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SUMMER TERM 2019 ECON0019: QUANTITATIVE ECONOMICS AND ECONOMETRICS TIME ALLOWANCE: 3 hours Answer ALL TWO questions from Part A and answer ONE question from Part B. Questions in Part A carry 60 per cent of the total mark and questions in Part B carry 40 per cent of the total. Tables for the normal and F-distribution are at the end of the examination paper. In cases where a student answers more questions than requested by the examination rubric, the policy of the Economics Department is that the student’s first set of answers up to the required number will be the ones that count (not the best answers). All remaining answers will be ignored. PART A Answer all questions from this section. A.1 You wish to quantify the effect of cannabis consumption on student performance. You carry out a survey asking a random sample of your fellow students about their average mark after two years of studies and number of times they have consumed cannabis in the last 30 days. Let AMi and SMi be student i’s self-reported average mark and number of times used, i = 1, …, n, where n is the number of students in the sample. (a) Suppose that AM is observed with measurement error while SM is observed without. That is, AMi = AM i +vi, where AM i is the actual average mark and vi is the measurement error. The measurement error is assumed to be fully independent of (SMi, ui) with E [vi] = 0, i = 1, …, n. Suppose that the actual average mark satisfies AM i = β0 + β1SMi + ui, (1) and that SLR.1-SLR.5 are satisfied in the above model. Derive the (conditional on SM1, …, SMn) mean and variance of the OLS estimator of β1 obtained by regressing AM on SM . ANSWER: With u = u v, AMi = β0 + β1SMi + u i. (2) SLR.1-SLR.5 combined with E [vi] = 0 and (SMi, ui) ⊥ vi yield E [u i|SMi] = 0 and Var (u i|SMi) = σ2u + σ2v . Thus, (2) also satisfies SLR.1-SLR.5 and we obtain E [ β 1|SM1, …, SMn ] = β1, Var(β 1|SM1, …, SMn) = σ 2 nσ 2SM , where σ2 =Var(u i|xi). ECON0019 1 TURN OVER (b) You use the following estimator of the variance of the OLS estimator β 1 as described in (a), V ar(β 1) = σ 2 nσ 2SM , σ 2 = 1 n 2 n∑ i=1 u 2i , σ 2 SM = 1 n n∑ i=1 ( SMi SM )2 , where u i = AMi β 0 β 1SMi, i = 1, …, n. Is this a consistent estimator of the variance of β 1 Explain. ANSWER: Since (2) satisfies SLR.1-SLR.5, we know from the lectures/Wooldridge that the above variance estimator is consistent. (c) Consider the reverse situation: You observe the actual mark average AM but now instead of SM you observe S M i = SMi + vi where vi still satisfies the assumptions stated in (a), i = 1, …, n. Derive the probability limit of the OLS estimator of β1 obtained by regressing AM on S M i. ANSWER: With u = u β1v, AM i = β0 + β1 ( S M i vi ) + ui = β0 + β1S M i + u i, (3) where SLR.1-SLR.5 combined with E [vi] = 0 and (SMi, ui) ⊥ vi yield E [ u iS M i ] = β1σ2v Thus, by the LLN, β 1 = β1 + 1 n ∑n i=1 ( S M i S M ) u i σ 2 S M →p β1 ( 1 σ 2 v σ2 S M ) . (d) You obtain a consistent estimator σ 2v of σ 2 v =Var(v). Use σ 2 v to develop a consistent esti- mator of β1. ANSWER: First compute the OLS estimator in (c), β 1, and then β 1 = β 1 ( 1 σ 2 v σ 2 S M ) 1 . Combining the answer to (c) with σ 2v →p σ2v, we obtain β 1 →p β1. (e) Still considering the scenario in (c), discuss how realistic the following two assumptions are, E [vi] = 0 and vi fully independent of (SMi, ui), when the measurement error is due to incorrect reporting of cannabis consumption. ECON0019 2 CONTINUED ANSWER: First, students who smoke may well likely understate their true consumption and so E [v] < 0. Second, even if they try to tell the truth, then most likely students who don’t smoke (SM = 0) are very likely to report SM = 0; but students who do smoke (SM > 0) are more likely to miscount number of times they smoked. This implies that v and SM are likely dependent. (f) Suppose that you observe SM and AM without measurement error. However, some of the students that you asked to participate in the survey refused. Is this a concern regarding the validity of SLR.1-SLR.5 ANSWER: SLR.4 may be violated in the sample. If the selection (reason