Ch. 19 Models of Nonstationary Time Series In time series analysis we do not confine ourselves to the analysis of stationary time series. In fact, most of the time series we encounter are nonstationary. How to deal with the nonstationary data and use what we have learned from stationary model are the main subjects of this chapter 1 Integrated Process Consider the following two process oXt-1+t,|o<1: where ut and ut are mutually uncorrelated white noise process with variance o and of, respectively. Both Xt and Yt are Ar(1) process. The difference between two models is that Yt is a special case of a Xt process when =1 and is called a random walk process. It is also refereed to as a ar(1) model with a unit root since the root of the AR(1)process is 1. When we consider the statistical behavior of the two processes by investigating the mean(the first moment), and the variance and autocovariance(the second moment ), they are completely different. Although the two process belong to the same AR(1) class, Xt is a stationary process, while yt is a nonstationary process Assume that tE T*, T*=10, 1, 2,1,I the two stochastic pr rocesses can be expressed ad Similarly. in the unit root case 0 Suppose that the initial observation is zero, Xo=0 and Yo=0. The means of the two (Xt) I This assumption is required to derive the convergence of integrated process to standard Brownian Motion. A standard Brown Motion is defined on t E0, 1
Ch. 19 Models of Nonstationary Time Series In time series analysis we do not confine ourselves to the analysis of stationary time series. In fact, most of the time series we encounter are nonstationary. How to deal with the nonstationary data and use what we have learned from stationary model are the main subjects of this chapter. 1 Integrated Process Consider the following two process Xt = φXt−1 + ut , |φ| < 1; Yt = Yt−1 + vt , where ut and vt are mutually uncorrelated white noise process with variance σ 2 u and σ 2 v , respectively. Both Xt and Yt are AR(1) process. The difference between two models is that Yt is a special case of a Xt process when φ = 1 and is called a random walk process. It is also refereed to as a AR(1) model with a unit root since the root of the AR(1) process is 1. When we consider the statistical behavior of the two processes by investigating the mean (the first moment), and the variance and autocovariance (the second moment), they are completely different. Although the two process belong to the same AR(1) class, Xt is a stationary process, while Yt is a nonstationary process. Assume that t ∈ T ∗ , T ∗ = {0, 1, 2, ...}, 1 the two stochastic processes can be expressed ad Xt = φ tX0 + X t−1 i=0 φ i ut−i . Similarly, in the unit root case Yt = Y0 + X t−1 i=0 vt−i . Suppose that the initial observation is zero, X0 = 0 and Y0 = 0. The means of the two process are E(Xt) = 0 and E(Yt) = 0, 1This assumption is required to derive the convergence of integrated process to standard Brownian Motion. A standard Brown Motion is defined on t ∈ [0, 1]. 1
and variances are var(x)=∑var(m-)→ Var Var(t-)=t·σ The autocovariance of the two series are E(XtXt-n out E[(u+2u-1+…+u-r+…+9-1u1)(u1-+2u-r-1+…+d-r-hu) ∑φ E(YYt-r)=E Ut-T-i Ut+Ut u1( (t-r) We may expect that the autocorrelation functions are and The means of Xt and Yt are the same, but the variances(including autoco- variance) are different. The important thing to note is that the variances and
and variances are V ar(Xt) = X t−1 i=0 φ 2iV ar(ut−i) −→ 1 1 − φ2 σ 2 u and V ar(Yt) = X t−1 i=0 V ar(vt−i) = t · σ 2 v . The autocovariance of the two series are γ X τ = E(XtXt−τ ) = E " X t−1 i=0 φ iut−i ! t−Xτ−1 i=0 φ iut−τ−i !# = E[(ut + φ 1ut−1 + ... + φ τut−τ + ... + φ t−1u1)(ut−τ + φ 1ut−τ−1 + ... + φ t−τ−1u1) = t−Xτ−1 i=0 φ iφ τ+i σ 2 u = σ 2 uφ τ ( t−Xτ−1 i=0 φ 2i ) −→ φ τ 1 − φ 2 σ 2 u = φ τ γ X 0 . and γ Y τ = E(YtYt−τ ) = E " X t−1 i=0 vt−i ! t−Xτ−1 i=0 vt−τ−i !# = E[(vt + vt−1 + ... + vt−τ + vt−τ−1 + ... + v1)(vt−τ + vt−τ−1 + ... + v1)] = (t − τ )σ 2 v . We may expect that the autocorrelation functions are r X τ = γ X τ γ X 0 = φ τ −→ 0 and r Y τ = γ Y τ γ Y 0 = (t − τ ) t −→ 1 ∀ τ. The means of Xt and Yt are the same, but the variances (including autocovariance) are different. The important thing to note is that the variances and 2
