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Unformatted text preview: University of California, Davis ARE 106 - Winter 2011 Homework 5 A quiz based on these homework problems will be held Thursday, February 10, 2011 at the start of class. Conceptual Questions 1. What is the unrestricted model? What is the restricted model? The unrestricted model is the full model, the restricted model has at least one fewer parameter in it than the unrestricted model does. 2. What is the difference between a single null hypothesis and a joint null hypothesis? A single null hypothesis involves one or multiple parameters but only one restriction. A joint null hypothesis involves two or more restrictions on one or more parameters. The rule of thumb: if you see only one equals sign: single. Two equals signs: joint. 3. Categorize the following as single or joint null hypotheses. How many restrictions are in each hypothesis? (a) H : 2 > One restriction, this is a single null hypothesis. (b) H : 4 = 3 = 9 = 5 Three restrictions, joint null hypothesis. (c) H : 3 > 4 One restriction, single null. (d) H : 3 = 0 and 4 = 9 Two restrictions, joint null. (e) H : 2 + 2 3 = 0 One restriction, single null. 4. When is an F-test equivalent to a t-test? When there is one restriction that is an equality. (Something like 2 = 3, will work with both an F or a t-test, so will 2- 4 = 3. However 2 = 4 = 9 will only work with an F-test, beta 3 0 will only work with a t-test.) 5. Why does the F-test look at the difference between SSE R and SSE U ? We want to understand how much more variation the unrestricted model explains com- pared to the restricted model. SSE, the sum of squared errors, is a measure of what the respective models do NOT explain. We compare whether or not the unexplained varia- tion in the restricted model is bigger than the unexplained variation in the unrestricted model. 1 University of California, Davis ARE 106 - Winter 2011 6. State the population regression function for the model where SSE = SST? How does knowing this fact change our use of the F-statistic? y i = 1 + e i . We use this to alter the formula of the F-statistic when we are testing the joint significance of all of the x variables in the model, replacing SSE R with SST , because the SSE R = SST and SST is the same for both the restricted and unrestricted models. (Remember from the formula for SST that so long as we use the same dependent variable, SST will be the same no matter which explanatory variables we use.) 7. What is the omitted variables problem? If you omit a variable that is a significant predictor of y and is correlated with one of your x variables, than your estimates for the coefficients on your x variables will be biased....
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