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Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: exercise 6.7 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/132/http://learningresources.lse.ac.uk/132/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/ http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

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6.7*A social scientist thinks that the level of activity in the shadow economy, Y, depends either positively on the level of the tax burden, X, or negatively on the level of government expenditure to discourage shadow economy activity, Z. Y might also depend on both X and Z. International cross- section data on Y, X, and Z, all measured in US $ million, are obtained for a sample of 30 industrialized countries and a second sample of 30 developing countries. The social scientist and regresses log Y (1) on both log X and log Y, (2) on log X alone, and (3) on log Z alone, for each sample, with the following results (standard errors in parentheses): 1 EXERCISE 6.7

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2 log X was positively correlated with log Z in both samples (0.81 and 0.75). Perform the appropriate statistical tests and write a short report advising how these results should be interpreted. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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3 We will start with the sample of 30 industrialized countries. In the multiple regression, the estimates of the elasticities of Y with respect to X and Z are both highly significant and have the expected signs. We will therefore adopt this as our preferred specification. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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4 In model (2), the estimate of the elasticity of Y with respect to X will be subject to omitted variable bias. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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5 It is reasonable to assume that 3 is negative, particularly since its estimate in the multiple regression is negative and highly significant. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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6 We are told that X and Z are positively correlated, so the numerator of the second factor in the bias term will be positive. The denominator must be positive. Hence the bias will be negative, accounting for the fall in the estimate of the elasticity. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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7 Similarly, in model (3), the estimate of the elasticity of Y with respect to Z will be subject to omitted variable bias. It is reasonable to assume that 2 is positive, and we know that the numerator and the denominator of the second factor are positive, so the bias is positive. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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8 The bias just about offsets the true value of the coefficient, with the consequence that the estimated coefficient is close to 0. It appears that log Z has very little effect on log Y, and this in turn accounts for the very low value of R 2. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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9 For similar reasons, R 2 in model (2) was also low. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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10 However, in the multiple regression, where the effect of omitted variable bias is eliminated, we see that in fact log X and log Z account for nearly half the variance in log Y. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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11 Next we come to the sample of 30 developing countries. In the multiple regression, the estimate of the elasticity of Y with respect to X is again highly significant, but that of the elasticity with respect to Z is not. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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12 If we drop log Z, the estimate of the elasticity with respect to X is a little lower. R 2 is virtually unchanged. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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13 If instead we drop log X, the estimate of the elasticity with respect to Z changes sign, suggesting that expenditure on enforcement actually encourages the growth of the shadow economy. Moreover, the effect appears to be highly significant. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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14 Obviously, this absurd result is attributable to omitted variable bias, because it is clear that log X belongs in the model. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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15 As in the case of the industrialized countries, the bias is positive, and it is so large that it dominates the estimate of the elasticity. log Z is acting as a proxy for log X. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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16 So what do you conclude in the case of the developing countries? EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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17 Clearly, we eliminate model (3). But the other two models both remain in contention. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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18 It is possible that enforcement expenditure has no effect and model (2) is the correct specification. If this is the case, the estimator of the elasticity of X in model (1) will be less efficient and this is the reason that the standard error is larger. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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19 Alternatively, model (1) might be the correct specification. Enforcement expenditure may have an effect, but bureaucracy and corruption undermine it and account for the low and insignificant estimate of its elasticity. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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20 In practice, in a situation like this, you would present the regression results for both models (1) and (2) in your report, giving the reader all the available information. EXERCISE 6.7 Industrialized Countries Developing Countries (1) (2) (3) (1) (2) (3) log X0.700.20–0.810.73– (0.15)(0.11)(0.14)(0.09) log Z–0.65––0.05–0.09–0.43 (0.16)(0.12)(0.12)(0.12) constant–1.14–1.071.23–1.12 – 1.022.82 (0.86)(1.07)(0.90)(0.87)(0.86)(0.84) R 2 0.440.100.010.710.700.33

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Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for personal use. 25.06.06

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