Week 8 (ii) Lecture PDF - Unit Roots and Stationarity

Lecture 5: Unit Roots and Staionarity Essential reading: Chapter 8 in Brooks. Dr Artur SemeyutinBIE0014 Econometrics Huddersfield Business School w/c 13/03/2023Dr Artur Semeyutin (BIE0014) UR Business School1 / 31 Stationarity and Unit Root Testing Why do we need to test for Non-Stationarity? The s...

Lecture 5: Unit Roots and Staionarity Essential reading: Chapter 8 in Brooks. Dr Artur SemeyutinBIE0014 Econometrics Huddersfield Business School w/c 13/03/2023Dr Artur Semeyutin (BIE0014) UR Business School1 / 31 Stationarity and Unit Root Testing Why do we need to test for Non-Stationarity? The stationarity or otherwise of a series can strongly influence its behaviour and properties - e.g. persistence of shocks will be infinite for nonstationary series Spurious regressions. If two variables are trending over time, a regression of one on the other could have a high R2 even if the two are totally unrelated If the variables in the regression model are not stationary, then it can be proved that the standard assumptions for asymptotic analysis will not be valid. In other words, the usual “ t-ratios” will not follow a t -distribution, so we cannot validly undertake hypothesis tests about the regression parameters. Dr Artur Semeyutin (BIE0014) UR Business School2 / 31 Value of R2 for 1000 Sets of Regressions of a Non-stationary Variable on another Independent Non-stationary Variable Dr Artur Semeyutin (BIE0014) UR Business School3 / 310.00 0.25 0.50 0.75 200 16 0 12 0 80 40 0 frequency 2 Value of t-ratio on Slope Coefficient for 1000 Sets of Regressions of a Non-stationary Variable on another Independent Non-stationary Variable Dr Artur Semeyutin (BIE0014) UR Business School4 / 31–750 –250 0 250 500 750–500 12 0 10 0 80 604020 0 frequency t-ratio Two types of Non-Stationarity Various definitions of non-stationarity exist In this chapter, we are really referring to the weak form or covariance stationarity There are two models which have been frequently used to characterise non-stationarity: the random walk model with drift: yt = µ+ y t− 1 + u t (1) and the deterministic trend process: yt = α+ βt + u t (2) where u t is iid in both cases. Dr Artur Semeyutin (BIE0014) UR Business School5 / 31 Stochastic Non-Stationarity Note that the model (1) could be generalised to the case where y t is an explosive process: yt = µ+ ϕy t− 1 + u t where ϕ >1. Typically, the explosive case is ignored and we use ϕ= 1 to characterise the non-stationarity because –ϕ > 1 does not describe many data series in economics and finance. – ϕ > 1 has an intuitively unappealing property: shocks to the system are not only persistent through time, they are propagated so that a given shock will have an increasingly large influence. Dr Artur Semeyutin (BIE0014) UR Business School6 / 31 Stochastic Non-stationarity: The Impact of Shocks To see this, consider the general case of an AR(1) with no drift: yt = ϕy t− 1 + u t (3) Let ϕtake any value for now. We can write: yt− 1 = ϕy t− 2 + u t− 1 y t− 2 = ϕy t− 3 + u t− 2 Substituting into (3) yields yt = ϕ(ϕ y t− 2 + u t− 1) + u t = ϕ2 y t− 2 + ϕu t− 1 + u t Dr Artur Semeyutin (BIE0014) UR Business School7 / 31 Stochastic Non-stationarity: The Impact of Shocks (Cont’d) Substituting again for y t− 2 y t = ϕ2 (ϕ y t− 3 + u t− 2) + ϕu t− 1 + u t = ϕ3 y t− 3 + ϕ2 u t− 2 + ϕu t− 1 + u t Successive substitutions of this type lead to: yt = ϕT y0 + ϕu t− 1 + ϕ2 u t− 2 + ϕ3 u t− 3 + · · · + ϕT u0 + u t Dr Artur Semeyutin (BIE0014) UR Business School8 / 31 The Impact of Shocks for Stationary and Non-stationary Series We have 3 cases: (1) ϕ < 1⇒ ϕT → 0 as T→ ∞ So the shocks to the system gradually die away. (2) ϕ= 1 ⇒ϕT = 1 ∀T So shocks persist in the system and never die away. We obtain yt = y 0 + ∞ X t =0 u t as T→∞ So the current value of yis just an infinite sum of past shocks plus some starting value of y 0. (3) ϕ > 1. Now given shocks become more influential as time goes on, since if ϕ >1,ϕ3 > ϕ 2 > ϕ , etc. Dr Artur Semeyutin (BIE0014) UR Business School9 / 31 Detrending a Stochastically Non-stationary Series Going back to our 2 characterisations of non-stationarity, the r.w. with drift: yt = µ+ y t− 1 + u t (4) and the trend-stationary process yt = α+ βt + u t (5)The two will require different treatments to induce stationarity. The second case is known as deterministic non-stationarity and what is required is detrending. Dr Artur Semeyutin (BIE0014) UR Business School10 / 31 Detrending a Stochastically Non-stationary Series (Cont’d) The first case is known as stochastic non-stationarity, where there is a stochastic trend in