Deck 9: Assessing Studies Based on Multiple Regression

Question 1

Panel data estimation can sometimes be used

A)to avoid the problems associated with misspecified functional forms.
B)in case the sum of residuals is not zero.
C)in the case of omitted variable bias when data on the omitted variable is not available.
D)to counter sample selection bias.

Answer 1

C

Question 2

The components of internal validity are

A)a large sample,and BLUE property of the estimator.
B)a regression R2 above 0.75 and serially uncorrelated errors.
C)unbiasedness and consistency of the estimator,and desired significance level of hypothesis testing.
D)nonstochastic explanatory variables,and prediction intervals close to the sample mean.

Answer 2

C

Question 3

The reliability of a study using multiple regression analysis depends on all of the following with the exception of

A)omitted variable bias.
B)errors-in-variables.
C)presence of homoskedasticity in the error term.
D)external validity.

Answer 3

C

Question 4

Misspecification of functional form of the regression function

A)is overcome by adding the squares of all explanatory variables.
B)is more serious in the case of homoskedasticity-only standard error.
C)results in a type of omitted variable bias.
D)requires alternative estimation methods such as maximum likelihood.

Question 5

By including another variable in the regression,you will

A)decrease the regression R2 if that variable is important.
B)eliminate the possibility of omitted variable bias from excluding that variable.
C)look at the t-statistic of the coefficient of that variable and include the variable only if the coefficient is statistically significant at the 1% level.
D)decrease the variance of the estimator of the coefficients of interest.

Question 6

A study based on OLS regressions is internally valid if

A)the errors are homoskedastic,and there are no more than two binary variables present among the regressors.
B)you use a two-sided alternative hypothesis,and standard errors are calculated using the heteroskedasticity-robust formula.
C)weighted least squares produces similar results,and the t-statistic is normally distributed in large samples.
D)the OLS estimator is unbiased and consistent,and the standard errors are computed in a way that makes confidence intervals have the desired confidence level.

Question 7

Applying the analysis from the California test scores to another U.S.state is an example of looking for

A)simultaneous causality bias.
B)external validity.
C)sample selection bias.
D)internal validity.

Question 8

Comparing the California test scores to test scores in Massachusetts is appropriate for external validity if

A)Massachusetts also allowed beach walking to be an appropriate P.E.activity.
B)the two income distributions were very similar.
C)the student-to-teacher ratio did not differ by more than five on average.
D)the institutional settings in California and Massachusetts,such as organization in classroom instruction and curriculum,were similar in the two states.

Question 9

The analysis is externally valid if

A)the statistical inferences about causal effects are valid for the population being studied.
B)the study has passed a double blind refereeing process for a journal.
C)its inferences and conclusions can be generalized from the population and setting studied to other populations and settings.
D)some committee outside the author's department has validated the findings.

Question 10

In the case of errors-in-variables bias,

A)maximum likelihood estimation must be used.
B)the OLS estimator is consistent if the variance in the unobservable variable is relatively large compared to variance in the measurement error.
C)the OLS estimator is consistent,but no longer unbiased in small samples.
D)binary variables should not be used as independent variables.

Question 11

In the case of a simple regression,where the independent variable is measured with i.i.d.error,

A)
B) .
C) .
D) .

Question 12

Sample selection bias

A)occurs when a selection process influences the availability of data and that process is related to the dependent variable.
B)is only important for finite sample results.
C)results in the OLS estimator being biased,although it is still consistent.
D)is more important for nonlinear least squares estimation than for OLS.

Question 13

A statistical analysis is internally valid if

A)its inferences and conclusions can be generalized from the population and setting studied to other populations and settings.
B)statistical inference is conducted inside the sample period.
C)the hypothesized parameter value is inside the confidence interval.
D)the statistical inferences about causal effects are valid for the population being studied.

Question 14

Sample selection bias occurs when

A)the choice between two samples is made by the researcher.
B)data are collected from a population by simple random sampling.
C)samples are chosen to be small rather than large.
D)the availability of the data is influenced by a selection process that is related to the value of the dependent variable.

