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business
statistical techniques in business

Questions and Answers of Statistical Techniques in Business

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
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
(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
(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).
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
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
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
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
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
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
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
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
(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
(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,
(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
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
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
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
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
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
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
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
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
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
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
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
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 +
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
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
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
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
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 −
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
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
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
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
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
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
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
(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
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
(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
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
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
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 #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
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 +
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
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 +
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
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
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
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
(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,
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
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
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
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)
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
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,...,

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