Long memory mean and volatility models of platinum and palladium price return series under heavy tailed distributions
 Edmore Ranganai^{1}Email author and
 Sihle Basil Kubheka^{1}
Received: 30 May 2016
Accepted: 30 November 2016
Published: 9 December 2016
Abstract
South Africa is a cornucopia of the platinum group metals particularly platinum and palladium. These metals have many unique physical and chemical characteristics which render them indispensable to technology and industry, the markets and the medical field. In this paper we carry out a holistic investigation on long memory (LM), structural breaks and stylized facts in platinum and palladium return and volatility series. To investigate LM we employed a wide range of methods based on time domain, Fourier and wavelet based techniques while we attend to the dual LM phenomenon using ARFIMA–FIGARCH type models, namely FIGARCH, ARFIMA–FIEGARCH, ARFIMA–FIAPARCH and ARFIMA–HYGARCH models. Our results suggests that platinum and palladium returns are mean reverting while volatility exhibited strong LM. Using the Akaike information criterion (AIC) the ARFIMA–FIAPARCH model under the Student distribution was adjudged to be the best model in the case of platinum returns although the ARCHeffect was slightly significant while using the Schwarz information criterion (SIC) the ARFIMA–FIAPARCH under the Normal Distribution outperforms all the other models. Further, the ARFIMA–FIEGARCH under the Skewed Student distribution model and ARFIMA–HYGARCH under the Normal distribution models were able to capture the ARCHeffect. In the case of palladium based on both the AIC and SIC, the ARFIMA–FIAPARCH under the GED distribution model is selected although the ARCHeffect was slightly significant. Also, ARFIMA–FIEGARCH under the GED and ARFIMA–HYGARCH under the normal distribution models were able to capture the ARCHeffect. The best models with respect to prediction excluded the ARFIMA–FIGARCH model and were dominated by the ARFIMA–FIAPARCH model under Nonnormal error distributions indicating the importance of asymmetry and heavy tailed error distributions.
Keywords
Platinum Palladium Long memory Structural breaks Volatility Heavy tailed distribution Heteroskedasticity ARFIMA–FIGARCH type modelsBackground
South Africa is a country rich in the platinum group metals (PGMs) particularly platinum and palladium and it is the largest producer of platinum and second largest producer of palladium (Matthey 2014) accounting for 96% of known PGMs global reserves. In addition to accounting for a significant proportion of global mineral production and resources, the contribution of the PGMs to South Africa economically and otherwise cannot be overemphasized. For instance, on average from 2008 to 2013, the percentage contribution to the South African GDP from this sector was 2.3% with a yearly increase of 3.3% and a head count of 191 781 in direct employment. Further, PGMs also play significant roles in the investment arena (Batten et al. 2010). Since platinum and palladium are two of the major precious metals that offer different volatility and returns of lower correlations with stocks at both sector and market levels, they are some of the attractive asset classes eligible for portfolio diversification (Arouri et al. 2012) which appear more likely to act as a financial instrument than gold. Recently, palladium has entered the Johannesburg Securities Exchange (JSE) as exchange traded funds (ETF). Two palladium funds, Standard Bank AfricaPalladium ETF and Absa Capital newPalladium ETF have been launched in March of 2014 on the JSE. These exchange traded funds are backed by the physical palladium metal. Also, the roles of the PMGs in the the medical field (e.g., their use in anticancer complexes) and industrial catalysis are everadvancing. Given this background, investigating the mechanisms which generate these data returns and their related dynamics are of paramount importance to policy makers, regulators, traders and investors globally.
It is well known that financial returns and hence volatility are dominated by the stylized facts. These include nonstationarity, volatility clustering, their returns are not normally distributed, i.e., the empirical distributions are more peaked and heavy tailed and sometimes asymmetrical and the autocorrelation functions (ACFs) of squared (absolute) returns and volatility exhibit persistence. Further, in precious metals returns and volatility, evidence of their respective ACFs exhibiting a hyperbolic decay, a phenomenon referred to as long memory (LM) (long range dependence) rather than an exponential one (short memory) exists in the literature. The LM phenomenon may be coupled with structural breaks which are shown to severely compromise LM tests as structural breaks induce spurious LM (Baneree and Urga 2005). Recent events that could result in structural breaks in the PGMs returns and volatility are the 2008/2009 global financial crisis and the occasional mining industry labour unrest since the 2012 Marikana incident which resulted in the death of 34 miner during a nationwide labour unrest. Such events bring extremes and jumps in data that may alter the underlying data generating mechanisms.
In the literature nonconstant variance (heteroskedasticity) is handled by autoregressive heteroskedastic (ARCH) models (Engle 1982) and generalized ARCH (GARCH) models (Bollerslev 1986) while LM in the mean is handled by autoregressive fractionally integrated moving average (ARFIMA) models (Tsay 2002). LM can be also inherent in the volatility and fractionally integrated GARCH (FIGARCH) models (Baillie et al. 1996) are proposed as appropriate models. ARFIMA and FIGARCH models generalize the ARIMA and integrated GARCH (IGARCH) to include noninteger (fractional) differencing. In recent times, LM memory has been observed both in the mean and volatility in precious metals, the socalled dual LM, see e.g., Arouri et al. (2012) and Diaz (2016). Using ARFIMA–FIGARCH type models in the article by the first authors did not address structural breaks and heavy tailed error distributions while that by the second author only addressed the dual LM and asymmetry phenomena. Further, their LM analysis was not detailed.

