Exam 9: Assessing Studies Based on Multiple Regression

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.

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\times,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 $\widehat { g 6090 }$ = 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, $\frac { \Delta _ { t } \ln Y _ { i , t } } { T }$ = ?₀ + ?₀ ln Y_i,0 + u_i,t, where ?_t ln Y_i,t = ln Y_i,0 - ln Y_i,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 Y_t = ?₀ + ?₁ ln Y₀ + u_t 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: $\tilde { Y } _ { t = \beta _ { 0 } + \gamma _ { 1 } } \tilde { Y } _ { 0 } + u _ { t }$ and assume that the "~" variables contain measurement errors of the following type, $\tilde { Y _ { i , t } } =Y _ { t }^* + v _ { i , t } \text { and } \tilde { Y _ { i , 0 } } = Y _ { 0 } ^ { * } + w _ { i , 0 } \text {. }$ 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 $\tilde { Y _ { i , t } } = \left( Y _ { t } ^ { * } - Y _ { 0 }^ { * } \right) + \tilde { Y _ { i , 0 } } + \left( v _ { i , t } - w _ { i , 0 } \right)$ 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: $\hat { \beta } _ { 1 } \xrightarrow{p} 1 - \frac { \sigma _ { v } ^ { 2 } } { \sigma _ { y } ^ { 2^* } + \sigma _ { v } ^ { 2 } }$ Discuss the implications for the convergence results above.

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

The analysis is externally valid if

Consider the one-variable regression model, Y_i = β₀ + β₁X_i + u_i, where the usual assumptions from Chapter 4 are satisfied. However, suppose that both Y and X are measured with error, $\tilde { Y } _ { i }$ = Y_i + z_i and $\widetilde { X } _ { i }$ = X_i + w_i. Let both measurement errors be i.i.d. and independent of both Y and X respectively. If you estimated the regression model $\tilde { Y } _ { i }$ = β₀ + β₁ $\widetilde { X } _ { i }$ + v_i using OLS, then show that the slope estimator is not consistent.

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 compare the slope coefficient from the California School data set with that of the Massachusetts School data set, you run the following two regressions: $\widehat {\text { TestScr }}$ _CA = 2.35 - 0.123×STR_{CA (0.54)(0.027) n = 420, R}^{2 = 0.051, SER = 0.98} $\widehat {\text { TestScr }}$ _MA = 1.97 - 0.114×STR_{MA (0.57)(0.033) n = 220, R}^{2 = 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?

Simultaneous causality bias

A study based on OLS regressions is internally valid if

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.

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 definition of internal validity is

The components of internal validity are

Threats to in internal validity lead to

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?

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

In the case of errors-in-variables bias,

Simultaneous causality

Consider a situation where Y is related to X in the following manner: Y_i = β₀ × $X _ { i } ^ { \beta _ { 1 } }$ × 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.

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 × $\text { Batavg } _ { i } ^ { 1997 }$ ; 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?