IS THE NORTH ATLANTIC OSCILLATION A RANDOM WALK? A COMMENT WITH FURTHER RESULTS

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 24: (24) Published online 11 February 24 in Wiley InterScience ( DOI: 1.12/joc.13 IS THE NORTH ATLANTIC OSCILLATION A RANDOM WALK? A COMMENT WITH FURTHER RESULTS TERENCE C. MILLS* Department of Economics, Loughborough University, Loughborough LE11 3TU, England, UK Received 3 June 23 Revised 28 November 23 Accepted 28 November 23 ABSTRACT Time series on the North Atlantic oscillation (NAO) have been subject to considerable analysis in recent years, with the consensus emerging that the data are not characterized by a random walk process. However, no consensus has yet emerged concerning the short run correlation structure of the data. This comment explores the time series properties of the NAO in more detail, using testing and modelling procedures that have found wide application in areas such as economics and finance. A structural time series model provides a comprehensive explanation of the index s rich dynamics, there being a slowly changing level component and a cyclical component having a period of approximately 7.5 years. These are dominated, however, by an irregular component, whose presence makes accurate forecasting of the index problematic. Copyright 24 Royal Meteorological Society. KEY WORDS: North Atlantic oscillation; NAO; stochastic process; stationarity; trends; cycles; structural time series models 1. INTRODUCTION Wunsch (1999) and Stephenson et al. (2) have recently investigated the time series properties of the North Atlantic oscillation (NAO) using annual data starting from Both reject the random walk model, but find that the data are not an uncorrelated sequence, for there is evidence of autocorrelation, which may have the property of long memory. This comment explores the time series properties of the NAO in more detail, using testing and modelling procedures that have found wide application in areas such as economics and finance. 2. THE DATA AND THEIR TIME SERIES PROPERTIES One of the NAO indices used by Stephenson et al. (2) is the winter sea-level pressure (SLP) index, based on the difference between standardized December March mean SLPs measured at Lisbon (Portugal) and Stykkisholmur/Reykjavik (Iceland). We focus on this series, as Stephenson et al. (2) fitted several simple time series models to it. Figure 1(a) shows this series for , and Figure 1(b) shows a plot of its sample autocorrelation function. This shows that only one autocorrelation, at lag 2, exceeds two standard errors, calculated using the formula 2T 1/2 = 2(139) 1/2 =.17 under the assumption that the series is an independent and identically distributed sequence, which is often referred to as white noise (e.g. see Box and Jenkins (1976)). Although this evidence suggests that the NAO SLP index is indistinguishable from white noise, it is based on asymptotic arguments that may not be entirely appropriate here, and thus should only be regarded as an * Correspondence to: Terence C. Mills, Department of Economics, Loughborough University, Loughborough LE11 3TU, UK; t.c.mills@lboro.ac.uk Copyright 24 Royal Meteorological Society

2 378 T. C. MILLS (a) SLP Index (b).2 Sample autocorrelation lag in years Figure 1. (a) Evolution from 1864 to 22 of the NAO SLP index difference in standardized December March mean SLPs between Lisbon (Portugal) and Stykkisholmur (Iceland). (b) Sample autocorrelation function with two standard errors bands (dashed lines), computed under the assumption that the index is white noise informal guide to model specification. Stephenson et al. (2), therefore, fit several stochastic models to the index: autoregressive (AR) models of orders 1 and 1 (AR(1) and AR(1)), a first-order fractional AR model (FAR(1)), and an AR integrated moving average (ARIMA(1,1,)) model. They compare the forecasting abilities of these models with two benchmarks: the climatology model, which is, in fact, a white noise sequence, and the persistence model, which is a pure random walk. If the index in year t is denoted as z t, then each of these six models is a special case of d z t = θ + ρz t 1 + p δ i d z t i + a t (1) where d z t = (z t z t 1 ) d and a t is white noise with variance σ 2. On setting d = 1, then if p = andρ<, Equation (1) produces the stationary AR(1) model z t = θ + φz t 1 + a t, with AR parameter φ = 1 + ρ<1,

