# Non-Stationary Time Series andunitroottests

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1 Econometrics 2 Fall 2005 Non-Stationary Time Series andunitroottests Heino Bohn Nielsen 1of25 Introduction Many economic time series are trending. Important to distinguish between two important cases: (1) A stationary process with a deterministic trend: Shocks have transitory effects. (2) A process with a stochastic trend or a unit root: Shocks have permanent effects. Why are unit roots important? (1) Interesting to know if shocks have permanent or transitory effects. (2) It is important for forecasting to know if the process has an attractor. (3) Stationarity was required to get a LLN and a CLT to hold. For unit root processes, many asymptotic distributions change! Later we look at regressions involving unit root processes: spurious regression and cointegration. 2of25

2 Outline of the Lecture (1) Difference between trend stationarity and unitrootprocesses. (2) Unit root testing. (3) Dickey-Fuller test. (4) Caution on deterministic terms. (5) An alternative test (KPSS). 3of25 Trend Stationarity Consider a stationary AR(1) model with a deterministic linear trend term: Y t = θy t 1 + δ + γt + t, t =1, 2,..., T, ( ) where θ < 1, andy 0 is an initial value. The solution for Y t (MA-representation) has the form Note that the mean, Y t = θ t Y 0 + µ + µ 1 t + t + θ t 1 + θ 2 t 2 + θ 3 t E[Y t ]=θ t Y 0 + µ + µ 1 t µ + µ 1 t for T, contains a linear trend, while the variance is constant: V [Y t ]=V [ t + θ t 1 + θ 2 t ] =σ 2 + θ 2 σ 2 + θ 4 σ = σ2 1 θ 2. 4of25

3 The original process, Y t,isnotstationary. The deviation from the mean, y t = Y t E[Y t ]=Y t µ µ 1 t is a stationary process. The process Y t is called trend-stationary. The stochastic part of the process is stationary and shocks have transitory effects We say that the process is mean reverting. Also, we say that the process has an attractor, namely the mean, µ + µ 1 t. We can analyze deviations from the mean, y t. From the Frisch-Waugh theorem this is the same as a regression including a trend. 5of25 Shock to a Trend-Stationarity Process 15.0 y t =0.8 y t 1 +ε t 12.5 Y t = y t t of25

4 Unit Root Processes Consider the AR(1) model with a unit root, θ =1: Y t = Y t 1 + δ + t, t =1, 2,..., T, ( ) or Y t = δ + t, where Y 0 is the initial value. Note that z =1is a root in the autoregressive polynomial, θ(l) =(1 L). θ(l) is not invertible and Y t is non-stationary. The process Y t is stationary. We denote Y t adifference stationary process. If Y t is stationary while Y t is not, Y t is called integrated of first order, I(1). A process is integrated of order d, I(d), if it contains d unit roots. 7of25 The solution for Y t is given by Y t = Y 0 + with moments tx tx Y i = Y 0 + (δ + i )=Y 0 + δt + i=1 i=1 E[Y t ]=Y 0 + δt and V [Y t ]=t σ 2 tx i, i=1 Remarks: (1) The effect of the initial value, Y 0, stays in the process. (2) The innovations, t,areaccumulatedtoarandom walk, P i. This is denoted a stochastic trend. Note that shocks have permanent effects. (3) The constant δ is accumulated to a linear trend in Y t. The process in ( ) is denoted a random walk with drift. (4) ThevarianceofY t grows with t. (5) The process has no attractor. 8of25

5 Shock to a Unit Root Process 2 y t =y t 1 +ε t of25 Unit Root Tests Agoodwaytothinkaboutunitroottests: We compare two relevant models: H 0 and H A. (1) What are the properties of the two models? (2) Do they adequately describe the data? (3) Which one is the null hypothesis? Consider two alternative test: (1) Dickey-Fuller test: H 0 is a unit root, H A is stationarity. (2) KPSS test: H 0 is stationarity, H A is a unit root. Often difficult to distinguish in practice (Unit root tests have low power). Many economic time series are persistent, but is the root 0.95 or 1.0? 10 of 25

