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Friction model and foreign exchange market intervention Jongbyung Jun ⁎ Department of Economics, Suffolk University, Boston, MA 02108, United States Received 23 May 2006; received in revised form 23 January 2007; accepted 21 March 2007 Available online 4 April 2007

Abstract The friction model is consistent with the hypothesis that a central bank intervenes in a foreign exchange market only if the necessity grows beyond certain thresholds. For this feature, the model is adopted in some recent studies as an attractive central bank reaction function. However, with official data on Federal Reserve and Bundesbank intervention, this paper shows that the friction model's advantage relative to a linear model may be negligible in terms of RMSE and MAE of in-sample fitting and out-of-sample forecasts. The implication is that intervention decisions are at the monetary authorities' discretion rather than dictated by a rule. © 2007 Elsevier Inc. All rights reserved. JEL classification: C24; E58; F31 Keywords: Central bank intervention; Friction model; Exchange rates

1. Introduction Under a floating exchange rate system, an exchange rate is supposed to be determined by market forces. However, as the survey results in Neely (2001) indicate, many central banks occasionally intervene to counter excessive exchange rate fluctuations by buying or selling a currency against another. In order to investigate how a central bank determines the amount of intervention or to evaluate whether observed intervention is consistent with policy objectives, it is necessary to specify and estimate the central bank's reaction function. A big challenge in modeling a reaction function is the infrequency of such intervention. The amount of intervention is zero for the majority of the observations, particularly with daily data, while explanatory variables are nonzero. One way to proceed is to model the probability rather than the quantity of intervention using a probit approach as in Baillie and Osterberg (1997), or a logit approach as in Frenkel and Stadtmann (2001) and Frenkel, Pierdzioch, and Stadtmann (2005). If the interest lies in the quantity of intervention rather than the probability, one may rely on a Tobit model as in Humpage (1999) and Almekinders and Eijffinger (1994). However, a Tobit model may take either buying or selling intervention as the dependent variable but not both. The friction model of Rosett (1959), on the other hand, allows us to consider both types of intervention simultaneously in a single reaction function. In addition, the friction model is consistent with the reasonable assumption that while a central bank tends to abstain from intervention most of the time, it does intervene when the necessity grows beyond certain thresholds.

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Almekinders and Eijffinger (1996) adopt the friction model for the first time to estimate the reaction functions of U.S. and German central banks. The model is also adopted more recently by Neely (2002, 2006). While these studies and the original model of Rosett assume that the dependent variable is insensitive to small values of explanatory variables, Kim and Sheen (2002) consider a case where the insensitivity is to extreme values of explanatory variables which are harder to counter with limited foreign reserves. Given that the underlying assumption of the friction model is consistent with the observed infrequency of foreign exchange market intervention, it seems reasonable to expect the nonlinear model to perform better than a simple linear model as a central bank's reaction function. But then one might raise the question of how much the friction model is better and in what sense. This is not a trivial matter in that the advantage of the nonlinear model may be insignificant if the reaction function is subject to frequent policy shifts, or if strategic approaches are taken in choosing the appropriate timing, such as occasional leaning with the wind rather than against the wind, to maximize the effects with limited resources. There is also the possibility that quite often the intervention decisions are made at the monetary authorities' discretion rather than dictated by a rule. To my best knowledge, no attempt has been made in the previous literature to answer this question. In this paper, the friction model is compared with a linear model in terms of in-sample fitting and out-of-sample forecast of the amounts of intervention measured by RMSE (Root Mean Squared Error) and MAE (Mean Absolute Error), with daily data on actual intervention in the Deutsche mark–U.S. dollar market between February 1987 and January 1993. The main finding is that the friction model tends to have lower MAE but higher RMSE than the linear model both within and out of sample. The lower MAE is from the fact that the residuals or forecast errors of the model are smaller for the majority of the observations. The higher RMSE, however, indicates that the large-size errors of the friction model, e.g. largest 10% of the absolute errors, tend to be bigger than those of the linear model, and also the advantage in terms of MAE is not significant enough to offset the impacts of these large-size errors when the average is taken with squared errors. As it turns out, these large-size errors occur due to intervention when the necessity is less than the estimated thresholds or lack of intervention when the necessity exceeds the thresholds, which is the opposite of what the friction model predicts. These violations are too frequent to be treated as outliers. On the whole, the friction model fits the data slightly differently from the linear model, but not necessarily better especially when the amount of intervention is not zero. The rest of this paper is organized as follows. In Section 2, the two reaction functions are presented with details about the methods of estimation, forecasting and performance comparison. The data set is described in Section 3 and the empirical results are in Section 4. Section 5 offers some concluding remarks. 2. A linear model and a friction model A linear model of a central bank reaction function can be written as yt ¼ b0 þ xt bl þ ut ;

