = 1 corresponds to linear utility, and < 1 corresponds to concave [convex] utility. Estimation was by nonlinear least squares. We also performed a mixed-effects estimation in which each individual parameter was estimated as the sum of a fixed effect, common to all subjects, and an individual-specific random effect. The mixed-effects estimation led to the same conclusions and will therefore not be reported.
A potential problem in estimating a model like binary PT is collinearity between utility and the event weights. The trade-off method avoids this problem. By keeping event weighting fixed during the elicitation of utility, the event weights drop from the equations and utility can be measured independent of event weighting. Hence, collinearity is completely excluded. This is an additional advantage of our method.

Loss aversion
There exist several definitions of loss aversion. Abdellaoui et al. 2007) concluded that the definitions of Kahneman and Tversky (1979) and Köbberling and Wakker (2005) were empirically most useful and we will use these. Other definitions (Wakker and Tversky 1993, Bowman et al. 1999, Neilson 2002) turned out to be too strict for empirical purposes, leaving many subjects unclassified. Kahneman and Tversky (1979)  more than half of the observations. Köbberling and Wakker (2005) defined loss aversion as the kink of utility at the reference point.

Results
For one subject the program crashed and we lost his data. Three subjects violated stochastic dominance in critical, early steps of the measurement procedure. Violations of stochastic dominance at these early measurements undermine subsequent answers and subjects committing them were removed from the analyses. For the remaining 71 subjects, we could determine the entire utility function, for both gains and losses and under both risk and ambiguity.

Consistency checks
We included a number of repetitions to test for consistency. First, in each of the six standard sequences (the short and the long gain sequences and the loss sequence for both risk and ambiguity), we repeated the final iteration in the elicitation of 2 , = +, −. Subjects made the same choice in 63.6% of the repeated choices. Reversal rates around ⅓ are common in the literature (Stott 2006). Moreover, our consistency test was strict as we repeated the final choice of the iteration process and subjects were close to indifference in this choice. There were no differences in consistency between risk and ambiguity.
Furthermore, at the end of eliciting the long gain sequence, we elicited 4 + again, both for risk and for ambiguity. The correlation between the original measurement and the repeated measurement of 4 + was almost perfect. 10 For risk, Kendall's  was 0.92, for ambiguity it was 0.94.
As a final indication of consistency, we compared whether the final answer provided by using the scrollbar fell within the interval as set up by the bisection procedure. Subjects provided answers that aligned with their original choices. Furthermore, when a subject's final answer was outside the bisection interval, it typically only violated the final choice, probably indicating that they were close to indifference at this point.

Sign-comonotonic trade-off consistency
As explained in Section 4, we elicited two sequences of gains, a longer one based on ℓ = −€300, which we use in the main analysis, and a shorter one based on ℓ = €0. If our subjects behaved according to binary PT and satisfied sign-comonotonic trade-off consistency, then the values of 2 + and 3 + in the short sequence should be equal to those obtained in the long sequence.
We could not reject binary PT, for both risk and ambiguity. The correlation between the obtained values was substantial. For risk, Kendall's  was 0.57 for 2 + and 0.51 for 3 + . For ambiguity, these values were 0.70 for 2 + and 0.64 for 3 + . All correlation coefficients differed from 0 ( < 0.001).
Moreover, for ambiguity, we could not reject the null hypotheses that the values of 2 + and 3 + obtained in the short sequence were equal to those obtained in the long sequence (Wilcoxon test, both > 0.72). For risk, the values of 2 + differed marginally ( = 0.08), but the values of 3 + did not differ ( = 0.19). Hence, even though 3 + was chained to 2 + , the marginal difference for 2 + did not carry over to 3 + .  To investigate these patterns more thoroughly, we move to the individual level analysis. Table 2 show s that the classification of subjects according to the shape of their utility function was very similar for risk and ambiguity and we could not reject the null hypothesis that the overall distribution of classific ations between the two conditions was the same (Fisher's exact test, = 0.97). The common pattern was S-shaped utility: concave for gains and convex for losses. Less than 20% of the subjects behaved according to the traditional assumption in decision theory that utility is concave throughout. The parametric results confirmed the above conclusions. Table 3 shows the medians of the estimated individual CRRA functions. Utility was mostly concave for gains and convex for losses. Under both risk and ambiguity, 31 subjects (44%) had S-shaped utility.  An advantage of our method is that it immediately reveals that there is loss aversion in the sense of Kahneman and Tversky (1979) when + > − − . 11 As Figure 2 clearly shows, this held true for all j, under both risk and ambiguity. We obtain an aggregate measure of loss aversion by regressing the + on (− − ) . The ′ in Figure 2 display the coefficients from this regression. Both ′ (for risk and ambiguity) exceeded one ( -test, < 0.01) and the values were close to those observed previously for risk (Fox and Poldrack 2014). We could not reject the hypothesis that the values of were the same for risk and ambiguity ( -test, = 0.32).  Moving to the individual level, we found that + > − − for all j (Wilcoxon test, all < 0.01), which is consistent with the existence of loss aversion à la Kahneman and Tversky (1979). Furthermore, + − − ⁄ did not differ between risk and ambiguity for any j (Wilcoxon test, all > 0.25), which is consistent with the hypothesis of prospect theory that loss aversion is the same under risk and under ambiguity.  Table 4 shows the results of the individual analyses of loss aversion based on Kahneman and Tversky's (1979) and Köbberling and Wakker's (2005) definitions. The table clearly shows evidence of loss aversion, irrespective of the definition used and regardless of whether we took response errors into account. According to both definitions, the median loss aversion coefficients for risk and ambiguity did not differ (Wilcoxon test, both > 0.26) and they were moderately correlated (both Kendall's  > 0.37, < 0.001).

Loss aversion
The two measures of loss aversion were substantially correlated. Kendall's  was 0.78 for risk and 0.82 for ambiguity (all < 0.001 ). It is comforting to observe that these two distinct measures, one of a local nature and relying on a single kink in the slope of the utility function, and the other global and relying on different absolute utilities associated with the same absolute money amounts in the positive and negative domain, showed a high degree of consistency in classifying subjects.

Discussion
Our data is consistent with prospect theory. Both utility and loss aversion were close for risk and ambiguity, as assumed by prospect theory. The results also supported sign-comonotonic trade-off consistency, the central condition of prospect theory. Finally, utility was S-shaped, concave for gains and convex for losses and there was substantial loss aversion.
An easy response strategy in measurements using the trade-off method is to let the outcomes of the standard sequence increase by the difference between the gauge outcomes (ℒ and ℓ in the sequence of gains and ℊ in the sequence of losses). This would bias the results in the direction of linear utility. We checked for this heuristic by counting the number of subjects for whom the outcomes of the standard sequence (approximately) increased by the difference between the gauge outcomes but found little evidence to support it.
We used large payoffs because we were interested in studying both utility curvature and loss aversion. Utility curvature is typically modest over small intervals (Luce 2000, Wakker andDeneffe 1996) and we were concerned that it would be hard to detect differences between utility under risk and ambiguity for small stakes. Because we used large losses, all choices were hypothetical. It is impossible to find subjects willing to participate in an experiment where they can lose substantial amounts of money. Because all but one of the questions involved losses, we could not play out one of the gain questions for real either, as subjects would know immediately which question would be played out for real. The literature on the importance of real incentives is mixed. Most studies found that for small to modest stakes there is little or no effect of using real instead of hypothetical choices for the kind of tasks that we asked our subjects to perform, except that hypothetical responses tend to be noisier (Bardsley et al. 2010).
Our method is chained (adaptive) in the sense that previous responses are used in the elicitation of subsequent choices. Chaining may lead to error propagation, where errors made in one particular choice affect later choices. We checked for the impact of error propagation using the simulation methods developed by Bleichrodt and Pinto (2000) and Abdellaoui et al. (2005). In both simulations, we confirmed the conclusions from those studies that the impact of error propagation on measurements using the trade-off method was negligible. 12 We also repeated the parametric analysis of utility accounting for serial correlation in the error terms. 13 The estimates were similar to the ones reported in Section 5. Hence, we conclude that the chained nature of our measurements did not affect the results.

Conclusion
In many real-world problems probabilities are unknown. To apply prospect theory to such decision situations requires methods to measure its parameters. This paper shows how utility and loss aversion can be measured in decision under ambiguity. Our method, for the first time, makes prospect theory completely observable. By combining our measurements with the method of Abdellaoui et al. (2005) all prospect theory's parameters can be measured without imposing simplifying assumptions. Our paper completes a program to make prospect theory empirically observable. Our method allows new tests of prospect theory's assumptions and an experimental implementation showed support for two of these assumptions: that sign-comonotonic trade-off consistency holds and that both utility and loss aversion are the same for risk and ambiguity. We hope that by providing a simple way to measure prospect theory our method will foster its applications.
12 Bleichrodt et al. (2010) also concluded that error propagation was negligible in their measurements using the trade-off method. 13 We assumed that the error terms followed an AR(1) process + −1 = with normally distributed wi th expectation 0 and variance 2 and estimated this using generalized least squares. Start value: -388 Interval: [-576,-200]