Question 15

A survey of earnings contains an unusually high fraction of individuals who state their weekly earnings in 100s,such as 300,400,500,etc.This is an example of

A)errors-in-variables bias.
B)sample selection bias.
C)simultaneous causality bias.
D)companies that typically bargain with workers in 100s of dollars.

Question 16

Errors-in-variables bias

A)is present when the probability limit of the OLS estimator is given by .
B)arises when an independent variable is measured imprecisely.
C)arises when the dependent variable is measured imprecisely.
D)always occurs in economics since economic data is never precisely measured.

Question 17

Correlation of the regression error across observations

A)results in incorrect OLS standard errors.
B)makes the OLS estimator inconsistent,but not unbiased.
C)results in correct OLS standard errors if heteroskedasticity-robust standard errors are used.
D)is not a problem in cross-sections since the data can always be "reshuffled."

Question 18

Errors-in-variables bias

A)is only a problem in small samples.
B)arises from error in the measurement of the independent variable.
C)becomes larger as the variance in the explanatory variable increases relative to the error variance.
D)is particularly severe when the source is an error in the measurement of the dependent variable.

Question 19

Simultaneous causality bias

A)is also called sample selection bias.
B)happens in complicated systems of equations called block recursive systems.
C)results in biased estimators if there is heteroskedasticity in the error term.
D)arises in a regression of Y on X when,in addition to the causal link of interest from X to Y,there is a causal link from Y to X.

Question 20

Simultaneous causality

A)means you must run a second regression of X on Y.
B)leads to correlation between the regressor and the error term.
C)means that a third variable affects both Y and X.
D)cannot be established since regression analysis only detects correlation between variables.

Question 21

A possible solution to errors-in-variables bias is to

A)use log-log specifications.
B)choose different functional forms.
C)use the square root of that variable since the error becomes smaller.
D)mitigate the problem through instrumental variables regression.

Question 22

The guidelines for whether or not to include an additional variable include all of the following,with the exception of

A)providing "full disclosure" representative tabulations of the results.
B)testing whether additional questionable variables have nonzero coefficients.
C)determining whether it can be measured in the population of interest.
D)being specific about the coefficient or coefficients of interest.

Question 23

You try to explain the number of IBM shares traded in the stock market per day in 2005.As an independent variable you choose the closing price of the share.This is an example of

A)simultaneous causality.
B)invalid inference due to a small sample size.
C)sample selection bias since you should analyze more than one stock.
D)a situation where homoskedasticity-only standard errors should be used since you only analyze one company.

Question 24

A definition of internal validity is

A)the estimator of the causal effect being unbiased and consistent
B)the estimator of the causal effect being efficient
C)inferences and conclusions being generalized from the population to other populations
D)OLS estimation being available in your statistical package

Question 25

The question of reliability/unreliability of a multiple regression depends on

A)internal but not external validity
B)the quality of your statistical software package
C)internal and external validity
D)external but not internal validity

Question 26

In the case of errors-in-variables bias,the precise size and direction of the bias depend on

A)the sample size in general.
B)the correlation between the measured variable and the measurement error.
C)the size of the regression R2.
D)whether the good in question is price elastic.

Question 27

You have decided to analyze the year-to-year variation in temperature data.Specifically you want to use this year's temperature to predict next year's temperature for certain cities.As a result,you collect the daily high temperature (Temp)for 100 randomly selected days in a given year for three United States cities: Boston,Chicago,and Los Angeles.You then repeat the exercise for the following year.The regression results are as follows (heteroskedasticity-robust standard errors in parentheses): = 18.19 + 0.75 × ;R2 = 0.62,SER = 12.33
(6.46)(0.10) = 2.47 + 0.95 × ;R2 = 0.93,SER = 5.85
(3.98)(0.05) = 37.54 + 0.44 × ;R2 = 0.18,SER = 7.17
(15.33)(0.22)
(a)What is the prediction of the above regression for Los Angeles if the temperature in the previous year was 75 degrees? What would be the prediction for Boston?
(b)Assume that the previous year's temperature gives accurate predictions,on average,for this year's temperature.What values would you expect in this case for the intercept and slope? Sketch how each of the above regressions behaves compared to this line.
(c)After reflecting on the results a bit,you consider the following explanation for the above results.Daily high temperatures on any given date are measured with error in the following sense: for any given day in any of the three cities,say January 28,there is a true underlying seasonal temperature (X),but each year there are different temporary weather patterns (v,w)which result in a temperature different from X.For the two years in your data set,the situation can be described as follows: = X + vt and = X + wt
Subtracting from ,you get = + wt - vt.Hence the population parameter for the intercept and slope are zero and one,as expected.Show that the OLS estimator for the slope is inconsistent,where (d)Use the formula above to explain the differences in the results for the three cities.Is your mathematical explanation intuitively plausible?