ð1Þ

where yt is the amount of daily intervention, xi is the vector of explanatory variables and ut is the error term. The amount of intervention is positive when a central bank purchases the U.S. dollar (USD), the numeraire currency in this paper, and negative when it sells the currency. A friction model can be written as y⁎ t ¼ xt b þ e t ; 8 þ < y⁎ t d yt ¼ 0 : ⁎ y t þ d

et jxt fN ð0; r2 Þ;

if if if

þ y⁎ t N d N 0; þ d V y⁎ t Vd ; ⁎ yt b d b 0:

ð2aÞ

ð2bÞ

The latent variable y⁎t , which is known to the central bank but unobservable to outsiders, measures the necessity of intervention or the amount of intervention when the central bank attempts to counter any market instability as in a fixed exchange rate system. However, as can be seen in Eq. (2b), the model assumes that actual intervention under a floating system takes place only if y⁎t is above the upper threshold (δ+) or below the lower threshold (− δ− ).

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In order to see which model is better as a central bank reaction function, the two models are compared in terms of insample fitting and out-of-sample forecast performance measured by RMSE and MAE. Given a sample of size T, the insample RMSE and MAE are computed as vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u T u1 X RMSE ¼ t ð3Þ ð yt yˆ t Þ2 T t¼1

MAE ¼

T 1X j yt yˆ t j T t¼1

ð4Þ

where yˆt is the in-sample fitted value. For the linear model, the fitted value is yˆt =βˆ 0 + xt βˆ l from Eq. (1), where the parameters can be estimated by OLS. For the friction model, yˆt is obtained by replacing the parameters in the conditional mean equation given below (see Appendix B for the derivation) with the estimates. Eð yt jxt Þ ¼ Pð yt N 0jxt ÞEð yt j yt N 0; xt Þ þ Pð yt b 0jxt ÞEð yt j yt b 0; xt Þ x t b dþ x t b dþ ¼ U ðxt b dþ Þ þ r/ r r xt b d xt b d U ðxt b d Þ þ r/ r r

ð5Þ

where ϕ(·) and Φ(·) are the PDF and CDF of the standard normal distribution. The parameters β, δ+, δ− and σ can be estimated by the method of maximum likelihood. The log-likelihood for observation t (see Appendix A for the derivation) is y t x t b þ dþ y t x t b d þ lt ðb; d ; d ; r; yt jxt Þ ¼ 1ð yt N 0Þd log / =r þ 1ð yt b 0Þd log / =r r r ð6Þ xt b þ dþ xt b d þ 1ð yt ¼ 0Þd log U U r r where 1(·) takes 1 if the expression inside is true and 0 otherwise. The likelihood function is notably characterized by its discontinuity at yt = 0, which reflects the fact that the model is deliberately designed for those data sets where yt = 0 more often than a continuous density function allows. Thus, even if the linear model is estimated by the ML method assuming a normal distribution for the errors, its likelihood is not directly comparable with that of the friction model. While βl in the linear model (1) represents the partial effects of each variable on the expected amount of intervention E( y|x), in the friction model (2a,b) β does not represent the partial effects on E( y|x). The partial effect of a variable xj can be obtained from Eq. (5) as AEð yjxÞ xb dþ xb d ¼ bj U þU ¼ bj ½Pð y N 0jxÞ þ Pð y b 0jxÞ: Axj r r

ð7Þ

Unlike in the linear model, this effect is not fixed but varying with the probability of intervention, P( y N 0| x) + P( y b 0| x) = P( y ≠ 0| x), which in turn depends on the values of x. Eq. (2b) indicates that the probability of intervention is close to zero when the values of x are small so that y⁎ stays between the two thresholds, but the probability grows with the values of x. As a corollary, this partial effect of x j, or equivalently the slope at a point on the nonlinear curve of E( y| x) given in Eq. (5), must be smaller than that of the linear model for small values of x but larger for sufficiently large values of x. That is, the nonlinear fitted line is closer to the horizontal axis than the linear fitted line for small x, but farther away for large x. This is in essential why the friction model is expected to fit or forecast both zero and nonzero amounts of intervention better than the linear model.

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The out-of-sample forecast performance of each model is measured in terms of RMSE and MAE of the forecast errors. Given the starting sample of size T0 for the estimation of the parameters, the k-step-ahead forecast error at time t is e½kt ¼

k 1X j ytþi yˆ tþi j; t ¼ T0 ; T0 þ 1; : : : ; T k; k i¼1

ð8Þ

where yˆt + i is the forecast value of intervention using the parameters estimated with t observations and xt + i. Note that e[k]t is the average of the absolute forecast errors for k days, which is more interesting to us than the forecast error for day t + k, yt + k − yˆt + k. For the sake of convenience, actual rather than forecast values of the explanatory variables xt + i are used to obtain yˆt + i provided that the explanatory variables are not the lags of the dependent variable. The corresponding RMSE and MAE are computed as vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u T k X u 1 RMSE½k ¼ t ðe½kt Þ2 T k T0 þ 1 t¼T0

MAE½k ¼

T k X 1 je½kt j: T k T0 þ 1 t¼T0

ð9Þ

ð10Þ

Since the linear model is estimated by OLS, which minimizes the sum of squared residuals, while the friction model is estimated by the method of maximum likelihood, the linear model has some advantage in terms of RMSE. However, it should be noted that although RMSE of MLE will never be smaller than that of OLS when the estimated models are the same, this is not necessarily the case when the models are different, in particular when the model of MLE is closer to the data generating process. On the other hand, it is also noteworthy that the friction model MLE has some advantage in terms of MAE. As explained above, the friction model MLE fits the days with no intervention better than the linear model if the value of x is relatively small. If, in addition, the majority of the observations in the data have y = 0, the friction model MLE may have smaller average size of errors (lower MAE) even if its fitting or forecasting on the days with y ≠ 0 is worse than the linear model OLS. Furthermore, if the sample contains significant number of observations inconsistent with the friction model, i.e. intervention with relatively small x or lack of intervention with large x, then the large-size errors of the friction model can be larger than those of the linear model, which may result in higher RMSE unless the advantage in MAE is large enough to offset the effect of those large-size errors. Therefore, if the friction model is indeed a significantly better specification, then it should beat the linear model in terms of both RMSE and MAE, within and out of sample. 3. The data The data set used for this study is the same as in Baillie and Osterberg (2000), which includes observations on the daily exchange rate of German mark per U.S. dollar (DM/USD) and the official data on the actual amount of daily intervention by the Federal Reserve and the German Bundesbank during 1/5/1987–1/22/1993. The exchange rates are observed at 9:30 AM in Paris and originally provided by Olsen and Associates of Zurich, Switzerland. Fig. 1 depicts the exchange rate movements and the amounts of intervention by the two central banks during the sample period. The central banks tend to intervene more frequently during periods of rapid appreciation or depreciation as expected. However, there are substantial variations in the direction and frequency of intervention over time. Most of the intervention operations between 1987 and mid-1988 are buying USD, but then mostly selling operations until early 1991. Also, the frequency of intervention dramatically drops since early 1990. Excluding holidays and weekends, the total available number of observations is 1464 business days. However, following Almekinders and Eijffinger (1996), the empirical part of this paper focuses on the period following the Louvre Accord, which starts from 2/23/1987 and thus has slightly less observations of 1430 days. The total number of days with the Federal Reserve intervention is 174 (65 days of buying U.S. dollar intervention and 109 days of selling

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481

Fig. 1. DM/USD exchange rate and central bank intervention.

intervention). The mean size of intervention is USD 124 million for buying intervention and USD 161 million for selling intervention. Bundesbank's intervention is a bit more frequent with total 205 days (50 buying and 155 selling days), and mean sizes of USD 157 million and USD 160 million, respectively. 4. Empirical results 4.1. Estimation of the reaction functions Table 1 reports the estimation results for the reaction functions of the two central banks with the same explanatory variables and sample period (2/23/1987–10/31/1989) as in Almekinders and Eijffinger (1996). The first explanatory variable (devt−1) is the percentage deviation of the exchange rate from its 7-day moving average. The moving average is a proxy for the central bank's time-varying target level of the exchange rate. The second explanatory variable (volt−1) is a measure of the volatility of the exchange rate estimated with a GARCH(1,1)

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Table 1 Reaction functions (2/23/87–10/31/89) Federal Reserve

Constant devt−1 volt−1

Bundesbank

Linear model

Friction model

Linear model

Friction model

OLS

MLE

OLS

MLE

− 16.86⁎⁎ (− 5.13) − 20.47⁎⁎ (− 7.07) − 48.58⁎⁎ (− 7.99)

δ+ − δ− σ

83.62

Q(10) Q(20) Observations RMSE MAE

113.66⁎⁎ 127.50⁎⁎ 651 83.42 46.30

− 89.88⁎⁎ (− 6.27) − 308.35⁎⁎ (− 9.41) 513.66⁎⁎ (11.05) − 347.62⁎⁎ (− 10.28) 255.03⁎⁎ (12.01)

651 85.85 43.95

− 18.20⁎⁎ (−4.21) − 30.16⁎⁎ (−7.92) − 46.74⁎⁎ (−5.85)

109.97 241.87⁎⁎ 291.35⁎⁎ 651 109.72 58.91

−113.07⁎⁎ (− 6.88) −219.84⁎⁎ (− 7.35) 502.39⁎⁎ (10.83) −303.01⁎⁎ (− 9.95) 282.64⁎⁎ (12.57)

651 112.35 58.20

In parentheses are the t-statistics and ⁎⁎ denotes significance at 1% level. Q(n) is the Ljung–Box statistic under the null hypothesis of no serial correlation in the residuals.

model for the log-return of the exchange rate. Since the estimated conditional variance is always positive while the dependent variable may be negative, a sign is assigned such that the variable is positive (negative) on days when the exchange rate is greater than or equal to 1.8255 (less than 1.8255), which is the proxy for the equilibrium level of the Louvre Accord. Note that both explanatory variables are lagged by 1 day so that they are uncorrelated with the regression error at time t. The two explanatory variables are highly significant as indicated by the large t-statistics, regardless of the model or the central bank considered. The t-statistics are computed with heteroscedasticity-robust standard errors (with White's correction for the linear model, and QML standard errors for the friction model). The estimated coefficients have negative signs as expected, which indicates that on average the banks ‘lean against the wind’. The results are also quite similar to those of Almekinders and Eijffinger (1996, Table 2). 4.2. Comparison of in-sample fitting Table 1 also presents RMSE and MAE of the two models on the bottom two rows. With the Federal Reserve intervention, the linear model has slightly lower RMSE (83.42 versus 85.85) while the friction model has slightly lower MAE (43.95 versus 46.30). The same pattern is repeated with the Bundesbank intervention. In order to obtain a better understanding of this result, the fitted values of the two models together with the actual amounts of Federal Reserve intervention are depicted in Fig. 2 against the values of xβˆ , which is an estimate of E( y⁎|x) and thus can be interpreted as a measure of the necessity of intervention. βˆ is the friction model MLE for β in Eqs. (2a,b). Note that the actual amounts marked by the symbol ‘x’ are mostly zero (514 out of 651) and thus fall on the horizontal axis with high density. As a result, the estimated thresholds (− 347.62 and 513.66) are quite large and 95% of the observations are between the two thresholds. When xβˆ lies inside the band formed by the two vertical lines, which is smaller than the band of the thresholds, the curve of friction model is slightly closer to most of the points with y = 0 but farther away from those with y ≠ 0 than the fitted line of the linear model. Outside the band, in contrast, the linear model fits zero amounts better while the friction model fits large amounts of intervention better, especially on the left-hand side of the band. Inside the band, y = 0 for 461 observations out of 549 and the residuals of the friction model are smaller for most of them (410 out of 461). Thus, the model has lower MAE within the band than the linear model (31.96 versus 36.89) and

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also marginally lower RMSE (69.58 versus 69.63) even though the residuals of the friction model are larger for the rest 88 observations with y ≠ 0. Outside the band, intervention is more frequent (49 out of 102) and the friction model fits the points with intervention slightly better than the linear model. However, the linear model fits the points of no intervention better in these outer regions and consequently has lower MAE (96.93 versus 108.48) and RMSE (135.37 versus 144.87) than the friction model. On the whole, Table 1 and Fig. 2 provide little evidence that the friction model has a significantly better explanatory power than the linear model. If anything, both models are biased toward the majority of the observations ( y = 0) and thus the days with intervention are poorly fitted. For the friction model to clearly dominate the linear model, the sample must have less points of y ≠ 0 inside the band and less points of y = 0 outside the band. The Ljung–Box Q statistics reported in Table 1 suggest that the errors of the linear model are serially correlated. When both models are re-estimated with lagged dependent variables to remove serial correlations, yt−1, yt−3, yt−5, yt−7, yt−9 for Federal Reserve and yt−1, yt−3, yt−4, yt−6, yt−10 for Bundesbank, similar pattern is still maintained and the results are not reported to save space. In order to see whether this result is due to structural breaks in intervention policy, a series of rolling regressions are run starting with the period from the Louvre Accord to the end of 1987 and then adding one more observation to each subsequent regression. The top row of Fig. 3, labeled as ‘Case 1’, depicts RMSE and MAE obtained from these rolling regressions with the data on Federal Reserve intervention. Until early 1989, about 2 years from the Louvre Accord, the friction model has slightly lower RMSE and MAE. This is not necessarily because the friction model performs better in these sub-periods, but rather because the number of intervention days is too small and thus the effects of large-size errors with y ≠ 0 are small relative to the outnumbering small-size errors with y = 0. As more observations are added, the friction model fails to maintain its advantage in RMSE although still better in MAE. One explanation for the decreasing relative explanatory power of the friction model is that the Federal Reserve might have changed its policy over time, presumably because the routinely sterilized intervention is criticized for being ineffective, so that intervention is gradually aimed at longer-run stability of the exchange rate with increased thresholds. To investigate this possibility, another set of rolling regressions is run with the deviation from a short-run target (dev [short]t−1) and the deviation from a long-run target (dev[ long ]t−1) as the two alternative explanatory variables.

