Forecasting performance of seasonal cointegration models
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International Journal of Forecasting 18 (2002) 3144www.elsevier.com/ locate / ijforecastForecasting performance of seasonal cointegration models* Marten Lof , Johan LyhagenDepartment of Economic Statistics, Stockholm School of Economics, P.O Box 6501 SE-113 83 Stockholm, SwedenAbstractForecasts from two different seasonal cointegration specifications are compared in an empirical forecasting example and ina Monte Carlo study. The two seasonal cointegration specifications are the one proposed by Lee [Journal of Econometrics 54(1992) 1], with a parameter restriction included at the annual frequency, and the model proposed by Johansen andSchaumburg [Journal of Econometrics 88 (1998) 301], with a general specification for the complex root frequency,respectively. In the empirical forecasting example we also include a standard cointegration model based on first differencesand seasonal dummies and analyze the effects of restricting or not restricting seasonal dummies in the seasonal cointegrationmodels. While the Monte Carlo results favor the specification suggested by Johansen and Schaumburg, and definitely so iflarger sample sizes are considered, we do not find such clear cut evidence in the empirical example. 2002 InternationalInstitute of Forecasters. Published by Elsevier Science B.V.Keywords: Seasonal cointegration; Monte Carlo1. Introduction about short and long run dynamics in thesystem. These negative effects may be seriousIt is common in applied work to assume an when the model is used in forecasting.approximately constant seasonal pattern, mod- Similar to the approach suggested by Johan-elled by seasonal dummies. Recently the unit sen (1988) and Johansen (1995) for the zeroroot and cointegration analysis at the zero frequency, Lee (1992) presents a maximumfrequency have been extended to seasonal fre- likelihood estimator for seasonally cointegratingquencies. Nonzero frequency unit roots imply a relations. The seasonal error correction modelnonstationary seasonal pattern. Assuming a [SECM], which allows for stochastic seasonalitydeterministic seasonal pattern in a multivariate is, however, only partially correct when itsetting, when it is in fact stochastically evolving comes to testing for the cointegrating rank at theover time, could lead to inappropriate inference annual frequency. A certain parameter restric-tion has been used in the literature to overcomethis problem. If this restriction is relaxed one*Corresponding author. Tel.: 146-8-736-9221; fax:cannot apply estimation methods based on146-8-348-161.E-mail address: email@example.com (M. Lof). canonical correlations, as suggested by Lee.0169-2070/02/$ see front matter 2002 International Institute of Forecasters. Published by Elsevier Science B.V.PI I : S0169-2070( 01 )00105-432 M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44Johansen and Schaumburg (1998) argue that casting effects of first deleting then eitherthis is a peculiar restriction as it constrains all restricting or not restricting the seasonal inter-coefficients at the annual frequency. They pres- cepts as discussed above. Using three empiricalent another estimation procedure for the param- data sets and various forecasting periods theyeters, corresponding to the annual frequency and show that the suggested restricted seasonalintroduce a general asymptotic theory for the dummy approach yields better forecasts in mostseasonal cointegration model. cases.Kunst and Franses (1998) argue that deter- In the present study we show results from anministic seasonal dummy variables, which are empirical forecasting example, but we alsooften included unrestrictedly in the SECM, conduct a Monte Carlo study where seasonallyshould be confined to the seasonal cointegrating cointegrated data generating processesrelations instead. If cointegration at seasonal [DGPs], with different parameter constellations,frequencies exists, an inclusion of unrestricted are investigated. In the empirical forecastingseasonal intercepts then implies a growing part we use macroeconomic data sets for Au-amplitude in the seasonal cycles, which may not stria, Germany and the United Kingdom, whichbe a realistic assumption in most cases. are also used in Kunst and Franses (1998). InIn the present study, various seasonal coin- the Monte Carlo study we analyze systemstegration specifications will be contrasted to the including three variables, whereas six variablesmodel suggested by Johansen and Schaumburg are used in the empirical example. The forecast-(1998) [henceforth JS]. We have found three ing performance of the more general specifica-studies on forecasting and seasonal cointegra- tion for the annual frequency, recently proposedtion: Kunst (1993a); Reimers (1997) and Kunst by JS is evaluated. We compare that specifica-and Franses (1998). Kunst (1993a) contrasts the tion with the original version of the seasonalSECM, where an intercept is included and the cointegration model, proposed by Lee (1992).restriction on the annual frequency is imposed, In the empirical forecasting example we alsoto a vector error correction model [VECM] and compare these two seasonal cointegration spe-also a VAR model in first differences and cifications with a VECM in first differences.seasonal dummies. Two examples based on real The remainder of this paper is organized asdata and a Monte Carlo experiment show that follows. Section 2 presents specifications of thethe benefits from accounting for seasonal coin- SECM, while Section 3 describes the data andtegration are quite limited. Reimers (1997) uses the estimation results for the models to be usedthe conventional seasonal cointegration model in the empirical forecasting example. Section 4suggested by Lee, with another simplifying presents forecast performance when using therestriction on the annual frequency and consi- empirical data sets. In Section 5 the Monteders a simulated two variable seasonal coin- Carlo setup is discussed. A comparison oftegration model, with a fixed lag length and no model forecasts from the Monte Carlo study isseasonal intercepts. The SECM is compared to a carried out in Section 6. The final sectiontraditional VECM in first differences with contains conclusions.seasonal dummies. The main conclusion is thatmodels in first differences produce smallerforecast errors for short horizons, but when 2. Seasonal cointegrationlonger forecasting periods are considered theseasonal cointegration model appears preferable. Lee (1992) suggests a maximum likelihoodKunst and Franses (1998) investigate the fore- estimator for seasonal cointegration relations.M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44 33This procedure extends the maximum likelihood cycles. If this restriction [henceforth denoted asapproach, which is summarized in Johansen the annual restriction] is relaxed one cannot(1995). Assuming quarterly data and that D Y apply the estimation method that uses canonical4 tis stationary, a seasonal VECM of order p of the correlations. There is also some evidence in thefollowing form is considered: literature that the absence of non-synchronousseasonal cycles should have little effect on thep3test for cointegration at frequency p /2, see Lee9Dn Y 5Oa b Z 1OGD Y 1 FD 1 4 t i i i,t j 4 t2j t t,i51 j51 (1992).JS, who argue that the above mentioned(1)restriction is peculiar and not justified from awhere the D are deterministic components andt theoretical point of view, refine the theory forwhere is i.i.d. N (0,V ). The process D Yt n 4 t seasonal cointegration in the general case. Theyabove is said to be seasonally cointegrated if propose the following error correction model for9and only if at least one of the a b matrices fori i quarterly data:i 5 2,3 on the right hand side has reduced,2non-zero rank. The linear filters which in (1) 99 9D Y 5Oa b X 1 2(a b 1 a b )X4 t i i i,t R R I I R,tremove all unit roots except those on the zero, i51biannual and annual frequencies, respectively 9 91 2(a b 2 a b )XR I I R I,tare:p2 3 4Z 5 (L 1 L 1 L 1 L )Y , 1OGD Y 1 FD 1 , (2)1,t t j 4 t2j t tj512 3 4Z 5 (L 2 L 1 L 2 L )Y ,2,t twhere the processes X , . . . ,X are:2 4 1,t I,tZ 5 (L 2 L )Y .3,t t1]X 5 Z ,These filters are the vector equivalents of the 1,t 1,t4univariate transformations used in the so-called 1]X 5 2 Z ,2,t 2,tHEGY-test for seasonal unit roots, see Hy- 4lleberg, Engle, Granger and Yoo (1990) 1]X 5 Z ,R,t 3,t[HEGY]. Furthermore, the regressors Z are 4i,t1asymptotically uncorrelated:]X 5 2 Z ,I,t 4,t4T22 9T OZ Z fi 0,i j, respectively. It can be seen that the annuali,t j,t Pt51 9 9restriction would imply that a b 2 a b 5 0R I I Rimplying that the cointegration vectors and in (2), which is a strong restriction on theadjustment coefficients can be found by remov- coefficients at the complex root frequency.9ing the reduced rank restriction on the other The estimation of a b for i 5 1,2 usesi ifrequencies by concentrating out the associated canonical correlations in analogy to the Johan-regressors. sen procedure and hence does not require anyOne filter that appears in the auxiliary regres- detailed explanation here. However the estima-sion of the HEGY-test at the complex frequency tion of b and b is nonstandard.R I3but is not present in (1) is Z 5 (L 2 L )Y . By JS propose the following estimation strategy.4,t timposing the restriction that this filtered vari- The first step involves regressing D Y , X , X ,4 t 1t 2table has no influence on D Y one assumes the X and X on lagged values of D Y and D , if4 t Rt It 4 t tabsence of so-called non-synchronous seasonal these are present in the model, and defining the34 M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44 residuals as R , R , R , R and R , respective- Since U is a vector and hence vec(U ) 50t 1t 2t Rt It 0t 0tly. If no lags or deterministic variables are U , this yields:0tpresent, X 5 R for j 5 1,2,R,I. The restrictionjt jt 9vec(b )R9of reduced rank on a b for i 5 1,2 is removed i i U 5 U 1 U (6)F G0t 2t t9vec(b )Iby regressing R , R and R on R and R . It0t Rt It 1t 2tNis shown in JS that the resulting residuals, 9where U 5 (U ^ a ) 2f2t Rt RN N Ndefined as U ,U and U , satisfy the following0t Rt It 9 9 9(U ^ a ) (U ^ a ) 1 (U ^ a ) . ThegIt I It R Rt Iequation: parameters in b and b can now be found from:R I9 9 T 21 TU 5 2(a b 1 a b )U0t R R I I Rt 9vec(b )R 995 OU U OU U . (7)S DF G 2t 2t 2t 0t9 91 2(a b 2 a b )U 1 U 9vec(b )R I I R It t I t51 t51b 2 b UR I Rt For a given value of b, which is generated5 2 a 2 a 9 1 Us dS D S DR I tb b UI R It randomly in the first iteration, estimates of a 521 21 (S b(b9S b ) ) /2 and V 5 S 25 ab9U 1 U , (3) 01 11 001t t21 21S b(b9S b ) )b9S are computed. In a sec-01 11 10asymptotically. Defining the product momentsond step U is normalized and vectorized asT 0t9as S 5 (1 /T )S U U for i, j 5 0,1, we haveij t51 it jt described above, yielding a new estimate of bthat for fixed values of b the concentrated for which we can compute a new likelihoodlikelihood function with respect to a and hence function. This procedure is repeated until athe variancecovariance matrix V take thesuitable convergence criterion is satisfied.form, apart from a constant: Kunst and Franses (1998) argue that deter-212 ministic seasonal dummy variables, which areub9(S 2 S S S )b u] 11 10 00 012T ]]]]]]]L (b ) 5 uS u . (4)max 00 often included unrestrictedly in (1) to handle theub9S b u11 deterministic part of seasonality, should beThe minimization of (4) cannot be solved as an confined to the seasonal cointegrating relationseigenvalue problem as in the zero and biannual instead. This is because unrestricted seasonalfrequency cases since b itself has complex intercepts in the SECM may lead to divergingstructure while the product matrices S 2 seasonal trends, which is unlikely in most cases.1121S S S and S do not. JS use a switching We restrict the seasonal dummy variables in a10 00 01 11algorithm proposed by Boswijk (1995) where similar way in the JS specification which leadsthe maximum likelihood estimator of b is to the extended variables X 5 (X , cos p(t)),2,t 2,t calculated iteratively: Isolate b and b by using X 5 (X , cos p /2(t)) and X 5 (X , 2 sin p /R I R,t R,t I,t I,t21 / 2a normalized form (V U 5 U ) of U , 2(t)). This gives the following augmented b0t 0t 0tnamely: matrix in (3):21 / 2 9 9U 5 2V (a b 1 a b )U b 2 b0t R R I I Rt R Ir 2 r21 / 2 21 / 2 R I9 91 2V (a b 2 a b )U 1 V UR I I R I,t tb bI R1 2N N N9 9 95 a b U 2 a b U 1 a b U r rR R Rt I R It R I It I RN92 a b U 1 U , (5)I I Rt t where r and r are the parameters corre-R Isponding to cos p /2(t) and 2 sin p /2(t),and then vectorize (5) by using:respectively.vec(ABC) 5 (C9 ^ A)vec(B) As in Kunst and Franses (1998) we alsoM. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44 35include an unrestricted intercept in the model. theory suggests various long-run relationsHenceforth we will denote the Lee specification among them. One example is the difference ofwith unrestricted seasonal intercepts and the output and consumption, in logarithms, whichannual restriction imposed as SCM1, which should be stationary according to the theory.means seasonal cointegration model of type 1. Another example is the logarithmic differenceThe same model with restricted seasonal inter- of output and investments, see Kunst and Neus-cepts, as proposed by Kunst and Franses (1998), ser (1990). However, it is still an open questionwill be denoted SCM2. Finally, SCM3 denote how these series should be related at seasonalthe SECM proposed by JS with restricted