Exam 9: Assessing Studies Based on Multiple Regression

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.

Sample selection bias occurs when

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?

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

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

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.

Simultaneous causality bias

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?

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

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).

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?

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

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

Correlation of the regression error across observations

A statistical analysis is internally valid if

A definition of internal validity is

Misspecification of functional form of the regression function

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

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?

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?