Experimetrics Econometrics For Experimental Economics 1st Edition Peter Moffatt - Solutions

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b. The sample size assumed in the Monte Carlo is 1,050. By altering the number in the obs option in the simulate command, obtain the power function for a lower sample size (e.g. 100) and for a higher sample size (e.g. 5,000).Draw all three power functions on the same plot. What happens to the power
a. By changing the number at the end of the command replace y_poor=y_bad if uniform() < 0.3, carry out the Monte Carlo separately for a grid of values of p ranging from 0 to 1. For each run, store the proportion of values of hausman2 that exceed 3.84 (i.e. store the power of the test, for each
2. In Section 9.4, a Monte Carlo study was conducted to investigate the performance of the Hausman test when testing the null hypothesis that subjects truthfully report strength of preference. This null hypothesis was expressed as p = 0 where p is the proportion of subjects who do not report
5. In Section 7.6, there was a variable s representing strength of preference from which an ordinal variable was constructed. The Hausman test was then performed which led to the conclusion that subjects were truthfully reporting“strength of preference”. This was not, in fact, surprising since
3. Use the ologit command in STATA to reproduce the results in Table 3 of Bosman & van Winden (2002).
2. The logit model for binary data was introduced in Exercise 1 of Chapter 6. A way of representing the logit model is:y i= xiβ + ui i = 1, ..., n (7.12)ui ∼ logistic(mean zero)where the logistic (mean zero) distribution for u is defined by the pdf f (u) = exp(u)[1 + exp(u)]2−∞< u < ∞
(g) Using the delta method, find a standard error, and confidence interval, for the “risk-neutral initial endowment” computed in (f). Do this for both probit and logit. Which model estimates this quantity with greater precision?
(f) In Section 6.2.2, the initial endowment necessary to induce risk-neutrality(i.e. a probability of 0.5 for the safe choice being chosen), under the probit model, was estimated to be $9.23. Find an estimate of the same quantity using the logit results.
(e) Do logit and probit lead to similar conclusions?
(d) Write a program in STATA that computes the logit log-likelihood, and maximise it using ML. It should give the same answer as (a).
(c) Use Excel to predict the probability of choosing S over the range of wealth levels.
(b) Test for the presence of a wealth effect.
(a) Estimate the logit model using the command:logit y w.
Consider again the house_money_sim data.
1. In Section 6.2, the probit model for binary data was introduced, and in Section 6.3, the procedure was described for writing a program to maximise the probit log-likelihood function. An alternative to the probit model is the logit model, defined as:P (yi = 1) = exp (β0 + β1wi )1 + exp (β0 +
3. Male proposers tend to offer 2.38 units more when the responder is female than when the responder is male, ceteris paribus. Eckel & Grossman (2001)refer to this effect as the“chivalry effect”. Note that the effect is not statistically significant in this sample
. Proposers tend to offer 3.7 units more when the responder is male, than when the responder is female, ceteris paribus. Note that this effect is only marginally significant.
1. Male proposers tend to offer 4.5 units less than female proposers, ceteris paribus.
Little (1949, p. 92) posed the following question: “how long must a person dither before he is pronounced indifferent?” How can the results presented in Section 5.6, and particularly Figure 5.7, be invoked to provide an answer to this question?
The coefficient of
The effect of experience is quite dramatic. Both τ and τ d are seen to have a strongly significant negative effect on decision time. As previously suggested, the first of these is interpretable as an experience effect, and the second as a boredom effect.
2. This might be interpreted as evidence that subjects are discouraged by complex tasks, and this interpretation could be extended to predict a reduction of effort when complexity reaches intolerable levels. However, there is no evidence that such levels of complexity are being encountered in this
3. Compute the standard deviation sB of the bootstrap test statistics ˆt∗j , j=1,. . . , B.This will be the bootstrap standard error.
but the key point is that they are drawn with replacement. For each bootstrap sample, compute the test statistic, ˆt∗j , j = 1, . . . , B.
