Efficiency testing revisited: a foreign exchange market with Bayesian learning

Efficiency testing revisited: a foreign exchange market with Bayesian learning

Pergamon PII: Journal o f International MoneT and Finance, Vol. 16, No. 3, pp. 367-385, 1997 © 1997 Elsevier Science Ltd. All rights reserved Printed...

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Pergamon PII:

Journal o f International MoneT and Finance, Vol. 16, No. 3, pp. 367-385, 1997 © 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0261-5606(97)00003-X 0261-5606/97 $17.00 + 0,(t0

Efficiency testing revisited: a foreign exchange market with Bayesian learning NICOS M C H R I S T O D O U L A K I S *

Athens University of Economics and Business, and CEPR, Athens 10434, Greece AND SARANTIS C KALYVITIS

Athens University of Economics and Business, Patission str. 76 and CEPR, Athens, Greece 10434 The paper re-examines the efficiency hypothesis in the foreign exchange market. The traditional efficiency-testing equations are reviewed and a more general model is developed that incorporates Bayesian revisions of devaluation expectations. A number of properties regarding the pattern of the coefficients in efficiency-testing equations are established. The empirical estimation of the model by using data from the emerging foreign exchange market in Greece confirms the properties of the theoretical model. (JEL F31). © 1997 Elsevier Science Ltd.

The question of efficiency, although important, has not yet been resolved in financial economics and continues to attract attention, since the existence of inefficiency implies opportunities for unexploited profits in capital markets. Inefficiency may be generated by non-rational traders, time-varying risk premia, or lack of knowledge about the full model of the economy or the intervention rule (Hsieh, 1984). The continuous adjustment of financial markets means that until full knowledge is acquired about the new instruments, inefficiency may be present; quoting the 'Economist' magazine: *The paper is part of the Research Project E-199 financed by the Center of Economic Research, Athens. Another part of research took place in Tinbergen Institute, Amsterdam, when the first author was a visiting fellow. We benefited from help by, and extensive discussions with, N. Karamouzis, comments by seminar participants in the Bank of Greece and the University of Cyprus, and valuable suggestions by, among others, N. Garganas, G. Hardouvelis, D. Moschos, N. Pittis and an anonymous referee of this journal. The authors are thankful to E. Diakoumakou and I. Segoura for data collection and processing. The responsibility for any remaining errors is strictly ours. 367

Efficiency testing revisited: N M Christodoulakis and S C Kalyvitis ...Although markets get steadily more efficient in terms of the speed and detail with which information reaches all traders, new inefficiencies continually appear in the form of new financial instruments. Learning how to price new instruments takes time and provides opportunities for fast learners to beat slow ones...New markets are inefficient markets. (Economist, October 9 1993, our emphasis).

Expectational errors may be of vital importance in order to explain the failure of the traditional efficiency hypothesis: for example, Froot and Frankel (1989) find that the forward discount bias can, to a large extent, be attributed to systematic expectational errors which 'are consistent with "peso problems" ... and learning on the part of investors' (p. 141). Lewis (1988), referring to the peso problem demonstrates how expectations about some discrete policy shift may affect efficiency tests; when a period of learning is required before the new policy becomes fully credible, then the foreign exchange market may appear irrational ex post. Kaminsky (1993) shows that the bias in the dollar forecasts during 1976-1987 can be consistent with rationality, if a regimeswitching process were underway. She also claims that 'investors can be rational and yet make repeated mistakes if the true model of the exchange rate is evolving over time' (ibid). In a recent series of papers, Evans and Lewis (1992, 1993, 1994) show that in the presence of an anticipated shift in the process of an asset price, trends in the forecast errors may appear ex post, inducing the coefficient of the forward rate to deviate from its theoretical value implied by the efficiency hypothesis; moreover, the authors show that the higher the magnitude of the expected shift, the larger the divergence. The present paper develops a generalization of the simple efficiency equation for the foreign exchange markets by introducing a mechanism of Bayesian learning about devaluation prospects and policy credibility. It examines the implications for the efficiency test of foreign exchange markets and finds that both the estimated values and the statistical significance of the parameters in the efficiency equation depend on the memory length of the learning process. The generalized model suggests that the coefficients of the simple efficiency equation are likely to be found statistically insignificant a n d / o r far from the expected size. This can, perhaps, explain why the abundant empirical literature finds no econometric evidence in favor of the simple efficiency equation. In contrast, econometric estimates are consistent with the hypothesis that traders follow a Bayesian pattern of learning. Moreover, if Bayesian learning takes place for a sufficient period of time, the estimates approach the theoretical coefficients suggested by the simple efficiency equation. The model collapses to the familiar efficiency equation, when learning is immediate and past information is of no additional value to the traders. The paper has two aims: first, to develop testable propositions regarding the pattern of coefficients estimated in a generalized efficiency equation and, second, to develop a model of policy credibility that can be empirically tested in exchange rate markets. To test the model, we consider an application in the Greek foreign exchange market which has been liberalized only during the last few years and, thus, poses all the issues of gradual learning by traders about the credibility of policy. 368

