THE ACCURACY OF UNEMPLOYMENT RATE FORECASTS IN ROMANIA AND THE ACTUAL ECONOMIC CRISIS

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1 Scietific Bulleti Ecoomic Scieces, Vol. / Issue THE ACCURACY OF UNEMPLOYMENT RATE FORECASTS IN ROMANIA AND THE ACTUAL ECONOMIC CRISIS Mihaela BRATU (SIMIONESCU) Faculty of Cyberetics, Statistics ad Ecoomic Iformatics Academy of Ecoomic Studies, Bucharest, Romaia mihaela_mb@yahoo.com Abstract: I this study, the problem of accuracy is aalysed o three differet forecastig horizos: durig the actual ecoomic crisis, i few years before the crisis ad o a large horizo. The accuracy of the made by Europea Commissio, Natioal Commissio for Progosis (NCP) ad Istitute for Ecoomic Forecastig (IEF) for uemploymet rate i Romaia is assessed. The most accurate predictios o the forecastig horizos 00-0 ad were provided by IEF ad the less accurate by NCP. These results were gotte usig U Theil s statistic ad a ew method that has ot bee used before i literature i this cotext. The multi-criteria rakig was applied to make a hierarchy of the istitutios regardig the accuracy ad five importat accuracy measures were take ito accout at the same time: mea errors, mea squared error, root mea squared error, U ad U statistics of Theil. I few years before crisis ( ) aother hierarchy of istitutios were gotte usig the accuracy criterio: NCP, IEF ad EC. The combied of istitutios predictios are the best strategy to improve the accuracy o overall ad before the crisis. Durig the ecoomic crisis IEF provided the most accurate predictios, the combied beig a good strategy of improvig oly the made by NCP ad EC usig iversely MSE scheme ad equally weighted scheme. The assessmet ad improvemet of accuracy have a importat cotributio i growig the quality of decisioal process. Key words:, predictios, accuracy, multi-criteria rakig, combied, combiig schemes. JEL Classificatio Codes: E, E7, C5, C53. INTRODUCTION The evaluatio of accuracy is ecessary for establishig the decisioal process. Whe more istitutios i a coutry provide for the same macroecoomic variable, the deciders have to choose the oe with the highest accuracy. The term of accuracy is put i correlatio with the errors that affect the forecastig process, because oly by hazard the predicted value of a idicator is exactly equal with its real value. The origial cotributio of this research is related to the proposal of a ew method of assessig the accuracy, takig ito accout more accuracy measures at the same time. The multi-criteria rakig let us make a classificatio of the istitutio accordig to more accuracy idicators. O the other had, the literature reports the ecessity of improvig the accuracy. We proposed as strategy of gettig better predictios tha the origial oes the combied ad we made comparisos with the origial predictios to measure the degree of improvemet. 56

