Now we turn back to the models without and with unit roots. The difference between the DT and ST models as discussed above is quite large, in the sense that in a DT model shocks have temporary effects that decay exponentially fast, while in an ST model shocks have permanent effects that do not die out at all. For many economic
time series, it seems that shocks have long-lasting but nevertheless temporary effects. A typical example concerns the effects of the oil crises in the 1970s on macro-economic variables such as output and inflation. Hence, there might be a need for time series models allowing for shocks to have temporary effects that decline at a rate that is slower than exponential. This can be achieved by means of so-called fractional differ- encing or fractional integration, by allowing d in the filter (1− L)dto take non-integer values.
The concept of fractional integration within the context of ARIMA models was first put forward byGranger and Joyeux(1980) andHosking (1981). A fractionally integrated model appears useful to describe time series with very long cycles for which it is difficult to estimate their mean. Such series may also concern variables that experience occasional level shifts, see Chapter6. Typical applications of such a model include inflation rates, foreign exchange rates, and volatility in financial markets, see, for example,Hassler and Wolters(1995),Cheung(1993), andAndersen et al.(2003). Baillie(1996) provides an extensive survey.
The simplest fractionally integrated time series model is given by
(1− L)dyt = εt, with 0< d < 1, (4.40) where the fractional differencing operator (1− L)d is defined by the binomial expansion (1− L)d = 1 − d L − d(1− d)L 2 2! − d(1− d)(2 − d)L3 3! − · · · −d(1− d)(2 − d) · · · (( j − 1) − 1 − d)Lj j ! − · · · . (4.41) which obviously becomes equal to 1 for d = 0 and equal to (1 − L) for d = 1. The expansion in (4.41) shows that yt can be described by an AR model of infinite order with a specific structure imposed on the autoregressive coefficients. Similarly, yt can be written as an MA model of infinite order. In this representation, the coefficient for εt−k is proportional to kd−1for large k, which demonstrates that the effects of a shock declines only at a hyperbolic rate. Furthermore, it can be shown that the ACF of yt does not decline towards zero at the familiar exponential rate but rather at a much slower hyperbolic rate, whereρkis proportional to k2d−1when d < 0.5. For that reason, fractionally integrated time series often are said to have “long memory.” When 0< d < 0.5, the time series is stationary, in the sense that the sum of the absolute autocorrelations is still finite. When 0.5 < d < 1, ytis non-stationary.
The so-called fractional white noise model (4.40) may not be sufficient to adequately describe the dynamics of the ytseries. It can be augmented with additional autoregres- sive and moving average terms though, which leads to the autoregressive fractionally
integrated moving average model of orders p, d, and q [ARFIMA( p,d,q)], that is,
φp(L)(1− L)dyt = θq(L)εt, (4.42)
whereφp(L) andθq(L) are lag polynomials of orders p and q, respectively, as defined in the previous chapter. The conditions for stationarity of this model are d < 0.5 together with the requirement that all roots of the AR-polynomial are outside the unit circle. Similarly, invertibility requires that d > −0.5 and that all solutions to θq(z)= 0 are outside the unit circle.
Several estimation methods for the parameters in ARFIMA( p,d,q) models are avail- able, including exact maximum likelihood developed bySowell (1992). In practice, the interest often centers on the value of the long memory parameter d, for which the semi-parametric estimator proposed byGeweke and Porter-Hudak(1983) is very pop- ular.Beran(1995) proposes an approximate maximum likelihood (AML) estimator for invertible and possibly non-stationary ARFIMA models based on least squares. The AML estimator amounts to minimizing the sum of squared residuals
QT(θ) = T
t=1
e2t(θ), (4.43)
whereθ = (φ, d, θ, μ) and the residuals et(θ) are computed as
et(θ) = (yt− μ) − t+p−1
j=1
πj(yt− j− μ),
where theπj’s are the autoregressive coefficients in the infinite order AR representation (yt− μ) − π1(yt−1− μ) − π2(yt−2− μ) − · · · = εt,
or π(L)(yt− μ) = εt with π(L) = θq−1(L)φp(L)(1− L)d. The AML estimator is asymptotically efficient if the errors εt are normally distributed. When normality ofεtdoes not hold, it is still consistent and asymptotically normal. We refer to Doornik and Ooms (2003, 2004) for a useful review and comparison of alternative estimation methods.
