1. Introduction to Econometrics, 5th edition
Type author name/s here
Dougherty
Introduction to Econometrics, 5th edition
Chapter heading
Chapter 11: Models Using Time Series Data
© Christopher Dougherty, All rights reserved.

2. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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. 1

3. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
We will assume | b2 | < 1. The OLS estimator is then consistent. This was demonstrated in the previous sequence. 2

4. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
10 million samples
The figure shows the distribution for samples of size 25, 50, 100, and 200 when b2 = It can be seen that the distribution contracts as the size increases, becoming progressively more concentrated around 0.6. 3

5. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
10 million samples
Here the distributions for even greater sample sizes are shown. The distribution is clearly collapsing to a spike at 0.6. 4

6. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
10 million samples
Standard theory tells us that 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 b2, so how can we say that has an asymptotically normal distribution? 5

7. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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. 6

8. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
As a first step, we multiply the estimator by This is sufficient to prevent the variance from tending to zero as T increases. However, does not have a limiting distribution, either, because tends to b2 and increases without limit with T. 7

9. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
So, instead, we consider For the model under discussion, it can be shown that, provided that │b2│< 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 – b22). 8

10. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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. 9

11. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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. 10

12. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
Hence, adding b2 to the statistic, we can say, that is distributed as shown, as an approximation, for sufficiently large samples. 11

13. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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. 12

14. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
Simulation reveals that the answer depends on the value of b2 itself. We will start by putting b2 = The figure shows the distributions of for T = 25, 50, 100, and 200. 13

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

16. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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. 15

17. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
Although the overall shape of 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. 16

18. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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 N( b2, (1 – b22) / T ) to be an accurate guide to the actual distribution of . 17

19. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
This is confirmed by the figure shown. It compares the actual distribution of for T = 100, obtained by simulation, with the theoretical distribution (still with b2 = 0.6). 18

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

21. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
The finite-sample bias is the stronger, the closer that b2 is to 1. The figure shows the distribution of when b2 = In this case, it is clear that, even for T = 200, the distribution is far from normal. 20

22. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
The left tail contracts towards the limiting distribution as the sample size increases, as it did for b2 = 0.6, but more slowly. 21

23. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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. 22

24. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
We have seen that, for finite samples, the tails of the distributions of and 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. 23

25. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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. 24

26. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
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. 25

27. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
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 H0: b2 = 0.9 when the null hypothesis is true, and T = 100. 26

28. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
The distribution is skewed, reflecting the fact that the distribution of is skewed. Further complexity is attributable to the fact that the standard error is also valid only asymptotically. 27

29. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
Nominal critical value of t, T = 100, is 1.98 at 5% level
Actual 2.5% tails start at –2.14 and 1.72
According to the tables, for a 5 percent two-sided test, with T = 100, the critical values of t are However, in reality the lower 2.5 percent tail of the distribution starts at –2.14 and the upper one at 1.72. 28

30. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
Nominal critical value of t, T = 100, is 1.98 at 5% level
p( t < –1.98 ) = 0.036
p( t > 1.98 ) = 0.013
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 is negative and less than 2.5 percent when it is positive. The figures are 3.6 percent and 1.3 percent. 29

31. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
The potential effect on Type II error is often of greater practical importance, for typically our null hypothesis is that b2 = 0 and if the process is truly autoregressive, the null hypothesis is false. 30

32. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
Fortunately, for this null hypothesis, the t test is unlikely to mislead us seriously. If the true value of b2 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. 31

33. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
If the true value of b2 is large, H0 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. 32

34. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
These remarks apply to the pure autoregressive model Yt = b2Yt–1 + ut . In practice, the model will include other explanatory variables, with unpredictable consequences. 33

35. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
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. 34

36. LIMITING DISTRIBUTION OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
H0: b2 = 0.9 is true
Caution is especially required when the sample is small and when there is evidence that b2 is large. 35

37. Copyright Christopher Dougherty 2016.
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The content of this slideshow comes from Section 11.5 of C. Dougherty, Introduction to Econometrics, fifth edition 2016, Oxford University Press.
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