2 The Accuracy of Uemploymet Rate Forecasts i Romaia ad the Actual Ecoomic. LITERATURE The accuracy evaluatio is oe of the curret cocers of may researchers. Oe purpose of this assessmet is related to the eed of improvig the predictios. The curret ecoomic ad fiacial crisis emphasized the struggles of ucertaity reductio. The accuracy is a very large domai of research, a exhaustive presetatio of it beig impossible. But, some of the recet results will be described. To assess the forecast accuracy, as well as their orderig, statisticias have developed several measures of accuracy. For comparisos betwee the MSE idicators of, Grager ad Newbold proposed a statistic. Aother statistic is preseted by Diebold ad Mariao (995) for compariso of other quatitative measures of errors. Diebold ad Mariao test proposed i 995 a test to compare the accuracy of two uder the ull hypothesis that assumes o differeces i accuracy. The test proposed by them was later improved by Ashley ad Harvey, who developed a ew statistic based o a bootstrap iferece. Subsequetly, Diebold ad Christofferse have developed a ew way of measurig the accuracy while preservig the co-itegratio relatio betwee variables. Meese ad Rogoff's paper, " Empirical exchage rate models of the seveties, remais the startig poit for may researches o the comparig of accuracy ad bias. Recet studies target accuracy aalysis usig as compariso criterio differet models used i makig predictios or the aalysis of forecasted values for the same macroecoomic idicators registered i several coutries. Alla (0) obtaied a good accuracy for the OECD combied with outtur values of GDP growth for G7 coutries betwee 984 ad 00. The same author metioed two groups of accuracy techiques used i assessig the predictios: quatitative accuracy statistics ad qualitative accuracy methods. Dover ad Weisser (0) used a broad set of idividual to aalyze four macroecoomic variables i G7 coutries. Aalyzig accuracy, bias ad efficiecy, resulted large discrepacies betwee coutries ad also i the same coutry for differet variables. Most iteratioal istitutios provide their ow macroecoomic. It is iterestig that may researchers compare the predictios of those istitutios (Melader for Europea Commissio, Vogel for OECD, Timmerma for IMF) with registered values ad those of other iteratioal orgaizatios, but it is omitted the compariso with official predictios of govermet. Abreu (0) evaluated the performace of macroecoomic made by IMF, Europea Commissio ad OECD ad two private istitutios (Cosesus Ecoomics ad The Ecoomist). The author aalized the directioal accuracy ad the ability of predictig a evetual ecoomic crisis. I Netherlads, experts made predictios startig from the macroecoomic model used by the Netherlads Bureau for Ecoomic Policy Aalysis (CPB). For the period was recostructed the model of the experts macroecoomic variables evolutio ad it was compared with the base model. The coclusios of Frases, Kraedok ad Laser (0) were that the CPB model are i geeral biased ad with a higher degree of accuracy. Gorr (009) showed that the uivariate method of predictio is suitable for ormal coditios of forecastig while usig covetioal measures for accuracy, but multivariate models are recommeded for predictig exceptioal coditios whe ROC curve is used to measure accuracy. Ruth (008), usig the empirical studies, obtaied with a higher degree of accuracy for Europea macroecoomic variables by combiig specific sub-groups predictios i compariso with based o a sigle model for the whole Uio. 57

3 Mihaela BRATU (SIMIONESCU) Heilema ad Stekler (007) explai why macroecoomic forecast accuracy i the last 50 years i G7 has ot improved. The first explaatio refers to the critic brought to macroecoometrics models ad to forecastig models, ad the secod oe is related to the urealistic expectatios of forecast accuracy. Problems related to the bias, data quality, the forecast process, predicted idicators, the relatioship betwee forecast accuracy ad forecast horizo are aalyzed. 3. COMPARISONS BETWEEN UNEMPLOYMENT FORECASTS MADE BY DIFFERENT INSTITUTIONS USING THE ACCURACY CRITERION I this study we used the forecasted values of the aual registered uemploymet rate made for Romaia by Europea Commissio, Natioal Commissio for Progosis ad Istitute for Ecoomic Forecastig. The forecastig horizo is The objective is to assess the accuracy, the biasess ad the efficiecy of these predictios ad determie the best istitutio with the highest performace. Armstrog ad Fildes (995) showed that it is ot sufficiet to use a sigle measure of accuracy. Therefore, more accuracy idicators were computed for the three types of o the specified horizo. To make comparisos betwee we propose to determie the hierarchy of istitutios accordig to the accuracy of their usig multi-criteria rakig. Two methods of multi-criteria rakig (raks method ad the method of relative distace with respect to the maximal performace) are used i order to select the istitutio that provided the best o the horizo 00-0 takig ito accout at the same time all computed measures of accuracy. The multi-criteria rakig ca be applied to make a hierarchy of istitutios takig ito accout the performace of i all its dimesios: accuracy, ubiasedess ad efficiecy. X t If we cosider (k) the predicted value after k periods from the origi time t, the the error at future time (t+k) is: e t ( t + k). This is the differece betwee the registered value ad the predicted oe. The idicators for evaluatig the accuracy that will be take ito cosideratio whe the multi-criteria rakig is used are: Root Mea Squared Error (RMSE) RMSE = j= e X ( T Mea error (ME) 0 + j, k) () ME = ex ( T0 + j, k) j= () The sig of idicator value provides importat iformatio: if it has a positive value, the the curret value of the variable was uderestimated, which meas expected average values too small. A egative value of the idicator shows expected values too high o average. Mea absolute error (MAE) 58

4 The Accuracy of Uemploymet Rate Forecasts i Romaia ad the Actual Ecoomic MAE = ex ( T0 + j, k) j= (3) These measures of accuracy have some disadvatages. For example, RMSE is affected by outliers. Armstrog ad Collopy stresses that these measures are ot idepedet of the uit of measuremet, uless if they are expressed as percetage. If we have two with the same mea absolute error, RMSE pealizes the oe with the biggest errors. A commo practice is to compare the forecast errors with those based o a radom-walk. Naïve model method assumes that the variable value i the ext period is equal to the oe recorded at actual momet. Theil proposed the calculatio of U statistic that takes ito accout both chages i the egative ad the positive sese of a idicator: U Theil s statistic ca be computed i two variats, specified also by the Australia Tresorery. The followig otatios are used: a- the registered results p- the predicted results t- referece time e- the error (e=a-p) - umber of time periods U = t= t= ( a p ) a t t + t t= p t A value close to zero for U implies a higher accuracy. pt+ at+ ( ) a t= t U = at+ at ( ) a t= t (5) If U ==> there are ot differeces i terms of accuracy betwee the two to compare If U <=> the forecast to compare has a higher degree of accuracy tha the aive oe If U >=> the forecast to compare has a lower degree of accuracy tha the aive oe Table : The accuracy of made by Europea Commissio, Natioal Commissio for Progosis ad Istitute for Ecoomic Forecastig for the uemploymet rate i Romaia (00-0) ACCURACY MEASURE INSTITUTION Europea Natioal Commissio (EC) Commissio for Progosis (NCP) ME MAE RMSE U U Source: ow computatios usig Excel (4) Istitute for Ecoomic Forecastig (IEF) 59

5 Mihaela BRATU (SIMIONESCU) Accordig to all accuracy idicators for made o the horizo 00-0, exceptig the mea error, the Istitute for Ecoomic Forecastig that used Dobrescu macromodel, provided the most accurate predictios for the uemploymet rate. Oly the of this istitutio outperformed the aïve predictios based o the radom walk. The egative values of the mea error imply too high i average predicted values for all istitutios. The less accurate are made by the Natioal Commissio for Progosis. We are iterested to see the accuracy durig the actual fiacial ad ecoomic crisis ad the accuracy i pre-crisis period. I Romaia the crisis started i 009, so the accuracy will be assessed o the forecastig horizo 009-0(i Table 3) ad before the crisis durig (i Table ). Table : The accuracy of made by Europea Commissio, Natioal Commissio for Progosis ad Istitute for Ecoomic Forecastig for the uemploymet rate i Romaia ( ) ACCURACY MEASURE Europea Commissio (EC) INSTITUTION Natioal Commissio for Progosis (NCP) ME MAE.9..7 RMSE U U Source: ow computatios usig Excel Istitute for Ecoomic Forecastig (IEF) I pre-crisis period, the best were provided by NCP, the istitutios with the lowest value for U. The NCP predictios have the lowest values for the other idicators (ME, RMSE ad MAE). The egative values for ME idicators show that all the istitutios overestimated the uemploymet rate. The multi-criteria rakig methods ad U will give the same hierarchy of istitutios: NCP, IEF ad EC. Table 3: The accuracy of made by Europea Commissio, Natioal Commissio for Progosis ad Istitute for Ecoomic Forecastig for the uemploymet rate i Romaia (009-0) ACCURACY MEASURE Europea Commissio (EC) INSTITUTION Natioal Commissio for Progosis (NCP) ME Istitute for Ecoomic Forecastig (IEF) MAE RMSE U U Source: ow computatios usig Excel Surprisigly, the U idicators shows a higher degree of accuracy i crisis period. But the U measure idicates that the durig the crisis are ot better tha the aive oes, while i pre-crisis years the predictios based o radom walk were less accurate. Exceptig U, all 60

6 The Accuracy of Uemploymet Rate Forecasts i Romaia ad the Actual Ecoomic the other accuracy idicators registered lower values durig the crisis for EC ad IEF. This meas that the two istitutios aticipated well the crisis effects, because it started i 008 i USA ad at mid 007 i Euroa Area. This time NCP uderestimated the uemploymet rate ad provided the higher values for all the accuracy measures. The IEF provided the most accurate predictios durig the crisis, this result beig gotte eve if we apply the multi-criteria rakig. Raks method applicatio supposes several steps:. Raks are assiged to each value of a accuracy idicator (the value that idicates the best accuracy receives the rak ); The statistical uits are the four istitutios that made. The rak for each istitutio is deoted by: ), i=,,3 ad accuracy idicator j. We chose 5 idicators: mea error, mea absolute error, root mea squared error, U ad U.. If the raks assiged to each istitutio are sum up, the score to each of them is computed., i=,,3 (6) 3. The istitutio with the lowest score has the highest performace ad it will get the fial rak. Table 4: The raks of istitutios accordig to the accuracy measures for the predictios durig 00-0 (raks method) INSTITUTION ACCURACY MEASURE Europea Commissio Natioal Commissio for Progosis Istitute for Ecoomic Forecastig ME 3 MAE 3 RMSE 3 U 3 U 3 Sum of raks Fial raks 3 Source: ow computatios usig Excel The results of the raks method are the same as those provided by most accuracy measures, especially U used i makig comparisos betwee. Actually, if all the calculated accuracy idicators are take ito accout at the same time, the followig hierarchy was gotte: Istitute for Ecoomic Forecastig, Europea Commissio ad Natioal Commissio for Progosis. 6

