TESTS OF NONSTATIONARITY: UNTRENDED DATA 1 In the previous slideshow we considered the conditions under which the process shown at the top of the slide would be stationary or nonstationary. We distinguished five cases arising from different assumptions relating to  1,  2 and . or General model Alternatives Case (a) Case (b) Case (c) Case (d) Case (e)

2 Case (e) has already been eliminated because it implies an implausible process (one with a quadratic time trend). TESTS OF NONSTATIONARITY: UNTRENDED DATA or General model Alternatives Case (e)

3 We then decided to proceed by plotting the data and assessing whether there is evidence of a trend. If there is not, then one limits the investigation to cases (a) and (b). If there is, one considers cases (c) and (d). TESTS OF NONSTATIONARITY: UNTRENDED DATA or General model Alternatives Case (a) Case (b) Case (c) Case (d) Evidence of a trend? No: (a) or (b). Yes: (c) or (d).

4 In this slideshow we will investigate how one might distinguish between Case (a), a stationary autoregressive process, and Case (b), a random walk. TESTS OF NONSTATIONARITY: UNTRENDED DATA or General model Alternatives Case (a) Case (b)

5 Note that there is an asymmetry in the specification under the two possibilities. TESTS OF NONSTATIONARITY: UNTRENDED DATA or General model Alternatives Case (a) Case (b)

6 If the process is a random walk, there must be no intercept. Otherwise the process would be a random walk with drift and therefore trended. If the process is stationary, the intercept may, and in general, will, be nonzero. TESTS OF NONSTATIONARITY: UNTRENDED DATA or General model Alternatives Case (a) Case (b)

7 For this reason, in the discussion below, it should be assumed that the fitted model includes an intercept, as in the general model, and that we are interested in testing the restrictions  2 = 1 and  1 = 0. TESTS OF NONSTATIONARITY: UNTRENDED DATA or General model Alternatives Case (a) Case (b) Include an intercept but no time trend and test,.

8 In Section 11.5 we investigated the properties of the OLS estimator of  2 when fitting the autoregressive process Y t =  2 Y t–1 +  t. This model has no intercept, but the analysis of the properties of b 2 is unchanged if we include one. TESTS OF NONSTATIONARITY: UNTRENDED DATA General model Case (a) Case (b) Case (b) is a special case of

9 We were not able to say anything analytically about the finite-sample properties of the estimator, but we saw that it is consistent and we could use a central limit theorem to demonstrate that the limiting distribution of √T(b 2 –  2 ) is N(0, 1 –  2 2 ). TESTS OF NONSTATIONARITY: UNTRENDED DATA General model Case (a) Case (b) Case (b) is a special case of

10 From this, we could assert that, as an approximation, for sufficiently large samples, b 2 would be distributed as shown and we investigated the closeness of the approximation using simulation methods. As noted, including an intercept does not affect the analysis. TESTS OF NONSTATIONARITY: UNTRENDED DATA General model Case (a) Case (b) Case (b) is a special case of

11 What happens if  2 = 1, as in Case (b)? If we try to apply the asymptotic analysis when  2 = 1, we immediately get into trouble. It implies that the limiting distribution of the scaled estimator has zero variance. TESTS OF NONSTATIONARITY: UNTRENDED DATA General model Case (a) Case (b) Case (b) is a special case of

12 This is actually correct. To understand what is happening, we will look at the distribution of the estimated values of b 2, when  1 = 0 and  2 = 1, for T = 25, 50, 100, and 200. The figure shows the distribution of b 2 for the different sample sizes when we fit Y t =  1 +  2 Y t–1 +  t. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of b 2 Random walk: distribution of b 2

13 As expected, the OLS estimator is biased for finite samples but consistent. The distribution collapses to a spike as the sample size increases, and the spike is at the true value  2 = 1. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of b 2 Random walk: distribution of b 2

14 This is as anticipated since OLS is a consistent estimator for all AR(1) processes. What is surprising is the rate at which the distribution contracts. In all applications so far, the variance of the OLS estimator has been inversely proportional to the size of the sample. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of b 2 Random walk: distribution of b 2

15 Hence the standard deviation is inversely proportional to √T, and since the area is constant at one, the height of the distribution is proportional to √T. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of b 2 Random walk: distribution of b 2

