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Use the data in KIELMC, only for the year $1981,$ to answer the following questions. The data are forhouses that sold during 1981 in North Andover, Massachusetts 1981 was the year construction began on a local garbage incinerator.(i) To study the effects of the incinerator location on housing price, consider the simple regressionmodellog $(price)=\beta_{0}+\beta_{1} \log (d i s t)+u$where price is housing price in dollars and dist is distance from the house to the incinerator measured in feet. Interpreting this equation causally, what sign do you expect for $\beta_{1}$ if the presence of the incinerator depresses housing prices? Estimate this equation and interpret the results.(ii) To the simple regression model in part (i), add the variables log( (intst), log(area), log(land), rooms, baths, and age, where intst is distance from the home to the interstate, area is square footage of the house, land is the lot size in square feet, rooms is total number of rooms, bath is number of bathrooms, and age is age of the house in years. Now, what do you conclude about the effects of the incinerator? Explain why (i) and (ii) give conflicting results.(iii) Add [log( intst) $]^{2}$ to the model from part (ii). Now what happens? What do you conclude about the importance of functional form?(iv) Is the square of log(dist) significant when you add it to the model from part (iii)?

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i.The causal (or ceteris paribus) effect of dist on price means that $\beta_{1} \geq 0$ : all other relevant factors equal, it is better to have a home farther away from the incinerator. The estimated equation is$$\log (\hat{p} r i c e)=\begin{array}{cc}{8.05} & {.365 \log (d i s t)} \\ {(0.65)} & {(.066)}\end{array}$$$$n=142, R^{2}=.180, \overline{R}^{2}=.174$$which means a 1$\%$ increase in distance from the incinerator is associated with a predicted pricethat is about. 37$\%$ higher.ii.When the variables log(inst), log(area), log(land), rooms, baths, and age are added to theregression, the coefficient on log(dist) becomes about. 055$($ se $\approx .058)$ . The effect is muchsmaller now, and it comes out to be statistically insignificant. This is because we have explicitlycontrolled for several other factors that determine the quality of a home (such as its size andnumber of baths and its location (distance to the interstate). This is consistent with thehypothesis that the incinerator was located near less desirable homes to begin with.iii.When [log(inst) $^{2}$ is added to the regression in part (ii), we obtain (with the results only partiallyreported)Equation not availableThe coefficient on log(dist) is now very statistically significant, with a $t$ statistic of about three. Thecoefficients on log(inst) and $[log(inst)]^{2}$ are both very statistically significant, each with $t$ statistics above four in absolute value. Just adding $[log(inst)]^{2}$important for policy purposes. This means that distance from the incinerator and distance fromthe interstate are correlated in some nonlinear way that also affects housing price.One can find the value of log(inst) where the effect on log(price) actually becomes negative:2.073$/[2(.193)] 8.69 .$ When one takes it as an exponent then one will obtain about $5,943$ feetfrom the interstate. Therefore, it is best to have home away from the interstate, for distances lessthan just over a mile. After that, moving farther away from the interstate lowers the predictedhouse price.iv.The coefficient on $[\log (d i s t)]^{2}$ when it is added to the model estimated in part (iii), is about$-.0365,$ and its $t$ statistic is come out to be $-0.33 .$ Therefore, it is not necessary to add thisvariable and make the analysis more complicate.

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Chapter 6

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Pierce N.

October 22, 2021

Sorry what is the formula to actually exponentiate a value. To better explain I got the answer 8.7 to determine when the marginal effect becomes negative, but I am lost on how you came to 5,943 feet.