autocovariance of Yt are function of t, while those of Xt converge to a constant asymptotically. Thus as t increase the variance of Yt increase, while the variance of Xt converges to a constant If we add a constant to the AR(1)process, then the means of two processes also behave differently. Consider the AR(1) process with a constant(or drift )as follows Xt=a+oXt-1+ut, o<1 Yt=a+Yt-1+Ut The successive substitution yield Xt=oXo+a>o+>but-i and Yt= Yo +at+>ut-1 (1) Note that Yt contains a(deterministic) trend t. If the initial observations are zero, Xo=0 and Yo=0, then the means of two process are E(XL E() but the variances and the autocovariance are the same as those derived from aR(1) model without the constant. By adding a constant to the Ar(1)pro- cesses, the means of two processes as well the variance are different. Both mean and variance of Yt are time varying, while those of Xt are constant Since the variance(the second moment)and even mean(the first moment)of the nonstationary series is not constant over time, the conventional asymptotic theory cannot be applied for these series(Recall the moment condition in CLT on p. 22 of Ch. 4)
autocovariance of Yt are function of t, while those of Xt converge to a constant asymptotically. Thus as t increase the variance of Yt increase, while the variance of Xt converges to a constant. If we add a constant to the AR(1) process, then the means of two processes also behave differently. Consider the AR(1) process with a constant (or drift) as follows Xt = α + φXt−1 + ut , |φ| < 1 and Yt = α + Yt−1 + vt . The successive substitution yields Xt = φ tX0 + α X t−1 i=0 φ i + X t−1 i=0 φ iut−i and Yt = Y0 + αt + X t−1 i=0 vt−i . (1) Note that Yt contains a (deterministic) trend t. If the initial observations are zero, X0 = 0 and Y0 = 0, then the means of two process are E(Xt) −→ α 1 − φ E(Yt) = αt but the variances and the autocovariance are the same as those derived from AR(1) model without the constant. By adding a constant to the AR(1) processes, the means of two processes as well the variance are different. Both mean and variance of Yt are time varying, while those of Xt are constant. Since the variance (the second moment) and even mean (the first moment) of the nonstationary series is not constant over time, the conventional asymptotic theory cannot be applied for these series (Recall the moment condition in CLT on p.22 of Ch. 4). 3
2 Deterministic Trend and stochastic Trend Many economic and financial times series do trended upward over time(such as GNP, M2, Stock Index etc. See the plots of Hamilton, p. 436. For a long time each trending (nonstationary) economic time series has been decomposed into a deterministic trend and a stationary process. In recent years the idea of stochas- tic trend has emerged, and enriched the framework of analysis to investigate economic time series 2.1 Detrending methods 2.1.1 Differencing-Stationary One of the easiest ways to analyze those nonstationary-trending series is to make those series stationary by differencing. In our example, the random walk series with drift Yt can be transformed to a stationary series by differencing once AY=Yi-Yi-1=(1-LY=a+ut Since Ut is assumed to be a white noise process, the first difference of Yt is sta- tionary. The variance of AYt is constant over the sample period. In the I(1) Yt=Yo +at+>ut-i i=0 at is a deterministic trend while 2i=o vt-i is a stochastic trend When the nonstationary series can be transformed to the stationary series by differencing once, the series is said to be integrated of order 1 and is denoted by I(1), or in common, a unit root process. If the series needs to be differenced d times to be stationery, then the series is said to be I(d). The I(d) series(d+0) is also called a dif ferencing- stationary process (DSP). When(1-L is a stationary and invertible series that can be represented by an aRMA(p, g) model. i.e (1-1L-2L2-…-qnDP)(1-DY1=a+(1+1L+2L2+…+L)et(3) o(L)△4Y1=a+6(L)t
