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yflK 336.745
QUALITY OF THE FORECASTS OF AND ITS IMPORTANCE FOR
1. Introduction
Forecasting the currency exchange rate is one of the most difficult and, at the same time, the most important tasks in business. To restrict the risk, industrial organizations in the global world are obliged to hedge their assets by future contracts and many other financial instruments. This requires forecasting the currency exchange rates of not only the trading partner countries but also of other global currencies. It is a well known fact that the best exchange rate forecast is produced by the random walk model (Meese and Rogoff, 1983). This simple approach shows that, the best possible forecast of a currency's value is its value today. Further studies (for example Wolff , 1988; Schinasi and Swamy, 1989; Canova, 1993; Meese and Rose, 1990; Mark, 1995; Groen, 2000), applying mostly fundamental as well as technical approaches, using advanced econometric techniques for time series and panel data cannot beat the Meese and Rogoff's general results, although in some cases the researchers obtained quite satisfactory forecasting results, especially in the long run. A summary of the mentioned research results was published by Neely and Sarno (2002).
A problem remains however how to forecast currencies exchange rates in a short run, when the exchange rates volatility plays the main role, and when managers are enforced to react quickly in response to market movements or to verify their long positions to avoid greater risk. Forecasting volatility is one of the most popular findings of financial market analysis and financial econometrics (Osin-ska, 2006). The volatility models based on Engle's ARCH construct (Engle, 1982) became widely developed and used in everyday practice. The measures of forecast quality were however underdeveloped in comparison with greater and greater needs for good forecasts. The researchers used mainly tra-
* Magdalena.osnska@byd.pl
Magdalena Osinska*
CURRENCIES EXCHANGE RATES INDUSTRIAL DEVELOPMENT
ditional measures of forecast accuracy, like mean square error (MSE) or mean absolute error (MAE). These measures were good enough for linear model specifications but became unsatisfactory for nonlinear or volatility models. The new idea of forecasting ability of econometric models starts with the article by Diebold and Mariano (1995) who proposed a formal test for models comparison in pairs, further developed by West (1996). The most important statistical procedures for determining predictive ability of the models were published by White (2000) and Hansen (2001 and 2005). They are connected with superior predictive ability of the model that is much more important than a simple comparison of the results. As a result, the SPA test was proposed as a contemporary measure appropriate for comparison of different types of models including volatility models.
The paper is aimed to forecast daily exchange rate volatility of such currencies as: the Russian ruble, Indian rupee, Brazilian real, Chinese yuan, Polish zloty and euro against US dollar, using different GARCH specification with different error distributions. Out-of-sample forecast accuracy was then examined using SPA test, as well as standard procedures, including Minzer and Zarnowitz regression (Minzer and Zarnowitz, 1969). The paper consists of four parts. In the first one the volatility models are discussed and in the second one the testing procedure as well as different accuracy criteria are discussed. In the third part empirical data are presented and analyzed, and in the last one some conclusions and recommendations for industrial development are drawn out.
2. Volatility models
A basis for formal theory of volatility modeling was presented by Engle (1982) who introduced the ARCH model, starting the new chapter in the financial time series analysis. This concept, awarded
with the Noble prize in 2003, resulted in numerous literature concerning volatility modeling and many modifications of the basic model, for example for univariate representations: the GARCH (Bollerslev, 1986), EGARCH (Nelson, 1991), TARCH (Rabe-manjara and Zakoian, 1993), APARCH (Ding, Granger and Engle, 1993), FIGARCH (Bailie, Bollerslev and Mikelsen, 1996) and many others.
The general GARCH(p,q) model is given as:
У, = X, Y+E,,
E,\U,_1~Ж(0, h,),
h, =ao + S a + S J -
i = 1 j=1
У,
h, = «0 + «1 У,'- 1 +ßlh, - 1,
J=1
where
eJ(/-i) = h,(/-i) for l- i > 0; e2(/ - j) = e2+ / - i and h,(/ - j) = h, + / - i for l - i < 0.
The representations presented above imply symmetric reactions for positive and negative information coming from the market. This is not always true, so models including asymmetry have been introduced. The most general representation of asymmetric univariate GARCH class is the Asymmetric Power ARCH model introduced by Ding, Granger and Engle in 1993. The conditional variance equation in the APARCH model is given as:
(1) (2)
(3)
l~s ^ s
4h, =a0 + S ai (l E-i I -YiE - i) +
i = 1
^ --s
J M ■ -I-J ,
(7)
+ S ß jjh,
j = 1
where xf - the vector of exogenous variables; y -
the vector of parameters; zt - the residual process; Ut _ i - the information set at time t - 1; ht - the conditional variance; ao > 0, a,- > 0 for i = 1, 2, ..., p and Pi > 0 for j = 1, 2, q.
Process generated by GARCH(p, q) for any value of p and q is covariance stationary if:
ai +a2 + k + aq +Pi + P2 + k + Pp <1.
In finance, the possibility of modelling the conditional mean of financial returns according to (1) is limited to ARMA-type models, especially when daily data are used. That is why the following representation is most often used in practice:
where 8 > 0, -1 < y < 1, i = 1, 2, ...,p, j = 1, 2, ..., q.
