Nonstationary Time Series:

Transcription

1 Nonstationary Time Series: Unit Roots Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board Summer School in Financial Mathematics Faculty of Mathematics & Physics University of Ljubljana September 14 19, 2009

2 Introduction Spurious Regression So far our analysis has been confined to covariance-stationary processes. However, many economic time series trend upward over time: Prices and nominal quantities Real economic variables (e.g., output, consumption, etc.) Financial asset prices (e.g., stock prices, nominal FX, etc.)

3 Example: Random Walk Model of Stock Prices Time series process {p t }: p t = p t 1 + ɛ t p t = log price of a particular stock at date t {ɛ t } = white noise process (covariance-stationary) p 0 = ln(ipo) 1-step ahead forecast at the forecast origing h: ˆp h (1) = E(p h+1 p h, p h 1,..., p 0 ) = p h k-step ahead forecast at the forecast origing h: ˆp h (k) = E(p h+k p h, p h 1,..., p 0 ) = p h, k > 0

4 Example: Random Walk Model of Stock Prices (cont.) Implications: {p t } is not mean-reverting MA representation: p t = ɛ t + ɛ t 1 + k-step ahead forecast error: e h (k) = ɛ h+k + + ɛ h+1 Var[e h (k)] = kσ 2 ɛ as k Usefulness of point forecast ˆp h (k) diminishes as k increases (i.e., stock prices are unpredictable) Unconditional variance of {p t } is unbounded (plausible for individual stocks; for stock price indexes?)

5 Example: Random Walk with Drift Model of Stock Prices Log return series of a market index tend to have a small and positive mean: p t = µ + p t 1 + ɛ t µ = drift of the model µ = E(p t p t 1 ) = E(r t ) = expected return Implications: Random walk with drift model implies: p t = µt + p 0 + ɛ t + ɛ t ɛ 1 µt = time trend component t i=1 ɛ i = random-walk component Conditional std. deviation of p t is tσ ɛ, which grows at a slower rate than the conditional expectation of p t.

6 Basic Nonstationary Processes Many trending economic/financial time series can be broadly characterized by one of the two following models: Trend Stationary: y t = γ 0 + γ 1 t + u t RW with Drift: y t = δ 1 + y t 1 + u t {u t } t= = covariance-stationary ARMA process Both series will exhibit trending behavior. In the trend-stationary case, detrending y t will produce a variable that is covariance stationary. In the random-walk with drift case, first differencing (i.e., y t = δ 1 + u t ) will produce a variable that is covariance stationary.

7 Spurious Regression Consider two trend-stationary series: y t = γ 0 + γ 1 t + u y,t x t = δ 0 + δ 1 t + u x,t {u s,t } = covariance-stationary ARMA processes Regression: y t = β 0 + β 1 x t + ɛ t Regression of y t on x t is very likely to find a significant relationship between the two series. The only thing they may have in common is the upward trend. There may be no correlation between the stochastic parts of y t and x t.

8 Spurious Regression Consider two random walk without drift processes: y t = y t 1 + u y,t x t = x t 1 + u x,t {u s,t } = independent Gaussian white noise processes Regression: y t = β 0 + β 1 x t + ɛ t Surely there is no significant relationship between the two series. They do not have any common trends. There is no correlation between the stochastic parts of y t and x t.

9 Integrated Series Spurious Regression A series that follows a random walk (with or without a drift) is integrated of order one, denoted by I(1). I(1) series must be differenced once in order to make it stationary. Covariance-stationary series is I(0). One might occasionally run into series that is I(2). If one mistakenly differences an I(0) series, the result will be an I( 1) series. Vast majority of the time, economists deal with time series that are either I(0) or I(1). How to decided whether or not a series needs to be differenced or detrended?

10 Testing for Unit Roots How to choose between the following two DGPs? Trend Stationary: y t = γ 0 + γ 1 t + u t RW with Drift: y t = δ 1 + y t 1 + u t {u t } t= = white noise process In the trend-stationary case, linear detrending will transform y t into a stationary process. In the random-walk with drift case, first-differencing will transform y t into a stationary process.