Question 28

Your textbook used the California Standardized Testing and Reporting (STAR)data set on test student performance in Chapters 4-7.One justification for putting second to twelfth graders through such an exercise once a year is to make schools more accountable.The hope is that schools with low scores will improve the following year and in the future.To test for the presence of such an effect,you collect data from 1,000 L.A.County schools for grade 4 scores in 1998 and 1999,both for reading (Read)and mathematics (Maths).Both are on a scale from zero to one hundred.The regression results are as follows (homoskedasticity-only standard errors in parentheses): = 6.967 + 0.919 ,R2 = 0.825,SER = 7.818
(0.542)(0.013) = 4.131 + 0.943, = R2 = 0.887,SER = 6.416
(0.409)(0.011)
(a)Interpret the results and indicate whether or not the coefficients are significantly different from zero.Do the coefficients have the expected sign and magnitude?
(b)Discuss various threats to internal and external validity,and try to assess whether or not these are likely to be present in your study.
(c)Changing the estimation method to allow for heteroskedasticity-robust standard errors produces four new standard errors: (0.539), (0.015), (0.452),and (0.015)in the order of appearance in the two equations above.Given these numbers,do any of your statements in (b)change? Do you think that the coefficients themselves changed?
(d)If reading and maths scores were the same in 1999 as in 1998,on average,what coefficients would you expect for the intercept and the slope? How would you test for the restrictions?
(e)The appropriate F-statistic in (d)is 138.27 for the maths scores,and 104.85 for the reading scores.Comparing these values to the critical values in the F table,can you reject the null hypothesis in each case?
(f)Your professor tells you that the analysis reminds her of "Galton's Fallacy." Sir Francis Galton regressed the height of children on the average height of their parents.He found a positive intercept and a slope between zero and one.Being concerned about the height of the English aristocracy,he interpreted the results as "regression to mediocrity" (hence the name regression).Do you see the parallel?

Question 29

The Phillips curve is a relationship in macroeconomics between the inflation rate (inf)and the unemployment rate (ur).Estimating the Phillips curve using quarterly data for the United States from 1962:I to 1995:IV,you find = 4.08 + 0.118 urt,R2 = 0.003,SER = 3.148
(1.11)(0.176)
(a)Explain why,at first glance,this is a surprising result.
(b)Do you think that there is omitted variable bias in the regression?
(c)What other threats to internal validity may be present?
(d)If you could find a proper specification for the Phillips curve using United States data,what external validity criteria would you suggest?

Question 30

Possible solutions to omitted variable bias,when the omitted variable is not observed,include the following with the exception of

A)panel data estimation.
B)nonlinear least squares estimation.
C)use of instrumental variables regressions.
D)use of randomized controlled experiments.

Question 31

Several authors have tried to measure the "persistence" in U.S state unemployment rates by running the following regression: where ur is the state unemployment rate,i is the index for the i-th state,t indicates a time period,and typically k ≥ 10.
(a)Explain why finding a slope estimate of one and an intercept of zero is typically interpreted as evidence of "persistence."
(b)You collect data on the 48 contiguous U.S.states' unemployment rates and find the following estimates: = 2.25 + 0.60 × ;R2 = 0.40,SER = 0.90
(0.61)(0.13)
Interpret the regression results.
(c)Analyzing the accompanying figure,and interpret the observation for Maryland and for Washington.Do you find evidence of persistence? How would you test for it? (d)One of your peers points out that this result makes little sense,since it implies that eventually all states would have identical unemployment rates.Explain the argument.
(e)Imagine that state unemployment rates were determined by their natural rates and some transitory shock.The natural rates themselves may be functions of the unemployment insurance benefits of the state,unionization rates of its labor force,demographics,sectoral composition,etc.The transitory components may include state-specific shocks to its terms of trade such as raw material movements and demand shocks from the other states.You specify the i-th state unemployment rate accordingly as follows for the two periods when you observe it, so that actual unemployment rates are measured with error.You have also assumed that the natural rate is the same for both periods.Subtracting the second period from the first then results in the following population regression function: It is not too hard to show that estimation of the observed unemployment rate in period t on the unemployment rate in period (t-k)by OLS results in an estimator for the slope coefficient that is biased towards zero.The formula is .
Using this insight,explain over which periods you would expect the slope to be closer to one,and over which period it should be closer to zero.
(f)Estimating the same regression for a different time period results in = 3.19 + 0.27 × ;R2 = 0.21,SER = 1.03
(0.56)(0.07)
If your above analysis is correct,what are the implications for this time period?