Fig. 2. Actual and fitted amounts of intervention (Federal Reserve, 2/23/87–10/31/89). yˆ L is the fitted line of the linear model and ˆy F is the fitted line of the friction model. The horizontal axis measures xβˆ of friction model.

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Table 2 Out-of-sample forecast performance (2/23/87–1/22/93) Forecast horizon

Federal Reserve

Bundesbank

RMSE Linear

MAE

RMSE

MAE

Friction

Linear

Friction

Linear

Friction

Linear

Friction

67.8 52.8 44.0

32.3 32.4 32.5

31.1 31.3 31.2

84.5 66.0 59.0

88.5 69.3 60.3

43.0 43.3 43.5

40.0 40.3 40.3

37.6 37.8 38.3

33.2 33.5 34.9

80.7 65.0 59.1

88.2 74.0 68.9

45.4 45.7 45.9

44.9 45.4 47.0

Case 3: xt = (dev[short]t−1, dev[long]t−1, lags of yt ) 58.5 64.1 28.5 1 day 1 week 48.8 56.5 32.0 1 month 45.0 151.4 35.5

26.7 28.9 41.3

78.8 63.1 57.8

83.7 70.3 86.3

37.4 40.4 42.8

35.0 36.9 44.4

Case 1: xt = (dev[7]t−1, volt−1 ) 1 day 63.2 1 week 49.2 42.4 1 month

Case 2: xt = (dev[short]t−1, dev[long]t−1 ) 1 day 64.2 65.2 1 week 52.2 52.2 1 month 46.9 47.4

Each RMSE/MAE is computed with 1200 forecast errors over 1988–1992, using observations of 1987 as the starting sample for parameter estimation. Underlined is the smaller of the two values compared between the linear model and the friction model.

Rather than fixing the targets for the entire sample, the order of moving average that attains the lowest level of RMSE is chosen for the friction model in each regression while the one that attains the lowest MAE is chosen for the linear model. Moving averages for 1 week and 1 month are considered for short-term targets, and 0.5, 1, 1.5 and 2 years for long-term targets. To avoid losing too many degrees of freedom for moving averages, the target exchange rate for the first m observations is assumed to be the Louvre Accord equilibrium level of 1.8255. Consequently, 8 if t V m < 100½logðSt Þ logð1:8255Þ m dev½mt ¼ 100½logðS Þ logðm1 X S Þ if t N m t ti : i¼1

where St is the spot exchange rate on day t. Case 2 in Fig. 3 depicts the RMSE and MAE obtained with the two alternative regressors. The friction model outperforms the linear model in terms of both RMSE and MAE for the entire sample period. Interestingly, however, when lagged dependent variables are added as regressors, the dominant performance of the friction model in terms of RMSE is again limited to the periods before early 1989, as depicted in Case 3. Although these lags are not really helpful in explaining why the central banks intervene, they seem to be useful in minimizing nonlinearity. With the data on the Bundesbank intervention, very similar results are obtained and thus not reported. 4.3. Comparison of out-of-sample forecasting performance RMSE and MAE obtained from the out-of-sample forecast errors, using Eqs. (9) and (10), are reported in Table 2. The forecast horizons considered are 1 day, 1 week and 1 month (k = 1, 5, 20). Out of the entire sample (2/23/1997–1/ 22/1993), observations of 1987 are used as the starting sample to estimate the parameters, and the last 20 observations (the largest value of k) are set aside for the forecast and thus not used in the parameter estimation. As a result, the RMSE and MAE for each forecast horizon of each model are computed with 1200 forecast errors. In Table 2, the out-of-sample RMSE and MAE of the two models are presented in columns 2 through 5 for the Federal Reserve intervention, and in the last four columns for the Bundesbank intervention. The rows of the table are grouped into the three cases as in Fig. 3, and each case consists of the three forecast horizons. The first row of Case 1 shows that the RMSE of the 1-day ahead forecasts for Federal Reserve intervention is smaller with the linear model (63.2) than with the friction model (67.8), while the corresponding MAE is smaller with the friction model (31.1) than with the linear model (32.3).

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Fig. 3. In-sample fitting from rolling regressions (Federal Reserve). The variable dev[m]t−1 measures the deviation of DM/USD rate on day t − 1 from previous m-day moving average. In Case 2 and 3, m is chosen to minimize RMSE for friction model and MAE for linear model in each regression. m = 5, 20 are considered for short-run targets, and m = 120, 240, 360, 480 for long-run targets.

By comparing the entire column 2 with column 3, one can see that the linear model has lower RMSE for the forecasts of Federal Reserve intervention regardless of the forecast horizons and the included explanatory variables. The same is also true for Bundesbank intervention as shown in columns 6 and 7. It is noticeable that the linear model beats the friction model in RMSE terms even in Case 2, which is in sharp contrast to the comparison result in in-sample

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Table 3 Forecasting performance in sub-periods Forecast horizon

Federal Reserve

Bundesbank

Case2 RMSE

Case 3

Case 2

Case 3

MAE

RMSE

MAE

RMSE

MAE

RMSE

MAE

I. First half of 1989 (frequent intervention) 1 day 1.09 1.10 1 week 1.22 1.08 1 month 1.51 1.11

1.20 3.00 108.72

1.11 1.15 2.58

1.17 2.09 1.85

1.05 1.05 1.04

1.14 2.15 2.97

1.05 1.05 1.12

0.70 0.37 0.14

0.60 0.47 0.37

0.68 0.42 0.61

0.69 0.68 0.71

0.48 0.14 0.17

0.50 0.44 0.40

II. First half of 1992 (no intervention) 1 day 0.90 0.81 1 week 0.98 0.81 1 month 0.96 0.84

This table shows either friction-model RMSE divided by linear-model RMSE, or friction-model MAE divided by linear-model MAE. If a value is greater than 1, it means that the friction model’s forecast errors have a larger RMSE or MAE than the linear model.

RMSE terms. In terms of MAE of the forecast errors, shown in columns 4 and 5 for Fed Reserve and in columns 8 and 9 for Bundesbank, the friction model is better but with a few exceptions. Except the RMSE for 1-month horizon in Case 3, what is remarkable in Table 2 is not the differences but the similarity of the values for the two models. Like the comparison of in-sample fitting, the comparison of out-of-sample forecasting performance fails to provide evidence that the friction model is significantly better than the linear model. While the values of RMSE and MAE in Table 2 are computed for the whole forecast period of 1988–1992, two subperiods are considered in Table 3; one with frequent intervention (first 6 months of 1989) and the other with no intervention (first 6 months of 1992). Also, instead of the values of RMSE and MAE of each model, the table contains quotients for easier comparison, i.e. each value of the friction model is divided by the corresponding value of the linear model such that the quotient is less (greater) than unity when the friction model has lower (higher) RMSE or MAE. In the first three rows of the table, all the quotients are greater than unity regardless of the bank, the explanatory variables or the forecast horizons, which indicates that the friction model's performance is worse in terms of MAE as well as RMSE in this forecast period. Note that while the relative performance of the friction model is not very sensitive to the forecast horizon in Case 2, it gets worse in Case 3 as the forecast horizon increases, in particular with the Federal Reserve intervention. On the other hand, the last three rows of the table, where all the quotients are less than unity, shows that the friction model clearly outperforms the linear model if there is no intervention during the forecast period. These results suggest that the relative performance of the friction model may get worse as intervention becomes more frequent during the forecast period. By referring to Fig. 2, we can see that this will be the case if more points with y ≠ 0 are added inside the band rather than outside the band. 5. Concluding remarks For a central bank reaction function under a floating exchange rate system, the friction model of Rosett (1959) has an attractive feature and thus in the previous literature it is considered to be a better specification than a simple linear model. In this paper, as a test for this conjecture, the friction model is compared with a linear model in terms of insample fitting and out-of-sample forecast performance, with actual data on daily intervention in the DM/USD market for about 6 years following the Louvre Accord (February 1987). The performance of each model is measured by RMSE and MAE of the residuals and the forecast errors, and it is analytically shown that the friction model must beat the linear model in both terms if it is indeed a better specification. From the comparison of in-sample fitting, it is found that the friction model tends to have lower MAE because it fits the days without intervention, the majority of the observations, slightly better than the linear model. The nonlinear reaction function also has lower RMSE when each model is allowed to choose the most favorable target exchange rate for different sample periods. However, when the lagged values of intervention are added as regressors to take care of serial correlation, the friction model's advantage becomes quite insignificant to the extent that the linear model performs better in terms of RMSE as the sample period is extended beyond about 2 years from the Louvre Accord.