2. Generate a healthy number, B, of “bootstrap samples”. These are samples of the same size as the original sample. They are also drawn from the original sample,
1. Apply the chosen parametric test on the data set, obtaining a test statistic, ˆt.
instructions, which, translated from Spanish into English, means roughly “Note that your recipient relies on you”. Dictators in the control group (1) did not see such a line. The experiment was performed both in a classroom (C
3. Branas-Garza (2007) investigate framing effects in dictator game giving. Dictators in the treatment group (2) had an additional line at the end of the
2. Eckel & Grossman (1998) present a large number of tests of the effect of gender on dictator game giving. Their data set is presented within the article. Reproduce as many of their results as you can.
. Burnham (2003) considers the binary decision to give nothing using the binomial test. Reproduce the results appearing in his Table 4.
5. Compare zB against the standard normal distribution in order to find the“bootstrap p-value”.
4. Obtain the new test-statistic zB = ˆt/sB.
3. Compute the standard deviation sB of the bootstrap test statistics ˆt∗j , j=1,. . . , B.
but the key point is that they are drawn with replacement. For each bootstrap sample, compute the test statistic, ˆt∗j , j = 1, . . . , B.
2. Generate a healthy number, B, of “bootstrap samples”. These are samples of the same size as the original sample. They are also drawn from the original sample,
1. Apply the parametric test on the data set, obtaining a test statistic, ˆt.
3. The seller chooses whether to accept the buyer’s proposal, realising the proposed split, or to reject it. If the seller rejects, both parties receive zero, leaving the seller with a net loss of 60.
2. If the seller has invested, the buyer proposes how to split the 100 units between the two players.
The seller is given 60 units, and has the opportunity to invest it in order to create the greater amount of 100 units. Note that this is a dichotomous choice: either he invests 60, or he does not invest at all.
2. What assumptions are required for the random lottery incentive system to be incentive compatible?
Burnham (2003) considers a “photograph” treatment in a dictator game. Given the effect sizes reported, find the optimal sample sizes.
Equation (14.35) gives the value of p3 that is required in order for the choice problem to have threshold risk attitude r ∗, assuming EU. Applying (14.35) to our three values of r ∗ gives rise to the three choice problems presented in Figure 14.2.
Let us (for the moment) restrict these possibilities by requiring that the safer lottery is a certainty of the middle outcome ($10), and the riskier lottery involves only the lowest ($0) and highest ($20) outcomes. When the choice problem is of this form,(14.18) becomes:r ∗ =
All that remains is to reverse-engineer choice problems with these values of r ∗.Of course, for any given value of r ∗, there is an infinity of possible choice problems.
Equation (14.35) gives the value of p3 that is required in order for the choice problem to have threshold risk attitude r ∗, assuming EU. Applying (14.35) to our three values of r ∗ gives rise to the three choice problems presented in Figure 14.2.As required the three probabilities computed in
where p3 is the probability of the highest outcome under the risky lottery. Inverting(14.34), we obtain:p3 = 2−r∗(14.35)
Let us (for the moment) restrict these possibilities by requiring that the safer lottery is a certainty of the middle outcome ($10), and the riskier lottery involves only the lowest ($0) and highest ($20) outcomes. When the choice problem is of this form,(14.18) becomes:r ∗ = −ln(p3)ln(2)(14.34)
All that remains is to reverse-engineer choice problems with these values of r ∗.Of course, for any given value of r ∗, there is an infinity of possible choice problems.
Equation (14.35) gives the value of p3 that is required in order for the choice problem to have threshold risk attitude r ∗, assuming EU. Applying (14.35) to our three values of r ∗ gives rise to the three choice problems presented in Figure 14.2.required the three probabilities computed in
where p3 is the probability of the highest outcome under the risky lottery. Inverting(14.34), we obtain:p3 = 2−r∗(14.35)
Let us (for the moment) restrict these possibilities by requiring that the safer lottery is a certainty of the middle outcome ($10), and the riskier lottery involves only the lowest ($0) and highest ($20) outcomes. When the choice problem is of this form,(14.18) becomes:r ∗ = −ln(p3)ln(2)(14.34)
All that remains is to reverse-engineer choice problems with these values of r ∗.Of course, for any given value of r ∗, there is an infinity of possible choice problems.
