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Use the data in VOTE1 for this exercise.(i) Consider a model with an interaction between expenditures:What is the partial effect of expendB on vote\Lambda, holding prtystrA and expendA fixed? What is the partial effect of expendA on voteA? Is the expected sign for $\beta_{4}$ obvious?(ii) Estimate the equation in part (i) and report the results in the usual form. Is the interaction termstatistically significant?(iii) Find the average of expendA in the sample. Fix expendA at 300 (for $\$ 300,000 ) .$ What is theestimated effect of another $\$ 100,000$ spent by Candidate $B$ on voteA? Is this a large effect?(iv) Now fix expendB at 100. What is the estimated effect of $\Delta$ expend $=100$ on voteA? Does thismake sense?(v) Now, estimate a model that replaces the interaction with shareA, Candidate A's percentage shareof total campaign expenditures. Does it make sense to hold both expend $A$ and expendB fixed, $\quad$ while changing shareA?(vi) (Requires calculus) In the model from part (v), find the partial effect of expendB on voteA,holding prtystrA and expend/ fixed. Evaluate this at expend $=300$ and expendB $=0$ and comment on the results.

(i) $\beta_3 + \beta_4*expendA$, $\beta_3 < 0$, $\beta_4$ ambiguous (ii) See video. Not significant. (iii) $\Delta voteA = -3.37$ (iv) $\Delta voteA = 3.36$ (v) See video. Partial effect is -1.64

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Chapter 6

Multiple Regression Analysis: Further Issues

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part one you can easily show from the model this equation. This is similar to the partial derivative of Vote A With respect to expand me. This ratio equals beta three plus beta. Four times expand A regarding the sign of the coefficient. Beta three would be less than zero with a three is negative because an increase in the spending by B lowers the share of the vote received by a Let me write that down. Yeah, for paid a four, the sign is AM biggest. We're not sure if beta four is positive or negative because we don't know when the spending of b No. Sorry. The spending of a increase would that lead to a greater or smaller effect of more spending by B mhm In part two of the question. Okay, Yeah, you will estimate the model and this is the regression result. Look at the interaction term. You can see that the standard error of the estimated coefficient 0.50 as 72 is greater in absolute value to the absolute value of the estimated coefficient. So this number is larger than this number, and so you can conclude that the interaction term is not statistically significant. So the T's got, which is the ratio of beta had over the standard error of data. This is a more formal way to prove which coefficient is significant. The T start of this interaction term is less than one. So this term is not significant. There is more information to the estimated equation. We have 173 observations. The our square is 1730.571 and they're adjusted. Our square is 0.561 Let's move to part three. This is what we get from part one. Now we can quantify the change of vote A given what we know from their regression result. So from the regression results, we have the value of Beta Three is Let me write this down. She This is minus 0.3 1/7 and Beta four is minus point. You wouldn't have five zeroes and 66. We want Thio measure the change in vote A when the spending of a is 300 mhm and when the change in the spending off B is 100 the unit is $1000. We can easily find out that the change in vote A is roughly minus 3.37 This is a quite large effect. We will continue with Part four. Again. We win. Calculate the change in thought a Now with different conditions. Again, we've been plug in the values the estimated values of beta two and beta four and the change in expanding spending off B. We wouldn't have the value of the bracket as 0.3 76 The change in the spending of a ISS 100 first. So what we got for the change in Vote A is three point 76 This result is, well, it's making sense. It is reasonable. Let's continue with Part five again. You will have to estimate a regression equation, and you will have a new variable here. Share a based on the are square and it just square. You can see that the model has an improvement in terms of goodness of fit. Sure, A is highly significant. It's coefficient is fairly large in magnitude, and the standard A roads are very small, as you can see here. Now we need to re calculate the partial derivative of Vote A. With respect to the spending of B. We will use this equation. Thio get the formula and get their values of the change in vote A. So, based on the previous equation, we have this partial derivative as beta three hat plus beta four act times the partial derivative of share A. We respect you the spending of speed and remember that share A is defined as the ratio of spending off A to the total spending off A and B, and it is in percent. So we have 100 times the ratio. We we need to resolve the partial derivative in the bracket based on this, um, definition of share A Yeah, so you wouldn't get Yeah, the partial derivative of share A with respect you the spending of b equals minus 100 times the spending of a invited by the total spending of A and B square. Okay, we will evaluate this partial derivative at spending off a equal 300 see and spending of B equal zero. The partial derivative is minus 100 times 300 over 300 square, and we will get minus one hurt. Now we know the value of the partial derivative of share, a respect you expand be. And we will plug this value of minus one third into this equation so we can find the change in what? A actually the partial derivative of Vote A. With respect to you, the spending of B we know, bigotry head and beta forehead from this estimated equation from here. And we know the value of the ratio inside the bracket as minus one third. Let me write that down. I have some space here, so I will finish this, uh, part five here. Okay. You know, these two values and also these values this is minus one third this one see below. Okay. After we gather all the numbers, we find the left hand side to be minus point 164 It means that what a well reduced by 0.164% points. Given the 1st $1000 of spending by Candidate B, where the spending of a is held fixed at 300 this is a quite large effect and noticed that the effect well reduced as the spending of B girls, it is so because because we have a minus sign here

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