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For the population of firms in the chemical industry, let $r d$ denote annual expenditures on research anddevelopment, and let sales denote annual sales (both are in millions of dollars).$\begin{array}{l}{\text { (i) Write down a model (not an estimated equation) that implies a constant elasticity between }} \\ {\text { rd and sales. Which parameter is the elasticity? }}\end{array}$$\begin{array}{l}{\text { (ii) Now, estimate the model using the data in RDCHEM. Write out the estimated equation in the }} \\ {\text { usual form. What is the estimated elasticity of } r d \text { with respect to sales? Explain in words what }} \\ {\text { this elasticity means. }}\end{array}$

$b_{1}=1.08$

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Chapter 2

The Simple Regression Model

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Set S.

July 28, 2021

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ain't moving on with the fifth computer exercise in Chapter two for the population of firms in the chemical industry. Let r D d note annual expenditures on research and development and let sales denote annual sales. And both variables are in millions of dollars. In part. A. When you write down the model, that implies a constant LSD city between RD in sales, which families the elasticity. Okay, so go back to basic micro one on one. Remember how the air hostesses defined is defined as the ratio ratio off 2% of changes, and its interpretation is if X If, um, X variable increases by 1% in the wild berry but will increase by whatever percent and which I find is whatever. And now remember what we talked about so far that in the context of the linear regression model, what information the with percent that changed? Yes, that's right is a natural logarithms because that luck difference of a variable is the approximate percentage change. So in the previous question, we've done in low level model, lug level model. Now it's time to do a lot of luck. Model, also called the Constant Elasticity model and we take the transformation of both are dependent in this case are the variable and the independent variable which is the sale. So that lug of rd clothes to intercept term plus the slope coefficient times a lot of sales plus an error or disturbance term you which, uh, this is not necessary to know right now. But when we reform testing another, um, kinds of things in the regression analyses, we won't need to assume that this is my i d and normally distributed with the main of zero and the constant said deviation. But you don't have to. This doesn't change anything. You know a question. No analysis here. So in this case, the coefficient B. The one will be the L S Stacy of RD. With respect to sales, we came All right, Now in part two, let's go back to ST. I'm gonna use it in part B. When you estimate this model using the data set already, Kem, and right out the estimated equation the usual form and ah, and explain what is the estimated listed Sephardi with respect to sales. Okay, let's first describe our data said to know that we have correct one as we can see, 32 observations, 800 lying variables. And here we are already in sales already spending infirm sales in millions of dollars. That's great. And not wanting to estimate are I love, like model. We already have the luck transformation here. We're gonna do, uh, r d our sailed, uh, love sales here. She's younger. All right. The correct number of observations before on extremely Ah, a statistically significant join f test. Uh, I don't remember if I've seen such a high, um, critical value recently, ofcourse the estimated value zero and extremely impressive R squared. Now we're talking about an exploiter of nine and 1%. So variation lug sales explains 91% of the variation log R and D. That's more than impressive. And unsurprisingly, are coefficients are very, very statistically significant at any level. Significance and the value for I understand his minus 4.10 meaning that if they're no sales right there, no lug sales, the logo for Indy will be negative. I guess we can. Yeah, right. You can see that. How that the dependencies capture doesn't make sense for they are the log ready to be negative or in this case plan. Excuse me. Does because it long is, uh it goes down to minus and Finn to get, so it's very, very close to zero. That's what it means here. Okay, Because it isn't love. Transformation is very close to zero. Remember that the natural law gave him the function goes a synthetically deserve it. Dissect the X axis at X equals one and down Thio it goes s synthetically the minus infinity One ex ghost zero So minus 4.10 is very close to zero. And now the estimate of interest the be the one coefficient is 1.75 Let's say 1.8 Right, So this is a value off our analysis t have written down the estimate equation here. A before the question is asking eyes to explain worth what the estimated elasticity of one foreign 076 means. Well, as we explained before, this means that 1% increase in sales in the independent variable will lead to an approximate increase of 1.8% in r and D. Okay, again, um, 1% increase in sales would lead to be the one increased be the 1% increase in Ah, aren't the an approximate increase, So Ah, here. What does this mean? That if sales increased by 1% loan guarantee will increase by more than 1%. Right. So in this sort of way to go back to basic Micro, we can say that RND is slightly elastic with respect to sales.

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