Non-Stationary Time Series and Unit Root Testing

Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor

Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity Level Shifts and Structural Breaks Variance Changes Unit Roots 2 An Autoregressive Unit Root Process Stationary AR(1) Autoregression with a Unit Root Deterministic Terms Stationary vs. unit root AR(1) 3 Dickey-Fuller Unit Root Testing Test in an AR(1) Test in an AR(p) Test with a Constant Term Test with a Trend Term 4 Empirical Examples 5 Further Issues Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 2/35

1. Stationarity and Deviation from Stationarity

Stationarity The main assumption on the time series data so far has been stationarity. Recall the definition: Weak stationarity A time series is called weakly stationary if E(y t) = µ V (y t) = E((y t µ) 2 ) = γ 0 Cov(y t, y t k ) = E((y t µ) (y t k µ)) = γ k for k = 1, 2,... This can be violated in different ways. Examples of non-stationarity: 1 Deterministic trends (trend stationarity). 2 Level shifts. 3 Variance changes. 4 Unit roots (stochastic trends). Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 4/35

Four Non-Stationary Four Non-Stationary Time Series Time Series 10 (A) Stationary and trend-stationary process (B) Process with a level shift 5 y t 5 0 z t 0 0 50 100 150 200 (C) Process with a change in the variance 0 50 100 150 200 (D) Unit root process 5 10 0 5-5 0 0 50 100 150 200 0 50 100 150 200 4of28 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 5/35

(A) Trend-Stationarity Observation: Many macro-economic variables are trending. How should we model the relationship between trending variables? Assume that z t is stationary and that y t is z t plus a deterministic linear trend, e.g., Remarks: z t = θz t 1 + ɛ t, θ < 1, and y t = z t + µ 0 + µ 1t. 1 y t has a trending mean, E(y t) = µ 0 + µ 1 t, and is non-stationary. 2 The de-trended variable, z t = y t µ 0 µ 1 t, is stationary. y t is trend-stationary. 3 We may analyze the OLS-detrended variable, ẑ t = y t µ 0 µ 1 t. Standard asymptotics apply to regressions with ẑ t. 4 This is equivalent to extending the regression with a deterministic trend, e.g., y t = β 0 + β 1 x t + β 3 t + ɛ t. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 6/35

(B) Level Shifts and Structural Breaks Another type of non-stationarity is due to changes in parameters, e.g., a level shift: { µ1 for t = 1, 2,..., T 0 E(y t) = µ 2 for t = T 0 + 1, T 0 + 2,..., T. If each sub-sample is stationary, then there are two modelling approaches: 1 Include a dummy variable in the regression model, D t = { 0 for t = 1, 2,..., T0 1 for t = T 0 + 1, T 0 + 2,..., T y t = β 0 + β 1 x t + β 3 D t + ɛ t. If y t β 3 D t is stationary, standard asymptotics apply. 2 Analyze the two sub-samples separately. This is particularly relevant if we think that more parameters have changed. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 7/35

(C) Variance Changes A third type of non-stationary is related to changes in the variance. An example is y t = 0.5 y t 1 + ɛ t, where ɛ t { N(0, 1) for t = 1, 2,..., T0 N(0, 5) for t = T 0 + 1, T 0 + 2,..., T The interpretation is that the time series covers different regimes. A natural solution is to model the regimes separately. Alternatively we can try to model the variance. We return to so-called ARCH models for changing variance later. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 8/35

(D) Unit Roots (D) Unit Roots If there is a unit root in an autoregressive model, no standard asymptotics apply! If Consider there is athe unitdgp root in an autoregressive model, no standard asymptotics apply! Consider the DGP y t = θy t 1 + ɛ t, ɛ t N(0, 1), = 1 + (0 1) for t =1 = 1, 2 500, 2,..., 500, and and 0 =0. y 0 = Consider 0. Consider the distribution the distribution of b. of θ. Note: the theshape, location location and and variance variance of theofdistributions. distributions. (C) Distribution of ^ for =0.5 (D) Distribution of ^ for =1 10.0 7.5 Distribution of ^ N(s=0.0389) 100 Distribution of ^ N(s=0.00588) 5.0 50 2.5 0.0 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.97 0.98 0.99 1.00 1.01 8of28 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 9/35

