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Use the data in RDCHEM to further examine the effects of outliers on OLS estimates and to see howLAD is less sensitive to outliers. The model is$$=\beta_{0}+\beta_{1} \text { sales }+\beta_{2} \text { sales }^{2}+\beta_{3} \text { profmarg }+u$$where you should first change sales to be in billions of dollars to make the estimates easier tointerpret.(i) Estimate the above equation by OLS, both with and without the firm having annual sales ofalmost $40 billion. Discuss any notable differences in the estimated coefficients.(ii) Estimate the same equation by LAD, again with and without the largest firm. Discuss anyimportant differences in estimated coefficients.(iii) Based on your findings in (i) and (ii), would you say OLS or LAD is more resilient to outliers?

(i) The coefficient of SALES $_{-} \mathrm{BN}_{-} \mathrm{DOL}$ (Sales figures in billion dollars) and SALESSQ _ DOL (the square of the sales figures in billion dollars) are less in Case 1 than in Case?It shall be noted that the coefficient of SALESSQ_DOL is statistically significant at 10$\%$ level of significance(p-value is less than the critical p-value of 0.1 at 10$\%$ level of significance) in themodel results with the inclusion of firm with annual sales of almost $\$ 40$ bn ( (Case $1 ),$ whereas, it is not statistically significant at 10$\%$ level of significance (p-value is greater than the critical p-value of 0.1 at 10$\%$ level of significance) in the model results with the exclusion of the firm withannual sales of almost $\$ 40$ bn (Case2) This indicates that the linear relationship between $r d i n t e n s$ and SALES $_{-} \mathrm{BN}_{-}$ DOL is evident in case when the firms with extremely large sales value is excluded from the model(ii) It shall be noted that the coefficient of SALES $_{-} \mathrm{BN}_{-} \mathrm{DOL}$ is positive $(0.263704)$ when the largest firm is included indicating that the sales has positive median effect on $R \& D$ spending as percentage of Sales and negative $(-0.222764)$ when the largest firm is excluded indicating that the sales has negative median effect on $R \& D$ spending as percentage of Sales Similarly SALESSQ_DOL also experiences change of sign when the largest firm is excluded.(iii) It shall be noted that OLS estimates showed a marginal change in magnitude, but not in signswhereas, LAD estimates showed change in sign and magnitude, indicating that OLS estimatesare more resilient to outliers

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all right. Hello, everybody. Today we're gonna be doing a fairly basic econometrics exercise, exploring the sensitivity of OLS and L A d estimates and how sensitive they are to outliers. So let's get right into it. First thing we want to do is install our World Ridge practice. You don't already have it installed. Um, I should already have this installed, but the fact that I don't slightly worries me. Okay. And there we go. The package is now installed, and I can open up the library. I said, um, for this one, we're also going to be using the l one pack package. This package has the l. A. D. Function. So it just makes it a little bit easier on us to actually perform the l A. Deflection. Uh, so breast set and really quick. In a comment here, I'm gonna write out our formula Our formula for today is gonna be aren't already intends. Vehicle The sales plus sales square Sail Square is, uh is a column already in the data set, so we don't have to worry about manually squaring sales plus Croft march. Okay, cool. So now let's get straight into it. So first we wanna estimate the equation using OLS. But before we do that, actually, we need to change sales and Sales Square to be in billions of dollars. Right now, they're in millions, so this is pretty easy. Um, it would be if I had to. Beta said, Well, let's look at this really quick. Right? So we have our data set here. You can see sales. Um, 4570. This is millions, right? All of these air millions and sales square is therefore million's squared. So we have 2.0 a nine times 10 to the seventh, right? Or the six? What? Etcetera, etcetera, etcetera. So, um, what we're gonna have to do is we are going to have to convert sales to billions and sales squares, two billions. So how do we do that? Well, to change something, what we're gonna want to do is we're gonna say rd come. That's art table or data said and then we're going to select the SEALS column. Then we're going to use our variable call the operator. Right? So rd kem sales is going to be equal to sales provided by 1000. Pretty simple to do this for sale squared. We can't divide by 1000. In fact, we have to divide by 1000 squared or one million. By doing that, you'll now see way come up to here. Our sales is now 4.570 right, That's 1000 times less. And our sales? The coefficient here went from 10 to the seventh to tend to the first. That's a difference of 10 to the sixth or one million. So you can see that this all worked out fun. All right, now we need to perform our OLS estimates. So we're gonna make our model called Olesz, using the L M Function right alum for linear model by OLS estimates, and we're going to sales for a sale squared plus profit margin are beta is equal to our camp. Now, for our second OLS estimate, we actually need to take a subset of the data because our second estimate has to exclude the data point with 40 billion in sales. So again, looking back at this for a sec, you'll see that you know, 4 to 000 1939. So this data point needs to be excluded. But nothing else even comes close to that, right? Except maybe this 19 1 So what we're gonna do is we're gonna say, All right, Rita, are new data sets. Name is gonna be called without 40 right? Without 40 billion. And we're gonna take a subset of Arctic. Um, that's dysfunction on. Our criteria for that subset is going to be Sales is less than 35. So wherever sales is less than 35 uh, that's going to be in on you think so? If I really couldn't do what? Count of rd come. Um, sorry, I don't have the library for the lab, but, um five. You without 40 b. You'll see that? Yeah. The row names up to nine or consistent, but then you get to 10 and this one is 11. So our 10th row right? The one with the 40 billion just got taken out of our without for TV data set. Okay, cool. So we have our OLS model are less without brat. 40 b model is going to be the same thing already intends sales plus sale squared, plus our profit margin, but our data, instead of taking from already, Kem, we're gonna take it from without 40 beat. And so if I think a summary of these two summary of oil s it out, you will see some key differences. So first thing to know, um, this these coefficients didn't change too much. It's a marginal difference in magnitude, but in terms of the probability, you can see the up here. Sales, um, is in the 5% confidence level, meaning it's statistically significant. And sales squared is not in the 5% but isn't the 10% level of significance So that is something to know. Meanwhile, down here, neither of these are within the five or the 10% significance level, indicating that neither of these is as significant when the 40 billion is taken out. Right. It also indicates that, um, that because the sale squared goes from slightly statistically significant to not significant at all, that when you take out these larger, larger sales values, uh, this follows more of a linear model. Right? Um, so you can use a linear model for smaller sales values companies with less sales basis. So that's an interesting point to know. Okay, now we want to do the same thing, but with the L A D. Estimates. So same thing function led in 10 sales. Facil square plus profit margin on our data is gonna be Arctic em and let me just and it our previous one Here, we're changing our data set to without 40 beat. And this is the lady without money. So cool. You just taking our two models. I've plucked some reason. Both of those. Now the l A B function won't show the, um they won't show the asterisks to show you significance. But you can still look at it yourself and find so we can see from here. This is an interesting one, because our estimate here is 10.264 positive. But here it's 0.2 to 4. Negative. So the entire sign has changed just by taking out that outlier and the same thing happens for sale square. Right? So what we're finding is that this outlier causes a huge change in the media in effect off our estimates of our independent variables. Rather on the already intense. So based on our findings, right, would we say that Oh, a lesser l A d is more resilient outliers? We would say that OLS is more resilient because OLS on Lee had a marginal change in magnitude. The reds up OLS was a marginal change in magnitude, whereas led was a change in magnitude and for chains in ah son. So obviously the out liar had more effect there. And Ola's estimates are more results. All right, that's it for today. Thank you very much. Have a good one.

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