6 The Dickey-Fuller (DF) Test Idea: Setupanautoregressivemodelfory t and test if θ(1) = 0. Consider the AR(1) regression model y t = θy t 1 + t. The unit root null hypothesis against the stationary alternative corresponds to H 0 : θ =1 against H A : θ<1. Alternatively, the model can be formulated as y t =(θ 1)y t 1 + t = πy t 1 + t, where π = θ 1=θ(1). The unit root hypothesis translates into H 0 : π =0 against H A : π<0. 11 of 25 The Dickey-Fuller (DF) test is simply the t test for H 0 : bτ = b θ 1 se( b θ) = bπ se(bπ). The asymptotic distribution of bτ is not normal! The distribution depends on the deterministic components. In the simple case, the 5% critical value (one-sided) is 1.95 and not Remarks: (1) The distribution only holds if the errors t are IID (check that!) If autocorrelation, allow more lags. (2) In most cases, MA components are approximated by AR lags. The distribution for the test of θ(1) = 0 alsoholdsinanarmamodel. 12 of 25

7 Augmented Dickey-Fuller (ADF) test The DF test is extended to an AR(p) model. Consider an AR(3): y t = θ 1 y t 1 + θ 2 y t 2 + θ 3 y t 3 + t. Aunitrootinθ(L) =1 θ 1 L θ 2 L 2 θ 3 L 3 corresponds to θ(1) = 0. The test is most easily performed by rewriting the model: y t y t 1 = (θ 1 1)y t 1 + θ 2 y t 2 + θ 3 y t 3 + t y t y t 1 = (θ 1 1)y t 1 +(θ 2 + θ 3 )y t 2 + θ 3 (y t 3 y t 2 )+ t y t y t 1 = (θ 1 + θ 2 + θ 3 1)y t 1 +(θ 2 + θ 3 )(y t 2 y t 1 )+θ 3 (y t 3 y t 2 )+ t y t = πy t 1 + c 1 y t 1 + c 2 y t 2 + t, where π = θ 1 + θ 2 + θ 3 1= θ(1) c 1 = (θ 2 + θ 3 ) c 2 = θ of 25 The hypothesis θ(1) = 0 again corresponds to H 0 : π =0 against H A : π<0. The t test for H 0 is denoted the augmented Dickey-Fuller (ADF) test. To determine the number of lags, k, we can use the normal procedures. General-to-specific testing:startwithk max and delete insignificant lags. Estimate possible models and use information criteria. Make sure there is no autocorrelation. Verbeek suggests to calculate the DF test for all values of k. This is a robustness check, but be careful! Why would we look at tests based on inferior/misspecified models? An alternative to the ADF test is to correct the DF test for autocorrelation. Phillips-Perron non-parametric correction based on HAC standard errors. Quite complicated and likely to be inferior in small samples. 14 of 25

8 Deterministic Terms in the DF Test The deterministic specification is important: We want an adequate model for the data. The deterministic specification changes the asymptotic distribution. If the variable has a non-zero level, consider a regression model of the form Y t = πy t 1 + c 1 y t 1 + c 2 y t 2 + δ + t. The ADF test is the t test, bτ c = bπ/se(bπ). The critical value at a 5% level is If the variable has a deterministic trend, consider a regression model of the form Y t = πy t 1 + c 1 y t 1 + c 2 y t 2 + δ + γt + t. The ADF test is the t test, bτ t = bπ/se(bπ). The critical value at a 5% level is of 25 The DF Distributions DF with a linear trend, τ t 0.5 DF with a constant, τ c DF, τ 0.4 N(0,1) of 25

