Key - Exam 1 - 2011

# Key Exam 1 2011 - Exam 1 Resource Economics 702 Econometrics I Complete all questions Point values for each question are at the left Point values

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Exam 1 Resource Economics 702 E c o n o m e t r i c s I Complete all questions. Point values for each question are at the left. Point values represent the amount of time for each question - you should finish within 100 minutes, plus or minus a small amount of sampling variability. I offer plenty of space for the first questions, but that does not mean you need to fill the space. (10) 1. Suppose we created a 90% confidence interval for the population parameter 1 (assume 2 is known). What do we mean when we say we are 90% confident? What are we confident about? What is this confidence based on? (Explain the theory behind confidence intervals.) We are confident, in this case 90% confident, that the interval we construct falls over or contains the true population parameter value 1 . This confidence is based on our knowledge of the sampling distribution for the OLS estimator, b 1 . We know that b 1 is distributed normally (given our CRM assumptions) and we know that in repeated sampling 90% of our estimates will fall within 1.645 σ b1 of the true population parameter value. If we know σ 2 , as is assumed here, then we can construct an interval that matches exactly the width of the symmetric interval around the true population parameter value that contains 90% of our estimates. And, if we were to construct a confidence interval ( 11 1.645 b b  ) around any of those estimates, that interval would contain or fall over the true population parameter. This is pictured below.

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MATH (10) 2. We talk a lot about “sampling variability of the estimators.” Explain clearly what sampling variability means for the OLS estimators and explain why it arises. What is the very basic cause of sampling variability? (This is a math-free zone, but you can include an equation for the estimator if it helps your explanation.) Sampling variability refers to the variation of the estimates we obtain from an estimator when we repeatedly sample from a population. Thus, sampling variability refers to the sampling distribution for an estimator, say b1. And, it refers more specifically to the shape of the sampling distribution and can be characterized by the standard error ( σ b1 ) if the sampling distribution is normally distributed. What causes this samping variability? When we repeatedly sample from the population, even when we hold X constant, we get different Y values and as a result different estimates using the same OLS estimator (problem set 2). The different values for Y depend upon the stochastic component, u. It helps to look at the OLS estimator: -1 -1 -1 b = (X'X) X'Y = (X'X) X'X β (X'X) X'u . Clearly, once we substitute for Y, we can see that the OLS estimator depends upon the disturbance. By assumption, all other parts of the expression for b are not random variables (X and β ). Thus, all randomness in the OLS estimator depends upon u and so the variance of b will depend upon the variance of the disturbance u . We’ve shown this in class as we found that 2 () Cov -1 b= (X 'X ) .
MATH (10) 3.What is the Gauss-Markov theorem - what does it tell us about the OLS estimators?

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## This note was uploaded on 12/08/2011 for the course ECON 702 taught by Professor Staff during the Spring '08 term at UMass (Amherst).

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Key Exam 1 2011 - Exam 1 Resource Economics 702 Econometrics I Complete all questions Point values for each question are at the left Point values

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