7 Mihaela BRATU (SIMIONESCU) The method of relative distace with respect to the maximal performace is the secod way of rakig. For each accuracy idicator the distace of each statistical uit (istitutio) with respect to the oe with the best performace is computed. The distace is calculated as a relative idicator of coordiatio:, i=,,3 ad j=,,..,5 (7) The relative distace computed for each istitutio is a ratio, where the deomiator is the best value for the accuracy idicator for all istitutios. The geometric mea for the distaces of each istitutio is calculated, its sigificace beig the average relative distace for istitutio i. =, i=,,3 (8) Accordig to the values of average relative distaces, the fial raks are assiged. The istitutio with the lowest average relative distace will take the rak. The positio (locatio) of each istitutio with respect to the oe with the best performace is computed as: its average relative distace over the lowest average relative distace. Table 5: The raks of istitutios accordig to the accuracy measures for the predictios made i the period 00-0 (method of relative distace with respect to the best istitutio) ACCURACY MEASURE Europea Commissio Natioal Commissio for Progosis Istitute for Ecoomic Forecastig ME MAE RMSE U U Average relative distace Raks 3 Locatio (%) Source: ow computatios usig Excel The method of relative distace with respect to the best istitutio gave the same results as the previous methods. The lowest average relative distace was registered by the Istitute for Ecoomic Forecastig (.059). The Diebold-Mariao test (DM test) is utilized to check if two have the same accuracy. The followig steps are applied: The differece betwee the squared errors of ( ) to compare ad the squared errors of referece ( ): (0) (9) 6

8 The Accuracy of Uemploymet Rate Forecasts i Romaia ad the Actual Ecoomic The followig model is estimated: () We test if a differs from zero, where the ull hypothesis is that a=0 (equal ). A p-value less tha 0.05 implies the rejectio of the ull hypothesis for a probability of 95% i guarateeig the results. The followig variables are computed: d, d ad d3 to make comparisos betwee EC ad NCP, EC ad IEF predictios, respectively NCP ad IEF expectatios. All the parameters are zero from statistical poit of view, so there are ot sigificat differeces betwee the provided by the three istitutios i terms of accuracy. The regressio models are estimated i EViews ad the results are preseted i Appedix. So, the accuracy test showed that there are ot sigificat differeces betwee the provided by the three istitutios. If we take ito accout the results based o accuracy idicators ad those of the DM test, we coclude the best predictios are those of IEF, followed by EC ad NCP, but the differeces betwee the uemploymet rate are ot too big. By applyig qualitative tests for directioal accuracy we check if there is a correct predictio of the chage. A test of idepedece betwee the effective values ad the directio of chage ca be applied i this situatio, the ull hypothesis showig the idepedece. A probability less tha 0.05 implies the rejectio of ull hypothesis. All the asymptotic sigificaces are greater tha 0.05, accordig to Appedix, fact that makes us to coclude that the directioal chages i the outtur are idepedet from the predictios. We ca coclude that we have differet hierarchies depedig o the forecastig horizo. The results are systematized i the followig table (Table 6). Table 6: The hierarchy of istitutios depedig o the forecastig horizo ad the accuracy criterio Forecastig horizo Hierarchy of istitutios accordig to accuracy criterio 00-0 IEF, EC ad NCP (pre-crisis period) NCP, IEF ad EC (crisis period) IEF, EC ad NCP As the table shows the same hierarchy was gotte for pre-crisis period ad for overall period. I the few years before the crisis NCP succeeded i providig the forecast with best accuracy for the uemploymet rate. The IEF seems to adapt more quickly to the chages i the ecoomic eviromet, makig a better aticipatio of the ecoomic crisis, despite the assumptios made i literature regardig the failure of Dobrescu macromodel i predictig the actual ecoomic crisis. 4. COMBINED FORECASTS TO IMPROVE THE ACCURACY OF UNEMPLOYMENT RATE PREDICTIONS Bratu (0) utilized some strategies to improve the accuracy (combied predictios, regressios models, historical errors method, applicatio of filters ad expoetial smoothig techiques). The combied are aother possible strategy of gettig more accurate predictios. The most utilized combiatio approaches are: optimal combiatio (OPT); equal-weights-scheme (EW); iverse MSE weightig scheme (INV). 63