the data. Letting ∆ y t = y t − y t− 1 and Ly t= y t− 1 so that (1 −L) y t = y t − Ly t= y t − y t− 1. If (4) is taken and y t− 1 subtracted from both sides yt − y t− 1 = µ+ u t ∆ y t = µ+ u t We say that we have induced stationarity “by differencing one”. Dr Artur Semeyutin (BIE0014) UR Business School11 / 31 Detrending a Series: Using the Right Method Although trend-stationary and difference-stationary series are both Ψrending” over time, the correct approach needs to be used in each case. If we first difference the trend-stationary series, it would flemove” the non-stationarity, but at the expense on introducing an MA(1) structure into the errors. Conversely if we try to detrend a series which has stochastic trend, then we will not remove the non-stationarity. We will now concentrate on the stochastic non-stationarity model since deterministic non-stationarity does not adequately describe most series in economics or finance. Dr Artur Semeyutin (BIE0014) UR Business School12 / 31 Sample Plots for various Stochastic Processes: A White Noise Process Dr Artur Semeyutin (BIE0014) UR Business School13 / 314 32 1 0 –1 –2 –3–4 14 0791181571 96 235 2743 13352 39 1430 469 Sample Plots for various Stochastic Processes: A Random Walk and a Random Walk with Drift Dr Artur Semeyutin (BIE0014) UR Business School14 / 3170 6050 40 30 2010 0 –10 –20 11 9375 573911091271 45 1631 81 19 9217235 253 27 1289 307 325 343 36 1379 397 415433 45 1469 487 Random wa lk Random wa lk with drift Sample Plots for various Stochastic Processes: A Deterministic Trend Process Dr Artur Semeyutin (BIE0014) UR Business School15 / 3130 25 2015 10 5 0 –5 14 01 181572 74313 352 391 430 46979 19 6 235 Autoregressive Processes with differing values of ϕ (0, 0.8, 1) Dr Artur Semeyutin (BIE0014) UR Business School16 / 311510 5 0 –5 –1 0 –1 5 –20 Phi =  1 Phi = 0.8 Phi =  0 15 3105 2092613 13 4175 21573 677 784 833 88 5 15 7 365469 625 729 937 989 Definition of Non-Stationarity Consider again the simplest stochastic trend model: yt = y t− 1 + u t or ∆y t = u t We can generalise this concept to consider the case where the series contains more than one “unit root”. That is, we would need to apply the first difference operator, ∆, more than once to induce stationarity. Definition If a non-stationary series, y t must be differenced dtimes before it becomes stationary, then it is said to be integrated of order d. We write y t ∼ I( d ). So if y t ∼ I( d ) then ∆ d yt ∼ I(0). An I(0) series is stationary An I(1) series contains one unit root yt = y t− 1 + u t Dr Artur Semeyutin (BIE0014) UR Business School17 / 31 Characteristics of I(0), I(1) and I(2) Series An I(2) series contains two unit roots and so would require differencing twice to induce stationarity. I(1) and I(2) series can wander a long way from their mean value and cross this mean value rarely. I(0) series should cross the mean frequently. The majority of economic and financial series contain a single unit root, although some are stationary and consumer prices have been argued to have 2 unit roots. Dr Artur Semeyutin (BIE0014) UR Business School18 / 31 How do we test for a unit root? The early and pioneering work on testing for a unit root in time series was done by Dickey and Fuller (Dickey and Fuller 1979, Fuller 1976). The basic objective of the test is to test the null hypothesis that ϕ =1 in: yt = ϕy t− 1 + u t against the one-sided alternative ϕ <1. So we have H 0: series contains a unit root versus H 1: series is stationary. We usually use the regression: ∆y t = ψy t− 1 + u t so that a test of ϕ= 1 is equivalent to a test of ψ= 0 (since ϕ− 1 = ψ). Dr Artur Semeyutin (BIE0014) UR Business School19 / 31 Different forms for the DF Test Regressions Dickey Fuller tests are also known as τtests: τ, τ µ , τ τ . The null (H 0) and alternative (H 1) models in each case are i. H 0: y t = y t− 1 + u t H 1: y t = ϕy t− 1 + u t, ϕ < 1 This is a test for a random walk against a stationary autoregressive process of order one (AR(1)) ii. H 0: y t = y t− 1 + u t H 1: y t = ϕy t− 1 + µ+ u t, ϕ < 1 This is a test for a random walk against a stationary AR(1) with drift. iii. H 0: y t = y t− 1 + u t H 1: y t = ϕy t− 1 + µ+ λt + u t, ϕ < 1 This is a test for a random walk against a stationary AR(1) with drift and a time trend. Dr Artur Semeyutin (BIE0014) UR Business School20 / 31 Computing the DF Test Statistic We can write ∆y t = u t where ∆ y t = y t − y t− 1, and the alternatives may be expressed as ∆ y t = ψy t− 1 + µ+ λt + u t with µ= λ= 0 in case i), and λ= 0 in case ii) and ψ= ϕ− 1. In each case, the tests are based on the t-ratio on the y t− 1 term in the estimated regression