Equation (14.35) gives the value of p3 that is required in order for the choice problem to have threshold risk attitude r ∗, assuming EU. Applying (14.35) to our three values of r ∗ gives rise to the three choice problems presented in Figure 14.2.As required the three probabilities computed in
where p3 is the probability of the highest outcome under the risky lottery. Inverting(14.34), we obtain:p3 = 2−r∗(14.35)
Let us (for the moment) restrict these possibilities by requiring that the safer lottery is a certainty of the middle outcome ($10), and the riskier lottery involves only the lowest ($0) and highest ($20) outcomes. When the choice problem is of this form,(14.18) becomes:r ∗ = −ln(p3)ln(2)(14.34)
All that remains is to reverse-engineer choice problems with these values of r ∗.Of course, for any given value of r ∗, there is an infinity of possible choice problems.
As required the three probabilities computed in (14.33) are in agreement with the three required percentage points.
4. Consider a population of subjects of whom a proportion p follow EU, and the remaining (1−p) follow RD, with weighting parameters varying continuously between them. Construct the likelihood function for the resulting mixture model. You may find it useful to consult Conte et al. (2011).
3. Starmer (2000) noted the growing evidence in the literature that choices under risk tend to change with experience, and he poses the question (p.376): “[are] individuals discovering expected utility preferences?” How could the models developed in this chapter be extended to allow a test of
2. How would you go about computing a posterior tremble probability for each individual decision?
1. Some risky choice experiments allow subjects to express indifference. Extend the likelihood function to deal with indifference. Hint: treat the data as ordinal with three outcomes (see Chapter 7) and introduce a threshold parameter κ.
Some readers may find that the annotation is not detailed enough. If this is the case, they are referred back to Chapter 9 for information relating to simulation, and to Chapter 10 for information relating to MSL estimation, including information about the use of Halton draws and use of the d0
Two sets of data are simulated: one assuming the Fechner model is the true model; the other assuming the RP model. Then, both models are estimated on both data sets, leading to four different sets of estimates.
Find an expression for the value of p at which this function crosses the 45 degree line. To do this, you need to set w(p) = p and then solve for p in terms of α and β.
3. Consider the Prelec (1998) weighting function introduced in Section 12.5.1:w(p) = exp−α(−ln p)γ α > 0;γ > 0
b. Show thatβ < 0 implies DRRA, andβ > 0 implies IRRA.
a. Show thatα < 1 implies DARA, α = 1 implies CARA, andα > 1 implies IARA.
2. Consider the expo-power utility function:U(x) = 1 − exp(−βxα) x ≥ 0; α = 0; β = 0; αβ > 0
e. Show that, as r → 1,U1(x) → ln(x).
d. For each of the utility functions, what is the impact of an increase in initial wealth on the propensity to choose the safer choice?
c. For each of U1 and U2, consider the effects on choice (i.e. on the individuals propensity to choose the safer choice) of:i. a doubling of all outcomes in the choice problem;ii. adding a fixed amount c to all outcomes in the choice problem.
b. Prove that: CRRA implies DARA; and CARA implies IRRA.
a. Derive the coefficient of relative risk aversion for U1, and the coefficient of absolute risk aversion for U2. Hence explain why the two utility functions are respectively labelled “CRRA” and “CARA”.
3. Apply the panel hurdle model to the public goods data of Bardsley (2000), contained in the file bardsley, that was analysed using different methods in Chapters 6, 8 and 10.
2. Consider the study of Eckel & Grossman (1998) on dictator game giving and gender. The data is presented within the article. Use their data to estimate a hurdle model with gender in both hurdles. Is the Andreoni & Vesterlund (2001)gender effect evident in this data?
1. Derive the likelihood function for the single hurdle model introduced in Section 11.4.3. Explain the logical problem that arises when one attempts to generalise the single hurdle model to a panel data setting.
