Pertanika J. Soc. Sci. & Hum. 10(2):131-141 (2002) ISSN: 0128-7702© Universiti Putra Malaysia Press

Performances of Non-linear Smooth Transition Autoregressive and LinearAutoregressive Models in Forecasting the Ringgit-Yen Rate

LIEW KHIM SEN & AHMAD ZUBAIDI BAHARUMSHAHDepartment of Economics,

Faculty of Economics and Management,Universiti Putra Malaysia,

Serdang, Selangar, Malaysia

Keywords: Autoregressive, smooth transition autoregressive, non-linear time series, forecastingaccuracy

ABSTRAK

Kajian ini membandingkan prestasi peramalan di antara model ketaklinearan AutoregresiBerperalihan Licin (Smooth Transition Autoregressive atau STAR) dengan model linear siri masaiaitu model Autoregresi (Autoregressive atau AR) sebagai peramal kadar pertukaran Ringgit-Yen.Berdasarkan kepada prosedur penentuan kelinearan piawai, kami memperoleh bukti empirikbahawa penyelarasan Ringgit-Yen ke arah keseimbangan pariti kuasa beli (Purchasing PowerParity) adalah secara tidak linear. Dari segi prestasi peramalan, keputusan empirik menunjukkanbahawa model STAR dan AR mempunyai purata mutlak ralat ramalan (mean absolute forecasterror atau MAFE), purata mutlak perarus ralat ramalan (mean absolute percentage forecast erroratau MAPFE) dan purata punca kuasa dua ralat ramalan (mean square forecast error atauRMSFE) yang lebih kecil jika dibanding dengan model SRW. Keputusan yang diperoleh jugamenunjukkan ramalan model STAR lebih baikjika dibandingkan dengan pesaing linearnya, iaitumodel AR Hasil kajian ini adalah konsisten dengan penyelidikan yang memberi penekanankepentingan membenarkan pelarasan yang tidak linear bagi kadar pertukaran asing ke arahkeseimbangan jangka panjang.

ABSTRACT

This study compares the performance of Smooth Transition Autoregressive (STAR) non-linearmodel and the conventional linear Autoregressive (AR) time series model in forecasting theRinggit-Yen rate. Based on standard linearity test procedure, we find empirical evidence that theadjustment of the Ringgit-Yen rate towards its long-run Purchasing Power Parity equilibriumfollows a non-linearity path. In terms of forecasting ability, results of this study suggest that boththe STAR and AR models exceed or match the performance of SRW model based mean absoluteforecast error (MAFE) mean absolute percentage forecast error (MAPFE) and mean squareforecast error (RMSFE). The results also show that the STAR model outperforms the AR model,its linear competitor. Our finding is consistent with the emerging line of research that emphasisedthe importance of allowing non-linearity in the adjustment of exchange rate toward its long runequilibrium.

INTRODUCTION

In 1926, Yule first formally introduced the timeseries model in the form of autoregressive (AR)model, which assumes that the future values ofa variable depend solely on its historical values.Since then, time series analysis has been viewedas a powerful forecasting tool. In the past twodecades or so the theory of time series

econometrics is progressing rapidly(Montogomery et at. 1990). As the methodologyprogresses, the issue of non-linearity wasincorporated into the analysis of time series.Smooth Transition Autoregressive (STAR) modelis one of the most recent models developedunder this concept. The STAR model is a non­linear time series model that allows the variable

Liew Khim Sen & Ahmad Zubaidi Baharumshah

under investigation to move within two differentstate spaces with a smooth transition process.STAR offers an alternative to the modelling oftime series variables that exhibit non-linearities.

In our review of the literature, we foundthat the application of the STAR model inempirical works is still very limited, and itsforecasting performance particularly withreference to exchange rates has yet to bedetermined. Taylor and Peel (2000), Sarno(2000a) and Baum et at. (2001) are among thefirst to demonstrate the usefulness of the STARin modelling exchange rate dynamics. However,all these authors did not evaluate the forecastingperformance of the model. Hence, the mainobjective of this study is to examine theapplicability of the STAR model to the MalaysianRinggit against the Japanese Yen (RM/YEN). Inaddition, the present article compares theforecasting performance of the STAR and ARmodels by using the simple random walk (SRW)model as the yardstick of comparison. This paperextends previous studies by evaluating theforecasting performance of the studied modelsusing mean absolute forecast error (MAFE),mean absolute percentage forecast error(MAPFE) and root mean square forecast error(RMSFE).

