Questions and Answers of Introductory Econometrics Modern

The ARCH model is sometimes presented in the following multiplicative form:\[\begin{aligned}& y_{t}=\beta_{0}+e_{t} \\& e_{t}=z_{t} \sqrt{h_{t}}, z_{t} \sim N(0,1) \\& h_{t}=\alpha_{0}+\alpha_{1}
The equations of an ARCH-in-mean model are shown below:\[\begin{aligned}& y_{t}=\beta_{0}+\theta h_{t}+e_{t} \\& e_{t} \mid I_{t-1} \sim N\left(0, h_{t}\right) \\& h_{t}=\delta+\alpha_{1}
Consider the following T-ARCH model:\[\begin{aligned}& h_{t}=\delta+\alpha_{1} e_{t-1}^{2}+\gamma d_{t-1} e_{t-1}^{2} \\& d_{t}=\left\{\begin{array}{lll}1 & e_{t}
The $\operatorname{GARCH}(1,1)$ model shown below can also be reexpressed as an $\operatorname{ARCH}(q)$ model, where $q$ is a large number (in fact, infinity). Derive the ARCH form of a GARCH
a. Let $I_{t-1}=\left\{e_{t-1}, e_{t-2}, \ldots\right\}$. Use the law of iterated iterations to show that $E\left(e_{t} \mid I_{t-1}\right)=0$ implies $E\left(e_{t}\right)=0$.b. Consider the
The estimates for the five models in Table 14.1 were obtained using monthly observations on returns to U.S. Nasdaq stock prices from 1985M1 to 2015M12. Use each of the models to estimate the mean and
The data file share contains time-series data on the Straits Times share price index of Singapore.a. Compute the time series of returns using the formula \(r_{t}=100 \ln \left(y_{t} /
The data file euro contains 204 monthly observations on the returns to the Euro share price index for the period 1988M1 to 2004M12. A plot of the returns data is shown in Figure 14.10(a), together
Monthly changes in the \$$US/\$AUS$ exchange rate $S_{t}$ for the period 1985M7 to 2010M6 are stored in the file exrate5.a. Plot the time series of the changes and their histogram. Are there
Figure 14.11 shows the weekly returns to the U.S. S\&P 500 for the sample period January 1990 to December 2004 (data file $s p$ ).a. Estimate an $\mathrm{ARCH}(1)$ model and check that you
Figure 14.12 shows the daily term premiums between a 180-day bank bill rate and a 90-day bank rate for the period July 1996 to December 1998 (data file term). Preliminary unit root tests confirm that
The data file gold contains 200 daily observations on the returns to shares in a company specializing in gold bullion for the period December 13, 2005, to September 19, 2006.a. Plot the returns data.
The seminal paper about ARCH by Robert Engle was concerned with the variance of UK inflation. The data file $u k$ contains seasonally adjusted data on the UK consumer price index (UKCPI) for the
The data file warner contains daily returns to holding shares in Time Warner Inc. The sample period is from January 3, 2008 to December 31, 2008 (260 observations), and a graph of the returns appears
Consider the quarterly rates of growth contained in data file $g f c$ used in Exercise 13.14. A researcher in the Euro Area (this is the group of countries in Europe where the Euro currency is the
The data file shanghai contains data on the daily returns to the Shanghai Stock Exchange Composite Index from July 7, 1995 to May 5, 2015.a. Plot the time series of returns and their histogram. For
Explain how a data panel differs from either a cross section or a time series of data.
Explain the different ways in which individual heterogeneity can be modeled using panel data, and the assumptions underlying each approach.
Explain how the fixed effects model allows for differences in the parameter values for each individual cross section in a data panel.
Compare and contrast the least squares dummy variable estimator and the fixed effects estimator.
Compare and contrast the fixed effects model and the random effects model. Explain what leads us to consider individual differences to be random.
Explain the error assumptions in the random effects model, and what characteristic leads us to consider generalized least squares estimation.
Describe the steps required to obtain generalized least squares estimates for the random effects estimator.
Explain the meaning of cluster-robust standard errors, and describe how they can be used with pooled least squares, fixed effects, and random effects estimators.