3 NORTH ATLANTIC OSCILLATION 379 so that white noise is obtained if ρ = 1. If p> then we have the stationary AR(p + 1) model p+1 z t = θ + φ i z t i + a t (2) where φ 1 = 1 + ρ + δ 1, φ i = (δ i δ i+1 ), i = 2,...,p,andφ p+1 = δ p.ifρ = andp = 1, then the ARIMA(1,1,) model results, and if p = we have the pure random walk. If <d<1, ρ = andp = 1 then we have the FAR(1) model. Conditional on d = 1, the AR lag order p may be determined by setting a maximum order p max and using the sequence of hypothesis tests H (1) : δ p max =, H (2) : δ p max 1 = δ pmax =, H (3) : δ pmax 2 = δ pmax 1 = δ pmax =,... (3) stopping at the first significant statistic. Conventional t-statistics from the sequential ordinary least-squares estimation of Equation (1) for p = p max, p max 1,... may be used for this. Alternatively, minimization of an information criterion, such as Akaike s information criterion, over the range of p values may be used to determine the optimal lag order ˆp. With this value of the lag order, the unit root hypothesis ρ = maybe tested against the stationary alternative ρ< by comparing the t-statistic on ˆρ with the critical value of the τ µ distribution provided by Dickey and Fuller (1979). The term unit root reflects the fact that, if ρ = and d = 1, Equation (1) becomes z t = θ + p δ i z t i + a t so that z t requires first differencing (i.e. filtering by a first-order autoregression with a root of unity) to render it stationary. Mills (1999), for example, provides a detailed development of the relevant statistical theory for testing for a unit root. By setting p max = 1 and employing a 5% significance level for each test in the sequence in Equation (3) we arrive at ˆp = 4; with this value, an estimate of ˆρ =.65 was obtained with an accompanying t-statistic of This may be compared with the 1% critical value of the τ µ distribution, which is 3.48 for the sample size available here, so that the unit root null is conclusively rejected in favour of the stationary alternative. A range of information criteria all selected ˆp =, for which value it was found that ˆρ =.86 and τ µ = 1.13, so that again the unit root null is conclusively rejected. There is thus no evidence that the NAO index contains a random walk component. There are also a variety of tests available for testing the null of d = against the fractional alternative d. Three tests are employed here: Lo s (1991) modification of the R/S statistic for long-range dependence to deal with short-run autocorrelation, the KPSS test of Lee and Schmidt (1996), and Agiakloglou and Newbold s (1994) Lagrange multiplier (LM) test. The R/S statistic is defined as R q =ˆσ 1 q [ max 1 i T i t=1 (z t z) min 1 i T ] i (z t z) t=1 where ˆσ q 2 =ˆσ T q ω qj r j ω qj = 1 j q + 1 j=1 q<t ˆσ 2 is the sample variance of z t and r j is the jth sample autocorrelation. On the null d =, T 1/2 R q converges in distribution to a well-defined random variable, whose upper 5% critical value is Lo

4 38 T. C. MILLS (1991) recommends setting q as the integer part of T.25.Thus,withq = 3, R 3 = 1.47, so providing no evidence against d =. The KPSS test is defined as η q = (T ˆσ q ) 2 T (z t z) 2 t=1 and has a limiting distribution that is again a well-defined random variable, although it is different, of course, to that for R q. The 5% critical value for testing the null d = is.463. Since η 3 =.199, there is again no evidence against this null. The LM test of d = isthet-ratio on the coefficient δ in the auxiliary regression p+1 â t = β i z t i + δk t (q) + u t where â t are the residuals from the fit of Equation (1) and K t (q) = q j 1 â t j j=1 With p = 4, setting q as before yielded a t-ratio on ˆδ of just.28, once again providing no evidence against the null of d =. These test results are consistent with the maximum likelihood estimate of ˆd =.13 provided by Stephenson et al. (2) for the slightly shorter sample ending in Although no standard errors are reported, it is unlikely that such an estimate is significantly different from zero. However, issues of power are likely to be important here. Lee and Schmidt (1996) investigated in detail the power of R q and η q against fractional alternatives, while Agiakloglou and Newbold (1994) looked at the power of the LM test. For samples of the size available here and for an alternative of the size estimated by Stephenson et al. (2), power is, unsurprisingly, quite low, so that it is difficult to discriminate reliably between the two values of d. Nevertheless, it is useful to estimate the two competing AR models implied by the above testing procedure: the AR(1) and AR(5) models. In both cases, estimation found that the constant θ was insignificant, implying that the index has zero mean and that, in the latter model, φ 2, φ 3 and φ 4 were also insignificant. Imposing these restrictions gave the following models: and z t =.15 (.8) z t 1 + a t ˆσ = z t =.15 (.8) z t (.8) z t 5 + a t ˆσ = 1.95 Standard errors are shown in parentheses and tests on the residuals of the AR(5) model revealed no significant autocorrelations and no evidence of misspecification in terms of non-constant variance or nonnormality. The AR(1) model, which is nearly identical to that reported by Stephenson et al. (2) for the shorter sample ending in 1998, had a lag five residual autocorrelation that was just significant, so that it is dominated by the AR(5) on goodness of fit. The AR(1) model s dynamics consist of a single damped exponential, with a damping factor of.15. In contrast, the AR(5) model has a single real root, producing a damped exponential with damping factor of.74, and two pairs of complex roots,.25 ±.67i and.54 ±.41i, which produce damped sine waves with damping factors.72 and.68 and periods of 5.2 years and 9.7 years respectively. A close examination of the sample autocorrelation function shown in