frequencies, see Wells (1997) for a discussion ofseasonal intercepts. these issues. The data set for Austria covers thetime period 1964:1 to 1994:4, whereas the timeseries for Germany run from 1960:1 to 1988:1and for the UK from 1957:1 to 1994:1. Unit3. Tests for seasonal integration androots in the univariate time series are testedcointegrationusing the HEGY-test. The test procedure in-Gross domestic product (Y), private con- vestigates whether the seasonal difference (1 24sumption (C), gross fixed investment (I), goods B ), which assumes the presence of four unitexports (X) and real wages (W) are transformed roots, is the appropriate filter compared withinto natural logarithms. The real interest rates other nested filters.(R) are given in percentage points. The three Results of the tests for seasonal and non-data sets were previously used in Kunst and seasonal unit roots appear in Table 1. AllFranses (1998), and the motivation for using auxiliary regressions include an intercept (I),these series is that the neoclassical growth seasonal dummies (D) and a deterministic trendTable 1Seasonal integration testsVar Austria Germany UKt t F t t F t t F p p p p p p p p p1 2 3,4 1 2 3,4 1 2 3,4Y 21.78 23.47* 1.81 21.52 22.83* 3.29 23.21 21.38 4.43C 21.28 21.83 3.14 21.21 21.45 3.41 23.49* 21.85 3.73I 22.29 24.96* 9.79* 22.83 22.68 6.21 22.68 22.53 13.3*X 21.49 25.18* 48.6* 20.96 25.13* 39.7* 21.12 22.12 6.82*W 21.69 23.31* 2.58 21.18 21.62 2.18 22.16 23.05* 19.9*R 23.01* 22.95* 7.20* 22.77 22.03 4.73 21.94 25.38* 22.3*Lags of dependent variable and additional regressors (F ).Lags F Lags F Lags FY 14 I,D,T 1 I,D,T 15 I,D,TC 14 I,D,T 14 I,D,T 15 I,D,TI 12 I,D,T 13 I,D,T 1 I,D,TX I,D,T I,D,T 14 I,D,TW 16 I,D,T 14 I,D,T 1 I,D,TR 13 I,D 1 I,D 15 I,D*Significant at the 5% level. I, D and T denote an intercept, seasonal dummies and a trend, respectively.36 M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44(T), except those for the real interest rates presented above for Austria, Finland, Germanywhere an exclusion of the trend seems to be a and the UK, mention that evidence of stochasticmore appropriate specification. All variables seasonality is quite weak in the UK series, whileseem to contain unit roots at the zero frequency, the evidence is rather strong in the German dataexcept UK consumption and the real interest set. Finally, Kunst and Neusser (1990) findrate in Austria. The results are more mixed at similar unit root results as here, for Austria.9the seasonal frequencies. All roots at the bian- The ranks of the matrices a b (i 5 1,2) andi inual frequency, except for consumption, are ab9 are determined using the trace test, whererejected for Austria. On the other hand we find the null hypothesis is: at most r cointegratingevidence of four unit roots at this frequency for vectors against the alternative of full rank. Tableboth Germany and UK. Turning to the annual 2 summarizes the result using the three differentfrequency we find evidence of three, one and model specifications SCM1, SCM2 and SCM3.four stationary variables for Austria, Germany We use the same lag lengths as Kunst andand UK, respectively. Since unit roots are most Franses (1998) for SCM1 and SCM2, indicatedfrequent in the German data set it is sensible to by p, and we use the same lag lengths for SCM3conclude that cointegration at all frequencies is as well. A* indicates significance at the 5%most likely to be found there. level. Critical values for columns two, three andKunst (1993b), who analyze the variables five are based on our own calculations withTable 2Tests for seasonal cointegration, LR-test statisticsRank Frequency0 p p /2SCM1,2,3 SCM1 SCM2,3 SCM1 SCM2 SCM3Austria, p500 161.3* 181.0* 161.4* 200.1* 187.9* 284.7*1 88.1* 116.5* 113.1* 122.8* 111.1* 173.7*2 48.8* 77.2* 73.8* 77.8* 67.2* 106.0*3 22.5 46.1* 42.7* 40.1* 41.7 64.5*4 6.8 21.0* 20.9* 18.9 21.3 29.75 1.5 9.7* 9.7* 6.2 7.8 8.3Germany, p500 132.2* 140.8* 128.9* 174.5* 171.2* 287.3*1 68.8* 94.1* 82.7* 95.1* 93.6* 179.2*2 38.2 52.3* 40.8 54.5 56.8 107.6*3 17.2 24.3 16.8 29.8 32.0 56.94 8.4 7.9 8.6 15.2 17.4 27.55 1.9 3.2 4.1 4.0 5.9 10.6UK, p550 140.7* 104.0* 95.9 130.7* 130.2 218.7*1 73.6* 64.5 58.2 71.7 71.5 143.4*2 47.7* 42.0 39.4 40.6 40.4 92.03 25.7 25.3 22.7 22.4 22.1 56.34 8.7 10.5 11.3 7.7 7.6 27.85 0.8 2.8 3.2 1.7 1.6 6.5*, Significant at the 5% level. p denotes the lag length.M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44 37100 000 replicates and with T 5 400. Lee and ing to the Schwarz Criterion, in addition toSiklos (1995) present