employing a wide spectrum of tests and methods which includes time domain, Fourier and wavelet domain techniques in exploring LM.

distinguishing whether nonstationarity is spurious due to structural breaks or authentic.

distinguishing whether nonstationarity is due to jumps in the mean or due to a trend.

using a wider range of model selection and forecasting diagnostics.

using a wider range of heavy tailed distributions.
Results from this method will assist in understanding if LM in the platinum and palladium returns are spurious or not. Lastly, we will compare different ARFIMA–FIGARCH type models under various distributional scenarios to find the models for platinum and palladium return and volatility series that best fit these data.
The outline of this paper is as follows. “Preliminary data exploration” section provides some preliminary data exploration aspects. “Long memory and structural breaks” section presents LM and structural breaks methods. “Volatility models” section discusses FIGARCH related volatility models. “Modelling of platinum and palladium returns series volatility” section gives empirical results of volatility models of the return series. “Conclusion” section gives the conclusion and further research work.
Preliminary data exploration
Descriptive statistics of returns
Returns statistic  Platinum returns  Palladium returns 

Q1  −0.0360  −0.0652 
Q2  0.0049  0.0000 
Q3  0.0450  0.0768 
Mean  0.0000  0.0004 
Kurtosis  11.9823  8.2040 
Skewness  0.6577  0.7207 
Statistical tests of returns
Test  Platinum (P value)  Palladium (P value) 

Kolmogorov–Smirnov  0.3655 (0.0001)  0.3569 (0.0001) 
Jarque–Bera  31640.87 (0.0001)  15107 (0.0001) 
Phillips–Perron (lags = 10)  −120.85 (0.01)  −220 (0.01) 
ArchLM (lags = 12)  1694.987 (<0.001)  1903.254 (<0.001) 
From these results, it is evident that these data are dominated by the stylized facts as well as LM. Since structural breaks usually induce spurious LM in financial time series, we discuss both LM and structural breaks in the next section.
Long memory and structural breaks
A stationary time series process \(X_t\) is a LM process if there exists a real number \(0<H<1\) such that the ACF, denoted by \(\rho (\tau )\), has a hyperbolic decay rate of the form \(\lim _{x\rightarrow \infty }\rho (\tau )=C^{2H2},\) where \(C>0\) is a finite constant and H is the Hurst exponent (Hurst 1951). In LM literature, the parameter d, called the long range dependence (long memory) parameter is associated to the Hurst exponent with the relationship, \(d=H1/2\). Although the ARFIMA model is stationary and invertible for d in the range \(1/2<d<1/2\) evidence of precious metals exhibiting strong persistence (\(0<d<1/2\)) as opposed to intermediate persistence (antipersistence) (\(1/2<d<0\)) is well documented in the literature, see e.g., Diaz (2016). The spectral density of a LM process will satisfy \(f(\omega )=C\omega ^{2d}\), \(0<d<1/2\). It is well known that this phenomenon can be spuriously induced by structural breaks. In this section we firstly dwell on LM and further elaborate on tests for structural breaks which confirm whether the inherent LM is authentic or spurious.
Long memory estimation methods
In the literature, methods for estimating the long range dependence parameter are divided into three classes, namely heuristic, semiparametric and maximum likelihood estimation (MLE) method. Heuristic (variancetype) methods are easy to compute and interpret but are both not accurate and robust. However, they are useful to test if LM exists and to obtain an initial estimate of d (or H). While on the other hand both semiparametric and MLE methods give more accurate estimates, parametric methods require prior knowledge of the true model which infact is always unknown. For a comparative study of these classes of methods see Boutahar et al. (2007). In the following sub sections, we discuss these methods.
Time domain estimation methods
In time domain analysis, a widely used heuristic method in estimating the Hurst exponent is the rescaled range estimator (R/S)(n) developed by Hurst (1951) and formerly introduced by Mandel (1971) in finance. This is mainly due to its simplicity and easy to estimate and interpret. For further details on this estimator see a paper by Kale and Butar (2010). The conclusions of Kristoufek and Lunackova (2013) and other authors in this field have recommended that this estimator must not be used in isolation, but rather be used in conjunction with other tests. Other time domain methods include aggregated variance, differenced aggregated variance and the aggregated absolute value estimators which are discussed by Teverovsky and Taqqu (1997) and Taqqu et al. (1995). The aggregated absolute value estimator only differ to aggregated variance one in that, instead of computing the sample variance the sum of absolute values of aggregated series is used. Another method very similar to this method that allows estimating the fractal dimension D such that \(D=1H\) for selfsimilar processes was suggested by Higuchi (1988). Also, another variancetype estimator based the variance of residuals was suggested by Peng et al. (1994). The differenced aggregated variance should be used together with the aggregated variance as the former can distinguish nonstationarity due to jumps in the mean from the one due to a slowly declining trend.

