Exam 9: Assessing Studies Based on Multiple Regression

Several authors have tried to measure the "persistence" in U.S state unemployment rates by running the following regression: $u r _ { i , t } = \beta _ { 0 } + \beta _ { 1 } \times u r _ { i , t - k } + z _ { i , t }$ where $u r$ is the state unemployment rate, $i$ is the index for the $i$ -th state, $t$ indicates a time period, and typically $k \geq 10$ . (a)Explain why finding a slope estimate of one and an intercept of zero is typically interpreted as evidence of "persistence."

Assume that a simple economy could be described by the following system of equations, =++ = 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 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,=0.825,SER=7.818 (0.542)(0.013) = 4.131+0.943 Re a,=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?

The Phillips curve is a relationship in macroeconomics between the inflation rate (inf) and the unemployment rate (ur).Estimating the Phillips curve using quarterly data for the United States from 1962:I to 1995:IV, you find $\widehat { \operatorname { Inf } _ { t } } = 4.08 + 0.118 u r _ { t } , R ^ { 2 } = 0.003 , \text { SER } = 3.148$ $(1.11)(0.176)$ (a)Explain why, at first glance, this is a surprising result.

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

Simultaneous causality

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.

The reliability of a study using multiple regression analysis depends on all of the following with the exception of

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,=0.00007,SER=0.016 (0.004)(0.0073), where $g 6090$ is the average annual growth rate of GDP per worker for the 1960-1990 sample period, and RelProd $_ { 60 }$ is GDP per worker relative to the United States in 1960. For the 24 OECD countries in the sample, the output is = 0.048-0.0404,=0.82,SER=0.0046 (0.004)(0.0063) Interpret the results and point out the difference with regard to unconditional convergence.

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

You have decided to analyze the year-to-year variation in temperature data.Specifically you want to use this year's temperature to predict next year's temperature for certain cities.As a result, you collect the daily high temperature (Temp)for 100 randomly selected days in a given year for three United States cities: Boston, Chicago, and Los Angeles.You then repeat the exercise for the following year.The regression results are as follows (heteroskedasticity-robust standard errors in parentheses): $\widehat { \operatorname { Temp } _ { t } ^ { B O S } } = 18.19 + 0.75 \times \operatorname { Temp } _ { t - 1 } ^ { B O S } ; R ^ { 2 } = 0.62 , S E R = 12.33$ $(6.46)(0.10)$ = 2.47+0.95\times;=0.93,SER=5.85 (3.98)(0.05) $\widehat { \text { Temp } _ { t } ^ { L A } } = 37.54 + 0.44 \times \text { Temp } _ { t - 1 } ^ { L A } ; R ^ { 2 } = 0.18 , \text { SER } = 7.17$ $(15.33)(0.22)$ (a)What is the prediction of the above regression for Los Angeles if the temperature in the previous year was 75 degrees? What would be the prediction for Boston?

Misspecification of functional form of the regression function

Think of three different economic examples where cross-sectional data could be collected.Indicate in each of these cases how you would check if the analysis is externally valid.

Keynes postulated that the marginal propensity to consume $\left( \mathrm { MPC } = \frac { \Delta C } { \Delta Y _ { p d } } \right)$ is between zero and one. He also hypothesized that the average propensity to consume (APC $= \frac { C } { Y _ { p d } }$ ) would fall as personal disposable income $Y _ { p d }$ increased. (a)Specify a linear consumption function.Show that the assumption of a falling APC implies the presence of a positive intercept.

Your professor wants to measure the class's knowledge of econometrics twice during the semester, once in a midterm and once in a final.Assume that your performance, and that of your peers, on the day of your midterm exam only measure knowledge imperfectly and with an error, $\widetilde { X _ { i } ^ { 1 } } = X _ { i } ^ { 1 } + w _ { i } ^ { 1 } ,$ where $\widetilde { X }$ is your exam grade, $X$ is underlying econometrics knowledge, and $w$ is a random error with mean zero and variance $\sigma _ { w } ^ { 2 } . w$ may depend on whether you have a headache that day, whether or not the questions you had prepared for appeared on the exam, your mood, etc. A similar situation holds for the final, which is exam two: $\widetilde { X _ { i } ^ { 2 } } = X _ { i } ^ { 2 } + w _ { i } ^ { 2 }$ . What would happen if you ran a regression of grades received by students in the final on midterm grades?

In macroeconomics, you studied the equilibrium in the goods and money market under the assumption of prices being fixed in the very short run.The goods market equilibrium was described by the so-called IS equation $R _ { i } = \beta _ { 0 } - \beta _ { 1 } Y _ { i } + u _ { i }$ where $R$ represented the nominal interest rate and $Y$ was real GDP. $\beta _ { 0 }$ contained variables determined outside the system, such as government expenditures, taxes, and inflationary expectations. The money market equilibrium was given by the so-called LM equation $R _ { i } = \gamma _ { 0 } + \gamma _ { 1 } Y _ { i } + v _ { i }$ and $\gamma _ { 0 }$ contained the real money supply and the intercept from the money demand equation. Show that there is simultaneous causality bias in this situation.

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: =++ =\rho+;|\rho|<1. Furthermore, assume that you had obtained consistent estimates for $\beta _ { 0 } , \beta _ { 1 } , \rho$ . If asked to make a prediction for $Y$ , given a value of $X \left( = X _ { j } \right)$ and $\hat { u } _ { j - 1 }$ , how would you proceed? Would you use the information on the lagged residual at all? Why or why not?

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

Panel data estimation can sometimes be used

Your textbook gives the following example of simultaneous causality bias of a two equation system: =++ =++ In microeconomics, you studied the demand and supply of goods in a single market. Let the demand $\left( Q _ { i } ^ { D } \right)$ and supply $\left( Q _ { i } ^ { S } \right)$ for the $i$ -th good be determined as follows, =-+, =++ where $P$ is the price of the good. In addition, you typically assume that the market clears. Explain how the simultaneous causality bias applies in this situation. The textbook explained a positive correlation between $X _ { i }$ and $u _ { i }$ for $\gamma _ { 1 } > 0$ through an argument that started from "imagine that $u _ { i }$ is negative." Repeat this exercise here.