16 We saw a typical example in the slideshow on spurious regressions. This figure shows the distribution of b 2 for the regression where Y t and X t were white noise processes drawn independently from a normal distribution with zero mean and unit variance. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of b 2 For comparison: distribution of b 2 when Y t and X t are independent white noise

17 You can see that, as the sample size doubles, the height is increased by a factor √2 = 1.41 TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of b 2 For comparison: distribution of b 2 when Y t and X t are independent white noise

18 However, when  2 = 1 and the true process is a random walk, it can be seen that the height is proportional to T. The height doubles from T = 25 to T = 50, it doubles again from T = 50 to T = 100, and it doubles once more from T = 100 to T = 200. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of b 2 Random walk: distribution of b 2

19 This means that the variance of the distribution of b 2 is inversely proportional to T 2, not T. Multiplying by √T is not enough to prevent the distribution contracting. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of b 2 Random walk: distribution of b 2

20 This figure shows the distribution of √T(b 2 – 1). Although we have scaled by a factor √T, the distribution still collapses to a spike instead of a limiting normal distribution. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of √T(b 2 – 1) Random walk: distribution of √T(b 2 – 1)

21 Because the distribution is contracting to the true value faster than the standard rate, the estimator is described as superconsistent. When  2 = 1, to obtain a limiting distribution related to the OLS estimator, we must consider T(b 2 – 1), not √T(b 2 – 1). TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of √T(b 2 – 1) Random walk: distribution of √T(b 2 – 1)

22 This figure shows the distribution of T(b 2 – 1). The statistic does have a limiting distribution, and reaches it quite quickly, since the distribution for T = 25 is almost the same as that for T = 200. (The distributions for T = 50 and T = 100 have been omitted, for clarity.) TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of T(b 2 – 1) Random walk: distribution of T(b 2 – 1)

23 However, we now have a further surprise. The distribution is not normal. This means that, if we work back from the limiting distribution to distributions in finite samples, we cannot claim that the estimator is approximately normal, even in large samples. TESTS OF NONSTATIONARITY: UNTRENDED DATA Random walk: distribution of T(b 2 – 1) Distribution of T(b 2 – 1)

24 We will now examine the tests for nonstationarity proposed by Dickey and Fuller in (1979) and (1981). These were the first tests for nonstationarity and they remain the starting point for any discussion of unit root tests. TESTS OF NONSTATIONARITY: UNTRENDED DATA Direct test using the distribution of T(b 2 – 1) Distribution of T(b 2 – 1)

25 Dickey and Fuller proposed three tests: a direct test using the distribution of T(b 2 – 1), a t test, and an F test. We will consider each in turn, starting with the first. TESTS OF NONSTATIONARITY: UNTRENDED DATA Direct test using the distribution of T(b 2 – 1) Distribution of T(b 2 – 1)

26 This test exploits the fact that  2 in Y t =  1 +  2 Y t–1 +  t is a pure number. Y t–1 must have the same dimensions as Y t, and so  2 is dimensionless. It follows that if we fit the model using OLS, b 2 logically must also be dimensionless. TESTS OF NONSTATIONARITY: UNTRENDED DATA Direct test using the distribution of T(b 2 – 1) Distribution of T(b 2 – 1)

27 The shapes of the distributions of b 2 and T(b 2 – 1) depend only on the sample size, and this means that we can use either distribution to make direct tests of hypotheses relating to  2. TESTS OF NONSTATIONARITY: UNTRENDED DATA Direct test using the distribution of T(b 2 – 1) Distribution of T(b 2 – 1)

28 It is convenient to use the distribution of T(b 2 – 1), shown in the figure, rather than that of b 2, for this purpose because it changes less with the sample size and indeed converges to a limiting distribution as the sample size increases. TESTS OF NONSTATIONARITY: UNTRENDED DATA Direct test using the distribution of T(b 2 – 1) Distribution of T(b 2 – 1)

29 Since we can rule out  2 > 1, we can perform a one-sided test with the null and alternative hypotheses being H 0 :  2 = 1 and H 1 :  2 < 1. TESTS OF NONSTATIONARITY: UNTRENDED DATA Direct test using the distribution of T(b 2 – 1) Distribution of T(b 2 – 1)

