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Christopher Dougherty EC220 - Introduction to econometrics (chapter 13) Slideshow: tests of nonstationarity: trended data Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 13). [Teaching Resource] © 2012 The Author This version available at: 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.

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General model Alternatives Case (a) Case (b) Case (c) Case (d) Case (e) TESTS OF NONSTATIONARITY: TRENDED DATA 1 In this slideshow we consider testing for nonstationarity when an inspection of the graph of a process reveals evidence of a trend. or

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General model Alternatives Case (a) Case (b) Case (c) Case (d) Case (e) TESTS OF NONSTATIONARITY: TRENDED DATA 2 Cases (a) and (b), considered in the previous slideshow, can be eliminated because they do not give rise to trends. Case (e) has been eliminated because it implies a quadratic trend. or

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General model Alternatives Case (a) Case (b) Case (c) Case (d) Case (e) TESTS OF NONSTATIONARITY: TRENDED DATA 3 So we are left with Cases (c) and (d). or

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TESTS OF NONSTATIONARITY: TRENDED DATA 4 We need to consider whether the process is better characterized as a random walk with drift, as in Case (c), or a deterministic trend, as in Case (d). or General model Alternatives Case (c) Case (d)

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TESTS OF NONSTATIONARITY: TRENDED DATA 5 or General model Alternatives Case (c) Case (d) To do this, we fit the general model, as in Case (d), with no assumption about the parameters. We can then test H 0 : 2 = 1 using as our test statistic either T(b 2 – 1) or the t statistic for b 2, as before.

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TESTS OF NONSTATIONARITY: TRENDED DATA 6 or General model Alternatives Case (c) Case (d) The inclusion of the time trend in the specification causes the critical values under the null hypothesis to be different from those in the untrended case. They are determined by simulation methods, as before.

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TESTS OF NONSTATIONARITY: TRENDED DATA 7 or General model Alternatives Case (c) Case (d) We can also perform an F test. We have argued that a process cannot combine a random walk with drift and a time trend, so we can test the composite hypothesis H 0 : 2 = 1, = 0. Critical values for the three tests are given in Table A.7 at the end of the text.

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TESTS OF NONSTATIONARITY: TRENDED DATA 8 or General model Alternatives Case (c) Case (d) If the null hypothesis is false, and Y t is therefore a stationary autoregressive process about a deterministic trend, the OLS estimators of the parameters are √T consistent, and the conventional test statistics are asymptotically valid.

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TESTS OF NONSTATIONARITY: TRENDED DATA 9 or General model Alternatives Case (c) Case (d) Two special cases should be mentioned, if only as econometric curiosities.

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TESTS OF NONSTATIONARITY: TRENDED DATA 10 or General model Alternatives Case (c) Case (d) In general, if a plot of the process exhibits a trend, we will not know whether it is caused by a deterministic trend or a random walk with drift, and we have to allow for both by fitting the general case, as in Case (d), with no restriction on the parameters.

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TESTS OF NONSTATIONARITY: TRENDED DATA 11 or General model Alternatives Case (c) Case (d) But if, for some reason, we know that the process is a deterministic trend or, alternatively, we know that it is a random walk with drift, and if we fit the model appropriately, there is a spectacular improvement in the properties of the OLS estimator of the slope coefficient.

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Special case where the process is known to be a deterministic trend TESTS OF NONSTATIONARITY: TRENDED DATA 12 In the special case where 2 = 0 and the process is just a simple deterministic trend, we encounter a surprising result. Distribution of b 2

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Special case where the process is known to be a deterministic trend TESTS OF NONSTATIONARITY: TRENDED DATA 13 If it is known that there is no autoregressive component, and the regression model is correctly specified with t as the only explanatory variable, the OLS estimator of is hyperconsistent, its variance being inversely proportional to T 3.

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Distribution of b 2 Special case where the process is known to be a deterministic trend TESTS OF NONSTATIONARITY: TRENDED DATA 14 This is illustrated for the case = 0.2 in the left chart in the figure. Since the standard deviation of the distribution is inversely proportional to T 3/2, the height is proportional to T 3/2, and so it more than doubles when the sample size is doubled.

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Distribution of b 2 Special case where the process is known to be a deterministic trend TESTS OF NONSTATIONARITY: TRENDED DATA 15 If Y t–1 is mistakenly included in the regression model, the loss of efficiency is dramatic. The estimator of reverts to being only √T consistent. Further, it is subject to finite-sample bias. This is illustrated in the right chart in the figure.

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Distribution of b 2 Special case where the process is known to be a deterministic trend TESTS OF NONSTATIONARITY: TRENDED DATA 16 In this special case, if the regression model is correctly specified, and the disturbance term is normally distributed, OLS t and F tests are valid for finite samples, despite the hyperconsistency of the estimator of .

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Distribution of d Special case where the process is known to be a deterministic trend TESTS OF NONSTATIONARITY: TRENDED DATA 17 If the disturbance term is not normal, but has constant variance and finite fourth moment, the t and F tests are asymptotically valid.

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Special case where the process is a random walk with drift TESTS OF NONSTATIONARITY: TRENDED DATA 18 Similarly, in the special case where the process is a random walk with drift, so that 2 = 1 and = 0, and the model is correctly specified with Y t–1 as the only explanatory variable, the OLS estimator of 2 is hyperconsistent. Distribution of b 2

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Special case where the process is a random walk with drift TESTS OF NONSTATIONARITY: TRENDED DATA 19 If a time trend is added to the specification by mistake, there is a loss of efficiency, but it is not as dramatic as in the other special case. The estimator is still superconsistent (variance inversely proportional to T 2 ). The distributions for the various sample sizes for this case are shown as the red lines in the figure. Distribution of b 2

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Special case where the process is a random walk with drift TESTS OF NONSTATIONARITY: TRENDED DATA 20 The conventional t and F tests are asymptotically valid, but not valid for finite samples because the process is autoregressive. Distribution of b 2

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Augmented Dickey–Fuller tests Second-order autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 21 We need to generalize the discussion to higher order processes. We will start with the second-order process shown. Main condition for stationarity:

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Augmented Dickey–Fuller tests Second-order autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 22 To be stationary, the parameters now need to satisfy several conditions. The most important in practice is | 2 + 3 | < 1. To test this, it is convenient to reparameterize the model. Main condition for stationarity:

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Augmented Dickey–Fuller tests Second-order autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 23 Subtract Y t–1 from both sides, add and subtract 3 Y t–1 on the right side, and group terms together. Main condition for stationarity:

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Augmented Dickey–Fuller tests Second-order autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 24 Thus we obtain a model where Y t = Y t – Y t–1 is related to Y t–1 and Y t–1, with 2 * = 2 + 3 and 3 * = 3. Main condition for stationarity:

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Augmented Dickey–Fuller tests Second-order autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 25 Under the null hypothesis H 0 : 2 * = 1, the process is nonstationary. Given the reparameterization, H 0 may be tested by testing whether the coefficient of Y t–1 is significantly different from zero. Main condition for stationarity:

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Augmented Dickey–Fuller tests Second-order autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 26 One may usually perform a one-sided test with alternative hypothesis H 1 : 2 * 1 implies an explosive process. Main condition for stationarity:

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Augmented Dickey–Fuller tests Second-order autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 27 Under the null hypothesis, the estimator of 2 * is superconsistent and the test statistics T(b 2 * – 1), t, and F have the same distributions, and therefore critical values, as before. Main condition for stationarity:

