Intraday Periodicity and Volatility Forecasting: Evidence from Indian Crude Oil Futures Market

2.1. Identification of Intraday Periodicities

In order to identify the intraday patterns in return volatility, we draw (Figure 1) the average of 10-minute returns and absolute returns across all intervals. We use absolute return as the proxy for return volatility. Pagan and Schwert (1990) among others, pointed out that absolute return reduces the influence of heavy tails and occasional shocks, which is in accordance with the literature (e.g., see Evans & Speight, 2010). ¹

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Figure 1.

Source: Author’s estimates.

Return volatility is high at the opening then it declines sharply and again starts rising around 1:00 pm with a high steep around 8:00 pm and then declines till the market closes. A sudden rise and spike in return volatility around 6:00 pm (12:30 GMT) may be from the contagion effect ² of US commodity markets as trading in the NYMEX, a leading exchange for crude oil in US, starts at around 12:00 GMT. Regularly scheduled macroeconomic announcements in the US are announced at 12:30 GMT, which may influence the Indian commodity futures markets. Decline in return volatility from 10:00 pm until market closing may be due to low business activities during this time as banks, local precious metals markets and other spot markets are closed.

Further, to characterize intraday periodicity in return volatility, we plotted a day-correlogram, which is shown in Figure 2.

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Figure 2.

Source: Author’s estimates.

The correlogram for 10-minute returns (left panels) in Figure 2 does not show any discernible patterns and violates the 95% confidence interval band at several lags, but it does not suggest any particular order for the return process. The day-correlogram for return volatility (right panel) shows a spike at first lag, declines sharply for higher lags and again starts rising until the end creating a distorted U shape. If these patterns are consistent across all the days, it may cause distortion in the long-run persistence of return volatility.

Several studies such as Poshakwale (1996), Murry and Zhu (2004), and Nath and Dalvi (2005) find a day-of-the-week effect in returns and return volatility. To be more conclusive and to capture the day-of-the-week effect, we plotted a 5-day correlogram.

1.3. Five-day Correlogram for 10-minute Absolute Returns

Figure 3 shows a correlogram for weekdays, that is, from Monday to Friday. The autocorrelation patterns in the correlogram show the presence of cyclical patterns in return volatility indicating the presence of deterministic intraday patterns in return volatility. The 5-day correlogram shows a small rise instead of decay in autocorrelations for Fridays. An increase in autocorrelations on Friday indicates the presence of a day-of-the-week effect. These results confirm the patterns observed in previous studies on foreign exchange markets (e.g., Andersen & Bollerslev, 1997, 1998; Bollerslev et al., 2000; Andersen et al., 2000).

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Figure 3.

Source: Author’s estimates.

2.1. Modeling Intraday Periodicity

From earlier studies by Wood et al. (1985), Ekman (1992) and Daigler (1997), periodicity in return, volatility is estimated as the average of return volatility across the trading days. In this study, we estimate intraday periodicity in return volatility by using two time series approaches—FFF and cubic spline. The advantage of the time series procedures such as FFF and cubic spline is that they use complete time series data in estimating the desired model and provides estimates that are more reliable.

2.1.1. Flexible Fourier Form

Andersen and Bollerslev (1998) pointed out that the volatility process can be assumed to be driven by the calendar effect, macroeconomic news announcement effect and volatility persistent factor. Intraday returns can be decomposed as

R t , n = E ( R t , n ) + σ t , n * s t , n * Z t , n

(1)

where R_t,n is the intraday return for the nth interval on day t. is the price of the asset at the end of nth interval on day t. E(R_t,n) denotes the expected return of nth interval on day t. N is the number of time intervals per day. σ_t,n is the remaining long memory volatility component for the nth interval on day t. S_t,n denotes the intraday periodicity component which includes the calendar effect and macroeconomic announcement effects observed during the nth interval on day t. Z_t,n is an i.i.d error term with a mean of zero and unit variance that is assumed to be independent from the daily volatility process.