2. An Autoregressive Unit Root Process

Properties of a Stationary AR(1) Consider the AR(1) model y t = θy t 1 + ɛ t, t = 1, 2,..., T. The characteristic polynomial is θ(z) = 1 θz, with characteristic root z 1 = θ 1 and inverse root φ 1 = z 1 1 = θ. The stationarity condition is that φ 1 = θ < 1. Recall the solution y t = ɛ t + θɛ t 1 + θ 2 ɛ t 2 +... + θ t 1 ɛ 1 + θ t y 0, where θ i 0 for i. Shocks have only transitory effects; y t has an attractor (mean reversion). Note the properties E(y t) = θ t y 0 0 V (y t) = σ 2 + θ 2 σ 2 + θ 4 σ 2 +... + θ t 1 σ ρ s = Corr(y t, y t s) = θ s σ2 1 θ 2 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 11/35

Simulated Example Simulated Example (A) Shock to a stationary process, = 0.8 (B) Shock to a unit root process, = 1 10 5 10 0 0 0 20 40 60 80 100 0 20 40 60 80 100 1.0 (C) ACF for stationary process, =0.8 1.0 (D) ACF for unit root process, =1 0.5 0.5 0 5 10 15 20 25 0 5 10 15 20 25 10 of 28 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 12/35

Autoregression with a Unit Root Consider the AR(1) with θ = 1, i.e., y t = y t 1 + ɛ t. The characteristic polynomial is θ(z) = 1 z. There is a unit root, θ(1) = 0. The solution is given by: y t = y 0 + y 1 + y 2 +... + y t 1 + ɛ t = y 0 + ɛ 1 + ɛ 2 +... + ɛ t t = y 0 + ɛ i. i=1 Note the remarkable differences between θ = 1 and a stationary process, θ < 1 : 1 The effect of the initial value, y 0, stays in the process. And E(y t y 0 ) = y 0. 2 Shocks, ɛ t, have permanent effects. Accumulate to a random walk component, ɛ i, called a stochastic trend. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 13/35

3 The variance increases, ( t V (y t y 0) = V The process is clearly non-stationary. i=1 4 The covariance, Cov(y t, y t s y 0), is given by ɛ i y 0 ) = t σ 2. E((y t y 0)(y t s y 0) y 0) = E((ɛ 1+ɛ 2+...+ɛ t)(ɛ 1+ɛ 2+...+ɛ t s) y 0) = (t s)σ 2. The autocorrelation is Corr(y t, y t s y 0) = which dies out very slowly with s. Cov(y t, y t s y 0) V (yt y 0) V (y = (t s)σ 2 t s y 0) tσ2 (t s)σ = t s, 2 t 5 The first difference, y t = ɛ t, is stationary. y t is called integrated of first order, I(1). Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 14/35

Deterministic Terms To model actual time series we include deterministic terms, e.g., y t = δ + θy t 1 + ɛ t. With a unit root, the terms accumulate! If θ < 1, the solution is y t = θ t y 0 + t θ i ɛ t i + (1 + θ + θ 2 +... + θ t 1 )δ, i=0 where the mean converges to (1 + θ + θ 2 +...)δ = δ/(1 θ). If θ = 1, the solution is y t = y 0 + t (δ + ɛ i) = y 0 + δt + i=1 t ɛ i. The constant term produces a deterministic linear trend: Random walk with drift. Note the parallel between a deterministic and a stochastic trend. i=1 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 15/35