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Hi, everyone. Hope you're doing great today. We're starting with chapter six from the textbook and into each other. We're gonna dig a bit deeper into the standard multiple regression analysis, which means that we're going to do a bit more involved and hence more exciting stuff. So we started with computer exercise number one, in which we're gonna use data set cold keel I'm seeing, and we're gonna need data only for the year 1981 and two. The following question. So before everything looks just describe our data said, All right, lots of variables here. Indeed. We have observations for multiple years. 321 to be exact. But we need only years from anyone. So we're gonna drop all the years apart from 1981. So drop if year is not equal to 90 81. Okay. All right. 18 7177179 Observations, lead. Let's go. The browser to check the Indeed, we have only a that four year, and I need one. And here days became, so we're done with that. Okay. The data for houses, though, sold during 1981 in North Andover, Massachusetts. 1981 was a year construction began on a local garbage incinerator. We're gonna need this piece of information. So in part one, we need to study the effects of the incinerator location on housing prices. So we consider the following simple regression model. The luck of the price equals to, ah, intercept plus Peter. One lug of the distance prices. Housing price in dollars and distance is a distance from the house, an incinerator measured in feet. So we needed wrong distribution and interpret this equation casually. What signed away. Expect for be the one if the presence of the incinerator depresses housing prices. All right, so let's think about it for a second. That's a very easy question. Ah, the standard causal or Cherries, parabolas effect of the distance in Syria or price. We expected to be what we expected to be positive, right? In the sense that the father's from the incinerator, the ah, higher price you're going to get if you ah, to your house to the market. So we expect me to want to be greater than zero, which means once again all other relevant factors being equal. It is better to have a house, a home further away from the incinerator. Now, let's see if this holds in practice. We're gonna run the regression look, Price. Uh, where is it? Here it is. How price, I guess. Yes. Um uh, constant term and log of distance. Log of distances l fist year. Just gonna new regression. All right, let's see. Okay, so we have. First of all, we have the correct number of observations nor squared of 18%. So 18% of the variation in the log price is explained by the luck of distance incinerator. They adjust their squared, which is not really needed here, because we only have one variable as living unless we have a very statistically statistically significant joint f test for the coefficients. And as we can see, both coefficients are very statistically significant at any level of significance. And here we report the estimated regression. So the luck of the price it was eight points there. Five Plus, Is there a 0.365 block distance? You hear a drill been So what does this mean? This means that at 1% increase in distance from the incinerator and 1% in Greece, this is Le Glaude model. So 1%. Increasing distance from the incinerator is associated with the predicted price. There is about 0.37% higher. Okay, does it look like we don't need to modify by 100? So 1% increase in luck distance associated with 0.36 50.37% increase in prices. So we get to be the one just great in zero exactly as we predicted. Ah, using our using our brain right before we're under aggression. Okay, now let's go back. You say that part two says, Do this simple regression model that we just try and add the variable slug. Ah, lug inst which is the luck of the distance from the home to the interstate. What's added here? I just added as we, uh of snow love inst was fined IDs three days. Ah, then log area, which is a log of the area of the house and square footage. Log area. Where is it? Right here. Mmm. What else? The lug of the land. This is love land here, which is that log of the of the of the large size and square feet. And we also add a room number of rooms over vast and the age. Let's see grooms, baths and the age of the house, the building that was once we do that, we need to run this regression whether we conclude about the effects of the incinerator. Now, let's see. All right, well, that's the difference. Look at that. This is the simple regression model. This he expanded multi very analysis here. Okay, so we have the correct number of observations. Very significant. Join F test and very, very high are square and adjust our squared. Look, let's look at the just one thio. Provide a more accurate, accurate Well, let's say Thio, just provide a penalty for the extra for the extra variables added. So it's, uh, 73.5% variation. Look, prices are explained by our model, and here we see that the love distance this is the very well. If interest the luck of the distance incinerator. Now it's amounts of this quiz significant? Not at all has it T statistic lesson one associated with people. You're hugely, Valerie said. This is not statistically significant, but even if Iwas look at that we're talking about. Before it was there a 0.365 but this 778 times less. Right. So we have Ah, very, very small estimated coefficient and ah, which is not statistically significant. And indeed the value of zeros within one standard error. So what does this mean? The effect is much, much smaller now, and statistically significant. Why? Because we have explicitly controlled for several other factors that determine the quality of the home. Rice is the size, number of rooms, number of best, its location, which is measured here in the distance from the interstate. So once we control for these very important factors in the distance of incinerator, it's not important here. And what does this mean? Does this mean that actually a distance? The incinerator is not that important? Maybe. But it was trying to be a bit more prudent with our analysis. The first thing we can say is that it's definitely not as important as we first thought. That many other factors that are way more important than that and those are the fundamentals. You know, the things that a buyer would look when he or she is