# Forecasting exchange rates out of sample: random walk vs Markov switching regimes

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Forecasting exchange rates out of sample: randomwalk vs Markov switching regimesDimitris G. KirikosPublished online: 06 Oct 2010.

To cite this article: Dimitris G. Kirikos (2000) Forecasting exchange rates out of sample: random walk vs Markovswitching regimes, Applied Economics Letters, 7:2, 133-136, DOI: 10.1080/135048500351979

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Forecasting exchange rates out of sample:random walk vs Markov switching regimes

DIMITRIS G. KIRIKOS

Department of Accounting, Technological Education Institute of Heraklion, Greece

Received 25 March 1998

A random walk is compared with a Markov switching regimes process in forecastingexchange rates out of sample, using quarterly data on three currencies relative to theUS dollar over the period 1973:3 1997:3. The results show that the relative perform-ance of the models varies with the length of the post-sample period suggesting thatthe availability of more past information may be useful in forecasting futureexchange rates.

I. INTRODUCTION

The results of Meese and Rogo (1983) that a randomwalk outperforms structural theories in forecastingexchange rates out of sample gave rise to extensive researchdealing with the empirical validity of the building blocksof the theories and the econometric methods used in testingstructural models of the exchange rate (Ho man andSchlagenhauf, 1983; Woo, 1985; MacDonald and Taylor,1994; Kirikos, 1993, 1996; Engel, 1996). This has culmi-nated in a general consensus that exchange rate assetmarket models are not empirically relevant and, in anycase, they cannot improve on random walk forecasts.However, the nding of Meese and Rogo (1983) that

the random walk is also a better predictor than other uni-variate time series models has been challenged by Engeland Hamilton (1990) who reported evidence in favour ofa Markov switching regimes process for exchange ratechanges. Indeed, these authors found that a model ofstochastic segmented trends produces better in-sampleand out-of-sample exchange rate forecasts on the basis ofmean square error.In this letter, we re-examine the forecasting performance

of a Markov switching process relative to that of randomwalk using a much larger data set. In particular, the twomodels are compared by means of the root mean squareerror (RMSE) of forecasts based on quarterly data for thecurrencies of the UK, Germany and France relative to theUS dollar over the period from 1973:3 to 1997:3. Theresults show that, contrary to the evidence reported in

Engel and Hamilton (1990), the random walk gives consis-tently better in-sample forecasts but the Markov switchingmodel predicts better for short out-of-sample horizonswhen the post-sample period is narrowed towards the endof the full sample. Thus, the result of Meese and Rogo(1983) that the forecasting performance of the randomwalk is not a ected by the length of the post-sample periodis not reproduced here.The forecasting methodology is discussed in the next

section and the results are presented in Section III. Thefourth section contains a summary and conclusions.

II. METHODOLOGY

Let et t 1; 2; . . . ; T be the natural logarithm of theexchange rate and st the rst di erence of et (i.e.st et et 1). Then, the k-period-ahead forecast of arandom walk with a drift parameter is given by:

et kj t et k s 1

where et k jt is the forecast of et k based on information attime t, s 1=n 1

Pn 1t 1 st is the sample mean of st, and n

is any sub-sample n T on which out-of-sample fore-casts are based.The alternative process is the following:

st ht ut u N0; 2ht 2

where ht is an unobserved state variable that takes onvalues in the set f1; 2g , and u is an error term. State 1

Applied Economics L etters ISSN 1350 4851 print/ISSN 1466 4291 online # 2000 Taylor & Francis Ltd

Applied Economics L etters, 2000, 7, 133 136

133

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will be referred to as the depreciation state while state 2 willbe associated with an appreciation of the relevant currencyrelative to the US dollar. Thus, Equation 2 allows for dif-ferent means and variances across regimes.The state variable ht is assumed to follow an irreducible

Markov chain with stationary transition probabilitymatrix:

P p11 p12p21 p22

3

where pij Pr ht j jht 1 i , i; j 1; 2. Under thisassumption the k-period-ahead forecast of st k , based ontime-t information is (Hamilton, 1993; Kirikos, 1996):

st k jt E st k jSt 0t P

kh 4

where St s1; s2; . . . ; st is the history of s up to time t,a 0t Pr ht 1jSt Prht 2jSt is the vector of prob-abilistic inferences about the state at date t (Hamilton,1990, 1993), and 0h 1 2 is the vector of statemeans. It should be noted that forecasts given byEquation 4 are nonlinear since the inferences 0t are pro-duced by a nonlinear lter.Maximum likelihood estimates of the means 1; 2 ,

variances 21; 22 , and the transition probabilities

p11; p22 can be obtained through the EM algorithmwhich is a method of maximizing the sample likelihoodfunction by iterating on the normal equations (Hamilton,1990). Also, a battery of speci cation tests, based onLagrange multiplier tests and on the autocorrelation prop-erties of the conditional scores, is reported in Hamilton(1996).Given Equation 4, forecasts of the logarithm of the

exchange rate are computed by:

et k j t et st 1j t st 2j t st k j t 5

Post-sample forecasts are computed through rolling esti-mation of the models. That is, initial k-period-ahead fore-casts are based on parameter estimates obtained with asub-sample of size n. Then, the models are re-estimatedby adding the next available observation and new forecastsare generated. This procedure continues until the sub-sample size becomes T-k, where T is the full sample size.The forecasting accuracy of the models is measured by

the root mean square error (RMSE) of forecasts:

RMSE 1

T n k 1

XT n k

i 0

en i kjn i en i k2

" #1=2

6

where n is the initial sub-sample size and k is the forecasthorizon.

III . EMPIRICAL RESULTS

The forecasting performance of the models is compared onquarterly data on the currencies of the UK (BP), Germany(DM), and France (FF) vis-a-vis the US dollar ($) coveringthe period from 1973:3 to 1997:3 (97 observations).1Some preliminary Wald tests of the Markovian

dynamics, based on the full sample, are reported in Table1. As in Engel and Hamilton (1990), we test the hypothesesHo: p11 1 p22 and Ho: 1 2. The former impliesthat the probability that the process is currently in state 1does not depend on the state of the previous period, that is,the state variable ht is not characterized by the Markovproperty. Under the second null hypothesis the processhas a non-shifting mean. In both cases, however, standardasymptotic theory applies only when 21 6

22 (Engel and

Hamilton, 1990; Hamilton, 1993). The results arefavourable to the Markov switching process since the nullHo: p11 1 p22 is strongly rejected for all currenciesconsidered. Also, the hypothesis of a non-shifting mean isrejected for the DM/$ and the FF/$ exchange rates but notfor BP/$.The above evidence is not fully corroborated by the fore-

casting performance of the Markov switching regimes pro-cess relative to that of a random walk with or without adrift parameter. Speci cally, the RMSE of 1 to 4-quarter-ahead in-sample forecasts of a random walk is consistentlylower than that of the Markov process given in Equation 2for all currencies.2 Besides, unlike the results reported inEngel and Hamilton (1990) who used data up to 1988:1, inour sample a random walk with drift performs somewhatbetter in terms of RMSE than a driftless random walk.The RMSE of out-of-sample forecasts at horizons of 1 to

4 quarters is reported in Table 2 for di erent post-sampleperiods. The rst period was selected to follow the sampleof Engel and Hamilton (1990) who found that Markov out-of-sample forecasts over the period 1984:1 1988:1 werebetter than those of a random walk with drift. However,the random walk model, with or without drift, appears to

134 D. G. Kirikos

1 The quarterly data are averages of monthly observations which are available from the Federal Reserve via the Internet site http://www.stls.frb.org/fred.2RMSEs of in-sample forecasts are not reported here but they are available from the author.

Table 1. Wald tests of Markovian dynamics

Ho: p11 1 p22 Ho: 1 2s 21 21

BP/$ 552.81 (0.000) 1.91 (0.167)DM/$ 8.05 (0.004) 45.75 (0.000)FF/$ 41.22 (0.000) 72.71 (0.000)

Notes: s denotes the rst di erences of the natural logarithm of theexchange rate. Numbers in parentheses next to the values of test statisticsare marginal signi cance levels.

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perform better in RMSE for all currencies and forecasthorizons, when the post-sample period is 1988:2 1997:3(see Table 2, panel A).3The forecasting accuracy of the Markov model improves

when the post-sample period is reduced to 1992:1 1997:3.Indeed, the Markov process outperforms the random walkat horizons of 1 and 2 quarters for the DM/$ and the FF/$exchange rates but not for the BP/$ rate (see Table 2,panel B). Furthermore, for the post-sample period from

1995:1 to 1997:3 Markov forecasts are better in RMSE athorizons of 1 and 2 quarters for the DM/$ rate and of 1 to4 quarters for the FF/$ rate. Also, the switching regimesmodel predicts BP/$ better than a randomwalk with a driftparameter (see Table 2, panel C). Thus, contrary to theevidence reported in Meese and Rogo (1983), the relativeout-of-sample forecasting performance of the randomwalkmodel seems to vary with the length of the post-sampleperiod.

