Economics 120c Name: _________________________ Professor Yongil Jeon Summer 2005 Student ID#: _________________________

Answer to Homework #2

(Final Exam, Summer 2004) (1) (20 points) To study the determinants of growth among the countries of the world,

researchers have used panels of countries and observations spanning long periods of time (e.g. 1965-1975, 1975-1985, 1985-1990). Some of these studies have focused on the effect that inflation has on growth, and found that although the effect is small for a given time period, it accumulates over time and therefore has an important negative effect.

(a) (5 points) Explain why the OLS estimator may be biased in this case.

Answer: The presence of simultaneous causality is highly likely since inflation may respond to growth. Depending on the list of regressors, omitted variables can also bias the estimator for the effect of the inflation rate.

(b) (10 points) Explain how methods using panel data could potentially alleviate the problem.

Answer: Country fixed effects or differencing the data can solve the problem if

inflation stays relatively constant over time from one country to the other. Unfortunately if the effect of inflation on growth is the focus of the study, then much of the cross-sectional information is lost using this approach.

(c) (5 points) Some authors have suggested using an index of central bank independence as an instrument, since the central bank independence is in general determined independently and separately by concerns on inflation. Discuss whether or not such an index would be a valid instrument.

Answer: For this index to be valid, central bank independence has to be relevant and exogenous. If inflation rates are correlated with the index, then central bank independence is a relevant instrument. Although there is a high correlation for developed countries, there is little to no correlation when data for all countries are considered. Whether or not the index is exogenous cannot be tested unless the coefficients of the equation are overidentified. Otherwise personal judgment is the only guide. An argument that central bank independence is exogenous would have to rely on it being based on institutional arrangements which are independent of inflation. Although the independence of central banks in many countries was initially determined by concerns independent of inflation, there have been many situations where the institutional arrangements were altered as a result of high inflation.

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2 Answer to HW #2, ECO 120C, SUMMER 2005

(2) (30 points) Two authors published a study in 1992 of the effect of minimum wages on teenage employment using a U.S. state panel. The paper used annual observations for the years 1977-1989 and included all 50 states plus the District of Columbia. The estimated equation is of the following type

Eit = β0 + β1 (Mit /Wit ) + γ2D2i + • • • + γnD51i + δ2B2t + • • • + δTB13t + uit, where E is the employment to population ratio of teenagers, M is the nominal

minimum wage, and W is average wage in the state. In addition, other explanatory variables, such as the prime-age male unemployment rate, and the teenage population share were included.

(a) (10 points) Briefly discuss the advantage of using panel data in this situation rather than pure cross-sections or time series.

Answer: There are likely to be omitted variables in the above regression. One way to deal with some of these is to introduce state and time effects. State effects will capture the influence of omitted variables that are state specific and do not vary over time, while time effects capture those of countrywide variables that are common to all states at a point in time. Furthermore, there are more observations when using panel data, resulting in more variation.

(b) (5 points) Estimating the model by OLS but including only time fixed effects results in the following output

Eit = β0 – 0.33× (Mit /Wit ) + 0.35× ( itSHY ) – 1.53× ituram ; _

2 0.20R = (0.08) (0.28) (0.13) where SHY is the proportion of teenagers in the population, and uram is the prime-age

male unemployment rate. Coefficients for the time fixed effects are not reported. Numbers in parentheses are (homoskedasticity-only) standard errors.

Comment on the above results. Are the coefficients statistically significant? Since

these are level regressions, how would you calculate elasticities?

Answer: There is a negative relationship between minimum wages and the employment to population ratio. Increases in the share of teenagers in the population result in a higher employment to population ratio, and increases in the prime-age male unemployment rate lower the employment to population ratio. 20 percent of employment to population of teenagers variation is explained by the above regression. The relative minimum wage and the prime-age male unemployment rate are significant using a 1% significance level, while the proportion of teenagers in the population is not. Elasticities vary with levels here. One possibility is to report elasticities at sample means.

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3 Answer to HW #2, ECO 120C, SUMMER 2005

(c) (5 points) Adding state fixed effects changes the above equation as follows: Eit = β0 + 0.07× (Mit /Wit ) – 0.19× ( itSHY ) – 0.54× ituram ; 2 0.69R = (0.10) (0.22) (0.11)

Compare the two results. Why would the inclusion of state fixed effects change the coefficients in this way?

Answer: The parameter of interest here is the coefficient on the relative minimum wage. While it was highly significant in the previous regression, it now has changed signs and is statistically insignificant. The explanatory power of the equation has increased substantially. The size of the other two coefficients has also decreased. The results suggest that omitted variables, which are now captured by state fixed effects, were correlated with the regressors and caused omitted variable bias.

(d) (10 points) The significance of each coefficient decreased, yet 2R increased. How is

that possible? What does this result tell you about testing the hypothesis that all of the state fixed effects can be restricted to have the same coefficient? How would you test for such a hypothesis?

Answer: The influence of the state effects is large. These are bound to be statistically significant and the hypothesis to restrict these same coefficients is bound to fail. Since these are linear hypotheses that are supposed to hold simultaneously, an F-test is appropriate here.

3. (30 points) Consider a simple model to estimate the effect of personal computer (PC) ownership on college grade point average for graduating seniors at a large public university

uPCGPA ++= 10 ββ where PC is a binary variable indicating PC ownership.

(a) (10 points) Why might PC ownership be correlated with u?

(Answer) It has been fairly well established that socioeconomic status affects student performance. The error term u contains, among other things, family income, which has a positive effect on GPA and is also very likely to be correlated with PC ownership.

(b) (10 points) Explain why PC is likely to be related to parents’ annual income. Does this mean parental income is a good IV for PC? Why or why not?

(Answer) Families with higher incomes can afford to buy computers for their children. Therefore, family income certainly satisfies the second requirement for an instrumental variable: it is correlated with the endogenous explanatory variables. But as we suggested in part (i), faminc has a positive effect on GPA, so the first requirement for a good IV,

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4 Answer to HW #2, ECO 120C, SUMMER 2005

fails for faminc. If we had faminc we would include it as an explanatory variable in the equation; if it is the only important omitted variable correlated with PC, we could then estimate the expanded equation by OLS.

(c) (10 points) Suppose that, four years ago, the university gave grants to buy computers to roughly one-half of the incoming students, and the students who received grants were randomly chosen. Carefully explain how you would use this information to construct an instrumental variable for PC?

(Answer) This is a natural experiment that affects whether some students own computers. Some students who buy computers when given the grant would not have without the grant. Define a dummy variable, grant, equal to one if the student received a grant, and zero otherwise. Then, if grant was randomly assigned, it is uncorrelated with u. In particular, it is uncorrelated with family income and other socioeconomic factors in u. Further, grant should clearly be correlated with PC: the probability of owning a PC should be significantly higher for students receiving grants. Incidentally, if the university gave grant priority to low income students, grant would be negatively correlated with u, and IV would be inconsistent.

4) (20 points) Describe the consequences of estimating an equation by OLS in the presence of an endogenous regressor. How can you overcome these obstacles? Present an alternative estimator.

(Answer) In the case of an endogenous regressor, there is correlation between the variable and the error term. In this case, the OLS estimator is inconsistent. To get a consistent estimator in this situation, instrumental variable techniques, such as 2SLS should be used. If one or more valid instruments can be found, meaning that the instrument must be relevant and exogenous, then a consistent estimator can be derived.

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