Questions and Answers of Introductory Econometrics Modern

The owners of a motel discovered that a defective product was used in its construction. It took seven months to correct the defects during which 14 rooms in the 100-unit motel were taken out of
How much does education affect wage rates? The data file cps5_small contains 1200 observations on hourly wage rates, education, and other variables from the 2013 Current Population Survey (CPS). a.
How much does education affect wage rates? The data file cps5_small contains 1200 observations on hourly wage rates, education, and other variables from the 2013 Current Population Survey (CPS). a.
In this exercise, we consider the amounts that are borrowed for single family home purchases in Las Vegas, Nevada, during 2010. Use the data file vegas5_small for this exercise.a. Compute summary
Discuss how "sampling theory" relates to interval estimation and hypothesis testing.
Explain why it is important for statistical inference that given $\mathbf{x}$ the least squares estimators $b_{1}$ and $b_{2}$ are normally distributed random variables.
Explain the "level of confidence" of an interval estimator, and exactly what it means in a sampling context, and give an example.
Explain the difference between an interval estimator and an interval estimate. Explain how to interpret an interval estimate.
Explain the terms null hypothesis, alternative hypothesis, and rejection region, giving an example and a sketch of the rejection region.
Explain the logic of a statistical test, including why it is important that a test statistic has a known probability distribution if the null hypothesis is true.
Explain the term $p$-value and how to use a $p$-value to determine the outcome of a hypothesis test; provide a sketch showing a $p$-value.
Explain the difference between one-tail and two-tail tests. Explain, intuitively, how to choose the rejection region for a one-tail test.
Explain Type I error and illustrate it in a sketch. Define the level of significance of a test.
Explain the difference between economic and statistical significance.
Explain how to choose what goes in the null hypothesis and what goes in the alternative hypothesis.
There were 64 countries in 1992 that competed in the Olympics and won at least one medal. Let MEDALS be the total number of medals won, and let GDPB be GDP (billions of 1995 dollars). A linear
There were 64 countries in 1992 that competed in the Olympics and won at least one medal. Let MEDALS be the total number of medals won, and let GDPB be GDP (billions of 1995 dollars). A linear
There were 64 countries in 1992 that competed in the Olympics and won at least one medal. Let MEDALS be the total number of medals won, and let GDPB be GDP (billions of 1995 dollars). A linear
Assume that assumptions SR1-SR6 hold for the simple linear regression model, $y_{i}=\beta_{1}+\beta_{2} x_{i}+e_{i}$, $i=1, \ldots, N$. Generally, as the sample size $N$ becomes larger,
If we have a large sample of data, then using critical values from the standard normal distribution for constructing a $\boldsymbol{p}$-value is justified. But how large is "large"?a. For a
We have data on 2323 randomly selected households consisting of three persons in 2013. Let ENTERT denote the monthly entertainment expenditure (\$) per person per month and let INCOME $(\$ 100)$ be
We have 2008 data on INCOME = income per capita (in thousands of dollars) and BACHELOR = percentage of the population with a bachelor's degree or more for the 50 U.S. States plus the District of
Using 2011 data on 141 U.S. public research universities, we examine the relationship between cost per student and full-time university enrollment. Let $A C A=$ real academic cost per student
Using data from 2013 on 64 black females, the estimated linear regression between WAGE (earnings per hour, in \$) and years of education, EDUC is $\widehat{W A G E}=-8.45+1.99 E D U C$.a. The
Using data from 2013 on 64 black females, the estimated log-linear regression between WAGE (earnings per hour, in \$) and years of education, EDUC is $\widehat{\ln (W A G E)}=1.58+0.09 E D U C$.