The components of equation (1) are not separately identifiable. Using the logarithm and rearranging the terms, we get

x ^ t , n ≡ 2 log ​ | ​ R t , n − E ( R t , n ) ​ | − log σ t , n 2 = 2 log s t , n + log Z t , n 2 ,

(2)

E(R_t,n) may be approximated with the average of intraday returns across trading days, whereas σ t , n 2 can be estimated by using the daily conditional variance of σ t , n 2 = σ t 2 / N , where N is the total number of intervals on tth day. x ^ t , n can be estimated through the following regression:

x ^ t , n = ∑ ​ ​ σ t j [ μ o j + μ 1 j n N 1 + μ 2 j n N 2 + ∑ ​ λ i j I n = d i + ∑ ​ ​ ( γ i j cos 2 π i n N + δ i j sin 2 π i n N ) ]

(3)

where P denotes the tuning parameter and refers to the order of expansion. P is chosen based on the information criteria such as Akike information criterion and Bayesian information criterion. N₁ = (N+1)/2 and N₂ = (N+1) (N+2)/2 are normalizing constants. Each of J+1 FFF are parameterized with the non-linear components in n (μ coefficients) and the number of sinusoids (γ and δ coefficients). J>0 allows the possible interaction between daily return volatility and periodic components. λ _ij is the coefficient corresponding to dummies which may be introduced in the time interval during a day, corresponds to calendar events or news announcements. The value of J is determined on the basis of the best fit of the intraday periodic component, information criteria and the parsimony of the model.

Intraday periodic component is estimated as follows:

s t , n = T × exp ( x t , n ^ 2 ) ∑ ​ ∑ ​ ( x t , n ^ 2 ) .

(4)

The filtered return series is obtained by normalizing the returns with the estimated periodic component as

R t , n ─ = R t , n s t , n ^ .

(5)

2.1.2. Cubic Spline

Although the FFF method provides a parsimonious model, but it is not flexible in its functional form and assumes equality at the starting and closing of the periodic cycle (Taylor, 2004a). On the other hand, cubic spline allows for sharp peaks and troughs observed in a time series (Evans & Speight, 2010)

In the cubic spline approach, intraday periodicity patterns are obtained by fitting a three-degree polynomial between the predefined points called ‘knots’. Knots are positioned in such a way so as it can capture the complexities in intraday periodicities, for example, knots may be positioned at the opening and closing of dominant markets to capture the opening and closing effects of those markets. Following Evans and Speight (2010), operational regression equation to estimate the intraday periodicity component can be written as

x ^ t , n = ∑ j = 0 J σ t j [ μ 0 , j + μ 1 , j n N 1 + μ 2 , j n N 2 + ∑ i = 1 D λ i j I i ( t , n ) + ∑ i = 1 P ( α 1 , i D i ( n − l i N ) + α 2 , i D i ( n − l i N ) 2 + α 3 , i D i ( n − l i N ) 3 ) ] ,

(6)

where N₁ = (N+1)/2 and N₂ = (N+1) (N+2)/2 are normalizing constants. l_i denotes the interval during the day in which ith knot (i=1,2,…,P) is placed. D_i’s are dummy variables assigned the value one if l_i > n, otherwise zero. N is the number of intervals during the trading day. α_1,i, α_2,i, α_3,i are the coefficients to be estimated. I_i(t,n) is the dummy variable taking the value of 1 for any calendar effect that occurs on day t and macroeconomic announcements that occurs at the nth interval on the tth day.

We employ five knots during the trading hours in order to capture the deterministic periodicity component. The first knot is positioned corresponding to the opening of the Indian commodity market at 10:00 am. The second knot is positioned at 11:30 am when the European markets open. ³ The third knot is positioned at 4:00 pm to capture the impact of closing the Indian equity and bond markets. The fourth knot is positioned at 5:30 pm to capture the impact of the opening of US markets ⁴ and finally we position a knot at around 8:00 pm to capture the impact of the closing/slowing down of business activities as banks and financial institutions close most of their business activities at around this time.