Efficiency testing revisited: N M Christodoulakis and S C Kalyvitis

The structure of the paper is the following: Section I outlines the traditional framework of efficiency testing and then introduces a mechanism of Bayesian learning, due to the gradual establishment of credibility. A number of propositions regarding the pattern of estimated parameters in the efficiency testing are established. In Section II the model is tested using two alternative estimation strategies. Section III draws the main conclusions.

I. Model specification 1.4. Efficient markets and risk premia In his seminal paper, Fama (1970) defines efficiency as a situation in which expected profits from speculation are zero. In terms of foreign exchange markets this amounts to assuming that the expected spot exchange rate at time t for time t + k equals the forward rate at time t with maturity k:

ft,t+k =Etst+k,

(1)

where all variables are expressed in logs, s t + k is the spot exchange rate at time t+k,f,,t+k is the forward rate at time t with maturity at t + k and E,(.) denotes the mathematical expectation at time t conditional on the information available in that period. Hsieh (1984) notes that for equation (1) to hold the conditions of risk neutral agents and competitive markets should be satisfied. In order to test for efficiency, we investigate whether the expected exchange rate is the best predictor of the actual exchange rate: (2)

st+ k = c~ + f l E , s,+~ + u,,

where u, is white noise and a and /3 denote parameter values. An efficient market implies the joint hypothesis ( a , / 3 ) = (0,1), i.e. that no excess profits can be made in the foreign exchange market. Substituting (1) into ( 2 ) w e get: (3)

st+k

= O[ -~ / 3 f t , t + k ~- U , ,

which provides a testable form for efficiency. To avoid problems associated with non-stationarity of exchange rates it is usual to employ a similar equation in first differences: (4)

s,+ k - s, = ol +/3[f,.,+k -- S,] + U,.

The null hypothesis requires the same condition as before, and implies that the forward premium is an unbiased predictor of the actual depreciation rate (the 'unbiasedness' hypothesis). Therefore, in (4) we test the joint hypothesis of market efficiency and no risk premia; failure to accept the null hypothesis could be attributed to either or both factors. Froot and Thaler (1990) discuss possible interpretations of obtaining parameters /3 different from unity. The case of /3 < 1 is interpreted either as ' . . . evidence of a time-varying risk premium on foreign exchange' when investors are risk averse, or as confirmation of (upwards) expectational errors when they are risk neutral. If, on the other hand, coefficient /3 exceeds unity, a 369

Efficiency testing reoisited: N M Christodoulakis and S C Kalyoitis

positive correlation between past and current depreciation is implied. This can be attributed to the presence of 'positive feedback' traders, who buy when prices start rising and sell when they begin to fall; for an analysis see, for example, Cutler et al. (1990). Other authors (e.g. Lewis, 1988) have mentioned the 'peso problem' as a potential source of departure from efficiency in a situation of partial credibility where policy makers attempt to defend a target exchange rate policy but the market is not fully convinced about either their intention or capability to achieve it. Learning the true intentions of authorities may require a considerable period of time and, in the meanwhile, this produces systematic bias in the efficiency-testing parameters. LB. A model with Bayesian revisions

To capture the presence of partial policy credibility, a model with a learning process in the foreign exchange market is required. Assume that the monetary authorities pursue an exchange rate policy which is not fully credible to the market. Agents fear that authorities might choose a different depreciation rate for various reasons: inability to defend the targeted parity from possible speculative attacks, wrong fundamentals, or simply because of time-inconsistency. In each period, credibility is eroded if the market realizes a departure from the announced policy, whilst it increases if no policy shift is observed. A model specification that takes into account such a learning process is obtained by assuming that the forward premium is formed on the basis of alternative beliefs about the future course of the exchange rate. With partial policy credibility, the expectation of actual depreciation takes the form: (5)