for not partici- pating) is mean-independent of u so that E[u|s,M ] = 0, where s is the selection dummy variable, SLR.4 will still hold for the selected sample. However, if the sample selection non-random (dependent on u) SLR.4 will fail to hold. A.2 You are interested in estimating the effect of per-student spending on math performance. For that purpose, you use a data set on 408 schools in the UK. For each school, the data set contains math, the percentage of students receving a passing mark in a standardized math test, together with spend, per-student spending, and enroll, number of students enrolled. (a) You obtain the following regression results, m ath = 69.24 + 11.13 log(spend) + 0.22 log (enroll) , R2 = .0297. (26.72) (3.30) (.615) If spend increases by 10% what is the (approximate) estimated percentage change in math ANSWER: We have m ath ≈ 11.13 100 (% spend) and so m ath ≈ 11.13 100 × 10 = 1.113. That is, we expect 1.113% more students to pass the math test if we increase spending by 10%. (b) Test the hypothesis that math does not change with spend against the alternative that it does increase with spend. Perform the test at a 5% and 1% level. Conclude. ANSWER: With β1 denoting the coefficient for log (spend), we wish to test H0 : β1 = 0 vs HA : β1 > 0. The t-stat is tobs = 11.13/3.30 = 3.37 which we then compare with critical val- ues 1.645 (5%) and 2.326 (1%). We reject the null at both levels and so conclude that there is strong statistical evidence of that school spending affects students’ math performance. ECON0019 3 TURN OVER (c) You conjecture that family background has an effect on student performance and would like to include poverty, the percentage of students in a given school that live in poverty, in your regression. However, this variable is not in the data set and you instead decide to include meal, the percentage of students eligible for free school meals, as an additional regressor. Is this a sensible strategy Explain. ANSWER: The usual proxy variable argument should be employed: First, eligibility for the free school meals is very tightly linked to being economically disadvantaged. Therefore, the percentage of students eligible for free school meals is very similar to the percentage of students living in poverty. Thus, we expect it to be a good proxy. Formally, we need that δ2 = δ3 = 0 to hold in the following regression for meal to be a valid proxy for poverty, poverty = δ0 + δ1meal + δ2 log(spend) + δ3 log(enroll) + v. Even after controlling for meal, school spending may predict level of poverty in which case using meal as a proxy will lead to biased results. If δ2 is not too big, the bias will be negiglible (d) Including meal you obtain the following results, m ath = 23.14 + 7.75 log(spend) 1.26 log (enroll) .324meal, R2 = .1893. (24.99) (3.04) (.580) (.036) Explain why the effect of spending on math is lower in this new regression compared to the one in (a). ANSWER: First note that meal is found to be relevant. We can then use our usual reason- ing on omitting important variables from a regression equation. The variables log(spend)and meal are negatively correlated: school districts with poorer children spend, on average, less on schools. Further, the coefficient on meal is negative ( .324). From Wooldridge we then find that omitting meal from the regression produces an upward biased estimator of β1 [ig- noring the presence of log (enroll) in the model]. So when we control for the poverty rate, the effect of spending falls. (e) Interpret the coefficients on log (enroll) and meal. ANSWER: Once we control for meal, the coefficient on log (enroll) becomes negative with t-stat of –2.17, which is significant at the 5% level against a two-sided alternative. The coefficient implies that m ath ≈ 1.26 100 (% enroll) Therefore, a 10% increase in enrollment leads to a drop in math of .126 percentage points, a small effect. Both math and meal are percentages. Therefore, a ten percentage point increase in meal leads to about a 3.23 percentage point fall in math, a sizeable effect. ECON0019 4 CONTINUED (f) What do you make of the increase in R2 from the regression in (a) to