For 8 = 2 equation (7) takes the form of the GJR-GARCH model (Glosten, Jagannathan and Runkle, 1993) and for 8 = 1 it corresponds to the Threshold GARCH model (Zakoian, 1994). Model (6) assumes that the lack of new information generates minimum volatility (Osinska, 2006). In further stages of the analysis we use the GJR-GARCH representation for modelling and forecasting asymmetric reactions of exchange rates changes. Forecasting formula is then the following:
h (/) = «0 + S (ai E2 (/ - i)+yA-+ / - ii, E2 (/ - i))+
(8)
(4)
(5)
+ s ßh(/ - J),
j=1
where ao > 0, ai > 0, ßi >0, e, ~ N(0,1),
yt = ln(pt / pt _ 1)100 denotes logarithmic financial
returns and h, is the conditional variance of yt. Model (5) is a GARCH(1,1) representation of (3).
Computing forecasts of volatility for /-horizon using GARCH(p, q) model is relatively simple and can be expressed by the following formula:
h, (/) = a0 +1 a, e2(/ _ /) + £ ß,h; (/ _ j), (6)
where St +, _ ^ = E (St +, _, | Ut) and the remained
symbols are the same as in (6).
S-/_, depends on chosen distribution of innovation in the GARCH model. When the distribution is symmetric (Gaussian, t-Student or GED) it takes the value of 1/2, because the probability of negative information is the same as of the positive one. In cases of non-symmetric distributions (Skewed t-Student, some stable distributions) it depends upon appropriate parameters.
GARCH-type models are estimated by using the maximum likelihood method, with the application of iterative methods like Berndt-Hall-Hall-Hausman algorithm (Berndt et al., 1974). The ARCH/GARCH effect can be a subject of testing. Two tests are usually applied: Engle's ARCH-LM
i = 1
test (Engle, 1982) and McLeod and Li test based on the Box-Ljung statistics for autocorrelation (McLeod and Li, 1983).
Many empirical findings (Doman and Doman, 2004) show that GARCH(1,1) model is very often quite satisfactory for modeling the financial returns, however the error distribution is not Gaussian. Thus other error distributions are applied, for example symmetric and skewed t-Student distributions, Generalized Error Distribution and stable distributions.
3. Measures of forecast accuracy and testing procedures
It is very problematic to measure the quality of forecasts based on volatility models because volatilities are not observable. In such a case a researcher has his forecast on the one hand and un-observable process on the other one. The proxy ej is very noisy and hitherto forecasts based on this measure exhibit large errors. The problem was, however, solved by taking realized volatility as a proxy of volatility process (Andersen and Boller-slev, 1998). The realized volatility measure is defined as a sum of squares of intraday returns and takes the form:
Reali, = X t
(9)
where r„d are intraday returns and ro,d corresponds to the period between the last quotation of the previous day and the first quotation of the current day t. Furthermore, t denotes the day number, d -a number of a given intraday return and D is a number of intraday returns of a given frequency within the day. Most often 5-minute returns are used, but also 10 or 30 minute returns are applied.
The most popular measures of forecast accuracy concentrate on computing the forecast errors. There are plenty of such measures, for example MSE, RMSE, MAE, MAPE among many others. Their usefulness consists in that they show the differences in accuracy of computed forecasts but say nothing about the method of forecasting. In 1995, Diebold and Mariano (Diebold and Mariano, 1995) derived a testing procedure of equal predictive accuracy. The hypothesis to be tested says that the alternative methods are equally accurate on the average. The
general idea of Diebold and Mariano's test relies on two time series including actual values and forecasts of a predicted variable, say yt and yit as well as on the loss function depending on the forecast and actual values only through the forecast error, defined as: g(yt, yit) = g(yu - yt) = g^). The loss function may take many different forms. What we compare is a loss differential between the two forecasts, coming from two competing models of the form: d(t) = g(eit) - g(e2t). The forecasting methods are equally accurate, if E(d(t)) = 0.
Further development of forecasting ability comparison took place in the article by White (2000) who introduced a procedure known as RC test. This procedure started a new stage in comparison of forecasts, because RC test was the first that enabled to check the superior predictive ability of the compared models, which gave some advantage over the previous equal predictive accuracy tests. The White test was asymptotically biased what was shown by Hansen (Hansen, 2001 and 2005). Hansen improved the previous test and developed the forecast comparison idea, giving quite a new impact for evaluating accuracy of economic forecasts. The idea of Hansen's SPA test is similar to Diebold and
Mariano's procedure. Let (hkl, k = 0,1,..., m) is
a finite set of forecasts computed from the models k = 0, 1, ..., m. The model corresponding to k = 0 is a special one, called a benchmark model. For
a given loss function L(h1 ¡,hk ,) ,, a relative loss
differential is defined as follows:
Xk.i=UKj . 4./)- L(hu . hk.i x
(10)
where h1 , is a forecasted variable, represented by the realized volatility, for example of the form defined in (9), k = 1, ..., m and I = 1,..., T*.
The null hypothesis in the SPA test says that the benchmark model is not worse than any of the competing models, that is:
Ho :E(Xk./)< 0 . k = 1, ...,
m.
(11)
The test statistics is then given as:
k = 1, ..., m, (12)
TS,FA = max
f maxk T4Xk 0Л
d = 0
- 1 T
where X, = — X XkJ and à2k
J i = i
is a consistent esti-
mator of the variance ak = var(T *~Xk). It is worth
noting that Tj,PA = 0, if the benchmark model is the
best for a given sample. The distribution of SPA test statistics is not standard and ^-values are computed using the appropriate bootstrap procedure (Hansen, 2005).
