Forecasting Exchange Rates 1

CHAPTER V One of the goals of studying the behavior of exchange rates is to be able to forecast exchange rates. Chapters III and IV introduced the main theories used to explain the movement of exchange rates. These theories fail to provide a good approximation to the behavior of exchange rates. Forecasting exchange rates, therefore, seems to be a difficult task. This chapter analyzes and evaluates the different methods used to forecast exchange rates. This chapter closes with a discussion of exchange rate volatility. I. Forecasting Exchange Rates International transactions are usually settled in the near future. Exchange rate forecasts are necessary to evaluate the foreign denominated cash flows involved in international transactions. Thus, exchange rate forecasting is very important to evaluate the benefits and risks attached to the international business environment. A forecast represents an expectation about a future value or values of a variable. The expectation is constructed using an information set selected by the forecaster. Based on the information set used by the forecaster, there are two pure approaches to forecasting foreign exchange rates: (1) The fundamental approach. (2) The technical approach. 1.A Fundamental Approach The fundamental approach is based on a wide range of data regarded as fundamental economic variables that determine exchange rates. These fundamental economic variables are taken from economic models. Usually included variables are GNP, consumption, trade balance, inflation rates, interest rates, unemployment, productivity indexes, etc. In general, the fundamental forecast is based on structural (equilibrium) models. These structural models are then modified to take into account statistical characteristics of the data and the experience of the forecasters. It is a mixture of art and science. (See Appendix IV.) Practitioners use structural model to generate equilibrium exchange rates. The equilibrium exchange rates can be used for projections or to generate trading signals. A trading signal can be generated every time there is a significant difference between the model-based expected or forecasted exchange rate and the exchange rate observed in the market. If there is a significant difference between the expected foreign exchange rate and the actual rate, the practitioner should decide if the difference is due to a mispricing or a heightened risk premium. If the practitioner decides the difference is due to mispricing, then a buy or sell signal is generated. V.1 1.A.1 Fundamental Approach: Forecasting at Work The fundamental approach starts with a model, which produces a forecasting equation. This model can be based on theory, say PPP, a combination of theories or on the ad-hoc experience of a practitioner. Based on this first step, a forecaster collects data to estimate the forecasting equation. The estimated forecasting equation will be evaluated using different statistics or measures. If the forecaster is happy with the model, she will move to the next step, the generation of forecasts. The final step is the evaluation of the forecast. As mentioned above, a forecast represents an expectation about a future value or values of a variable. In this chapter, we will forecast a future value of the exchange rate, 𝑆 . The expectation is constructed using an information set selected by the forecaster. The information set should be available at time t. The notation used for forecasts of 𝑆 is: Et[𝑆 ], where Et[.] represent an expectation taken at time t. Each forecast has an associated forecasting error, t+T. We will define the forecasting error as: 𝜀 =𝑆 - Et[𝑆 ] The forecasting error will be used to judge the quality of the forecasts. A typical metric used for this purpose is the Mean Square Error or MSE. For 𝑇 =1, the MSE is defined as: MSE = [(𝜀 )2 + (𝜀 )2 + (𝜀 )2 + ... + (𝜀 )2]/Q, where Q is the number of forecasts. We will say that the higher the MSE, the less accurate the forecasting model. There are two kinds of forecasts: in-sample and out-of-sample. The first type of forecasts works within the sample at hand, while the latter works outside the sample. In-sample forecasting does not attempt to forecast the future path of one or several economic variables. In-sample forecasting uses today's information to forecast what today's spot rates should be. That is, we generate a forecast within the sample (in-sample). The fitted values estimated in a regression are in-sample forecasts. The corresponding forecast errors are called residuals or in-sample forecasting errors. On the other hand, out-of-sample forecasting attempts to use today’s information to forecast the future behavior of exchange rates. That is, we forecast the path of exchange rates outside of our sample. In general, at time t, it is very unlikely that we know the inflation rate for time t+1. That is, in order to generate out-of-sample forecasts, it will be necessary to make some assumptions about the future behavior of the fundamental variables. V.2 Summary: Fundamental Forecasting Steps (1) Selection of Model (for example, PPP model) used to generate the forecasts. (2) Collection of St, Xt (in the case of PPP, exchange rates and CPI data needed.) (3) Estimation of model, if needed (regression, other methods) (4) Generation of forecasts based on estimated model. Assumptions about Xt+T may be needed. (5) Evaluation. Forecasts are evaluated. If forecasts are very bad, model must be changed. Example V.1: In-sample Forecasting Exchange Rates with PPP Suppose you work for a U.S. firm. You are given the following quarterly CPI series in the U.S. and in the U.K. from 2008:1 to 2009:3. The exchange rate in 2008:1 is equal to 1.9754 USD/GBP. You believe that this exchange rate, 1.5262 USD/GBP, is an equilibrium rate. Your job is to generate equilibrium exchange rates using PPP. In order to do this, you do quarterly in-sample forecasts of the USD/GBP exchange rate using relative PPP. That is, Et[𝑠 ]=𝑠 = 𝑺𝒕 –1 𝐼 , 𝐼 , The forecasted level of the exchange rate USD/GBP for next period is given by Et[St+1]=SFt+1=: Et[𝑆 ] = 𝑺𝒕 * [1 + 𝑠 ] = 𝑺𝒕 * [1 + (𝐼 , – 𝐼 , )] Date 2008.1 2008.2 2008.3 2008.4 2009.1 2009.2 2009.3 CPI U.S. 108.6 111.0 112.3 109.1 108.6 109.7 110.5 CPI U.K. 106.2 108.2 109.3 108.4 106.1 106.9 107.8 Inflation U.S. (IUS) Inflation U.K. (IUK) - - 0.0221 0.0117 -0.0285 -0.0046 0.0101 0.0073 In-Sample Forecast (SFt+1) 0.019091 0.009813 -0.00795 -0.02137 0.007279 0.009033 1.9813 1.9951 1.7341 1.4619 1.4422 1.6452 Actual (St) 1.9754 1.9914 1.7705 1.4378 1.4381 1.6481 1.5990 Forecast Error t+1=SFt+1-St+1 -0.0100 0.2246 0.2964 0.0237 -0.2059 0.0463 Some calculations for SF2008:2 and SF2008:3: 1. Forecast SF2008:2. IUS,2008:2 = (USCPI2008:2/USCPI2008:1) - 1 = (111.0/108.6) - 1 = 0.0221. IUK,2008:2 = (UKCPI2008:2/UKCPI2008:1) - 1 = (108.2/106.2) - 1 = 0.0191. sF2008:2 = IUS,2008:2 - IUK,2008:2 = 0.0221 - 0.0191 = 0.0030. SF2008:2 = SF2008:1 x [1 + sF2008:2] = 1.9754 USD/GBP x [1 + (0.0030)] = 1.9813 USD/GBP. 2008:2 = SF2008:2-S2008:2 = 1.9813 – 1.9914 = -0.01. 2. Forecast SF2008:3. SF2008:3 = S2008:2 x [1 + sF2008:3] = 1.9914 USD/GBP x [1 + (0.0019)] = 1.9951 USD/GBP. 2008:3 = SF2008:3-S2008:3 = 1.9951 – 1.7705 = 0.2246. 3. Evaluation of forecasts. MSE: [(-0.01)2 + (0.2246)2 + (0.2964)2 + .... + (0.0463)2]/6 = 0.0306 Now, you can generate trading signals. According to this PPP model, the equilibrium exchange rate in 2008:2 V.3 should be 1.9813 USD/GBP. The market price, however, is 1.9914 USD/GBP. That is, the market is valuing the GBP higher than your fundamental model. Suppose you believe that the difference (1.9813-1.9914) is due to miss-pricing factors, then you will generate a sell GBP signal. ¶ In general, practitioners will divide the sample in two parts: a longer sample (estimation period) and a shorter sample (validation period). The estimation period is used to select the model and to estimate its parameters. Suppose we are interested in one-step-ahead forecasts. The one-step-ahead forecasts made in this period are in-sample forecasts, not “true forecasts.” These one-step-ahead forecasts are just fitted values. The corresponding forecast errors are called residuals. The data in the validation period are not used during model and parameter estimation. One-stepahead forecasts made in this period