The above authors suggested it for testing for unit roots in the economic time series, i.e., testing for both levelnonstationarity and trend nonstationarity.

Lee and Schmidt (1996) used it to distinguish between short and LM processes.
Fourier and wavelet based estimation methods
In this section we consider Fourier based and wavelet based methods for estimating the LM parameter. We first dwell on the fourier based methods. These methods are the socalled frequency domain techniques based on the log of the periodogram (logperiodogram). Various fourier based LM parameter estimators have proliferated since Geweke and PorterHudak (1983) (GPH) first suggested one such logperiodogram estimator.
Since the periodogram is an unbiased but inconsistent estimator of the spectrum, a consistent estimator can be achieved by smoothing it (use of lag windows or averaging). One such consistent estimator is the modified (boxed) periodogram. Actually, Robinson (1994) proved that the averaged periodogram estimator was consistent under very mild conditions. It involves dividing the log of the periodogram into equally spaced boxes and then averaging the values inside each of the boxes leaving out very low frequencies. Further, to address the scattered nature of the periodogram, a robustified least squares (leasttrimmed squares of regression) which minimises approximately T / 2 smallest squared residuals can be employed.
One and a half decade after the advent of the GPH Fourier based estimator, Abry and Veitch (1998) ushered in the wavelet methodology in estimating the LM memory parameter. Wavelet based estimators have desirable properties, i.e., they capture the scaledependent properties of data directly via the coefficients of a joint scaletime wavelet decomposition, require very little assumptions of the data generating process, are asymptotically unbiased and efficient and are robust to deterministic trends. Thus it is recommended that time domain and fourier based methods should be complemented by wavelet based ones.
Testing for LM and estimating the LM parameter may not be adequate in addressing the LM memory phenomenon as the presence of structural breaks can result in spurious LM. Therefore we attend to this aspect in the next section.
Structural breaks diagnosis
Volatility models
If the AR polynomial in the above has unit roots such that \(\sum _{i=1}^{max(m,s)} (\alpha _i + \beta _i) \approx 1\), the resulting model becomes an integrated GARCH (IGARCH) model. A key feature of this model outlined in Tsay (2002) is that the impact of past squared shocks \(\eta _{ti} = a_{ti}^2  \sigma _{ti}^2\) on \(a_t^2\) are persistent. When the return series contains LM, its ACF is not summable as it declines hyperbolically as the lag increases. In this case, the fractional IGARCH (FIGARCH) model is used.
 1.
An asymmetric response of volatility to positive and negative shocks,
 2.
The data to determine the power of returns for which the predictable structure in the volatility pattern is strongest, and
 3.
Long range volatility dependence.
Modelling of platinum and palladium returns series volatility
In this section we discuss the results from structural breaks diagnosis. This will assist with the identification of breaks inherent in data. We then discuss the results from LM tests to examine LM properties of the series. Lastly, we report of the results of volatility models used under various distributional scenarios and the evaluation of forecasts.
Structural breaks diagnosis
Test results of platinum squared returns
m  \(\hat{d}\)  \(\hat{d_2}\)  \(\hat{d_4}\)  \(W_2\)  \(W_4\)  KPSS  P value (KPSS) 

