statistics question and need guidance to help me learn.

nonlinear regression

Requirements: answer with explanation

1 American University Department of Economics Econ 623 Amos Golan Fall 2023 TA: Hoa Vo Problem Set 4: Multiple Regression – Theoretical Exercises, Nonlinear Regression and a Case Study (Due in Three Weeks: October 25) Note: Start working on this PS now and leave questions from Chapter 8 for later (after we discuss it in class.) Special Notes Related to All Problem Sets: A) Please type what you can. B) All math problems do not have to be typed, but please write very clear and explain all derivations. C) Submission: A hard copy in class. Part I 1. Read SW, Chapters 7- 9. 2. Please answer problems from SW: 6.3, 6.4, 6.5, 6.8, 6.12; 7.2, 7.5, 7.6, 7.9; 8.2, 83, 8.5, 8.7, 8.8, 8.10. Part II Search in a recent newspaper (but not before September 1, 2023) for an article that discusses a certain economic policy/issue. (It can be an online newspaper.) Discuss in a few sentences (no more than half a page) the problem/issue and how you would analyze it using the tools of linear and/or nonlinear regression (and related hypothesis tests) we developed/discussed in class. Please write clear and remember to provide exact reference to your article. Part III 1. Explain in no more than 100 words (and in your own words) the advantages of using nonlinear regression methods with real data. 2. Explain in no more than 100 words (and in your own words) under what conditions you will use control variable in a regression analysis. 3. Provide an example of a regression model with interaction among two continuous variables. (You need to specify the complete model.) Explain.

2 4. Derive and show (in equation) the marginal effect of one of the two continuous interacting variables in problem 3 above. 5. Under what conditions will you use a nonlinear regression? Why? Explain. Part IV – Case Study- Concealed Handgun Laws and Crime A number of U.S. states have enacted laws that allow citizens to carry concealed weapons. These laws are known as “shall-issue” laws because they instruct local authorities to issue a concealed weapons permit to all applicants who are citizens, mentally competent, and have not been convicted of a felony (a number of states have additional restrictions). Proponents argue that, if more people carry concealed weapons, crime will decline because criminals are deterred. Opponents argue that crime will increase because of accidental or spontaneous use of the weapon. In this exercise, you are asked to analyze the effect of concealed weapons laws on three different categories of crimes: violent crimes; robberies (such as the robbery of a convenience store); and murder (many of which are spontaneous acts of passion). The data set “Econ 623 Fall 2023 PS 4 Concealed Handguns.xlsx” is on Canvas. The detailed definitions of all the variables are at the end of this problem set. 1. Estimate (i) a regression of ln(vio) against shall and (ii) a regression of ln(vio) against shall, incarc_rate, density, avginc, pop, pb1064, pw1064, and pm1029. a) Interpret the coefficient on shall in regression (ii). Is this estimate large or small in a “real-world” sense? Explain. b) Does adding the control variables in regression (ii) change the estimated effect of a shall-carry law in regression (i), as measured by statistical significance? as measured by the “real-world” significance of the estimated coefficient? Explain. c) Think of a possible variable which varies across states but plausibly varies little, or not at all, over time, and which plausibly could cause omitted variable bias in regression (ii). Motivate your choice. d) Describe how you would test the null hypothesis: States with a higher population of young male have a higher level of violent crimes? Could you prove causality? 2. Read the notes to Tables 1–3 below and then fill in the tables. (Note, in some of these regressions you are asked to use “fixed effects” which are just additional dummies either for the states or for time. See book for more.) 3. Write a short essay (200 words max) summarizing the conclusions and caveats from this study about the effect on crime of concealed weapons laws. Please address the following issues: • Do the results change when you add fixed state effects? If so, which set of regression results are more credible, and why? • Do the results change when you add fixed time effects? If so, which set of regression results are more credible, and why? • Based on what you consider to be the most credible specifications, what are the estimated effects of concealed weapons laws on crime rates, by type of crime;

3 • Are the differences in these estimates (if any) across crime rates consistent with differences in the nature of the crimes and how they might be affected by concealed weapons laws; and • In your view, what are the most important remaining threats to the internal validity of this regression analysis? • Based on your analysis, what conclusions would you draw about the effects of concealed weapons laws on these crime rates? Make sure to write clear or type!

4 Table 1 The Effect of Concealed Handgun Laws on Violent Crime: Regression Results Dependent variable: ln(vio) (1) (2) (3) (4) Coefficient on shall State characteristic control variablesa? no yes yes yes State fixed effects? no no yes yes Year fixed effects? no no no yes F-statistic testing the hypothesis that the state fixed effects are zero – – F-statistic testing the hypothesis that the year fixed effects are zero – – – n Notes: All regressions include an intercept. Heteroskedasticity-robust standard errors appear in parentheses below estimated coefficients; p-values appear in parentheses beneath heteroskedasticity-robust F-statistics. aRegressions with “state characteristic control variables” include the following regressors: incarc_rate, density, avginc, pop, pb1064, pw1064, pm1029. Table 2 The Effect of Concealed Handgun Laws on Robberies: Regression results Dependent variable: ln(rob) (1) (2) (3) (4) Coefficient on shall State characteristic control variablesa? no yes yes yes State fixed effects? no no yes yes Year fixed effects? no no no yes F-statistic testing the hypothesis that the state fixed effects are zero – – F-statistic testing the hypothesis that the year fixed effects are zero – – – n Notes: See the notes to Table 1.

5 Table 3 The Effect of Concealed Handgun Laws on Murders: Regression results Dependent variable: ln(mur) (1) (2) (3) (4) Coefficient on shall State characteristic control variablesa? no yes yes yes State fixed effects? no no yes yes Year fixed effects? no no no yes F-statistic testing the hypothesis that the state fixed effects are zero – – F-statistic testing the hypothesis that the year fixed effects are zero – – – n Notes: See the notes to Table 1. DATA DESCRIPTION, FILE: Econ 623 Fall 2023 PS 4 Concealed Handguns.xlsx This dataset is a balanced panel of data on 50 US states, plus the District of Columbia (for a total of 51 “states”), by year for 1977 – 1999, so an observation is a given state in a given year. There are a total of 51 states 23 years = 1173 observations. Crime rates from the Bureau of Justice Statistics. The data set contains more variables than those listed below; brief definitions of those variables are given in the data file. Variable Definition vio violent crime rate (incidents per 100,000 members of the population) rob robbery rate (incidents per 100,000) mur murder rate (incidents per 100,000) shall = 1 if the state has a shall-carry law in effect in that year = 0 otherwise incarc_rate incarceration rate in the state in the previous year (sentenced prisoners per 100,000 residents; value for the previous year) density population per square mile of land area, divided by 1000 avginc real per capita personal income in the state, in thousands of dollars pop state population, in millions of people pm1029 percent of state population that is male, ages 10 to 29 pw1064 percent of state population that is white, ages 10 to 64 pb1064 percent of state population that is black, ages 10 to 64 stateid ID number of states (Alabama = 1, Alaska = 2, etc.)