Statistical Models Theory And Practice 2nd Edition David A. Freedman - Solutions

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Is v in equation (9) the variance of a data variable, or a random variable?What about σ2?
Check that the left hand side of (9) is the standardized slope. Hint: work out the correlation coefficient between the weights and the lengths.
What happens to (9) if σ2 .= 0? What would that tell us about springs and weights?
Is the −0.35 in figure 2 a parameter or an estimate? How is it related to equation (10)?
The correlation between mass and elite tolerance scores is 0.52; between mass tolerance scores and repression scores, −0.26; between elite tolerance scores and repression scores, −0.42. Compute the path coefficients in figure 2.
Estimate the SD of δ in equation (10). You may assume the correlations are based on 36 states but you need to decide if p is 2 or 3. (See text for Gibson’s sample sizes.)
Find the SEs for the path coefficients and their difference.
The repression scale is lumpy: scores go from 0 to 3.5 in steps of 0.5(table 1 in the paper). Does this make the linearity assumption more plausible, or less plausible?
Suppose we run a regression of Y on U and V , getting Y = ˆa + bUˆ + ˆcV + e,where e is the vector of residuals. Express the standardized coefficients in terms of the unstandardized coefficients and the sample variances of U,V, Y.
(This is a hypothetical; SAT stands for Scholastic Achievement Test, widely used for college admissions in the US.) Dr. Sally Smith is doing a study on coaching for the Math SAT. She assumes the response schedule Yi,x = 450 + 3x + δi. In this equation, Yi,x is the score that subject i would get on
(This continues exercise 1; it is still a hypothetical.) After thinking things over, Dr. Smith still believes that the response schedule is linear:Yi,x = a + bx + δi, the δi being IID N (0, σ2). But she decides that her values fora, b, and σ2 are unrealistic. (They probably are.) She wants to
In the path diagram below, free arrows are omitted. How many free arrows should there be, where do they go, and what do they mean? What does the curved line mean? The diagram represents some regression equations. What are the equations? the parameters? State the assumptions that would be needed to
With the assumptions of this section, show that a regression of Yi on Xi gives unbiased estimates, conditionally on the Xi’s, of a and b in (17).Show also that a regression of Zi on Xi and Yi gives unbiased estimates, conditionally on the Xi’s and Yi’s, ofc, d, and e in (18). Hints. What are
Suppose you are only interested in the effects of X and Y on Z; you are not interested in the effect of X on Y . You are willing to assume the response schedule (18), with IID errors i, independent of the Xi’s and Yi’s. How would you estimatec, d, e? Do the estimates have a causal
True or false, and explain.(a) In figure 1, father’s education has a direct influence on son’s occupation.(b) In figure 1, father’s education has an indirect influence on son’s occupation through son’s education.(c) In exercise 1, U has a direct influence on Y .(d) In exercise 1, V has a
Suppose Dr. Arbuthnot’s models are correct; and in his data, X77 =12, Y77 = 2, Z77 = 29.(a) How much bigger would Y77 have been, if Dr. Arbuthnot had intervened, setting X77 to 13?(b) How much bigger would Z77 have been, if Dr. Arbuthnot had intervened, setting X77 to 13 and Y77 to 5?
An investigator writes, “Statistical tests are a powerful tool for deciding whether effects are large.” Do you agree or disagree? Discuss briefly.
