On the measurement of risk aversion from experimental data

Discussion Paper #D-16/2004 On the Measurement of Risk Aversion from Experimental Data by Mette Wik, Tewodros Aragie Kebede, Olvar Bergland and Stein Holden Department of Economics and Resource Management Agricultural University of Norway PO Box 5033, NO-1432 Ås, Norway http:/www.nlh.no/ior/ e-mail: [email protected] This series consists of papers intended to stimulate discussion. Some of the papers are of a preliminary character and should thus not be referred to or quoted without permission from the author(s). The interpretations and conclusions in this paper are those of the author(s) and do not necessarily represent the views of the Department of Economics and Resource Management nor of the Agricultural University of Norway. 1 On the Measurement of Risk Aversion from Experimental Data Mette Wik Tewodros Aragie Kebede Olvar Bergland Stein T. Holden Department of Economics and Resource Management, Agricultural University of Norway, P.O. Box 5003, 1432 Ås, Norway E-mail address: [email protected] Abstract Attitudes towards risk are measured for households in Northern Zambia using an experimental gambling approach with real payoffs that at maximum were equal to 30 percent of average total annual income per capita. The results of the experiment show decreasing absolute risk aversion and increasing partial risk aversion. Determinants of risk aversion are investigated using random effects interval regression model exploiting the panel data structure of the repeated experiments. Wealth indicator variables are found to be significant and partial relative risk aversion decreases as wealth increases. Females are found to be more risk averse than males. Keywords risk aversion, random effects interval regression, expected utility, Zambia 2 I. Introduction Knowledge of how subsistence farmers make decisions is important in determining strategies for agricultural development. There is evidence that poor peasant farmers are risk averse (Moscardi and de Janvry, 1977; Dillon and Scandizzo, 1978; Binswanger, 1980, 1981, 1982 and Binswanger and Sillers, 1983), and that their production and economic environments are characterized by a high degree of uncertainty (Roumasset, 1976). These general conclusions and observations have stimulated extensive research into the effects of risk on peasants' adaptation. Considerable research has attempted to provide empirical evidence of individuals' risk attitudes. These attempts can be classified into two general categories: econometric approach, and experimental elicitation techniques. The econometric approach is based on individuals' actual behaviour. The pioneering work by Moscardi and de Janvry (1977) used a safety-first rule approach. Later, individual risk attitudes have been elicited assuming expected utility maximization. One of the best-known methods is that of Antle (1983, 1987, 1989), but Bardsley and Harris (1987), Love and Buccola (1991), Pope and Just (1991), Saha et al. (1994), Chavas and Holt (1996) and Bar-Shira et al. (1997) are all examples of econometric approaches assuming expected utility maximization. The econometric approach is criticized for confounding risk behaviour with other factors such as resource constraints faced by decision makers (Eswaran and Kotwal 1990). Thus, it may appear as if individuals are more risk averse than they truly are (Binswanger 1982, Wik and Holden, 1996). This is particularly important in developing countries where market imperfections are prominent and production and consumption decisions, therefore, are non-separable (Singh et al., 1986, de Janvry et al., 1991, Sadoulet and de Janvry 1995). 3 The experimental approach is based on hypothetical questions regarding risk alternatives or risky games with or without real payments. Dillon and Scandizzo (1978) used hypothetical questions to elicit risk attitudes of subsistence farmers in northeast Brazil. Binswanger (1980, 1981), Binswanger and Sillers (1983) and Quizon et al. (1984) used risk games with real payments to measure peasants' risk preferences in a large-scale experiment in rural India. Others who have applied the same general experimental method are Sillers (1980) in the Philippines, Walker (1980) in El Salvador, Grisley and Kellog (1987) in Thailand, and Miyata (2003) in Indonesia. The existing experimental methods have one limitation in that they have not explored fully the potential of the available experimental data. Since the studies have conducted the experiments repeatedly at different levels of risk, the availability of repeated data on individuals can be explored in panel data structure to understand the interactions and dynamics of risk aversion when individuals are exposed to different experimental levels of risk. In this study, using the experimental approach carried out in northern Zambia, we