5 NORTH ATLANTIC OSCILLATION 381 Figure 1(b) does reveal evidence of an approximately cyclical pattern with peaks and troughs around 7 years apart (close to the average of the two periods) after the damped exponential has died away. Given the difficulty of discriminating between a stationary, short-memory, AR process and a stationary, but long-memory, FAR process just on the basis of statistical tests, it is interesting to follow an alternative modelling route. Harvey (1989, 1997) argues for imposing more structure on the data, rather than just letting it speak for itself. Given the evidence of cyclicality in the NAO index, a structural time series model may thus be considered. The term structural signifies that the observed index z t is decomposed into the sum of unobserved components modelling various features of the data, in particular a level component µ t, a cyclical component ψ t, and an irregular component ε t : z t = µ t + ψ t + ε t (4) The level is modelled as µ t = µ t 1 + ξ t so that it is allowed to vary over time through the presence of the random variable ξ t. Although this random walk specification will introduce nonstationarity into the model, it is unlikely to be detected in the data by standard tests when the variation in the level is dominated by the variation in the other components (simulation evidence for a closely related problem that supports this line of argument is provided by Markellos and Mills (21)). The cycle is defined as [ ] [ ][ ] [ ] ψt cos λ sin λ ψt 1 ςt ψt = ρ sin λ cos λ ψt 1 + ςt (5) ψt appears by construction and ς t and ςt are assumed to be independent zero-mean white-noise processes with common variances σς 2. The parameter ρ is the damping factor and λ measures the frequency of the cycle in radians, so that the period is 2π/λ. Both the irregular component ε t and the innovation to the level ξ t are assumed to be zero-mean white noises with variances σε 2 and σ ξ 2 respectively. Fitting the model in Equation (4) using the software routines provided by Koopman et al. (1999) produced the following maximum likelihood estimates of the level and irregular variances: ˆσ ξ 2 =.424 and ˆσ ε 2 = Since ˆσ ξ 2/ ˆσ ε 2 =.14, the level is very much smoother than the irregular component. The estimated components ˆµ t and ˆε t are shown in Figure 2(a) and (c), and confirm this feature. The level is characterized by long swings between a minimum of.77, reached in the mid-196s, and a maximum of 1.32, achieved in the early 199s. The irregular component fluctuates in the range 4.5 to 4, with little evidence of any changing volatility. The cycle component has ˆλ =.819 and ˆρ = 1. These imply that ψ t reduces to the deterministic cycle ˆψ t = ψ cos.819t + ψ sin.819t where ψ and ψ are uncorrelated random variables with zero means and a common variance estimated to be.315. The cycle is bounded by ±.74, has a period of 7.7 years and is shown as Figure 2(b). The period is consistent with the average period of the two implied cycles from the AR(5) fit and the evidence from the sample autocorrelation function shown in Figure 1(b). The predictable component z t ε t = µ t + ψ t thus goes through long swings overlain with a shorter cycle, and is shown superimposed on z t in Figure 3. The irregular component, however, dominates the variation in the index, with the signal-to-noise variance ratio being.21. This necessarily makes the index difficult to forecast accurately from its past history. 3. CONCLUSIONS Using a variety of tests from the time series literature, it has been established that the NAO SLP index is certainly stationary and, within the class of ARIMA models, is best fitted by a restricted fifth-order

6 382 T. C. MILLS (a) (b) (c) Figure 2. Components of the NAO SLP index estimated from the structural time series model in Equation (4): (a) level component; (b) cycle with period 7.7 years; (c) irregular component autoregression. Although there is no evidence in favour of fractional integration, as suggested by Stephenson et al. (2), it is difficult to discriminate between the two classes of model on the basis of statistical tests alone. When structural time series models are considered, it is found that the dynamics of the index have a rich structure. Three components are identified: a slowly oscillating level exhibiting long swings, a stable cycle having a period of about 7.5 years (consistent with the cycle found by Wunsch (1999) using spectral techniques), and an irregular component that dominates the variation of the index. The presence of this

7 NORTH ATLANTIC OSCILLATION Figure 3. NAO SLP index with its predictable component superimposed component makes accurate forecasting of the index difficult, since the structural model is found to explain less than 15% of the variation in the index itself. REFERENCES Agiakloglou C, Newbold P Lagrange multiplier tests for fractional difference. Journal of Time Series Analysis 15: Box GEP, Jenkins GM Time Series Analysis: Forecasting and Control, revised edition. Holden-Day: San Francisco. Dickey DA, Fuller WA Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74: Harvey AC Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press: Cambridge. Harvey AC Trends, cycles and autoregressions. Economic Journal 17: Koopman SJ, Harvey AC, Doornik JA, Shephard N STAMP: Structural Time Series Analyser, Modeller and Predictor. Timberlake Consultants Ltd: London. Lee D, Schmidt P On the power of the KPSS test of stationarity against fractionally integrated alternatives. Journal of Econometrics 73: Lo AW Long-term memory in stock market prices. Econometrica 59: Markellos RN, Mills TC. 21. Unit roots in the CAPM? Applied Economics Letters 8: Mills TC The Econometric Modelling of Financial Time Series, 2nd edn. Cambridge University Press: Cambridge. Stephenson DB, Pavan V, Bojariu R. 2. Is the North Atlantic oscillation a random walk? International Journal of Climatology 2: Wunsch C The interpretation of short climate records, with comments on the North Atlantic and southern oscillations. Bulletin of the American Meteorological Society 8: