Using the Divergence Information Criterion for the Determination of the Order of an Autoregressive Process

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1 Using the Divergence Information Criterion for the Determination of the Order of an Autoregressive Process P. Mantalos a1, K. Mattheou b, A. Karagrigoriou b a.deartment of Statistics University of Lund Sweden b.deartment of Mathematics and Statistics University of Cyrus Cyrus Version 17/04/008 Abstract This article comares traditional Model Selection Criteria with the recently roosed Divergence Information Criterion which is based on the BHHJ measure of divergence. We use Monte Carlo methods and different Data Generating Processes for small, medium and large samle sizes. The Divergence Information Criterion shows remarkable good results by choosing the correct model more frequently than the known traditional Information Criteria. Key words: divergence AR rocess, Information Criterion, Model selection, Measure of 1 Corresonding author: Docent Panagiotis Mantalos. Panagiotis.Mantalos@stat.lu.se 1

2 1. Introduction. Consider a discrete-time stationary AR() rocess of the form X = ax + a X + + a X + e (1) t 1 t 1 t t t where { t e } is a sequence of uncorrelated ( 0, ) N σ random variables and all the roots of the autoregressive olynomial are outside the unit circle, that is, Az ( ) = 1 az a z 0 1 for all z such that z < 1. One of the crucial roblems in time series modeling is selecting the otimal model order and secifying a arsimonious model for the data generating rocess (DGP). Many techniques have been develoed for the order selection of linear models, namely a) Grahical methods with reresentative the Box and Jenkins methods (1970); b) Hyothesis test rocedures with the ioneering work of Whittle (195) as reresentative. Akaike (1974) viewed the roblem of model fitting in the context of time series analysis as a multile decision rocedure rather than a hyothesis testing roblem. However the same author (Akaike, 1973) initiated the research on automatic model selection techniques by develoing the oular Akaike Information Criterion (AIC) which is considered as a significant contribution to statistical modeling. Schwarz (1978), Hannan and Quinn (1979) and others followed the ioneering work of Akaike. Most of the criteria roosed in the literature consist in minimizing, with resect to the model order, a function of the given observations lus a enalty term for the introduction of additional arameters, which

3 generally deends on the model order and samle size. Thus, according to automatic criteria, the arameters of a variety of cometing models are estimated and the model chosen is the one with the smallest criterion value. In a resent Ph.D thesis, Mattheou (007) and in a resulting article Mattheou, Lee and Karagrigoriou (008) introduced a new model selection criterion, the Divergence Information Criterion (DIC). This gives the signal to study the new DIC criterion and exlore its caabilities in autoregressive rocesses and in general in a time series context. The aer is organised as follows. In Section, we resent and briefly discuss the different Information Criteria including the Divergence Information Criterion (DIC). In Section 3, we resent a comarative study of various Information Criteria for different time series examles by using a Monte Carlo method. Finally in Section 4 we summarize our results. For the simulations we make use of the Gauss 8 rogram.. Information Criterion One of the most interesting and imortant roblems in Statistics is the roblem of selecting a satisfactory model for a given set of observations. It might aear at first sight that the higher the order of the selected rocess (model) chosen, the better the fitted model will be. Such a thought may be true for fitting uroses but not for redicting ones. Indeed, the fit may be erfect for the given data but the use of the selected model for redictive uroses may result in gross errors. Numerous selection criteria have been develoed over the last 40 years which attemt to revent overfitting by assigning a enalty to the introduction of an unnecessary large order and consequently to an unnecessary number of arameters. The measures of divergence are used as indices of similarity or dissimilarity between oulations and for measuring mutual information concerning two variables and as such they 3