the regression in (d) ANSWER: The regression in (a) explains less than 3% of the variation in math while the one in (d) explains almost 19%. Most of the (explained) variation in math must therefore be due to meal. This seems to indicate that family income (or related factors, such as living in poverty) are much more important in explaining student performance than are spending per student or other school characteristics. ECON0019 5 TURN OVER PART B Answer ONE question from this section. B.1 Schumpeterian growth theory implies that the threat of technologically advanced entry spurs innovation incentives in sectors close to the technology frontier, where successful innovation allows incumbents to survive the threat, but discourages innovation in laggard sectors, where the threat reduces incumbents’ expected rents from innovating. In “The Effects of Entry on Incumbent Innovation and Productivity,” (The Review of Economics and Statistics, Vol.91, No.1, 2009), Philippe Aghion, Richard Blundell, Rachel Griffith, Peter Howitt and Susanne Prantl study the effects of firm entry on labour productivity — more specifically, the real output per employee in the firm — and innovation — more specifically, the count of patents issued to the firm — taking into account how far the industry of interest is from the technological frontier. The authors use data from the United Kingdom and measure distance to the technological frontier by comparing the labour productivity in the industry in the United Kingdom to labour productivity in the same industry in the United States. (a) To study the relationship between entry, distance to the frontier and patent counts, the authors use a Poisson model. Suppose you decide to estimate a similar (i.e., Poisson model) where the expected number of patents is given by: E(Pj |Dj , EFj ) = exp(β0 + β1EFj + β2Dj + β3Dj × EFj ), where Pj is the count of patents for firm j in a given year, E F j measures the entry rate of foreign firms in firm j’s industry in the previous year and Dj measures the distance from the technological frontier. Both Dj and E F j are continuous. Write down the expression for the (log-)likelihood used to compute the Maximum Likelihood Estimator. In their estimates (which uses a somewhat more sophisticated version of the model above), the authors estimate β2 to be between 0.582 and 0.852 (depending on the specification used). Does this imply that the partial effect at the average (PEA) for distance to the technological frontier is positive Please elaborate on your answer. Hint: If Y follows a Poisson distribution with parameter λ > 0, its probability mass function is P(Y = k) = λk exp( λ) k! for k = 0, 1, 2, . . . . ANSWER: The log-likelihood function is: n∑ j=1 {pj(β0 + β1eFj + β2dj + β3dj × eFj ) exp(β0 + β1eFj + β2dj + β3dj × eFj )} ECON0019 6 CONTINUED (omitting terms that do not depend on teh coefficients of interest). The PEA with respect to Dj is: (β 2 + β 3EFj )× exp(β 0 + β 1EFj + β 2Dj + β 3Dj × EFj ). Its sign is thus that of (β 2 + β 3EFj ) which may differ from the sign of β 2. (b) The authors note that “entry can be endogenous to innovation and productivity growth” and consider a set of instrumental variables related to policy reforms related to entry: “reforms at the European level and reforms at the U.K. level that changed the entry costs and effected entry differentially across industries and time.” The European reforms were undertaken as part of the Single Market Programme and deemed to reduce medium or high entry barriers. The U.K. reforms include, for instance, privatization cases which resulted in opening up markets to firm entry. Consider then the following simple linear regression model for labour productivity growth, LPj , as it relates to entry, E F j : LPj = α0 + α1E F j + Uj , (4) where Uj is an unobserved error. Suppose you have at your disposal one instrumental vari- able Zj that consolidates information about the implementation of the reforms alluded to above. Describe how you would implement the TSLS estimator in this context. How would you argue for the validity of this instrument ANSWER: TSLS: ( 1 ) Regress EFj on Zj. ( 2 ) Regress LPj on E F j . The in- strumental variable is valid if cov(zj , uj) = 0. This means that any unobserved variables that affect the number of patents do not vary systematically with this variable. This will be the case if Thatcher era privatisations and the EU Single Market Programme only influence ilabour productivity through entry and do not affect it directly. (c) How can you use the estimates from (4) above to test whether EFj is endogenous ANSWER: Describe Hausman regression-based test for endogeneity. (d) Let E Fj = p i0 + p i1Zj , where p i0 and p i1 are OLS estimates from a regression of E F j on a constant and Zj . If one uses E F j as an instrumental variable instead of Zj how would the estimates compare with those obtained in the previous item Elaborate. Hint: Since p i0 and p i1 are obtained by OLS, E F j = E F j + Vj = p i0 + p i1Zj + Vj and ECON0019 7 TURN OVER ∑n j=1(E F j E Fj )Vj = 0. Furthermore, EFj = E Fj . ANSWER: The first stage using E Fj is obtained by the OLS estimates of a regression of EFj on E F j . The slope coefficient of this regression is given by∑n j=1(E F j E Fj )(EFj EFj )∑n j=1(E F j E Fj )2 = 1 + ∑n j=1(E F j E Fj )Vj∑n j=1(E F j E Fj )2 = 1 and the intercept is given by EFj E Fj = 0. Consequently the forecast for EFj used in the second stage in the item above is exactly the same. (e) Imagine you have time series data for a single firm and estimate the following time-series regression by OLS: LPt = α0 + α1E F t + α2 LPt 1 + Ut, Would the estimator be unbiased Under what conditions would it be consistent Elabo- rate on your answers. ANSWER: The estimator would be biased since the model does not satisfy strict exo- geneity. It will be consistent if Ut is not correlated with E F t or LPt 1. B.2 To study alcohol consumption in the UK, James Collis, Andrew Grayson and Surjinder Johal (“Econometric Analysis of Alcohol Consumption in the UK”, HRMC Working Paper 10, Decem- ber 2010) use data from the Expenditure and Food Survey (2001-2006) to estimate the following model: Y j = X > j β + j Yj = max{Y j , 0} where Yj is the proportion of total expenditure on a particular category of alcohol by household j and the explanatory variables Xj include (log) prices for all alcohol categories, (log) income and other controls. The alcohol categories analyzed were beer, wine, spirits, cider and ready-to-drink (RTDs, also known as ‘alcopops’). Each category was also subdivided into on-trade (pubs and restaurants) and off-trade (supermarkets and off-licences). (a) Assume that ～ N (0, σ2). Provide the (log-)likelihood function for the above model. Hint: The cummulative distribution function for is F(e) = Φ(e/σ) and its probability density function f(e) = φ(e/σ)/σ where Φ(·) and φ(·) are, respectively, the cummulative ECON0019 8 CONTINUED distribution function and the probability density function for the standard normal distribu- tion. ANSWER: TOBIT likelihood. (b) To assess the adequacy of the Tobit, the authors compare estimates of β/σ (where σ is the standard deviation of j) to estimates of the coefficients from a Probit where the dependent variable is whether expenditure on alcohol (for particular categories) is zero or positive. Part of the table is reproduced below (for the purposes of the exam, it is irrelevant whether the table cells or numbers are shaded or not): Explain why this comparison might be useful. ANSWER: The model for y = 0 or 6= 0 is a Probit and its coefficients correspond to β/σ. As indicated by the authors: “Wooldridge proposes an informal evaluation of the general ap- propriateness of the Tobit model. This is conducted by comparing the estimated coefficients from a probit regression to those from the Tobit model. The estimated Tobit coefficients, β , must be divided by the estimated parameter σ to make this comparison possible. As we saw in section 4, whilst this parameter does not affect the sign of the estimated marginal effect, it does impact its magnitude. If the assumptions of the Tobit model are valid, then the probit coefficients should be largely equivalent to the modified Tobit coefficients β /σ .” (c) The researchers are ultimately