Question 32

A statistical analysis is internally valid if

A)all t-statistics are greater than |1.96|
B)the regression R2 > 0.05
C)the population is small,say less than 2,000,and can be observed
D)the statistical inferences about causal effects are valid for the population studied

Question 33

Keynes postulated that the marginal propensity to consume (MPC = )is between zero and one.He also hypothesized that the average propensity to consume (APC = )would fall as personal disposable income increased.
(a)Specify a linear consumption function.Show that the assumption of a falling APC implies the presence of a positive intercept.
(b)Using annual per capita data,estimation of the consumption function for the United States results in the following output for the years 1929-1938: = 981.35 + 0.735 ,R2 = 0.98,SER= 50.65
(158.65)(0.038)
Can you reject the null hypothesis that the slope is less than one? Greater than zero? Test the hypothesis that the intercept is zero.Should you be concerned about the sample size when conducting these tests? What other threats to internal validity may be present here?
(c)Given the GDP identity for a closed economy, show why economists saw important policy implications in finding an APC that would decrease over time.
(d)Simon Kuznets,who won the Nobel Prize in economics,collected data on consumption expenditures and national income from 1869 to 1938 and found,using overlapping period averages,that the APC was relatively constant over this period.To reconcile this finding with the regression results,Milton Friedman,who also won the Nobel Prize,formulated the "permanent income" hypothesis.In essence,Friedman hypothesized that both actual consumption and income are measured with error, and where and were called "permanent" consumption and income,respectively,and and ,the two measurement errors,were labeled transitory consumption and income.Friedman hypothesized that the transitory components were purely random error terms,uncorrelated with the permanent parts.
Let permanent consumption and income be related as follows: so that the APC and MPC are the same and constant over time.Furthermore,let both transitory and permanent income be independent of the error term.Show that by regressing actual consumption on actual income,the MPC will be downward biased,and the intercept will be greater than zero,even in large samples (to simplify the analysis,assume that permanent income and all of the errors are i.i.d.and mutually independent).

Question 34

Sir Francis Galton (1822-1911),an anthropologist and cousin of Charles Darwin,created the term regression.In his article "Regression towards Mediocrity in Hereditary Stature," Galton compared the height of children to that of their parents,using a sample of 930 adult children and 205 couples.In essence he found that tall (short)parents will have tall (short)offspring,but that the children will not be quite as tall (short)as their parents,on average.Hence there is regression towards the mean,or as Galton referred to it,mediocrity.This result is obviously a fallacy if you attempted to infer behavior over time since,if true,the variance of height in humans would shrink over generations.This is not the case.
(a)To research this result,you collect data from 110 college students and estimate the following relationship: = 19.6 + 0.73 × Midparh,R2 = 0.45,SER = 2.0
(7.2)(0.10)
where Studenth is the height of students in inches and Midparh is the average of the parental heights.Values in parentheses are heteroskedasticity-robust standard errors.Sketching this regression line together with the 45 degree line,explain why the above results suggest "regression to the mean" or "mean reversion."
(b)Researching the medical literature,you find that height depends,to a large extent,on one gene ("phog")and on environmental influences.Let us assume that parents and offspring have the same invariant (over time)gene and that actual height is therefore measured with error in the following sense, where is measured height,X is the height given through the gene,v and w are environmental influences,and the subscripts o and p stand for offspring and parents,respectively.Let the environmental influences be independent from each other and from the gene.
Subtracting the measured height of offspring from the height of parents,what sort of population regression function do you expect?
(c)How would you test for the two restrictions implicit in the population regression function in (b)? Can you tell from the results in (a)whether or not the restrictions hold?
(d)Proceeding in a similar way to the proof in your textbook,you can show that for the situation in (b).Discuss under what conditions you will find a slope closer to one for the height comparison.Under what conditions will you find a slope closer to zero?
(e)Can you think of other examples where Galton's Fallacy might apply?

