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Unformatted text preview: Practice Questions for Midterm 1 Introduction to Econometrics - Fall 2006 1. The random variables X and Y are related by the following relationship: Y = a + bX: Show that corr ( X;Y ) = 1 if b > and corr ( X;Y ) = & 1 if b < . 2. The distribution of the temperature in Madison is characterized by parameter & . The value of & is unknown. However, we know it & = 1 . After several measurements of the weather, an estimator ^ & is obtained. It is known that ^ & can only take 4 possible values. Given that the true value of the parameter is & = 1 , the likelihood of each value of ^ & is given in the following table: ^ & 1 2 3 Pr [^ & ] : 20 : 50 : 25 : 05 For instance, Pr [^ & = 2] = 0 : 25 . (a) Write the cumulative distribution of ^ & and show it on a graph. (b) Suppose that you obtain an estimate ^ & = 3 Given the distribution for ^ & , would you reject the null hypothesis that & = 1 at a 90% con&dence level? Explain. (c) Suppose that you obtain an estimate ^ & = 2 Given the distribution for ^ & , would you reject the null hypothesis that & = 1 at a 90% con&dence level? Explain. (d) Compute the mean and the variance of ^ & . Is ^ & a biased estimator of & ? 3. You want to study the relationship between the level of education of a person and his/her parents income. (a) You decide to consider the following linear regression model: edu i = + 1 inc i + u i where E ( u i j inc i ) = 0 : In this model, edu i is the number of years individual i went to school and inc i is the average annual income (measured in thousand of dollars) of his/her parents in the &rst 25 years of his/her life. You use the simple random sampling scheme to obtain a sample of 5000 observation from the population and from this sample you obtain the following OLS regression line. d edu i = b + b 1 inc i Are the three least squares assumption holding? What does this imply in terms of unbiasedness and consistency of the OLS estimators b and b 1 ? What is the distribution of b and b 1 ? (b) Now you decide to consider an alternative linear regression model: edu i = alt 1 inc i + u alt i Using the same sample as in a., you obtain the following OLS regression line: d edu i = b alt 1 inc i Are the three least squares assumption holding? What does this imply in terms of unbiasednessAre the three least squares assumption holding?...
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This note was uploaded on 07/28/2010 for the course ECON 410 taught by Professor Staff during the Fall '08 term at Wisconsin.
- Fall '08