apply both a panel data technique and a pooled interval regression to measure risk attitude, and identify how these estimates are correlated with individual and socio-economic characteristics such as sex, age, wealth etc. In Section 2, we briefly discuss expected utility (EU) theory and define some measures for risk aversion. Section 3 presents the methodological approach for the experiment. In section 4, we present the results of the experiment. Section 5 discusses the empirical framework. Empirical results are discussed in section 6. Section 7 concludes. II. Measures of Risk Aversion 4 Risk aversion is defined with reference to the von Neumann-Morgenstern expected utility function. The three measures most commonly used are: (1) absolute risk aversion A(W) (Pratt, 1964); (2) relative risk aversion R(W) (Arrow, 1971); and (3) partial risk aversion P(Wo ,π ) (Menezes and Hanson, 1970 and Zeckhauser and Keeler, 1970): A(W ) = − U ′′(W ) U ′′(Wo ) U ′′(π ) =− =− U ′(W ) U ′(Wo ) U ′(π ) R(W ) = −W U ′′(W ) = WA(W ) U ′(W ) P(Wo ,π ) = −π U ′′(Wo + π ) = πA(Wo + π ) U ′(Wo + π ) where W indicates total wealth, U ′ and U ′′ indicate the first and second derivatives of the expected utility function, W0 denotes initial wealth and π denotes stochastic income. At the point ( W = Wo + π ) the three measures are related to each other as: R(W ) = Wo A(W ) + P(Wo ,π ) Absolute risk aversion is an appropriate measure to describe situations in which income or gain is fixed, and initial wealth is variable. It is commonly assumed that absolute risk aversion decreases as wealth rises, which implies that wealthier individuals should be more willing to accept a given fair gamble. When both income and initial wealth change proportionally, relative risk aversion is the appropriate measure. Arrow (1971) hypothesised increasing relative risk aversion. This implies that an individual's willingness to accept a gamble decreases when both wealth and all outcomes of a fair gamble are increased proportionally. The measure of partial 5 risk aversion is appropriate to describe situations when initial wealth is fixed and income is variable. Bar-Shira et al. (1997) show that decreasing absolute risk aversion (DARA) implies decreasing partial risk aversion (DPRA) with respect to initial wealth, and that increasing relative risk aversion (IRRA) implies increasing partial risk aversion (IPRA) with respect to income. The opposite does not necessarily hold. It is possible to have DRRA and IPRA at the same time. III. Experimental Procedure The setting The study was carried out in 110 randomly sampled households in 6 different villages in Northern Zambia in 1994. Three villages were situated in a fairly densely populated area (26-82 persons/km2) close to the province capital, Kasama. The remaining three villages were located in a rather sparsely populated area (<6 persons/km2) approximately 60 km further north. All households except 4 were cultivating land. Farmers in this area of Zambia participate in two main production systems. They produce several crops in a traditional slash and burn system called chitemene, and they produce monoculture maize in “permanent” fields. On average, each household cultivates 1.4 ha of land, out of which 0.6 ha is used for maize production. Average total income per capita (including own consumption of major subsistence crops) was very low, approximately 65,000 Kwi, or roughly 110 US $. Survey procedure In the experiment, subjects were confronted with a series of choices among sets of alternative prospects (gambles) involving real money payment. The amounts of money were relatively large compared to average income in the area. (The highest 6 possible gain was approximately 30 percent of average total annual income per capita.) The subject's choice among these alternative prospects is taken as an indication of the degree of her risk aversion. The experiment follows a method developed by Binswanger (1980). A series of schedules of prospects (called games) similar to those shown in Table 1 were presented to each subject. Each game lists six prospects, each with 50% probability of winning. Each subject was asked to select one of the six prospects: O, A, B, C, D, or E. Once chosen, a coin was tossed and the subject received the left hand amount if the coin showed heads and the right hand amount if the coin showed tails. If the subject selected alternative O, she received 100 Kw whether she got a head or a tail. If she chose alternative A instead of O, her expected gain increased by 35 Kw, but a bad luck alternative (heads) would now give her 10 Kw less in return than she would have received with the safe alternative O. In other words, in choosing A instead of O, the standard deviation in gain is increased from 0 to 45 Kw. For the successive alternatives, A to B, B to C, and C to D, the same is true: the