A regression of wife’s educational level (years of schooling) on husband’s educational level gives the equation WifeEdLevel = 5.60 + 0.57×HusbandEdLevel + residual.(Data are from the Current Population Survey in 2001.) If Mr. Wang’s company sends him back to school for a year to catch up on
In equation (10), δ is a random error; there is a δ for each state. Gibson finds that βˆ1 is statistically insignificant, while βˆ2 is highly significant(two-tailed). Suppose that Gibson computed his P-values from the standard normal curve; the area under the curve between −2.58 and +2.58
Timberlake and Williams (1984) offer a regression model to explain political oppression (PO) in terms of foreign investment (FI), energy development (EN), and civil liberties (CV). High values of PO correspond to authoritarian regimes that exclude most citizens from political participation. High
(Hard.) There is a population of N subjects, indexed by i = 1,...,N.Associated with subject i there is a number vi. A sample of size n is chosen at random without replacement.(a) Show that the sample average of the v’s is an unbiased estimate of the population average. (There are hints below.)(b)
(This continues question 8.) Let Xi = x if subject i is assigned to treatment at level x. A simple regression model says that given the assignments, the response Yi of subject i is α +Xiβ +i, where α, β are scalar parameters and the i are IID with mean 0 and variance σ2. Does randomization
(This continues questions 8 and 9.) Let Yi be the response of subject i. According to a multiple regression model, given the assignments, Yi = α+Xiβ+wiγ +i, where wi is a vector of personal characteristics for subject i (question 8); α, β are scalar parameters, γ is a vector of parameters,
Suppose (Xi, i) are IID as pairs for i = 1,...,n, with E(i) = 0 and var(i) = σ2. Here Xi is a 1×p random vector and i is a random variable (unobservable). Suppose E(XiXi) is p×p positive definite.Finally, Yi = Xiβ +i where β is a p×1 vector of unknown parameters.Is OLS biased or
To demonstrate causation, investigators have used (i) natural experiments, (ii) randomized controlled experiments, and (iii) regression models, among other methods. What are the strengths and weaknesses of methods (i), (ii), and (iii)? Discuss, preferably giving examples to illustrate your points.
True or false, and explain: if the OLS assumptions are wrong, the computer can’t fit the model to data.
An investigator fits the linear model Y = Xβ + . The OLS estimate for β is βˆ, and the fitted values are Yˆ. The investigator writes down the equation Yˆ = Xβˆ + ˆ. What is ˆ?
Suppose the Xi are IID N (0, 1). Let i = 0.025(X4 i − 3X2 i ) and Yi =Xi + i. An investigator does not know how the data were generated, and runs a regression of Y on X.(a) Show that R2 is about 0.97. (This is hard.)(b) Do the OLS assumptions hold?(c) Should the investigator trust the usual
Assume the response schedule Yi,x = a + bx + i. The i are IID N (0, σ2). The variables Xi are IID N (0, τ 2). In fact, the pairs (i, Xi)are IID in i, and jointly normal. However, the correlation between(i, Xi) is ρ, which may not be 0. The parametersa, b, σ2, τ 2, ρ are unknown. You
A statistician fits a regression model (n = 107, p = 6) and tests whether the coefficient she cares about is 0. Choose one or more of the options below. Explain briefly.(i) The null hypothesis says that β2 = 0.(ii) The null hypothesis says that βˆ2 = 0.(iii) The null hypothesis says that t =
Doctors often use body mass index (BMI) to measure obesity. BMI is weight/height2, where weight is measured in kilograms and height in meters. A BMI of 30 is getting up there. For American women age 18–24, the mean BMI is 24.6 and the variance is 29.4. Although the BMI for a typical woman in this
An epidemiologist says that “randomization does not exclude confounding ... confounding is very likely if information is collected—as it should be—on a sufficient number of baseline characteristics....” Do you agree or disagree? Discuss briefly.Notes. “Baseline characteristics” are
A political scientist is studying a regression model with the usual assumptions, including IID errors. The design matrix X is fixed, with full rank p = 5, and n = 57. The chief parameter of interest is β2 −β4. One possible estimator is βˆ2 −βˆ4, where βˆ = (XX)−1XY . Is there
In example 1, the log likelihood function is a sum—as it is in examples 2, 3, and 4. Is this a coincidence? If not, what is the principle?
(a) Suppose X1, X2,...,Xn are IID N (µ, 1). Find the mean and variance of the MLE for µ. Find the distribution of the MLE andcompare to the theorem. Show that −Ln(µ)/n ˆ → Iµ. Comment: for the normal, the asymptotics are awfully good.(b) If U is N (0, 1), show that U is symmetric:
Repeat 2(a) for the binomial in example 2. Is the MLE normally distributed? Or is it only approximately normal?
Repeat 2(a) for the Poisson in example 3. Is the MLE normally distributed? Or is it only approximately normal?
Find the density of θU/(1−U ), where U is uniform on [0,1] and θ > 0.