Etst+ k - s t =pz,.t+ ~ + (1 - p ) x t . t + k ,

where zt, t + k is the target depreciation rate for period t + k , xt, t+ k is the

depreciation rate 'feared' by the market to take place against the official policy, and p is the probability assigned to the announced policy being actually implemented. At period t, the target depreciation rate for period t + k is either explicitly announced by the authorities, or otherwise discerned by the market. In general, probability p may be time-varying as the market adjusts its beliefs in each period. In the case of full credibility, we have p = 1. Under the assumption that the target depreciation rate is reflected in the forward premium, equation (5) collapses to (4) when the null hypothesis is confirmed. Equation (5) has been widely used in modelling the 'peso' problem, named after the case of the Mexican currency that was for a long period traded at a forward discount despite the fact that the monetary authorities announced and supported a fixed parity policy before the crisis. Lewis (1988) used this formulation to examine how agents form expectations in a situation where one of two different monetary policies is likely to be chosen. Andersen and Risager (1991) and Driffill and Miller (1993), among others, apply equation (5) in connection with a mechanism of building-up reputation: in that case authorities steadily pursue the announced exchange rate policy, but expectations are formed through a learning process as the market is only gradually convinced about the true policy. In the end, authorities enjoy full credibility and probabil370

Efficiency testing revisite& N M Christodoulakis and S C Kalyvitis

ity p approaches unity. In all the above cases coefficients in (5) vary over time, and converge through a learning mechanism. To model this behavior we assume that in each period of time agents compare actual depreciation to the previously announced target, and if there is no recognizable deviation, they adjust the odds following a Bayesian pattern. If at starting period t = 0, the initial probability of sticking with the target were P0, the posterior odds Pt at period t would be determined by the expression: (6)

Pt 1 -Pt

L,

Po 1 -Pll '

where L t is the likelihood ratio at period t. To calculate L t we need additional assumptions about the nature of the monetary authorities, which are not known with certainty by traders. Monetary authorities are assumed to be either of the 'strong' type who do not intend to depart from the announced foreign exchange policy, or of the 'weak' type in the sense that they are prone to indulge in an unexpected depreciation. The first type of monetary authorities have true preferences with are compatible with keeping the announced target; authorities of the second type pretend to adhere to an exchange rate target simply in order to increase the impact of a surprise devaluation. The market has prior beliefs 7r1 and 7r2 concerning the probability that authorities are going to behave as 'strong' or 'weak' types, respectively. If monetary authorities belong to the 'strong' type, prior 7r1 is the probability that they will not depart from the announced policy. If, on the other hand, authorities are of the 'weak' type, 7r2 is the probability that they devalue further than the announced target. If the identity of the policy maker is known with certainty, we have the symmetric binomial case with ~l + ~r2 = 1. Using the priors, we can calculate how posterior odds evolve over time. If up to period t, there have been n, instances in which policy did actually follow the announced target and ( t - n t) incidents of eventual departure, one can apply the 'likelihood principle' (Edwards, 1982) and derive the log-likelihood ratio as: (7)

logLt = n t l o g ( T r j / T r 2) + ( t - nt)log[(1 - 7rl)/(1 - 372)].

The above expression leads to a time-varying pattern of posterior odds, p,, the direction of which is determined by the following proposition: Proposition 1. When an announced policy is confirmed, the posterior odds in favor of the assumption that the policy will be maintained rises (falls) whenever 7r~ > ~'2 (Tr~ < -n-z), i.e. when it is more likely that authorities are of the strong (weak) type. The opposite happens when the anticipated policy is not implemented and a departure is observed.

Proposition 1 follows easily from (6) and (7). When the policy is confirmed we have that n,+l = n t -t-1, and (t + 1 - nt+ 1) = ( t - n t ) . T h u s , the change in the likelihood ratio is given by: logLt+ l - logLt = l°g(Trl/Tr2). Clearly, the likelihood ratio rises from one period to the next when 7r~ > 7r2. From (6) one easily obtains the result that the posterior odds ( p , ) is an 371

Efficiency testing revisited: N M Christodoulakis and S C Kalyvitis

increasing function of the likelihood ratio ( L t) and the Proposition follows. In a similar vein, we can treat the case of ~-~ < 7rz and show that posterior odds will diminish whenever a policy confirmation is observed. Using this Proposition, one can discern how the 'peso psychology' is intensified or subsides over time. If authorities are believed to be of the 'strong' type (rr 1 > 7r2), then policy confirmations increase credibility, posterior odds gradually approach unity, and the fear of a sudden devaluation subsides. Alternatively if authorities are more likely to be 'weak' (TrI < rr2), sticking with policy targets reduces the posterior odds that the policy will be in place in the future. The crisis erupts when the posterior odds fall below a threshold level determined by the risk attitudes prevailing in the market. Using (6), one can also examine the case in which the log-likelihood ratio and, thus, the posterior odds remain constant over time. Setting logL, = A, we obtain after some manipulation the result that the number of times in which the policy target should be validated is given by: (8)

IIt =

[A + tlog(1

-

7r2/1

-

7rl)]//log[(~l/Tr2)(1

-

"rr2/1

-- 7"/'1) ].

In the symmetric binomial case, ~'1 + 722 = 1, the above expression is simplified to (9)

nt

t A ~- -~ + -~log-1 ( ~ 1 / 7 r 2 ) .

This expression can be used to assess the number of instances by which confirmations should exceed half of the number of periods, for the posterior odds to remain constant over time. For A > 0, (i.e. L t > 1), and 7rl > 7r2, there is an excess of policy confirmations over departures, and analogously for other cases. Using the above framework, we now move to examine how expectations evolve over time, given that monetary authorities do pursue the announced policy and credibility is gradually established. In each period, actual depreciation may differ from the target rate by a white noise deviation: (10)

s t + k - s, = z , , , + ~ - d t ,

where d is assumed to be a N(0, tr 2) process. At the same time, the market continues to fear that the exchange rate may reach a level different from what the official policy implies. This happens when the market feels that the exchange rate is overvalued and needs a corrective depreciation (say, c) determined by 'fundamentals'. In the absence of permanent shocks in the economy, fundamentals do not change and the extent of corrective action would remain constant over time. However, if in a particular period there has been an actual depreciation according to the policy pursued, the extent of correction is adjusted and the depreciation 'feared' to take place after k periods is revised accordingly, ff we assume that the fear of a corrective depreciation was developed m periods ago, we have: (11)

372

xt.t+ k = c m -

(st - St_m).

Efficiency testing revisited: N M Christodoulakis and S C Kalyvitis

The second term in the r.h.s, implies that, in the absence of permanent shocks, depreciation fears subside over time by the extent of actual depreciation that took place between period t and (t - m). Setting: (12)

a m =

-(

[3" -

1)Cm,

and

[3m = 1 / p

and substituting (1) and (11) into (5), we obtain the result that the actual depreciation is the following function of the forward discount:

St+k--St=ozm+[3m(ft,k--St)+([3"~--l)(st--St_,,,)+dt,

(13)

which when rearranged becomes (14)

st+ k - s , _ m = a "

+ [3m(ft.k

-- S t _ m ) + d , .

Superscript m is put on parameters to indicate that, in general, probability p depends on the length of memory m , over which policy is observed and adjustment takes place. Obviously, equation (14) is a generalization of equation (4), to which it collapses by assuming no memory (rn = 0) and that the coefficients a and [3 are invariant with time.l The Bayesian specification offers a possible interpretation of the cases in which [3 is estimated to be above unity. This may be due to two possible reasons: (i) asymmetric information between agents and monetary authorities that leads to a lack of credibility and thus p < 1 for most of the estimation period, or (ii) a reduction of the log-likelihood ratio, L,, in (7) due to an expected policy not validated in practice (n t = n t_ l)" With 7r 1 > 7r2, this makes L t and Pt fall and, consequently, [3m rise. The following properties of the estimators are readily established. 2. The estimate of coefficient [3m is positive, above unity, and decreasing in the lag m. Its statistical significance increases with m.

Proposition

The first two assertions are obvious, since the coefficient denotes a probability inverse. For the third, it suffices to show this property for the estimator of the term ( [3m __ 1) in equation (13). The estimate and its standard error are inversely proportional 2 to the variance of (s t -st_m). The latter can be written as the sum of period-by-period depreciations, s t - s t _ m = ~ ' ~ - 1 ( s t _ j _ s t _ r - ~), and if we recall that these depreciations have a variance cr 2, we obtain the result that V a r ( s t - s t _ m ) =mtr 2, and thus that the estimator of [3m is inversely proportional to m. Consequently, the estimator of [3 m and its standard error decrease with the lag m. 3. The estimate of coefficient ofm is negative and decreases in absolute value with the lag m.

Proposition

With the corrective depreciation, c m, positive, the parameter a m is negative by virtue of Proposition 2. It also follows that the estimator of ~ m decreases in absolute value with m. The Bayesian model can be used in two ways: (i)

Assuming that no deviation from official policy is observed, one can estimate equation (14) and check whether coefficient [3m approaches 373

Efficiency testing revisited: N M Christodoulakis and S C Kalyvitis

(ii)

unity as the memory length, m, increases. If such empirical evidence is found, it suggests that the market needs a certain period of time to learn about policy credibility and adjust expectations. The deviation of /3 m from unity and its speed of convergence, can provide information on how quickly this adjustment takes place. For any given m, the Bayesian model will give estimators above unity, denoting the inverse of policy credibility. By employing time-varying estimation methods for/3 t 1/pt, one can investigate how credibility has evolved over time in the face of external shocks or events that might have been considered as affecting exchange rate policy. =

II. Model estimation

In this section, we test the Bayesian model in the Greek foreign exchange market during the period of transition from capital controls to a more liberalized regime. The Greek forward exchange market began its operation on January 1992 and lived through a variety of uncertainties until transactions were suspended during the EMS crisis on September 18th, 1992. During that period, the government pledged a policy of target depreciation below the inflation differential with competitors, as a means to contain price rises and signal a tight monetary stance. Traders not only had to adapt to the new institutional circumstances, but also discover the true intentions of the monetary authorities. 3 Thus, the model of equation (14) is likely to capture the foreign exchange market behavior better than the simple efficiency equation.

11.4. Simple efficiency tests First, we investigate the standard efficiency equation (4>. Unit root tests on the levels of spot and forward rates reveal non-stationarity of the series, but the depreciation rates and the forward discount series are found to be stationary. 4 Thus, equation (4) can be jointly tested for efficiency and the absence of risk premia. Since the daily sampling interval is different from the forecasting interval (weekly or monthly) the errors in equation (4) exhibit a moving average pattern, and, consequently, estimation via OLS is inappropriate, because it requires serially uncorrelated residuals. Thus, we opt for the Generalized Method of Moments (GMM) estimators (Hansen and Sargent, 1982; Hansen, 1982) designed for cases in which errors are found to be serially correlated and heteroscedastic, and the instrumental variables may not be strictly exogenous but simply have to satisfy certain orthogonality conditions. The results of the Deutschemark and US dollar exchange rates of the Greek drachma (GRD) are displayed in Table 1. The only relatively significant coefficient appears in the DM weekly rate, but it enters with the wrong sign. In general, results are not consistent with the unbiasedness hypothesis in any of the markets, adding another case in the long list of similar findings for other markets; surveys of the literature on exchange market efficiency tests include Boothe and Longworth (1986), Hodrick (1987) and MacDonald and Taylor (1992). However, it must be pointed out that such tests constitute a weak-form 374

Efficiency testing revisite& N M Christodoulakis and S C Kalyvitis

of efficiency testing, since the information set is limited to one exchange market each time (Geweke and Feige, 1979), and, therefore, a joint estimation for the two markets is more appropriate. The results, as shown in Table 1, do not indicate any noticeable change in the coefficients pattern, even if interaction between the Deutschemark and US dollar markets is allowed. We conelude that we are unable to accept the joint hypothesis that markets were efficient and no risk premia existed in the Greek foreign exchange market during the estimation period.

ll.B. Parameter stability Given the failure of the simple efficiency equation, we move to examine the question of whether the parameter estimates are stable across the sample, by applying the Cumulative Sum of Squares (CUSUM) test as described in Brown et al. (1976). This test uses recursive residuals from the re-estimated regression to construct stability bounds, which are parallel lines to the mean-value of the parameter. The probability that the path crosses one (or both) of the bounds is the required significance level: Parameter instability of the Deutschemark equation is demonstrated in Figure l(a), particularly during the last weeks of the sample period. This is easily explained by the aforementioned EMS crisis and the concurrent resurgence of devaluation expectations. Parameters of the US dollar equation [see Figure l(b)] exhibit a less unstable pattern during the critical last period.

II. C. Estimates of the Bayesian model In the preceding subsections, we showed that the simple efficiency model was TABLE 1. Deutschemark Parameter

Weekly

Monthly

Single market estimates a of equation (4) a 0.0056 0.0700 (0.0018) (0.0528) /3 - 1.3439 - 5.0753 (0.7672) (4.6478) Multimarket estimates a of equation a /3

0.0058 (0.0017) - 1.5008 (0.6884)

U S dollar Weekly

Monthly

- 0.0198 (0.0132) 5.8605 (3.9085)

- 0.3707 (0.3511) 23.2911 (21.8491)

- 0.0176 (0.0117) 5.1154 (3.3902)

- 0.3977 (0.2538) 25.0661 (15.7520)

(4) 0.0342 (0.0190) - 2.0732 (1.7629)

a P a r a m e t e r values are G M M estimators with [ f t - 5 - s,_ 5] and [ ft-22 - st- 2 2 ] as instrumental variables for the weekly and monthly series, respectively. Standard errors are shown in parentheses, and are robust to heteroscedasticity according to the White (1980) procedure.

375

Efficiency testing revisited: N M Christodoulakis and S C Kalyvitis

(al 125-

1.000.75

0.50-

025-

0.00-

-025WEEK

[

CUSUM of

squares

. . . . . 5x significance

'i5 ....

~'o....

___

I

(b) 125-

1.00 0.75

0...,-500.250.00.-03..5 t,

....

"io ....

.~5 ....

~

....

,~

WEEK CUSUM of squares

. . . . . 5~ significance

___

]

FIGURE 1. (a) C u s u m test o n D e u t s c h e m a r k . (b) C u s u m test o n U S dollar.

376

Efficiency testing revisited." N M Christodoulalds and S C Kalyvitis

not accepted by the data, and that its parameters were unstable across the estimation period. These findings suggest that the market was characterized by a long adjustment process due either to a gradual conviction about policy credibility or to some other type of learning. To examine these issues, we employ two alternative versions of the Bayesian model described in Section I: (i)

(ii)

For a given lag m, obtain time-varying estimates of parameters otm and /3 " reflecting the process of gradual learning. Coefficient /3 m should be above one, peaking at periods of devaluation fears and approaching unity when markets calm and policy regains credibility. For a variety of lags m, obtain time-invariant estimates of coefficients o~" and /3" to see whether they approach the theoretical pair (0, 1), as the length of memory increases. If the market believes that authorities are of the 'strong' rather than the 'weak' type (i.e. ~r~ > 7r2) and observes a series of policy confirmations, then coefficient /3" will approach unity from above, whilst o~m will be diminishing to zero.

First, we obtain time-varying estimates of coefficients ~m and /3 m. In order to test how they respond to policy changes or how extraneous information affects the credibility of policy we choose a lag m sufficiently large so that coefficients have converged to a steady state level. 6 The weekly data set used for the parameter stability tests in Section II.B. is again applied here, due to the moving average pattern of errors in the daily data; thus, to estimate equation (14) we choose m = 5 (i.e. 1 week lag). Here we only report the case for the Deutschemark exchange rate for which a specific and announced policy was pursued by the monetary authorities for a long time. As Figure 2 shows, coefficient /3 " undergoes considerable variation during the period. According to the theoretical model of Section I, coefficient /3" should be above unity and rise whenever an event makes the observance of the announced policy less likely and diminishes the log-likelihood ratio L, defined in (7). During the estimation period there appear to be four incidents (El, E2, E3 and E4) which increased the odds that a shift in policy might take place. These events are reported in Table 2. All can reasonably be viewed as having induced devaluation fears in the market, 7 in agreement with the pattern of coefficient f l ' . This finding suggests that the market was taking into account developments likely to affect the resolve of authorities to maintain the promised exchange rate targets against the Deutschemark. One further way to investigate empirically the learning process is to allow for a memory long enough to take into account a sufficient amount of past policy developments. To this end, equation (14) is specified for various lags m and estimates are obtained concerning the drachma weekly and monthly forward rates of Deutschemark and US dollar, respectively, using the daily data set and the estimation method described in Section I.B. Figures 3(a) and 3(b) display the estimates of coefficient /3m and the associated t-statistics vs the order of lag, m, for the Deutschemark rate of weekly and monthly maturities, respectively. In the case of the Deutschemark, both results suggest that coefficient /3" approaches unity from above and its significance increases with the order of lag, 8 thus confirming Proposition 2. 377

Efficiency testing revisited: N M Christodoulakis and S C Kalyvitis

1.9

1.7

1.6 1.5 1.4 1.3 1.2 1.1

E2 1.0 lo

. . . .

2o

z;o

WEEK I.... ~ t o Coefficient I FIGURE 2. Time-varying Beta coefficient in equation (14).

Figures 4(a) and 4(b) are for the US dollar rate. In the case of US dollar, the coefficient is approaching unity very slowly for both maturities. The statistical significance rises slowly with m in the monthly rate but more substantially in the weekly rate, especially after the tenth lag. Such a finding suggests that even after a long period of time, markets do not disregard the possibility of an

TABLE 2. List of events affecting L, Date

Event

E1

2 April

E2

4 June

E3

25-27 August

E4

20 September

Byelections in Athens are won by the opposition and shake Government prospects. The Danish referendum on June 2nd rejected the Maastricht Treaty. Nervousness in international markets that realignments in EMS are imminent. Polls suggest rejection of the Maastricht Treaty is likely in forthcoming French referendum, a Speculation increases sharply and foreign exchange market is suspended.

aRose and Svensson (1994, Section VII), and Eichengreen and Wyplosz (1993) for a description of those events and their effects in the exchange rate market. 378

Efficiency testing revisited: N M Christodoulakis and S C Kalyvitis

(a) 12.50

10.00 f

7.50

so • o0•

o•o0

5.00 o •

/

2.50

0.00

6

8

i0 [

i2

i4

i8

~a 50

22

~'4

LAG m Beto ..... tStot I

(b) 5.0 4.5"

j/

¢.0 3.5 3.0 2.5

/ oi S

2.01.51.0 LAG m Beta . . . . . tSt~ I FIGURE 3. (a) Beta coefficient and t-statistic on weekly Deutschemark. (b) Beta coefficient and t-statistic on weekly US dollar.

379

Efficiency testing revisited: N M Christodoulakis and S C Kalyvitis

(a) 15.0012.50 10.00 7.50 5.00 2.50 0.00

8

io

~.

~

1'6

~

2'0

22

2'4

LAG m

[

Bata . . . . . tSt.at I

(b) 9

8 7 6 5 43 2

4

6

a

io i2

~

is

ia 2'0 2~~'4

LAG m

FIGURE 4. (a) Beta coefficient and t-statistic on monthly Deutschemark. (b) Beta coefficient and t-statistic on monthly US dollar.

380

Efficiency testing reuisited: N M Christodoulakis and S C Kalyoitis

(a) 0.50

0.25

0.00 l

"

i.

-025

~ I

oS

t t

0 !

V

-0.50

-0.75 LAG m

(b) 0.000-0.025 •-0.050

- " ' " ...... oo

-0.075

/ a

--O.lOO



t -0.125

0

-0.150

2

4-

6

8

10

12

14-

16

18

20

22

24

LAG m

F]GURE 5. (a) Alpha coefficient on weekly and monthly Deutschemark. (b) Alpha coefficient on weekly and monthly US dollar.

381

Efficiency testing revisited: N M Christodoulakis and S C Kalyoitis

unexpected devaluation against the dollar. This should come as no surprise, given that monetary authorities in Greece were by no means committed to a dollar exchange rate target and, moreover, the uncertainty in the EMS was feeding widespread fears of a further rise of the US dollar. Figures 5(a) and 5(b) display coefficient a m in the cases of the Deutschemark and US dollar, respectively. Again with the exception of the first three lags in the Deutschemark case, the coefficients are found to be negative and approaching zero from below, in agreement with Proposition 3.

II.D. Forecastingperformance Given the empirical support found for the generalized model of equation (14), we move to compare its forecasting performance against the alternative random walk model (see Meese and Rogoff, 1983). The first-two thirds of the sample were used for an estimation of the model of equation (14) with a lag-length m chosen is such a way as to render coefficient ~m statistically equal to one. Then one-step-ahead daily forecasts are generated by continuously updating the sample and re-estimating the model as new observations are incorporated in each period. The results are displayed in Table 3 and it is clear that the devaluation model outperforms the random walk model for both currencies and both maturities, despite the fact that the period has been excessively turbulent. IH. Conclusions

The bulk of studies that so far have rejected the efficiency hypothesis in foreign exchange markets were concentrating on the simple efficiency hypothesis, namely that the forward exchange rate is an unbiased predictor of the spot TABLE 3. Forecasting performance of equation (4) Currency/Horizon

Criterion

Bayesian model

Random Walk

Score a

Deutschemark/Weekly

RMSE b MAE c RMSE MAE RMSE MAE RMSE MAE

0.0058 0.0035 0.0104 0.0081 0.0197 0.0119 0.0425 0.0330

0.0070 0.0042 0.0158 0.0118 0.0213 0.0151 0.0587 0.0472

1.21 1.20 1.52 1.46 1.08 1.27 1.38 1.43

Deutschemark/Monthly US dollar/Weekly US dollar/Monthly

aThe score is obtained as the ratio of errors found in the random walk and the Bayesian model, respectively. The higher the score, the more the Bayesian model outperforms the former. bRMSE denotes Root Mean Square Error. CMAE denotes Mean Absolute Error. 382

Efficiency testing revisite& N M Christodoulakis and S C Kalyoitis

exchange rate. In this formulation, expectations are identified with the forward exchange rate. Despite the fact that extraneous depreciation expectations and gradual learning about the regime of monetary policy are frequently mentioned as potential reasons for departure from efficiency, surprisingly little effort has been put towards either modelling how expectations are formed under such circumstances, or deriving implications for efficiency testing. The aim of this paper was to show that if agents form expectations using Bayesian revisions, then simple efficiency tests may reject the joint hypothesis of efficiency and no risk premia. In such a case, the correct specification would be to test the effÉciency equation for various lags in the depreciation and forward discount rates, in order to see whether a learning process is underway in the formation of expectations. Propositions regarding the pattern of parameters in the efficiency equation that incorporates such a learning mechanism are derived and tested empirically. The Bayesian model was tested by using data from the Greek forward market during the transition period from capital controls to a liberalized regime. We expect agents in a newly developed market to require a period of time to discern the credibility of monetary authorities. While simple efficiency was rejected by a battery of standard tests, the learning model was found to support the properties and pattern of parameters in the generalized efficiency equation. The model was estimated in two alternative ways: first, we obtained time-varying estimates of the coefficients and saw that they responded to the occurrence of events likely to affect exchange rate policy. Second, the model was estimated for different time windows to check whether the formation of expectations takes into account past exchange rate behavior and here we confirmed a pattern of coefficients consistent with the theoretical implications of the Bayesian model. Finally, the forecasting performance of this model was found to outperform the random walk model for both currencies (Deutschemark, US dollar) at both weekly and monthly horizons. Thus, the Bayesian model seems capable not only of explaining the empirical failure of the simple efficiency tests, but also of suggesting what kind of parameter estimates can be expected in markets with a learning process underway. A learning process can start by the initial lack of credibility of monetary authorities in defending announced target, in which case the market requires some time to be convinced about the true intentions of policy makers. The probability attached to a shift in exchange rate policy may lead to 'long swings' in the exchange rate, like those described by Engel and Hamilton (1990) for the Deutschemark/US dollar rate. In such cases the simple efficiency model is unworkable and should be augmented to include the gradual learning process as a plausible explanation for the long swings.

Data Appendix All daily G R D / D e u t s c h e m a r k and G R D / U S dollar spot and forward rates used in this study were obtained from the database of the Athens branch, Midland Bank. Forward rates of 1 week and 1 month maturity are matched with future spot rates by appropriately 383

Efficiency testing revisited: N M Christodoulakis and S C Kalyoitis counting value dates and taking into account the occurrence of holidays in Greece, Germany and the USA. The data sample covers the period January 1, 1992 to September 17, 1992. Due to the suspension of forward exchange transactions with maturity of less than 3 months imposed by the Bank of Greece in mid-September, and resultant lack of reliable and consistent data series afterwards, our sample could not be extended beyond this date. Notes 1. Ayuso et al. (1992) estimate equation (14) for m = k, though no theoretical justification is given for this choice. 2. This is a trivial result for the OLS estimator. Both properties survive also in the Generalized Method of Moments which is employed in Section II of the paper. This is because the GMM estimator is obtained through a linear transformation of the explanatory variables; see, for example, Newey (1985). 3. A detailed description of the events surrounding the period under consideration is available from the authors. 4. The results of stationarity tests are available upon request. 5. As the CUSUM test involves OLS estimates, it is not applicable to the full frequency of the sample due to the overlapping observations problem. Therefore, we had to create weekly series (covering the 37 weeks for the period January 1, 1992 to September 18, 1992 by choosing one observation (namely Thursday's) from each week and then estimating equation (4) using the 1-week forward rates. 6. The estimation method is the Kalman filter, as implemented in the REG-X package by Hall (1991). The first eight estimates are omitted to insure convergence of the parameter. A number of misspecification tests were carried out and the model appears to be well specified; further details are available from the authors. 7. To our knowledge, the only study that attempts to provide a systematic link between political news (in the form of political risk associated with the outcome of elections) and the forward bias is that of Bachman (1992). 8. Some observations corresponding to large and negative, though statistically insignificant, values of/3 for low lags m were dropped.

Unlinked reference Levin, 1989

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