Question 35

Threats to in internal validity lead to

A)perfect multicollinearity
B)the inability to transfer data sets into your statistical package
C)failures of one or more of the least squares assumptions
D)a false generalization to the population of interest

Question 36

The true causal effect might not be the same in the population studied and the population of interest because

A)of differences in characteristics of the population
B)of geographical differences
C)the study is out of date
D)all of the above

Question 37

Macroeconomists who study the determinants of per capita income (the "wealth of nations")have been particularly interested in finding evidence on conditional convergence in the countries of the world.Finding such a result would imply that all countries would end up with the same per capita income once other variables such as saving and population growth rates,education,government policies,etc. ,took on the same value.Unconditional convergence,on the other hand,does not control for these additional variables.
(a)The results of the regression for 104 countries was as follows, = 0.019 - 0.0006 × RelProd60,R2= 0.00007,SER = 0.016
(0.004)(0.0073),
where g6090 is the average annual growth rate of GDP per worker for the 1960-1990 sample period,and RelProd60 is GDP per worker relative to the United States in 1960.
For the 24 OECD countries in the sample,the output is = 0.048 - 0.0404 RelProd60,R2 = 0.82,SER = 0.0046
(0.004)(0.0063)
Interpret the results and point out the difference with regard to unconditional convergence.
(b)The "beta-convergence" regressions in (a)are of the following type, = β0 + β0 ln Yi,0 + ui,t,
where △t ln Yi,t = ln Yi,0 - ln Yi,0,and t and o refer to two time periods,i is the i-th country.
Explain why a significantly negative slope implies convergence (hence the name).
(c)The equation in (b)can be rewritten without any change in information as (ignoring the division by T)
ln Yt = β0 + γ1 ln Y0 + ut
In this form,how would you test for unconditional convergence? What would be the implication for convergence if the slope coefficient were one?
(d)Let's write the equation in (c)as follows: and assume that the "~" variables contain measurement errors of the following type, where the "*" variables represent true,or permanent,per capita income components,while v and w are temporary or transitory components.Subtraction of the initial period from the current period then results in Ignoring,without loss of generality,the constant in the above equation,and making standard assumptions about the error term,one can show that by regressing current per capita income on a constant and the initial period per capita income,the slope behaves as follows: Discuss the implications for the convergence results above.

Question 38

A study of United States and Canadian labor markets shows that aggregate unemployment rates between the two countries behaved very similarly from 1920 to 1982,when a two percentage point gap opened between the two countries,which has persisted over the last 20 years.To study the causes of this phenomenon,you specify a regression of Canadian unemployment rates on demographic variables,aggregate demand variables,and labor market characteristics.
(a)Assume that your analysis is internally valid.What would make it externally valid?
(b)If one of the determinants of Canadian unemployment is aggregate United States economic activity (or perhaps shocks to it),what variable would you suggest as its replacement if you did a similar study for the United States?
(c)Certain Canadian geographical areas,such as the prairies and British Columbia,seem particularly sensitive to commodity price shocks (Edmonton's NHL team is called the Edmonton Oilers).Having collected provincial data,you establish a relationship between provincial unemployment rates and commodity price changes (shocks).How would you address external validity now?

Question 39

Until about 10 years ago,most studies in labor economics found a small but significant negative relationship between minimum wages and employment for teenagers.Two labor economists challenged this perceived wisdom with a publication in 1992 by comparing employment changes of fast-food restaurants in Texas,before and after a federal minimum wage increase.
(a)Explain how you would obtain external validity in this field of study.
(b)List the various threats to external validity and suggest how to address them in this case.

