Econometrics HW 8 solutions
Econometrics HW 8 solutions Econ 4400
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This 3 page Class Notes was uploaded by Noah Scovill on Sunday March 20, 2016. The Class Notes belongs to Econ 4400 at Ohio State University taught by Anthony Bradfield in Fall 2015. Since its upload, it has received 26 views. For similar materials see Elementary Econometrics in Economcs at Ohio State University.
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Date Created: 03/20/16
Homework 8 10/21/2013 Name: _____________________________ Econ 4400-101 Due: 10/28/2013 Show all necessary work. Instructor: Anthony J. Bradfield All work should be done by hand, unless stated otherwise. 1.  The following is a demand equation for beef. lnQ = β + β lnPi+ ε0 wher1 lniis tie natural log, Q is the quantity of beef demanded, P is the price of beef, and β represents the 1rice elasticity of demand when written in this form. Previous results have shown that the demand for beef is unit elastic. a. Write out the set of hypotheses that relate to demand for beef being unit elastic. H 0 β 1 -1 H A β 1 -1 b. Suppose that you estimate this equation with the result that ???? = -0.3, which ▯s significantly different than the point in your null hypothesis. What is the likely reason this occurred? How would you go about correcting this problem (be specific)? There is likely an omitted variable. Since ???? = -0.3 > -1 = ???? , we should try to include a variable with a ▯ ▯ ▯ positive bias. If we cannot find one, we should present the result as 1 is likely less than -0.3 on average (-0.3 is an upper bound for an unbiased estimate of β ). 1 c. Suppose that you now estimate the equation lnQ = β + β lnP + β lnPC + β lnPP + β lnYD + ε, where i 0 1 i 2 i 3 i 4 i i PC is the price of chicken, PP is the price of potatoes, and YD is disposable income. The estimate of ????▯now changes such that it is not significantly different from your null hypothesized value. Which variable(s) could have caused this change? For PC, chicken and beef are substitutes, so as PC increases, you consume more beef, so β should2be positive, and P and PC are negatively correlated. Since Bias = β *α 2 t1is means that the bias on β is 1 positive, so including lnPC could cause this change. Similarly, potatoes and beef are compliments, so if PP increases, then you consume less beef (β is 3 negative), and P and PP are positively correlated; this leads to positive bias as well. Finally, if disposable income increases, then you should consume more beef (assuming it is a normal good), so β 4hould be positive, and disposable income and prices are positively correlated, so this also leads to positive bias in 1β . Therefore, all three variables could be responsible for the change in β . 1 2.  Suppose you are about to estimate Cars = β + β Price + β PriceGas + β PriceTires + β Prec + ε. i 0 1 i 2 i 3 i 4 i i Classify each independent variable as relevant or irrelevant. Also, list one major variable that is omitted. The data comes from one observation from every state in a given year. Cars is the number of cars sold in that state and year, Price is the price of the Car, PriceGas is the average gas price, PriceTires is the price of a set of tires, and Prec is the amount of precipitation in that year. Relevant: Price, PriceGas Irrelevant: PriceTires, Prec Omitted: any of disposable income, GDP growth, the price of certain cars (SUVs, hybrids, electrics, compressed natural gas), the price of diesel fuel, etc. 3.  Suppose that (1) is the true equation, but you write out (2) and estimate it. Write out the relevant exogeneity condition(s) for the true equation, and show whether ???? estimated from (2) will be ▯▯ unbiased by checking if the exogeneity condition for (2) is violated. (1) Y = i + β0X + β1 1i ε 2 2i i (2) Y = i + β0X + β1 1i ε* 3 3i i E[X 1i i 0 and E[X ε] = 02i i First, β 2 2i ε = i X + 3 3i so εi = β X +i ε - β2 2i i 3 3i E[X 1i i= E[X (β X1i β 2 2i ε)]3 3i E[X i ] - β 2[X X1i 2iE[X ε]3= β 1i 3i 0*X X + 1i iβ X X2 1i 2i 1i 3i 2 1i 2i The first 0 is from X being 3irelevant, so β = 0, and the s3cond is an exogeneity condition from (1). Since the result is not zero, which occurs unless β happens to be zero or X is independent of X , there 2 1i 2i is bias in β . 1 4.  Determine the sign of the bias introduced by leaving a variable out of an equation in each of the following cases: (1)In an annual equation for corn yields per acre (in year t), the impact on the coefficient of rainfall in year t of omitting average temperature that year. (Hint: Drought and cold weather both hurt corn yields.) β 2s positive, but α is 1ikely close to 0, so there may be a small amount of bias, but we wouldn’t know what direction it would be in in general. (If it rains more when it is cold, then α is negati1e, so there would be negative bias). (2)In an equation for daily attendance at Log Angeles Lakers’ home basketball games, the impact on the coefficient of the winning percentage of the opponent (as of the game in question) of omitting a dummy variable that equals 1 if the opponent’s team includes a superstar. β 2s positive, and α is p1sitive, so there is positive bias. (3)In an equation for annual consumption of apples in the United States, the impact on the coefficient of the price of bananas of omitting the price of oranges. β2is positive, and α1is positive, so there is positive bias. This is the same as in #1 since oranges and bananas are substitutes. (4)In an equation for student grades on the first midterm in this class, the impact on the coefficient of total hours studied (for the test) of omitting hours slept the night before the test. β is positive, and α is negative since you must make a tradeoff with your time, so there is negative bias. 2 1 (5)In an equation for U.S. sales of a college textbook, the impact on the coefficient on price of the textbook of omitting the price of an international edition sold in other countries. β2is positive, and 1 is positive, so there is positive bias. These are substitutes, so the same idea holds. More specifically, if the price of an international edition goes up, the arbitrage opportunity shrinks, so less international editions will be sold in the U.S., and if there is less competition from international editions, then the quantity and price of U.S. textbooks will rise.
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