We chose to apply the model to theexchange rate for the following reasons: First,the bulk of the literature shows that structuralmodels failed to outperform a simple randomwalk (SRW) model, and attempts by analystsusing more elaborate models have also failed toimprove the forecast performances at short orlong run horizons significantly. Second, severalauthors have argued that the failure of existingexchange rate models to yield superior forecastis because these models ignore the non-linearityadjustment of exchange rates towards itsequilibrium value (Micheal et at. 1997; Taylorand Peel 1997; Sarno 2000a,b; Coakley andFuertes 2001). These studies also argued thatthe classical unit root tests may not be able todetect mean reverting behaviour of exchangerate if the variable is a stationary non-linearprocess. Finally, in applying this model to theRinggit, we intend to broaden our understandingon the appropriateness of the STAR model as aforecasting tool in the currency markets.

This study models the adjustment process ofthe deviations of Yen-based Ringgit movementfrom its fundamental equilibrium as determinedby the Purchasing Power Parity (PPP) hypothesis.Simply, the PPP hypothesis postulates that thenominal exchange rate is given by the ratio ofthe domestic and foreign price levels. It statesthat exchange rates should tend to equalizeprices for identical goods in different countries.Recent studies based on careful application oftime series econometrics methods, are moresupportive of the mean reverting behaviour ofexchange rates towards the long-run PPPequilibrium value (M. Azali et at. 2001; Baum etat. 2001) I.

To anticipate our results, we find that RM/YEN rate adjusts in a non-linear fashion towardsits long-run equilibrium path using the LagrangeMultilpier (LM)-type test developed by Teriisvirta(1994). Unlike most of the earlier studies, wefind that both the STAR and AR modelsoutperform a random walk forecast for nominalRM/YEN rate. In addition, the empirical resultssuggest that the non-linear STAR model performsbetter than the linear AR model in the out­sample forecasts. The set out of the paper isorganised as follows. The first two sections offera brief review on the development of STARmodels and a discussion on the data used in theanalysis. The section that follows immediatelydescribes the linearity test and the test results.We report and interpret the results of forecastaccuracy comparison just before offering someconcluding remarks in the last section.

THE STAR MODELS

The origin of the non-linear Smooth TransitionThreshold Autoregressive or just SmoothTransition Autoregressive (STAR) model couldbe traced back to the Threshold Autoregressive(TAR) model first proposed by Tong in 1977(see Tong and Lim 1980). The TAR modelassumes that a variable has different behaviourwithin different regimes. The basic ideaunderlying the TAR model is piecewiselinearisation of non-linear models over the statespace by the introduction of the thresholds. Anexample of the TAR model is the Self-excitedTAR (or SETAR) model, which assumes that a

The stylized fact that emerged from this literature is that exchange rate adjusts non-linearly towards its long-run PPPequilibrium (Mah,yan and Wagner 1999; Sarno 2000a; Baum et at. 2001 and Coakley and Fuertes 2001).

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variable (say, exchange rate), y, is a linearautoregression within regime, but may movebetween regimes depending on the value takenby a lag of Y" say YHt where d is known as delayparameter. For two-regime case (q = 2) where Y,follows an AR (PI) process in the one regimeand AR (P2) process in the other, the SETAR (2;PI' P2

) representation of can be written compactlyas:

Y, = fJo + ~ fJiY'-i + elt + I,_Ar};..]

(1)

where ~(r) = 1 if Y, > rand 0 otherwise is thethreshold. For h = 1, 2, e

h, - G (0, di, ) where G

(-) may be a Gaussian distribution but this is notnecessarily the case. fJ i for i = 0, .. , PI and rr.. fori = 0, .. , P

2are parameters to be estimated.