considering to buy a house. But another thing. But when my consider is that, remember that this data set is a dame from houses salt in 1981. But the other hand, we think that less desirable homes are located near the incinerator. It's very likely that those houses were not salt, so we don't have information. And this data set the boats house there were not sold. So maybe the houses that were considering here there were there were very, uh, desirable in the first place, not very close to the incinerator. So this could also be an explanation. All right, In part three, now that this regression model, we need to add a quadratic term, that lug of instant into distance, Interstate Square. And now what happens when we conclude about the importance of the functional form? Okay, let's see, Here is the love institution, and we're gonna do the love Insulation Square is already here. The Davis, that data said luck. Instance square and hit. Enter and look at that. All right, look at that. Okay, So the first thing we notice is that the adjusted our square is significantly higher than before, given the penalty that we had an extra very boy is much, much higher now. What happens? Wow, look at that. This thing become the distance from incinerate. It becomes very well. There. It becomes a disco significant. The 1% level the T value almost looks three. Not bad. Now it's the fact is, around 0.18% you know, nearly halfway from the original one. And most importantly, the the lug of the difference. Interstate and the quadratic terms are very, very statistically significant. Okay, look at that. Let's report the results here. Of course, I haven't reported the whole thing. That would take me like time to write. We're here. The point is of interest. So again, the coefficients and lug inst logging square both very statistically significant with, uh each with t status sticks above four and absolute value. So just adding the quadratic term has had a very big effect on the coefficient that is important for policy purposes, right? The difference. Both of the the distance to the incinerator and the location, as in the different and the distance from interstate. So this means two things, first of all, could see that the relationship between the lug off the location, the distance from interstate and the luck of price have concave relationship. Sort of like this. My right, this Let's stay here long. Okay. Well, let me do it again. Like this, right? It's an increasing here. Let's save love of the price. I'm gonna write. Look here. Here's a slug of that, uh, inst. And somewhere here is the maximum. Here's the maximum point, which is also turn around point. After that, it goes down. Would say it goes down after that. So it's a conch, a function. And we know that a relation between them is increasing, but the first, every one of his negative meaning that it every step along the way it gets less and less right. So this means that it's better to be further from the interstate. But the further you are from the interstate, it's not the relative, you know. Increasing price gets less and less. This is the first thing we learned from this aggression. The other thing and very important is that since adding this quadratic term changes the relationship in a positive way and increase the adjusted R squared and many coefficient becomes a statistical significant now, this means that the the distance from the incinerator and the distance from the Interstate are correlated in some lot nonlinear way that also affects the housing price. Okay, this is exactly what this means. They're coordinating some nonlinear way that also fix the housing price. We don't know exactly what this means, but, you know, theory, real estate or economic theory could give answers to that. Now. The next thing we can do is that we can find the value of the law against where the effect of love price actually becomes negative. This is a turnaround point, but this is also the point where the where the effect is maximal, meaning that this is a point where the distant from the interstate yields the higher highest increase in lung price. And as we learned from our chapter, we can find it. We can find this number, the exact number by dividing the coefficient. Let's say it's better to hear the coefficient on, uh, the Lena term divide by two times the coefficient of the quadratic terms. So here we get 2.73 divided by two points 0 11 93 which is 8.698 point 70 now. I've written the approximately here because I did Ah rounding. But this gives the exact number right is not approximation, and this doesn't eyes never easy to interpret. What is this log of them, feet or whatever days so exponentially ate it, and we obtain about 5943 feet from the interstate, which means now one mile has 5280 feet. So this is just about in my 1.5 miles. So this means it is best to have your home away from the interstate for distant up to just over one mile after that, moving further, moving further away from the interstate actually lowers. Predict house prices like this makes a lot of sense, like it's nice to be close to the interstate, but not too close, not too far. So here, from the analysis, the estimated more efficiently, finally, is the best. It's just to be around one mile, not more. Okay, right? No, for parts for let's see, four for us is the square of luck distance significant when you add it to the model. Well, let's see. Ah, here it is. Look, this is legit. Let's add Looked into squared. We have it here. Ah, well we don't have it, We're gonna create it. Let's generate Was the eldest squared equals? But how is it is look, distance square easy. So we're gonna add it to the model here and see what happens. No game. Well, first thing when you look at is that he adjusted r squared, actually dropped. Look at that. Stifled by did drop. Or we can say that they are square just increases the marginally a noticeable difference. And of course, the luck of the a distant square is not siddiq physics to disclose that at all. All right, so we don't have any evidence that this coefficient is that basically significant from zero. So adding this term to the model Ah, it's not necessary to add this level of complication. It doesn't provide anything. In fact, it makes our model slightly worse. So there's no need to add it to our model

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