Chapter 2: Unit Roots

Transcription

1 Chapter 2: Unit Roots 1

2 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and undeconometrics II. Unit Roots... 3 II.1 Integration Level... 3 II.2 Nonstationarity and Dickey-Fuller Tests... 8 II.2.1 Random Walk... 8 II.2.2 Unit Root Tests (Dickey-Fuller and Augmented Dickey-Fuller Test)

3 II. Unit Roots Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and undeconometrics II.1 Integration Level A time series is called integrated of order d (= I(d)), if after d differencing the series follows a stable and invertible ARMA process and thus an I(0) process. The property of I(0) implies stationarity whereas the reverse does not hold. To highlight this issue we have to remind briefly, the properties of stability and invertibilty of an ARMA process. In the simplest case of an ARMA(1,1) process without a constant and with a white noise error term ε we obtain: Y t = γy t-1 + θε t-1 + ε t. This process is stationary and invertible, if γ and θ in absolute values is lesser than 1. 3

4 Stability implies that we receive a MA term with coefficients tending against zero by repeating the substitution of the lagged Y variable. Y ε γ θ γ ε Y is non-stationary if γ = 1 and we get a typical situation of a random walk where the error term exhibits a permanent influence on the time series. Conversely, by repeating substitution of lagged error terms ε, we obtain an AR representation with against zero tending coefficients: Y γ θ θ Y ε Thus it appears that if θ = -1 the coefficients of the AR representation independent of the lag are always (θ + γ) and therefore the time series cannot be approximated by an AR process. 4

5 The difference of Y is under the assumption θ < 1 a I(0) series. But differencing does not always result in a I(0) time series. For example, let us consider a trend stationary (or for a = 0 stationary) process: Y t = c + at + ε t. By differencing we obtain: Y t - Y t-1 = Y t = a + ε t - ε t-1. For this reason the trend of the time series is indeed eliminated and the series is stationary with mean a. But, at the same time a noninvertible MA(1) process was generated with θ = -1 so that we obtain no I(0) series. In this context we speak about overdifferencing. 5

6 Please note that: - trend stationary time series are not mean stationary but include a trend. This trend can be eliminated by including a trend component into the regression model Y t = a + bt + βx t + ε t. As we can see above differencing is not appropriate to eliminate trends because the variance of the error term would increase. - difference stationary time series (which are most of economic time series) contain a stochastic trend, i.e. a non stationarity in the variance component so that with the length of forecasting horizon the uncertainty increases to endless. In this context only differencing results in a stationary time series. 6

7 For stationarity of the error terms of the estimation equation Y t = a + βx t + ε t the following rules are observed: Y t ~ I(0) and X t ~ I(0) Y t ~ I(1) and X t ~ I(0) Y t ~ I(1) and X t ~ I(1) Y t ~ I(1) and X t ~ I(1) ε t ~ I(0), ε t ~ I(1), ε t ~ I(1), if Y and X are not cointegrated, ε t ~ I(0), if Y and X are cointegrated. The residuals are only then I(0) if both variables Y and X either are I(0) or I(1) and cointegrated. The simplest case of cointegration is given when Y and X are I(1) and the linear combination of both variables is I(0), i.e. the residuals are stationary. 7

8 II.2 Nonstationarity and Dickey-Fuller Tests II.2.1 Random Walk A simple example for a stochastic (non-stationary) time series is a random walk: Yt Yt-1 εt with a white noise error term ε t and the properties as follows: E ε 0 E ε,ε 0 E Y Y E ε Y process is mean stationary Var Y Var ε σ n σ process is not variance stationary (the variance changes depending on time) Cov Y,Y σ t n s process is not covariance stationary 8

9 A pure (simple) random walk (without drift) is defined as follows: Y t = Y t-1 + ε t with ε t ~ i.i.d.(0,σ 2 ) The random variable Y t equals to value of the past period (Y t-1 ) plus the realisation of an pure random process; i.i.d. means identically and independently distributed. The values of the random process are of the same distribution, which must not essentially be a normal distribution, and the realisations have to be independently from each other. Thus, ε is a special stationary stochastic process where the autocovariance equals zero. These properties of a distribution of ε are called white noise. Whereas the variable Y t is not stationary as we have seen from above. The mean is indeed constant, but the variance increases over all limits if n tends to infinity. 9