All the tests mentioned here rely on the loss function as a measure of the forecast quality. That is why the choice of the loss function is getting more and more importance. As it was mentioned above the most frequently used loss functions are usually based on mean square error (MSE) or mean absolute error (MAE). They are fully appropriate for forecasting the conditional mean, but they are not very precise when volatility is predicted, mainly in case of forecasting asymmetry (see: Lopez, 2001 for further discussion). That is why some other loss functions, penalizing non-positive variance forecasts were constructed and examined. Some of them, used for the evaluation of currencies volatility forecasts, are presented below, but the list does not cover all the possible loss functions available in the literature. The first to be mentioned is logarithmic loss (LL) function used by Pagan and Schwert (1990), which takes the form:
1 J-
LL = -J* X |_ln (hi,i )-ln (( +,)
(13)
where hi, / is a forecasted variable, represented by the realized volatility, for example of the form defined in (9), hj +/ is the volatility forecast /-periods
ahead, J is the number of forecasts, T is the number of observations in the sample.
The second one is heteroscedasticity-adjusted mean squared error (HMSE) used by Bollerslev and Ghysels (1996):
1 r
HMSE = ^ £ (hi,/ / hj + / -1)2. (14)
J / = 1
Bollerslev et al. (1994) suggested the loss function based on the Gaussian maximum likelihood function used for estimation GARCH models of the form:
1 J
GMLE = — У T * à
Another loss function practically used for volatility forecasts comparison was given by Angelidis and Degiannakis (2006). They suggested that mean absolute error (MAE) and mean squared error (MSE) may be computed in a conventional way. Then the loss function (LF) for each model is constructed as the sum of these errors.
It is important to note that each of the mentioned loss functions possesses its own characteristics, useful for different decision problems. The user has plenty to choose, depending upon his individual needs.
One of the popular indicators of accuracy of the volatility forecasts is the R2 coefficient of Minzer and Zarnowitz regression (1969) of the form:
h1 ! = a + bhT +
(16)
ln (( +1 ) + hu / hj +1
(15)
where l = 1, ..., T*. Equation (16) is also often presented in logarithmic form. Its interpretation has very practical implications for investors wishing to hedge their portfolios. High values of the coefficient show that the dynamics of future volatility is properly generated by the model, i.e. increases and falls in volatility are correctly predicted. However in that case 'high' values of R2 should be thought of more than 10 per cent.
4. Forecasting volatility of exchange rates
Typical characteristics of financial returns were the most precisely described by Cont (2001). These are: the lack of serial correlation, heavy tails and lep-tokurtosis, asymmetry of rises and falls, volatility clustering, mean reversion, the leverage effect and many others. Volatility modeling gives just one of the best theoretical and practical answers for the need of the financial time series analysis. It is strongly supported by the fact that the volatility models can take into account error distributions other than Gaussian that better fit the empirical characteristics of the financial time series. The above findings concern in the same level capital as well as currency markets.
The reasons for such a state of the art are the following:
- Non-continuous trading of financial instruments, non-trading periods cause that information comes to investors in packages and their reactions differ in such cases;
- Volatility of currencies often depends on expectations of the events, which occurs with a relatively high probability;
- Strong correlations (co-movements in volatilities) between rates of return of different instruments can be observed;
- The variance of currency exchange rates depends on macroeconomic characteristics (BrzeszczyDski and Kelm, 2002).
To model the exchange rates volatility daily data from Jul 20, 2005 till Aug 27, 2010 were taken, while the last 30 observations were left for
forecasting. These observations cover partly a period of raising volatility of currencies, caused by a great financial crisis started in 2007 in the USA and spreading all over the world in 2008. The following currencies were the subjects of the analysis: the Russian ruble (rub), Indian rupee (inr), Brazilian real (blr), Chinese yuan (cny), Polish zloty (pln) and euro (eur) against the US dollar (see figure).
USD_ CNY
USD_ CNY
USD_INR
55
50
45
40
35
30
USD_RUB
40 35 30 25 20
0,9 0,8 0,7 0,6 0,5
USD_EUR
Currencies exchange rates in Jul. 20, 2005 till Aug. 27, 2010 (daily observations). The period of the economic crisis
was rounded
The data were transformed to logarithmic rates of change as follows: rt = (ln pt _ ln pt _ 1) 100. The
examined time series, apart from the euro, are rarely the subject of econometric research because trading these currencies often happens in packages and it is made for speculative reasons rather than for hedging or diversifying portfolios. Nevertheless their dynamics and volatility forecast are of great interest for different reasons. The Polish zloty is interesting because of the soon expected decision of the Polish government to enter the euro-zone. The Brazilian real, Chinese yuan, Indian rupee and Russian ruble belong to the new economic force on the global market that is created by huge economies of Brazil, Russia, India and China, called the BRIC group.
The descriptive statistics values presented in table 1 show that the mean values of the rates of changes of the currency exchange rates under investigation were around zero, while standard deviations took values around one. The only exception is the Chinese yuan, for which standard deviation in this period was extremely low while kurtosis was extremely high. This series also exhibited a unit root, and for that reason it was excluded from further studies. It should be emphasized that the Chinese yuan was stable against the US dollar, registered on the level of around 8,3 for over 10 years. Since 2007 appreciation of yuan against USD has been observed and now, at the end of 2010, its rate is equal to 6,8. It is related to the financial crisis that has been observed in the USA since mid 2007. However, the Chinese currency can-
not be considered as market driven. The great reserve of the US currency in China makes it possible to peg cny against the US dollar at almost unchanged level. The rate of change of the Russian ruble was an exception considering ARCH effect, but we used it for later examination because the null hypothesis of no ARCH was rejected at 5,5 % significance level.