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When the comparison is about the out-of-sample forecast performance, the linear model shows lower RMSE regardless of the chosen bank, explanatory variables and forecast horizons. The friction model maintains its advantage in terms of MAE but with exceptions. When the forecast errors are compared for sub-periods, it is also revealed that the friction model is dominated in both RMSE and MAE terms during a sub-period of active intervention although the opposite is true for a period with no intervention. Admittedly, the comparison could have been more comprehensive, especially with respect to the types of explanatory variables and the methods of forecast. However, the empirical results above seem to be sufficient to demonstrate that the friction model may not perform significantly better than a linear model if the decision-making process of intervention is much more complicated than the one implied by the friction model. What remains to be seen is whether the results can be overturned in future studies through the identification of certain explanatory variables that are more consistent with the friction model's assumptions. Acknowledgements I am grateful to Richard Baillie for his valuable advice and also for providing the data, and to Jeffrey Wooldridge, Christine Amsler and two anonymous referees for their helpful comments. All errors are mine. Appendix A. Log likelihood of the friction model The following shows the steps to derive Eq. (6) in the main text. By removing yt⁎ in Eq. (2b) using Eq. (2a), xt b dþ þ et if yt N 0; yt ¼ ðA1Þ xt b þ d þ et if yt b 0: With the normality assumption for the errors, εt|xt ∼ N(0,σ2), the conditional distribution of yt is given as N ðxt b dþ ; r2 Þ if yt N 0; yt jxt f N ðxt b þ d ; r2 Þ if yt b 0: Let Lt(θ) ≡ Lt(θ;yt|xt) be the likelihood function for observation t. From Eq. (A2), /ðð yt xt b þ dþ Þ=rÞ=r if yt N 0; Lt ðhÞ ¼ /ðð yt xt b d Þ=rÞ=r if yt b 0;

ðA2Þ

ðA3Þ

where ϕ(·) is the standard normal probability density function (PDF). In the case of yt = 0, Lt ðhÞ ¼ Pð yt ¼ 0jxt Þ ¼ 1 Pð yt N 0jxt Þ Pð yt b 0jxt Þ:

ðA4Þ

The relationship among P ( yt = 0|xt), P( yt N 0|xt) and P( yt b 0|xt) is illustrated in Fig. A1 assuming β b 0. In this figure, P ( yt = 0|xt) is the shaded area under the PDF of N(xt β,σ2). Note that P( yt N 0|xt) is larger than P( yt b 0|xt) when xt b 0. Also note that P ( yt b 0|xt) ≠ 0 even if xt b 0. From Eq. (A1), Pð yt N 0jxt Þ ¼ Pðxt b dþ þ et N 0jxt Þ ¼ Pðet N xt b þ dþ jxt Þ ¼ 1 Pðet V xt b þ dþ jxt Þ ¼ 1 Uððxt b þ dþ Þ=rÞ;

ðA5Þ

Pð yt b 0jxt Þ ¼ Pðxt b þ d þ et b 0jxt Þ ¼ Pðet b xt b d jxt Þ ¼ Uððxt b d Þ=rÞ;

ðA6Þ

where ϕ(·) is the standard normal cumulative density function (CDF). By plugging Eqs. (A5) and (A6) into Eq. (A4) and rearranging, Lt ðhÞ ¼ Uððxt b þ dþ Þ=rÞ Uððxt b d Þ=rÞ

if yt ¼ 0:

ðA7Þ

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J. Jun / International Review of Economics and Finance 17 (2008) 477–489

Fig. A1. Conditional distribution of y⁎ and relative probability of intervention. Assuming that y⁎ ∼ N(xβ,σ2), this figure demonstrates how the conditional probabilities of buying intervention P( y N 0 | x), selling intervention P( y b 0 | x) and no intervention P( y = 0 | x) are determined given x and the two thresholds δ+ and − δ−.