9 Mihaela BRATU (SIMIONESCU) Bates ad Grager (969) started from two f;t ad f;t, for the same variable Xt, derived h periods ago. If the are ubiased, the error is calculated as: e = X f. i, t i, t i, t The errors follow a ormal distributio of parameters 0 ad σ i. If ρ is the correlatio betwee the errors, the their covariace is σ = ρ σ σ. The liear combiatio of the two predictios is a weighted average: c = m f + ( m) f.the error of the combied forecast t t t is: e c, t = m et + ( m) e t.the mea of the combied forecast is zero ad the variace is: σ = m σ + ( m) σ + m ( m) t σ. By miimizig the error variace, the c optimal value for m is determied ( m opt ): σ σ m opt = σ + σ σ () The idividual are iversely weighted to their relative mea squared forecast error (MSE) resultig INV. I this case, the iverse weight ( m iv ) is: σ m iv = σ + σ (3) Equally weighted combied predictios (EW) are gotte whe the same weights are give to all models. The U Theil s statistics were computed for the combied based o the three schemes, the results beig show i the followig table (Table 7): Table 7: The accuracy of combied for uemploymet rate (00-0) Accuracy idicator EC+NCP EC+IEF NCP+IEF U (optimal U (optimal U (iverse MSE U (iverse MSE U (equally weighted U (equally weighted Author s computatios usig Excel The combied proved to be a good strategy of improvig the accuracy whe EC ad NCP, respectively EC ad IEF predictios are combied usig OPT ad INV schemes. Oly if equally weighted scheme is utilized we got better for the combied predictios of NCP ad IEF. The most accurate are those resulted from combiig EC 64

10 The Accuracy of Uemploymet Rate Forecasts i Romaia ad the Actual Ecoomic ad IEF expectatios. All the combied predictios are better tha the aïve oes exceptig those of NCP ad IEF usig OPT scheme. We tested if the combied are a good strategy of gettig better i precrisis period ad durig the crisis. The results are preseted i Table 8 ad Table 9. Table 8: The accuracy of combied for uemploymet rate ( ) Accuracy idicator EC+NCP EC+IEF NCP+IEF U (optimal U (optimal U (iverse MSE U (iverse MSE U (equally weighted U (equally weighted Author s computatios usig Excel All the combied predictios are less accurate tha the aïve i pre-crisis period, but a great improvemet i accuracy was made. Exceptig the combied of EC ad IEF usig OPT scheme, all the are more accurate tha the oes made idepedetly by the three istitutios. Table 9: The accuracy of combied for uemploymet rate (009-0) Accuracy idicator EC+NCP EC+IEF NCP+IEF U (optimal U (optimal U (iverse MSE U (iverse MSE U (equally weighted U (equally weighted Author s computatios usig Excel Oly i some cases the combied predictios are better tha those made by IEF: the combied based o equally weighted scheme, the combied predictios of EC ad NCP, respectively NCP ad IEF whe INV scheme is used. Noe of the combied outperformed the predictios made by IEF durig the crisis. Exceptig the combied of EC ad NPC usig OPT scheme, all the other predictios are less accurate tha the aïve oes. 65

