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Use the data in WAGE2 to estimate a simple regression explaining monthly salary (wage) in terms ofIQ score $(I Q) .$$\begin{array}{l}{\text { (i) Find the average salary and average IQ in the sample. What is the sample standard deviation of }} \\ {\text { IQ? (1Q scores are standardized so that the average in the population is } 100 \text { with a standard de- }} \\ {\text { viation equal to } 15 . \text { ) }}\end{array}$$\begin{array}{l}{\text { (ii) Estimate a simple regression model where a one-point increase in } 1 Q \text { changes wage by a con- }} \\ {\text { stant dollar amount. Use this model to find the predicted increase in wage for an increase in }}\end{array}$IQ of 15 points. Does $I Q$ explain most of the variation in wage?$\begin{array}{l}{\text { (iii) Now, estimate a model where each one-point increase in } 1 Q \text { has the same percentage effect on }} \\ {\text { wage. If } 1 Q \text { increases by } 15 \text { points, what is the approximate percentage increase in predicted wage? }}\end{array}$

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Chapter 2

The Simple Regression Model

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Okay, we are in the fourth computer exercise and Chapter two, and we need to use the data in the data set called Wage to to estimate simple regression explaining monthly salary in terms of I Q score. So right off the bat, what we know is that are dependent or explain. Variable will be monthly salary called Wage from the Data said no explanatory or independent Variable will be the you score. Let's go ahead and describe our data set. First of all, we have 935 observations, 1700 lying variables, and the two variables we're gonna use with this question is wage monthly earnings. And I like you, which is a A Q score. Again part, eh? Find the ever Sally and average I Q. In the sample. What is this sample standard deviation of I Q. And we also given the information that I Q scores were standardised so that the average in the population is 100 with standard deviation equal to 15. Well, now remember that whenever we've been getting were given information about the population, this means information about the quote unquote true or riel honored lying distribution of the population right. So if we can if we could have the means to sample to get information about everyone in the U. S, then this would form the population distribution. But now, of course, here we have just a hopefully representative sample, hopefully random sample that we need to make inference space from that sample about the population. Okay, let's start with summarizing our variables Wage. I Sorry, wage and I to all right, first of what? We have no missing observation. That's good. 9 35 both. And we can see that the average the sample average Sally is 957.94 $95. Remember, this is not in today's dollars is probably like something like 1980 and the average I Q scores 100 and one point 28th. The sample sand deviation is 15 points verified. So those two numbers here a very fine example of what we call in statistics and probability theory. The law of large numbers. This law states that it would have a random sample with finally finite moments, but you need to think about it right now when we have a random sample and when this the number of observations in a random sample grows indefinitely, tends to infinity, then the sample mean of some of our quantities were interesting convergence and probability to the rial underlying, uh, expectation. Right? So in this case, assemble mean well conversion probability to the real expectation, which is 100. This is lose. You consume pretty close, and the same happens for the standard deviation. Again it gets it's very close to 15. And if we had even more data, it would be even closer to 50. Okay, Part B. Estimate the symbol regression model with a where at one point increase in i Q changes wage by a constant dollar amount. Ah, uses model to find the predicted increasing wage for an increase of I Q. 15 points. All right, so we've dealt with questions like that before. Whenever we see a constant dollar amount or constant whatever constant extremes, then we need to think of it level, level, model, level, level model. So both the dependent independent variables will not be subject to any transformation will be levels. So here the regression want estimate is wage equals to some constantly see beat a zero plays plus beata one times like you, a simple linear regression. We run it and we get Craig. Number of observation. The F join F test. It is very highly significant. You can see there's a very, very large a critical value, and the associative be Valley zero. The R squared is a bit less than 10% UH, 9.55% bad, and here are estimates we get an estimate where the concert is almost 117. It is not statistically significant, though. Can see very low level for the T test and the quantity of interests. The main one is a slope coefficient. I Q is equal to 8.3, and it's very, very statistically significant at any level of significance. So right here I've written down the estimated equation of the level of a model. Predicted wage equals 116.998 30 i Q. And so if we plug in 15 into like you hear an increase of Nike, you're 15. Increases predicted that monthly salary by 8.30 times 15 $124.50 so that this is our answer. And now there's like you explain most of the variation wage Well, no causes explains around 10% of it less than 10%. Is this most of the variation? No, of course, most would be more than 50%. However, I don't want you to fall into the trap of saying OK, so I Q is not important. Of course it's important. Just one variable explains almost 10% of the variation and something that is a function of maybe hundreds of variables. So it is very important. Should be including the analysis. However, it does not explain most of the variation. Okay, no one part. See, we need to estimate the model where each one point increase in like you has the same percentage effect on the wage. So what we did before. But now we want a constant percentage change and what we know about that we know that when you estimated lug level model so the lug of ways was indeed we also have here The natural log of wage equals to some constant beauty. Zero plus b, the one times Thank you. Simple lug level model. We run it again, we stick or IQ number of observation and even more statistically significant F test, which means that our motto is much better than a model with just intercept term with no slope coefficient. Us quite is slightly better than before. He's even closer to 10% and ah, Cynthia root me square is also lower than before. Actually, I mean, we can compare the two, but it has to be lower. Since the r squared, it's higher. And here are estimates for the coefficients, both of them extremely statistically significant at least under the assumption we're imposing. The concentration is equal to 5.8 e a and the coefficient and the slope coefficient term is equal to 0.0 88. I've written down here the estimated locked level model. As we just said Le Goff s means luck of wage equal to 5 49 0.88 I too. So if we plug in 15 right here, then the, um, the difference in the lug wait will be 0100 88 times 15 which is 0.13 to And since we know that the luck differences are the approximate percentage differences for small changes, it's almost identical for larger changes just being approximation. Then we know that the percentage increase it will be approximately 13.2%

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