500  0.0125  0.0095  −0.0494  0.6151  5.6580  0.0077  0.1000 
1000  0.1275  0.0125  0.0095  7.9680  19.4500  0.0184  0.1000 
1500  0.0763  −0.0016  0.0091  0.0001  10.9500  0.0185  0.1000 
2000  0.0781  0.1275  0.0125  22.6700  31.5300  0.0089  0.1000 
2500  0.0670  0.1156  0.0120  38.5800  15.0100  0.0077  0.1000 
3000  0.0722  0.0763  −0.0016  3.0550  11.5700  0.0131  0.1000 
3500  0.0672  0.1054  0.1547  16.3000  46.6100  0.0250  0.1000 
4000  0.1085  0.0781  0.1275  2.8320  30.0600  0.0093  0.1000 
4500  0.1076  0.0712  0.1175  5.3880  33.0400  0.0153  0.1000 
5000  0.1076  0.0670  0.1156  8.0600  53.4300  0.0145  0.1000 
Test results of palladium squared returns
m  \(\hat{d}\)  \(\hat{d_2}\)  \(\hat{d_4}\)  \(W_2\)  \(W_4\)  KPSS  P value (KPSS) 

500  −0.1426  −0.1589  −0.1550  1.4070  0.1183  0.0942  0.1000 
1000  0.0839  −0.1426  −0.1589  43.1200  3.7230  4.5200  0.1000 
1500  0.0947  −0.0862  −0.1659  24.3500  67.1800  2.4780  0.1000 
2000  0.0769  0.0839  −0.1426  1.7850  81.4900  6.4480  0.1000 
2500  0.0755  0.0943  −0.1140  4.0330  50.0400  3.5400  0.1000 
3000  0.0770  0.0947  −0.0862  4.2810  31.0300  2.1110  0.1000 
3500  0.0806  0.1183  0.0938  8.3180  9.2730  1.6800  0.1000 
4000  0.0755  0.0769  0.0839  0.0544  1.9680  1.6640  0.1000 
4500  0.0721  0.0764  0.0918  0.5930  5.7750  1.3290  0.1000 
5000  0.0762  0.0755  0.0943  0.0134  6.6750  0.9774  0.1000 
From Table 3, it is evident that the platinum return series contain breaks as the long range dependence parameter is not consistent between subsamples and hence, between samples and the full data set. This is further shown by the rejection of parameter consistency by \(W_2\) and \(W_4\) tests. The KPSS test statistic does not reject the presence of LM.
Results of palladium return series in Table 4 show that the series contain breaks as well. However, the Wald test statistics \(W_2\) and \(W_4\) do not reject parameter consistency in as many subsamples as seen in platinum return series results in Table 3. Further, the KPSS statistic does not reject the presence of LM as well. This is indicative of the fact that not all LM maybe spurious, i.e., due to structural breaks. In the next sub section, we further carry out more tests for LM and estimate the long range dependence parameter using different estimation methods.
Long memory tests
Platinum and palladium log squared returns LM tests
Method  Hurst  Standard error  t value  P value 