Suppose the Xi > 0 are independent, and their common density isθ/(θ + x)2 for i = 1,...,n, as in example 4. Show that θLn(θ ) =−n + 2 ni=1 Xi/(θ + Xi). Deduce that θ → θLn(θ ) decreases from n to −n as θ increases from 0 to ∞. Conclude that Ln has a unique maximum. (Reminder: Ln
What is the median of X in example 4?
Show that the Fisher information in example 4 is 1/(3θ 2).
Suppose Xi are independent for i = 1,...,n, with a common Poisson distribution. Suppose E(Xi) = λ > 0, but the parameter of interest isθ = λ2. Find the MLE for θ. Is the MLE biased or unbiased?
As in exercise 9, but the parameter of interest is θ = √λ. Find the MLE for θ. Is the MLE biased or unbiased?
Let β be a positive real number, which is unknown. Suppose Xi are independent Poisson random variables, with E(Xi) = βi for i =1, 2,..., 20. How would you estimate β?
Suppose X, Y, Z are independent normal random variables, each having variance 1. The means are α+β, α+2β, 2α+β, respectively: α,β are parameters to be estimated. Show that maximum likelihood and OLS give the same estimates. Note: this won’t usually be true—the result depends on the
Let θ be a positive real number, which is unknown. Suppose the Xi are independent for i = 1,...,n, with a common distribution Pθ that depends on θ: Pθ {Xi = j } = c(θ )(θ + j )−1(θ + j + 1)−1 for j =0, 1, 2,.... What is c(θ )? How would you estimate θ? Hints on finding c(θ ). What is
Suppose Xi are independent for i = 1,...,n, with common density 12 exp(−|x − θ|), where θ is a parameter, x is real, and n is odd. Show that the MLE for θ is the sample median. Hint: see exercise 2B18.
Let Z be N (0, 1) with density function φ and distribution function #(section 3.5). True or false, and explain:(a) The slope of # at x is φ(x).(b) The area to the left of x under φ is #(x).(c) P (Z = x) = φ(x).(d) P (Z < x) = #(x).(e) P (Z ≤ x) = #(x).(f) P (x < Z < x + h) .= φ(x)h if h is
In brief, the probit model for reading says that subject i read a book last year if Xiβ + Ui > 0.(a) What are Xi and β?(b) The Ui is a variable. Options (more than one may be right):data random latent dummy observable(c) What are the assumptions on Ui?(d) The log likelihood function is a , with
As in example 5, suppose we know β1 = −0.35, β2 = 0.02, β3 =1/100,000, β4 = −0.1. George has 12 years of education and makes$40,000 a year. His brother Harry also has 12 years of education but makes $50,000 a year. True or false, and explain: according to the model, the probability that
If X is N (µ, σ2), show that µ is estimable and σ2 is identifiable.
Suppose X1, X2, and X3 are independent normal random variables.Each has variance 1. The means are α, α+9β, and α+99β, respectively.Are α and β identifiable? estimable?
Suppose Y = Xβ + $, where X is a fixed n × p matrix, β is a p × 1 parameter vector, the $i are IID with mean 0 and variance σ2. Is βidentifiable if the rank of X is p? if the rank of X is p − 1?
Suppose Y = Xβ + $, where X is a fixed n×p matrix of rank p, andβ is a p × 1 parameter vector. The $i are independent with common variance σ2 and E($i) = µi, where µ is an n×1 parameter vector. Is βidentifiable?
Suppose X1 and X2 are IID, with Pp(X1 = 1) = p and Pp(X1 = 0) =1−p; the parameter p is between 0 and 1. Is p3 identifiable? estimable?
Suppose U and V are independent; U is N (0, σ2) and V is N (0, τ 2), where σ2 and τ 2 are parameters. However, U and V are not observable.Only U +V is observable. Is σ2 +τ 2 identifiable? How about σ2? τ 2?
If X is distributed like the absolute value of an N (µ, 1) variable, show that:(a) |µ| is identifiable. Hint: what is E(X2)?(b) µ itself is not identifiable. Hint: µ and −µ lead to the same distribution for X.
For incredibly many bonus points: suppose X is N (µ, σ2). Is |µ| estimable? What about σ2? Comments. We only have one observation X, not many observations. A rigorous solution to this exercise might involve the dominated convergence theorem, or the uniqueness theorem for Laplace transforms.