An example of evaluating the predictive accuracy of currency exchange rates volatilities can be found in Lopez (2001). He applied a loss function based on probability forecasts as well as the Die-bold and Mariano test and calibration test. He found out that the GARCH model with general exponential distribution (GED) fitted the dynamics of the analysed time series best.
In the presented study the following volatility models are considered: the GARCH(1,1) with Gaussian distribution, GARCH(1,1) with Z-Student distribution, GARCH(1,1) with GED distribution and GJR-GARCH(1,1) with GED distribution of innovations. When ARMA component was significant for modelling conditional mean it was included into the analysis. The out-of-sample forecasts were computed for one day ahead and for five days ahead. Realized volatility is represented by Real1 measure for 5-, 15-and 30- minute returns and denoted as Real1_5, Rea1_15 and Real1_30, respectively. The computed loss functions as well as the test procedures were discussed in section 3. The results of the analysis are collected in tables 2-6. Each table concerns the investigated currency exchange rate against the US dollar.
Table 1
Descriptive statistics and basic tests results for exchange rates logarithmic percentage changes
Characteristics brl cny inr rub pln eur
Mean -0.0266 -0.0179 0.0072 0.0062 -0.0091 -0.0045
Median -0.0954 -0.0058 0 -0.0195 -0.0527 -0.0143
Standard deviation 1.3088 0.1152 0.5142 0.6684 1.1967 0.7594
Kurtosis 26.6161 98.0246 12.9191 40.5772 10.5475 14.6608
Skewness 2.0226 -5.3779 -0.4944 3.3328 0.4697 1.3097
Minimum -7.2741 -2.0823 -4.0948 -3.7556 -7.4386 -3.4081
Maximum 16.3656 0.8235 3.9545 9.0186 10.91873 8.0548
Unit root KPSS** 0.063 0.760* 0.130 0.215 0.181 0.178
ARCH(5) LM test 50.987* 28.208* 69.288* 10.641 62.191* 15.962*
* Denotes rejection of the null hypothesis at 5 % significance level. ** KPSS unit root test Kwiatkowski, Phillips, Schmidt and Shin (1992).
Table 2
The values of the accuracy measures of the one- and five-day ahead forecasts of the brl/usd volatility
in the period Jul. 28 - Aug. 27, 2010
Model GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH
Gauss ¿-Student GED GED Gauss GED GED GED Gauss ¿-Student GED GED
Real1_5 Real1_15 Real1_30
Minzer-Zarnowitz regression R2
one day 0.022784 0.0236883 0.0225715 0.0203285 0.00546 0.0060781 0.0048874 0.0005462 0.0213121 0.0229226 0.0223114 0.0193307 0.0020053
five days 0.0353367 0.0362391 0.0344776 0.0116258 0.0394102 0.0408529 0.0390568 0.0210802 0.0589949 0.0584469 0.0553371
MSE
one day 13.706943 13.58327 13.657612 13.462479 63.076294 62.838255 62.982467 62.525667 27.242696 27.082596 27.181275 26.928751 25.728003
five days 13.117474 12.842785 12.906174 12.586989 61.942468 61.350556 61.488817 60.815897 26.547548 26.166518 26.251423
RMSE
one day 3.7022888 3.6855487 3.6956207 3.6691252 7.9420585 7.9270584 7.9361494 7.9073173 5.2194536 5.2040941 5.2135665 5.1892919 5.0722779
five days 3.6218054 3.5836831 3.5925163 3.5478147 7.8703538 7.8326596 7.8414805 7.7984548 5.1524313 5.1153218 5.1236143
MAE
one day 2.3498633 2.3263053 2.3415629 2.3036387 4.8998229 4.8762649 4.8915225 4.8535983 3.0705403 3.0469823 3.0622398 3.0243157 2.8466807
five days 2.2063728 2.1494144 2.1630002 2.1062485 4.7541415 4.6972851 4.7110017 4.6553823 2.9251227 2.872919 2.8861613
LF
one day 16.056806 15.909575 15.999175 15.766118 67.976116 67.71452 67.873989 67.379265 30.313237 30.129578 30.243515 29.953066 28.574684
five days 15.323847 14.992199 15.069174 14.693238 66.69661 66.047842 66.199818 65.47128 29.472671 29.039437 29.137585
LL
one day 1.4300569 1.20891 1.3289958 1.0358932 2.9583243 2.5465052 2.7768757 2.1400581 2.0291393 1.7238958 1.8981806 1.5052614 0.4656403
five days 0.7128077 0.4894297 0.5335299 0.3249064 1.6824064 1.1455895 1.2566037 0.7981793 1.0826225 0.7395716 0.8067182
HMSE
one day 167.36207 134.25355 151.17703 110.13069 713.94888 585.74282 652.06958 478.05881 320.25884 260.85974 293.90047 226.983 77.037041
five days 76.494444 55.062296 58.904329 40.35237 356.81344 260.40077 277.6395 195.25645 155.31657 112.63812 119.73441
GLME
one day 7.6879573 6.9736811 7.3629973 6.4272364 15.545344 14.248444 14.966557 13.034739 10.044625 9.1334592 9.6521658 8.5036355 5.4558302
five days 5.4159103 4.7090952 4.8526511 4.2132273 11.454584 9.8960532 10.215971 8.8305036 7.2417467 6.2566913 6.4513765
Table 3
The values of the accuracy measures of the one- and five-day ahead forecasts of the inr/usd volatility
in the period Jul. 28 - Aug. 27, 2010
Model GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH
Gauss ¿-Student GED GED Gauss GED GED GED Gauss ¿-Student GED GED
Real1_5 Real1_15 Real1_30
Minzer-Zarnowitz regression R2
one day 0.003942 0.0984667 0.0192407 0.0361138 0.0037424 0.1845872 0.0568954 0.0043821 0.048603 0.0032872 0.0238477 0.0639102
five days 0.0242401 0.0004687 0.026727 0.0054558 0.0474357 0.0043776 0.0616295 0.0072834 0.0366063 0.024286 0.0015354 0.0282915
MSE
one day 8.5147101 5.5485735 8.9310505 8.8088245 4.6710207 2.8153522 4.9658877 4.8715102 2.8912404 1.5103898 3.1003605 3.038675
five days 8.4187684 5.5064102 8.8291586 8.6787375 4.6002621 2.7445634 4.8840108 4.7827104 2.8429329 1.4969963 3.0475588 2.9774561
RMSE
one day 2.9179976 2.355541 2.9884863 2.9679664 2.1612544 1.6779011 2.2284272 2.2071498 1.7003648 1.2289792 1.7607841 1.7431796
five days 2.9015114 2.3465741 2.97139 2.9459697 2.1448222 1.6566724 2.2099798 2.1869409 1.6860999 1.223518 1.7457259 1.7255307
MAE
one day 2.6701805 2.017539 2.7454041 2.7221168 1.8387318 1.2173464 1.9139553 1.890668 1.3835301 0.8664823 1.4546011 1.4313137
five days 2.65312 2.0279421 2.72912 2.7011036 1.8216712 1.2322876 1.8976712 1.8696549 1.3676629 0.870172 1.4383169 1.410858
LF
one day 11.184891 7.5661125 11.676455 11.530941 6.5097524 4.0326985 6.8798431 6.7621782 4.2747704 2.3768721 4.5549615 4.4699887
five days 11.071888 7.5343523 11.558279 11.379841 6.4219333 3.976851 6.781682 6.6523653 4.2105958 2.3671683 4.4858757 4.3883141
LL
one day 1.9778597 0.0348014 3.1281754 2.761683 1.0697635 0.0299083 1.7402945 1.4999559 0.8548194 0.0088477 1.3130827 1.2032024
five days 1.7814663 0.0198784 2.7641972 2.3802312 0.9470329 0.0108402 1.4724617 1.2804628 0.7728697 0.0059236 1.1710181 1.0239322
HMSE
one day 116.35134 7.5902936 239.66212 198.89695 63.373968 4.4222133 139.20477 108.49338 41.728269 1.7994044 84.473482 68.595582
five days 100.22753 6.7030261 192.43236 154.5587 53.270497 3.4881015 104.05716 85.229538 35.733915 1.7150844 67.272301 55.30274
GLME
one day 9.5280029 3.1945431 13.447448 12.092196 6.4631188 2.3065391 9.2739949 8.212522 4.850061 1.7218044 6.8618273 6.1466898
five days 8.9166724 3.1516164 12.229353 10.917369 6.0126484 2.2429268 8.2775224 7.4189821 4.5293839 1.7104145 6.2098083 5.5916574
Table 4
The values of the accuracy measures of the one- and five-day ahead forecasts of the rub/usd volatility
in the period Jul. 28 - Aug. 27, 2010
Model GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH
Gauss ¿-Student GED GED Gauss GED GED GED Gauss ¿-Student GED GED
Real1_5 Real1_15 Real1_30
Minzer-Zarnowitz regression R2
one day 4.91E-06 0.00148 0.000594 0.244887 0.000536 0.002545 0.001489 0.273257 7.97E-07 0.000116 1.28E-06 0.243502
five days 0.037699 0.015766 0.019129 0.043791 0.014404 0.009666 0.009788 0.063936 0.043553 0.01147 0.015369 0.057437
MSE
one day 1.4034134 1.3261675 1.3261675 1.4103203 0.1478933 0.1399829 0.142134 0.1540531 0.0718484 0.081664 0.0736848 0.078038
five days 1.3758121 1.3015759 1.3261675 1.3849437 0.1426359 0.1391992 0.1387164 0.1496445 0.0677027 0.08515 0.0740845 0.0780669
RMSE
one day 1.1846575 1.1515935 1.3015759 1.1875691 0.3845689 0.3741429 0.3770067 0.392496 0.2680454 0.2857692 0.2714495 0.2793527
five days 1.1729502 1.1408663 1.3015759 1.1768363 0.3776716 0.373094 0.3724465 0.3868391 0.2601975 0.2918048 0.2721847 0.2794046
MAE
one day 0.5416507 0.5192833 0.5275014 0.5372949 0.2274965 0.2616588 0.2333554 0.2352813 0.1810375 0.2366626 0.2003136 0.1961932
five days 0.5300168 0.5148807 0.5169007 0.5233015 0.2296676 0.2764892 0.2458781 0.2445601 0.1799822 0.2512901 0.2131267 0.2068183
LF
one day 1.945064 1.8454509 1.8536689 1.9476153 0.3753898 0.4016417 0.3754895 0.3893344 0.2528859 0.3183266 0.2739985 0.2742312
five days 1.9058289 1.8164567 1.8430682 1.9082452 0.3723035 0.4156884 0.3845945 0.3942047 0.2476849 0.3364401 0.2872112 0.2848852
LL
one day 2.1105102 1.3674211 1.7794698 1.8780667 3.1829422 2.0319393 2.6484367 2.7484569 4.0325081 2.5463904 3.3237772 3.4894002
five days 1.9439615 1.2042995 1.5761858 1.7769775 2.9534538 1.7908837 2.3495872 2.5613621 3.8055508 2.2572723 2.9719026 3.240482
HMSE
one day 15.589704 8.8755022 12.087401 17.053157 1.6330067 0.9365614 1.250947 1.8354028 0.7923566 0.5464448 0.6487083 0.9019673
five days 12.153902 7.3636992 9.6885267 13.996445 1.3278418 0.804958 1.0276772 1.4787748 0.6038386 0.4969647 0.54961 0.7467398
GLME
one day 1.3680718 1.0282226 1.1863556 1.4048365 0.0559858 0.0195232 0.026789 0.0910236 -0.280096 -0.2402524 -0.271160 -0.2445029
five days 1.2350449 0.9538554 1.0769997 1.2696471 0.0337731 0.0184057 0.0147117 0.0623337 -0.299788 -0.2283178 -0.268249 -0.2508375
Table 5
The values of the accuracy measures of the one- and five-day ahead forecasts of the pln/usd volatility
in the period Jul. 28 - Aug. 27, 2010
Model GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH
Gauss t- Student GED GED Gauss GED GED GED Gauss t-Student GED GED
Real1_5 Real1_15 Real1_30
Minzer-Zarnowitz regression R2
one day 0.00028 0.002093 0.013531 0.003548 0.157655 0.188845 0.029977 0.056213 0.00154 0.00335 0.010128 0.008381
five days 0.013813 0.035494 0.123126 0.03318 0.06368 0.038123 0.000139 0.031434 0.153051 0.033052 0.166647 0.054478
MSE
one day 0.251677 0.244677 0.268199 0.289451 0.228135 0.218592 0.210609 0.267025 0.551271 0.536483 0.550615 0.640541
five days 0.262057 0.939224 0.282867 0.317755 0.166729 0.891716 0.188199 0.202826 0.618573 1.276204 0.61538 0.659206
RMSE
one day 0.501674 0.494649 0.517879 0.538007 0.477634 0.467539 0.458922 0.516744 0.742476 0.73245 0.742035 0.800338
five days 0.511915 0.969136 0.531852 0.563698 0.408325 0.944307 0.433819 0.450362 0.786494 1.129692 0.784462 0.811915
MAE
one day 0.348109 0.345578 0.383088 0.390101 0.308878 0.304268 0.303923 0.336683 0.485695 0.481476 0.505384 0.513854
five days 0.414999 0.582139 0.42246 0.436697 0.3112 0.478037 0.314589 0.315983 0.544198 0.710295 0.550941 0.575044
LF
one day 0.513734 1.183901 0.551065 0.607206 0.394864 1.110308 0.398809 0.469851 1.169843 1.812687 1.165996 1.299748
five days 0.677055 1.521363 0.705327 0.754452 0.477929 1.369753 0.502789 0.518809 1.162771 1.986499 1.166321 1.23425
LL
one day 0.010306 0.008009 0.00649 0.023695 0.008052 0.006017 0.008656 0.021187 0.019322 0.014904 0.010013 0.045505
five days 0.003216 0.008027 0.006777 0.010394 0.005632 0.023674 0.004176 0.019724 0.01145 0.012522 0.016972 0.021229
HMSE
one day 0.409176 0.369777 0.417217 0.677084 0.387136 0.34383 0.363115 0.662362 0.754118 0.686088 0.7082 1.268339
five days 0.31602 0.355797 0.370954 0.525277 0.211149 0.249229 0.248966 0.382985 0.891747 0.809627 0.843977 1.042343
GLME
one day 1.16656 1.159361 1.175866 1.215059 1.117668 1.109052 1.107273 1.165619 1.270631 1.259535 1.271851 1.351316
five days 1.164319 1.207247 1.177961 1.213651 1.075251 1.124843 1.08752 1.108617 1.306776 1.33448 1.30326 1.346073
Table 6
The values of the accuracy measures of the one- and five-day ahead forecasts of the eur/usd volatility
in the period Jul. 28 - Aug. 27, 2010
Model GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH GARCH(1.1) GJR-GARCH
Gauss ¿-Student GED GED Gauss GED GED GED Gauss ¿-Student GED GED
Real1_5 Real1_15 Real1_30
Minzer-Zarnowitz regression R2
one day 0.2068349 0.2099054 0.2337162 0.1594049 0.1264574 0.1302885 0.1610774 0.0936448 0.0755524 0.0777884 0.0960373 0.0609134
five days 0.0141596 0.0154004 0.0209393 0.0166008 0.0948827 0.099025 0.1135176 0.1085497 0.047831 0.0499977 0.0578525 0.0625627
MSE
one day 0.0891536 0.0881187 0.0793344 0.0618754 0.0912944 0.0902729 0.0818316 0.0616404 0.1029082 0.1018674 0.0933643 0.0730266
five days 0.098777 0.1000423 0.0891674 0.068952 0.1111403 0.112522 0.099956 0.077919 0.1225232 0.1239197 0.1113638 0.0891094
RMSE
one day 0.298586 0.2968479 0.2816637 0.2487478 0.3021497 0.3004545 0.2860622 0.2482749 0.3207931 0.3191667 0.3055558 0.2702343
five days 0.3142882 0.3162946 0.2986091 0.2625871 0.3333771 0.3354429 0.3161583 0.2791397 0.3500331 0.3520223 0.3337122 0.298512
MAE
one day 0.2498828 0.2481673 0.2344465 0.2033057 0.2669235 0.2652297 0.2509065 0.2230145 0.293896 0.2921737 0.2780908 0.2441897
five days 0.270011 0.271371 0.2548786 0.2256867 0.28314 0.284826 0.2685404 0.2373731 0.3103336 0.3120196 0.2957339 0.2650677
LF
one day 0.3390364 0.336286 0.3137809 0.2651812 0.3582179 0.3555026 0.3327381 0.2846549 0.3968042 0.3940411 0.3714551 0.3172163
five days 0.368788 0.3714133 0.344046 0.2946387 0.3942804 0.397348 0.3684964 0.3152921 0.4328568 0.4359393 0.4070978 0.3541771
LL
one day 0.0819447 0.0829666 0.0916066 0.1397863 0.0976877 0.0985858 0.1047459 0.1588718 0.1302063 0.1311868 0.1363995 0.2043858
five days 0.1101167 0.1073579 0.1111108 0.1725408 0.0743426 0.0724465 0.079401 0.1242036 0.0975883 0.0951424 0.1023844 0.1594517
HMSE
one day 0.1577689 0.1569766 0.1497834 0.139395 0.1717009 0.1707492 0.1622672 0.1477957 0.2035745 0.2026091 0.1934155 0.1853856
five days 0.1697825 0.1706822 0.1617526 0.1467152 0.1792544 0.1803478 0.1713479 0.1515935 0.2063305 0.2072163 0.1982491 0.1837706
GLME
one day 0.3606224 0.3597962 0.3525983 0.3352909 0.3226493 0.3217934 0.3145885 0.2938748 0.3148888 0.3140053 0.3066872 0.2858543
five days 0.3671665 0.3682966 0.3601378 0.3400729 0.3379393 0.3391444 0.3296732 0.308211 0.3299958 0.3312221 0.3217098 0.2999376
The tables 2-6 show that the obtained results are rather diversified. They differ between currencies, loss functions as well as the chosen frequency as a basis of computing the realized volatility. For the brl/usd volatility the forecasting performance of the applied models is rather poor that is confirmed by low values of the Minzer and Zarnowitz regression R2 coefficient. It means that the realized and forecasted values differ as the direction of changes is concerned. By comparing the values representing the loss functions one can state that the forecasts computed on the basis of GJR-GARCH model are mostly accurate. The best results of the forecasts were achieved using 5-minute returns to compute the realized volatility. A very similar picture can be observed concerning the inr/usd. In this case the best forecasting models were GARCH(1,1) with Z-Student error and GED distributions. The volatility forecasts computed for the rub/usd are character-
ized by quite satisfactory (greater than 0.24) R2 values for one-day ahead forecasts when GJR-GARCH model with GED error distribution was applied. This fact confirms the forecasting asymmetry in volatility. Positive and negative facts are perceived in a different way by the market. The values of the loss functions indicate different models for different frequencies of realized volatility. For the Polish zloty it is very hard to indicate the best forecasting model on the basis of the loss functions. Generally in this case GJR-GARCH model occurred to generate the worse forecasts. The best empirical results are obtained for eur/usd forecasts. The R2 of the Minzer and Zarnowitz regression for one-day ahead forecast is greater the 0.2 for Real1_5 and about 0.10 for Real1_15. The values of the loss functions are minimal in comparison with the remained currencies. The GJR-GARCH model was indicated as the best tool of volatility forecasting.
Table 7
The results of the SPA test for the models of volatility of the exchange rates
Loss function Real1_5 Real1_15 Real1_30
Brl
MSE GJR GARCH GED GJR GARCH GED GJR GARCH GED
SPAc* 0.00000 0.00040 0.00050
MAD GJR GARCH GED GJR GARCH GED GJR GARCH GED
SPAc 0.00000 0.00000 0.00000
Inr
MSE GARCH t-Stud GARCH t-Stud GARCH t-Stud
SPAc 0.00000 0.00000 0.00000
MAD GARCH Stud GARCH t-Stud GARCH t-Stud
SPAc 0.00000 0.00000 0.00000
Rub
MSE GARCH t-Stud GARCH GED GARCH GAUSS
SPAc 0.02860 0.03350 0.05930
MAD GARCH GED GARCH GAUSS GARCH GAUSS
SPAc 0.00000 0.00000 0.00760
Pln
MSE GARCH t-Stud GARCH GAUSS GARCH t-Stud
SPAc 0.00000 0.00000 0.00000
MAD GARCH t-Stud GARCH GED GARCH t-Stud
SPAc 0.00000 0.00000 0.00000
Eur
MSE GJR GARCH GED GJR GARCH GED GJR GARCH GED
SPAc 0.00010 0.00000 0.00110
MAD GJR GARCH GED GJR GARCH GED GJR GARCH GED
SPAc 0.00000 0.00000 0.00000
* SPAc denotes p-values of the consistent TSPA statistic (Hansen, 2005).
In table 7 the results of the SPA tests are presented. To compute the SPA test statistics a special computer subroutine in Ox provided by Hansen, Kim and Lunde (2003) has been used. It should be noted that the results of the SPA test depend on two important factors:
- the choice of the loss function,
- the choice of the benchmark model.