interested in the elasticities with respect to prices (own- and cross-) and to income. Since those variables are entered as logarithms, you decide to ECON0019 9 TURN OVER estimate those as: = E(Y |X = x)/ xk/y where xk is the relevant variable for the elasticity of interest (i.e., log of own price, log of substitute category or log of income). Explain how you would estimate the elasticity for a particular household. Suggest a measure of elasticity for the general population and explain how you would estimate it. ANSWER: Given the model with normal residuals, E(y|x)/ xk = βkΦ(x>i β/σ) which can be estimated by β kΦ(x > i β /σ) once ML estimates are produced. One can then estimate the “average elasticity” (simlar to APE) as: N 1 N∑ i=1 β kΦ(x > i β /σ )/yi or the “elasticity at the average” (similar to PEA) as: β kΦ(x >β /σ )/y. (d) Suppose that instead of individual data, you have access to data on the market shares for off-trade beer (i.e., beer bought in supermarkets and off-licences) and prices for each of the alcohol categories in several local markets in the United Kingdom. Consider then the following model for the market share for off-trade beer: logSm = β0 + β1 logEm + β2 logPm + m (5) where Sm is the market share for off-trade beer in market m, Em is the expenditure on alcohol in market m and Pm is the price for off-trade beer in market m. (Assume that off- trade beer prices are uniform within a market.) Since the market share for off-trade beer depends not only on the variables above, but also on other variables not included in the model (e.g., prices for other alcohol categories), you decide to use a variable Zm encoding the distribution costs of supermarkets or off-licences for beer (e.g., average distance to beer producers) as an instrumental variable for logPm. (Assume that logEm is uncorrelated with m.) Describe how you would implement the TSLS estimator in this context. How would you argue for the validity of this instrument ANSWER: TSLS: ( 1 ) Regress logPm on Zm and logEm. ( 2 ) Regress logSm on logPm and logEm. The instrumental variable is valid if cov(Zm, m) = 0. This means that distribution costs for beer are not correlated with the unobservable m. If on-trade beer prices are also related to Zm, the validity may be threatened. ECON0019 10 CONTINUED (e) Consider now equation (5) for a single market but across many periods t and suppose there are no endogeneity issues: logSt = β0 + β1 logEt + β2 logPt + t Explain how you would test whether there is serial correlation in t. Would serial correla- tion imply that OLS is inconsistent ANSWER: Explain Durbin-Watson. Serial correlation would not necessarily imply incon- sistency. ECON0019 11 TURN OVER 5 % Critical values for the Fν1,ν2 distribution ν2ν1 1 2 3 4 5 6 7 8 10 12 15 20 30 50 ∞ 1 161 199. 216. 225. 230. 234. 237. 239. 242. 244. 246. 248. 250. 252. 254. 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.4 19.5 19.5 19.5 3 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.79 8.74 8.70 8.66 8.62 8.58 8.53 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 5.96 5.91 5.86 5.80 5.75 5.70 5.63 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.74 4.68 4.62 4.56 4.50 4.44 4.36 10 4.96 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.38 2.31 2.23 2.16 2.07 2.00 1.88 20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.35 2.28 2.20 2.12 2.04 1.97 1.84 30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.16 2.09 2.01 1.93 1.84 1.76 1.62 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 1.99 1.92 1.84 1.75 1.65 1.56 1.39 80 3.97 3.11 2.72 2.49 2.33 2.21 2.13 2.06 1.95 1.88 1.79 1.70 1.60 1.51 1.32 100 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.93 1.85 1.77 1.68 1.57 1.48 1.28 120 3.91 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.91 1.83 1.75 1.66 1.55 1.46 1.25 ∞ 3.85 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.83 1.75 1.67 1.57 1.46 1.35 1.00 ECON0019 12 CONTINUED NORMAL CUMULATIVE DISTRIBUTION FUNCTION (Prob(z < za) where z ～ N(0, 1)) za 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7703 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 ECON0019 13 END OF PAPER