Stationary vs. unit root AR(1) Properties of the processes Stationary AR(1) Unit root AR(1) Process y t = δ + θy t 1 + ɛ t, θ < 1, y t = δ + y t 1 + ɛ t, ɛ t IID(0, σ 2 ) ɛ t IID(0, σ 2 ) (1) MA Repr. y t = θ t y 0 + t 1 i=0 θi δ + t 1 i=0 θi ɛ t i y t = y 0 + δt + t i=1 ɛi (2) Role of y 0 Dies out Stays in the process (3) Role of δ Effect on the level Accummulates to a linear trend (4) Shocks Transitory effects Permanent effects (5) Variance Converges to a constant Increases with t (6) Autocorrelation Coverges exponentially to zero Converges slowly to zero Does not depend on t Depends on t (7) OLS estimator Consistent (Super)Consistent for θ Asymptotically normal Non-normal asymp. distribution Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 16/35

3. Dickey-Fuller Unit Root Testing

Unit Root Testing Estimate an autoregressive model and test whether θ(1) = 0, i.e., whether z = 1 is a root in the autoregressive polynomial. This is a straightforward hypothesis test! 1 Careful statistical model What kinds of deterministic components are relevant: constant or trend? 2 Hypothesis We compare two models, H 0 and H A. What are the properties of the model under the null and under the alternative. Are both models relevant? 3 Test statistic What is a relevant test statistic for H 0 H A? 4 Asymptotic distribution The asymptotic distribution for a unit root test is non-standard. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 18/35

Dickey-Fuller Test in an AR(1) Consider an AR(1) model y t = θy t 1 + ɛ t. The unit root hypothesis is θ(1) = 1 θ = 0. The one-sided test against stationarity: H 0 : θ = 1 against H A : 1 < θ < 1. An equivalent formulation is y t = πy t 1 + ɛ t, where π = θ 1 = θ(1). The hypothesis θ(1) = 0 translates into H 0 : π = 0 against H A : 2 < π < 0. The Dickey-Fuller (DF) test statistic is simply the t-ratio, i.e., τ = θ 1 = π se( θ) se( π) The asymptotic distribution is Dickey-Fuller, DF, and not N(0, 1). Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 19/35

Quantiles Distribution 1% 2.5% 5% 10% Quantile Distribution N(0, 1) 2.33 1% 1.96 2 5% 1.64 5% 1.28 10% (0 DF 1) 2 33 2.56 2.23 1 96 1.94 1 64 1.62 1 28 DF DF c 2 56 3.43 3.12 2 23 2.86 1 94 2.57 1 62 DF DF l 3 43 3.96 3.66 3 12 3.41 2 86 3.13 2 57 DF 3 96 3 66 3 41 3 13 0.6 (A) Dickey-Fuller distributions 0.4 DF l DF c DF N(0,1) 0.2 0.0-4 -2 0 2 4 16 of 28 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 20/35

Dickey-Fuller Test in an AR(p) For the AR(p) process we derive the Augmented Dickey-Fuller (ADF) test. Consider the case of p = 3 lags: y t = θ 1y t 1 + θ 2y t 2 + θ 3y t 3 + ɛ t. A unit root in θ(z) = 1 θ 1z θ 2z 2 θ 3z 3 corresponds to θ(1) = 0. To avoid testing a restriction on 1 θ 1 θ 2 θ 3, the model is rewritten as y t y t 1 = (θ 1 1)y t 1 + θ 2y t 2 + θ 3y t 3 + ɛ 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) + ɛ 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) + ɛ t y t = πy t 1 + c 1 y t 1 + c 2 y t 2 + ɛ t, where π = θ 1 + θ 2 + θ 3 1 = θ(1), c 1 = (θ 2 + θ 3), c 2 = θ 3. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 21/35

Dickey-Fuller Test in an AR(p) The hypothesis for θ(1) = 0 is unchanged: H 0 : π = 0 against H A : 2 < π < 0. The t test statistic τ π=0 again follows the DF-distribution. Remarks: 1 It is only the test for π = 0 that follows the DF distribution. Tests on c 1 and c 2 are N(0, 1). 2 We use the normal tools to determine the appropriate lag-length: general-to-specific testing or information criteria. Lag-length cannot be decided by Box-Jenkins identification (PACF). 3 Verbeek suggests to calculate the DF test for all values of p.... but why should we look at inferior or misspecified models? Find the best model and test in that. 4 The tests are based on the assumption that ɛ t IID(0, σ 2 ). This is checked by applying the usual misspecification tests. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 22/35

