Testing and Modelling Autoregressive Conditional Heteroskedasticity of Streamflow Processes W

Testing and modelling autoregressive conditional heteroskedasticity of streamflow processes W. Wang, P. H. A. J. M. van Gelder, J. K. Vrijling, J. Ma

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W. Wang, P. H. A. J. M. van Gelder, J. K. Vrijling, J. Ma. Testing and modelling autoregressive conditional heteroskedasticity of streamflow processes. Nonlinear Processes in Geophysics, European Geosciences Union (EGU), 2005, 12 (1), pp.55-66. hal-00302471

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Testing and modelling autoregressive conditional heteroskedasticity of streamﬂow processes

W. Wang1, 2, P. H. A. J. M. Van Gelder2, J. K. Vrijling2, and J. Ma3 1Faculty of Water Resources and Environment, Hohai University, Nanjing, 210098, China 2Faculty of Civil Engineering & Geosciences, Section of Hydraulic Engineering, Delft University of Technology, P.O.Box 5048, 2600 GA Delft, The Netherlands 3Yellow River Conservancy Commission, Hydrology Bureau, Zhengzhou, 450004, China Received: 24 May 2004 – Revised: 15 December 2004 – Accepted: 5 January 2005 – Published: 21 January 2005

Part of Special Issue “Nonlinear deterministic dynamics in hydrologic systems: present activities and future challenges”

Abstract. Conventional streamflow models operate under dition in terms of statistical modelling of daily streamflow the assumption of constant variance or season-dependent processes for the hydrological community. variances (e.g. ARMA (AutoRegressive Moving Average) models for deseasonalized streamflow series and PARMA (Periodic AutoRegressive Moving Average) models for sea- 1 Introduction to autoregressive conditional het- sonal streamflow series). However, with McLeod-Li test eroskedasticity and Engle’s Lagrange Multiplier test, clear evidences are When modelling hydrologic time series, we usually focus on found for the existence of autoregressive conditional het- modelling and predicting the mean behaviour, or the first eroskedasticity (i.e. the ARCH (AutoRegressive Conditional order moments, and are rarely concerned with the condi- Heteroskedasticity) effect), a nonlinear phenomenon of the tional variance, or their second order moments, although variance behaviour, in the residual series from linear models unconditional season-dependent variances are usually con- fitted to daily and monthly streamflow processes of the up- sidered. The increased importance played by risk and un- per Yellow River, China. It is shown that the major cause certainty considerations in water resources management and of the ARCH effect is the seasonal variation in variance of flood control practice, as well as in modern hydrology the- the residual series. However, while the seasonal variation ory, however, has necessitated the development of new time in variance can fully explain the ARCH effect for monthly series techniques that allow for the modelling of time varying streamflow, it is only a partial explanation for daily flow. variances. It is also shown that while the periodic autoregressive mov- ARCH-type models, which originate from econometrics, ing average model is adequate in modelling monthly flows, give us an appropriate framework for studying this prob- no model is adequate in modelling daily streamflow pro- lem. Volatility (i.e. time-varying variance) clustering, in cesses because none of the conventional time series mod- which large changes tend to follow large changes, and els takes the seasonal variation in variance, as well as the small changes tend to follow small changes, has been well ARCH effect in the residuals, into account. Therefore, an recognized in financial time series. This phenomenon is ARMA-GARCH (Generalized AutoRegressive Conditional called conditional heteroskedasticity, and can be modeled by Heteroskedasticity) error model is proposed to capture the ARCH-type models, including the ARCH model proposed ARCH effect present in daily streamflow series, as well as to by Engle (1982) and the later extension GARCH (general- preserve seasonal variation in variance in the residuals. The ized ARCH) model proposed by Bollerslev (1986), etc. Ac- ARMA-GARCH error model combines an ARMA model cordingly, when a time series exhibits autoregressive condi- for modelling the mean behaviour and a GARCH model for tionally heteroskedasticity, we say it has the ARCH effect or modelling the variance behaviour of the residuals from the GARCH effect. ARCH-type models have been widely used ARMA model. Since the GARCH model is not followed to model the ARCH effect for economic and financial time widely in statistical hydrology, the work can be a useful ad- series. Correspondence to: W. Wang The ARCH-type model is a nonlinear model that includes ([email protected]) past variances in the explanation of future variances. ARCH- Discharge (cms) Discharge 0 1000 2000 3000 4000 5000

0 5000 10000 15000 Day

56 W. Wang et al.:Figure Testing 1 Daily and modelling streamflow autoregressive (m3/s) of the conditional upper Yellow heteroskedasticity River at Tangnaihai

1600 1400 daily mean /S)

3 1200 standard deviation 1000 800 600 400

Discharge (m Discharge 200 0 Discharge (cms) Discharge 1-Jan 2-Mar 1-May 30-Jun 29-Aug 28-Oct 27-Dec Date

Fig. 2. Variation in daily mean and standard deviation of the stream- Figure 2 Variationﬂow in at daily Tangnaihai. mean and standard deviation of the streamflow at Tangnaihai 0 1000 2000 3000 4000 5000

0 5000 10000 15000 Day tributing watershed, including a permanently snow-covered 3 area of 192 km2. The length of the main channel of this wa- Figure 1 DailyFig. 1.streamflowDaily streamﬂow (m3/s) (m of/s) ofthe the upper upper YellowYellow River River at Tang- at Tangnaihai naihai. tershed is over 1500 km. Most of the area is 3000∼6000 meters above sea level. Snowmelt water composes about 5%

1600 ACF of total runoff. Most rain falls in summer. Because the water-

type1400 models can generate accurate forecasts of future volatil- shed is partly permanently snow-covered ACF Partial and sparsely pop- daily mean /S)