The introduction of non-linear time seriesmodel such as SETAR model is motivated by thefact that linear time series model should giveplace to a much wider class of models if we wereto gain more understanding into the morecomplicated phenomena such as limit cycles,time irreversibility, amplitude-frequencydependency and jump resonance (Tong andLim 1980). Since its introduction, few attemptshave been made in applying and validating theSETAR mode, and hence the usefulness of themodel in empirical work is yet to be determined.For instance, Diebold and ason (1990) pointout that there is no guarantee that SETAR modelwill perform better than linear AR model. Asimilar view is expressed in Clements and Smith(1997), where they note that neither in-sample,nor the rejection of null of linearity in a formaltest in favour of non-linearity guarantees thatSETAR predicts more accurately than AR models.

The deficiency in SETAR is deemed due tothe unrealistic fixed threshold in the model.The fixed threshold of SETAR model is laterreplaced with a smooth function and thus leadsto the formation of STAR model in the early1990s. STAR model allows the variable understudy to alternate between two different regimeswith a smooth transition function between theseregimes, so that there can be a continuum ofstates between extreme regimes. STAR

representation is given by (Terasvirta andAnderson 1993):

p

Y, = fJo + 2: fJiY'-i +i-]

(2)

where Y, is mean-corrected, fJo' and f3'o areconstants, fJ i and fJ:, i = 1, ... , P are autoregressiveparameters, FO is the transition functiondepending on the lagged level, YHt ' where d isknown as the delay length or delay parameter,and e, is a white noise with zero mean andconstant variance cr.

E

For the application of STAR model, Grangerand Teriisvirta (1993) have proposed exponentialfunction as one of the plausible transitionfunctions, thus resulting in the exponential STARor ESTAR model. The exponential function isdefined as:

where f is the unstandardized transitionparameter, u is the equilibrium or thresholdvalue of the mean corrected Y, series and hence

E( u) = 0, and a~, is the estimated variance of Y,.

Note that f I a~, is the standardized

transition parameter and the speed of transitionbetween the two regimes is positively related tothe value of the transition parameter f. In otherwords, higher values of f imply much fasterspeed of transition. Taylor and Peel (2000) use

a version of transition function, F(-) with a~, =

1. Nevertheless, Granger and Tecisvirta (1993,p. 124) argue that scaling the exponential termby the sample variance speeds the convergenceand improves the stability of the non-linear leastsquares estimation algorithm. It also makes itpossible to compare estimates of transitionparameter across equations.

The exponential transition function isbounded between zero and one. Judging fromEquation 3, when Y

Htequals its equilibrium value

'2u or when il G y, goes to zero, F(-) = 0 and

Equation 2 reverts to a standard linear AR(P)representation:

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Liew Khim Sen & Ahmad Zubaidi Baharumshah

(4)exception is Teriisvirta and Anderson (1993),which evaluates the forecast performance ofESTAR model, in the context of business cycles.

In such case, the conventional restriction of

pL 13; < 1 applies so that Y. is mean-reverting.;=1

For extreme deviations from the fundamental'2 hequilibrium, FO = 1 (when '1/0y. approac es

infinity), and Equation 2 becomes a non-linearAR (p) model:

Yt = (130 + 130) + f (~ + 13;)YI-i H t (5)i=1

If the non-linear model as in Equation 5 is the

correct specification, it is expected that I~ fJ, s 11

such that Y may exhibit unit root behaviour butthe requi:ement for global stability is that

I f (~ + 13;)1 < I must be met.i-I

The exponential FO allows a smoothtransition between two regimes symmetryadjustment for deviations above and bel~w th.efundamental equilibrium. This function IS

considered suitable for the non-linear modellingof exchange rate as it has a number of attractiveproperties. For instance, it.:an capture ~e

symmetrical response to pOSltlVe and ~~g~tlve

deviations from its fundamental eqUihbnum(Baum et at. 2001) by its inverted bell-shap~d

distribution around zero. Despite the potentialusefulness of the ESTAR model in modellingthe exchange rate specifically and other non­linearities in general, much more empirical workhas yet to be done to fill up the related literature.To this end, there are only few published articleson the STAR model. Moreover, most of themare theoretical in nature and the application isonly for illustration purposes. Earlier works, forinstance by Chan and Tong (1986), Luukkonnenet at. (1988), Saikkonen and Luukkonen (1988),Luukonnen (1990), Terasvirta (1994) andEirtheim and Terasvirta (1996) discuss thetheoretical issues on the linearity tests and modelspecification of the STAR models. One notable