expected gain increases, but so does the spread between the two outcomes. Alternative D and E have the same expected gain, but alternative E has larger spread. (Table 1 approx. here) When risk is viewed in terms of uncertainty in gains, income, or wealth, as in utility-based choice theories, the prospects involve more risk the further down one gets in Table 1. This means that a decision maker possessing a utility function concave in wealth, income or gain would demand a higher risk premium to accept prospect B rather than its expected outcome than she would demand to accept prospect A rather than its expected outcome. Whether she prefers prospect B to prospect A depends on the degree of concavity of her utility function. The different 7 prospects classified from extreme risk aversion (alternative O) to neutral to preferring (alternative E). To get a unique risk aversion coefficient for each game level, we used the utility function with constant partial risk aversion (CPRA): U = (1 − γ )w(1−γ ) where w is the certainty equivalent of the prospect. The parameter γ will then be equivalent to the constant partial risk aversion coefficient. The fifth column of Table 1 shows the end points of the constant partial risk aversion coefficients implied by each possible choice. The games were played at different levels. Each individual played games 1 to 7 (Table 2) during the first visit, and games 8 through 14 during a visit two weeks later. All gains-only games (game 1 to 11) were derived from 100 Kw game by multiplying all amounts by 10, 50 or 100. The 100 Kw game was played five times, the 1,000 Kw three times, the 5,000 Kw game two times, and the 10,000 Kw game was played once. (Table 2 approx. here) Most of the games were real, i.e. the individual actually received the payment. Because of budget restrictions, we also included hypothetical games. Game 10 was played with real payment for one individual in the household and as a hypothetical game for the other. Before starting the experiments the individuals were told that a few of the games would be hypothetical, but they were not told which games would be hypothetical and which would be real. After each game they were paid the gain if the game was real, or otherwise told the game was hypothetical. In this way we hoped the individuals would play all games as if they were real. The three gain-and-losses games were played at 100, 1000 and 5000 Kw level. The first two games were real, and the last game was hypothetical. In this case, 8 the subjects knew that the game was hypothetical. It is however, difficult to ask poor peasants to participate in games with real losses and put their own money at risk. First, it is hard to defend morally. Second, it would often not be possible to carry out such an experiment, because many would not have sufficient cash to allow them to select one of the riskier alternatives if they so desired. In this case the analyst would no longer be able to distinguish between pure risk aversion and the effect of liquidity constraints. To reduce this effect, we played the gains-and-losses games immediately after having played four rounds of gains-only games. The gains and losses games were only administered to those individuals who had won sufficient money in the previous four games. IV Experimental Results The risk aversion distributions corresponding to different game levels are given in Table 3. The first panel shows the distribution for the games with gains-only, while the third panel shows the distribution for games with gains and losses. Note that the distribution was rather widely spread over all classes of risk aversion, but as the game level rose, the distribution shifted to the left, i.e., risk aversion increased. Even at the lowest level of the game, more than 20 percent chose the alternatives representing severe to extreme degree of risk aversion. When the game level increased to Kw. 10,000, more than 35 percent chose the two most risk averse alternatives. Considering the slight-to-neutral and neutral-to-preferring alternatives, the percentage choosing these alternatives was reduced from 28 percent at the Kw. 100 game level to only 11 percent in the Kw. 10,000 level. The share of responses falling into the intermediate and moderate risk aversion categories remained fairly stable at 45 to 50 percent for all game levels. It appears that the majority of people 9 with initially low levels of risk aversion increased to moderate to intermediate risk aversion when game levels rise. For individuals who initially had moderate to intermediate levels of risk aversion, the level seemed to increase more slowly. For games with gains and losses we see the tendency of increased risk aversion when game levels increase. In addition, we see an inclination of people being more risk averse in these games than in games with gains-only. At the Kw. 1,000 level, more than 45 percent chose the two most risk averse alternatives, while only 12 percent chose the two most risky alternatives. For the gains-only game at Kw. 1,000 level, we found 29 and 24 percent in the respective categories. (Table 3 approx. here) The data in Table 3 suggest that individuals are risk averse and they tend to get more risk averse when the size of the gamble increases. Later in the study we will formally test the hypothesis of increasing partial risk aversion with respect to the size of the gamble. We will also test whether people are more risk averse in games with losses than in games with gains only. V Empirical Framework The participants have, through choosing between six different prospects, revealed which alternative gives the highest utility. Assuming constant partial risk aversion, each observed response represents a category of risk aversion in which the true but unobservable risk aversion falls within. The games were played repeatedly at different game levels. Then by creating a panel data structure the availability of repeated information can be exploited. Suppose the latent dependent variable, risk aversion, y *gm satisfies a classical linear model: 10 y *gm = x g β + v gm where g is the index of individual g in the experiment, and m is the index of game number m. Mg is the size of the experiment, and G is the number of individuals. The 1× K vector x g contains explanatory variables. We also assume v gm = c g + u gm , m = 1,..., M g where c g is an unobserved individual effect and u gm is the idiosyncratic error. y *gm is never observed, but only the range that it falls into, i.e. the data is interval coded. More specifically, the observed variable y gm is now defined as: ( ) y gm ≡ t gm y *gm =O =A if =B if =D if =C =E if if if y *gm ≥ 7.5 7.5 > y *gm ≥ 2.0 2.0 > y *gm ≥ 0.812 0.812 > y *gm ≥ 0.316 0.316 > y *gm ≥ 0 and 0 > y *gm When risk aversion is grouped into intervals, we say that we have intervalcoded data. If we observed risk aversion y * we would just use Ordinary Least Squares (OLS) to estimate β . However, because we only observe whether risk aversion falls into one of several cells, we have a data-coding problem (Wooldridge 2002). We can still consistently estimate β using the method of maximum likelihood if we make some distributional assumptions. values are calculated across groups of observations. Assuming cg ~ N (0,σ c2 ) and Since the observations in the panel data set are not independent, likelihood ( ) u gm ~ N 0,σ u2 , the likelihood for the gth group of observations is 11    7.5 − x gm β − c g  M 1( y gm = O )Φ1 − σu  g   Lg = ∫ ∏   − x gm β − c g  − ∞ m = 1    + 1( y gm = E )Φ  σu     ∞   U − x gm β − c g  D  + ∑ 1( y gm = i )Φ i  σu   = i 1   L − x gm β − c g   − Φ i   σu     2    1  c g   −   2  σ c2    1   e dc g     2π σ c     where Ui and Li are the upper and lower bounds of interval i, i=A,…D; Φ (.) is the cumulative density of normal distribution. For a comparison, we also propose estimating an interval regression model reduces to y *gm = xgm β + u gm with u gm ~ N (0,σ u2 ) . The pooled interval regression is disregarding the panel data structure, i.e. on the pooled data. Hence our model also estimated using the method of maximum likelihood. The likelihood function for the pooled interval data is    U − xgm β − cg   7.5 − xgm β − cg  D  L − xgm β − cg   + ∑1( y gm = i )Φ i  − Φ i  1( y gm = O )Φ1 − G σu σu σu     i =1     = ∏∏   g =1 m =1  + 1( y = E )Φ − xgm β − cg  gm     σ u     Mg Lgm where Ui and Li are the upper and lower bounds of interval i, i=A,…D; Φ(.) is the cumulative density of normal distribution. These models can be easily estimated using the standard software, STATA 8.2. Risk Aversion and personal characteristics In this section we discuss why and how some socio-economic variables and personal characteristics might be correlated with some of the variation in risk aversion. From theory and the common assumption of DARA, we would expect wealthier individuals to be less risk averse. We have included several different wealth variables in the model (income per capita, cash liquidity per capita and education). This is because asset market imperfections severely constrain substitution between different categories of wealth (Reardon and Vosti, 1995; Holden et al., 1998). Under such conditions each asset category may have an 12 independent correlation with risk aversion. All the wealth variables are expected to be negatively correlated with risk aversion. Household size can have two opposing effects on risk aversion. On the one hand, household size can be viewed as a wealth variable. A larger family could represent an increased labour force for the household and thus have a negative effect on risk aversion. Furthermore, household size may also have an indirect negative effect on risk aversion in terms of providing insurance, diversification and coping opportunities. On the other hand, a larger family means more people to feed, which may increase risk aversion. We believe that in poor peasant societies, where children start to work at a young age, the first effect is more prominent; while in a developed economy the second would be more common. Earlier studies are inconclusive. Moscardi and deJanvry (1977) and Feinermann and Finkelshtain (1996) found that increased family size is leading to more cautious and conservative behaviour, while Dillon and Scandizzo (1978) found that farmers with larger households were less risk averse. In most cases, total farm area would represent a type of household wealth. In Northern Zambia, however, there was abundance of land and access to land was not considered a binding constraint. Since access to land is free (except for labour costs of cultivating and transactions costs due to distance from the homestead), we suppose cultivating more land could be a risk-coping strategy for the risk averse. Farmers in the study area participate in two main production systems. They produce several crops in a slash and burn system called chitemene, and they produce monoculture maize in «permanent» fields. Maize production is dependent upon use of chemical fertilisers. This production requires access to credit and a supply of fertilizer. Maize production is considered a more risky activity by the farmers than 13 the production of other crops in chitemene (Wik and Holden, 1996). We would therefore expect more risk averse farmers to have a relatively smaller area for maize-production than would less risk averse farmers. Both total farmland and area of maize are included in the model as per-capita variables. Due to very different roles of men and women in this society, we included a gender dummy-variable in the model (men=0, women=1). We believe that different attitudes to risk might reflect gender differences in the society. In this society we expect women to be more risk averse than men. Women have more responsibilities for providing and preparing the food and for feeding and caring for the children. Traditionally, men were warriors, and supposed to engage in dangerous and risky activities. They should, according to Richards (1939), be brave and willing to take risk to achieve status in the society. We also included a variable for the age of the decision-maker. Moscardi and deJanvry (1977) writes, «it is generally assumed that older farmers tend to be less prone to take risks than younger ones...» We find this to be an assumption without any theoretical grounding, and include the variable, without any a priori expectation of the sign. To believe that past experience with a random process (such as tossing a coin), would influence a person’s next choice, is not common in economic theory. Psychologists, on the other hand, would find it surprising to think that such past experience would not influence future choices. To check whether previous luck had a significant effect on subject’s choices, we included a dummy variable defined as ∑Xi, where i is the game number of previous games (for game 11, i would be 1, 2, 3... 10), and X takes the value of 1 when the person wins (tails) and -1 when he looses (heads). Binswanger (1980) found the effect of previous luck to be highly 14 significant. We expect to find that previous luck does affect people’s choices. Subjects who have experienced previous luck will be more willing to take risk than subjects who have experienced previous losses. As we saw in the section on theory, it has been common in economics to consider risk by looking at changes in total wealth. We saw that another aspect of asset integration is that opportunity losses should be treated in the same way as real losses. Empirical evidence has shown that this is usually not the case. As the famous basketball player and coach Larry Bird remarked, «Loosing hurts more than winning feels good». Research in both economics and psychology has found this to be true (Markowitz, 1952; Kahneman and Tversky, 1979; Hershey and Schoemaker, 1980; Sillers, 1980). People are more risk averse when it comes to losses than to gains. In their prospect theory, Kahneman and Tversky (1979) proposed a convex utility for losses that was steeper for losses than the concave utility for gains. Markowitz (1952) and Hershey and Schoemaker (1980) proposed a utility function in gains and losses, which was concave for small losses and convex for larger ones. To test for a difference between gains-only and gains and losses games we have included a dummy-variable for losses where 0 is equal to gains-only games and 1 is equal to games with gains and losses. We expect to find that people are more risk averse in games with losses. To test whether the level of gains in the games influenced individuals’ risk aversion, we