critical values for these equation by equation diagnostic tests. The em-cases, but they only consider smaller systems. pirical data set for Germany, which covers theCritical values for columns four and six are time period 1960:11988:1 is used. We imposefrom Table 1af in Franses and Kunst (1999). the ranks according to the results found inFor the last column we use the critical values Section 3 for the different models. All forecastfrom Table 2 in JS. Notice that the results are errors correspond to level variables. In the firstidentical for the three model specifications at step we save eight observations at the end to bethe zero frequency and for SCM2 and SCM3 at forecasted ex ante. In the next step, the estima-the biannual frequency. Three long-run cointeg- tion period is extended by one quarter and werating vectors are found at the zero frequency reestimate the parameters in the models. How-for Austria and the UK and two for Germany. ever, we do not change lag orders or the chosenOn frequency p we find no evidence of unit cointegration ranks. These extensions of theroots for Austria but identify two cointegrating sample are through 1987:4, where we generatevectors for Germany. For the UK we find no the last one-step ahead forecast error. Henceevidence of cointegrating vectors at frequency p there are eight one-step ahead forecast errors,using SCM2 or SCM3, but we identify one if seven two-step ahead forecast errors and so on.SCM1 is used. The results are more mixed at The eight-step ahead forecasts are then based onfrequency p /2. One observation is that SCM3 a single observation for each equation. In totalsuggests more cointegrating vectors. Based on we have 36 forecasts for each model and datathese results, where the biannual frequency for set. The results are summarized by the RMSEAustria has full rank and the rather weak statistic (root mean squared error) for 1, 2 andevidence of cointegration at the seasonal fre- 4-step ahead forecasts. In Table 3 below we alsoquencies in the UK data set, only the German consider the RMSE based on all the 36 forecastsdata set is used in Section 4. from each model, denoted All. Finally, weHaving chosen the rank for SCM3 at the present the average performance rank of thedifferent frequencies we test for further reduc- four model specifications at the different fore-tion of the model. A hypothesis that simplifies cast horizons considered, denoted AR.the cointegration analysis on frequency p /2 is Comparing the four cointegration specifica-the one implying real structure (H ), i.e. tions we see that the result is quite clear, whenREALb 5 0 in (2). However this hypothesis is strong- looking at the average rank of the models. TheIly rejected with LR-test statistics 43.5, 72.5 and version of the seasonal cointegration model with22.4 for Austria, Germany and the UK respec- the annual restriction and unrestricted seasonaltively. dummies included (SCM1) is better than theother three specifications if one step aheadforecasts are considered. However, for longerforecast horizons the two versions of the SECM4. An empirical forecasting examplewith restricted seasonal dummies seems to be aIn this section we investigate the forecasting better choice. This result is in line with thatperformance of the models presented in previ- found in Kunst and Franses (1998). Comparingous sections, namely SCM1, SCM2, SCM3. We SCM2 and SCM3 one can see that SCM2 isalso include a VECM in first differences, de- clearly better for forecast horizons shorter thannoted CM. For the seasonal cointegration four steps. The model in first differences (CM)models we use the same lag lengths as in generates better forecasts than the seasonalSection 3 and we choose p 5 3 for CM accord- cointegration models at all horizons for goods38 M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44Table 3aForecast performance. The ranks at different frequencies according to trace test result in Table 2 for the German data setHorizon: 1 2 4 All Horizon: 1 2 4 AllSCM1 SCM2Y 1.09 1.05 1.42 1.16 Y 1.33 0.97 1.29 1.24] ] ] ]C 0.84 0.99 0.90 0.84 C 0.91 0.80 0.84 0.77] ] ] ]I 3.94 2.87 3.62 3.57 I 4.07 3.03 3.64 3.73] ] ]X 2.56 3.17 4.76 4.04 X 2.65 3.02 4.61 3.97W 1.49 1.70 1.93 2.18 W 1.61 1.58 1.79 2.14] ]R 1.35 1.54 1.84 1.52 R 1.40 1.40 1.72 1.45] ] ]AR 1.33 2.17 3.00 2.17 AR 2.67 1.50 2.00 2.00] ] ] ]SCM3 CMY 1.30 1.20 1.39 1.27 Y 1.30 1.75 1.79 1.60C 1.23 1.12 0.86 0.98 C 1.19 1.47 1.29 1.27I 4.28 3.26 3.37 3.58 I 4.74 4.55 3.77 4.08]X 2.29 2.53 4.35 3.73 X 1.85 2.46 4.14 3.55] ] ] ]W 1.61 1.77 1.74 2.09 W 1.86 2.25 2.09 2.48] ]R 1.94 1.93 1.89 1.88 R 2.39 2.55 1.49 2.03]AR 2.83 2.83 2.00 2.33 AR 3.17 3.50 3.00 3.50]a Forecast errors evaluated with RMSE3100. Unrestricted constant and seasonal intercepts included. All models includestwo vectors at the zero frequency. SCM2 and SCM3 includes two cointegration vectors at the biannual frequency, whileSCM1 include three. Models SCM1 and SCM2 includes two vectors at the annual frequency, while SCM3 include three. ARdenote the average rank of the models, at different forecast horizons, and the smallest is underlined.exports (X). Note that the result of the HEGY- i.e. no deterministic variables are included. Thetests indicate no seasonal unit roots for goods systems include three variables in each case.exports in this data set, which may explain why Four different DGPs are investigated, see Tablethe model in first differences generates the best 4, and they are denoted DGP1, DGP2, DGP3forecasts for this time series. However, CM and DGP4. The matrices a and b concern the1 1generates the poorest forecasts on average. If zero frequency and include the adjustmentthe forecast accuracies are evaluated using the parameters and the cointegrating vectors, re-mean absolute error (MAE) instead, we come to spectively. The matrices a and b concerns the2 2the same conclusions (not shown here). biannual frequency, whereas a , b and a , bR R I Iconcerns the annual frequency. The cointegrat-ing vectors and adjustment parameters are dif-ferent across frequencies for all DGPs and the5. The Monte Carlo setupadjustment parameters are always different ac-The seasonal cointegration specifications we ross DGPs for the annual frequency. We let thecompare are those of Lee (SCM1) and JS columns in all the adjustment parameter ma-(SCM3), respectively and we set p 5 0 for both trices for a certain DGP sum to the samespecifications. We compare the forecast accura- absolute value, just to have a similar parametercy for the seasonal cointegration models when structure across frequencies. For example theseasonality is viewed as being purely stochastic, first column sum is 0.9 in absolute value acrossM. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44 39Table 4Cointegrating vectors (b ) and adjustment parameters (a) used in the Monte Carlo studyDGP1 DGP2 DGP3 DGP41.00 0.00 21.00 0.00 1.00 0.00 1.00 0.00b 21.00 1.00 21.00 1.00 20.80 1.00 20.80 1.0010.00 0.50 0.00 0.50 0.00 0.30 0.00 2.301.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00b 20.90 1.00 20.90 1.00 20.70 1.00 20.70 1.0020.00 0.30 0.00 0.30 0.00 0.60 0.00 0.601.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00b 20.70 1.00 20.70 1.00 20.90 1.00 20.90 1.00R0.00 0.60 0.00 0.60 0.00 0.30 0.00 0.301.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00b 20.80 1.00 20.80 1.00 21.00 1.00 21.00 1.00I0.00 0.30 0.00 0.30 0.00 0.50 0.00 0.5020.80 0.00 20.40 0.00 20.40 0.00 20.80 0.00a 20.10 20.30 20.05 20.15 20.20 20.20 20.10 20.3010.00 20.20 0.00 20.10 0.00 20.10 0.00 20.2020.60 0.00 20.30 0.00 20.35 0.00 20.60 0.00a 20.30 20.40 20.15 20.20 20.25 20.15 20.30 20.4020.00 20.10 0.00 20.05 0.00 20.15 0.00 20.100.10 0.00 0.05 0.00 0.50 0.00 0.80 0.00a 0.80 0.10 0.40 0.05 0.10 0.10 0.10 0.40R0.00 0.40 0.00 0.20 0.00 0.20 0.00 0.100.80 0.00 0.40 0.00 0.40 0.00 0.85 0.00a 0.10 0.40 0.05 0.20 0.20 0.25 0.05 0.42I0.00 0.10 0.00 0.05 0.00 0.05 0.00 0.08Table 5frequencies in DGP1, whereas the second col-Diagonal and non-diagonal variancecovariance matricesumn sum is 0.5. The adjustment parameters areused in the Monte Carlo study An identity matrix is alsolower in DGP2 than in DGP1. In DGP3 the included i the analysis, denoted S3values in a and a lead to lower parameterR I9 9values in the P 5 (a b 2 a b ) matrix, so 0.7 0.0 0.0 0.7 0.3 20.2I R I I RS 5 0.0 0.8 0.0 S 5 0.3 0.8 20.1we may expect a lower impact of non-syn- 1 2F G F G0.0 0.0 0.5 20.2 20.1 0.5chronous cointegration. Finally, in DGP4 thefilter X should play an even smaller role, whenItforecasting.We investigate the following true rank cases: are used in each case (see Table 5) and fourdifferent simple sizes, namely T 5 40,80,1201: r 5 1, r 5 1, r 5 1,0 p p / 2 and 200. Furthermore, 100 presample observa-2: r 5 2, r 5 2, r 5 1,0 p p / 2 tions and 12 postsample observations are gener-3: r 5 1, r 5 1, r 5 2,0 p p / 2 ated. These last 12 observations are saved and4: r 5 2, r 5 2, r 5 2.0 p p / 2 used for ex-ante forecasting. In each case 5000Three different variancecovariance matrices replications (N) are generated.40 M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44The data is generated in the following way: graphically when DGP1 is used, because SCM3gives the best forecasts in this case, regardlessX 5 X 1 P X 1 P Xt t24 1 1,t 2 2,t of which sample size or true rank is used.Moreover, the true rank cases 2 and 3 produce1 P X 1 P X 1 ,R R,t I I,t tsimilar results as compared to cases 1 and 4,where | N (0,S ) and where the different respectively, so we restrict the presentation tot 3 1,2,3P matrices are changed according to which the latter two cases (rank51 and 2 at allrank is chosen at a particular frequency. For frequencies). Furthermore, if the sample size isexample P equals the first r columns in a larger than or equal to 80 SCM3 always gener-1 0 19times the first r rows in b . The matrices at the ates lower forecast errors, at all horizons, than0 1biannual and the annual frequency are con- SCM1, if DGP2 is used. As the result for DGP1structed in the same manner, but the matrices the difference between the two specifications isP and P at the annual frequency equal sometimes substantial, when using DGP2.R I9 9 9 9(a b 1 a b ) and (a b 2 a b ), respective- Note that there seems to be a seasonal patternR R I I R I I Rly. in the forecast results, see Figs. 1 and 2. This isFor both models we calculate squared fore- due to the estimated variances used in (8) but2cast errors (e ) for each horizon, k 5 1, . . . ,12, also because the results are for data in levels,jkand for each equation, j 5 1, 2 and 3. Then we hence transformed from fourth differences.take the equation by equation average of these SCM3 produces poorer forecasts for smallerover the number of replications (R). We then sample sizes if DGP3 and DGP4 are considered,weigh the resulting mean squared forecast errors see for example (c), (d), (e) and (f) in Fig. 1.with a variance estimate of the 12 first observa- This is because the values in a and a result inR Itions generated from each equation, denoted V . a lower impact of so-called non-synchronousjkThe variances are calculated after 5000 replica- seasonal cycles. Some results indicate thattions and they are useful when we, in the last SCM3 may yield worse forecasts if more coin-step, average the mean squared forecast errors tegrating relations are included in the model.over the three equations. With this relative One example is (a) and (b) in Fig. 1, for DGP2measure we avoid to give high weight to and T 5 40. Looking at the rest of the figuresvariables with large variances. To summarize, increasing sample size works in favor of SCM3,for each model and horizon we have the follow- so that for T 5 200 SCM3 dominate SCM1. Weing mean squared error [MSE] measure: also find in a smaller complimentary study (notall 192 combinations considered) that these3 R1 1 2 results hold when various lag lengths are select-] ]MSE 5 O Oe YV . (8)FS D Gjk jk3 Rj51 r51 ed, using the Akaike Information Criterion.6. Monte Carlo results7. ConclusionsIn total we have 192 combinations to consi-The forecasting performance of the seasonal-der if we look at all sample sizes, true rankly cointegrated model of Johansen and Schaum-cases, DGPs and variancecovariance matrices.burg (1998) is compared to a related specifica-However, the results are not different dependingtion, with a restriction at the annual frequency.on which variancecovariance matrix is used.In the empirical forecasting part, we also in-Because of that only results based on S are1clude a model in first differences with co-considered. We choose not to report any resultsM. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44 41Fig. 1. Mean squared errors weighted with variances. Dashed line: SCM1, solid line: SCM3. Rank concerns all frequencies,T540 and 80.42 M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44Fig. 2. Mean squared errors weighted with variances. Dashed line: SCM1, solid line: SCM3. Rank concerns all frequencies,T5120 and 200.M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44 43integration restrictions. We examine data sets This sample size effect may explain the result inthe empirical example which is based on fewerfrom Austria, Germany and the UK, eachobservations.containing six variables: gross domestic prod-uct, private consumption, gross fixed invest-ment, goods exports, real wages and the realAcknowledgementsinterest rate. We examine the integration andcointegration properties for the three data setsWe thank Lars-Erik Oller, the Editor andand consider the effect of including restricted orthree anonymous referees for helpful comments.unrestricted seasonal dummies in the seasonal We also thank Robert Kunst who generouslycointegration models. The biannual frequencyallowed us to use the data sets. Financialfor Austria seems to have full rank and we findsupport for Lyhagen from Tore Browaldhsrather weak evidence of cointegration at the Foundation for Scientific Research and Higherseasonal frequencies in the UK data set, so that Education and The Swedish Foundation foronly the German data set is used in the empiri- International Cooperation in Research andcal forecasting example. Higher Education (STINT) is gratefully ack-The seasonal cointegration model with the nowledged. Lyhagen is also grateful to Prof. M.annual restriction and unrestricted seasonal H. Pesaran who acted as his sponsor at a Post-dummies included is better than the other three Doc stay with the Faculty of Economics andspecifications if one step ahead forecasts are Politics, University of Cambridge.considered. However, for longer forecasthorizons the two versions of the seasonal coin-tegration model with restricted seasonal dum- Referencesmies seems to be a better choice. When compar-Boswijk, H. P. (1995). Identifiability of cointegrateding the two restricted seasonal dummy spe-systems. Discussion paper, Tinbergen Institute, TI 95,cifications we find that the version with the78. (http: / /www.fee.uva.nl /ke /boswijk /main.htm)annual restriction is better for forecast horizons Franses, P. H., & Kunst, R. M. (1999). On the role ofshorter than four steps. The cointegration model seasonal intercepts in seasonal cointegration. Oxfordin first differences generates the poorest fore- Bulletin of Economics and Statistics 61, 409434.Hylleberg, S., Engle, R. F., Granger, C. W. J., & Yoo, B. S.casts on average.(1990). Seasonal integration and cointegration. JournalIn the Monte Carlo study we analyze fourof Econometrics 44, 215238.different DGPs and four different sample sizes, Johansen, S. (1988). Statistical analysis of cointegrationnamely T 5 40, 80, 120 and 200 when using a vectors. Journal of Economic Dynamics and Controlthree variate system. We do not include any 12, 231254.Johansen, S. (1995). Likelihood-based inference in co-deterministic variables and seasonality is thusintegrated vector autoregressive models, Oxford Uni-generated as being purely stochastic and we doversity Press, Oxford.not consider any model in first differences. Johansen, S., & Schaumburg, E. (1998). Likelihood analy-When the smaller sample sizes are considered, sis of seasonal cointegration. Journal of Econometricswe find some evidence that the specification 88, 301339.Kunst, R. M. (1993a). Seasonal cointegration, commonproposed by Johansen and Schaumburg (1998)seasonals, and forecasting seasonal series. Empiricalmay yield worse forecasts if more cointegratingEconomics 18, 761776.relations are included in the model. IncreasingKunst, R. M. (1993b). Seasonal cointegration in macro-sample size works in favor of this model, so that economic systems: case studies for small and largefor T 5 200 it dominate the seasonal cointegra- European countries. Review of Economics and Statisticstion model with the annual restriction included. LXXV, 325330.44 M. Lof, J. Lyhagen / International Journal of Forecasting 18 (2002) 31 44Kunst, R. M., & Franses, P. H. (1998). The impact of Wells, J. M. (1997). Business cycles, seasonal cycles andseasonal constants oncasting seasonally cointegrated common trends. Journal of Macroeconomics 19, 443time series. Journal of Forecasting 17, 109124. 469.Kunst, R. M., & Neusser, K. (1990). Cointegration in amacroeconomic system. Journal of Applied Economet- Biographies: Marten LOF is a Ph.D. student at therics 5, 351365. Department of Economic Statistics, Stockholm School ofLee, H. S. (1992). Maximum likelihood inference on Economics. His research focuses on seasonality, cointegra-cointegration and seasonal cointegration. Journal of tion and forecasting.Econometrics 54, 147.Lee, H. S., & Siklos, P. (1995). A note on the critical Dr. Johan LYHAGEN obtained his Ph.D. in Statistics fromvalues for the maximum likelihood (seasonal) cointegra- Uppsala University in 1997. In his thesis he examinedtion tests. Economic Letters 49, 137145. different aspects of fractional ARMA models. His currentReimers, H. -E. (1997). Forecasting of seasonal cointe- research interests are time series analysis, cointegrationgrated processes. International Journal of Forecasting and the analysis of financial data.13, 369380.
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