Explain why endogeneity is a potential problem in random effects models, and how it affects our choice of estimator.
Test for the existence of fixed and/or random effects, and use the Hausman test to assess whether the random effects estimator is inconsistent.
Explain how the Hausman-Taylor estimator can be used to obtain consistent estimates of coefficients of time-invariant variables in a random effects model.
Use your software to estimate fixed effects models and random effects models for panel data.
Consider the model\[y_{i t}=\beta_{1 i}+\beta_{2} x_{i t}+e_{i t}\]a. Show that the fixed effects estimator for $\beta_{2}$ can be written as\[\hat{\beta}_{2, F E}=\frac{\sum_{i=1}^{N}
Consider the panel data regression model with unobserved heterogeneity, $y_{i t}=\beta_{1}+\beta_{2} x_{i t}+v_{i t}=\beta_{1}+$ $\beta_{2} x_{i t}+u_{i}+e_{i t}$. Given that assumptions RE1-RE5
In the random effects model, under assumptions RE1-RE5, suppose that the variance of the idiosyncratic error is $\sigma_{e}^{2}=\operatorname{var}\left(e_{i t}\right)=1$.a. If the variance of the
Consider the regression model $y_{i t}=\beta_{1}+\beta_{2} x_{2 i t}+\alpha_{1} w_{1 i}+u_{i}+e_{i t}, i=1, \ldots, N, t=1, \ldots, T$, where $x_{2 i t}$ and $w_{1 i}$ are explanatory
Table 15.9 contains some simulated panel data, where $i d$ is the individual cross-section identifier, $t$ is the time period, $x$ is an explanatory variable, $e$ is the idiosyncratic error,
Using the NLS panel data on $N=716$ young women, we consider only years 1987 and 1988 . We are interested in the relationship between $\ln (W A G E)$ and experience, its square, and indicator
Using the NLS panel data on $N=716$ young women, we consider only years 1987 and 1988 . We are interested in the relationship between $\ln (W A G E)$ and experience, its square, and indicator
Using the NLS panel data on $N=716$ young women, we are interested in the relationship between $\ln (W A G E)$ and experience, its square, and indicator variables for living in the south and
Examples 15.7 and 15.8 estimate a production function by OLS and fixed effects, respectively, with both conventional nonrobust standard errors and cluster-robust standard errors for $N=1000$
This exercise uses the simulated data $\left(y_{i t}, x_{i t}\right)$ in Table 15.9.a. The fitted least squares dummy variable model, given in equation (15.17), is $\hat{y}_{i t}=5.57 D_{1 i}+$
Several software companies report fixed effects estimates with an estimated intercept. As explained in Example 15.6, the value they report is the average of the coefficients of the indicator
Do larger universities have lower cost per student or a higher cost per student? A university is many things and here we only focus on the effect of undergraduate full-time student enrollment
Consider the panel data regression in equation (15.1) for $N$ cross-sectional units with $T=3$ time-series observations. Assume that FE1-FE5 hold.a. Apply the first-difference transformation to
Using the NLS panel data on $N=716$ young women for years 1982, 1983, 1985, 1987, and 1988, we are interested in the relationship between $\ln (W A G E)$ and education, experience, its square,
Using 352 observations on 44 rice farmers in the Tarlac region of the Phillipines for 8 years from 1990 to 1997, we estimated the relationship between tonnes of freshly threshed rice produced (PROD),
The data file liquor contains observations on annual expenditure on liquor (LIQUOR) and annual income (INCOME), (both in thousands of dollars) for 40 randomly selected households for three
The data file liquor contains observations on annual expenditure on liquor (LIQUOR) and annual income (INCOME) (both in thousands of dollars) for 40 randomly selected households for three consecutive
The data file mexican contains data collected in 2001 from the transactions of 754 female Mexican sex workers. There is information on four transactions per worker. ${ }^{17}$ The labels ID and
This exercise uses the data and model in Exercise 15.18.a. Estimate the model assuming random effects and with the characteristics of the sex workers included in the model. Carry out a test of the
This exercise uses data from the STAR experiment introduced to illustrate fixed and random effects for grouped data. In the STAR experiment, children were randomly assigned within schools into three