Forecasting exchange rates out of sample 135

Table 2. RMSE of out-of-sample forecasts

Forecast horizon (quarters)

1 2 3 4

A. Post-sample period 1988:2 1997:3BP/$Markov model 0.0553 0.0875 0.1071 0.1277Random walk with drift 0.0508 0.0751 0.0882 0.0992Random walk without drift 0.0505 0.0742 0.0863 0.0965

DM/$Markov model 0.0510 0.0827 0.1012 0.1259Random walk with drift 0.0497 0.0764 0.0913 0.1071Random walk without drift 0.0490 0.0746 0.0887 0.1038

FF/$Markov model 0.0488 0.0818 0.1035 0.1306Random walk with drift 0.0476 0.0744 0.0916 0.1087Random walk without drift 0.0473 0.0731 0.0885 0.1034

B. Post-sample period 1992:1 1997:3BP/$Markov model 0.0489 0.0762 0.0893 0.1034Random walk with drift 0.0467 0.0693 0.0797 0.0898Random walk without drift 0.0465 0.0689 0.0793 0.0899

DM/$Markov model 0.0417 0.0658 0.0855 0.1075Random walk with drift 0.0435 0.0678 0.0878 0.1035Random walk without drift 0.0427 0.0655 0.0840 0.0973

FF/$Markov model 0.0384 0.0614 0.0806 0.1015Random walk with drift 0.0409 0.0630 0.0815 0.0937Random walk without drift 0.0408 0.0626 0.0805 0.0921

C. Post-sample period 1995:1 1997:3BP/$Markov model 0.0192 0.0331 0.0461 0.0542Random walk with drift 0.0201 0.0346 0.0467 0.0565Random walk without drift 0.0189 0.0311 0.0406 0.0460

DM/$Markov model 0.0367 0.0643 0.0894 0.1133Random walk with drift 0.0432 0.0739 0.0990 0.1187Random walk without drift 0.0414 0.0689 0.0887 0.1031

FF/$Markov model 0.0294 0.0528 0.0728 0.0853Random walk with drift 0.0352 0.0601 0.0788 0.0874Random walk without drift 0.0361 0.0618 0.0817 0.0916

3 The result of Yoon (1998) that in the presence of a structural change the randomwalk model gives forecasts with lower mean square error does not applyhere because the Markov model accounts for regime shifts and allows for di erent means and variances across regimes.

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IV. CONCLUSIONS

Based on a large data set, we conclude that the out-of-sample forecasting performance of a Markov switchingregimes model for the exchange rate improves relative tothat of a random walk as the post-sample period is nar-rowed towards the end of the full sample. That is, as weexclude forecasts based on less information, the Markovmodel improves on the random walk in forecastingexchange rates out of sample. This result, coupled withthe in-sample forecasting superiority of the random walk,shows that the availability of more past information isuseful in forecasting exchange rates since Markov forecastsare based on probabilistic inferences about the state, whichdepend on information through the date of the forecast.Thus, in spite of the relatively good performance of naiveforecasts over certain post-sample periods, our evidencesuggests that the natural logarithm of the exchange ratedoes not follow a random walk.

REFERENCES

Engel, C. (1996) The forward discount anomaly and the risk pre-mium: a survey of recent evidence, Journal of EmpiricalFinance, 3, 123 91.

Engel, C. and Hamilton, J. D. (1990) Long swings in the dollar:Are they in the data and do markets know it?, AmericanEconomic Review, 80, 689 713.

Hamilton, J. D. (1990) Analysis of time series subject to changesin regime, Journal of Econometrics, 45, 39 70.

Hamilton, J. D. (1993) Estimation, inference, and forecasting oftime series subject to changes in regime, in Handbook ofStatistics, G. S. Maddala, C. R. Rao and H. D. Vinod(Eds), Vol. 11, North-Holland, New York.

Hamilton, J. D. (1996) Speci cation testing in Markov switchingtime series models, Journal of Econometrics, 70, 127 57.

Ho man, D. L. and Schlagenhauf, D. E. (1983) Rational expec-tations and monetary models of exchange rate determination,Journal of Monetary Economics, 11, 247 60.

Kirikos, D. G. (1993) Testing asset market models of theexchange rate: a VAR approach, Applied Economics, 25,1197 216.

Kirikos, D. G. (1996) The role of the forecast-generatingprocess in assessing asset market models of the exchangerate: a non-linear case, European Journal of Finance, 2,125 44.

MacDonald, R. and Taylor, M. P. (1994) Reexamining the mone-tary approach to the exchange rate: the dollar-franc, 197690, Applied Financial Economics, 4, 423 29.

Meese, R. A. and Rogo , K. (1983) Empirical exchange ratemodels of the seventies: do they t out of sample? Journalof International Economics, 14, 3 24.

Woo, W. T. (1985) The monetary approach to exchange ratedetermination under rational expectations, Journal ofInternational Economics, 18, 1 16.

Yoon, G. (1998) Forecasting with structural change: why is therandom walk model so damned di cult to beat? AppliedEconomics L etters, 5, 41 2.

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