30 For the limiting distribution, the critical values are –14.1 and –20.6 at the 5 percent and 1 percent levels. Finite-sample distributions, established through simulation, have thinner left tails and so the critical values are smaller. TESTS OF NONSTATIONARITY: UNTRENDED DATA Direct test using the distribution of T(b 2 – 1) Distribution of T(b 2 – 1)

31 This figure shows the rejection regions for one-sided 5 percent (limit –12.1) and 1 percent (limit –16.6) tests for T = 25. Table A.6 gives the critical values of for finite samples. TESTS OF NONSTATIONARITY: UNTRENDED DATA T = 25 5% limit = –12.1 1% limit = –16.6 Direct test using the distribution of T(b 2 – 1) Distribution of T(b 2 – 1)

32 The second test proposed by Dickey and Fuller uses the conventional OLS t statistic for H 0 :  2 = 1. However, the distribution of the t statistic, established by simulation, is nonstandard. Distribution of OLS t statistic TESTS OF NONSTATIONARITY: UNTRENDED DATA t test

33 The figure shows the distributions for n = 25 and n = 200 when, as throughout this discussion, the fitted model includes an intercept. It can be seen that the mode of the distribution is close to –2. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of OLS t statistic t test

34 As with the previous approach, we are justified in using a one-sided test, and hence the critical value for any given significance level will be much more negative than the conventional critical value. TESTS OF NONSTATIONARITY: UNTRENDED DATA Distribution of OLS t statistic t test

Distribution of OLS t statistic t test 35 For example, the critical value at the 5 percent level for T = 25 is –2.99. The conventional critical value for a one-sided test with 23 degrees of freedom is Table A.6 gives critical values for different sample sizes, established by simulation. T = 25 t crit, 5% = –2.99 (standard: –1.71) TESTS OF NONSTATIONARITY: UNTRENDED DATA

36 The third Dickey–Fuller test exploits the fact that, under H 0, the model is a restricted version of that under H 1 with two restrictions:  2 = 1 and, also,  1 = 0. The first two tests do not make use of this second restriction. TESTS OF NONSTATIONARITY: UNTRENDED DATA F test

37 To test the joint restrictions, it is sufficient to construct the OLS F statistic in the usual way. However, as might be anticipated, it does not have a standard distribution. Table A.6 provides critical values. TESTS OF NONSTATIONARITY: UNTRENDED DATA F test

38 We now have three ways of testing H 0 against H 1 and there is no guarantee that they will lead to the same conclusion. There are two obvious questions that we should ask. How good are these tests, and which is the best? Value of  2 TESTS OF NONSTATIONARITY: UNTRENDED DATA Power of the tests

39 To answer these questions, we need to be more specific about what we mean by ‘good’ and ‘best’. For any given significance level, we would like a test to have the least risk of a Type II error, that is, of failing to reject the null hypothesis when it is false. TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests

40 The power of a test is defined to be probability of rejecting the null hypothesis when it is false, so we want our tests to have high power, and the best test will be that with the highest power. TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests

41 TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests We fit the model Y t =  1 +  2 Y t–1 for |  2 | < 1 and test H 0 :  2 = 1.

42 The figure shows the power of the three tests as a function of  2, using a 5 percent significance level for each of them. TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests

43 The figure seems to answer our questions. The power of the test based on T(b 2 – 1) is greatest for all values of  2 < 1. The t test is next, and the F is test least powerful. TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests

44 The tests all have high power when  2 < 0.8. For values above 0.8, the power rapidly diminishes and becomes low. TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests

45 For  2 = 0.9, for example, even the test based on T(b 2 – 1) will fail to reject the null hypothesis 50 percent of the time. This demonstrates the difficulty of discriminating between nonstationary processes and stationary processes that are highly autoregressive. TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests

46 The inferiority of the F test in this comparison may come as a surprise, given that it is making more use of the data than the other two. TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests

47 However, there is a simple reason for this. The other two tests are one-sided and this increases their power, holding the significance level constant. TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests

48 In the next slideshow we will consider the case where the process appears to contain a trend. TESTS OF NONSTATIONARITY: UNTRENDED DATA Value of  2 Power of the tests

Copyright Christopher Dougherty 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 13.4 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 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics