1. 1 We will now consider the distributional properties of OLS estimators in models with a lagged dependent variable. We will do so for the simplest such model of all, where the right side is just the lagged dependent variable and an error term. and the estimator is as shown. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

2. 2 We will assume |  2 | < 1. The OLS estimator is then consistent. This was demonstrated in the previous sequence.

3. 3 The figure shows the distribution for samples of size 25, 50, 100, and 200 when  2 = 0.6. It can be seen that the distribution contracts as the size increases, becoming progressively more concentrated around 0.6. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES T = 25 T = 50 T = 100 T = 200 10 million samples

4. 4 Here the distributions for even greater sample sizes are shown. The distribution is clearly collapsing to a spike at 0.6. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES T = 25 T = 100 T = 400 T = 1,600 10 million samples

5. 5 LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES T = 25 T = 100 T = 400 T = 1,600 Standard theory tells us that b 2 is asymptotically normally distributed. What does this mean? We have just shown that, as the sample size increases, the distribution degenerates to a spike at  2, so how can we say that b 2 has an asymptotically normal distribution? 10 million samples

6. 6 We encountered this problem when determining the asymptotic properties of IV estimators in Chapter 8. To deal with it, we again use the technique involving the use of a central limit theorem discussed in Section R.15. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

7. 7 As a first step, we multiply the estimator by √T. This is sufficient to prevent the variance from tending to zero as T increases. However, (√T)b 2 does not have a limiting distribution, either, because b 2 tends to  2 and (√T)b 2 increases without limit with T. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

8. 8 So, instead, we consider √T(b 2 –  2 ). For the model under discussion, it can be shown that, provided that │  2 │< 1, the conditions for the application of a central limit theorem are satisfied and that the limiting distribution is normal with zero mean and variance (1 –  2 2 ). LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

9. 9 This asymptotic result is all that we have in analytical terms. We are not entitled to say anything analytically for finite samples. However, given the limiting distribution, we can start working back tentatively to finite samples and make some plausible assertions. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

10. 10 We can say, that for large T, the relationship may hold approximately. If this is the case, dividing the statistic by √T, we obtain the result shown, as an approximation, for sufficiently large samples. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

11. 11 Hence, adding  2 to the statistic, we can say, that b 2 is distributed as shown, as an approximation, for sufficiently large samples. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

12. 12 Of course, there remains the question of what might be considered to be a ‘sufficiently large’ sample. To answer this question, we turn to simulation. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

13. 13 Simulation reveals that the answer depends on the value of  2 itself. We will start by putting  2 = 0.6. The figure shows the distributions of √T(b 2 – 0.6) for T = 25, 50, 100, and 200. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

14. 14 For the simulation, the disturbance term was drawn randomly from a normal distribution with zero mean and unit variance. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

15. 15 According to the theory, the distribution of ought to converge to a normal distribution with mean zero and variance. This limiting normal distribution is shown as the red curve in the figure. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

16. 16 Although the overall shape of √T(b 2 – 0.6) is not far from normal, even for T as small as 25, there are serious discrepancies in the tails, and it is the shape of the tails that matters for inference. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

17. 17 Even for T = 200, the left tail is far too fat and the right tail far too thin. This implies that we should not expect the N(  2, (1 –  2 2 ) / T ) to be an accurate guide to the actual distribution of b 2. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

18. 18 This is confirmed by the figure shown. It compares the actual distribution of b 2 for T = 100, obtained by simulation, with the theoretical distribution (still with  2 = 0.6). LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

19. 19 (Since (√100)(b 2 –0.6) is just a linearly scaled function of b 2, the relationship between the actual distribution of b 2 and its theoretical distribution is parallel to that between (√100)(b 2 – 0.6) and its limiting normal distribution in the previous figure.) LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

20. 20 The finite-sample bias is the stronger, the closer that  2 is to 1. The figure shows the distribution of √T(b 2 –  2 ) when  2 = 0.9. In this case, it is clear that, even for T = 200, the distribution is far from normal. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

21. 21 The left tail contracts towards the limiting distribution as the sample size increases, as it did for  2 = 0.6, but more slowly. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

22. 22 The right tail actually shifts in the wrong direction as the sample size increases from T = 25 to T = 50. However, it then starts moving back in the direction of the limiting distribution, but there is still a large discrepancy even for T = 200. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

23. 23 We have seen that, for finite samples, the tails of the distributions of √T(b 2 –  2 ) and b 2 differ markedly from their approximate theoretical distributions, even for T = 200, and this can be expected to cause problems for inference. This is indeed the case. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

24. 24 We know that inference is asymptotically valid in a model with a lagged dependent variable. However, as always, we have to ask how large the sample should be in practice. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

25. 25 We need to consider the effect on Type I error when the null hypothesis is true, and the effect on Type II error when the null hypothesis is false. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

26. 26 We will start with the effect on Type I error and we again will illustrate the issue with the simple autoregressive model. The figure shows the distribution of the t statistic for H 0 :  2 = 0.9 when the null hypothesis is true, and T = 100. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES H 0 :  2 = 0.9 is true

27. 27 The distribution is skewed, reflecting the fact that the distribution of b 2 is skewed. Further complexity is attributable to the fact that the standard error is also valid only asymptotically. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES H 0 :  2 = 0.9 is true

28. 28 According to the tables, for a 5 percent two-sided test, with T = 100, the critical values of t are 1.98. However, in reality the lower 2.5 percent tail of the distribution starts at –2.14 and the upper one at 1.72. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES Nominal critical value of t, T = 100, is 1.98 at 5% level Actual 2.5% tails start at –2.14 and 1.72

29. H 0 :  2 = 0.9 is true Nominal critical value of t, T = 100, is 1.98 at 5% level 29 This means that, if one uses the critical values from the table, the risk of a Type I error when the null hypothesis is true is greater than 2.5 percent when b 2 is negative and less than 2.5 percent when it is positive. The figures are 3.6 percent and 1.3 percent. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES p( t < –1.98 ) = 0.036 p( t > 1.98 ) = 0.013

30. 30 The potential effect on Type II error is often of greater practical importance, for typically our null hypothesis is that  2 = 0 and if the process is truly autoregressive, the null hypothesis is false. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES H 0 :  2 = 0.9 is true

31. 31 Fortunately, for this null hypothesis, the t test is unlikely to mislead us seriously. If the true value of  2 is low, the distorting effect of the failure of Assumption C.7 part (2) can be expected to be minor and our conclusions valid, even for finite samples. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES H 0 :  2 = 0.9 is true

32. 32 If the true value of  2 is large, H 0 is likely to be rejected anyway, even though the t statistic does not have its conventional distribution and the nominal critical values of t are incorrect. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES H 0 :  2 = 0.9 is true

33. 33 These remarks apply to the pure autoregressive model Y t =  2 Y t–1 + u t. In practice, the model will include other explanatory variables, with unpredictable consequences. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES H 0 :  2 = 0.9 is true

34. 34 The most that one can say is that, if there is a lagged dependent variable in the model, one should expect point estimates, standard errors, and t statistics to be subject to distortion and therefore one should treat them with caution. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES H 0 :  2 = 0.9 is true

35. 35 Caution is especially required when the sample is small and when there is evidence that  2 is large. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES H 0 :  2 = 0.9 is true

36. Copyright Christopher Dougherty 2013. 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 11.5 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 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. 2013.01.27