Dickey-Fuller Test with a Constant Term We need deterministic variables to model actual time series, E(y t) 0. The DF regression with a constant term (and p = 3 lags again) is y t = δ + πy t 1 + c 1 y t 1 + c 2 y t 2 + ɛ t. ( ) The hypothesis is unchanged H 0 : π = 0, and as a test statistic we can use Remarks: τ c = π se( π). 1 The constant term in the regression changes the asymptotic distribution. The relevant distribution, DF c, is shifted to the left of DF. 2 Under the null hypothesis, π = 0, the constant gives a trend in y t. We have { µ + stationary process for θ < 1 y t = y 0 + random walk + δt for θ = 1 That is not a natural comparison. We implicitly assume that δ = 0 if θ = 1. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 23/35

Dickey-Fuller Test with a Constant Term A more satisfactory hypothesis is H 0 : π = δ = 0, i.e., compare ( ) with y t = c 1 y t 1 + c 2 y t 2 + ɛ t. ( ) The joint hypothesis can be tested by a LR test, LR(π = δ = 0) = 2 (log L 0 log L A ), where log L 0 and log L A denote the log-likelihood values from ( ) and ( ). The LR statistic follows a non-standard distribution, DF 2 c, under the null. Quantiles Distribution 1% 2.5% 5% 10% χ 2 (2) 9.21 7.38 5.99 4.61 DF 2 c 12.73 10.73 9.13 7.50 DF 2 l 16.39 14.13 12.39 10.56 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 24/35

Dickey-Fuller Test with a Trend Term For trending variables, the relevant alternative is often trend-stationarity. We use y t = δ + γt + πy t 1 + c 1 y t 1 + c 2 y t 2 + ɛ t. (#) The hypothesis is still H 0 : π = 0, and the DF t test is τ l = π se( π). The presence of a trend shifts the asymptotic distribution, DF l, further to the left. For the AR(1) with θ = 1, y t = δ + γt + y t 1 + ɛ t = ( δ + γ ) t + γ 2 2 t2 + t ɛ i + y 0. To avoid the accumulation of the trend under π = 0, we may consider the joint hypothesis, H 0 : π = γ = 0, i.e., to compare (#) with i=1 y t = δ + c 1 y t 1 + c 2 y t 2 + ɛ t. (##) The LR test is LR(π = γ = 0) = 2 (log L 0 log L A ), which follows a DF 2 l. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 25/35

4. Empirical Examples

Empirical Example: Danish Bond Rate 0.200 0.175 0.150 0.125 Bond rate r First difference r 0.100 0.075 0.050 0.025 0.000-0.025 1970 1975 1980 1985 1990 1995 2000 2005 2010 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 27/35

Empirical Example: Danish Bond Rate An AR(4) model for 1972 : 1 2010 : 4 gives (t values): r t = 0.0069r t 1 + 0.3945 r t 1 0.0152 r t 2 0.0795 r t 3 + 0.0003. ( 0.66) (4.86) ( 0.17) ( 0.97) (0.29) Removing insignificant terms produces a model r t = 0.0091r t 1 + 0.3832 r t 1 + 0.0006, ( 0.88) (5.08) (0.52) with log L A = 567.72. The DF t test is τ c = 0.88. We do not reject the hypothesis of a unit root (DF c = 2.86 at a 5% level). To test the joint hypothesis, H0 : π = δ = 0, we estimate the model under the null, r t = 0.3787 r t 1, (5.09) with log L 0 = 567.14. The LR test is given by LR(π = δ = 0) = 2 (log L 0 log L A ) = 2 (567.14 567.72) = 1.16, where the 5% critical value is 9.13 in DF 2 c. The LR test does not reject the null of a unit root (1.16 9.13 = DF 2 c at 5%) Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 28/35