11 Mihaela BRATU (SIMIONESCU) 5. CONCLUSIONS I additio to ecoomic aalysis, the elaboratio of is a essetial aspect that coducts the way of developig the activity al macroecoomic level. But ay forecast must be accompaied by macroecoomic explaatios of its accuracy. The purpose of this evaluatio is related to differet aspects: the improvemet of the model o which the forecast was based, adjustmet of gouvermet policies, the plaig of results. Basically, accuracy evaluatio i this cotext refers directly to the degree of trust cofered to the predictio. Although the literature o forecastig methods ad techiques used i describig the evolutio of a ecoomic pheomeo is particularly rich, surprisigly, few researchers have dealt with the methods used to improve the measuremet of forecast ucertaity. The aspect is importat, because the macroecoomic predictios must ot be easily accepted, takig ito accout the egative cosequeces of macroecoomic failures, cosequeces that affect the state policies. The decisios of ecoomic policy are based o these. Hece, there is a evidet iterest of improvig their accuracy. I our study, we assessed the uemploymet performace for the predictios provided durig 00-0 by three istitutios: Europea Commissio, Natioal Commissio for Progosis ad Istitute of Ecoomic Forecastig. The best accuracy is provided by IEF, followed by EC ad NCP. This hierarchy resulted from the applicatio of the multi-criteria rakig, but also from the measuremet of accuracy idicators, as U, used i makig comparisos betwee. The same hierarchy was gotte durig the crisis period, while i the few years before it The combied usig the three classical schemes are a good strategy of improvig the accuracy, most of the combied predictios beig better tha the iitial oes. I crisis period oly some of the combied are better tha the NCP ad EC oes, oe of them succeeded i outperformig the IEF predictios. Before the crisis the combied predictios are a very good way of improvig almost all the made by the three istitutios. The accuracy should be a priority for the public that uses these predictios i uderlyig the decisioal process. The combied are a very good strategy of gettig improvemets i accuracy for the uemploymet rate predictios. REFERENCES. Abreu I., Iteratioal orgaisatios vs. private aalysts : a Evaluatio, Baco de Portugal, 0 article available at: US/BdP%0Publicatios%0Research/wp00.pdf. Alla G., Evaluatig the usefuless of of relative growth, Strathclyde, Discussio Papers i Ecoomics, No. -4, 0 3. Armstrog, J. S., Fildes, R., O the selectio of Error Measures for Comparisos Amog Forecastig Methods, Joural of Forecastig, 4, pp. 67-7, Bates, J., Grager C. W. J. (969), The Combiatio of Forecasts, Operatios Research Quarterly, 0(4), pp , Bratu M., Strategies to Improve the Accuracy of Macroecoomic Forecasts i USA, LAP LAMBERT Academic Publishig, ISBN-0: , ISBN-3: , 0 6. Diebold, F.X., Mariao, R., Comparig Predictive Accuracy, Joural of Busiess ad Ecoomic Statistics, 3, pp , Dover, J. ad Weisser J., Accuracy, ubiasedess ad efficiecy of professioal macroecoomic : A empirical compariso for the G7, Iteratioal Joural of Forecastig, 7 (), pp , 0 8. Frases, P.H., McAleer, M., Legerstee, R., Evaluatig Macroecoomic Forecasts: A Cocise Review of Some Recet Developmets, Workig paper/ Departmet of Ecoomics ad Fiace, Uiversity of Catebury, 0 66

12 The Accuracy of Uemploymet Rate Forecasts i Romaia ad the Actual Ecoomic 9. Gorr, W. L., Forecast accuracy measures for exceptio reportig usig receiver operatig characteristic curves, Iteratioal Joural of Forecastig, Volum 5, Issue, pp. 48-6, Heilema, U., Stekler, H., Itroductio to The future of macroecoomic forecastig, Iteratioal Joural of Forecastig, 3(), pp , 007. Ruth, K, Macroecoomic forecastig i the EMU: Does disaggregate modelig improve forecast accuracy?, Joural of Policy Modelig, Volume 30, Issue 3,pp , 008 APPENDIX The results of Diebold-Mariao test i EViews Depedet Variable: D Method: Least Squares Date: // Time: 3:0 Sample: 00 0 Icluded observatios: Variable Coefficiet Std. Error t-statistic Prob. C Depedet Variable: D Method: Least Squares Date: // Time: 3:0 Sample: 00 0 Icluded observatios: Variable Coefficiet Std. Error t-statistic Prob. C Log likelihood Depedet Variable: D3 Method: Least Squares Date: // Time: 3:03 Sample: 00 0 Icluded observatios: stat Durbi-Watso.5367 Variable Coefficiet Std. Error t-statistic Prob. C APPENDIX The results of tests for directioal accuracy Test Statistics ur Ec Chi-Square.88 a.73 b Df 9 8 Asymp. Sig Test Statistics ur Ief Chi-Square 88 a.73 b Df 9 8 Asym.0.99 p. Sig Test Statistics ur Ncp Chi-Square.88 a.000 b Df 9 0 Asymp. Sig