Exam 9: Assessing Studies Based on Multiple Regression

Your textbook uses the following example of simultaneous causality bias of a two equation system: =++ =++ 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 $D ^ { 0 }$ and $S ^ { 0 }$ . 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?

The textbook derived the following result: $\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$ Show that thisis sthe same as $\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \beta _ { 1 } - \frac { \sigma _ { w } ^ { 2 } } { \sigma _ { w } ^ { 2 } + \sigma _ { X } ^ { 2 } } \beta _ { 1 }$

A statistical analysis is internally valid if

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

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

The errors-in-variables model analyzed in the text results in $\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$ 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.

By including another variable in the regression, you will a. decrease the regression $R ^ { 2 }$ 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.

The components of internal validity are a. a large sample, and BLUE property of the estimator. b. a regression $R ^ { 2 }$ 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.

In the case of errors-in-variables bias,

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: $\widehat { \text { Studenth } } = 19.6 + 0.73 \times \text { Midparh, } R ^ { 2 } = 0.45 , \text { SER } = 2.0$ $(7.2) \quad(0.10)$ 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."

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.

To analyze the situation of simultaneous causality bias, consider the following system of equations: =++ =++ Demonstrate the negative correlation between $X _ { i }$ and $u _ { i }$ for $\gamma _ { 1 } < 0$ , either through mathematics or by presenting an argument which starts as follows: "Imagine that $u _ { i }$ is negative."

Correlation of the regression error across observations

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

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?

Errors-in-variables bias

A study based on OLS regressions is internally valid if

Sample selection bias occurs when

The analysis is externally valid if

Errors-in-variables bias