Platinum log squared returns  
Aggregated variance method  0.9358 (0.4358)  0.0421  22.2319  <0.0001 
Differenced aggregated variances  1.1907 (0.6907)  0.1813  6.5684  <0.0001 
Aggregated absolute value method  0.9909 (0.4909)  0.0253  39.2360  <0.0001 
Higuchi method  0.9739 (0.4739)  0.0358  27.1544  <0.0001 
Peng method  0.6836 (0.1836)  0.1127  6.0681  <0.0001 
R/S method  0.6667 (0.1667)  0.0754  8.8486  <0.0001 
Periodogram method (GPH)  0.9665 (0.4665)  0.0382  25.2618  <0.0001 
Boxed (modified) periodogram method  0.8313 (0.3313)  0.0463  17.9453  <0.0001 
Wavelet estimator  0.5248 (0.0248)  0.1005  5.2199  0.0034 
Whittle estimator  0.6080 (0.1080)  0.0089  68.2000  <0.0001 
Palladium log squared returns  
Aggregated variance method  0.9340 (0.4340)  0.0606  15.4258  <0.0001 
Differenced aggregated variances  0.9175 (0.4175)  0.1656  5.5416  <0.0001 
Aggregated absolute value method  0.9625 (0.4625)  0.0342  28.1158  <0.0001 
Higuchi method  0.9754 (0.4754)  0.0380  25.6083  <0.0001 
Peng method  0.5451 (0.0451)  0.1256  4.3386  <0.0001 
R/S method  0.4038 (−0.0962)  0.1471  2.7450  0.0087 
Periodogram method (GPH)  0.9103 (0.4103)  0.0384  23.7393  <0.0001 
Boxed (modified) periodogram method  0.7707 (0.2707)  0.0448  17.2098  <0.0001 
Wavelet estimator  0.6145 (0.1145)  0.1510  4.0691  0.0096 
Whittle estimator  0.5779 (0.0779)  0.0088  65.4000  <0.0001 
On platinum squared log returns, all of the tests used suggest LM as all the P values are less than 0.01. Note that the differenced aggregated variances method violates the condition \(0<H<1\). This should not be a concern as its main purpose is to distinguish nonstationarity due to jumps (\(H\approxeq 0.5\)) to that due to actual trend (\(H\gg 0.5\)). So in this case, trend is not due to jumps in the data. It is clear that platinum squared returns have high persistence and it appears they could be explained by a fractionally integrated model.
Like platinum, palladium log squared returns also suggest a high degree of LM as confirmed by very low P values, hence they can be explained by a fractionally integrated model. In the next sub section, we fit LM mean models and conditional volatility models on both platinum and palladium return series to investigate the dual LM of mean returns and volatility.
Empirical results of volatility models
To explain the dual LM of the mean and volatility of platinum and palladium return series, we fitted ARFIMA–FIGARCH type models under heavy tailed error distributions including the Normal distribution. We used an ARFIMA model for modelling squared log returns and for volatility we used FIGARCH, FIEGARCH, FIAPARCH and HYGARCH models under heavy tailed error distributions bench marking them with the Normal distribution. Distributions considered are the Normal, Student, Generalized extreme distribution (GED), and the skewed Student distribution.
ARFIMA–FIGARCH parameter estimation of models
Parameters  Normal platinum  Student  GED  Skewed student  Normal palladium  Student  GED  Skewed student 

Cst(M)  0.0002**  0.0001***  0.0001**  0.0003***  0.0001  0.0001  0.0002***  
\(d_m\)  −0.1027***  −0.0893**  −0.0838  −0.5276***  −0.1019***  −0.0601*  −0.4513***  −0.5062*** 
AR(1)  0.4062***  0.4158***  0.4193***  0.1356***  0.3908***  0.4217***  0.1172***  0.0887 
MA(1)  −0.9327***  −0.9353***  −0.9361***  0.0699*  −0.9225***  −0.9302***  0.0191  0.0818* 
Cst(V)  0.0013**  0.0011*  0.0011*  0.0006  0.0095  0.0104*  0.0087***  0.0085* 
\(d_v\)  0.6492***  0.6486***  0.6473***  0.6545***  0.7227***  0.7040***  0.6419***  0.6258*** 
ARCH(\(\alpha _1\))  0.3686***  0.3567***  0.3543***  0.2138***  0.3907***  0.3678***  0.3088***  0.3094*** 
GARCH(\(\beta _1\))  0.8721***  0.8730***  0.8720***  0.8680***  0.9016***  0.8986***  0.8712***  0.8708*** 
Akaike  –2.4717  –2.4720  –2.4715  –2.4017  −1.2415  −1.2534  −1.1814  −1.1740 
Schwarz  –2.4591  –2.4578  –2.4574  –2.3860  −1.2289  −1.2393  −1.1688  −1.1583 
ARCHLM  4.0565**  4.7917***  4.6751***  4.5570**  0.97785  4.1599**  7.8780***  31.069*** 
ARFIMA–FIEGARCH parameter estimation of models
Parameters  Normal platinum  Student  GED  Skewed student  Normal palladium  Student  GED  Skewed student 