ity,3 1200 especially over short horizons, therefore providing a bet- standard deviation ulated, without any large-scale hydraulic works, it is fairly ter1000 estimate of the forecast uncertainty which is valuable for pristine. The average annual runoff volume (during 1956– water800 resource management and flood control. And they take 2000) at Tangnaihai gauging station is 20.4 billion cubic me- 0.00.20.40.60.81.0 into600 account excess kurtosis (i.e. fat tail behaviour), which ters, about 35% of the whole Yellow-0.2 -0.0 River 0.2 0.4 0.6Basin, 0.8 1.0 and it is the 400 0 20406080100 0 20406080100 is common in hydrologic processes. Therefore, ARCH- majorLag runoff producing area of the Yellow River basin. DailyLag Discharge (m Discharge 200 type models could be very useful for hydrologic time se- average streamflow at Tangnaihai has been recorded since 1 0 ries modelling. Some authors propose new models to repro- January 1956. Monthly series is obtained from daily data by 1-Jan 2-Mar 1-May 30-Jun 29-Aug 28-Oct 27-Dec Figure 3 ACF and PACF of deseasonalized daily flow series duce the asymmetric periodicDate behaviour with large fluctua- taking the average of daily discharges in every month. In this tions around large streamflow and small fluctuations around study, data from 1 January 1956 to 31 December 2000 are small streamflow (e.g. Livina et al., 2003), which basically used. The daily streamflow series from 1956 to 2000 is plot- Figure 2 Variationcan in daily be handled mean withand standard those conventional deviation time of the series streamflow mod- atted Tangnaihai in Fig. 1, and variations in the daily mean discharge and els that have taken season-dependent variance into account, daily standard deviation of the streamflow at Tangnaihai are such as PARMA models and deseasonalized ARMA models. shown in Fig. 2. However, little attention has been paid so far by the hydrologic community to test and model the possible presence of the ARCH effect with which large fluctuations tend to follow 3 Tests for the ARCH effect of streamflow process 18 large fluctuations, and small fluctuations tend to follow small fluctuations in streamflow series. The detection of the ARCH effect in a streamflow series is ACF In this paper, we will take the daily and monthly stream- actually a test of serial independence applied to the serially flow of the upper Yellow River ACF Partial at Tangnaihai in China as uncorrelated fitting error of some model, usually a linear au- case study hydrologic time series to test for the existence toregressive (AR) model. We assume that linear serial depen- of the ARCH effect, and propose an ARMA-GARCH error dence inside the original series is removed with a well-fitted, pre-whitening model; any remaining serial dependence must

0.00.20.40.60.81.0 model for daily flow series. The paper is organized as fol- -0.2 -0.0 0.2 0.4 0.6 0.8 1.0 be due to some nonlinear generating mechanism which is 0 20406080100lows. First, the method of testing conditional0 20406080100 heteroskedas- ticityLag of streamflow process is described. Then, the causesLag of not captured by the model. Here, the nonlinear mechanism the ARCH effect and the inadequacy of commonly used sea- we are concerned with is the conditional heteroskedastic- Figuresonal 3 ACF time seriesand PACF models of for deseasonalized modelling streamflow daily flow are dis- series ity. We will show that the nonlinear mechanism remaining cussed. Finally, an ARMA-GARCH error model is proposed in the pre-whitened streamflow series, namely the residual for capturing the ARCH effect existing in daily streamflow series, can be well interpreted as autoregressive conditional series. heteroskedasticity.

3.1 Linear ARMA models ﬁtted to daily and monthly ﬂows 2 Case study area and data set Three types of seasonal time series models are commonly The case study area is the headwaters of the Yellow River, used to model hydrologic processes which usually have located in the northeastern Tibet Plateau. In this area, the strong seasonality18 (Hipel and McLeod, 1994): 1) seasonal discharge gauging station Tangnaihai has a 133 650 km2 con- autoregressive integrated moving average (SARIMA) mod- Discharge (cms) Discharge 0 1000 2000 3000 4000 5000

0 5000 10000 15000 Day

Figure 1 Daily streamflow (m3/s) of the upper Yellow River at Tangnaihai

1600 1400 daily mean /S)

3 1200 standard deviation 1000 800 600 400

Discharge (m Discharge 200 0 1-Jan 2-Mar 1-May 30-Jun 29-Aug 28-Oct 27-Dec Date

W. Wang et al.: Testing and modelling autoregressive conditional heteroskedasticity 57 Figure 2 Variation in daily mean and standard deviation of the streamflow at Tangnaihai ACF Partial ACF Partial 0.00.20.40.60.81.0 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0 0 20406080100 0 20406080100 Lag Lag Fig. 3. ACF and PACF of deseasonalizedFigure 3 daily ACF ﬂow and series. PACF of deseasonalized daily flow series ACF

Partial ACF 18 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.8 1.0

0 102030405060 0 102030405060 Lag Lag Fig. 4. ACF and PACF of deseasonalizedFigure 4 ACF monthly and flow PACF series. of deseasonalized monthly flow series els; 2) deseasonalized ARMA models; and 3) periodic Firstly, we inspect the ACF of the residuals. It is well- ARMA models. The deseasonalized modelling approach is known that for random and independent series of length n, adopted in this study. The procedure of fitting deseasonalized the lag k autocorrelation coefficient is normally distributed ARMA models to daily and monthly streamflow at Tang-(a) with a mean of zero and a variance of√ 1/(bn), and the 95% naihai includes two steps. First, logarithmize both flow se- confidence limits are given by ±1.96/ n. The ACF plots ries, and deseasonalize them by subtracting the seasonal (e.g. in Fig. 6 show that there is no significant autocorrelation left daily or monthly) mean values and dividing by the seasonal in the residuals from both ARMA-type models for daily and standard deviationsResiduals of the logarithmized series. To alleviate monthly flow. Residuals the stochastic fluctuations of the daily means and standard Then, more formally, we apply the Ljung-Box test (Ljung deviations, we smooth them with first 8 Fourier harmonics and Box, 1978) to the residual series, which tests whether ˆ2 2 before using them for standardization. Then, according to the the first L autocorrelations rk (ε ) (k = 1, ..., L) from a pro- ACF (AutoCorrelation Function) and PACF (Periodic Auto- cess-1.5 -1.0 -0.5are 0.0 0.5 collectively 1.0 1.5 small in magnitude. Suppose we have -1.0 -0.5 0.0 0.5 1.0 Correlation Function)0 structures 200 of 400 the two 600 series,as 800 well as1000 the0 first L 20406080100120autocorrelations rˆ (ε) (k = 1, ..., L) from any Day Month k the model selection criterion AIC, two linear ARMA-type ARMA(p, d, q) process. For a fixed sufficiently large L, models (one ARMA(20,1) and one AR(4)) are fitted to the the usual Ljung-Box Q-statistic is given by logarithmizedFigure and 5 deseasonalizedSegments of dailythe residual and monthly series flow from se- (a) ARMA(20,1) for daily flow and (b) AR(4) L ries, respectively, following the model construction proce- X rˆ2(ε) for monthly flow at Tangnaihai. Q = N(N + 2) k , (1) dures suggested by Box and Jenkins (1976). Figures 3 and − = N k 4 show the ACF and PACF of the deseasonalized daily and k 1 monthly series. Figure 5 shows parts of the two residual se- where N = sample size, L= the number of autocorrelations (a) ˆ2 (b) ries obtained from the two models. included in the statistic, and rk is the squared sample auto- Before applying ARCH tests to the residual series, to en- correlation of residual series {εt } at lag k. Under the null sure that the null hypothesis of no ARCH effect is not re- hypothesis of model adequacy, the test statistic is asymp- jected due to the failure of the pre-whitening linear models, totically χ 2(L−p−q) distributed. Thus, we would reject ACF we must check the goodness-of-fit of the linear models. theACF null hypothesis at level α if the value of Q exceeds the 0.0 0.1 0.2 0.00 0.05 0.10