The test indicates the best model for each currency as well as gives p-values of the consistent TSPA statistic. The test results are generally stable for a given currency, apart from pln and rub. For the two currencies the test indicates different models concerning both: the loss function and real values represented by three time frequencies of the realized volatility.
While analysing the forecasting performance of the applied models on the basis of the loss functions and SPA test a general and very positive conclusion can be formulated. Namely, there is no contradiction between the results obtained for the currencies under investigation. For forecasting the brl and eur volatility the GJR-GARCH model was selected, while for the inr - the GARCH with i-Student error distribution was mostly proper. For forecasting the pln only the GJR-GARCH occurred to be inappropriate but the remained models were occasionally more or less correct. For prediction of the volatility of the Russian ruble no leading model can be indicated.
5. Conclusions
It is difficult to say how long current monetary system in the world is going to preserve. The BRIC group countries representing the most dynamically developing economies make their efforts to eliminate the US dollar hegemony and establish a new monetary system based on the basket of 20-30 currencies, including among others the Russian ruble, Chinese yuan, Indian rupee and Brazilian real. It is very likely that in the next ten years the new system will be introduced.
The results shown in the article indicate that the international currencies exchange rates like euro/usd are better predictable in a systematic way, because the transactions in this segment of the currency market are frequent. Transactions concerning rub/usd, cny/usd, inr/usd and brl/usd occur much more incidentally during the trading day, taking for example 5-minute or 15 minute intervals, and fore-
casting them is difficult, because one has to predict not only the future value of the currency or its volatility but also the moment when the transaction occurs. The time gaps between sequent transactions concerning the Polish zloty at the forex market are generally bigger than for eur/usd and smaller than the BRIC countries currencies.
It has been shown that the GARCH(1,1) model is capable to predict the volatility of exchange rates, but it usually requires non-Gaussian error distributions. Sometimes it has to be transformed into an asymmetric version in order to catch asymmetric reactions of the market. Although the mentioned changes cannot be seen at the standard goodness-of-fit analysis level, they become very clear at the forecasting level.
The obtained empirical results bring one to a general conclusion that there is no contradiction between the results obtained for the currencies under investigation between SPA test as well as the computed loss function. Forecasting the brl/usd and eur/usd volatility using the GJR-GARCH model gave the best results and in the case of the inr/usd the GARCH model with i-Student error distribution was mostly proper. For forecasting the pln/usd only the GJR-GARCH occurred to be inappropriate and the remained models occurred generally to be correct. The most ambiguous results concern the Russian ruble. Nevertheless, the results should be considered very positively, because the financial econometrics provides the efficient tools for model comparison.
Considering markets microstructure it could be observed that for such currencies like brl, inr and rub the changes of their exchange rates against the US dollar were not very frequent, and there were many zero returns during the trading day. The cases of pln as well as euro were quite opposite. One of the possible reasons is that the eur/usd exchange rate may dominate the volume of transactions on the market. On the other side, the speculative capital flows among the markets trying to find the best way for earning money and to limit the risk. In this way global capital and global institutions may exploit the mentioned currencies. This also explains why it is so difficult to forecast them. One should predict not only the future value of an exchange rate but also the time moment when risk is raising or falling.
Further industry development of any country cannot be done in isolation. The global economic and technological processes have already become the fact. In case of different monetary systems in the world one
is enforced to foresee future changes on the currency markets as well as to apply different financial instruments, limiting the risk. Acquaintance of best forecasting models gives a practical tool for achieving this goal.
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УДК 658,6:35.078.3
С.И. Шаныгин
СТАТИСТИЧЕСКИЙ АНАЛИЗ ДОСТОВЕРНОСТИ РЕЗУЛЬТАТОВ КОНТРОЛЯ СОСТОЯНИЯ ОРГАНИЗАЦИОННО-ЭКОНОМИЧЕСКОЙ СИСТЕМЫ
Характерной чертой современного уровня развития экономической науки является широкое применение статистических методов анализа. Они могут быть использованы при выявлении влияния различных факторов на деятельность производственного или иного предприятия как организационно-экономической системы (ОЭС). Необходимость применения этих методов обусловлена также тем, что изучение взаимодействия ОЭС с факторами внутренней и внешней среды требует анализа не только основных закономерностей целевых процессов, но и возможных случайных отклонений от них. В частности, применение статистических методов позволяет существенно повышать обоснованность управленческих решений при оценке состояния ОЭС.
Цель статистического анализа состояния ОЭС заключается в выявлении законов, управляющих процессами ее целевой деятельности. Статистическое изучение таких закономерностей позволяет более точно описывать целевые процессы ОЭС, ограничивать влияние случайных факторов, анализировать и прогнозировать деятельность ОЭС, целенаправленно влиять на
процесс ее развития, решать задачи текущего управления и контроля ее состояния. Методы, основанные на использовании математической статистики, являются эффективным инструментом сбора, анализа и интерпретации информации о состоянии и функционировании ОЭС. Применение этих методов не требует значительных затрат, позволяет с известной точностью и достоверностью анализировать состояние исследуемых объектов и процессов в организации и на основе этого вырабатывать оптимальные управленческие решения. Основой для статистических решений и оптимальной обработки информации о состоянии организации служат аналитические методы математической статистики, исходными данными для которых являются результаты наблюдения или экспериментальные данные, имеющие случайный характер. Традиционно любая функция экспериментальных (опытных) данных, которая не зависит от неизвестных статистических характеристик, называется статистикой. Оценкой статистической характеристики называется статистика, реализация которой принимается за неизвестное истинное значение параметра [5].