PREUMINARY DATA ANALYSIS

The data used in this paper are end-of-periodnominal bilateral exchange rate for the MalaysianRinggit vis-a.-vis the Japanese Yen (RM/YEN)and relative price (P.), which is constructed asthe ratio Consumer Price Index (CPI) of Malaysiato CPI of Japan. The data are mainly fromInternational Monetary Fund's InternationalFinancial Statistics (IMF/IFS), comprising ofseasonally unadjusted observations. Our sampleperiod ranges from 1980:1 to 2000:2. The fullsample period is divided into two portions. Thefirst sub-period, which starts from 1980:1 andends in 1997:2 is used for the model estimationpurpose while the remaining observations arekept for assessing the out-sample forecastperformance of the studied mode1s2•

To test whether RM/YEN rate exhibits meanreverting behaviour to its long-run PPPequilibrium, we check for the cointegratingrelationship between the two price series.However, prior to any cointegration test, theseries involved should be tested for stationarityand order of integration beforehand. This isimportant as only variables of the same order ofintegration may provide a meaningfulrelationship. The commonly used AugmentedDickey-Fuller (ADF) and non-parametric Philips­Perron (PP) unit root tests are employed for thispurpose. The results of the unit root tests assummarised in Table 1 strongly suggest that thevariables are first difference stationary, whichimplies they are all integrated of the same order,that is, 1(1). These results hold whether trend orwithout trend.

Next, we proceed to investigate whether ornot the long-run PPP conditions hold using theJohansen and Juselius (1990) multivariatecointegration test. Results of the trace test aredepicted in Table 2. The test result providesstrong evidence that RM/YEN rate and relativeprice are cointegrated at standard significancelevels, thereby verifying that RM/YEN rateexhibits mean reverting behaviour to its long­run PPP equilibrium. The results so far obtained

The estimation period ends in 1997: 2 and the forecasting horizon includes the 1997/98 Asian financial crises. Thevolatility of the exchange rates during the currency turmoil allows us to evaluate the robustness of OUf forecastsduring the crisis and post-erisis periods.

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TABLE 1Unit root tests results

Tests Intercept Without Trend Intercept with Trend

X ~X P ~P X ~ P ~P

ADF -0.362 -4.958* 1.989 -5.537* -2.696 -4.953* -0.040 -6.260*PP -0.528 -8.653* 3.726 -11.67* -2.987 -8.653* 0.233 -12.86*

Notes. ADF and PP refer to Augmented Dickey-Fuller test and Phillip-Perrontest respectively. X and P denote exchange rate and relative price respectively.Variable with ~ in front means its first difference. Test-statistics with asterisk (*)imply reject null hypothesis of unit-root at 1% significance level.

TABLE 2Johansen and Juselius cointegration test result

Likelihood Ratio of Eigen Value

Optimal Lag r = 0 r s 1

10 24.369* 5.061

Critical Values5% 19.90 9.241% 24.60 12.97

Notes. r denotes the hypothesized number of cointegrating equation. Optimumlag-length is determined by the Akaike Information Criterion (AlC). The singleasterisk (*) denotes rejection of hypothesis at 5% significance level.

are consistent with those reported inBaharumshah and Ariff (1997) and M. Azali etat. (2001).

This fmding enables us to estimate theequilibrium values of RM/YEN rate based onthe PPP hypothesis. Deviations of RM/YEN ratefrom its equilibrium (Y,)can then be deduced bysubtracting its observed values from the estimatedequilibrium values. The nature of adjustmentprocess of these deviations towards theequilibrium position is not known yet. Todetermine the linearity (or non-linearity) of thisadjustment process, we employ the linearity testsagainst the STAR models as described below.

LINEARfIY TESTS

The minimum requirement for the estimationof STAR models is to reject the linearity of thevariable under study (Tong and Lim 1980).Various linearity tests have been developed basedon the idea of testing the null hypothesis that allf3* s in Equation 2 are simultaneously zero,against the alternative hypothesis that at leastone f3* is not zero. ote that if the null hypothesiscannot be rejected, Equation 2 would simply be

reduced to the linear AR (p) model. By thesame token, rejection of null hypothesis impliesthe presence of non-linearity in favour of STAR(p) model. As the properties of the transitionparameter (i), the coefficients of non-linearterms (f3* s) and the mean value (IJ) of thevariable under estimation are not identifiedunder the null hypothesis, linearity is tested inthe context of auxiliary model instead of theoriginal STAR specification as in Equation 2.Theoretical issues on linearity tests against STARmodels are found in Luukkonen et at. (1988),Saikkonen and Luukkonen (1998), Terasvirtaand Anderson (1993), Terasvirta (1994) andEirtheim and Terasvirta (1996). Interestedreaders may refer to these articles for moredetailed discussion on the tests.

This study only highlights a specification oflinearity test with alternative hypothesis in favourof the ESTAR model, a variant of STAR modelrelevant to this study. This specification asproposed by Terasvirta (1994), is based on thefollowing auxiliary regression:

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Liew Khim Sen & Ahmad Zubaidi Baharumshah

estimated variance of it (8)

(SSRo - SSR)LM = '2 where a: is the

at

In practice, Terasvirta's Lagrange Multiplier(LM) linearity tests can be performed byfollowing these steps:(1) Regress Y, on II, Y

0; j = 1, ... , pl. Obtain the

estimated residuals i, and compute the residual

is usually unknown even if the true model islinear, and it has to be determined from thedata. Model selection criteria such as FinalPrediction Error (FPE), Schwarz InformationCriterion (SIC) and Akaike's InformationCriterion (AlC) are normally used for thispurpose. However, these criteria are of coursenot without any shortcomings (see for instance,Liew and Shitan (2002) for a brief review of theproperties of these selection criteria). In general,these information criteria tend to penalise high­order lags.

On the other hand, if the selected p is toolow, the estimated AR (P) model may sufferfrom autocorrelated residuals. Terasvirta andAnderson (1993) pointed out that neglectingthe autocorrelation structure of the residualsmay lead to false rejection of the linearityhypothesis in favour of the non-linearitiesalternative, because often the test also has lowpower against serially correlated errors. As such,one may think that over-parameterisation of thelinear AR (P) is preferable to under­parameterisation. However, selecting a maximumlag-length greater than the true order pmay alsoweaken the power of the test compared to thecase where the maximum lag is known (Terasvirtaand Anderson 1993). Thus, it is important toselect order p sufficient enough to eliminateautocorrelation.

In this study, the optimal lag length p oflinear AR (P) model is selected based on theAkaike's biased Corrected Information Criterion(AlCC). This criterion selects the minimum AlCCmodel, among a class of models with no serialcorrelation. Liew and Shitan (2002) examinethe behaviour of AlCC through a simulationstudy and find that it has little tendency tounderestimate the true order p. Thus the use ofAlCC avoids the problem of too parsimoniousmodel being selected. Based on AleC, theoptimal p is determined as 2 in the present case.

Having selected p, d needs to be determined.In order to specify d, linearity test is carried outfor the range of values considered appropriate,in this case, Isds5. If the linearity is rejected formore than one value of d, then d is determinedsuch that Z(d)= sup Z(d) for Isds5 where Z isthe selected test LM or LST. The argumentbehind this rule of maximising the test statisticis that the test has maximum power if d ischosen correctly, whereas an incorrect choice ofd weakens the power of the test. Ljung-Box

(9)

(6)

(7)

r(SSRo - SSR)LST =

SSRo

Ho : a~ = T~ = 0; i, j = 1, ... , P

Under the null hypothesis the LM statistic isasymptotically distributed as a chi-squared (X2

)

with 2p degrees of freedom, given that the delayparameter d is known. For unknown d, thedegrees of freedom would be as large as0.5P(ptI)+2p. This LM linearity test is actuallysimilar to that of Luukkonen, Saikkonen andTerasvirta (LST) (1988), which is given as:

LST is also asymptotically distributed as a X2

with O.5P(pt1)+p degrees of freedom. Luukkonenet al. (1988) point out that if the delay parameterd in Equation 2 is assumed known, the numberof degrees of freedom for LST statistic wouldreduce largely to pt1 only. This shows that priorknowledge about d is thus very useful in testinglinearity against ESTAR model.

