The CSS and The Two-Staged Methods for Parameter Estimation in SARFIMA Models

Seasonal Autoregressive Fractionally Integrated Moving Average SARFIMA models are used in the analysis of seasonal long memory-dependent time series. Two methods, which are conditional sum of squares CSS and two-staged methods introduced by Hosking 1984 , are proposed to estimate the parameters of SARFIMA models. However, no simulation study has been conducted in the literature. Therefore, it is not known how these methods behave under different parameter settings and sample sizes in SARFIMA models. The aim of this study is to show the behavior of these methods by a simulation study. According to results of the simulation, advantages and disadvantages of both methods under different parameter settings and sample sizes are discussed by comparing the root mean square error RMSE obtained by the CSS and two-staged methods. As a result of the comparison, it is seen that CSS method produces better results than those obtained from the two-staged method.


Introduction
In the recent years, there have been a lot of studies about Autoregressive Fractionally Integrated Moving Average ARFIMA models in the literature.However, most of time series in real life may have seasonality, in addition to long-term structure.Therefore, SARFIMA models have been introduced to model such time series.Generally, SARFIMA p, d, q P, D, Q s process is given in the following form: 1.1 where X t is a time series, B is the back shift operator, such as B i X t X t−i , s is the seasonal lag, d and D represent the nonseasonal and seasonal fractionally differences; respectively, e t is The outline of this study is as follows.Section 2 contains brief information related to SARFIMA models.The CSS method and two staged methods are explained in Sections 3 and 4, respectively.The outline of the simulation study and the results are given in Section 5. Finally, the results obtained from the simulation study are summarized in the last section.

SARFIMA Models
When p, q, d, P , and Q are set to zero in model 1.1 , this model is called as Seasonal Fractionally Integrated SFI model.The SFI model was firstly introduced by Arteche and Robinson 4 , and basic information about the model can be found in Baillie 1 .SFI model can be given by 1 − B s D X t e t .

2.1
Infinite moving average presentation of the model 2.1 is as follows: where Infinite autoregressive presentation of the model 2.1 is as follows: where For model 2.1 , autocovariance and autocorrelation functions can be, respectively, written as follows: For model 2.1 , spectral density function is as follows: Note that the spectral density function is infinite at the frequencies 2πν/s, ν 1, . . ., s/2 .
When p, q, d, D, P , and Q are different from zero in model 1.1 , closed form for autocovariances cannot be determined.However, some methods, such as the splitting method presented by Bertelli and Caporin 27 , employed to calculate autocovariances of ARFIMA models, can also be used for those in SARFIMA models.

2.10
To generate series, which are appropriate for SARFIMA P, D, Q s models, the following algorithm is applied.
Step 1. Generate Z z 1 , . . ., z n T random variable vector with standard normal distribution.
Step 3. Split the covariance matrix as follows: Σ LL T where, L is a lower triangular matrix.
This splitting is called Cholesky.It is possible to obtain Cholesky decomposition of positive definite and symmetric matrices.Note that matrix Σ n is positive definite and symmetric.
Step 5. Generate series according to SARMA P, Q s model by taking X X 1 , . . ., X n as error series.By this way, the new generated series have the structure of SARFIMA P, D, Q s .This algorithm is easily extended to SARFIMA p, d, q P, D, Q s model.

The Two-Staged Method
The two-staged method can be used to estimate the parameters of SARFIMA P, D, Q s model.In the first phase of this method, it is assumed that the time series has a suitable structure to use the SARFIMA 0, D, 0 s model, and seasonal fractionally difference parameter D is estimated.In the second phase, estimation of the parameter, EML method given below, can be employed.
Theoretical autocovariance and autocorrelation functions for SARFIMA 0, D, 0 s model are shown in 2.4 and 2.5 respectively.Let time series X t have n observations x 1 , . . ., x n , and let Ω represent the autocorrelation matrix of x 1 , . . ., x n .Therefore, the likelihood function of x 1 , . . ., x n is as follows: Cholesky decomposition is used for the matrix Ω as multiplication of lower and upper triangular matrices in calculation of the likelihood function.Instead of calculating the inverse of matrix Ω n×n , inverses of lower and upper triangular matrices are calculated by using the decomposition.Thus, the decomposition decreases computational difficulty and calculation time.Cholesky decomposition of the matrix Ω is written as follows: Ω LL .

3.2
Let W L −1 X, and it can be written

3.3
Thus, 3.1 can be rewritten as The likelihood function given in 3.4 is maximized in terms of seasonal fractionally difference parameter by using an optimization algorithm.After seasonal fractionally difference parameter is estimated by using EML, the rest of the parameters of SARMA P, Q s model are estimated in the second phase by using the classic method.In the second phase, the order of the seasonal model can be determined by using the Box-Jenkins approach.Therefore, the two-staged method can be summarized as follows.
Phase 1. Estimate the parameter D by assuming the time series suitable for SARFIMA 0, D, 0 s .
Phase 2. Estimate seasonal autoregressive and moving average parameters by using the Box-Jenkins methodology.