4 can be used for the construction of model selection criteria. The distance between a candidate model and the true but unknown model could be measured by a measure of divergence. The candidate model for which the measure of divergence is minimized will be selected. The well known Kullback-Leibler measure of divergence (Kullback and Leibler, 1951) was the one used by Akaike (1973) to develo the Akaike Information Criterion (AIC). Akaike roosed the evaluation of the fit of the candidate model using minus twice the mean exected loglikelihood (also known as exected overall discreancy). Furthermore, he rovided an unbiased estimator of the exected loglikelihood so that the resulting AIC criterion is given by AIC = n ˆ +, ( ) log( σ ) where n the samle size, the order of the candidate model and ˆ σ the estimator of the variance of the th -order candidate model. For fitting autoregressive rocesses, Jones (1975) suggested that AIC has a tendency to overestimate the order of the rocess and Shibata (1976) showed that the robability of overestimation for a large samle size is nonzero. To correct this tendency, the SIC criterion was roosed indeendently by Akaike (1978) and Schwarz (1978): ˆ SIC( ) = n log( σ ) + log n. The above two criteria can be considered as members of the General Information Criterion given by ˆ GIC( ) = n log( σ ) + c n 4

5 where cn a quantity that may deend on n. For c n = the criterion reduces to AIC while for cn = log n to SIC. The case c = clog log nwith c > corresonds to the Hannan and Quinn n criterion (HQ, Hannan and Quinn, 1979) which has been found to be equivalent to SIC in the sense that they are both consistent (Hannan, 1980). The roblem of avoiding overfitting esecially for urely autoregressive rocesses can also be dealt with the minimization of the final rediction error criterion (FPE) of Akaike (1969). The FPE is an estimate of the one-ste ahead rediction mean squared error for a realization of the rocess indeendent of the one observed. If we fit AR rocesses of steadily increasing order the maximum likelihood estimator (MLE) of the white noise variance will usually decrease with. However, FPE will decrease to a minimum value and then increase as will increase. According to FPE, we choose the order of the candidate rocess to be the value of for which FPE is minimized. The FPE is given by n+ FPE( ) = ˆ σ n. In all the above cases, any arameter estimation is handled through the maximum likelihood method. A general class of criteria has been introduced by Konishi and Kitagawa (1996) which also estimates the Kullback-Leibler measure where the estimation is not necessarily based on maximum likelihood. One of the most recently roosed measures of divergence is the Basu-Harris-Hjort-Jones ower divergence between the candidate model f θ (.) and the true model g (Basu et. al, 1998) which is denoted by BHHJ, indexed by a ositive arameter α, and defined as: a (, ) a ( ) 1 ( ) a I g f f z g z f ( z) g 1 a = + + ( z) dz θ θ + + a θ a θ. 5

6 This family of measures was roosed by Basu et al. (1998) for the develoment of a minimum divergence estimating method for robust arameter estimation. The index α controls the trade-off between robustness and asymtotic efficiency of the arameter estimators which are the values of θ that minimize the measure over a arametric saceθ. It should be also noted that the BHHJ family reduces to the Kullback-Leibler measure of divergence for a 0 (see Mattheou, 007) and as it can be easily seen, to the square of the standard L distance between the candidate and the true model for α=1. Mattheou et al. (008) alying the same methodology used for AIC to the BHHJ divergence develoed a new criterion, the Divergence Information Criterion (DIC) which for a set of observations x 1,...,xnis given by / / ( ) ˆ ( ) a DIC nq π (1 a) + θ = + +, where n 1+ a 1 1 a Qˆ = f ˆ ( z) dz 1 + f ˆ ( x ) and ˆ θ θ i a n θ i= 1 θ a consistent and asymtotically normal estimator of θ. Preliminary simulation studies for regression models (Mattheou, 007) showed a very good medium samle size erformance for DIC for values of α close to zero. Although the DIC criterion was constructed so that it will be an asymtotically unbiased estimator of the BHHJ divergence measure between the candidate and the true model, the calculation of the first art of Q ˆ θ, namely the integral 1+ a f ˆ ( zdz ) is not comutationally θ attractive for ractitioners. Furthermore, the simulation study shows that the difference in the calculations of the above integrals, for the different candidate models is negligible comared with the difference in the calculation for the entire quantity Q ˆ θ. In other words the integral term does not affect the selection of the correct model and therefore can be removed from the 6

