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microeconomics

Questions and Answers of Microeconomics

Suppose a sample yields estimates \(\widehat{\theta}_{1}=5, \widehat{\theta}_{2}=3\) with asymptotic variance estimates 4 and 2 and the correlation coefficient between \(\widehat{\theta}_{1}\) and
Consider NLS regression for the model \(y=\exp (\alpha+\beta x)+\varepsilon\), where \(\alpha, \beta\), and \(x\) are scalars and \(\varepsilon \sim \mathcal{N}[0,1]\). Note that for simplicity
Suppose we wish to choose between two nested parametric models. The relationship between the densities of the two models is that \(g(y \mid x, \beta, \alpha=0)=f(y \mid x, \beta)\), where for
Consider test of \(H_{0}: \mu=0\) against \(H_{a}: \mu eq 0\) at nominal size 0.05 when the dgp is \(y \sim \mathcal{N}[\mu, 100]\), so the standard deviation is 10 , and the sample size is \(N=10\).
Use the health expenditure data of Section 16.6. The model is a probit regression of DMED, an indicator variable for positive health expenditures, against the 17 regressors listed in the second
Consider the exponential-gamma mixture. This model is a special case of a MPH model. The survivor function, conditional on a multiplicative heterogeneity factor \(v\), for the exponential model is
For each of the following situations develop an appropriate expression for the joint likelihood of N observations in terms of the duration density f(t∣x,θ) and survivor function S(t∣x,θ).(a) A
(a) Using a 50\% random sample of the McCall data set estimate the KaplanMeier nonparametric survival and integrated hazard function estimates by type of censoring, that is, by whether transition is
Consider a latent variable modeled by \(\boldsymbol{y}_{i}^{*}=\mathbf{x}_{i}^{\prime} \boldsymbol{\beta}+\varepsilon_{i}\), with \(\varepsilon_{i} \sim \mathcal{N}[0,1]\). Suppose we observe only
Consider the logit model with \(\operatorname{Pr}\left[y=1 \mid x_{1}, x_{2}\right]=\Lambda\left(\beta_{0}+\beta_{1} x_{1 i}+\beta_{2} x_{2}\right)\), where \(\Lambda(z)=e^{z} /\left(1+e^{2}\right)
Suppose we use an index formulation for a discrete choice model but it is felt that the latent variable is strictly positive. This is accommodated by supposing that the latent variable \(y^{*}\) has
Use the health expenditure data of Section 16.6. The model is a probit regression of DMED, an indicator variable for positive health expenditures, against just one regressor for simplicity, NDISEASE,
Continue the analysis of Exercise 14.5.(a) Compare the three binary models on the basis of statistical significance of NDISEASE.(b) Compare the three binary models on the basis of the estimated
Consider a latent variable modeled by y∗=x′β+ε, with ε∼N[0,1]. Suppose we observe only y=2 if y∗
Use a 50\% subsample of the fishing mode choice data of Section 15.2.(a) Estimate the conditional logit model of Section 15.2.1.(b) Comment on the statistical significance of parameter estimates.(c)
Use a 50\% subsample of the fishing mode choice data of Section 15.2.(a) Estimate the multinomial logit model of Section 15.2.2.(b) Comment on the statistical significance of parameter estimates.(c)
Use a 50% subsample of the fishing mode choice data of Section 15.2. Suppose we collapse the model to three alternatives and order the alternatives, with y=0 if fishing from a pier or beach, y=1 if
This question considers the impact of different degrees of truncation in the Tobit model.(a) Generate 200 draws of a latent variable y∗=k+3x+u, where u∼N[0,3] and the regressor x∼ uniform
Consider a latent variable modeled by yi∗=xi′β+εi with εi∼N[0,σ2]. Suppose yi∗ is censored from above so that we observe yi=yi∗ if yi∗
This question considers the consequences of misspecification in the Tobit model. The starting point is the model of Exercise 16.1.(a) Generate y∗ with heteroskedasticity by letting u∼N[0,σ2z],
Consider a Poisson regression model where y∗ has density f∗(y∗)= e−μμy/y∗!!,yi∗=0,1,2,…, and we have independence over i. Because of coding error we only fully observe y∗ when
Suppose \(y=\mathbf{x}^{\prime} \boldsymbol{\beta}+u\), where \(u \sim \mathcal{N}\left[0, \sigma^{2}\right]\), with parameter vector \(\theta=\left[\boldsymbol{\beta}^{\prime},
Consider the multinomial version of the PCGF test given in (8.23) with \(p_{j}\) replaced by \(\widehat{p}_{j}=N^{-1} \sum_{i} F_{j}\left(\mathbf{x}_{i}, \widehat{\boldsymbol{\theta}}\right)\). Show
For the Hausman test given in Section 8.4.1 let \(\mathrm{V}_{11}=\mathrm{V}[\widehat{\theta}], \mathrm{V}_{22}=\mathrm{V}[\tilde{\theta}]\), and
Suppose that two models are non-nested and there are \(N=200\) observations. For model 1 , the number of parameters \(q=10\) and \(\ln L=-400\). For model 3 , \(q=10\) and \(\ln L=-380\).(a) Which
Use the health expenditure data of Section 16.6. The model is a probit regression of DMED, an indicator variable for positive health expenditures, against the 17 regressors listed in the second
Suppose we obtain a kernel density estimate using the uniform kernel (see Table 9.1) with \(h=1\) and a sample of size \(N=100\). Suppose in fact the data \(x \sim \mathcal{N}[0,1]\).(a) Calculate
Suppose we obtain a kernel regression estimate using a uniform kernel (see Table 9.1) with \(h=1\) and a sample of size \(N=100\). Suppose in fact the data \(x \sim \mathcal{N}[0,1]\) and the
This question assumes access to a nonparametric density estimation program. Use the Section 4.6.4 data on health expenditure. Use a kernel density estimate with Gaussian kernel (if available).(a)
This question assumes access to a kernel regression program or other nonparametric smoother. Use the complete sample of the Section 4.6.4 data on natural logarithm of health expenditure \((y)\) and
Consider calculation of the MLE in the logit regression model when the only regressor is the intercept. Then E[y]=1/(1+e−β) and the gradient of the scaled log-likelihood function
Consider the nonlinear regression model y=αx1+γ/(x2−δ)+u, where x1 and x2 are exogenous regressors independent of the iid error u∼N[0,σ2].(a) Derive the equation for the Gauss-Newton
Suppose that the pdf of y has a C-component mixture form, f(y∣π)= ∑j=1cπjfj(y), where π=(π1,…,πC),πj>0,∑j=1cπj=1. The πj are unknown mixing proportions whereas the parameters of the
Let (y1i,y2i),i=1,…,N, have a bivariate normal distribution with mean (μ1,μ2) and covariance parameters (σ11,σ12,σ22) and correlation coefficient ρ. Suppose that all N observations on y1 are
Consider the model \(y=\alpha+\beta x+\varepsilon\), where \(\alpha, \beta\), and \(x\) are scalars and \(\varepsilon \sim\) \(\mathcal{N}\left[0, \sigma^{2}\right]\). Generate a sample of size
Generate a sample of size 20 according from the following dgp. The two regressors are generated by \(x_{1} \sim \chi^{2}(4)-4\) and \(x_{2} \sim 3.5+\mathcal{U}[1,2]\); the error is from a mixture of
Use 200 observations from the Section 4.6.4 data on natural logarithm of health expenditure \((y)\) and natural logarithm of total expenditure \((x)\). Obtain OLS estimates of the model
To estimate the integral I=∫t(x)g(x)dx by Monte Carlo, the sum I^=N−1∑ t(xi)g(xi)/p(xi) is used, where xi are draws from the importance sampling distribution p(x). Show that plim I^=1.
For f(θ)=|Σ|−1/2[1+1v(θ−μ)′Σ−1(θ−μ)]−(ν+α)/2, consider the d-dimensional integral ∫Rdf(θ)dθ. The integrand is the kernel of a multivariate- t density, so the correct answer
For the MSM estimator in Section 12.5.3 suppose that the simulator is the frequency simulator.(a) Show that Vy,u[m^(θ0)]=(1+1/S)∨y[m(θ0)].(b) Hence show that the effect of simulation using the
(a) Write an algorithm for drawing a pseudo-random sample from a threedimensional multivariate normal distribution N[0,Σ] with σjj=1,j=1,2.3, and covariances σ12=σ13=σ23=0.5. Draw a sample of
Write a computing procedure to make draws from a univariate truncated normal density TN[a,b][μ,σ2] using the inverse transform method given in Section 12.8.2. Here [a,b] are lower and upper
Consider the standard binary logit regression model (see Section 14.3).(a) Write down the log-likelihood function.(b) Introduce a random intercept assumption in which the intercept is drawn from a
Some computing packages allow you to draw both Poisson and Gamma pseudorandom numbers directly. It is also known that the negative binomial distribution can be derived as a mixture of Poisson and
Show that if β∣λ∼N[μ,λ−1Σ], and λ∼Gamma⁡[α/2,α/2], then the unconditional distribution of β is a multivariate t-distribution with parameters (μ,Σ,α).
For the gamma regression model of Exercise 5.2, E[y∣x]=exp⁡(x′β) and V[y∣x]= (exp⁡(x′β))2/2(a) Show that these conditions imply that E[x{(y−x′β)2−(exp⁡(x′β))2/2}]=0.(b) Use
Consider the linear regression model for data independent over i with yi= xi′β+ui. Suppose E[ui∣xi]≠0 but there are available instruments zi with E[ui∣zi]=0 and V[ui∣zi]=σi2, where
A nonnegative integer variable \(y\) that is geometric distributed has density (or more formally probability mass function) \(f(y)=(y+1)(2 \lambda)^{y}(1+2 \lambda)^{-(y+0.5)}, y=\) \(0,1,2, \ldots,
Consider the linear regression model yi=xi′β+ui with nonstochastic regressors xi and error ui that has mean zero but is correlated as follows: E[uiuj]=σ2 if i=j,E[uiuj]=ρσ2 if |i−j|=1, and
Consider IV estimation (Section 4.8) of the model y=x′β+u using instruments z in the just-identified case with Z an N×K matrix of full rank.(a) What essential assumptions must z satisfy for the
Select a 50% random subsample of the Section 4.9.6 data on earnings and education, and reproduce as much of Table 4.5 as possible and provide appropriate interpretation. Table 4.5. Returns to
Suppose we obtain model estimates that yield predicted conditional mean \(\widehat{E}[y \mid x]=\exp (1+0.01 x) /[1+\exp (1+0.01 x)]\). Suppose the sample is of size 100 and \(x\) takes integer
Consider the following special one-parameter case of the gamma distribution, \(f(y)=\left(y / \lambda^{2}\right) \exp (-y / \lambda), y>0, \lambda>0\). For this distribution it can be shown that
Continue with the gamma model of Exercise 5-2.(a) Show that \(\partial Q_{N}(\boldsymbol{\beta}) / \partial \boldsymbol{\beta}=\boldsymbol{N}^{-1} \sum_{i} 2\left[\left(y_{i}-\exp
Suppose a sample yields estimates \(\widehat{\theta}_{1}=5, \widehat{\theta}_{2}=3\), se \(\left[\widehat{\theta}_{1}\right]=2\), and se \(\left[\widehat{\theta}_{2}\right]=1\) and the correlation
Consider the nonlinear regression model \(y=\exp \left(\mathbf{x}^{\prime} \boldsymbol{\beta}\right) /\left[1+\exp \left(\mathbf{x}^{\prime} \boldsymbol{\beta}\right)\right]+u\), where the error term
This question presumes access to software that allows NLS and ML estimation. Consider the gamma regression model of Exercise 5-2. An appropriate gamma variate can be generated using \(y=-\lambda \ln
Suppose we estimate the model yi=μ+ui, where ui∼N[0,σi2].(a) Show that the OLS estimator of μ simplifies to μ^=y¯.(b) Hence directly obtain the variance of y¯. Show that this equals White's
Consider the linear regression model with scalar regressor yi=βxi+ui with data (yi,xi) iid over i though the error may be conditionally heteroskedastic.(a) Show that
Consider the linear regression model y=Xβ+u.(a) Obtain the formula for β^ that minimizes Q(β)=u′Wu, where W is of full rank. [The chain rule for matrix differentiation for column vectors x and z
Consider the three-equation model, y= βx+u;x=λu+ε;z=γε+v, where the mutually independent errors u,ε, and v are iid normal with mean 0 and variances, respectively, σu2,σε2, and σv2.(a) Show
Using information from the Application "Flight Insurance," calculate the price of fair insurance if the probability were as high as 0.00000077 , the frequency in 2001 when many people died in the
Using the information in the Application "Botox and Price Discrimination," determine how much Allergan loses by being a single-price monopoly rather than a perfectly price-discriminating monopoly.