Question 40

One of the most frequently used summary statistics for the performance of a baseball hitter is the so-called batting average.In essence,it calculates the percentage of hits in the number of opportunities to hit (appearances "at the plate").The management of a professional team has hired you to predict next season's performance of a certain hitter who is up for a contract renegotiation after a particularly great year.To analyze the situation,you search the literature and find a study which analyzed players who had at least 50 at bats in 1998 and 1997.There were 379 such players.
(a)The reported regression line in the study is = 0.138 + 0.467 × ;R2= 0.17
and the intercept and slope are both statistically significant.What does the regression imply about the relationship between past performance and present performance? What values would the slope and intercept have to take on for the future performance to be as good as the past performance,on average?
(b)Being somewhat puzzled about the results,you call your econometrics professor and describe the results to her.She says that she is not surprised at all,since this is an example of "Galton's Fallacy." She explains that Sir Francis Galton regressed the height of offspring on the mid-height of their parents and found a positive intercept and a slope between zero and one.He referred to this result as "regression towards mediocrity." Why do you think econometricians refer to this result as a fallacy?
(c)Your professor continues by mentioning that this is an example of errors-in-variables bias.What does she mean by that in general? In this case,why would batting averages be measured with error? Are baseball statisticians sloppy?
(d)The top three performers in terms of highest batting averages in 1997 were Tony Gwynn (.372),Larry Walker (.366),and Mike Piazza (.362).Given your answers for the previous questions,what would be your predictions for the 1998 season?

Question 41

Your professor wants to measure the class's knowledge of econometrics twice during the semester,once in a midterm and once in a final.Assume that your performance,and that of your peers,on the day of your midterm exam only measure knowledge imperfectly and with an error, where is your exam grade,X is underlying econometrics knowledge,and w is a random error with mean zero and variance .w may depend on whether you have a headache that day,whether or not the questions you had prepared for appeared on the exam,your mood,etc.A similar situation holds for the final,which is exam two: .What would happen if you ran a regression of grades received by students in the final on midterm grades?

Question 42

Assume that you had found correlation of the residuals across observations.This may happen because the regressor is ordered by size.Your regression model could therefore be specified as follows:
Yi = β0 + β1Xi + ui
ui = ρui-1 + vi; < 1.
Furthermore,assume that you had obtained consistent estimates for β0,β1,ρ.If asked to make a prediction for Y,given a value of X(= Xj)and j-1,how would you proceed? Would you use the information on the lagged residual at all? Why or why not?

Question 43

Assume that a simple economy could be described by the following system of equations,
Ct = β0 + β1Yt + ui
It = ,
where C is consumption,Y is income,and I is investment.(This may be a primitive island society which does not trade with other islands.There is no government,and the only good consumed and invested (saved)is sunflower seeds. )
Assume the presence of the GDP identity,Y = C + I.If you estimated the consumption function,what sort of problem involving internal validity may be present?

Question 44

Your textbook states that correlation of the error term across observations "will not happen if the data are obtained by sampling at random from the population." However,in one famous study of the electric utility industry,the observations were listed by the size of the output level,from smallest to largest.The pattern of the residuals was as shown in the figure. What does this pattern suggest to you?

Question 45

Your textbook uses the following example of simultaneous causality bias of a two equation system:
Yi = β0 + β1Xi + ui
Xi = + Yi + vi
To be more specific,think of the first equation as a demand equation for a certain good,where Y is the quantity demanded and X is the price.The second equation then represents the supply equation,with a third equation establishing that demand equals supply.Sketch the market outcome over a few periods and explain why it is impossible to identify the demand and supply curves in such a situation.Next assume that an additional variable enters the demand equation: income.In a new graph,draw the initial position of the demand and supply curves and label them D0 and S0.Now allow for income to take on four different values and sketch what happens to the two curves.Is there a pattern that you see which suggests that you might be able to identify one of the two equations with real-life data?

Question 46

Explain why the OLS estimator for the slope in the simple regression model is still unbiased,even if there is correlation of the error term across observations.

Question 47

A professor in your microeconomics lectures derived a labor demand curve in the lecture.Given some reasonable assumptions,she showed that the demand for labor depends negatively on the real wage.You want to put this hypothesis to the test ("show me")and collect data on employment and real wages for a certain industry.You try to estimate the labor demand curve but find no relationship between the two variables.Is economic theory wrong? Explain.