\(d_m\)  −0.4668***  −0.4012***  −0.3903***  −0.0488  −0.1190***  −0.4738***  0.0010  0.4776*** 
AR(1)  −0.2576***  −0.0012  0.0300  0.4407***  0.3981***  0.0399  0.4509***  0.0857* 
MA(1)  0.3411***  0.1230  0.0863  −0.9404***  −0.9183***  0.1032***  0.9381***  0.0711* 
\(d_v\)  0.8975***  0.9173***  0.9173***  0.6623***  0.2624***  0.9061***  0.8884***  0.9088*** 
ARCH(\(\alpha _1\))  1.1344***  1.2337***  1.2557***  −0.6664***  −0.8026***  −0.3899***  0.5881***  0.4803*** 
GARCH(\(\beta _1\))  −0.8946***  −0.8977***  −0.9047***  0.8725***  0.9845  −0.2092*  0.0432  0.2842*** 
EGARCH(\(\theta _1\))  0.0234***  0.0645***  0.0433***  0.0414  0.0212  −0.0092  0.0387*  0.0061 
EGARCH(\(\theta _2\))  0.2187***  0.2279***  0.2122***  0.1367  0.4046***  0.4426***  0.4089***  0.3307*** 
Akaike  –2.2483  –2.2809  –2.2885  –2.4602  −1.1975  −1.1087  −1.2389  −1.1452 
Schwarz  –2.2358  –2.2668  –2.2743  –2.4413  −1.1817  −1.0945  −1.2216  −1.1278 
ARCHLM  0.8213  14.202***  14.175***  1.6987  4.0757**  3.5115**  0.60530  2.2058 
ARFIMA–FIAPARCH parameter estimation of models
Parameters  Normal platinum  Student  GED  Skewed student  Normal palladium  Student  GED  Skewed student 

Cst(M)  0.0001  0.0001  0.0001  0.0003***  0.0001  0.0001  0.0001  0.0001 
\(d_m\)  −0.0793***  −0.0556  −0.0410  −0.5176***  −0.1001***  −0.0582***  −0.00968  −0.0587** 
AR(1)  0.3920***  0.4053***  0.4137***  0.1308***  0.3884***  0.4202***  0.4516***  0.4186*** 
MA(1)  −0.9327***  −0.9361***  −0.9381***  0.0687*  −0.9229***  −0.9306***  −0.9382***  −0.9306*** 
Cst(V)  0.0041  0.0043*  0.0046*  0.0005  0.0106  0.0156*  0.0166*  0.0150* 
\(d_v\)  0.6512***  0.6462***  0.6378***  0.6432***  0.7306***  0.6980***  0.6877***  0.6987*** 
ARCH(\(\alpha _1\))  0.3974***  0.3929***  0.3919***  0.2325***  0.3872***  0.3703***  0.3551***  0.3716*** 
GARCH(\(\beta _1\))  0.8743***  0.8744***  0.8718***  0.8607***  0.9035***  0.8973***  0.8900***  0.8974*** 
APARCH(\(\gamma _1\))  −0.2928***  −0.3716***  −0.4169*  −0.2012*  −0.0863*  −0.0859  −0.09485  −0.0914 
APARCH(\(\delta\))  1.8067***  1.7569***  1.7309***  2.0904***  1.9933***  1.9222***  1.9228***  1.9234*** 
Akaike  –2.4773  –2.4783  –2.4781  –2.4030  −1.2420  −1.2532  −1.2589  −1.2533 
Schwarz  –2.4616  –2.4610  –2.4608  –2.3841  −1.2262  −1.2359  −1.2416  −1.2344 
ARCHLM  2.2572  2.6098*  2.7374*  2.8892*  0.67543  3.8257*  2.6699*  3.5548** 
ARFIMA–HYGARCH parameter estimation of models
Parameters  Normal platinum  Student  GED  Skewed student  Normal palladium  Student  GED  Skewed student 