Briefly, the optimum lag length p in the firststep of the above auxiliary regression procedure

T ,

sum of squares, SS~ = L £, , where r is the,-1sample size;

(2) Regress i, on II, Y,-i' Y,_)~j' Y~fJ- Obtain theestimated residuals W, and compute the residual

T~ '2sum of squares SSR = L. W, ;,-)

(3) Compute the test statistic:

The null hypothesis to be tested is that:

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Performances of Non-linear Smooth Transition Autoregressive and Linear Autoregressive Models

portmanteau Q test is also employed to confIrmthe absence of serial correlation up to 20 lags.

Results of the linearity tests are summarisedin Table 3. It is clear from the table that linearityis rejected at 1% signifIcance level for thedeviations of RM/YEN rate from the equilibriumvalue and hence in favour of the ESTAR model.The optimal values of pand d are determined as2 and 1 respectively. The Q statistic suggests thatthe combination of p and d selected for themodel yields residuals that are free fromautocorrelation problem up to 20 lags. Thisimplies ESTAR (2) process with a delayparameter, d =1 is the appropriate representationof the adjustment of deviations towards the long­run PPP equilibrium for the RM/YEN rate duringthe sample period.

ESTIMATED MODELS

In this study, two versions of ESTAR model,namely the unrestricted and restricted ESTARmodels are estimated for Y" the deviations ofRM/YEN rate. The unrestricted ESTAR model,is actually the model as specifIed in Equation 2with a transition function given in Equation 3.The word "unrestricted" is given to differentiateit from a special case whereby certain restrictions

p(}: f3i = 1 P; = -f3i and (Taylor and Peel 2000»

i-1

are imposed on it, thus resulting in a so-calledrestricted ESTAR model. The results obtainedfrom these models are reported in Table 4.

Several features for the estimatedunrestricted model are noteworthy here: First,the non-linear parameters (~ and y) of theunrestricted ESTAR (2) model are statistically

Test statistics

TABLE 3Linearity tests results

Sup Z (d) Delay parameter, d 1% Critical value

LM testLST testLjung-Box Q statistic

19.62719.92015.912

11

r(4) = 13.28r(3) = 11.34r(20)=37.57

Notes. Null hypothesis, Ho: Linear model is correct. Rejection of Ho impliesnonlinearity in favour of ESTAR model.

TABLE 4Estimated ESTAR (2) models

Estimated Values (Standard Errors)

Parameters

"2Oy

Unrestricted model

- 0.031 (0.19)1.799 (0.33)**-0.340 (0.23)-1.170 (0.40)**0.349 (0.50)1.364 (0.25)**

0.171

Restricted model

-0.025 (0.02)1.442 (0.19)**-0.253 (0.16)

0.357 (0.05)**

0.171

Diagnostic Tests (Marginal Significance Values)

akARakAR/a1Q (20)WHITEARCH (4)GARCH (1, 1)LR (3)Adjusted R2

0.002

0.769

17.522 [0.619]5.276 [0.809]1.563 [0.816]0.546 [0.761]

0.882

0.002

0.819

20.492 [0.428]8.270 [0.507]1.595 [0.809]0.531 [0.767]8.952 [0.030]0.878

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Liew Khim Sen & Ahmad Zubaidi Baharumshah

significant at 1% level. Second, the residualvariance ratio of this unrestricted ESTAR (2)model to the linear AR (2) model is 0.769indicating that the former has a much smallervariance. This implies that the non-linear modelhas the ability to produce smaller forecast errorsthan the linear model. Third, the model passeda battery of diagnostic tests at conventionalsignificance levels. Fourth, the adjusted R2 value(0.882), the explanatory power of this non-linearmodel on the adjustment of deviations, is fairlyhigh. Fifth, the sum of linear parameters f31 andf3

2equals 1.459 > 1, suggesting that Y, exhibits

unit root behaviour and therefore linear AR (2)itself is inadequate representation of Y,. On the

other hand, f (f3i + f3i·) = 0.638 <1 implies that.-1

the requirement for global stability is met. ThisconfIrms that Y, is mean-reverting in the non­linear specification (Baum et at. 2001; Taylorand Peel 2000). Lastly, our results seem to besupportive of the following relationships: ~ =

-f31 and f3; = -f32•

The conclusion that can be drawn from thefirst five findings is that the above non-linearmodel is appropriate representation of Yf thedeviations of RM/YEN rate; whereas the two lastresults enable us to estimate the restricted ESTARmodel, with the following restrictions imposedon the unrestricted model: f31 + f32 =1, ~ = -f31

and f3; = -f32

(Taylor and Peel 2000). We utilised

the likelihood ratio (LR) test to determine thevalidity of these restrictions. The computed LRstatistic is compared with chi-squared criticalvalue with 3 degrees of freedom. The LR statisticof 8.952 suggests that the above restrictionscannot be rejected at the 5% significance level.The adequacy of the model is verified by theabsence of serial correlation (Ljung-Box Qstatistic) and heteroscedasticity (WHITE statistic).Moreover, neither ARCH (4) nor GARCH (1, 1)suggests the presence of ARCH effect. Thus, therestricted ESTAR (2) model passes a battery ofdiagnostic tests and thereby can be used as aforecasting model, as its unrestricted version.

The linear AR model is also estimated forthe purpose of forecast accuracy comparison.The estimation of AR model requires that thevariables must be stationary; otherwiseinterpretation from the outcome would bespurious. We employed classical ADF and PPstationarity tests to check whether the stationaryrequirement is met. The stationarity tests resultsare summarized in Table 5. The results in Table5 postulate that instead of Y" which is notstationary, we should estimate linear AR modelfor stationary series, I::.Y" the first difference of Y,.Strategically, we need to estimate AR (1) modelfor I::.Y" in order to obtain the requiredbenchmark model, namely AR (2) model for Y,.The estimated AR (1) model is reported inTable 6.

Tests

TABLE 5Stationarity tests results

Intercept Without Trend

y,

Intercept with Trend

y,

ADFPP

-0.858-1.244

-4.081 *-8.657*

-0.034-0.493

4.081*-8.627*

Note. * denotes the variable is stationary at 1% significance level.

TABLE 6Estimated AR (1) model

Series Coefficient of Diagnostic Tests (Marginal Significance Values)

~y.. (t statistic) Q (20) WHITE ARCH (4) GARCH

~y, 0.024 (-0.96) 0.857 16.919(0.657)

2.267(0.757)

1.149(0.362)

0.037(0.982)

138 PertanikaJ. Soc. Sci. & Hum. Vol. 10 No.2 2002

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From Table 6, our AR (1) model for t'1y, is givenby:

which is exactly the AR (2) specification of y"the deviation of MYR/JPY rate from its PPPequilibrium position. Ifwe take into account theinsignificance of the autoregressive parameter,Equation 10 would effectively be reduced to:

Diagnostic tests results show that Equation10 is free from all the autocorrelation,heteroscedasticity and ARCH effects. Theadjusted R2 values also suggest that the AR (1)model fits fairly well for the t'1y, series. To sumup, the estimated AR (1) model is an appropriaterepresentation of first difference series of y,. Theonly shortcoming of this model is that the linearautoregressive parameter is not significant basedon the standard t-test. Equation 10 can berewritten as:

FORECAST PERFORMANCE

The models are used to generate out-sampleforecasts and the forecasting performances ofthese models are evaluated. The out-sampleperformance of the estimated forecasting modelsover the forecast horizon of n = 4, 8 and 12quarters during the period 1997:3 to 2000:2 areevaluated based on mean absolute forecast error(MAFE), mean absolute percentage forecast error(MAPFE) and root mean square forecast error(RMSFE). The overall forecasting performancesare reported in Table 7. Generally, all accuracycriteria consistently suggest that all threeforecasting models, namely the unrestrictedESTAR (2), restricted ESTAR (2) and AR (2)models outperformed the SRW model at allhorizons. This implies that both the linear andnon-linear time series models under this studyimprove over the SRW model in the short- and

Equation 13 is simply the commonly usedbenchmark for the evaluation of exchange rateforecasting models, namely the simple randomwalk (SRW) models. With the availability of twobenchmarks model, in particular the linear AR(2) model (Equation 11) and SRW model(Equation 13), this study proceeds to comparethe forecast accuracy of the unrestricted andrestricted ESTAR (2) models with the benchmark(AR(2) and SRW) models.