Question 48

Your textbook has analyzed simultaneous equation systems in the case of two equations,
Yi = β0 + β1Xi + ui
Xi = + Yi + vi,
where the first equation might be the labor demand equation (with capital stock and technology being held constant),and the second the labor supply equation (X being the real wage,and the labor market clears).What if you had a a production function as the third equation
Zi = + Yi + wi
where Z is output.If the error terms,u,v,and w,were pairwise uncorrelated,explain why there would be no simultaneous causality bias when estimating the production function using OLS.

Question 49

Give at least three examples where you could envision errors-in-variables problems.For the case where the measurement error occurs only for the explanatory variable in the simple regression case,derive .

Question 50

Think of three different economic examples where cross-sectional data could be collected.Indicate in each of these cases how you would check if the analysis is externally valid.

Question 51

Your textbook gives the following example of simultaneous causality bias of a two equation system:
Yi = β0 + β1Xi + ui
Xi = + Yi + vi
In microeconomics,you studied the demand and supply of goods in a single market.Let the demand ( )and supply ( )for the i-th good be determined as follows, = β0 - β1Pi + ui, = - Pi + vi,
where P is the price of the good.In addition,you typically assume that the market clears.
Explain how the simultaneous causality bias applies in this situation.The textbook explained a positive correlation between Xi and ui for > 0 through an argument that started from "imagine that ui is negative." Repeat this exercise here.

Question 52

Suppose that you have just read a review of the literature of the effect of beauty on earnings.You were initially surprised to find a mild effect of beauty even on teaching evaluations at colleges.Intrigued by this effect,you consider explanations as to why more attractive individuals receive higher salaries.One of the possibilities you consider is that beauty may be a marker of performance/productivity.As a result,you set out to test whether or not more attractive individuals receive higher grades (cumulative GPA)at college.You happen to have access to individuals at two highly selective liberal arts colleges nearby.One of these specializes in Economics and Government and incoming students have an average SAT of 2,100;the other is known for its engineering program and has an incoming SAT average of 2,200.Conducting a survey,where you offer students a small incentive to answer a few questions regarding their academic performance,and taking a picture of these individuals,you establish that there is no relationship between grades and beauty.Write a short essay using some of the concepts of internal and external validity to determine if these results are likely to apply to universities in general.

Question 53

To analyze the situation of simultaneous causality bias,consider the following system of equations:
Yi = β0 + β1Xi + ui
Xi = + Yi + vi
Demonstrate the negative correlation between Xi and for < 0 ,either through mathematics or by presenting an argument which starts as follows: "Imagine that ui is negative."

Question 54

Your textbook only analyzed the case of an error-in-variables bias of the type i= Xi + wi.What if the error were generated in the simple regression model by entering data that always contained the same typographical error,say i= Xi + a or i= bXi,where a and b are constants.What effect would this have on your regression model?

Question 55

Your textbook compares the results of a regression of test scores on the student-teacher ratio using a sample of school districts from California and from Massachusetts.Before standardizing the test scores for California,you get the following regression result: = 698.9 - 2.28×STR
n = 420,R2 = 0.051,SER = 18.6
In addition,you are given the following information: the sample mean of the student-teacher ratio is 19.64 with a standard deviation of 1.89,and the standard deviation of the test scores is 19.05.
a.After standardizing the test scores variable and running the regression again,what is the value of the slope? What is the meaning of this new slope here (interpret the result)?
b.What will be the new intercept? Now that test scores have been standardized,should you interpret the intercept?
c.Does the regression R2 change between the two regressions? What about the t-statistic for the slope estimator?

Question 56

You have been hired as a consultant by building contractor,who have been sued by the owners' representatives of a large condominium project for shoddy construction work.In order to assess the damages for the various units,the owners' association sent out a letter to owners and asked if people were willing to make their units available for destructive testing.Destructive testing was conducted in some of these units as a result of the responses.Based on the tests,the owners' association inferred the damage over the entire condo complex.Do you think that the inference is valid in this case? Discuss how proper sampling should proceed in this situation.