included a variable of overall expected outcome of each particular game. Other researchers (Binswanger, 1980, 1981; Sillers, 1980) have found evidence of people getting more risk averse when there are bigger gains/losses at stake, i.e., that their utility function exhibits increasing partial risk aversion (IPRA) 15 with respect to the possible income of the games. We therefore expect the sign of this variable to be positive. VI Empirical Results In our econometric analysis, we have included all 11 games in the data set. In the regression, we have assigned the same categories for the choices in the “gains and losses” alternatives as for the gains-only alternatives. To check whether individuals actually treat real losses in the same way as opportunity gains and losses, we included a dummy variable for games with losses. A random effects interval regression model and a pooled interval regression model are estimated on the data, using the lower and upper boundaries of the interval for the unobserved risk aversion level, corresponding to the observed choice of the individual, as the dependent variable. Table 4 shows the results from estimating these models. (Table 4 approx. here) The estimates of the random effects model and pooled OLS regression are similar in sign and magnitude. The advantage of using the random effects model is that it gives more efficient and reliable estimates. The Breusch and Pagan (1980) Lagrange multiplier (LM) test provides a test of random effects model against the pooled interval regression model. The specific hypothesis under investigation is the following: H0 : σ c = 0 HA :σc ≠ 0 . The p-value of this test, shown in the table, is found to be highly significant suggesting that random effects model is more suitable in analysing the data at hand. The consistency of this test may be understood by looking at the education and maize 16 per capita variables. These variables are highly significant in the pooled interval regression model but not in the random effects model. This comes from having a more reliable and efficient estimate of the standard errors in using the random effects model. Therefore, the discussion here after will be based on the results from the random effects model. Most of the variables are significant in the random effects model. The wealth variables- log of income per capita and household size are significant at 5% significance levels. All these variables have negative signs indicating that higher wealth is correlated with lower degrees of risk aversion. This is consistent with the common assumption of Decreasing Absolute Risk Aversion (DARA). The cash per capita variable is not significant. The result on the household size variable indicates that the effect of household size as providing labour and improved possibilities of insurance, diversification and coping opportunities are more important than the effect of more people to feed. In this way it seems like farm households in this area behave as if household size is a wealth variable. We also find that the total farm area per capita variable is positively correlated with risk aversion, supporting our hypothesis that when land is abundant, cultivating more land would be a risk coping strategy. Gender differences seem to be significantly correlated (1% level) with differences in risk aversion. We found that women were more risk averse than men. We found no significant correlation between age and risk aversion. Previous luck in random processes seems to influence a person's next choice. The prior luck variable was highly significant (1% level). This suggests a strong impact of prior luck, and may thus imply that people are correcting their subjective probabilities as the game progresses. 17 The dummy variable for loss is also significant at 1% level, and strongly supports the hypothesis that people are more risk averse in games with losses than in games with gains only. This result is also an indication that people do not treat opportunity losses in the same way as real losses, and thus, that asset integration does not hold. We also found that the level of the game has a strong, significant impact on how large a risk people prefer to take in these types of games. People seem to be willing to take less risk when higher gains are at stake. This implies that people are revealing increasing partial risk aversion (IPRA) with respect to the game income level. Another important indicator in the regression is a dummy variable that indicate whether the games is hypothetical or real. The dummy variable, type, is found to be insignificant indicating that there is no significant difference if the game is designed as a hypothetical or real game. This result may be induced from mixing real and hypothetical games in the design of the experiment. If this result is replicable in other studies, it may suggest a cost effective way of carrying out such experiments. VII Conclusion This study measured attitudes towards risk for individuals in Northern Zambia using an experimental gambling approach, similar to that of Binswanger (1980) and Sillers (1980), with real payoffs. The games were defined as both “gains only” and “gainsand-losses” games. Highest possible real gain was approximately equal to 30 percent of average total annual income per capita. 