of ∆ y t on y t− 1, plus a constant in case ii) and a constant and trend in case iii). The test statistics are defined as test statistic=ˆ ψ ˆ SE (ˆ ) ψ Dr Artur Semeyutin (BIE0014) UR Business School21 / 31 Computing the DF Test Statistic (Cont’d)The test statistic does not follow the usual t-distribution under the null, since the null is one of non-stationarity, but rather follows a non-standard distribution. Critical values are derived from Monte Carlo experiments in, for example, Fuller (1976). Relevant examples of the distribution are shown in table 4.1 below Dr Artur Semeyutin (BIE0014) UR Business School22 / 31 Critical Values for the DF Test Significance level 10% 5% 1% CV for constant but no trend −2.57 −2.86 −3.43 CV for constant and trend −3.12 −3.41 −3.96 The null hypothesis of a unit root is rejected in favour of the stationary alternative in each case if the test statistic is more negative than the critical value. Dr Artur Semeyutin (BIE0014) UR Business School23 / 31 The Augmented Dickey Fuller (ADF) Test The tests above are only valid if u t is white noise. In particular, u t will be autocorrelated if there was autocorrelation in the dependent variable of the regression (∆ y t) which we have not modelled. The solution is to “augment” the test using plags of the dependent variable. The alternative model in case (i) is now written: ∆y t = ψy t− 1 + p X i =1 α i∆ y t− i+ u t The same critical values from the DF tables are used as before. A problem now arises in determining the optimal number of lags of the dependent variable There are 2 ways –use the frequency of the data to decide –use information criteria Dr Artur Semeyutin (BIE0014) UR Business School24 / 31 Testing for Higher Orders of Integration Consider the simple regression: ∆y t = ψy t− 1 + u t We test that H 0: ψ = 0 vs. H 1: ψ < 0. If H 0is rejected, we simply conclude that y t does not contain a unit root. But what do we conclude if H 0is not rejected? The series contains a unit root, but is that it? No! What if y t ∼ I(2)? We would still not have rejected. So we now need to test H0: y t ∼ I(2) vs. H 1: y t ∼ I(1) We would continue to test for a further unit root until we rejected H 0. We now regress ∆ 2 y t on ∆ y t− 1 (plus lags of ∆ 2 y t if necessary). Dr Artur Semeyutin (BIE0014) UR Business School25 / 31 Testing for Higher Orders of Integration (Cont’d)Now we test H 0: ∆ y t ∼ I(1) which is equivalent to H 0: y t ∼ I(2). So in this case, if we do not reject (unlikely), we conclude that y t is at least I(2). Dr Artur Semeyutin (BIE0014) UR Business School26 / 31 The Phillips-Perron Test Phillips and Perron have developed a more comprehensive theory of unit root nonstationarity. The tests are similar to ADF tests, but they incorporate an automatic correction to the DF procedure to allow for autocorrelated residuals. The tests usually give the same conclusions as the ADF tests, and the calculation of the test statistics is complex. Dr Artur Semeyutin (BIE0014) UR Business School27 / 31 Criticism of Dickey-Fuller and Phillips-Perron-type tests Main criticism is that the power of the tests is low if the process is stationary but with a root close to the non-stationary boundary. e.g. the tests are poor at deciding if ϕ=1 or ϕ=0.95, especially with small sample sizes. If the true data generating process (dgp) is yt = 0 .95 y t− 1 + u t then the null hypothesis of a unit root should be rejected. One way to get around this is to use a stationarity test as well as the unit root tests we have looked at. Dr Artur Semeyutin (BIE0014) UR Business School28 / 31 Stationarity tests Stationarity tests have H0: y t is stationary versus H 1: y t is non-stationary So that by default under the null the data will appear stationary. One such stationarity test is the KPSS test (Kwaitowski, Phillips, Schmidt and Shin, 1992). Thus we can compare the results of these tests with the ADF/PP procedure to see if we obtain the same conclusion. A Comparison ADF / PP KPSS H 0: y t ∼ I(1) H 0: y t ∼ I(0) H 1: y t ∼ I(0) H 0: y t ∼ I(1) Dr Artur Semeyutin (BIE0014) UR Business School29 / 31 Stationarity tests (Cont’d)4 possible outcomes Reject H 0 and Do not reject H 0 Do not Reject H 0and Reject H 0 Reject H 0 and Reject H 0 Do not reject H 0and Do not reject H 0Dr Artur Semeyutin (BIE0014) UR Business School30 / 31 Essential Reading Please read the text book chapter: Chris Brooks - Introductory Econometrics for Finance, 4th Edition (2019) Cambridge University Press, Chapter 8. Or read: Jeffrey Wooldridge - Introductory Econometrics, 7th Edition (2019) Cengage, Chapters 10. Dr Artur Semeyutin (BIE0014) UR Business School31 / 31