This exercise uses data from the STAR experiment introduced to illustrate fixed and random effects for grouped data. It replicates Exercise 15.20 with teachers (TCHID) being chosen as the cross
What is the relationship between crime and punishment? This important question has been examined by Cornwell and Trumbull ${ }^{18}$ using a panel of data from North Carolina. The cross sections
Macroeconomists are interested in factors that explain economic growth. An aggregate production function specification was studied by Duffy and Papageorgiou. ${ }^{19}$ The data are in the data
This exercise illustrates the transformation that is necessary to produce GLS estimates for the random effects model. It utilizes the data on investment $(I N V)$, value $(V)$ and capital $(K)$
Consider the production relationship on Chinese firms used in several chapter examples. We now add another input, MATERIALS. Use the data set from the data file chemical3 for this exercise. (The data
The data file collegecost contains data on cost per student and related factors at four-year colleges in the U.S., covering the period 1987 to 2011. In this exercise, we explore a minimalist model
The data file collegecost contains data on cost per student and related factors at four-year colleges in the U.S., covering the period from 1987 to 2011. In this exercise, we explore a minimalist
The data file college cost contains data on cost per student and related factors at four-year colleges in the U.S., covering the period 1987 to 2011. In this exercise, we explore a minimalist model
In this exercise, we re-examine the data in Exercise 15.22, a panel of data from North Carolina. Consider a model in which the log of crime rate (LCRMRTE) is a function of the log of police per
In this exercise, we extend Exercise 15.29 by also considering the possibility that the probability of arrest is jointly determined with the crime rate and the number of police per capita. The idea
Give some examples of economic decisions in which the observed outcome is a binary variable.
Explain why probit, or logit, is usually preferred to least squares when estimating a model in which the dependent variable is binary.
Give some examples of economic decisions in which the observed outcome is a choice among several alternatives, both ordered and unordered.
Compare and contrast the multinomial logit model to the conditional logit model.
Give some examples of models in which the dependent variable is a count variable.
Discuss the implications of censored data for least squares estimation.
Describe what is meant by the phrase "sample selection."
Explain why estimation of a supply and demand model requires an alternative to ordinary least squares (OLS).
Explain the difference between exogenous and endogenous variables.
Define the "identification" problem in simultaneous equations models.
Define the reduced form of a simultaneous equations model and explain its usefulness.
Explain why it is acceptable to estimate reduced-form equations by least squares.
Describe the two-stage least squares estimation procedure for estimating an equation in a simultaneous equations model, and explain how it resolves the estimation problem for least squares.
Our aim is to estimate the parameters of the simultaneous equations modelWe assume that $x_{1}$ and $x_{2}$ are exogenous and uncorrelated with the error terms $e_{1}$ and $e_{2}$.a. Solve
Consider a supply and demand model written in its most general implicit form, using capital Greek letters for the unknown parameters and $E_{i}$ for the random errors,a. Multiply each equation by 3
Consider a supply and demand model written in its most general implicit form, using capital Greek letters for the unknown parameters and $E_{i}$ for the random errors:a. Find the reduced-form
Consider the supply and demand model below:a. Find the reduced-form equations for $p$ and $q$ as a function of the exogenous variable $x$.b. Now suppose that the demand equation is \(q=-5
Consider the supply and demand model below:a. Find the reduced-form equations for $p$ and $q$ as a function of the exogenous variable $x$.b. Now suppose that the demand equation is \(q=-5
Consider the supply and demand model below, where $x$ is exogenous.a. Find the reduced-form equations for $p$ and $q, q=\pi_{11}+\pi_{21} x+v_{1}$ and $p=\pi_{12}+\pi_{22} x+v_{2}$,