Empirical Empirical Examples: Examples: Trend-Stationarity Trend-Stationarity 1.00 (A) Log of Danish productivity 6.4 (B) Log of Danish private consumption 0.75 6.2 0.50 0.25 6.0 1970 1980 1990 2000 (C) Productivity, deviation from trend 1970 1980 1990 2000 (D) Consumption, deviation from trend 0.05 0.05 0.00 0.00-0.05 1970 1980 1990 2000-0.05 1970 1980 1990 2000 24 of 28 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 29/35

Empirical Example: Danish Productivity To test whether log-productivity is trend-stationary we use an AR(1) regression LPROD t = 0.091 (6.58) + 0.0024 (6.15) t 0.439LPROD t 1 + ɛ t, ( 6.22) with log L A = 366.09. The DF t test is given by τ l = 6.22 3.96 (1% critical value). Here we reject the hypothesis of a unit root and conclude that productivity is trend-stationary. To test the joint hypothesis, H0 : π = γ = 0, we run the regression under the null LPROD t = 0.0057 + ɛ t, (3.48) with log L 0 = 348.63. The LR test is LR(π = γ = 0) = 2 (log L 0 log L A ) = 2 (348.63 366.09) = 34.92. This is larger than the critical value of 12.39 in DF 2 l the unit root hypothesis. at 5%, so we reject Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 30/35

Empirical Example: Danish Consumption To test if log-consumption is trend-stationary we use the regression LCONS t = 0.764 (2.57) + 0.0004 (2.58) t 0.129LCONS t 1 0.209 LCONS t 1 + ɛ t, ( 2.56) ( 2.43) with log L A = 359.23. The Dickey-Fuller t test is given by τ l = 2.56, which is not significant in the DF l distribution (5% critical value is 3.41). We conclude that private consumption seems to have a unit-root. To test the joint hypothesis, H0 : π = γ = 0, we use the regression under the null, LCONS t = 0.0046 0.274 LCONS t 1 + ɛ t, (2.97) ( 3.29) with log L 0 = 355.87. The LR test for a unit root is given by LR(π = γ = 0) = 2 (log L 0 log L A ) = 2 (355.87 359.23) = 6.72, which is smaller than 12.39 (critical value DF 2 l at 5% level), so we cannot reject the hypothesis of a unit root in consumption. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 31/35

5. Further Issues

The Problem of Low Power The Problem of Low Power It isitdifficult difficult to distinguish to distinguish unit unit roots roots from from large large stationary stationary roots. roots. Always Always be careful be careful in conclusions. in conclusions. Consider time series generated from the two models Consider time series generated from the two models y t = 0.2 y t 1 + 0.05 t + ɛ t x t = 0 2 0.25 + 1 ɛ t. +0 05 + = 0 25 + Hard to tell apart in practice. We need many observations to be sure. Hard to tell apart in practice. We need many observations to be sure. Graphs of Y t: 30 20 (A) Trend-stationary and unit root process Y t = 0.2 Y t 1 0.05 t t Y t = 0.25 t 100 (B) Trend-stationary and unit root process Y t = 0.2 Y t 1 0.05 t t Y t = 0.25 t 10 50 0 0 0 20 40 60 80 100 0 100 200 300 400 500 27 of 28 Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 33/35

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. That may bias the conclusion towards a unit root. 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. Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 34/35

Summary of the Tests Testing for a unit root in an autoregressive model: Model Test Statistic DF LR y t = πy t 1 + ɛ t DF DF 2 y t = πy t 1 + c 1 y t 1 + c 2 y t 2 + ɛ t DF DF 2 y t = δ + πy t 1 + c 1 y t 1 + c 2 y t 2 + ɛ t DF c DF 2 c y t = δ + γ t + πy t 1 + c 1 y t 1 + c 2 y t 2 + ɛ t DF l DF 2 l The null hypothesis is always: For the (augmented) Dicky-Fuller (DF) test: H 0 : π = 0 vs. H A = 2 < π < 0. For the likelihood-ratio (LR) test: H 0 : π = 0, or H 0 : π = δ = 0, or H 0 : π = γ = 0. Tests on the remaining coefficients c 1 and c 2 are N(0, 1). Econometrics II Non-Stationary Time Series and Unit Root Testing Slide 35/35