Using Eqs. (A3) and (A7), the log-likelihood for observation t is obtained as lt ðhÞulog½Lt ðhÞ ¼ 1ð yt N 0Þd log½/ðð yt xt b þ dþ Þ=rÞ=r þ 1ð yt b 0Þd log½/ðð yt xt b d Þ=rÞ=r þ 1ð yt ¼ 0Þd log½Uððxt b þ dþ Þ=rÞ Uðxt b d Þ=r

ðA8Þ

where 1(·) is the indicator function. Appendix B. Conditional mean of y in the friction model This appendix shows the steps to derive Eq. (5) in the main text. Since yt may be positive, negative or zero, E( y|x) is given, after dropping the time subscript for notational simplicity, as Eð yjxÞ ¼ Pð y N 0jxÞEð yjy N 0; xÞ þ Pð y b 0jxÞEð yjy b 0; xÞ:

ðB1Þ

The two conditional probabilities in this equation, P( y N 0|x) and P( y b 0|x), are given in Eqs. (A5) and (A6), respectively. The two expectation terms can be derived from Eq. (A1) with the normality assumption for the errors. From the first line of Eq. (A1), Eð yj y N 0; xÞ ¼ Eðxb dþ þ eje N xb þ dþ ; xÞ ¼ xb dþ þ Eðeje N xb þ dþ ; xÞ ¼ xb dþ þ rEððe=rÞjðe=rÞ N ðxb þ dþ Þ=r; xÞ ¼ xb dx þ r/ððxb þ dþ Þ=rÞ½1 Uððxb þ dþ Þ=rÞ

ðB2Þ

where the last equality follows from the fact that (ε / σ)| x ∼ N(0,1), and E(z|z N c) = ϕ(c) / [1 − Φ(c)] when z ∼ N(0,1). Similarly from the second line of Eq. (A1), Eð yj y b 0; xÞ ¼ Eðxb þ d þ eje b xb d ; xÞ ¼ xb þ d þ Eðeje b xb d ; xÞ ¼ xb þ d þ rEððe=rÞjðe=rÞ b ðxb d Þ=r; xÞ ¼ xb þ d rEððe=rÞjðe=rÞ N ðxb þ d Þ=r; xÞ ¼ xb þ d r/ððxb þ d Þ=rÞ½1 Uððxb þ d Þ=rÞ:

ðB3Þ

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After plugging Eqs. (A5), (A6), (B2) and (B3) into Eq. (B1) and rearranging, we get Eð yjxÞ ¼ ½Uððxb dþ Þ=rÞðxb dþ Þ þ r/ððxb dþ Þ=rÞ ½Uððxb d Þ=rÞðxb d Þ þ r/ððxb d Þ=rÞ

ðB4Þ

where two properties of the standard normal distribution due to its symmetry, i.e. ϕ(c) = ϕ(− c) and Φ(c) = 1 − Φ(− c), are used to simplify the equation. References Almekinders, G.J., & Eijffinger, S.C.W. (1994). The ineffectiveness of central bank intervention. In S.C.W. Eijffinger (ed.), 1998, Foreign Exchange Intervention: Objectives and Effectiveness. (Cheltenham and Northampton: Edward Elgar Publishing Limited). Almekinders, G. J., & Eijffinger, S. C. W. (1996). A friction model of daily Bundesbank and Federal Reserve intervention. Journal of Banking and Finance, 20, 1365−1380. Baillie, R. T., & Osterberg, W. P. (1997). Why do central banks intervene? Journal of International Money and Finance, 16, 909−919. Baillie, R. T., & Osterberg, W. P. (2000). Deviations from daily uncovered interest rate parity and the role of intervention. Journal of International Financial Markets, Institutions, and Money, 10, 363−379. Frenkel, M., & Stadtmann, G. (2001). Intervention reaction functions in the dollar–deutschmark market. Financial Markets and Portfolio Management, 15, 328−343. Frenkel, M., Pierdzioch, C., & Stadtmann, G. (2005). Japanese and U.S. interventions in the yen/U.S. dollar market: estimating the monetary authorities reaction functions. The Quarterly Review of Economics and Finance, 45, 680−698. Humpage, O. F. (1999). U.S. intervention: Assessing the probability of success. Journal of Money, Credit and Banking, 31, 731−746. Kim, S. -J., & Sheen, J. (2002). The determinants of foreign exchange intervention by central banks: Evidence from Australia. Journal of International Money and Finance, 21, 619−649. Neely, C. J. (2001). The practice of central bank intervention: Looking under the hood. Federal Reserve Bank of St. Louis Review, 83, 1−10. Neely, C. J. (2002). The temporal pattern of trading rule returns and exchange rate intervention: Intervention does not generate technical trading profits. Journal of International Economics, 58, 211−232. Neely, C. J. (2006). Identifying the effects of U.S. intervention on the levels of exchange rates. Working Paper 2005-031C : Federal Reserve Bank of St. Louis. Rosett, R. N. (1959). A statistical model of friction in economics. Econometrica, 27, 263−267.