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Unformatted text preview: ECON 140, Fall 2008 - 10/23 Alex Rothenberg Practice Problems: Omitted Variables, Errors in Variables, Randomized Experiments Problem 1 Suppose you were interested in understanding how corruption affects the provision of drivers licenses in India. There are two competing hypotheses that wed like to distinguish empirically: Hypothesis I : Paying bribes might help to eliminate red tape and grease the wheels of a slow, inefficient bureaucracy. Bribes represent a transfer between good drivers who are willing to pay for a license and bureaucrats. There is no social loss. Hypothesis II : Corruption has damaging effects to society. Those who pay bribes to obtain a license are typically less likely to be good drivers. As a result, corruption makes the streets unsafe for everyone who drives. How might we test these hypotheses? Suppose we collected data on a random sample of people who tried to obtain drivers licenses in India. Let D i be an indicator equal to 1 if individual i got a license, 0 otherwise. Let B i denote the amount of bribes paid by individual i in trying to obtain the license. Let A i denote the driving ability of individual i , measured by their score on an objective driving exam. 1. Suppose we were trying to decide between estimating two possible linear probability models: D i = + 1 B i + i (1) D i = + 1 B i + 2 A i + i (2) Which of these models would be most interesting to estimate for the purpose of determining which of our hypotheses is more plausible? ANSWER: It would clearly be more interesting to estimate model (2). In model (1), we could be observing a positive effect of bribes on the probability of obtaining a license simply because people who pay more bribes are more willing to pay and are more able, better drivers. Or, it could be that lower ability drivers are more likely to pay bribes. Theres no way of distinguishing without controlling for ability. In model (2), 1 is the effect of bribes on the probability of getting a license, holding constant ability. If this is positive, this is pretty good (but not perfect, as we shall see) evidence in favor of hypothesis II. If 1 = 0 and 2 > , then this is evidence consistent with hypothesis I. 1 ECON 140, Fall 2008 - 10/23 Alex Rothenberg 2. Suppose the true model is (2) but we estimate (1) with ordinary least squares. Derive an expression for the omitted variable bias. ANSWER: Assume that the regressors are non-random, so that they can taken outside of the expectations operators without concern. From the usual least squares formula, when we estimate (1) using ordinary least squares, we have: 1 = N i =1 ( D i- D )( B i- B ) N i =1 ( B i- B ) 2 If the true model is (2), then a bit of algebra gives us the following: D i- D = + 1 B i + 2 A i + i- ( + 1 B + 2 A + ) = 1 ( B i- B ) + 2 ( A i- A ) + ( i- ) So, applying the expectation operator to our expression for 1 , we have:...
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