0 100 200 300 024681012 Lag Lag Figure 6 ACFs of residuals from (a) ARMA(20,1) model for daily flow and (b) AR(4) model for monthly flow at Tangnaihai

19 ACF Partial ACF ACF Partial ACF 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.8 1.0

0 102030405060 0.00 0.2 102030405060 0.4 0.6 0.0 0.2 0.4 0.6 0.8 1.0 Lag Lag 0 102030405060 0 102030405060 Figure 4 ACFLag and PACF of deseasonalized monthly flowLag series

Figure 4 ACF and PACF of deseasonalized monthly flow series

58 W. Wang et al.: Testing and modelling autoregressive conditional heteroskedasticity (a) (b)

(a) (b) Residuals Residuals Residuals Residuals -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.0 -0.5 0.0 0.5 1.0 0 200 400 600 800 1000 0 20406080100120

Day -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Month

-1.0 -0.5 0.0 0.5 1.0 0 200 400 600 800 1000 0 20406080100120 Figure 5 Segments of the residualDay series from (a) ARMA(20,1) forMonth daily flow and (b) AR(4) Fig. 5. SegmentsforFigure monthly of5 theSegments residualflow at series Tangnaihai.of the from residual(a) ARMA(20,1) series forfrom daily (a) ﬂow ARMA(20,1) and (b) AR(4) for for monthly daily ﬂow flow at Tangnaihai. and (b) AR(4) for monthly flow at Tangnaihai.

(a) (b) (a) (b) ACF ACF 0.0 0.1 0.2 ACF ACF 0.00 0.05 0.10 0.0 0.1 0.2 0.00 0.05 0.10

0 100 200 300 024681012 Lag Lag 0 100 200 300 024681012 Lag Lag Fig. 6. ACFsFigure of residuals6 ACFs from of (a)residualsthe ARMA(20,1) from (a) model ARMA(20,1) for daily flow and model(b) the for AR(4) daily model flow for monthly and (b) flow AR(4) at Tangnaihai. model forFigure monthly 6 ACFs flow of at residuals Tangnaihai from (a) ARMA(20,1) model for daily flow and (b) AR(4) model − 2 − − (1 α)-quantilefor monthly of the χflow(L atp Tangnaihaiq) distribution. The Ljung- Fig. 6, the squared residual series are autocorrelated, and the Box test results for ARMA(20,1) and AR(4) are shown in ACF structures of both squared residual series exhibit strong Fig. 7. The p-values’ exceedance of 0.05 indicates the ac- seasonality. This indicates that the variance of residual series ceptance of the null hypothesis of model adequacy at signif- is conditional on its past history, namely, the residual series icance level 0.05. may exhibit an ARCH effect. However, while the residuals seem statistically uncorre- There are some formal methods to test for the19 ARCH effect of a process, such as the McLeod-Li test (McLeod lated according to ACF and PACF shown in Fig. 6, they 19 are not identically distributed from visual inspection of the and Li, 1983), the Engle’s Lagrange Multiplier test (Engle, Fig. 5, that is, the residuals are not independent and identi- 1982), the BDS test (Brock et. al., 1996), etc. McLeod- cally distributed (i.i.d.) through time. There is a tendency, Li test and Engle’s Lagrange Multiplier test are used here especially for daily flow, that large (small) absolute values of to check the existence of an ARCH effect in the streamflow the residual process are followed by other large (small) val- series. ues of unpredictable sign, which is a common behaviour of GARCH processes. Granger and Andersen (1978) found that 3.2 McLeod-Li test for the ARCH effect some of the series modelled by Box and Jenkins (1976) exhibit autocorrelated squared residuals even though the resid- McLeod and Li (1983) proposed a formal test for ARCH uals themselves do no seem to be correlated over time, and effect based on the Ljung-Box test. It looks at the auto- therefore suggested that the ACF of the squared time series correlation function of the squares of the pre-whitened data, could be useful in identifying nonlinear time series. Boller- and tests whether the first L autocorrelations for the squared slev (1986) stated that the ACF and PACF of squared process residuals are collectively small in magnitude. are useful in identifying and checking GARCH behaviour. Similar to Eq. (1), for fixed sufficiently large L, the Ljung- Figure 8 shows the ACFs of the squared residual series Box Q-statistic of Mcleod-Li test is given by from the ARMA(20,1) model for daily flow and the AR(4) L X rˆ2(ε2) model for monthly flow at Tangnaihai. It is shown that al- Q = N(N + 2) k , (2) N − k though the residuals are almost uncorrelated, as shown in k=1 (a) 0. 3 (b) 1 0. 25 0. 8 0. 2 0. 6 W. Wang et al.:0. 15 Testing and modelling autoregressive conditional heteroskedasticity 59 p-value p-value 0. 4 0. 1 0. 05 0. 2 0. 3 1 (a)0 (b)0 0. 25 21 26 31 36 0.4 8 6 8 10 12 14 16 18 20 0. 2 Lag Lag 0. 6 0. 15 p-value p-value 0. 4 0. 1 Figure 7 0.Ljung-Box 05 lack-of-fit tests for (a) ARMA(20,1)0. 2 model for daily flow and (b) AR(4) 0 0 model for monthly21 flow. 26 31 36 4 6 8 10 12 14 16 18 20 Lag Lag

Fig. 7. Ljung-Box Figure lack-of-fit 7 Ljung-Box tests for (a) lack-of-fitthe ARMA(20,1) tests model for (a) for ARMA(20,1) daily flow and (b) modelthe AR(4) for modeldaily forflow monthly and flow.(b) AR(4) model for monthly flow. (a) (b)