\(d_m\)  −0.0802***  −0.4125***  −0.3947***  −0.4247***  −0.4988***  −0.4660***  −0.4508***  −0.4691*** 
AR(1)  0.3788***  0.0254  0.0684  0.0899**  −0.3287***  0.0543  0.1039***  0.0521 
MA(1)  −0.9355***  0.0985*  0.0540  0.0587  0.4119***  0.0889*  0.0294***  0.0935*** 
\(d_v\)  0.8199***  0.7930***  0.7968***  0.7983***  0.7266***  0.6475***  0.6741***  0.6474*** 
ARCH(\(\alpha _1\))  0.2981***  0.1091***  0.1086***  0.0960***  0.3045***  0.2995***  0.2968***  0.3002*** 
GARCH(\(\beta _1\))  0.9142***  0.909***2  0.9088***  0.9113***  0.8925***  0.8787***  0.8799***  0.8790*** 
Akaike  –2.4590  –2.3467  –2.3547  –2.3710  −1.1248  −1.1650  −1.1786  −1.1649 
Schwarz  –2.4480  –2.3341  –2.3421  –2.3568  −1.1138  −1.1524  −1.1660  −1.1508 
ARCHLM  1.9986  1.4914  0.72470  0.72483  0.064012  28.207***  5.9269***  27.107*** 
Model selection results for platinum
Based on the Akaike information criterion, the best model is the ARFIMA–FIAPARCH under the Student distribution. However, the ARCHeffect is slightly significant (*). Based on the Schwarz information criterion, the best model is the ARFIMA–FIAPARCH under the Normal distribution and has no ARCHeffect. Although the ARFIMA–FIEGARCH under the Skewed Student distribution and ARFIMA–HYGARCH under the Normal distribution were not selected based on the two information criteria they have no ARCHeffect.
Model selection results for palladium
In the case of palladium based on both Akaike and Schwarz information criteria selected the ARFIMA–FIAPARCH under the GED. However, the ARCHeffect is slightly significant (*). Although the ARFIMA–FIEGARCH under the GED and ARFIMA–HYGARCH under the Normal distribution were not selected based on the two information criteria they have no ARCHeffect.
The results of the two metals agree with the results of Diaz (2016) who found that platinum and palladium returns volatility are characterized by asymmetric response to negative and positive shocks as explained by the FIAPARCH model. From the results for models, \(\gamma<\)0 which illustrates that positive shocks have relatively more impact on volatility than negative shocks. Thus, although these metals respond to negative and positive news the same, positive news have a higher impact and thus making these metals a good investment vehicle as outlined in Arouri et al. (2012). We discuss forecasting performance of these models in the following sub section.
Forecast evaluation methods
Tables 10, 11, 12 and 13 show forecast evaluation results. For platinum return series, the MSE gives low prediction errors for all models except ARFIMA–FIEGARCH which has slightly high errors. Further, based on the MAE, the ARFIMA–FIAPARCH under the Normal and Student distribution and the ARFIMA–HYGARCH model under the Normal distribution gives less prediction errors. Lastly, based on the TIC, the ARFIMA–FIEGARCH under Student distribution gives less prediction error
ARFIMA–FIGARCH forecast evaluation
Parameters  Normal platinum  Student  GED  Skewed student  Normal palladium  Student  GED  Skewed student 

MSE  0.0003  0.0003  0.0003  0.0003  0.0037  0.0037  0.0038  0.0038 
MAE  0.0082  0.0082  0.0083  0.0087  0.0256  0.0257  0.0272  0.0269 
TIC  0.6515  0.6517  0.6517  0.6331  0.6308  0.6407  0.6356  0.6391 
Alpha (MZ)  0.0002  0.0002  0.0002  0.0006  −0.0029  −0.0026  −0.0022  −0.0029 
Beta (MZ)  1.2062  1.1974  1.1948  1.0028  1.3566  1.3449  1.1784  1.2280 
\(R^2\) (MZ)  0.0689  0.0670  0.0665  0.0551  0.1443  0.1274  0.0972  0.1000 
ARFIMA–FIEGARCH forecast evaluation
Parameters  Normal platinum  Student  GED  Skewed student  Normal palladium  Student  GED  Skewed student 

MSE  0.0004  0.0004  0.0004  0.0004  0.0037  0.0043  0.0037  0.0039 
MAE  0.0097  0.0115  0.0107  0.0085  0.0260  0.0362  0.0284  0.0327 
TIC  0.6012  0.5626  0.5773  0.6657  0.5460  0.5479  0.5959  0.5725 
Alpha (MZ)  0.0034  0.0035  0.0037  0.0028  −0.0005  0.0055  0.0026  0.0019 
Beta (MZ)  0.5231  0.3826  0.4132  0.7664  1.2041  0.5204  0.8858  0.7139 
\(R^2\) (MZ)  0.0317  0.0320  0.0307  0.0220  0.1277  0.0832  0.1067  0.0861 
ARFIMA–FIAPARCH forecast evaluation
Parameters  Normal platinum  Student  GED  Skewed student  Normal palladium  Student  GED  Skewed student 