Consider the supply and demand model below, where $x$ and $w$ are exogenous.a. Find the reduced-form equations for $p$ and $q, q=\pi_{11} x+\pi_{21} w+v_{1}$ and \(p=\pi_{12} x+\pi_{22}
In macroeconomics, the simple "consumption function" relates national expenditure on consumption goods, CONSUMP $_{t}=$ aggregate consumption, in period $t$ to national income, \(I N C O M
Consider the simultaneous equations model, where $x$ is exogenous.Assume that \(E\left(e_{i 1} \mid \mathbf{x}_{1}\right)=E\left(e_{i 2} \mid \mathbf{x}_{1}\right)=0, \operatorname{var}\left(e_{i
Reconsider the Truffle supply and demand model in Example 11.1. Modify the demand equation as $Q_{i}=\alpha_{1}+\alpha_{2} P_{i}+\alpha_{3} P S_{i}+e_{i}^{d}$, keeping the supply equation
Reconsider the Truffle supply and demand model in Example 11.1. Suppose we modify the supply equation to be $Q_{i}=\beta_{1}+\beta_{2} P_{i}+e_{i}^{s}$, keeping the demand equation unchanged.a. Are
Suppose you want to estimate a wage equation for married women of the formwhere WAGE is the hourly wage, $H O U R S$ is number of hours worked per week, EDUC is years of education, and EXPER is
Australian wine is popular in Australia and worldwide. Using annual data on wine grape transactions $Q(10,000$ tonne units) and price $P(\$ \mathrm{AU} 100$ per tonne) of wine produced in warm
Consider the supply and demand for labor, and in particular that for married women. Wages and hours worked are jointly determined by supply and demand. Let the supply equation beKIDSL6 are the number
Consider the following supply and demand modelwhere $Q$ is the quantity, $P$ is the price, and $W$ is the wage rate, which is assumed exogenous. Data on these variables are in Table 11.7.a.
Example 11.3 introduces Klein's Model I.a. Do we have an adequate number of IVs to estimate each equation? Check the necessary condition for the identification of each equation. The necessary
Example 11.3 introduces Klein's Model I. Here we examine a simplified model that excludes the government sector and allows further practice with simultaneous equations models. Suppose the model is
The labor supply of married women has been a subject of a great deal of economic research. The data file is $m r o z$, and the variable definitions are in the file mroz.def. The data file contains
This exercise examines a supply and demand model for edible chicken, which the U.S. Department of Agriculture calls "broilers." The data for this exercise are in the file newbroiler, which is adapted
This exercise examines a supply and demand model for edible chicken, which the U.S. Department of Agriculture calls "broilers." The data for this exercise are in the file newbroiler, which is adapted
This exercise examines a supply and demand model for edible chicken, which the U.S. Department of Agriculture calls "broilers." The data for this exercise are in the file newbroiler, which is adapted
Use your computer software for two-stage least squares or IVs estimation, and the 30 observations in the data file truffles to obtain 2SLS estimates of the system in equations (11.4) and (11.5).
Estimate equations (11.4) and (11.5) by OLS, ignoring the fact that they form a simultaneous system. Use the data file truffles. Compare your results to those in Table 11.3. Do the signs of the least
Supply and demand curves as traditionally drawn in economics principles classes have price $(P)$ on the vertical axis and quantity $(Q)$ on the horizontal axis.a. Rewrite the truffle demand and
Example 11.3 introduces Klein's Model I. Use the data file klein to answer the following questions.a. Estimate the consumption function in equation (11.17) by OLS. Comment on the signs and
Example 11.3 introduces Klein's Model I. Use the data file klein to answer the following questions.a. Estimate the investment function in equation (11.18) by OLS. Comment on the signs and
Consider the ARDL $(2,1)$ modelwith auxiliary AR(1) model $x_{t}=\alpha+\phi x_{t-1}+v_{t}$, where \(I_{t}=\left\{y_{t}, y_{t-1}, \ldots, x_{t}, x_{t-1}, \ldots\right\}, E\left(e_{t} \mid
Let $e_{t}$ denote the error term in a time series regression. We wish to compare the autocorrelations from an AR(1) error model $e_{t}=ho e_{t-1}+v_{t}$ with those from an MA(1) error model
This question is designed to clarify some of the results used to explain HAC standard errors.a. Given that \(\widehat{\operatorname{var}}\left(\hat{q}_{t}\right)=(T-2)^{-1}

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