Write down the first few numbers of the Halton sequence with p = 5. Check that your answers are correct using the STATA commands:mat p=5 mdraws, neq(1) dr(1) prefix(h) primes(p)
Draw all three power functions on the same plot. What happens to the power function as the sample size increases?
b. The sample size assumed in the Monte Carlo is 1,050. By altering the number in the obs option in the simulate command, obtain the power function for a lower sample size (e.g. 100) and for a higher sample size (e.g. 5,000).
a. By changing the number at the end of the command replace y_poor=y_bad if uniform() < 0.3, carry out the Monte Carlo separately for a grid of values of p ranging from 0 to 1. For each run, store the proportion of values of hausman2 that exceed 3.84 (i.e. store the power of the test, for each
2. In Section 9.4, a Monte Carlo study was conducted to investigate the performance of the Hausman test when testing the null hypothesis that subjects truthfully report strength of preference. This null hypothesis was expressed as p = 0 where p is the proportion of subjects who do not report
How serious are the consequences of neglecting heterogeneity in this setting?
Conduct a similar exercise for the two-limit Tobit model, of the type used in Sections 6.6 and 8.5 for the modelling of contributions in public goods experiments.
1. In Section 9.3.4, a Monte Carlo study was conducted in which the impact of unobserved heterogeneity on the estimation of the binary probit model was assessed.
which an ordinal variable was constructed. The Hausman test was then performed which led to the conclusion that subjects were truthfully reporting“strength of preference”. This was not, in fact, surprising since the variable s was generated as the actual strength of preference in the
5. In Section 7.6, there was a variable s representing strength of preference from
4. Derive the log-likelihood function for the panel ordered probit model. You might find it useful to consult Fréchette (2001).
. Use the ologit command in STATA to reproduce the results in Table 3 of Bosman & van Winden (2002).
Consider the ordered logit model, defined in exactly the same way as the ordered probit model, except for the assumption concerning the distribution of u. Derive the likelihood function for the ordered logit model. Note that the ordered logit model can be estimated in STATA with the command ologit.
where the logistic (mean zero) distribution for u is defined by the pdf f (u) = exp(u)[1 + exp(u)]2−∞< u < ∞ (7.13)
2. The logit model for binary data was introduced in Exercise 1 of Chapter 6. A way of representing the logit model is:y i= xiβ + ui i = 1, ..., n (7.12)ui ∼ logistic(mean zero)
1. Verify that when the ordered probit model is applied to binary data (i.e. data with only two possible outcomes) it becomes equivalent to the binary probit model.
(b) Show that, for a model with continuous outcomes (e.g. a linear regression model), it is possible for the sample log-likelihood to be positive.
4. (a) Show that, for a discrete data model (e.g. a binary data model), the sample log-likelihood is always a negative quantity.
By the way, methods for introducing tremble parameters to models will be covered in much more detail in later chapters.
(c) Consider the public goods game analysed in Section 6.6.3. There, the decision variable was the number of tokens to contribute to the public fund(y, 0 ≤ y ≤ 10). How would you introduce a tremble parameter into this sort of model?
(b) Do the same for the heteregeneous agent model introduced in Section 6.5.1.
. In Section 6.4, we derived the probability of the safe choice being made, and then constructed the sample log-likelihood. What is the probability of the safe choice in the presence of the tremble probability? Amend the ML program in Section 6.4 to include estimation of this parameter.
(a) Consider the Fechner model outlined in Section 6.4. That is, individuals all have CRRA utility, with the same risk aversion parameter, r . For any individual, the safe choice is made if:EU (S) − EU (R) + > 0 where is a Fechner error term, ∼ N0, σ2
Now assume that in any task, with probability ω, the individual loses concentration, and chooses randomly between the two alternatives. The parameter ωhas come to be known as the “tremble probability”. It is usually found to be between 0.01 and 0.10.

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Experimetrics Econometrics For Experimental Economics 1st Edition Peter Moffatt - Solutions

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