Exam 9: Assessing Studies Based on Multiple Regression

In the simple, one-explanatory variable, errors-in-variables model, the OLS estimator for the slope is inconsistent.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 the OLS estimator for the intercept behaves as follows in large samples: $\hat { \beta } _ { 0 } \stackrel { p } { \rightarrow } \beta _ { 0 } + \mu _ { \bar { X } } \frac { \sigma _ { w } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 } ,$ where $\widetilde { \widetilde { X } } \stackrel { p } { \rightarrow } \mu _ { \tilde { X } }$

Consider the one-variable regression model, $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$ assumptions from Chapter 4 are satisfied. However, suppose that both Y and X are measured with error, $\widetilde { Y } _ { i } = Y _ { i } + z _ { i } \text { and } \widetilde { X } _ { i } = X _ { i } + w _ { i }$ and independent of both Y and X respectively. If you estimated the regression model $\widetilde { Y } _ { i } = \beta _ { 0 } + \beta _ { 1 } \widetilde { X } _ { i } + v _ { i }$ \quad using OLS, then show that the slope estimator is not consistent.

Simultaneous causality bias

In the case of a simple regression, where the independent variable is measured with i.i.d. error, a. $\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$ b. $\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } }$ . c. $\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { w } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$ . d. $\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \beta _ { 1 } + \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } }$ .

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 $\widehat { \text { Batavg } _ { i } ^ { 1998 } } = 0.138 + 0.467 \times \text { Batavg } _ { i } ^ { 1997 } ; R ^ { 2 } = 0.17$ 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?

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 $\text { simple regression case, derive } \hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 } \text {. }$

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

Your textbook has analyzed simultaneous equation systems in the case of two equations, =++ =++ 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 $Z _ { i } = \delta _ { 0 } + \delta _ { 1 } Y _ { i } + w _ { i }$ 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.

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,

Consider a situation where Y is related to X in the following manner: $Y _ { i } = \beta _ { 0 } \times X _ { i } ^ { \beta _ { 1 } } \times e ^ { u _ { i } }$ 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.

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 $R ^ { 2 }$ . d. whether the good in question is price elastic.

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 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?

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.

Sample selection bias