2 Deterministic Trend and Stochastic Trend Many economic and financial times series do trended upward over time (such as GNP, M2, Stock Index etc.). See the plots of Hamilton, p.436. For a long time each trending (nonstationary) economic time series has been decomposed into a deterministic trend and a stationary process. In recent years the idea of stochastic trend has emerged, and enriched the framework of analysis to investigate economic time series. 2.1 Detrending Methods 2.1.1 Differencing-Stationary One of the easiest ways to analyze those nonstationary-trending series is to make those series stationary by differencing. In our example, the random walk series with drift Yt can be transformed to a stationary series by differencing once 4Yt = Yt − Yt−1 = (1 − L)Yt = α + vt . Since vt is assumed to be a white noise process, the first difference of Yt is stationary. The variance of 4Yt is constant over the sample period. In the I(1) process, Yt = Y0 + αt + X t−1 i=0 vt−i , (2) αt is a deterministic trend while Pt−1 i=0 vt−i is a stochastic trend. When the nonstationary series can be transformed to the stationary series by differencing once, the series is said to be integrated of order 1 and is denoted by I(1), or in common, a unit root process. If the series needs to be differenced d times to be stationery, then the series is said to be I(d). The I(d) series (d 6= 0) is also called a differencing − stationary process (DSP). When (1 − L) dYt is a stationary and invertible series that can be represented by an ARMA(p, q) model, i.e. (1 − φ1L − φ2L 2 − ... − φpL p )(1 − L) dYt = α + (1 + θ1L + θ2L 2 + ... + θqL q )εt (3) or φ(L)4dYt = α + θ(L)εt , 4
where all the roots of o(L)=0 and 0(L)=0 lie outside the unit circle, we say that Yt is an autoregressive integrated moving-average ARIMA(p, d, q) process In particular an unit root process, d= l or an ARIMA(p, 1, g) process is therefore o(L)△Y=a+b(L)et (1- LY=a+v(LEt where v(l)=o(L)e(L) and is absolutely summable Successive substitution yields Y=Y0+at+v(L)∑= 2.1.2 Trend-Stationary Another important class is the trend- stationary process(TSP). Consider the series v(L)Et where the coefficients of v(L)is absolute summable The mean of Xt is E(X,=u+at and is not constant over time, wh the variance of Xt is Var(Xt)=(1+1+v2 +.o2 and constant. Although the mean of Xt is not constant over the period, it can be forecasted perfectly whenever we know the value of t and the parameters a and d. In the sense it is stationary around the deterministic trend t and Xt can be transformed to stationarity by regressing it on time. Note that both DSP model equation (5) and the TSP model equation(6) exhibit a linear trend, but the appropriated method of eliminating the trend differs.(It can be seen that the DsP is trend nonstationary from the definition of TSP. Most economic analysis is based the variance and covariance among the vari- bles. For example, The OLS estimator from the regression Yt on Xt is the ratio of the covariance between Y and X, to variance of Xt. Thus if the variance of the
where all the roots of φ(L) = 0 and θ(L) = 0 lie outside the unit circle, we say that Yt is an autoregressive integrated moving-average ARIMA(p, d, q) process. In particular an unit root process, d = 1 or an ARIMA(p, 1, q) process is therefore φ(L)4Yt = α + θ(L)εt or (1 − L)Yt = α + ψ(L)εt , (4) where ψ(L) = φ −1 (L)θ(L) and is absolutely summable. Successive substitution yields Yt = Y0 + αt + ψ(L) X t−1 i=0 εt−i . (5) 2.1.2 Trend-Stationary Another important class is the trend − stationary process (TSP). Consider the series Xt = µ + αt + ψ(L)εt , (6) where the coefficients of ψ(L) is absolute summable. The mean of Xt is E(Xt) = µ + αt and is not constant over time, while the variance of Xt is V ar(Xt) = (1+ψ 2 1 + ψ 2 2 + ...)σ 2 and constant. Although the mean of Xt is not constant over the period, it can be forecasted perfectly whenever we know the value of t and the parameters α and δ. In the sense it is stationary around the deterministic trend t and Xt can be transformed to stationarity by regressing it on time. Note that both DSP model equation (5) and the TSP model equation (6) exhibit a linear trend, but the appropriated method of eliminating the trend differs. (It can be seen that the DSP is trend − nonstationary from the definition of TSP.) Most economic analysis is based the variance and covariance among the variables. For example, The OLS estimator from the regression Yt on Xt is the ratio of the covariance between Yt and Xt to variance of Xt . Thus if the variance of the 5