ACF ACF (a) 0.0 0.1 0.2 (b) 0.0 0.1 0.2 ACF ACF 0.0 0.1 0.2 0 100 200 300 024681012 Lag Lag 0.0 0.1 0.2 Fig. 8. ACFsFigure of the 8 squared ACFs residuals of the from squared(a) the ARMA(20,1)residuals from model (a) for dailyARMA(20,1) flow and (b) themodel AR(4) for model daily for monthly flow and flow at(b) Tangnaihai. 0 100 200 300 024681012 AR(4) model for monthly flowLag at Tangnaihai Lag ˆ2 where N is the sample size, and rk is the squared sample lags less than 4 are removed by the AR(4) model, when we autocorrelationFigure of squared 8 ACFs residual of the series squared at lag residualsk. Under from take(a) ARMA(20,1) autocorrelations model at longer for lagsdaily into flow consideration, and (b) sig- the null hypothesis of a linear generating mechanism for the nificant autocorrelations remain and the null hypothesis of data, namely, noAR(4) ARCH model effect for in monthly the data, theflow test at statistic Tangnaihai is no ARCH effect is rejected. Because it is required for the asymptotically χ 2(L) distributed. Figure 9 shows the results McLeod-Li test to use sufficiently large L, namely, a suf- of the McLeod-Li(a) 0. 06 test for daily and monthly flow. It illus- (b)ficient 0. 8 number of autocorrelations to calculate the Ljung- 0. 7 0. 05 trates that the null hypothesis of no ARCH effect is rejected Box0. 6 statistic (typically around 20), we still consider that the for both daily and0. 04 monthly flow series. monthly0. 5 flow has the ARCH effect. 0. 03 0. 4 0. 06 On0. the 8 whole, evidences are clear with the McLeod-Li test p-value 0. 3 p-value (a) (b) 3.3 Engle’s Lagrange0. 02 Multiplier test for the ARCH effect 0. 7 0. 05 and0. 2 Engle’s LM test about the existence of conditional het- 0. 6 0. 01 eroskedasticity0. 1 in the residual series from linear models fitted 0. 04 0. 5 Since the ARCH0 model has the form of an autoregres- 0. 0 0. 03 to the0. logarithmized 4 and deseasonalized daily and monthly sive model, Engle (1982)0 proposed 5 10 the 15 Lagrange 20 Multiplier 25 30 0 5 10 15 20 25 30 p-value 0. 3 p-value 0. 02 streamflow processes of the upper Yellow River at Tangnai- (LM) test, in order to test for the existenceLag of ARCH be- 0. 2 Lag 0. 01 hai. haviour based on the regression. The test statistic is given 0. 1 0. 0 by TR2, where R is the0 sample multiple correlation coef- 0 5 10 15 20 25 30 Figure 9 McLeod-Li0 5test for 10 the 152 residuals 20 25 from 30 (a) ARMA(20,1) model for daily flow and ficient computed from the regression of εLagt on a constant 4 Discussion of the causesLag of ARCH effects and inade- 2 2 and εt−1,. . . ,εt−q , and T is the sample size. Under the null quacy of commonly used seasonal time series models hypothesis(b) thatAR(4) there model is no for ARCH monthly effect, flow the test statistic Figure 9 McLeod-Li test for the residuals from (a) ARMA(20,1) model for daily flow and is asymptotically distributed as chi-square distribution with 4.1 Causes of ARCH effects in the residuals from ARMA- q degrees of freedom. As Bollerslev (1986) suggested, it type models for daily and monthly flow should also have(b) powerAR(4) against model GARCH for monthly alternatives. flow Figure 10 shows Engle’s LM test results for the residu- From the above analyses, it is clear that although20 the resid- als from the ARMA(20,1) model for daily flow and from the uals are serially uncorrelated, they are not independent AR(4) model for monthly flow. The results also firmly in- through time. At the mean time, we notice that seasonal- dicate the existence of an ARCH effect in both the residual ity dominates autocorrelation structures of squared20 residual series. series for both daily and monthly flow processes (as shown One point that should be noticed is that although Figs. 8b, in Fig. 8). This suggests that there are seasonal variations in 9b and 10b show that for monthly flow, autocorrelations at the variance of the residual series, and we should standardize (a) 0. 3 (b) 1 0. 25 0. 8 0. 2 0. 6 0. 15 p-value p-value 0. 4 0. 1 0. 05 0. 2 0 0 21 26 31 36 4 6 8 10 12 14 16 18 20 Lag Lag

Figure 7 Ljung-Box lack-of-fit tests for (a) ARMA(20,1) model for daily flow and (b) AR(4) model for monthly flow.

(a) (b) ACF ACF 0.0 0.1 0.2 0.0 0.1 0.2

0 100 200 300 024681012 Lag Lag Figure 8 ACFs of the squared residuals from (a) ARMA(20,1) model for daily flow and (b) AR(4) model for monthly flow at Tangnaihai

60 W. Wang et al.: Testing and modelling autoregressive conditional heteroskedasticity

(a) 0. 06 (b) 0. 8 0. 7 0. 05 0. 6 0. 04 0. 5 0. 03 0. 4

p-value 0. 3 p-value 0. 02 0. 2 0. 01 0. 1 0 0. 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Lag Lag

Fig. 9. McLeod-LiFigure test 9 forMcLeod-Li the residuals test from for(a) thethe ARMA(20,1)residuals from model (a) for ARMA(20,1) daily ﬂow and (b) modelthe AR(4) for daily model flow for monthly and ﬂow.

(b)(a) AR(4) 0. 06 model for monthly flow (b) 0. 8 0. 05 0. 7 0. 6 0. 04 0. 5 0. 03 0. 4 p-value p-value 0. 02 0. 3 0. 2 20 0. 01 0. 1 0 0. 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Lag Lag

Fig. 10. Engle’s LM test for residuals from (a) the ARMA(20,1) model for daily flow and (b) the AR(4) model for monthly flow. Figure 10 Engle’s LM test for residuals from (a) ARMA(20,1) model for daily flow and the residual(b) series AR(4) from model linear models for monthly with seasonal flow standard From the above analyses, it is clear that the ARCH effect is deviations of the residuals first, then look at the standard- fully caused by seasonal variances for monthly flow, but only ized series to check(a) whether0.4 seasonal variances can explain partly(b) 1 for daily flow. Other causes, besides the seasonal vari- ARCH effects. ation in variance, of the ARCH effect in daily flow may in- 0. 8 Seasonal standard0.3 deviations of the residual series from clude the perturbations of the temperature fluctuations which 0. 6 the ARMA(20,1) model0.2 for daily flow and the AR(4) model is an influential factor for snowmelt, as well as evapotran- SD SD for monthly flow are calculated and shown in Figs. 11a and spiration,0. 4 and the precipitation variation which is the domi- 0.1 11b. They are used to standardize the residual series from nant0. factor 2 for streamflow processes. As reported by Miller the ARMA(20,1) model0 and the AR(4) model. Figure 12 (1979), when modelling a daily average streamflow series, 0 shows the ACFs of the squared0 60 standardized 120 180 residual 240 300 series 360 the residuals from a fitted AR(4) model signaled white-noise 036912 of daily and monthly flow. It is illustratedDay that, after sea- errors, but the squared residualsMonth signaled bilinearity. When sonal standardized autocorrelation, as well as the seasonality precipitation covariates were included in the model, Miller in the squared standardized residual series for monthly flow found that neither the residuals nor the squared residuals sig- is basicallyFigure removed 11 (Fig. Seasonal 12b), thestandard significant deviations autocorrela- (SD) of thenaled residuals any problems. form (a) While ARMA(20,1) we agree thatmodel the for autocorrelation still existsdaily in theflow squared and (b) standardized AR(4) model residual for seriesmonthly for flowtion existing in the squared residuals is basically caused by daily flow (Fig. 12a), despite the fact that the autocorrela- a precipitation process, we want to show that the autocorre- tions are significantly(Note: the reduced smoothed compared line within Figure Fig. 8a and11(a) the is givenlation by in the squaredfirst 8 residualsFourier harmonics can be well of described the by an seasonality inseasonal the ACF SD structure series.) is removed. This means that ARCH model, which is very close to the bilinear model (En- the seasonality, as well as the autocorrelation in the squared gle, 1982). residuals from the AR model of monthly flow series is basically caused by seasonal variances. But seasonal variances 4.2 Inadequacy of commonly used seasonal time series only explain partly the autocorrelation in the squared residu- models for modelling streamflow processes als of daily flow series. (a) (b) The residual series of daily flow and monthly flow stan- As mentioned in Sect. 3.1, SARIMA models, deseasonal- dardized by seasonal standard deviation are also tested for ized ARMA models and periodic models are commonly used ARCH effects with the McLeod-Li test and Engle’s LM to model hydrologic processes (Hipel and McLeod, 1994). Given a time series (xt ), the general form of SARIMA model, test. Figure 13 shows that the seasonally standardized ACF ACF residual series of daily flow still cannot pass the LM test denoted by SARIMA(p,d,q)×(P,D,Q)S, is 0.0 0.1 0.2 (Fig. 13a), whereas the seasonally standardized residual se- s ∇d ∇D = s ries of monthly flow pass the LM test with high p-values φ(B)8(B ) s xt θ(B)2(B )εt , (3) (Fig. 13b). The McLeod-Li0.0 0.1 test 0.2 gives 0.3 similar results. where φ(B) and θ(B) of orders p and q represent the ordi- s 0 100 200 300 nary autoregressive0510 and moving average components; 8(B ) Lag Lag Figure 12 ACFs of squared seasonally standardized residuals from (a) ARMA(20,1) model for daily flow and (b) AR(4) model for monthly flow

(a) 0. 06 (b) 0. 8 0. 05 0. 7 0. 6 0. 04 0. 5 0. 03 0. 4 p-value p-value 0. 02 0. 3 0. 2 0. 01 0. 1 0 0. 0 0. 06 0. 8 (a) 0 5 10 15 20 25 30 (b) 0 5 10 15 20 25 30 0. 7 0. 05 Lag Lag 0. 6 0. 04 0. 5 0. 03 0. 4 p-value Figure p-value 100. 02Engle’s LM test for residuals from (a) ARMA(20,1)0. 3 model for daily flow and 0. 2 0. 01 0. 1 (b) AR(4) model0 for monthly flow 0. 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Lag Lag (a) 0.4 (b) 1 0.3 0. 8 Figure 10 Engle’s LM test for residuals from (a) ARMA(20,1) model for daily flow and 0. 6 0.2 SD SD W. Wang et(b) al.: AR(4) Testing model and modelling for monthly autoregressive flow conditional heteroskedasticity0. 4 61 0.1 0. 2 0 (a) 0.4 (b) 0 1 0 60 120 180 240 300 360 036912 0.3 Day 0. 8 Month 0. 6 0.2 SD SD Figure 11 Seasonal standard deviations (SD) of the residuals0. 4 form (a) ARMA(20,1) model for 0.1 0. 2 daily flow and0 (b) AR(4) model for monthly flow 0 0 60 120 180 240 300 360 (Note: the smoothed line in Figure 11(a) is given by the036912 first 8 Fourier harmonics of the Day Month seasonal SD series.) Fig. 11. Seasonal standard deviations (SD) of the residuals from (a) the ARMA(20,1) model for daily flow and (b) the AR(4) model for monthly flowFigure (note: 11 the smoothedSeasonal line standard in (a) is givendeviations by the first (SD) 8 Fourier of the harmonics residuals of the form seasonal (a) SDARMA(20,1) series). model for daily flow and (b) AR(4) model for monthly flow (Note: the smoothed line in Figure (a)11(a) is given by the first 8 Fourier harmonics(b) of the seasonal SD series.)

ACF ACF 0.0 0.1 0.2 (a) (b) 0.0 0.1 0.2 0.3

0 100 200 300 0510 Lag Lag ACF ACF 0.0 0.1 0.2 Fig. 12. FigureACFs of 12 squared ACFs seasonally of squared standardized seasonally residuals standardized from (a) the ARMA(20,1) residuals model from for (a) daily ARMA(20,1) flow and (b) the model AR(4) model for monthly flow. for daily flow0.0 and 0.1 (b) 0.2 AR(4) 0.3 model for monthly flow and 2(Bs) of orders P and Q represent the seasonal autore- resent a day, week, month or season), we have the following 0 100 200d 300d 0510 gressive and moving average components;Lag∇ =(1−B) and PAR(p) model (Salas, 1993):Lag 21 ∇D − s D S =(1 B ) are the ordinary and seasonal difference com- p ponents. X Figure 12 ACFs of squared seasonally standardizedxn,s = residualsµs + φ j,sfrom(xv,s (a)−j −ARMA(20,1)µs−j ) + εn,s, model (5) The general form of the ARMA(p, q) model fitted to de- j=1 seasonalizedfor seriesdaily is flow and (b) AR(4) model for monthly flow where εn,s is an uncorrelated normal variable with mean zero 2 and variance σ s . For daily streamflow series, to make the model parsimonious, we can cluster the days in21 the year φ(B)xt = θ(B)εt . (4) into several groups and fit separate AR models to separate groups (Wang et al., 2004). Periodic models would per- From the model equations we know that although the sea- form better than the SARIMA model and the deseasonal- sonal variation in the variance present in the original time ized ARMA model for capturing the ARCH effect, because series is basically dealt with well by the deseasonalized ap- it takes season-varying variances into account. However, proach, the seasonal variation in variance in the residual se- as analyzed in Sect. 4.1, while considering seasonal vari- ries is not considered by either of the two models, because in ances could be sufficient for describing the ARCH effect in both cases the innovation series εt is assumed to be i.i.d. N(0, monthly flow series because the ARCH effect in monthly 2 σ ). Therefore, both SARIMA models and deseasonalized flow series is fully caused by seasonal variances, it is still models cannot capture the ARCH effect that we observed in insufficient to fully capture the ARCH effect in daily flow the residual series. series. In contrast, the periodic model, which is basically a group In summary, while the PARMA model is adequate for of ARMA models fitted to separate seasons, allows for sea- modelling the variance behaviour for monthly flow, none of sonal variances in not only the original series but also the the commonly used seasonal models is efficient enough to residual series. Taking the special case PAR(p) model (pe- describe the ARCH effect for daily flow, although PARMA riodic autoregressive model of order p) as an example of can partly describe it by considering seasonal variances. It a PARMA model, given a hydrological time series xn,s, in is necessary to apply the GARCH model to achieve the pur- which n defines the year and s defines the season (could rep- pose. 1. 0 (a) 0. 06 (b) 0. 05 0. 8 0. 04 0. 6 0. 03 62 W. Wang et al.: Testing and modelling autoregressive conditional heteroskedasticity 0. 4 p-value p-value 0. 02 1. 0 0. 01 (a) 0. 06 0. 2 (b) 0. 05 0. 8 0 0. 0 0. 04 0 5 10 15 20 25 30 0 5 10 150. 6 20 25 30 0. 03 Lag Lag 0. 4 p-value p-value 0. 02 0. 01 0. 2 0 0. 0 Figure 13 Engle’s LM test for seasonally standardized0 5 10 residuals 15 20 25from 30 (a) ARMA(20,1)0 5 10model 15 20 25 30 Lag Lag for daily flow and (b) from AR(4) model for monthly flow Fig. 13. Engle’sFigure LM test 13 for Engle’s seasonally LM standardized test for seasonally residuals from standardized(a) the ARMA(20,1) residuals model from for daily(a) ARMA(20,1) flow and (b) from model the AR(4) model for monthly flow. for daily flow and (b) from AR(4) model for monthly flow

GARCH(p, q) model has the form (Bollerslev, 1986)  ε |ψ ∼ N(0, h )  t t−1 t q p , (6) h = α + P α ε2 + P β h  t 0 i t−i i t−i i=1 i=1

where, εt denotes a real-valued discrete-time stochastic process, and ψt the available information set, p ≥0, q >0, α0 >0, αi ≥0, βi ≥0. When p=0, the GARCH(p,q) model Partial ACF

Partial ACF reduces to the ARCH(q) model. Under the GARCH(p, q) model, the conditional variance of εt , ht , depends on the squared residuals in the previous q time steps, and the condi- 0.00 0.05 0.10 0.15 0.20

0.00 0.05 0.10 0.15 0.20 tional variance in the previous p time steps. Since GARCH 0 20406080100models can be treated as ARMA models for squared residu- 0 20406080100Lagals, the order of GARCH can be determined with the method Lag for selecting the order of ARMA models, and traditional Figure 14 PACF of the squared seasonally starndardizedmodel selection residual criteria, series such asfrom Akaike ARMA(20,1) information criterion Fig. 14. PACF of the squared seasonally starndardized residual se- (AIC) and Bayesian information criterion (BIC), can also be Figure 14 PACF of the squaredries from ARMA(20,1) seasonally for daily starndardized flow. residual series from ARMA(20,1) for daily flow used for selecting models. The unknown model parameters for daily flow αi (i = 0, ··· , q) and βj (j= 1, ··· , p) can be estimated using (conditional) maximum likelihood estimation (a) (b) 1/2 5 Modelling the daily steamflow with ARMA-GARCH (MLE). Estimates of the conditional standard deviation ht error model are also obtained as a side product with the MLE method. (a) When there is(b) obvious seasonality present in the residuals (as in the case of daily streamflow at Tangnaihai), to preserve 5.1 Model building the seasonal variances in the residuals, instead of fitting the Residuals ARCH model to the residual series directly, we fit the ARCH model to the seasonally standardized residual series, which -4-20246

is deviation standard Conditional obtained by dividing the residual series by seasonal stan- Weiss (1984) proposed ARMA models with ARCH errors. 1.0 1.5 2.0 2.5 3.0

Residuals This approach is adopted and extended by many researchers dard deviations (i.e. daily standard deviations for daily flow). for modelling economic0 time 200 series 400 (e.g. Hauser 600 and 800 Kunst, 1000 Therefore,0 the 200 general 400 ARMA-GARCH 600 800 model 1000 with seasonal Day 1998; Karanasos, 2001). In the field ofDay geo-sciences, Tol standard deviations we propose here has the following form (1996) fitted a GARCH model for the conditional variance -4-20246 Conditional standard deviation standard Conditional 1.0 1.5 2.0 2.5 3.0 φ(B)x = θ(B)ε and the conditionalFigure 15 standard A segment deviation, of (a) in the conjunction seasonally with standardized residualst fromt ARMA(20,1) and (b)  εt = σszt , zt ∼N(0, ht ) an AR(2) model for the mean, to model daily mean temper- q p , (7) 0 200 400ature. 600 In thisits paper,800 corresponding we 1000 propose conditional to use0 ARMA-GARCH standard 200 deviation 400 er-  600sequenceh = α 800 estimated+ P α 1000z 2with+ ARCH(21)P β h model Day Day t 0 i t−i i t−i ror (or, for notation convenience, called ARMA-GARCH) i=1 i=1 model for modelling daily streamflow processes. where σ s is the seasonal standard deviation of εt , s is the sea- Figure 15 A segment of (a) Thethe ARMA-GARCH seasonally standardized model may be interpreted residuals as a from com- ARMA(20,1)son number depending and (b) on which season the time t belongs to. bination ofan ARMA model which is used to model mean For daily series, s ranges from 1 to 366. Other notations22 are its corresponding conditionalbehaviour, standard and an deviation ARCH model sequence which is usedestimated to model withthe ARCH(21) same as in Eqs. model (4) and (6). the ARCH effect in the residual series from the ARMA The model building procedure proceeds in the following model. The ARMA model has the form as in Eq. (4). The steps:

22 1. 0 (a) 0. 06 (b) 0. 05 0. 8 0. 04 0. 6 0. 03 0. 4 p-value p-value 0. 02 0. 01 0. 2 0 0. 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Lag Lag

Figure 13 Engle’s LM test for seasonally standardized residuals from (a) ARMA(20,1) model for daily flow and (b) from AR(4) model for monthly flow

Partial ACF 0.00 0.05 0.10 0.15 0.20

0 20406080100 Lag Figure 14 PACF of the squared seasonally starndardized residual series from ARMA(20,1) W. Wangfor et al.: daily Testing flow and modelling autoregressive conditional heteroskedasticity 63

(a) (b) Residuals -4-20246 Conditional standard deviation standard Conditional 1.0 1.5 2.0 2.5 3.0

0 200 400 600 800 1000 0 200 400 600 800 1000 Day Day

Fig. 15. A segment of (a) the seasonally standardized residuals from ARMA(20,1) and (b) its corresponding conditional standard deviation sequence estimatedFigure 15 with A the segment ARCH(21) of model.(a) the seasonally standardized residuals from ARMA(20,1) and (b) its corresponding conditional standard deviation sequence estimated with ARCH(21) model

(a) (b)

22 ACF ACF 0.0 0.1 0.2 0.0 0.1 0.2

0 20406080100 0 20406080100 Lag Lag Fig. 16.FigureACFs of16(a) ACFsthe standardized of (a) the residuals standardized and (b) squared residuals standardized and (b) residuals squared from standardized the ARMA(20,1)-ARCH(21) residuals from model. The standardization is accomplished by dividing the seasonally standardized residuals from ARMA(20,1) by the conditional standard deviation estimatedARMA(20,1)-ARCH(21) with ARCH(21). model. The standardization is accomplished by dividing the seasonally standardized residuals from ARMA(20,1) by the conditional standard deviation 1. Logarithmize and deseasonalize the original flow series; the squared seasonally standardized residuals are shown in estimated with ARCH(21). Fig. 12a and Fig. 14, respectively. According to the AIC, as 2. Fit an ARMA model to the logarithmized and deseason- well as the ACF and PACF structure, a GARCH(0,21) model, alized flow series; i.e. ARCH(21) model, which has the smallest AIC value is 3. Calculate seasonal standard deviations of the residuals selected. Therefore, the prelimilary ARMA-GARCH model obtained from ARMA model, and seasonally standard- fitted to the daily streamflow series at Tangnaihai is com- ize the residuals with the first 8 Fourier harmonics of posed of an ARMA(20,1) model and an ARCH(21) model. the seasonal standard deviations; The model is constructed with statistics software S-Plus (Zivot and Wang, 2003).

4. Fit a GARCHACF model to the seasonally standardized

residual series. 5.2Partial ACF Model diagnostic and modiﬁcation

For forecasting0.0 and simulation, 0.1 inverse 0.2 transformation (in- If the ARMA-GARCH model is successful in modelling the 0.0 0.1 0.2 cluding logarithmization and deseasonalization) is needed. serial correlation structure in the conditional mean and con- When forecasting, the ARMA part of the ARMA-GARCH ditional variance, then there should be no autocorrelation left model forecasts future0 mean 20406080100 values of the underlying time se- in both0 the 20406080100residuals and the squared residuals standardized Lag Lag ries following the traditional approach for ARMA prediction, by the estimated conditional standard deviation. whereas the GARCH part gives forecasts of future volatility, A segment of the seasonally standardized residual se- especiallyFigure over short 17 horizons.ACF and PACF of seasonally standardizedries from residuals the ARMA(20,1) from ARMA(20,1) model and itsmodel corresponding Following the above-mentioned steps, a preliminary conditional standard deviation sequence estimated with the ARMA-GARCH model is ﬁtted to the daily streamﬂow ARCH(21) model are shown in Figs. 15a and 15b. We series at Tangnaihai. The ACF and PACF structure(a) of standardize the seasonally standardized(b) residual series from ACF ACF 0.0 0.1 0.2 0.0 0.1 0.2

0 20406080100 0 20406080100 Lag Lag Figure 18 ACFs of (a) the second-residuals and (b) the squared second-residuals from the ARMA(20,1)-AR(16) model

23 (a)(a) (b)(b) ACF ACF ACF ACF 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2

0 20406080100 00 20406080100 20406080100 Lag LagLag Figure 16 ACFs of (a) the standardized residuals andand (b)(b) squaredsquared standardizedstandardized residualsresiduals fromfrom ARMA(20,1)-ARCH(21) model. The standardizationstandardization isis accomplishedaccomplished byby dividingdividing thethe seasonally standardized residuals from ARMA(20,1)ARMA(20,1) byby thethe conditionalconditional standardstandard deviationdeviation

64estimated with ARCH(21). W. Wang et al.: Testing and modelling autoregressive conditional heteroskedasticity ACF ACF Partial ACF Partial ACF 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2

0 20406080100 00 20406080100 20406080100 Lag LagLag Fig. 17. ACF and PACF of seasonally standardized residuals from the ARMA(20,1) model. Figure 17 ACF and PACF of seasonally standardizedstandardized residualsresiduals fromfrom ARMA(20,1)ARMA(20,1) modelmodel

(a)(a) (b)(b) ACF ACF ACF ACF 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2

0 20406080100 00 20406080100 20406080100 Lag Lag Lag Lag Fig. 18. ACFsFigure of (a) 18the ACFs second-residuals of (a) the and (b)second-residualsthe squared second-residuals andand (b)(b) fromthethe thesquaredsquared ARMA(20,1)-AR(16) second-residualssecond-residuals model. fromfrom thethe ARMA(20,1)-AR(16) model the ARMA(20,1) model by dividing it by the estimated flow of the Umpqua River near Elkton and the Wisconsin conditional standard deviation sequence. The autocorrela- River near Wisconsin Dells, available on the USGS website tions of the standardized residuals and squared standardized http://water.usgs.gov/waterwatch), which have strong2323 sea- residuals are plotted in Fig. 16. It is shown that although sonality in the ACF structures of their original series, as well there is no autocorrelation left in the squared standardized as their residual series. To handle the problem of the weak residuals, which means that the ARCH effect has been re- correlations, an additional ARMA model is needed to model moved (Fig. 16b), however, in the non-squared standardized the mean behaviour in the seasonally standardized residuals, residuals of daily flow significant autocorrelation remains and a GARCH is then fitted to the residuals from this ad- (Fig. 16a). ditional ARMA model. Therefore, we obtain an extended Because the GARCH model is designed to deal with the version of the model in Eq. (7) as conditional variance behavior, rather than mean behavior, the φ(B)x = θ(B)ε autocorrelation in the non-squared residual series must arise  t t  εt = σsyt from the seasonally standardized residuals obtained in step 3  0 0 φ (B)yt = θ (B)zt , zt ∼N(0, ht ) , (8) of the ARMA-GARCH model building procedure. Therefore  q p  = + P 2 + P we revisit the seasonally standardized residuals. It is found ht α0 αizt−i βiht−i that although the residuals from the ARMA(20,1) model i=1 i=1 present no obvious autocorrelation as shown in Fig. 6a, weak where yt is the seasonally standardized residuals from the but significant autocorrelations in the residuals are revealed first ARMA model, zt is the residuals (for notation conve- after the residuals are seasonally standardized, as shown by nience, we call it second-residuals) from the second ARMA the ACF and PACF in Fig. 17. We refer to this weak autocor- model fitted to yt . relation as the hidden weak autocorrelation. An AR(16) model, whose autoregressive order is cho- The mechanism underlying such weak autocorrelation is sen according to AIC, is fitted to the seasonally standard- not clear yet. Similar phenomena are also found for some ized residuals from the ARMA(20,1) model of the daily flow other daily streamflow processes (such as the daily stream- series at Tangnaihai, and we obtain a second-residual se-

(a) (b) ACF W. Wang et al.: Testing and modelling autoregressive conditional heteroskedasticityACF 65 0.0 0.1 0.2 0.0 0.1(a) 0.2 (b)

0 20406080100 020406080100 Lag Lag ACF ACF

0.0 0.1 0.2 0.0 0.1 0.2 Figure 19 ACFs of the (a) standardized second-residuals and (b) squared standardized second- 0 20406080100 020406080100 residualsLag from ARMA(20,1)-AR(16)-ARCH(21) Lag model. The second-residuals are obtained Fig. 19. ACFs of (a) the standardized second-residuals and (b) the squared standardized second-residuals from the ARMA(20,1)-AR(16)- ARCH(21) model. The second-residualsfrom areAR(16) obtained fitted from AR(16) to the ﬁtted seasonally to the seasonally standardized standardized residuals residuals from ARMA(20,1). form ARMA(20,1).

Figure 19 ACFs of the (a) standardized second-residuals and (b) squared standardized second- ries from this AR(16) model. The autocorrelations of the 1 second-residual series and the squared second-residual series residuals from ARMA(20,1)-AR(16)-ARCH(21) model.0. 8 The second-residuals are obtained from the ARMA(20,1)-AR(16) combined model are shown in Fig. 18.from From AR(16) visual inspection,fitted to the we seasonally find that no autocorre-standardized residuals0. 6 form ARMA(20,1). lation is left in the second-residual series, but there is strong 0. 4 autocorrelation in the squared second-residual series which p-value 1 0. 2 indicates the existence of an ARCH effect. 0. 8 0 0. 6 0 5 10 15 20 25 30 0. 4 Lag Because the squared second-residualp-value series has similar 0. 2 ACF and PACF stucture to the seasonally standardized resid- Fig. 20. Engle’s LM test for the standardized second-residuals from uals from the ARMA(21,0) model, the same0 structure of the ARMA(20,1)-AR(16)-ARCH(21) model. the GARCH model, i.e. an ARCH(21)Figure model, 200 Engle’s is 5 fitted 10 toLM 15test for 20 the 25 standardized 30 second-residuals from the ARMA(20,1)- Lag the second-residual series. Therefore, the ultimate ARMA- GARCH model fitted to the daily streamflow at Tangnai- 6 ConclusionsAR(16)-ARCH(21) model hai is ARMA(20,1)-AR(16)-ARCH(21),Figure 20 Engle’s LM test for composed the standardized of an second-residuals from the ARMA(20,1)- ARMA(20,1) model fitted to logarithmized and deseasonal- The nonlinear mechanism conditional heteroskedasticity in ized series, an AR(16) model fitted to the seasonallyAR(16)-ARCH(21) stan- hydrologic model processes has not received much attention in the dardized residuals from the ARMA(20,1) model, and an literature so far. Modelling data with time varying condi- ARCH(21) model fitted to the second-residuals from the tional variance could be attempted in various ways, includ- AR(16) model. ing nonparametric and semi-parametric approaches (see Lall, 1995; Sankarasubramanian and Lall, 2003). A parametric approach with ARCH model is proposed in this paper to de- We standardize the second-residual series with the con- scribe the conditional variance behavior. ARCH-type mod- ditional standard deviation sequence obtained with the els which originate from econometrics can provide accurate ARCH(21) model. The autocorrelations of the standard- forecasts of variances. As a consequence, they can be ap- ized second-residuals and the squared standardized second- plied to such diverse fields as water management risk anal- residuals are shown in Fig. 19. Compared with Fig. 16, the ysis, prediction uncertainty analysis and streamflow series autocorrelations are basically removed for both the squared simulation. and non-squared series, although the autocorrelation at lag The existence of conditional heteroskedasticity is verified 1 of the standardized second-residuals slightly exceeds the in the residual series from linear models fitted to the daily 5% significance level. The McLeod-Li test and the LM- and monthly streamflow processes of the upper Yellow River test (shown in Fig. 20) for standardized second-residuals also with the McLeod-Li test and the Engle’s Lagrange Multi- confirm that the ARCH(21) model fits the second-residual se- plier test. It is shown that the ARCH effect is fully caused ries well. The small lag-1 autocorrelation in the standardized by seasonal variation in variance for monthly flow, but sea- second-residual series (shown in Fig. 19) is a hidden autocor- sonal variation in variance only partly explains the ARCH relation covered by conditional heteroskedasticity. This au- effect for daily streamflow. Among three types of conven- tocorrelation can be further modeled with another AR model, tional seasonal time series model (i.e. SARIMA, deseasonal-24 but because the autocorrelation is very small, it could be ne- ized ARMA and PARMA), none of them is efficient enough 24 glected. to describe the ARCH effect for daily flow, although the 66 W. Wang et al.: Testing and modelling autoregressive conditional heteroskedasticity

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P.: On a measure of lack of fit in time dicting weather two to three days in advance (for example, series models, Biometrika, 65, 297–303, 1978. see http://weather.gov/rivers tab.php), the use in developing McLeod, A. I. and Li, W. K.: Diagnostic checking ARMA time an ARMA-GARCH model would be limited. However, be- series models using squared residual autocorrelations, Journal of Time Series Analysis, 4, 269–273, 1983. cause (1) on the one hand, the relationship between runoff Miller, R. B.: Book review on “An Introduction to Bilinear Time and rainfall and temperature is hard to capture precisely by Series Models” by Granger, C. W. and Andersen, A. P., J. Amer. any model so far; (2) on the other hand, usually there are Statist. Ass., 74, 927, 1979. not enough rainfall data available to fully capture the rain- Hauser, M. A. and Kunst, R. M.: Fractionally Integrated Models fall spatial pattern, especially for remote areas, such as Ti- With ARCH Errors: With an Application to the Swiss 1-Month bet Plateau, and (3) the accuracy of the weather forecasts for Euromarket Interest Rate, Review of Quantitative Finance and these areas are very limited, the ARCH effect cannot be fully Accounting, 10, 95–113, 1998. removed even after limited rainfall data and temperature data Salas, J. D.: Analysis and modelling of hydrologic time series. In: are included in the model. Therefore, the ARMA-GARCH Handbook of Hydrology, edited by Maidment, D. R., McGraw- model would be a very useful addition in terms of statistical Hill, 19.1–19.72, 1993. modelling of daily streamflow processes for the hydrological Sankarasubramanian, A. and Lall, U.: Flood quantiles in a chang- ing climate: Seasonal forecasts and causal relations, Water Re- community. sources Research, 39, 5, Art. No. 1134, 2003. Tol, R. J. S.: Autoregressive conditional heteroscedasticity in daily Acknowledgements. We are very grateful to I. McLeod and an temperature measurements, Environmetrics, 7, 67–75, 1996. anonymous reviewer. Their comments, especially the detailed Wang, W., Van Gelder, P. H. A. J. M., and Vrijling, J. K.: Peri- comments from the anonymous reviewer, are very helpful to odic autoregressive models applied to daily streamflow. In: 6th improve the paper considerably. International Conference on Hydroinformatics, edited by Liong, S. Y., Phoon, K. K. and Babovic, V., Singapore, World Scientific, Edited by: B. Sivakumar 1334–1341, 2004. Reviewed by: I. McLeaod and another referee Weiss, A. A.: ARMA models with ARCH errors, Journal of Time Series Analysis, 5, 129–143, 1984. Zivot, E. and Wang, J.: Modelling Financial Time Series with S- PLUS, New York, Springer, 2003.