11 Testing for Unit Roots (cont.) Nested specification: y t = γ 0 + γ 1 t + α[y t 1 γ 0 γ 1 (t 1)] + u t If α < 1 trend-stationary model If α = 1 random-walk with drift model Convenient reparametrization: y t = β 0 + β 1 t + αy t 1 + u t, β 0 γ 0 (1 α) + γ 1 α; β 1 γ 1 (1 α). Subtracting y t 1 from both sides: y t = β 0 + β 1 t + (α 1)y t 1 + u t, If α < 1 trend-stationary specification if α = 1 random walk with drift specification

12 Testing for Unit Roots (cont.) Unit root test: H 0 : (α 1) = 0 H A : (α 1) < 0 y t = β 0 + β 1 t + (α 1)y t 1 + u t, Standard t-test of the null hypothesis that α = 1 against a one-sided alternative that α < 1. Key points: Cannot use the ordinary t-statistic for (α 1) = 0. Under the null hypothesis, α = 1, and the process generating y t is I(1)! This means that the regressor y t 1 does not satisfy the standard assumptions needed for asymptotic analysis. t-statistic does not have a N(0, 1) distribution asymptotically.

13 Dickey Fuller DF unit root test: y t = (α 1)y t 1 + u t DGP: y t = u t y t = β 0 + (α 1)y t 1 + u t DGP: y t = γ 0 + u t y t = β 0 + β 1 t + (α 1)y t 1 + u t DGP: y t = γ 0 + γ 1 t + u t H 0 : (α 1) = 0 H A : (α 1) < 0 Test of the null hypothesis that α = 1 against a one-sided alternative that α < 1. Tests based on DF τ-statistics. Critical values for τ nc, τ c, and τ ct tabulated using using Monte Carlo simulations.

14 Serial Correlation Spurious Regression One important feature of the asymptotic results for unit root processes is that they do not depend on the assumption that the variances of the error terms u t are constant: The asymptotic distributions of the unit root τ-statistics are the same under heteroscedasticity of unknown form as under homoscedasticity. But it is essential that there be no correlation between u t and u t j for all j 0. Thus, the τ-statistics are not valid when the disturbances are serially correlated. Untenable assumption when using economic or financial variables, because these are often highly autocorrelated, thus, making it very likely that the error terms will be serially correlated. Need unit root tests that are (asymptotically) valid in the presence of serial correlation.

15 Augmented Dickey-Fuller Tests Standard DF regression: y t = x tβ + (α 1)y t 1 + u t x t = all nonstochastic regressors (i.e., constant, trend, etc.) u t = ρu t 1 + ɛ t stationary AR(1) process Quasi-differencing (and adding & subtracting αρy t 1 ): y t = x tβ + [(α 1)(1 ρ)]y t 1 + αρ y t 1 + ɛ t We were able to replace x tβ ρ(x tβ) by x tβ for some choice β, because x t contains only deterministic variables, so that each element of β is a linear combination of the elements of β. Adding y t 1 to the regression causes the serially correlated term u t to be replaced by the serially uncorrelated term ɛ t. Obviously generalizes to higher-order AR processes.

16 Serial Correlation (cont.) But what if the error terms followed an MA or an ARMA process? Cannot add an infinite number of lagged values of y t to the test regression. One can validly use ADF tests even when there is a MA component in the error process, provided that the number of lags of y t that are included tends to infinity at an appropriate rate (no faster than T 1/3 ). In practice, T is fixed and does not tend to infinity, so knowing the critical rate T 1/3 if no help.

17 Choosing the Lag Length p for the ADF Test If lag length p is too small, the remaining serial correlation in the errors will bias the test. If lag length p is to big, the power of the test will be adversly affected. Monte Carlo evidence indicates that it is better to err on the side of including too many lags.

18 Ng & Peron Procedure (JASA 1995) Set an upper bound p max for p. Estimate the ADF regression with p = p max. If absolute t-statistic for testing the significance of the last lagged difference is greater than 1.6, set p = p max and perform the unit root test. Otherwise, reduce the lag length by one and repeat the process.

19 Some Problems with ADF ADF tests have very low power against I(0) alternatives that are close to being I(1). The power of ADF tests diminishes significantly as deterministic terms are added to the test regression. For maximum power against very persistent alternatives the efficient unit root tests should be used: GLS-ADF test of Eliot, Rothenberg & Stock (Econometrica 1996) Useful to also test for the null hypothesis of stationarity: KPSS test of Kwiatowski, Phillips, Schmidt & Shin (J. of Econometrics 1992)