Nelson (1991) proposed an exponential-GARCH (EGARCH) model where conditional variance is a function of past residuals and conditional variance. Nelson assumed that the errors follow a generalized error distribution (GED). He also pointed out that the use of GED improves the estimates of models in the presence of high kurtosis and fat-tails, which are the stylized facts of the return series. The conditional variance equation of the EGARCH model can be expressed as

log ( σ t 2 ) = ω + β log ( σ t − 1 2 ) + γ ε t − 1 ( σ t − 1 2 ) + α [ | ​ ε t − 1 ​ | ( σ t − 1 2 ) − 2 π ] .

(7)

The EGARCH model has several advantages over the GARCH models. First, there is no need to impose restrictions to ensure non-negativity in the parameters since it always provides a positive conditional variance (Brooks, 2008). Second, the asymmetric effect or leverage effect, that is, higher volatility levels observed following the negative shocks as compared to positive shocks, can be tested through this model. Haniff and Pok (2010) pointed out that EGARCH produced superior results compared to other GARCH and threshold GARCH (TGARCH) models. In this study, we have implemented two approaches—a sequential estimation approach and PGARCH models augmented with the FFF and cubic spline for volatility forecasting.

2.2.2. Periodic Models

Bollerslev and Ghysels (1996) proposed periodic-GARCH (PGARCH) allow the time dependency and periodicity in the conditional variance terms of GARCH model. Variance equation for the PGARCH model can be expressed as

σ t 2 = ω s ( t ) + α s ( t ) ε t − 1 2 + β s ( t ) ,

(8)

where s(t) denotes the stage of the periodic cycle. Within periodic cycles, a periodic model allows the coefficients in the variance equation to take a different value in each period where the change in volatility is expected. One way to do this is with the inclusion of a set of dummy variables. Taylor (2004a) mentioned that, ‘these models are computationally expensive and are required to estimate a large number of parameters if there are many time periods within the periodic cycle’.

Haniff and Pok (2010) used a periodic version for the EGARCH models and found that EGARCH models provide a better fit than the GARCH and TGARCH models. The FFF and cubic spline versions of the PGARCH models are discussed in the following two sections:

2.2.2.1. Periodic EGARCH Models with FFF

The intraday periodic component can be introduced in the variance equation of Bollerslev and Ghysels’ (1996) periodic models without complicating the volatility estimation process (Haniff & Pok, 2010; Taylor, 2004a). The conditional variance equation in this framework can be expressed as

log ( σ t 2 ) = ω ′ F t + β log ( σ t − 1 2 ) + γ ε t − 1 ( σ t − 1 2 ) + α [ | ​ ε t − 1 ​ | ( σ t − 1 2 ) − 2 π ] ,

(9)

w h e r e F ′ t = [ 1 , f ′ 1 , t , … … , f ′ j , t , … .. , f ′ j , t ]

a n d f ′ j , t = sin ( 2 π j s ( t ) N ) + cos ( 2 π j s ( t ) N ) ,

(10)

jϵ (1, 2, … J) denotes the intraday periodic cycle which may be chosen on the basis of the information criteria. N is the number of intervals during the trading day. S (t) denotes the tth interval. ⁵

2.2.2.2. Periodic EGARCH Models with Cubic Spline

The conditional volatility equation of cubic spline version of the periodic model in the EGARCH framework can be expressed as

log ( σ t 2 ) = ω ′ G t + β log ( σ t − 1 2 ) + γ ε t − 1 ( σ t − 1 2 ) + α [ | ​ ε t − 1 ​ | σ t − 1 2 − 2 π ] ,

(11)

w h e r e G ′ t = [ 1 , g ′ 1 , t , … , g ′ j , t , . .. , g ′ j , t ]

and g ′ j , t

= ∑ m = 1 M [ α 1 , m D m ( n − l m N ) + α 2 , m D m ( n − l m N ) 2 + α 3 , m D m ( n − l m N ) 3 ] ,

(12)

where l_m denotes the interval of the day in which knot m (m=1,2, …, M) is placed, the knots are chosen by a priori-based criteria on the underlying intraday pattern in volatility or according to the expected change in intraday return volatility. ⁶

We determine the best model among the all five fitted models by comparing the in-sample model-fit and out-of-sample forecasting performance. ⁷ For the in-sample fit, we use three criteria: log likelihood (LL), Akaike information criterion (AIC) and Bayesian information criterion (BIC). The model which produces the maximum LL and minimum AIC and BIC among the competing models is considered best.

Accurate forecasting of the return volatility is important given its application in risk management and asset pricing. The model that provides forecasts of return volatility nearest to the realized volatility is considered the best. We compared all five models with the realized volatility. We considered absolute returns as a measure of realized volatility since it is less affected with the outliers. These outliers occur occasionally and they cannot be assumed repetitive over time (Martens, Chang, & Taylor, 2002).

We use several performance measures to ensure the robustness of the results. First, we compare the correlation between realized volatility and forecasted volatility. Second, we compare the mean forecast with the average forecast in the forecasted period. Third, we compare the mean absolute error (MAE) for all the models. The model that produces the minimum MAE is considered the best forecasting model. The MAE for N*T forecasts can be expressed as

M A E = 1 N * T ∑ ​ ​ ∑ ​ ​ | | R n , t | − σ n , t ^ |

where σ n , t ^ is the estimated volatility for nth interval on day t, n=1,2, …, N and t=1, 2, … T. |R_n,t| is the realized volatility.

Fourth, we estimate the root mean squared forecast error (RMSE), the N*T forecasts RMSE is given as

R M S E = 1 N * T ∑ ​ ​ ∑ ​ ​ · ( | R n , t | − σ n , t ^ ) 2 .

Finally, we compare the model-based forecasts by using Diebold and Mariano’s (1995) asymptotic test. The Diebold and Mariano test provides a measure of predictive accuracy for two competing predictions for a given loss function, by comparing them with the actual time series (we use the MAE as the loss function). This test verifies the null hypothesis that forecasts if two competing models are equally accurate. The advantage of the Diebold and Mariano test is that it provides robust results in case of non-normal and serially correlated forecasts errors (Taylor, 2004a).

3.1. Estimation of Intraday Periodicities

Economic theories do not provide any theoretical prediction about the intraday periodicities. While modelling intraday periodicities using FFF, the choice of the number of sinusoids is important. We choose appropriate models based on information criteria such as the AICs and BICs. We compare the information criteria for intraday periodicities estimated by FFF and explained in a regression equation (3) in conjunction with equation (2) for different values of P from a grid (1 to 10) and for J=0. ⁸ For models with J=1, the model cannot be applied directly for forecasting intraday volatility since the future daily return also needs to be estimated. We ignore the interaction between daily return volatility and intraday periodicity for the parsimony of the model. ⁹

Figure 4 shows the intraday patterns in absolute returns estimated by using the FFF (left panel) and cubic spline (right panel) approach. It is apparent that both FFF and cubic spline provides a close fit over average intraday periodicity patterns. Table 2 provides the estimate of FFF and cubic spline coefficients, the calendar effect and macroeconomic announcements. ¹⁰ A high F-statistic indicates the presence of recurring and consistent patterns in volatility. The adjusted R-squares obtained by the cubic spline (0.107) method is higher than the FFF (0.099), which indicates that cubic spline provides a better fit for the intraday periodicities.

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Figure 4.

Source: Author’s estimates.

Figure 5 shows the 5-day correlogram for raw absolute returns |R_t,n| (left panels) and filtered absolute returns | R t , n | / s t , n ^ (right panels), where s t , n ^ denotes the normalized estimate for deterministic periodic component estimated by using FFF ¹¹ whereas Figure 6 shows the 5-day correlogram for the raw returns and filtered returns obtained by using cubic spline approach. The implied hyperbolic decay (j^2d–1) patterns are also shown in the correlogram for the raw and filtered returns in Figures 5 and 6.

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Table 2.

Estimated Coefficients by Using FFF and Cubic Spline

Coefficients	FFF	Coefficients	CS
λ11 δ11 λ21 δ21 λ31 δ31	–0.731*** 0.071*** –0.466*** –0.180*** –0.164*** 0.094***	α11 α21 α31 α12 α22 α32 α13 α23 α33 α14 α24 α34 α15 α25 α35	–46.74*** 1284.97*** –8578.10*** 9.50*** –38.31*** 33.15*** –6.05*** 53.00 –199.01 –4.82** 83.16** –380.91* –0.78 –65.44*** 208.85***
Macroeconomic announcement Effect
Government auction Export import Index of industrial production Monetary policy Quarterly GDP T-Bills yield Wholesale price index Oil & natural gas production	0.008 0.22 0.198 0.035 0.103 0.108 0.757*** 0.252		–0.050.220.13–0.010.070.080.66***0.21
NF-StatAdj. R-Sq	107,250619.330.099		107,250429.5520.108

Source: Author’s estimates.

Notes: The table reports the estimated coefficients with the p value for equation (2), using equations (3) and (6) as FFF and cubic spline specifications for the intraday periodicities. For the brevity of space, we have not reported the coefficients of the weekdays. Returns are estimated from 10-minute logarithmic bid-ask quotes from June 2005 to December 2010.

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Figure 5.

Source: Author’s estimates.

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Figure 6.

Source: Author’s estimates.

The correlogram for the raw returns (left panels) shows the presence of strong cyclical patterns. These cyclical patterns almost disappeared in the correlogram for filtered returns (right panels). The correlogram for filtered returns (right panels) shows that the 5-day correlogram is very close to the theoretical hyperbolic decaying patterns indicating that high-frequency returns exhibit high volatility persistence.

The filtration of high-frequency absolute returns determined by using FFF and cubic spline reveal interesting characteristics. After the adjustments for intraday periodicities, a long memory feature in the intraday return process emerges as an inherent feature of the return process. This reflects the importance of intraday periodicities in revealing the true characteristics of the returns. Furthermore, it is also important to test if the removal of intraday periodicities in return volatility results in a consistent parameter estimation of the conditional volatility models.

The presence of strictly positive autocorrelations and hyperbolic declining patterns in the correlation allow for estimating the degree of volatility persistence or the parameter of fractional integration (denoted by ‘d’). The relationship between autocorrelations ρ_j, of a long-memory process for large lags j, can be expressed as ρ_j ≈ cj^2d–1 where c denotes the factor of proportionality (Bollerslev et al., 2000). The operational regression equation to estimate the fractional integration parameter is

log ( ρ ​ j ) ≈ log ( c ) + ( 2 d − 1 ) log ( j )

Estimates of the fractional integration parameter are between (–0.5, 0.5) which shows that returns are neither stationary I (0) nor of order I (1). The value of the parameter of fractional integration d’s comes out to be 0.475.

3.3. Out-of-Sample Forecast

The estimated coefficients associated with non-periodic (M1), sequential models (M2 and M3) and periodic models (M4 and M5) are used to generate 10-minute return volatility forecasts for all the intervals during a trading day. Table 3 reports the estimated coefficients of FFF and cubic spline for the in-sample period. Intraday and interday periodicities in volatility estimated for the in-sample period are assumed consistent for the out-of-sample period.

The EGARCH (1,1) parameters for each model (M1 to M5) are re-estimated after each trading day (i.e., after 78, 10-minute intervals) using a rolling window of one year which constitutes around 250 days, starting from July 1, 2009 to June 30, 2010. The volatility forecast for 100 days starting from July 1, 2010 has been forecasted. This generates 7,800 forecasts.

The relative performances of all the models are compared by using four measures, which include correlation between forecasted volatility and realized volatility, mean forecast, MAE and root mean squared error (RMSE). We also make use of the asymptotic test proposed by Diebold and Mariano (1995). Table 4 reports the outcome of the mean forecast, correlation with the realized volatility, MAE and RMSE.

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Table 4.

Out-of-sample Forecasting Performance

	GARCH	Sequential Approach		PGARCH
		FFF	Cubic Spline	FFF	Cubic Spline
	(M1)	(M2)	(M3)	(M4)	(M5)
Correlation with realized volatility [R(t,n)]	0.254	0.322	0.337	0.342	0.339
Mean forecast (realized: 0.0963)	0.143	0.149	0.148	0.145	0.146
Mean absolute error	0.086	0.088	0.086	0.084	0.085
Root mean squared error	0.108	0.113	0.111	0.108	0.109

Source: Author’s estimates.

Notes: The table reports the out-of-sample errors associated with the volatility forecast for the 10-minute volatility for the five models M1 to M5. 100 days prediction is made for the 10-minute volatility forecast by using a rolling window of one year. Model M1 is estimated for raw returns through EGARCH(1,1) discussed in equation (7). Models M2 and M3 are estimated by using sequential estimation approach, where in the first step, returns are filtered from the intraday periodicities by using FFF discussed in equations (5) in conjunction with equations (2) to (4) and cubic spline given in equation (6). In the second step filtered returns are used to forecast volatility with the EGARCH (1,1). Models M4 and M5 provide the volatility forecast based on the FFF and cubic spline version of PGARCH model provided in equations from (9) to (12).

The non-periodic EGARCH model as evidenced by MAE, RMSE provide the worst out-of-sample forecasts compared to the sequential and periodic models. Forecasts obtained by the sequential estimation approach, particularly the cubic spline version (M3), is far better than the non-periodic model (M1), since the correlation between the estimated volatility and realized volatility increases from 0.25 to 0.33. The periodic models (M4 and M5) even perform marginally better than the sequential models (M2 and M3). The FFF version of the periodic model (M4) produces a minimum MAE and RMSE. However, the MAE and RMSE are very close across the models; therefore, to conclude which model is better, we need to test the statistical significance of difference between the MAE produced by the different models. Finally, we employ the Diebold and Mariano asymptotic test and to compare the forecasting performance of all the five competing models, where MAE is considered as the loss function. The main advantage of the Diebold and Mariano test is that it is robust enough to account for serially correlated and non-normally distributed errors (Taylor, 2004a). The results obtained from the Diebold–Mariano test are given in Table 5 with the corresponding level of significance (p values). It is apparent that the FFF and cubic spline versions of the PGARCH model outperform the non-periodic (M1) models and sequential models (M2 & M3).

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Table 5.

Diebold and Mariano Test to Measure Relative Forecasting Performance

	(M1)	(M2)	(M3)	(M4)	(M5)
M1	––	0.0021(0.0190)	0.0008(0.4101)	–0.0014(0.0693)	–0.0008(0.3561)
M2	–0.0021(0.0000)	––	–0.0013(0.0000)	–0.0035(0.0000)	–0.0029(0.0000)
M3	–0.0008(0.0000)	0.0013(0.0000)	––	–0.0022(0.0000)	–0.0016(0.0000)
M4	0.0014(0.0000)	0.0035(0.0000)	0.0022(0.0000)	––	0.0006(0.0013)
M5	0.0008(0.0000)	0.0029(0.0000)	0.0016(0.0000)	–0.0006(0.0013)	––

Source: Author’s estimates.

Notes: Table reports the Diebold and Mariano test statistics for all pair-wise combinations of five models estimated. In model M1 forecast is predicted by using EGARCH (1,1) for the raw returns. M2 and M3 are the FFF and cubic spline version of the sequential estimation approach discussed in equations (2) to (6). Model M4 and M5 are the FFF and cubic spline version of the PGARCH given in equations from (9) to (12). For the comparison of the predictive accuracy of the two competing models, mean absolute error is taken as a loss function with respect to the realized volatility. Difference in the average loss is reported in the cross cell of the models and p values are given in the parentheses. For example, corresponding to model M2 and other models M*, difference in the corresponding loss function is reported in respective cells (M*–M2). Positive and significant values of the difference show that the model given in row is better than those given in corresponding column. Cell at the intersection of the first row and the second column provides that model M1 provides better forecasting performance than model M2 at less than 2% level of significance.