As mentioned above, long memory is often found in the volatility of financial asset returns. Figure 4.6shows the first 250 empirical autocorrelations for absolute first differences of the log gold price series, over the period January 1, 1978–December 31, 2012 (9131 observations). As discussed in Chapter 2, first differences of logs are approximately equal to returns, but absolute returns often are considered as a measure of volatility. While the first-order autocorrelation is not particularly large at
ˆ
ρ1= 0.251, Figure4.6suggests that the decay of the EACF is very slow. Given the
fact that the standard error of ˆρkis equal to 0.01, even ˆρ200 = 0.074 is still significantly
.04 .08 .12 .16 .20 .24 .28 25 50 75 100 125 150 175 200
Figure 4.6: Empirical autocorrelation function for absolute daily gold returns, 1/1/1978–31/12/2012.
where the MA(1) component is included to handle residual autocorrelation. This gives the following estimation results, with standard errors of the estimated parameters in parentheses:
(1− L)dˆ(yt− 0.877) = (1 + 0.256L)ˆεt with ˆd= 0.340 (4.44) (0.122) (0.025) (0.019)
The estimate of d is significantly positive but also significantly less than 0.5, suggesting that volatility of gold returns indeed is fractionally integrated but stationary.
Although often an ARFIMA model may provide an improvement over (possibly lengthy) AR models in terms of in-sample fit, the evidence on the usefulness of long memory models for out-of-sample forecasting is mixed. On the one hand,Crato and Ray (1996) show that simple AR models outperform ARFIMA models in out- of-sample forecasting, also because selecting the appropriate AR and MA orders p and q is difficult and because estimation of d can be quite complicated. By contrast, Brodsky and Hurvich(1999) provide simulation evidence that a fractionally integrated model provides substantially more accurate forecasts than an approximating ARMA model, especially at long forecast horizons.Bhardwaj and Swanson(2006) conduct an extensive empirical analysis of a large number of macroeconomic and financial time series, also finding that often the forecasts from ARFIMA models are more accurate than those from a variety of AR, MA, and ARMA models.
4.2
Unit root tests
r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r As already indicated before, one of the specific features of a time series that is governed by a stochastic trend is that its AR representation (or the AR part of an ARMA model) contains the component (1− L). In other words, the AR polynomial can then be decomposed as φp(L)= 1 − φ1L− φ2L2− · · · − φpLp (4.45) = (1 − φ∗ 1L− φ2∗L 2− · · · − φ∗ p−1L p−1)(1− L),
or written more compactly
φp(L)= φp−1(L)(1− L). (4.46)
From (4.46), it is easy to see that having a component (1− L) in the AR-polynomial
φp(L) is equivalent to saying thatφp(L) contains a unit root, that is, z= 1 is a solution of the characteristic equation
φp(z)= 0.
From (4.45) it follows that in that case the AR-parameters φ1, . . . , φp sum to one, because
φp(1)= 1 − φ1− φ2− · · · − φp = 0. (4.47) For an I(1) series,φp(L) contains a single unit root, such that the polynomialφp−1(L) in (4.46) has all roots outside the unit circle. For an I(2) time series, it holds that the AR-polynomial contains two unit roots, that is z= 1 also is a solution to φp−1(z)= 0. Equivalently, when a time series is I(2), it holds true that
φp(L)= φp−2(L)(1− L)2, (4.48)
where all the roots ofφp−2(L) are outside the unit circle.