10 Hence, Y is also not stationary, whereas the first difference of Y is a stationary random variable: Yt - Yt-1 Yt εt Time series that follow a random walk are integrated of order one I(1). By adding a constant to the random walk equation, we obtain a random walk with drift: Yt a Yt-1 εt with εt ~ i.i.d. 0,σ 2. If the constant is positive then Y exhibits an upward tendency and when the constant is negative, Y has a downward tendency. The first and second moments of a random walk with drift are: E Y Y n a E ε Y n a process is not mean stationary Var Y Var ε σ n σ process is not variance stationary (the variance changes depending on time) 10

11 Lehrstuhl für Departme Empiris ent sche of Wirtschaftsforschung Empirical Research and undeconometrics Dr. Roland Füss? St SStatistik S 2007 Financial II: Schließende Data Analysis Statistik Winter Term 2007/08 Random walk without drift 11

12 Lehrstuhl für Departme Empiris ent sche of Wirtschaftsforschung Empirical Research and undeconometrics Dr. Roland Füss? St SStatistik S 2007 Financial II: Schließende Data Analysis Statistik Winter Term 2007/08 Random walk with drift Trend stationary 12

13 Accordingly, a random walk with drift defines a non-stationary random variable. The variance is unlimited when n tends to infinity and the expected value change per time unit of value a. The first difference yields, Y Y a ε, which is a stationary sequence. In comparison to a trend stationary process, Yt c at εt with εt ~ i.i.d. 0, σ 2, which alters deterministic in time, a random walk with drift exhibits a stochastic trend. For both processes the expected change in t periods are indeed the same of t a, but completely different. Whereas for a random walk with drift the first difference is stationary, for a trend stationary process the deviations from the trend (Y t at = c + ε t ) are stationary. 13

14 II.2.2 Unit Root Tests (Dickey-Fuller and Augmented Dickey-Fuller Test) The most frequently used test of the null hypothesis of a I(1) series against a I(0) alternative hypothesis is the Dickey-Fuller t-test. First, let us consider this test in the simplest AR(1) case without a constant. The initial equation is: Y t = γy t-1 + ε t H 0 : γ = 1 random walk without drift H 1 : γ < 1 stationary AR(1) process It concerns a one-sided test with the null hypothesis of nonstationarity and the alternative hypothesis of stationarity. For an AR(1) process the t-statistic of the OLS estimation is biased, i.e. it do not follow a t-distribution under the null hypothesis. Hence, Dickey and Fuller derive the distribution and of this test statistic and determine the critical values. 14

15 If we want to test a random walk with drift against the alternative of a stationary AR(1) process with a mean which does not equal zero, we have to insert a constant term into the regression equation: Y t = µ + γy t-1 + ε t H 0 : γ = 1 random walk with drift H 1 : γ < 1 stationary AR(1) process with mean µ 0 Finally, the alternative hypothesis of a trend stationary process is of interest. For this purpose we insert additionally a trend variable into the regression equation: Y t = µ + βt + γy t-1 + ε t H 0 : γ = 1 random walk with drift and trend H 1 : γ < 1 trend stationary 15

16 Up to now we did not take into consideration the case of the null hypothesis of autocorrelated I(0) differences. For this, we have to expand the test equation by subtracting the lagged variable on both sides. Subsequently we complete the equation by p-1 lagged differences: Y Y Y µ βt γ Y Φ Y ε with γ γ 1 H 0 : γ * = 0 in differences stationary AR(p-1) process H 1 : γ * < 0 trend stationary AR(p) process For this so-called Augmented Dickey-Fuller test, the same critical values are valid as for the simple model of this test. This means, if we do not consider any lag we obtain the simple Dickey-Fuller test. 16

17 Let us turn to an AR(p) process: y a a y a y a y a y a y a y ε add and subtract: a y y a a y a y a y a y a a y a Δy ε add and subtract: a a y y a a y a y a y a a Δy a Δy ε 17

18 We can take the difference and receive: Δy a γy β Δy ε 1 0 in the first difference has a unit root when 1. 18

19 Sequential test procedure: 1. Starting with a relatively high number of 10 lags, 2. Subsequently, reduce the number of lags until the last coefficient is significant different from zero on the 10 % level. 3. Compare the three different models (without drift and trend, with drift, and with drift and trend) by looking at the Akaike criterion. Then choose the model with the lowest Akaike criterion. 4. If the value of the test statistic is greater than (or in absolute values lesser than) the critical value, you cannot reject the I(1) null hypothesis on conventional significant levels. 19