Question 57

Consider the one-variable regression model,Yi = β0 + β1Xi + ui,where the usual assumptions from Chapter 4 are satisfied.However,suppose that both Y and X are measured with error, = Yi + zi and = Xi + wi.Let both measurement errors be i.i.d.and independent of both Y and X respectively.If you estimated the regression model = β0 + β1 + vi using OLS,then show that the slope estimator is not consistent.

Question 58

In the simple,one-explanatory variable,errors-in-variables model,the OLS estimator for the slope is inconsistent.The textbook derived the following result .
Show that the OLS estimator for the intercept behaves as follows in large samples: where .

Question 59

The textbook derived the following result: .Show that this is the same as .

Question 60

The errors-in-variables model analyzed in the text results in so that the OLS estimator is inconsistent.Give a condition involving the variances of X and w,under which the bias towards zero becomes small.

Question 61

In macroeconomics,you studied the equilibrium in the goods and money market under the assumption of prices being fixed in the very short run.The goods market equilibrium was described by the so-called IS equation
Ri = β0 - β1Yi + ui
where R represented the nominal interest rate and Y was real GDP.β0 contained variables determined outside the system,such as government expenditures,taxes,and inflationary expectations.
The money market equilibrium was given by the so-called LM equation
Ri = + Yi + vi
and contained the real money supply and the intercept from the money demand equation.
Show that there is simultaneous causality bias in this situation.

Question 62

Assume the following model of the labor market:
Nd = β0 + β1 + u
Ns = γ0 + γ1 + v
Nd = Ns = N
where N is employment, (W/P)is the real wage in the labor market,and u and v are determinants other than the real wage which affect labor demand and labor supply (respectively).Let
E(u)= E(v)= 0;var(u)= ;var(v)= ;cov(u,v)= 0
Assume that you had collected data on employment and the real wage from a random sample of observations and estimated a regression of employment on the real wage (employment being the regressand and the real wage being the regressor).It is easy but tedious to show that > 0
since the slope of the labor supply function is positive and the slope of the labor demand function is negative.Hence,in general,you will not find the correct answer even in large samples.
a.What is this bias referred to?
b.What would the relationship between the variance of the labor supply/demand shift variable have to be for the bias to disappear?
c.Give an intuitive answer why the bias would disappear in that situation.Draw a graph to illustrate your argument.

Question 63

To compare the slope coefficient from the California School data set with that of the Massachusetts School data set,you run the following two regressions: CA = 2.35 - 0.123×STRCA
(0.54)(0.027)
n = 420,R2 = 0.051,SER = 0.98 MA = 1.97 - 0.114×STRMA
(0.57)(0.033)
n = 220,R2 = 0.067,SER = 0.97
Numbers in parenthesis are heteroskedasticity-robust standard errors,and the LHS variable has been standardized.
Calculate a t-statistic to test whether or not the two coefficients are the same.State the alternative hypothesis.Which level of significance did you choose?

Question 64

You have read the analysis in chapter 9 and want to explore the relationship between poverty and test scores.You decide to start your analysis by running a regression of test scores on the percent of students who are eligible to receive a free/reduced price lunch both in California and in Massachusetts.The results are as follows: CA = 681.44 - 0.610×PctLchCA
(0.99)(0.018)
n = 420,R2 = 0.75,SER = 9.45 MA = 731.89 - 0.788×PctLchMA
(0.95)(0.045)
n = 220,R2 = 0.61,SER = 9.41
Numbers in parenthesis are heteroskedasticity-robust standard errors.
a.Calculate a t-statistic to test whether or not the two slope coefficients are the same.
b.Your textbook compares the slope coefficients for the student-teacher ratio instead of the percent eligible for a free lunch.The authors remark: "Because the two standardized tests are different,the coefficients themselves cannot be compared directly: One point on the Massachusetts test is not the same as one point on the California test." What solution do they suggest?

Question 65

Consider a situation where Y is related to X in the following manner: Yi = β0 × × eui.Draw the deterministic part of the above function.Next add,in the same graph,a hypothetical Y,X scatterplot of the actual observations.Assume that you have misspecified the functional form of the regression function and estimated the relationship between Y and X using a linear regression function.Add this linear regression function to your graph.Separately,show what the plot of the residuals against the X variable in your regression would look like.