MSE  0.0003  0.0003  0.0003  0.0003  0.0036  0.0037  0.0038  0.0037 
MAE  0.0081  0.0081  0.0082  0.0087  0.0257  0.0255  0.0255  0.0255 
TIC  0.6699  0.6721  0.6727  0.6384  0.6186  0.6383  0.6475  0.6369 
Alpha (MZ)  0.0004  0.0005  0.0005  −0.0009  −0.0034  −0.0036  −0.0032  −0.0036 
Beta (MZ)  1.2716  1.2550  1.2415  1.1667  1.3477  1.4073  1.4054  1.4052 
\(R^2\) (MZ)  0.0634  0.0584  0.0554  0.0510  0.1587  0.1401  0.1270  0.1416 
ARFIMA–HYGARCH forecast evaluation
Parameters  Normal platinum  Student  GED  Skewed student  Normal palladium  Student  GED  Skewed student 

MSE  0.0003  0.0003  0.0003  0.0003  0.0038  0.0038  0.0038  0.0038 
MAE  0.0081  0.0088  0.0087  0.0092  0.0268  0.0270  0.0268  0.0271 
TIC  0.6581  0.6319  0.6351  0.6152  0.6345  0.6363  0.6403  0.6353 
Alpha (MZ)  0.0007  0.0013  0.0013  0.0014  −0.0023  −0.0025  −0.0014  −0.0025 
Beta (MZ)  1.2053  0.9064  0.9233  0.8067  1.2148  1.1995  1.1788  1.1942 
\(R^2\) (MZ)  0.0737  0.0491  0.0495  0.0470  0.1076  0.0998  0.0966  0.0998 
For the selected models the platinum model has intercept estimate of 0.0005 and the palladium model has intercept estimate of −0.00032, hence the platinum model underestimates volatility while the palladium model overestimates volatility. The null hypothesis of a unit slope is not rejected at 5% level of significance for all models. This tells us that our forecasts from the models explains the observed values. In summary, the ARFIMA–FIGARCH type models under heavy tailed error distributions show an improvement of forecasts as compared to the assumption of Normally distributed errors, and further ARFIMA–FIAPARCH models proved to explain platinum and palladium return series better under non Normal error distributions.
Conclusion
With the current South African economic conditions and volatile commodity markets, it is of interest to understand the distribution of platinum group metals and inherent volatility overtime. As it is widely known in literature that financial returns do not follow Normal distributions, we used different heavy tailed error distributions.
Recently LM has been a phenomena of interest in econometrics and financial markets. LM is summarized by the long range dependence parameter. Since spurious LM can also result from structural breaks in data, we used the subsample methodology to test long range dependence parameter consistency to establish whether the LM is spurious or not. From the results, we found that both platinum and palladium log squared returns contain structural breaks. This was identified by long range dependence parameter estimates not being consistent in subsample estimation. To further analyze LM, we used the fact that the \(d\text {th}\) difference of an I(d) process should yield an I(0) process (based on KPSS test statistic.) This further confirmed results of high persistence in platinum and palladium as documented in the literature.
To understand and model volatility inherent in log squared returns of platinum and palladium, we fitted ARFIMA–FIGARCH related models under heavy tailed error distributions bench marking these distributions with the Normal distribution. These models are able to capture LM and the stylized facts in returns and volatility. In forecasting volatility using these models, adjustments from the Mincer–Zarnowitz regression needs to the factored in as these models will slightly underestimate/overestimate volatility.
Results from the paper points to the need for more empirical analysis on the platinum group metals. For further research, we will compare time varying ARFIMA–FIGARCH type models that will factor in structural breaks to structural breaks adjusted ARFIMA–FIGARCH type models.
Declarations
Authors’ contributions
Although all the software programming was entirely done by SBK, both authors carried out the work in this manuscript in equal proportion. Both authors read and approved the final manuscript.
Competing interests
Both authors declare that they have no competing interests.
Funding
The research was supported by the University of South Africa’s Research Department.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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