1. Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: asymptotic properties of estimators: the use of simulation Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/141/http://learningresources.lse.ac.uk/141/ 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/

2. 1 ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION Consistency Why are we interested in consistency, when in practice we have finite samples? As a first approximation, the answer is that if we can show that an estimator is consistent, then we may be optimistic about its finite sample properties, whereas is the estimator is inconsistent, we know that for finite samples it will definitely be biased.

3. 2 Consistency Why are we interested in consistency, when in practice we have finite samples? As a first approximation, the answer is that if we can show that an estimator is consistent, then we may be optimistic about its finite sample properties, whereas is the estimator is inconsistent, we know that for finite samples it will definitely be biased. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

4. 3 Consistency However, there are reasons for being cautious about preferring consistent estimators to inconsistent ones. First, a consistent estimator may be biased for finite samples. Second, we are usually also interested in variances. If a consistent estimator has a larger variance than an inconsistent one, the latter might be preferable if judged by the mean square error or similar criterion that allows a trade-off between bias and variance. How can you resolve these issues? Mathematically they are intractable, otherwise we would not have resorted to large sample analysis in the first place. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

5. 4 Consistency However, there are reasons for being cautious about preferring consistent estimators to inconsistent ones. First, a consistent estimator may be biased for finite samples. Second, we are usually also interested in variances. If a consistent estimator has a larger variance than an inconsistent one, the latter might be preferable if judged by the mean square error or similar criterion that allows a trade-off between bias and variance. How can you resolve these issues? Mathematically they are intractable, otherwise we would not have resorted to large sample analysis in the first place. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

6. 5 Consistency However, there are reasons for being cautious about preferring consistent estimators to inconsistent ones. First, a consistent estimator may be biased for finite samples. Second, we are usually also interested in variances. If a consistent estimator has a larger variance than an inconsistent one, the latter might be preferable if judged by the mean square error or similar criterion that allows a trade-off between bias and variance. How can you resolve these issues? Mathematically they are intractable, otherwise we would not have resorted to large sample analysis in the first place. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

7. 6 Simulations The answer is to conduct a simulation experiment, directly investigating the distributions of estimators under controlled conditions. We will do this for the example in the previous subsection. We will generate Z as a random variable with a normal distribution with mean 1 and variance 0.25. We will set equal to 5, so the value of Y in any observation is 5 times the value of Z: Y = 5Z. We will generate the measurement error as a normally distributed random variable with zero mean and unit variance. The value of X in any observation is equal to the value of Z plus this measurement error: X = Z + w. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

8. 7 Simulations The answer is to conduct a simulation experiment, directly investigating the distributions of estimators under controlled conditions. We will do this for the example in the previous subsection. We will generate Z as a random variable with a normal distribution with mean 1 and variance 0.25. We will set equal to 5, so the value of Y in any observation is 5 times the value of Z: Y = 5Z. We will generate the measurement error as a normally distributed random variable with zero mean and unit variance. The value of X in any observation is equal to the value of Z plus this measurement error: X = Z + w. We then use  Y /  X as an estimator of. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

9. 8 Simulations We then use as an estimator of. We have already seen that the estimator is consistent. The question now is how well it performs in finite samples. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

10. 9 We will start by taking samples of size 20. The figure shows the distribution of l, the estimator of, for one million samples. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

11. 10 Although the mode of the distribution, 4.65, Is lower than the true value, the estimator is actually upwards biased, the mean estimate in the million samples being 5.33. n = 20 mode = 4.65 mean = 5.33 ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

12. 11 If we increase the sample size to 100, the distribution of the estimates obtained with one million samples is less skewed. The mode is 4.94 and the mean is 5.05. n = 100 mode = 4.94 mean = 5.05 ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

13. 12 n = 1000 mode = 4.99 mean = 5.01 If we increase the sample size to 1000, the distribution of the estimates obtained with one million samples is less skewed. The mode is 4.99 and the mean is 5.01. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

14. 13 n = 1000 mode = 4.99 mean = 5.01 We demonstrated analytically that the estimator is consistent, but this is a theoretical property relating to samples of infinite size. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

15. 14 n = 1000 mode = 4.99 mean = 5.01 The simulation shows us that for sample size 1000, the estimator is almost unbiased. However for smaller sample sizes the estimator is biased, especially when the sample size is as small as 20. ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

16. Copyright Christopher Dougherty 2011. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section R.14 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 11.07.25