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ps1_sp12_sol

# ps1_sp12_sol - Department of Economics Columbia University...

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Department of Economics W3412 Columbia University Spring 2012 SOLUTIONS to Problem Set 1 Introduction to Econometrics Profs: Seyhan E Arkonac and Christopher C Conlon for all sections 1. You are hired by the governor to study whether a tax on liquor has decreased average liquor consumption in New York. You obtain from a random sample of n individuals in New York, each person’ s liquor consumption both for the year before and for the year after the introduction of the tax. From this data, you compute Y i ="change in liquor consumption" for individual i = 1 ,…. n. Y i is measured in ounces so if, for example, Y i = 10, then individual i increased his liquor consumption by 10 ounces. Let the parameters μ y and σ y 2 Y denote the population mean and variance of Y. a) You are interested in testing the hypothesis H 0 that there was no change in liquor consumption due to the tax. State this formally in terms of the population parameters. b) The alternative, H 1 , is that there was a decline in liquor consumption; state the alternative in terms of the population parameters. c) Suppose that your sample size is n = 900 and you obtain estimates μ y = - 32.8 and σ y 2 = 466.4. Report the t-statistic for testing H 0 against H 1 . Obtain the p-value for the test [use Table 1 in Stock and Watson, p. 749-750]. Do you reject at a 5% level? At 1% level? d) Would you say that the estimated fall in consumption is large in magnitude? Comment on the practical versus statistical significance of this estimate. e) In your analysis, what has been implicitly assumed about other determinants of liquor consumption over the two-year period in order to infer causality from the tax change to liquor consumption? Solution a) The formal statement of the null hypothesis is H 0 : μ y = 0. b) The formal statement of the alternative hypothesis is H 1 : μ y < 0. c) With s y = 466.4 and n = 900, we obtain SE ( ) = s y / √n = 466.4/√900 = 15.5467. Thus, the observed value of the t-statistic is t obs = (-32.8-0)/15.5467 = -2.1098. The p-value is then given the probability of observing something more extreme. That is, p = P (t < t obs ) = Φ (-2:1098) = 0.0174 Thus, we reject the null hypothesis at a 5% but do not reject at a 1% level since 0.01 < p < 0.05. d) The estimated fall is quite large at 33 ounces of liquor. So people on average drink about one bottle less per year due to the new tax. However, due to the relatively large standard error of , we cannot draw any strong conclusions from the data regarding the effect of the tax. e) We have implicitly assumed that all other determinants of liquor consumption have remained unchanged over the two-year period. If this is not the case, then we are comparing two different populations in a sense. Imagine for example, that people’s preferences have changed and they

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(regardless of the new tax) decided to drink less liquor and more wine. Then we would also have observed a decrease in the consumption of liquor, but that would not have been because of the tax.
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