7 formula. As a result, we roose now a modified new criterion, the Modified Divergence Information Criterion (MDIC) which is given by / / ( ) * ( ) a + MDIC = n MQ + π (1 + a) ˆ θ 1 MQ 1 f ( x ) θ n 1 where ˆ = ( + ) n a α ˆ. θ i i= 1 Note that a model selection criterion can be considered as an aroximately unbiased estimator of the exected overall discreancy, a nonnegative quantity which measures the distance between the true unknown model and a fitted aroximating model. Observe also that, as it was mentioned in the introduction, a criterion consists of two terms, the first of which is a biased estimator of the exected overall discreancy. As a result, if we choose the model with the smallest estimator of the exected overall discreancy we may end u with a selection with an unnecessarily large order. The estimator becomes asymtotically unbiased by introducing the aroriate correction term. 3. Simulations. 3.1 Selection of the index α Before the Monte Carlo exeriment is erformed one has to decide which ositive value of the index α is aroriate for ractical uroses. As a result, we simulate a 100 observation series for 5 different models with α between 0.01 and 0.5 and we rovide in Figure 1, the ower of the selection, namely the roortion of times the correct model is selected as a function ofα. In Figure 1, [Dot dash] line is the AR(1) model defined as Model 1 in section 3.; [Whole] line shows the AR() aearing as Model 5 in section 3.; [Dot close] line refers to the AR(3) rocess: xt = 1+ 0.xt xt 0.35xt 3+ et; [Dash] line is the AR(4) rocess: 7

8 x = 1+ 0.x + 0.5x 0.35x 0.x + e ; finally [Dot] is the AR(5) rocess defined as t t 1 t t 3 t 4 t Model 6 in section 3.. Figure 1 shows that for small lags (models AR(1) and AR()) the ower increases as the value of α increases. In all the other cases the ower increases u to a value of α and then decreases. More secifically, for the comlete AR(3) and AR(4) rocesses the ower stays high aroximately u to the value α = 0.5. For the AR(5) rocess the ower decreases after the value α = Although an otimum value of α for all tyes of autoregressive models may be considered to be the value of 0.10, in the Monte Carlo study of section 3. we choose the value of 0.5 since it aears to serve as a fair balance between small and large lag models. Figure 1 8

9 3. The Monte Carlo Exeriment In this section we rovide the characteristics of the Monte Carlo exeriment undertaken. We calculate the estimated ercent by simly observing how many times the correct AR() model is selected in reeated samles. By varying factors such as the number of observations 50 (small samle) 75, 100 (medium samle) and 00, 500 (large samle); and the order of AR() model we obtain a succession of estimated ercent of the correct selection model under different conditions. The Monte Carlo exeriment has been erformed by generating data according to the following Data generating rocesses: Model 1: xt = xt 1 + et Model : xt = xt + et Model 3: xt = xt xt + et Model 4: xt = xt xt + et Model 5: xt = xt 1 0.5xt + et Model 6: xt = xt 1 0.xt xt 5 + et where { t e } is a sequence of uncorrelated ( 0, ) N σ random variables. The criteria used in this exeriment are the ones defined in Section, namely AIC, SIC, HQ, FPE, and MDIC with α = 0.5. For the imlementation of the criteria we use f 1 eˆ ( x) = ex ˆ σ, ( πσˆ ) ˆ θ i where ê the estimated residuals and ˆ σ the estimated variance. For each time series 0 resamle values are generated with zero initial conditions, taking net samle sizes of n = 50,75,100,00,500 in order to cover small, medium and large samles. 9

10 The number of relications er model used is The calculations were erformed using GAUSS Results In this section we resent the results of the Monte Carlo exeriment concerning the ercent of correct selected model. Table 1 shows the results for Model 1. It is not difficult to see the good erformance of the MDIC criterion for all samle sizes. The rate of success of MDIC is almost 89% for the small samle with 50 observations, while the other criteria have a rate of success between 73.3% and 83.6%. We also observe the samle effect. Indeed, by increasing the number of observations, the only criterion for which the ercent of correct model selection increases significantly is the Hannan and Quinn criterion (HQ). A smaller increase is observed for the SIC criterion. For large samle sizes HQ comes first with a rate of success of aroximately 91% and MDIC close second with a success rate almost equal to 90%. At the same time, as the samle size increases, the difference between the MDIC and the remaining Information Criteria decreases, although it is still in favor of MDIC. The good erformance of the MDIC criterion is evident from the high ercent of selecting the correct AR(1) model in reeated samles. While in Table 1 MDIC erforms quite well with though some cometition from the HQ criterion, in the case of the second model, as seen in Table, the MDIC is suerior to all other criteria for all samle sizes. Observe that MDIC is the best criterion among the cometing criteria with the HQ coming second. The worst erformance was observed by SIC. The 10

11 samle effect shows the tendency of the best criteria, namely MDIC and HQ, to aroach each other in terms of their rate of success as n increases. Observe the imressive success rate of MDIC even for small samle sizes where for n=75 rreaches a remarkable success rate of 95%. Our attemt to see if the incomlete AR() rocess misleads the Information criteria does not seem to work since the rate of success is not significantly affected by the articular form of the underlying rocess. In case of Models 3 and 5 as Tables 3 and 5 show the results are not different from the revious cases. Indeed, MDIC is suerior to all other criteria with success rate as high as 95% even for relatively small samle sizes. In case of Model 4 as Table 4 shows all criteria for small samle sizes (n=50) have a high tendency of selecting the simlest model (AR(1)) which is due to the fact that the coefficient of lag is relatively close to zero (0.35). As the samle size increases the rate of success imroves. MDIC is the best criterion among the criteria examined with HQ coming second. Finally for Model 6, Table 6 shows that for small samle sizes the rate of success is less than 50% with SIC being the best, MDIC being the worse, and HQ coming second to last. As the samle size increases, MDIC and HQ have the higher imrovement so that for n 100 they become again the best models with MDIC reaching an imressive success rate of 98.5% for n=00 and 99.1% for n=500. To summarize the findings we could safely conclude that the MDIC criterion erforms much better than the other selection criteria for AR rocesses and for various samle sizes. In articular, the magnitude of the sueriority of MDIC is extremely high for AIC, SIC, and FPE for all samle sizes. In reference to the HQ criterion the sueriority of MDIC is of lesser magnitude as comared to the other criteria. 11

12 4. Conclusions In this aer we attemted a comarative study of model selection criteria for autoregressive rocesses for small, medium, and large samle sizes. Based on the results of our study we conclude that for regular AR models, the erformance of MDIC is excellent with very high rate of success, for all samle sizes. The rate of success of MDIC increases with samle size but usually not as much as the HQ and SIC criteria which is exected though since they are both consistent. It is imortant to oint out that in most cases MDIC aears to have a high rate of success (arox. equal to 90%) even for small samle sizes. For irregular models (like Model 6), MDIC needs a sufficient number of observations for erforming well. More secifically, a medium samle size of order 75 or 100 seems to be enough in order to ick u high rates of success. Finally observe that MDIC is the only criterion that never selects too large models. In fact, in all cases the robability of overestimation is at most 8% and for lags at most or 3 higher than the true lag while the other criteria may select lags as large as 7 lags higher than the true one. 1

13 Table nr 1: Model 1 Lag1 Lag Lag3 Lag4 Lag5 Lag6 Lag7 Lag8 50 obs AIC 0,769 0,108 0,049 0,0 0,018 0,01 0,013 0,009 SIC 0,733 0,114 0,051 0,08 0,0 0,017 0,018 0,017 HQ 0,836 0,097 0,08 0,015 0,009 0,008 0,005 0,00 FPE 0,769 0,11 0,049 0,01 0,017 0,01 0,013 0,009 MDIC 0,889 0,095 0,014 0, obs AIC 0,775 0,14 0,04 0,05 0,016 0,009 0,005 0,006 SIC 0,744 0,14 0,047 0,09 0,013 0,017 0,011 0,015 HQ 0,873 0,085 0,04 0,008 0,004 0,00 0,001 0,003 FPE 0,777 0,14 0,04 0,05 0,017 0,008 0,003 0,006 MDIC 0,894 0,086 0,018 0,001 0, obs AIC 0,774 0,107 0,057 0,03 0,019 0,01 0,004 0,006 SIC 0,75 0,106 0,065 0,08 0,019 0,013 0,009 0,01 HQ 0,881 0,071 0,07 0,009 0,009 0,00 0,001 0 FPE 0,774 0,107 0,057 0,03 0,019 0,01 0,005 0,005 MDIC 0,896 0,08 0,017 0,004 0, obs AIC 0,797 0,116 0,038 0,03 0,01 0,007 0,005 0,00 SIC 0,785 0,1 0,04 0,06 0,014 0,007 0,006 0,00 HQ 0,916 0,06 0,013 0,005 0,001 0, FPE 0,797 0,116 0,038 0,03 0,01 0,007 0,005 0,00 MDIC 0,896 0,088 0,01 0,00 0, obs AIC 0,776 0,117 0,049 0,019 0,016 0,013 0,008 0,00 SIC 0,773 0,118 0,049 0,01 0,016 0,013 0,007 0,003 HQ 0,918 0,07 0,01 0,001 0, FPE 0,776 0,117 0,049 0,019 0,016 0,013 0,008 0,00 MDIC 0,893 0,098 0, The shading indicates best erformance Information Criterion. 13

14 Table nr : Model Lag1 Lag Lag3 Lag4 Lag5 Lag6 Lag7 Lag8 50 obs AIC 0,018 0,748 0,103 0,05 0,06 0,0 0,015 0,0 SIC 0,015 0,688 0,111 0,06 0,036 0,031 0,031 0,08 HQ 0,034 0,805 0,08 0,034 0,0 0,014 0,006 0,007 FPE 0,018 0,751 0,103 0,047 0,06 0,0 0,015 0,0 MDIC 0,037 0,894 0,06 0,008 0, obs AIC 0,001 0,777 0,104 0,051 0,08 0,017 0,009 0,013 SIC 0,001 0,743 0,11 0,056 0,033 0,03 0,01 0,0 HQ 0,00 0,878 0,076 0,04 0,01 0,006 0,001 0,003 FPE 0,001 0,777 0,104 0,051 0,08 0,017 0,009 0,013 MDIC 0,003 0,948 0,041 0,007 0, obs AIC 0,001 0,775 0,1 0,04 0,031 0,011 0,009 0,011 SIC 0,001 0,759 0,11 0,05 0,08 0,016 0,011 0,014 HQ 0,001 0,874 0,086 0,019 0,013 0,004 0,003 0 FPE 0,001 0,776 0,1 0,04 0,031 0,011 0,008 0,011 MDIC 0,001 0,93 0,063 0, obs AIC 0,001 0,814 0,104 0,07 0,03 0,007 0,004 0,011 SIC 0,001 0,804 0,106 0,03 0,034 0,008 0,005 0,01 HQ 0,001 0,915 0,061 0,013 0,007 0,00 0,001 0 FPE 0,001 0,814 0,104 0,07 0,03 0,007 0,004 0,011 MDIC 0,001 0,945 0,051 0, obs AIC 0,001 0,793 0,114 0,04 0,04 0,008 0,011 0,007 SIC 0,001 0,788 0,115 0,041 0,06 0,01 0,011 0,008 HQ 0,001 0,915 0,066 0,01 0,005 0, FPE 0,001 0,793 0,114 0,04 0,04 0,008 0,011 0,007 MDIC 0,001 0,939 0,057 0, The shading indicates best erformance Information Criterion. 14

