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Use the data in COUNTYMURDERS to answer this questions. Use only the data for 1996.$\begin{array}{l}{\text { (i) How many counties had zero murders in } 1996 ? \text { How many counties had at least one execution? }} \\ {\text { What is the largest number of executions? }}\end{array}$$$\begin{array}{r}{\text { (ii) Estimate the equation }}\end{array}$$$$=\beta_{0}+\beta_{\text { lexecs }}+u$$by OLS and report the results in the usual way, including sample size and $R$ -squared.$\begin{array}{l}{\text { (iii) Interpret the slope coefficient reported in part (ii). Does the estimated equation suggest a deter- }} \\ {\text { rent effect of capital punishment? }} \\ {\text { (iv) What is the smallest number of murders that can be predicted by the equation? What is the }} \\ {\text { residual for a county with zero executions and zero murders? }}\end{array}$$\begin{array}{l}{\text { (v) Explain why a simple regression analysis is not well suited for determining whether capital pun }} \\ {\text { ishment has a deterrent effect on murders. }}\end{array}$

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Chapter 2

The Simple Regression Model

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Brandon S.

April 25, 2020

Use Wooldridge’s “wage2.dta” to estimate the following model:

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Use the data in COUNTYMURD…

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Use the data in MURDER onl…

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Use the state-level data o…

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Use the data in MURDER for…

moving on with computer exercise number nine. We were going to use the data set called Country Murders. Went to this question. We We use this data Celso in chapter one, one of the questions and the first thing we're gonna do is hit describe, to see what we're dealing with here we consume, been, have lot of observations too many. And indeed, we're gonna drop some of them. We're going to see here that some of the variables we're gonna use our executions, the number of executions in every county and the number of murders. But now, here's the deal. We have data pull data for, um, from 1980 to 90 96. The question says to use on Lee data for 1996. So one thing we could do, it's just never command. Include, um, restriction. Such as if years it won't 9 96 But because we want to make our lives easier, not harder. We're gonna say drop everything. If year is not equal to 99 6 right? Became before I hit entered. Look at that. Have thousands of data many years. Okay, Once I hit, enter 35,152 observations that lead it now. They knew. They just said we'll be just 2100. Name 97 variables. And as you can see, the only year remaining is 1996. Okay, um, party says how many countries had zero murders in 1996 And how many counties had at least one execution? What is the largest number of executions? Okay, various questions we're going to account if murders able. Zero 1051 counties had zero murders in 1996. Now, how many counties had at least one seclusion? At least one means that we're gonna do count if executions where greater or equal to one 31 counties is the answer. What is the largest number of executions? Well, gonna just summarize. Uh, execution variable only. Look at the maximum Maximum is three. Okay, Minimum is zero. And the overwhelming majority of Countess had zero executions and weaken. We can see that from the mean that is extremely close to zero. Just a few observation. Other than zero basically in part B. We need to estimate the equations. Murders equals B, the zero plus B. The one executions buzz it disturbance term and report the results in the usual way. Very easy. Regress murders and executions hit. Enter. All right, Here we can see we have the correct number of observations. The joint F test. The it's, um, critical value. It's very high. And so the ironed lying pig bellies practically zero. This means of the model is better in explaining murders in just intercept model with no variables. However, the are square is not that highs around for 4.4% of variation in murders is explained by at varies and executions. Um, no high at all. Here we can see the constant terms is highly significant is 5.45 and our slow coefficient is 58.55. That's a huge number. Okay, why am I saying is huge? Because in part three, when you interpret the slope coefficient reported, and, um and see if it's a just a deterrent effect of capital punishment. Well, this coefficient right here, since we're dealing with that log log model right here, I almost forgot to show you the the output since we're doing with that log love model. The interpretation is if there's one more execution in a county. This caused this is, um, associated with 55 points. 58.55 more murders. Isn't that crazy? I guess the analysis I was hoping to suggest that executions actually deter crime, but are simple by very digression. Says there's one more execution a county disassociated with 59 more murders in the county. So one thing we we sees that this is a positive number of slope coefficient, whereas I guess, Ah, the original idea was they were supposed to be negative, right? If capital punishment is more strict than we have less murders, but here and did we have more? Right. So the estimate equation does not suggest a return effect of capital punishment. Now, go back to stay them in Part four asks what is the smallest number of murders that can be predicted by the equation was the residual for a county with zero executions and zero murders. All right, so for the smallest number that can be predicted, we're gonna hit exit. You're gonna replace execution with zero. And this is, of course, our constant term. So our modern implies that they're no counters with lesson 5.45 murders off course. That's false. We know that many counties have zero murderers. In fact, 1051. Whatever we found around half and again in the residual, uh, there's Digital's for a counter with zero execution, zero murders again with zero here. And, ah, the residual will be fitted minus the, um, the actual one. So again it will be zero. It will be five point. The intercept minus zero will be. The intercept came. And so the answer to both questions in part for is the intercept term for four five. And take my word for granted. You conceived here in list. Okay, let's just create the values predict fitted values. So this command is gonna create a vector of values, Ana, Right away you can see here. For example, these are all observations with zero executions the fitted Bali's air for, uh, 00.457 which is our intercept. Okay. And let's find an observation that has zero murders. Your execution plus a right here. Observation 17 0 Where's the zero murders? Their execution again. The fitted value is, um, pipe 1 47 The residual now, uh, this conflict, the residuals. All right. Residual as well 00 There's YSL would be negative. Of course. Excuse me, I I forgot to say negative, Right. So the residual food will be minus the intercept. That plus plus is a faded valleys became There's a very good way to see it here. For example, if a country has zero county has their executions and 14 murders. Look at that. The fit, of course, is gonna be the same as, uh, as any number of murders doesn't matter just the execution matter, but the residual will be even higher. Kind of a terrible progression results. Now, in Part five, when you explain why simple regression analysis like this is not well suited for determining whether couple punishment has a deterrent effect on murders. Well, there are. I can think of atleast five reasons, but I'm gonna tell you the two main ones. The first reason is that as you consume from the r squared execution, executions do not really explain much of the variation in murders. And of course, it makes sense because murders in a county, our function off too many variables, maybe hundreds of variables, for example, I would think, uh, variables. That would be potentially more important than executions. For example, the rial income, average income in a county, the unemployment rates the ah, you know, percentage of people that have been incarcerated before so many variables. So again, just by burying analysis is not a good idea. And this is again captured by the very small are square now, second and maybe more. Most importantly, is that we haven't dodge ineighty problem here. So we're trying to explain murders in terms of executions. So in some way, we're maybe implying that might be a quiz ality going from executions, murders, but because L. A. That would be the other way. More murders caused more executions, right? Or there might be a two way causalities. One thing you know there's indulgently by is we don't know what explains the other. And of course, it might be even more variables that are correlated with those two variables, and that would make the analysis even wars, especially when we just have two variables. So just illustrate how misleading this could be. So so far, we found this Ah coefficient of interest are estimated to be No. One is 58. Let's just run aggression like before, right murders except. But then let's agree more variables. Okay? All the things we have fearlessly flew. Let's include density, population. Ah, rial Per capita income. Ah, really? Because the unemployment. Ah, yeah, that's it. Okay. For more variables. What's his enter? Oh, look at that. We're talking R squared one from 1 4.4% to 6 76% Okay, all the estimates are statistically significant of the 1% level. And let's without executions now. Ah, now one more execution in place. 13 more murders as opposed to 58. We're talking about, um, 1/4 overestimate them before even less. So again, you see that we need to, um, incorporate more variables in our analysis and concentrate on a multi varied regression framework for difficult questions like this one

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