(13)

(12)

(11)

(10)

t'1y, = 0

or its equivalent:

TABLE 7Comparison of out-sample forecast accuracy

Models Unrestricted ESTAR (2) Restricted ESTAR (2) AR (2)

Forecast 4 8 12 4 8 12 4 8 12Horizon

Simple Random Walk model as benchmarkMAFE 0.721 0.919 0.954 0.671 0.880 0.934 0.721 0.919 0.954RatioMAFPE 0.587 0.838 0.890 0.668 0.876 0.923 0.712 0.901 0.935RatioRMSPE 0.731 0.846 0.851 0.804 0.903 0.908 0.825 0.950 0.962Ratio

Linear AR (2) model as benchmarkMAFE 0.590 0.844 0.903 0.931 0.958 0.979RatioMAFPE 0.824 0.930 0.952 0.938 0.972 0.988RatioRMSPE 0.886 0.891 0.885 0.975 0.950 0.944Ratio

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Liew Khim Sen & Ahmad Zubaidi Baharumshah

medium-run.It is interesting to know whether the non­

linear models yield more accurate forecasts thanthe linear model. As a matter of fact, it isrational for one to expect the former to improveupon the latter, since the former is moreinvolved. In view of the fact that the actualanswer to this issue is important to the applicationof time series analysis, this empirical studyproceeds further to contrast the forecastperformance between these two competingmodels directly. The conclusion from thisexercise is that both the unrestricted andrestricted ESTAR (2) models predict better thanlinear AR (2) model on the basis of all accuracycriteria, for forecast horizon equals 4, 8 and 12quarters. For instance, MAFE of unrestricted(restricted) ESTAR (2) model are 0.590 (0.931),0.844 (0.958) and 0.903 (0.979) times smallerthan the MAFE of AR (2) model, for n = 4, 8and 12, in that order. These results areoverwhelmingly supported by the MAFPE andRMSPE criteria. As such, our extra resourcespent on the modelling of ESTAR model is atleast paid off. Perhaps, more importantly, thisstudy has provided evidence that theperformance of ESTAR is superior to its linearcompetitor, the AR model.

CONCLUSION

The empirical performance of foreign exchangerate models has been frequently criticized inrecent years. These critiques come from studiesthat have found exchange rate models predictpoorly out of sample periods. In the foreignexchange market, central banks often intervene,in an effort either to attenuate or to amplifyvariations in the exchange rate. The Ringgit isno exception and this explained partly the poor­out-of sample prediction of the exchange ratemodels in previous studies.

In this study, we demonstrate that theadjustment of the RM/YEN rate is in factpredictable based on time series model. TheSTAR model and AR model both outperformthe random walk in the out sample forecastingat all horizons. The results also show that theSTAR model outperforms the AR model, itslinear competitor. Importantly, we demonstrateformally that the adjustment of the Ringgit to itslong run equilibrium follows a non-linearity path.This suggests that there is a systematic predictablecomponent in the movement of nominal RM/

YEN exchange rate.Two major implications of this finding are:

(1) non-linear ESTAR model should be givenpriority in modelling exchange rate time series;and (2) exchange rate forecasters could rely onthe non-linear model as a more reliableforecasting tool.

ACKNOWLEDGEMENT

This paper is part of the ongoing project ofIRPA 2000 (Project No: 05-02-04-0046) andFundamental Project 2002 (06-02-02-0098S). Theauthors are grateful to the referees for thecomments and suggestions on the earlier versionof the paper. All remaining errors are theresponsibility of the authors.

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(Received: 7 March 2002)

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