The theory of labor supply indicates that more labor services will be offered at higher wages. Suppose that HRSWK is the usual number of hours worked per week by a randomly selected person and WAGE
Consider a log-linear regression for the weekly sales (number of cans) of a national brand of canned tuna $(S A L 1=$ target brand sales $)$ as a function of the ratio of its price to the price
Consider the following estimated area response model for sugar cane (area of sugar cane planted in thousands of hectares in a region of Bangladesh), as a function of relative price (100 times the
What is the meaning of statistical significance and how valuable is this concept? A $t$-statistic is $t=(b-c) / \operatorname{se}(b)$, where $b$ is an estimate of a parameter $\beta, c$ is
In a capital murder trial, with a potential penalty of life in prison, would you as judge tell the jury to make sure that we accidently convict an innocent person only one time in a hundred, or use
A big question in the United States, a question of "cause and effect," is whether mandatory health care will really make Americans healthier. What is the role of hypothesis testing in such an
Consider the regression model $W A G E=\beta_{1}+\beta_{2} E D U C+e$. Where $W A G E$ is hourly wage rate in US 2013 dollars. EDUC is years of schooling. The model is estimated twice, once using
A life insurance company examines the relationship between the amount of life insurance held by a household and household income. Let INCOME be household income (thousands of dollars) and INSURANCE
The owners of a motel discovered that a defective product was used during construction. It took 7 months to correct the defects during which approximately 14 rooms in the 100 -unit motel were taken
The owners of a motel discovered that a defective product was used during construction. It took seven months to correct the defects during which approximately 14 rooms in the 100 -unit motel were
The capital asset pricing model (CAPM) is described in Exercise 2.16. Use all available observations in the data file capm 5 for this exercise.a. Construct $95 \%$ interval estimates of
The data file collegetown contains data on 500 single-family houses sold in Baton Rouge, Louisiana, during 2009-2013. The data include sale price (in $\$ 1000$ units), PRICE, and total interior
The data file collegetown contains data on 500 single-family houses sold in Baton Rouge, Louisiana, during 2009-2013. The data include sale price in $\$ 1000$ units, PRICE, and total interior area
We introduced Professor Ray C. Fair's model for explaining and predicting U.S. presidential elections in Exercise 2.23. Fair's data, 26 observations for the election years from 1916 to 2016, are in
Using data on the "Ashcan School," we have an opportunity to study the market for art. What factors determine the value of a work of art? Use the data in the file ashcan_small. [Note: the file ashcan
How much does experience affect wage rates? The data file cps5_small contains 1200 observations on hourly wage rates, experience, and other variables from the March 2013 Current Population Survey
Is the relationship between experience and wages constant over one's lifetime? We will investigate this question using a quadratic model. The data file cps5_small contains 1200 observations on hourly
The owners of a motel discovered that a defective product was used during construction. It took 7 months to correct the defects during which approximately 14 rooms in the 100 -unit motel were taken
We introduced Tennessee's Project STAR (Student/Teacher Achievement Ratio) in Exercise 2.22. The data file is star5_small. [The data file star5 contains more observations and more variables.] Three
We introduced Tennessee's Project STAR (Student/Teacher Achievement Ratio) in Exercise 2.22. The data file is star5_small. [The data file star 5 contains more observations and more variables.] Three
Data on weekly sales of a major brand of canned tuna by a supermarket chain in a large midwestern U.S. city during a mid-1990s calendar year are contained in the data file tuna. There are 52
What is the relationship between crime and punishment? We use data from 90 North Carolina counties to examine the question. County crime rates and other characteristics are observed over the period
Explain how to use the simple linear regression model to predict the value of $y$ for a given value of $x$.
Explain, intuitively and technically, why predictions for $x$ values further from $\bar{x}$ are less reliable.
Explain the meaning of SST, SSR, and SSE, and how they are related to $R^{2}$.
Explain the relationship between correlation analysis and $R^{2}$.
Report the results of a fitted regression equation in such a way that confidence intervals and hypothesis tests for the unknown coefficients can be constructed quickly and easily.
Describe how estimated coefficients and other quantities from a regression equation will change when the variables are scaled. Why would you want to scale the variables?
Appreciate the wide range of nonlinear functions that can be estimated using a model that is linear in the parameters.
Write down the equations for the log-log, log-linear, and linear-log functional forms.
Explain the difference between the slope of a functional form and the elasticity from a functional form.
Explain how you would go about choosing a functional form and deciding that a functional form is adequate.
Explain how to test whether the equation "errors" are normally distributed.
Explain how to compute a prediction, a prediction interval, and a goodness-of-fit measure in a log-linear model.
Explain alternative methods for detecting unusual, extreme, or incorrect data values.
Let Y denote a Bernoulli(θ) random variable with 0 < θ < 1. Suppose we are interested in estimating the odds ratio, γ = θ/ (1 – θ) , which is the probability of success over the
Consider the population modely = xβ + uE(u|x) = 0,where the 1 × (k + 1) vector x isx = (1, x1, x2, . . . , xk).Let {(xi,yi): i = 1, 2, . . . , n} be a random sample. Show that Assumptions E.3 and
Assume that the model y = Xβ + u satisfies the Gauss-Markov assumptions, let β̂ be the OLS estimator of β. Let Z = G(X) be an n × (k + 1) matrix function of X and assume that Z’X[a(k + 1) ×
Assume that the model y = Xβ + u satisfies the Gauss-Markov assumptions, let G be a (k + 1) × (k + 1) nonsingular, nonrandom matrix, and define δ = Gβ, so that d is also a (k + 1) × 1 vector.
Let β̂ be the OLS estimate from the regression of y on X. Let A be a (k + 1) × (k + 1) nonsingular matrix and define zt ; xtA, t = 1, . . . , n. Therefore, zt is 1 × (k + 1) and is a nonsingular
Let β̂ be the (k + 1) × 1 vector of OLS estimates.(i) Show that for any (k + 1) × 1 vector b, we can write the sum of squared residuals as(ii) Explain how the expression for SSR(b) in part (i)
Let xt be the 1 × (k + 1) vector of explanatory variables for observation t. Show that the OLS estimator β̂ can be written asDividing each summation by n shows that β̂ is a function of sample
(i) Let A be an n × n symmetric matrix such that A and In – A are both positive semi-definite. Show that 0 ≤ aii ≤ 1 for i = 1, . . . , n, where αii is the ith diagonal element of A.(ii)
Let X be an n × k matrix partitioned as X = (X1 X2), where X1 is n × k1 and X2 is n × k2.(i) Show thatWhat are the dimensions of each of the matrices?(ii) Let b be a k × 1 vector, partitioned
Take as given the properties of the chi-square distribution listed in the text. Show how those properties, along with the definition of an F random variable, imply the stated property of the F
Let a be an n × 1 nonrandom vector and let u be an n × 1 random vector with E(uu’) = In. Show that E(tr (auu‘a’)] = Σni=1α2i.