15 Table nr 3: Model 3 Lag1 Lag Lag3 Lag4 Lag5 Lag6 Lag7 Lag8 50 obs AIC 0,001 0,749 0,11 0,05 0,04 0,07 0,015 0,011 SIC 0,001 0,709 0,11 0,06 0,031 0,03 0,04 0,0 HQ 0,001 0,85 0,106 0,031 0,011 0,0 0,004 0,00 FPE 0,001 0,749 0,1 0,05 0,04 0,07 0,015 0,01 MDIC 0,001 0,9 0,07 0, obs AIC 0,001 0,78 0,094 0,059 0,034 0,01 0,01 0,01 SIC 0,001 0,749 0,097 0,065 0,038 0,0 0,014 0,016 HQ 0,001 0,88 0,068 0,04 0,016 0,003 0,003 0,003 FPE 0,001 0,781 0,094 0,058 0,034 0,01 0,01 0,01 MDIC 0,001 0,944 0,048 0, obs AIC 0,001 0,764 0,11 0,046 0,07 0,0 0,013 0,006 SIC 0,001 0,736 0,18 0,051 0,09 0,08 0,017 0,01 HQ 0,001 0,879 0,08 0,019 0,014 0,005 0,00 0 FPE 0,001 0,765 0,11 0,046 0,07 0,0 0,01 0,006 MDIC 0,001 0,933 0,061 0, obs AIC 0,001 0,776 0,116 0,04 0,031 0,018 0,009 0,007 SIC 0,001 0,763 0,118 0,043 0,034 0,019 0,01 0,01 HQ 0,001 0,913 0,066 0,009 0,009 0, FPE 0,001 0,777 0,116 0,04 0,03 0,018 0,009 0,007 MDIC 0,001 0,949 0,047 0, obs AIC 0,001 0,795 0,095 0,058 0,05 0,01 0,011 0,003 SIC 0,001 0,795 0,095 0,058 0,05 0,01 0,011 0,003 HQ 0,001 0,931 0,046 0,018 0, FPE 0,001 0,795 0,095 0,058 0,05 0,01 0,011 0,003 MDIC 0,001 0,945 0,048 0, The shading indicates best erformance Information Criterion. 15

16 Table nr 4: Model 4 Lag1 Lag Lag3 Lag4 Lag5 Lag6 Lag7 Lag8 50 obs AIC 0,159 0,607 0,111 0,046 0,06 0,06 0,013 0,01 SIC 0,144 0,575 0,113 0,06 0,03 0,09 0,0 0,05 HQ 0,14 0,643 0,078 0,036 0,009 0,013 0,00 0,005 FPE 0,161 0,61 0,109 0,045 0,06 0,05 0,01 0,01 MDIC 0,68 0,673 0,05 0, obs AIC 0,075 0,71 0,11 0,045 0,03 0,015 0,007 0,006 SIC 0,067 0,684 0,11 0,054 0,036 0,0 0,01 0,017 HQ 0,13 0,756 0,074 0,04 0,014 0,003 0,00 0,004 FPE 0,077 0,71 0,11 0,045 0,09 0,014 0,008 0,005 MDIC 0,144 0,8 0,049 0, obs AIC 0,033 0,739 0,11 0,054 0,01 0,04 0,01 0,009 SIC 0,03 0,716 0,116 0,061 0,04 0,08 0,013 0,01 HQ 0,059 0,818 0,07 0,033 0,009 0,006 0,00 0,001 FPE 0,033 0,739 0,11 0,054 0,01 0,04 0,01 0,009 MDIC 0,07 0,863 0,051 0,013 0, obs AIC 0,001 0,779 0,114 0,047 0,033 0,014 0,005 0,007 SIC 0,001 0,771 0,11 0,053 0,034 0,014 0,006 0,009 HQ 0,00 0,909 0,06 0,016 0,007 0,005 0,001 0 FPE 0,001 0,779 0,114 0,047 0,033 0,014 0,005 0,007 MDIC 0,003 0,938 0,048 0,009 0, obs AIC 0,001 0,795 0,106 0,053 0,0 0,013 0,008 0,00 SIC 0,001 0,791 0,105 0,053 0,05 0,014 0,009 0,00 HQ 0,001 0,917 0,061 0,018 0, FPE 0,001 0,795 0,106 0,053 0,0 0,013 0,008 0,00 MDIC 0,001 0,933 0,059 0, The shading indicates best erformance Information Criterion. 16

17 Table nr 5: Model 5 Lag1 Lag Lag3 Lag4 Lag5 Lag6 Lag7 Lag8 50 obs AIC 0,05 0,71 0,137 0,045 0,07 0,07 0,013 0,014 SIC 0,03 0,677 0,14 0,05 0,031 0,033 0,0 0,06 HQ 0,04 0,78 0,106 0,031 0,01 0,016 0,004 0,007 FPE 0,06 0,716 0,136 0,045 0,06 0,07 0,01 0,01 MDIC 0,058 0,861 0,073 0,007 0, obs AIC 0,005 0,763 0,17 0,046 0,08 0,014 0,008 0,009 SIC 0,005 0,734 0,13 0,05 0,031 0,01 0,01 0,015 HQ 0,009 0,861 0,086 0,07 0,011 0,003 0,001 0,00 FPE 0,005 0,764 0,17 0,046 0,08 0,014 0,007 0,009 MDIC 0,011 0,97 0,053 0,008 0, obs AIC 0,001 0,759 0,119 0,055 0,04 0,05 0,009 0,008 SIC 0,001 0,739 0,1 0,057 0,07 0,09 0,009 0,016 HQ 0,00 0,859 0,089 0,07 0,013 0,009 0,001 0 FPE 0,001 0,76 0,118 0,055 0,04 0,05 0,009 0,008 MDIC 0,00 0,97 0,06 0,007 0, obs AIC 0,001 0,775 0,1 0,044 0,033 0,013 0,009 0,005 SIC 0,001 0,765 0,1 0,045 0,034 0,015 0,011 0,007 HQ 0,001 0,898 0,071 0,019 0,004 0,004 0,003 0 FPE 0,001 0,775 0,1 0,044 0,033 0,013 0,009 0,005 MDIC 0,001 0,941 0,05 0, obs AIC 0,001 0,789 0,107 0,05 0,06 0,011 0,01 0,004 SIC 0,001 0,788 0,108 0,05 0,06 0,011 0,01 0,004 HQ 0,001 0,917 0,055 0,018 0,007 0, FPE 0,001 0,789 0,107 0,05 0,06 0,011 0,01 0,004 MDIC 0,001 0,936 0,056 0, The shading indicates best erformance Information Criterion. 17

18 Table nr 6: Model 6 Lag1 Lag Lag3 Lag4 Lag5 Lag6 Lag7 Lag8 50 obs AIC 0,149 0,0 0,096 0,033 0,498 0,114 0,049 0,041 SIC 0,11 0,019 0,076 0,03 0,499 0,16 0,057 0,07 HQ 0,37 0,0 0,113 0,033 0,466 0,08 0,09 0,0 FPE 0,15 0,0 0,097 0,033 0,497 0,114 0,048 0,041 MDIC 0,495 0,051 0,05 0,06 0, 0, obs AIC 0,059 0,001 0,031 0,014 0,70 0,1 0,047 0,06 SIC 0,047 0,001 0,05 0,011 0,683 0,136 0,058 0,039 HQ 0,11 0,008 0,051 0,013 0,701 0,068 0,06 0,01 FPE 0,059 0,001 0,031 0,014 0,704 0,119 0,046 0,06 MDIC 0,84 0,06 0,146 0,016 0,54 0, obs AIC 0,01 0,001 0,008 0,004 0,751 0,139 0,05 0,035 SIC 0,009 0,001 0,007 0,003 0,73 0,147 0,057 0,046 HQ 0,038 0,004 0,013 0,005 0,813 0,087 0,05 0,015 FPE 0,01 0,001 0,008 0,004 0,75 0,139 0,049 0,035 MDIC 0,18 0,015 0,08 0,011 0,758 0, obs AIC 0,001 0,001 0,001 0,001 0,801 0,111 0,047 0,037 SIC 0,001 0,001 0,001 0,001 0,78 0,13 0,05 0,041 HQ 0,001 0,001 0,001 0,001 0,901 0,078 0,015 0,00 FPE 0,001 0,001 0,001 0,001 0,801 0,111 0,047 0,037 MDIC 0,00 0,001 0,001 0,001 0,985 0, obs AIC 0,001 0,001 0,001 0,001 0,813 0,117 0,034 0,03 SIC 0,001 0,001 0,001 0,001 0,807 0,1 0,034 0,035 HQ 0,001 0,001 0,001 0,001 0,9 0,067 0,005 0,00 FPE 0,001 0,001 0,001 0,001 0,813 0,117 0,035 0,031 MDIC 0,001 0,001 0,001 0,001 0,991 0, The shading indicates best erformance Information Criterion. 18

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