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Augmented Dickey–Fuller tests Second-order autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 28 Main condition for stationarity: If a deterministic time trend is suspected, it may be included and the critical values are those for the first-order specification with a time trend.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 29 Main condition for stationarity: Generalizing to the case where Y t depends on Y t–1,..., Y t–p, a condition for stationarity is that | p+1 | < 1 and it is convenient to reparameterize the model as shown, where 2 * = p+1 and the other * coefficients are appropriate linear combinations of the original coefficients.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 30 Main condition for stationarity: Under the null hypothesis of non-explosive nonstationarity, the test statistics T(b 2 * – 1), t, and F asymptotically have the same distributions and critical values as before. In practice, the t test is particularly popular and is generally known as the augmented Dickey–Fuller (ADF) test.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 31 Main condition for stationarity: There remains the issue of the determination of p. Two main approaches have been proposed and both start by assuming that one can hypothesize some maximum value p max.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 32 Main condition for stationarity: In the F test approach, the reparameterized model is fitted with p = p max and a t test is performed on the coefficient of Y t–pmax. If this is not significant, this term may be dropped.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 33 Main condition for stationarity: Next, an F test is performed on the joint explanatory power of Y t–pmax and Y t–pmax–1. If this is not significant, both terms may be dropped.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 34 Main condition for stationarity: The process continues, including further lagged differences in the F test until the null hypothesis of no joint explanatory power is rejected.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 35 Main condition for stationarity: The last lagged difference included in the test becomes the term with the maximum lag. Higher order lags may be dropped because the previous F test was not significant.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 36 Main condition for stationarity: Provided that the disturbance term is iid, the normalized coefficient of Y t–1 and its t statistic will have the same (non-standard) distributions as for the Dickey–Fuller test.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 37 Main condition for stationarity: The other method is to use an information criterion such as the Bayes Information Criterion (BIC), also known as the Schwarz Information Criterion (SIC). This requires the computation of the BIC statistic shown and choosing p so as to minimize the expression.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 38 Main condition for stationarity: The first term falls as p increases, but the second term increases, and the trade-off is such that asymptotically the criterion will select the true value of p.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 39 Main condition for stationarity: A common alternative is the Akaike Information Criterion (AIC) shown. This imposes a smaller penalty on overparameterization and will therefore tend to select a larger value of p, but simulation studies suggest that it may produce better results in practice.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 40 Main condition for stationarity: Whether one uses the F test approach or information criteria, it is necessary to check that the residuals are not subject to autocorrelation, for example, using a Breusch–Godfrey lagrange multiplier test.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 41 Main condition for stationarity: Autocorrelation would provide evidence that there remain dynamics in the model not accounted for by the specification and that the model does not include enough lags.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 42 Main condition for stationarity: The 1979 and 1981 Dickey–Fuller papers were truly seminal in that they have given rise to a very extensive research literature devoted to the improvement of testing for nonstationarity and of the representation of nonstationary processes.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 43 Main condition for stationarity: The low power of the Dickey–Fuller tests was acknowledged in the original papers and much effort has been directed to the problem of distinguishing between nonstationary processes and highly autoregressive stationary processes.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 44 Main condition for stationarity: Remarkably, the original Dickey–Fuller tests, particularly the t test in augmented form, are still widely used, perhaps even dominant.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 45 Main condition for stationarity: Other tests with superior asymptotic properties have been proposed, but some underperform in finite samples, as far as this can be established by simulation.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 46 Main condition for stationarity: The augmented Dickey–Fuller t test has retained its popularity on account of robustness and, perhaps, theoretical simplicity.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 47 Main condition for stationarity: However, a refinement, the ADF–GLS (generalized least squares) test due to Elliott, Rothenberg, and Stock (1996) appears to be gaining in popularity and is implemented in major regression applications.

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Augmented Dickey–Fuller tests General autoregressive process TESTS OF NONSTATIONARITY: TRENDED DATA 48 Main condition for stationarity: Simulations indicate that its power to discriminate between a nonstationary process and a stationary autoregressive process is uniformly closer to the theoretical limit than the standard tests, irrespective of the degree of autocorrelation.

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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 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics

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