The CSS Method
Chung and Baillie 20 proposed a method based on minimization of conditional sum of square.This method can be used for SARFIMA p, d, q P, D, Q s models.Conditional sum of square method for SARFIMA model is as follows:

4.1
In the CSS method, firstly, seasonal fractionally difference procedure is executed for X t .Secondly, fractionally difference procedure is executed for 1 − B s D X t .Thirdly, SARMA filtering is applied to 1 conditional sum of square is calculated for a fixed value of σ 2 e and D. Chung and Baillie 20 also emphasize that the estimations of parameters obtained by the CSS method have less bias when the mean value of the series is known.It is easy to use the CSS method because it does not need to calculate autocovariances.In the literature, the CSS method for the SARFIMA P, D, Q s model has been used only by Darné et al. 19 .

Simulation Study
In this section, the parameters of SARFIMA P, D, Q s model are estimated by using the CSS and the two-staged methods separately under different parameter settings and sample sizes.Also, the advantages and the disadvantages of both methods are discussed.
The algorithm, whose steps are given in Section 2, is used to generate various SARFIMA P, D, Q s models.SARFIMA 1, D, 0 s and SARFIMA 0, D, 1 s models are emphasized in the simulation study.For SARFIMA 1, D, 0 s model, 36 different cases are examined such as seasonal fractionally difference D 0.1, 0.2, 0.3, seasonal autoregressive parameter Φ 0.3, 0.7, −0.3 , −0.7 , sample sizes n 120, 240, 360, and period s 4. Similarly, the same parameters are also used for SARFIMA 0, D, 1 s model by taking Θ 0.3, 0.7, −0.3 , −0.7 .For each case, 1000 time series are generated, so totally we generate 72000 time series.The parameters of the generated time series are estimated by using both the CSS and two-staged methods whose results are summarized in Tables 1 and 2. For each 1000 time series, the mean, standard deviation, and root mean square error RMSE values of estimated parameters are exhibited in these tables.RMSE values are computed by where β i and β i denote the real and estimated values of parameter, respectively.In Table 1, for SARFIMA 1, D, 0 s model, the simulation results for different values of Φ and sample size n are shown when the CSS and the two-stage methods are executed.From this table, for CSS method, we observe that RMSE values have sharply decreased for the estimated parameters of seasonal fractional difference and seasonal autoregressive, when the sample size increases.It is also seen that the values of RMSE do not change much whether values when the seasonal autoregressive parameter is negative.Therefore, for the negative values of seasonal autoregressive parameters, we can say that the estimation error gets bigger while the order of seasonal fractional difference is increasing.
In Table 2, for the SARFIMA 0, D, 1 s model, the simulation results for different values of parameter Θ and sample size n are shown for the CSS and two-staged methods.From this table, we observe that RMSE values have decreased for the estimated parameters of seasonal fractional difference and seasonal moving average, when the sample size increases, the CSS method is executed.It is also seen that the values of RMSE do not change much whether the sign of parameter of seasonal moving average is positive or not.In the case of having larger value of seasonal moving average parameter for the negative values, RMSE values for seasonal fractionally difference RMSE D are smaller.When we compare D 0. When Table 2 is examined, it is observed that the values of RMSE Θ decrease when sample size increases for two-staged method.However, there is no positive or negative relations between the value of seasonal moving average parameter and the values of RMSE D and RMSE Θ when two-stage method is executed.We would like to remark that RMSE Θ has the smallest value in each sample size for Θ 0.7 and that values of RMSE D are quite big for the negative values of seasonal moving average parameter with respect to its positive values when D 0.2, and 0.3 in Table 2.

Discussions
In the literature, the two-staged method is a widely used method to estimate parameters of SARFIMA models.Although there is another method called CSS, this method has not been employed to estimate the parameters of SARFIMA model.In this study, the CSS and the twostaged methods are employed to estimate parameters of the SARFIMA models by conducting a simulation study, and by this way the properties of these two methods are examined under different parameter settings and sample sizes.
1 with D 0.2, the values of RMSE D in D 0.1 are smaller than those in D 0.2, whereas the values of RMSE Θ among D 0.1, D 0.2, and D 0.3 are close with each other.Note that the values of RMSE D in D 0.3 are larger than those in D 0.1.

Table 1 :
Simulation results of the CSS and two-stage method in SARFIMA 1, D, 0 4 .According toTable1, when the two-staged method is executed, it is observed that the sample size does not affect significantly the values of RMSE, especially for RMSE Φ when Φ −0.7.However, when the absolute value estimated of seasonal autoregressive parameter increases, the values of RMSE Φ increase dramatically in D 0.1 and D 0.2.The values of RMSE D are not affected by both the sign and magnitude of seasonal autoregressive parameter, especially in D 0.1.It is worth to point out that the values of RMSE D are quite larger for the negative values of seasonal autoregressive parameters in both D 0.2 and D 0.3.It can be inferred from the comparison between D 0.1 and D 0.2 that for the negative values of seasonal autoregressive parameter, both the values of RMSE D and RMSE Φ increase gradually while D is increasing.Especially, the values of RMSE Φ in D 0.3 get the biggest

Table 2 :
Simulation results of CSS and two-stage method in SARFIMA 0, D, 1 4 .