18 In the low levels of games, individuals' choice of alternatives is evenly spread from severe to slight-to-neutral risk aversion. When game level rose, the distribution shifted towards a more risk averse attitude, and 80% of the people revealed moderate to extreme risk aversion. We also found that people were more risk averse in games with gains and losses than in games with gains only. The main contribution of the study is to exploit the panel data structure arising from the repeated nature of the experiments. Random effects interval regression and pooled interval regression models are estimated. Existence of random effects is tested and is found to be highly significant relative to the pooled interval regression model. Hence, using random effects interval regression model, we obtained more efficient and reliable estimates in investigating determinants of partial risk aversion. The results indicate that utility functions should exhibit increasing partial risk aversion and that they should be defined over gains and losses. Wealth does, however, also have a significant effect on partial risk aversion. 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(2002) Econometric analysis of cross section and panel data. Cambridge, MA: MIT Press. 23 Zeckahuser, R. and Keeler E. (1970) Another type of risk aversion, Econometrica, 38, 661-665. 24 Table 1. Payoffs and corresponding risk aversion for ”gains-only” games at 100 Kw level. Bounds on Risk Aversion parameter of Prospect Bad Luck Payoff Good Luck Payoff class CPRA function O 100 100 Extreme >7.5 A 90 180 Severe 7.5-2.0 B 80 240 Intermediate 2.0-0.812 C 60 300 Moderate 0.812-0.316 D 20 380 Slight to neutral 0.316-0 0 400 Neutral to <0 E preferring 25 Table 2. Sequence and Level of Games Day Game Number Game Level in Type Kwacha (Kw) I) Gains Only Day 1 1 100 real ’’ 2 100 real ’’ 3 100 real ’’ 4 1,000 hypothetical ’’ 5 100 real ’’ 6 1,000 real ’’ 7 5,000 hypothetical Day 2 8 100 real ’’ 9 1,000 real ’’ 10 5,000 real/hypothetical ’’ 11 10,000 hypothetical II) Gains and Losses ’’ 12 100 real ’’ 13 1,000 real ’’ 14 5,000 hypothetical 26 Table 3. Percentage distribution of experimental results of risk aversion Severe Extreme Intermediate Slight-to- Neutral-to- No. of neutral preferred observations D E C B A O Moderate II) Percentages of choices at different levels: Gains only games 100 Kw real 6.4 15.6 26.8 22.8 20 8.4 250 1000 Kw real 11.83 18.82 22.31 22.04 16.13 8.87 372 5000 Kw real/hyp 19.33 21.01 26.05 19.33 12.61 1.68 119 10000 Kw hyp 16.39 19.67 28.69 25.41 8.20 1.64 122 III) Alternatives at 100 Kw. level for Gains and Losses Bad Luck (50%) 0 -10 -20 -40 -80 -100 Good Luck (50%) 0 80 140 200 280 300 IV) Percentage of choices at different levels: Gains and Losses games 100 Kw real 21.31 14.75 25.41 9.84 6.56 22.13 122 1000 Kw real 23.77 24.59 22.13 18.03 4.92 6.56 122 5000 Kw real/hyp 20.49 45.90 19.67 8.20 2.46 3.28 122 27 Table 4. Random effects interval regression and pooled OLS regression of partial risk aversion on personal and socio economic characteristics Random effects interval Pooled interval regression regression Variables Coefficients p-value (standard error) sex .734188 .0000687 0.026** -1.445223 0.003*** -.1420829 0.000*** .2882947 0.000*** .0030181 0.191 -.0636008 0.774 -.5397986 0.244 -6.17e-06 0.036** -.1002088 0.000*** .3090606 0.184 .0023044 0.755 -.0730698 0.057* -.5149827 0.004*** (.1792722) 0.590 (.0000114) household size -.1473983 (.0383408) (.2567814) cash per capita 0.000*** (.0073843) (.054627) Logincome -1.428155 (.2323843) (.0105279) education 0.006*** (.033019) (.2203867) age .0000666 (.2169856) (.0406036) type of game 0.003*** (.0000244) (.2064366 previous luck .6953377 (.2307544) (.0000232) dummy for loss P-value (standard error) (.3296883) game level Coefficients -5.87e-06 0.466 (8.04e-06) 0.023** -.0976302 0.001*** 28 (.0439862) 1.874412 farm area per cap (.030593) 0.025** (.8350999) maize area per -2.423155 capita (1.505832) constant 8.544398 Sigma u ( σ u ) Sigma c ( σ c ) ( 1.896847 0.001*** (.5902671) 0.108 -2.525291 0.018** (1.064618) 0.002*** 8.376101 (2.819178) (1.974654) 1.082802 3.126343 (.1235847) (.0752042) 0.000*** 2.93294 Rho= σ c2 σ c2 + σ u2 ) (.0739185) .1199497 - (.0250848) Log-Likelihood Number -2840.4511 of 1353 -2864.88 1353 Observations * significance at 10% level; ** significance at 5% level; *** significance at 1% level 29 Footnotes i 1US Dollar ≅ 600 Kw in 1994 30