Prove Property 5 of variances for vectors, using Property 3.
Let A be an n × n symmetric, positive definite matrix. Show that if P is any n × n nonsingular matrix, then PrAP is positive definite.
(i) Show that if A is an n × n symmetric, positive semi-definite matrix, then A must have nonnegative diagonal elements.(ii) Show that if A is an n × n symmetric, positive definite matrix, then A
(i) Use the definition of inverse to prove the following: if A and B are n × n nonsingular matrices, then (AB)-1 = B-1A-1.(ii) If A, B, and C are all n × n nonsingular matrices, find (ABC2)-1 in
(i) Use the properties of trace to argue that tr (A'A) = tr (AA') for any n × m matrix A.(ii) [2 0 verify that tr(A'A) = tr(AA'). For A = 3
Let X be any n × k matrix. Show that XrX is a symmetric matrix.
If A and B are n × n diagonal matrices, show that AB = BA.
(i) Find the product AB using(ii) Does BA exist? Го 1 B = | 1 6. -1 2 A = -4 8 5 3 0
A count random variable, say Y, takes on nonnegative integer values, {0, 1, 2, . . .}. The most common distribution for a count variable is the Poisson(θ) distribution, where the parameter u is the
Suppose that between their first and second years in college, 400 students are randomly selected and given a university grant to purchase a new computer. For student i, yi denotes the change in GPA
Before a strike prematurely ended the 1994 major league baseball season, Tony Gwynn of the San Diego Padres had 165 hits in 419 at bats, for a .394 batting average. There was discussion about whether
Suppose that a military dictator in an unnamed country holds a plebiscite (a yes/no vote of confidence) and claims that he was supported by 65% of the voters. A human rights group suspects foul play
The New York Times (2/5/90) reported three-point shooting performance for the top 10 three-point shooters in the NBA. The following table summarizes these data:For a given player, the outcome of a
The new management at a bakery claims that workers are now more productive than they were under old management, which is why wages have “generally increased.” Let Wbi be Worker i’s wage
For positive random variables X and Y, suppose the expected value of Y given X is E(Y|X) = θX. The unknown parameter u shows how the expected value of Y changes with X.(i) Define the random variable
Let Y̅ denote the sample average from a random sample with mean m and variance s2. Consider two alternative estimators of m:(i) Show that W1 and W2 are both biased estimators of m and find the
This is a more general version of Problem C.1. Let Y1, Y2,c, Yn be n pairwise uncorrelated random variables with common mean m and common variance s2. Let Y denote the sample average.(i) Define the
Let Y1, Y2, Y3, and Y4 be independent, identically distributed random variables from a population with mean m and variance s2.Letdenote the average of these four random variables.(i) What are the
The geometric distribution can be used to model the number of trials before a certain event occurs. For example, we might flip a coin repeatedly until the first head appears. If the coin is fair, the
(i) Let X be a random variable taking on the values – 1 and 1, each with probability 1/2. Find E(X) and E(X2).(ii) Now let X be a random variable taking on the values 1 and 2, each with probability
Suppose that at a large university, college grade point average, GPA, and SAT score, SAT, are related by the conditional expectation E(GPA|SAT) = .70 + .002 SAT.(i) Find the expected GPA when SAT =
Let X denote the annual salary of university professors in the United States, measured in thousands of dollars. Suppose that the average salary is 52.3, with a standard deviation of 14.6. Find the
Suppose that a college student is taking three courses: a two-credit course, a three-credit course, and a four-credit course. The expected grade in the two-credit course is 3.5, while the expected
If a basketball player is a 74% free throw shooter, then, on average, how many free throws will he or she make in a game with eight free throw attempts?
Let X denote the prison sentence, in years, for people convicted of auto theft in a particular state in the United States. Suppose that the pdf of X is given byF(x) = (1/9)x2, 0 < x <
Just prior to jury selection for O. J. Simpson’s murder trial in 1995, a poll found that about 20% of the adult population believed Simpson was innocent (after much of the physical evidence in the
For a randomly selected county in the United States, let X represent the proportion of adults over age 65 who are employed, or the elderly employment rate. Then, X is restricted to a value between
Much is made of the fact that certain mutual funds outperform the market year after year (that is, the return from holding shares in the mutual fund is higher than the return from holding a portfolio
Let X be a random variable distributed as Normal(5,4). Find the probabilities of the following events:(i) P(X ≤ 6).(ii) P(X > 4).(iii) P(|X - 5| > 1).

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