9 Empirical Example: Danish Bond Rate Bond rate, r t First difference, r t of 25 An AR(4) model gives r t = r t r t r t r t ( 0.672) (4.49) ( 0.199) ( 0.817) (0.406) Removing insignificant terms produce a model r t = r t r t ( 0.911) (4.70) (0.647) The 5% critical value (T = 100) is 2.89, so we do not reject the null of a unit root. We can also test for a unit root in the first difference. Deleting insignificant terms we find a preferred model 2 r t = r t ( 7.49) ( 0.534) Here we safely reject the null hypothesis of a unit root ( ). Basedonthetestweconclucethatr t is non-stationary while r t is stationary. That is r t I(1) 18 of 25

10 A Note of Caution on Deterministic Terms The way to think about the inclusion of deterministic terms is via the factor representation: y t = θy t 1 + t Y t = y t + µ It follows that (Y t µ) = θ(y t 1 µ)+ t Y t = θy t 1 +(1 θ)µ + t Y t = θy t 1 + δ + t which implies a common factor restriction. If θ =1, then implicitly the constant should also be zero, i.e. δ =(1 θ)µ =0. 19 of 25 The common factor is not imposed by the normal t test. Consider Y t = θy t 1 + δ + t. The hypotheses H 0 : θ =1 against H A : θ<1, imply H A : Y t = µ + stationary process H 0 : Y t = Y 0 + δt + stochastic trend. We compare a model with a linear trend against a model with a non-zero level! Potentially difficult to interpret. 20 of 25

11 A simple alternative is to consider the combined hypothesis H 0 : π = δ =0. The hypothesis H 0 can be tested by running the two regressions H A : Y t = πy t 1 + δ + t H0 : Y t = t, and perform a likelihood ratio test µ RSS0 τ LR = T log = 2(loglik RSS 0 loglik A ), A where RSS 0 and RSS A denote the residual sum of squares. The 5% critical value is of 25 Same Point with a Trend Thesamepointcouldbemadewithatrendterm Y t = πy t 1 + δ + γt + t. Here, the common factor restriction implies that if π =0then γ =0. Since we do not impose the restriction under the null, the trend will accumulate. A quadratic trend is allowed under H 0,butonlyalinear trend under H A. A solution is to impose the combined hypothesis H 0 : π = γ =0. This is done by running the two regressions H A : Y t = πy t 1 + δ + γt + t H0 : Y t = δ + t, and perform a likelihood ratio test. The 5% critical value for this test is of 25

12 Special Events Unit root tests assess whether shocks have transitory or permanent effects. The conclusions are sensitive to a few large shocks. Consider a one-time change in the mean of the series, a so-called break. This is one large shock with a permanent effect. Even if the series is stationary, such that normal shocks have transitory effects, the presence of a break will make it look like the shocks have permanent effects. Thatmaybiastheconclusiontowardsaunitroot. Consider a few large outliers, i.e. a single strange observations. The series may look more mean reverting than it actually is. That may bias the results towards stationarity. 23 of 25 A Reversed Test: KPSS Sometimes it is convenient to have stationarity as the null hypothesis. KPSS (Kwiatkowski, Phillips, Schmidt, and Shin) Test. Assume there is no trend. The point of departure is a DGP of the form Y t = ξ t + e t, where e t is stationary and ξ t is a random walk, i.e. ξ t = ξ t 1 + v t, v t IID(0,σ 2 v). Ifthevarianceiszero,σ 2 v =0,thenξ t = ξ 0 for all t and Y t is stationary. Use a simple regression: Y t = bµ + be t, ( ) to find the estimated stochastic component. Under the null, be t is stationary. That observation can be used to design a test: H 0 : σ 2 v =0 against H A : σ 2 v > of 25

13 The test statistic is given by KPSS = 1 P T T t=1 S2 t 2 bσ 2, where S t = P t s=1 be s is a partial sum; bσ 2 is a HAC estimator of the variance of be t. (This is an LM test for constant parameters against a RW parameter). Theregressionin( ) can be augmented with a linear trend. Critical values: Deterministic terms Critical values in regression ( ) Constant Constant and trend Can be used for confirmatory analysis: KPSS: Rejection of I(0) Non-rejection of I(0) DF: Rejection of I(1)? I(0) Non-rejection of I(1) I(1)? 25 of 25

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