Using the figure in the Application "A Semiconductor Integrated Circuit Isoquant," show that as the firm employs additional fixed-proportion technologies, the firm's overall isoquant approaches a
In the Old Testament, God promised riches to Israel if Israel kept God’s commandments. But in the New Testament, Jesus says that it is easier for a camel to go through the eye of a needle than for
If the firm described in Figure 13-4 is producing 4 units, what would you advise it to do, and why?
A business firm is thinking of buying a capital good, which will earn \($2,000\) a year for the next four years and cost \($7,000.\) The interest rate is 8 percent. Should the firm buy the capital
Suppose college students are given two options. With option A, the price a student pays for a class is always the equilibrium price. For example, if the equilibrium price to take Economics 101 is
If people pay for good weather, who ultimately receives the “good-weather payment”?
For this exercise, your instuctor may assign you to work with another student.a. With your partner, survey 10 people (perhaps fellow students in other classes or family members and friends) and ask
Jason Furman served as the chairman of the White House Council of Economic Advisers under President Obama. In an opinion column in the Wall Street Journal discussing President Trump’s tax reform
Evaluate the following statement: “Policies to redistribute income are desperately needed in the United States. Without such policies, the roughly 13 percent of the population that is currently
Imagine that the next time the New England Patriots play the Miami Dolphins at Gillette Stadium in Foxborough, Massachusetts, Patriots star quarterback Tom Brady has a temporary lack of judgment and
The chapter mentions that in 1965, married women with children did an average of 32 hours of housework per week, while men did an average of only 4 hours of housework—a total of 36 hours of
Graphs simplify economic ideas and make them more concrete. Consider the data available on these two Web sites:• www.eia.gov/dnav/pet/pet_pri_gnd_a_epmr_pte_dpgal_w.htm provides data on the retail
José has $55 to spend on apples and oranges. Given the information in the following table, is José maximizing utility? Briefly explain. Total Marginal Utility of Last Price Quantity Utility Unit
Use the following graph to answer the questions.a. If the wage rate and the rental price of ovens are both $100 and total cost is $2,000, is the cost-minimizing point A, B, or C? Briefly explain.b.
Explain why it is true that for a firm in a perfectly competitive market, the profit-maximizing condition MR = MC is equivalent to the condition P = MC.
If a politician running for office does not speak in general terms, does not try to move to the middle of the political spectrum, and does not take polls, is the median voter model therefore wrong?
Voters often criticize politicians running for office who do not speak in specific terms (i.e., who do not specify which spending programs will be cut, whose taxes will be raised, etc.). If voters
Would bad weather be something that could affect voter turnout? Explain your answer.
In Example 2 with equal taxes, did the outcome of the vote make anyone worse off? If so, whom and by how much?
The average farmer is likely to be better informed about federal agricultural policy than the average food consumer is. Why?
Consider special-interest legislation that will transfer$40 million from group A to group B, a group with 10,000 persons. Is this legislation more likely to pass when group A consists of (a) 10,000
Give an example of public-interest talk spoken by a specialinterest group.
Why is rent-seeking activity socially wasteful?
A national survey poll found that two-thirds of Americans polled could not name all three branches of the United States government nor a single Supreme Court justice. Another poll found that 91
What does it mean to say that a theory is falsifiable?
What is the five-step process to building a theory?
Women in some countries have more children than women in other countries because they have a stronger preference for having children. Would an economist be likely to use different preferences to
Consider Jack who attends both his son’s high school football games and a few professional football games too. At which games is Jack more likely to be well behaved? Why?
Kidnappers can’t be rational because no rational person would be a kidnapper. Comment.
Can evidence prove that a theory is correct?
Identify two observations that are inconsistent with the baseball-cap-wearing theory of cheating on an exam.
Ocean views (from a house) are not free. Do you agree or disagree? Explain your answer.
What does it mean if you are included in another person’s utility function in a positive way?
I’ve heard that economists build a lot of theories. That’s why I think that economics might be too abstract a course of study. Theories come with a lot of mathematics, diagrams, and complex
Suppose the United States can produce 120 units of X at an opportunity cost of 20 units of Y and the United Kingdom can produce 40 units of X at an opportunity cost of 80 units of Y. Identify
If a country can produce more of all of its goods than any other country can, would it benefit from specializing and trading? Explain your answer.
Do government officials analyze data to determine what their country can produce at a comparative advantage?

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