Suppose the random variable X has a continuous, strictly increasing distribution function F. Show that F (X) is uniform on [0,1]. Hints. Show that F has a continuous, strictly increasing inverse F −1. So F (X) < y if and only if X
Conversely, if U is uniform on [0,1], show that F −1(U ) has distribution function F. (This idea is often used to simulate IID picks from F.)On the logit model
Check that the logistic distribution function / is monotone increasing.Hint: if 1 − / is decreasing, you’re there.
Check that /(−∞) = 0 and /(∞) = 1.
Check that the logistic distribution is symmetric, i.e., 1 − /(x) =/(−x). Appearances can be deceiving....
(a) If P (Yi = 1|X) = /(Xiβ), show that logitP (Yi = 1|X) = Xiβ.(b) If logitP (Yi = 1|X) = Xiβ, show that P (Yi = 1|X) = /(Xiβ).
What is the distribution of log U − log (1 − U ), where U is uniform on [0, 1]? Hints. Show that log u − log (1 − u) is a strictly increasing function of u. Then compute the chance that log U − log (1 − U) > x.
For θ > 0, suppose X has the density θ/(θ +x)2 on the positive half-line(0,∞). Show that log(X/θ ) has the logistic distribution.
Show that ϕ(x) = − log(1+ex ) is strictly concave on (−∞,∞). Hint:check that ϕ(x) = −ex /(1 + ex )2 < 0.
Suppose that, conditional on the covariates X, the Y ’s are independent 0–1 variables, with logit P (Yi = 1|X) = Xiβ, i.e., the logit model holds. Show that the log likelihood function can be written as Ln(β) = −n i=1 log 1 + exp(Xiβ)+n i=1 XiYiβ.
(This continues exercises 9 and 10: hard.) Show that Ln(β) is a concave function of β, and strictly concave if X has full rank. Hints. Let the parameter vector β be p × 1. Let c be a p × 1 vector with c > 0. You need to show cLn(β)c ≤ 0, with strict inequality if X has full rank. Let
Let # be the standard normal distribution function (mean 0, variance 1).Let φ = # be the density. Show that φ(x) = −xφ(x). If x > 0, show that ∞x zφ(z) dz = φ(x) and 1 − #(x) < ∞x zxφ(z) dz.Conclude that 1 − #(x) < φ(x)/x for x > 0. If x < 0, show that#(x) < φ(x)/|x|. Show
(This continues exercise 12: hard.) Show that the log likelihood for the probit model is concave, and strictly concave if X has full rank. Hint:this is like exercise 11.
In table 3 of Evans and Schwab, is 0.777 a parameter or an estimate? How is this number related to equation (1)? Is this number on the probability scale or the probit scale? Repeat for 0.041, in the FEMALE line of the table. (The paper is reprinted at the back of the book.)
What does the −0.204 for PARENT SOME COLLEGE in table 3 mean?
Here is the two-equation model in brief: student i goes to Catholic school(Ci = 1) if IsCatia + Xib + Ui > 0, and graduates if Ciα + Xiβ + Vi > 0.(a) Which parameter tells you the effect of Catholic schools?(b) The Ui and Vi are variables. Options (more than one may be right):data random latent
In line (2) of table 6 in Evans and Schwab, is 0.859 a parameter or an estimate? How is it related to the equations in exercise 3? What about the −0.053? What does the −0.053 tell you about selection effects in the one-equation model?
In the two-equation model, the log likelihood function is a , with one for each . Fill in the blanks using one of the options below, and explain briefly.sum product quotient matrix term factor entry student school variable
Student #77 is Presbyterian, went to public school, and graduated. What does this subject contribute to the likelihood function? Write your answer using φ in equation (15).
Student #4039 is Catholic, went to public school, and failed to graduate.What does this subject contribute to the likelihood function? Write your answer using φ in equation (15).
Does the correlation between the latent variables in the two equations turn up in your answers to exercises 6 and 7? If so, where?
Table 1 in Evans and Schwab shows the total sample as 10,767 in the Catholic schools and 2527 in the public schools. Is this reasonable?Discuss briefly.
Table 1 shows that 0.97 of the students at Catholic schools graduated.Underneath the 0.97 is the number 0.17. What is this number, and how is it computed? Comment briefly.
For bonus points: suppose the two-equation model is right, and you had a really big sample. Would you get accurate estimates for α? β? the Vi?
Is the MLE biased or unbiased?
In the usual probit model, are the response variables independent from one subject to another? Or conditionally independent given the explanatory variables? Do the explanatory variables have to be statistically independent? Do they have to be linearly independent? Explain briefly.
Here is the two-equation model of Evans and Schwab, in brief. Student i goes to Catholic school (Ci = 1) if IsCatia + Xib + Ui > 0, (selection)otherwise Ci = 0. Student i graduates (Yi = 1) if Ciα + Xiβ + Vi > 0, (graduation)otherwise Yi = 0. IsCati is 1 if i is Catholic, and 0 otherwise; Xi is a
Is the MLE biased or unbiased?
In the usual probit model, are the response variables independent from one subject to another? Or conditionally independent given the explanatory variables? Do the explanatory variables have to be statistically independent? Do they have to be linearly independent? Explain briefly.
Here is the two-equation model of Evans and Schwab, in brief. Student i goes to Catholic school (Ci = 1) if IsCatia + Xib + Ui > 0, (selection)otherwise Ci = 0. Student i graduates (Yi = 1) if Ciα + Xiβ + Vi > 0, (graduation)otherwise Yi = 0. IsCati is 1 if i is Catholic, and 0 otherwise; Xi is a
Shaw (1999) uses a regression model to study the effect of TV ads and candidate appearances on votes in the presidential elections of 1988, 1992, and 1996. With three elections and 51 states (DC counts for this purpose), there are 153 data points, i.e., pairs of years and states. Each variable in
The Nurses’ Health Study wanted to show that hormone replacement therapy (HRT) reduces the risk of heart attack for post-menopausal women. The investigators found out whether each woman experienced a heart attack during the study period, and her HRT usage: 6,224 subjects were on HRT and 27,034
People often use observational studies to demonstrate causation, but there’s a big problem. What is an observational study, what’s the problem, and how do people try to get around it? Discuss. If possible, give examples to illustrate your points.
There is a population of N subjects, indexed by i = 1,...,N. Each subject will be assigned to treatment T or control C. Subject i has a response yT i if assigned to treatment and yC i if assigned to control. Each response is 0 (“failure”) or 1 (“success”). For instance, in an experiment to
(This continues question 10.) The assignment variable Xi is defined as follows: Xi = 1 if i is in treatment, else Xi = 0. The probit model says that given the assignments, subjects are independent, the probability of success for subject i being #(Xiα + wiβ), where # is the standard normal
Malaria is endemic in parts of Africa. A vaccine is developed to protect children against this disease. A randomized controlled experiment is done in a small rural village: half the children are chosen at random to get the vaccine, and half get a placebo. Some epidemiologists want to analyze the
As in question 12, but this time, the epidemiologists have 20 isolated rural villages. They choose 10 villages at random for treatment. In these villages, everybody will get the vaccine. The other 10 villages will serve as the control group: nobody gets the vaccine. Can the epidemiologists use the
Suppose we accept the model in question 10, but data are collected on Xi and Yi in an observational study, not a controlled experiment. Subjects assign themselves to treatment (Xi = 1) or control (Xi = 0), and we observe the response Yi as well as the covariates wi . One person suggests separating
Paula has observed values on four independent random variables with common density fα,β (x) = c(α, β)(αx − β)2 exp[−(αx − β)2], whereα > 0, −∞
Jacobs and Carmichael (2002) are comparing various sociological theories that explain why some states have the death penalty and some do not. The investigators have data for 50 states (indexed by i) in years t = 1971, 1981, 1991. The response variable Yit is 1 if state i has the death penalty in
Ludwig is working hard on a statistics project. He is overheard muttering to himself, “Ach! Schrecklich! So many Parameters! So little Data!”Is he worried about bias, endogeneity, or non-identifiability?
Garrett (1998) considers the impact of left-wing political power (LPP)and trade-union power (TUP) on economic growth. There are 25 years of data on 14 countries. Countries are indexed by i = 1,..., 14; years are indexed by t = 1,..., 25. The growth rate for country i in year t is modeled as a

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Statistical Models Theory And Practice 2nd Edition David A. Freedman - Solutions

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