are “true forecasts,” often called backtests. These true forecasts and their error statistics are representative of errors that will be made in forecasting the future. A forecaster will use the results from this validation step to decide if the selected model can be used to generate outside the sample forecasts. Figure V.1 shows a typical partition of the sample. Suppose that today is March 2015 and a forecaster wants to generate monthly forecasts until January 2016. The estimation period covers from February 1978 to December 2009. Different models are estimated using this sample. Based on some statistical measures, the best model is selected. The validation period covers from January 2010 to March 2015. This period is used to check the forecasting performance of the model. If the forecaster is happy with the performance of the forecasts during the validation period, then the forecaster will use the selected model to generate out-of-sample forecasts. Figure V.1: Estimation, Validation & Out-of-sample Periods. Example V.2: Out-of-sample Forecasting Exchange Rates with PPP Go back to Example V.1. Now, you want to generate out-of-sample forecasts. You need to make some assumptions about the future behavior of the inflation rate. (A) Naive assumption: Et[It+1] = IFt+1 = It. V.4 You can generate out-of-sample forecasting by assuming that today's inflation is the best predictor for tomorrow's inflation. That is, Et[𝐼 ] = 𝐼 . This "naive" forecasting model leads us to a simplified version of the Relative PPP: Et[𝑠 ]=𝑠 = (Et[𝑆 ]/𝑆 ) – 1  𝐼 , 𝐼 , With the above information we can predict S2008:3: and, then, calculate the forecast error, 2008:3: sF2008:3 = IUS,2008:2 - IUK,2008:2 = 0.0221 - 0.0191 = 0.0030. SF2008:3 = S2008:2 x [1 + sF2008:3] = 1.9914 USD/GBP x [1 + (0.0030)] = 1.99735 USD/GBP. 2008:3 = SF2008:3-S2008:3 = 1.99735 – 1.7705 = 0.2269. (B) Autoregressive model: E[It+1] = 0 + 1 It. More sophisticated out-of-sample forecasts can be achieved by estimating regression models, using survey data on expectations of inflation, etc. For example, consider the following regression model: IUS,t = US0 + US1 IUS,t-1 + US.t. IUK,t = UK0 + UK1 IUK,t-1 + UK,t. This autoregressive model can be estimated using historical data, say 1978:1-2008:1. Then, we have 119 quarterly inflation rates for both series. We estimate both equations. (1) Excel output for autoregressive model for the US. Regression Statistics Multiple R 0.715136 R Square 0.51142 Adjusted R Square 0.507244 Standard Error 0.005517 Observations 119 ANOVA df Regression Residual Total Intercept X Variable 1 1 117 118 SS 0.003727 0.003561 0.007288 Coefficients 0.00292 0.700106 Standard Error 0.00082 0.063263 MS 0.00373 3.04E-05 F 122.469 Significance F 6.48E-20 t Stat 3.55993 11.0666 P-value 0.00054 6.48E-20 Lower 95% 0.001295 0.574817 V.5 (2) Excel output for autoregressive model for the UK. Regression Statistics Multiple R 0.416771 R Square 0.173698 Adjusted R Square 0.166635 Standard Error 0.011067 Observations 119 ANOVA df Regression Residual Total Intercept X Variable 1 1 117 118 SS 0.003013 0.014331 0.017344 Coefficients 0.007128 0.414376 Standard Error 0.001455 0.083555 MS 0.00301 0.00012 F 24.5947 Significance F 2.42E-06 t Stat 4.89963 4.95930 P-value 3.12E-06 2.42E-06 Lower 95% 0.004247 0.248899 That is, we obtain the following estimated coefficients: US0=.00292, US1=.7001, UK0=.00713, and UK1=.4144. First, you evaluate the regression by looking at the t-statistics and the R2. The t-statistic is used to test the null hypothesis that a coefficient is equal to zero. The R2 measures how much of the variability of the dependent variables is explained by the variability of the independent variables. That is, the R2 measures the explanatory power of our regression model. Both R2 coefficients are far from zero, relatively high for the U.S. inflation rate (51%). All coefficients have a t-stats higher than 1.96. That is, you will say that they are significant at the 5% level –i.e., with p-values smaller than .05. That is, all the coefficients are statistically different from zero. Second, you use the regression to forecast inflation rates. Then, you will use these inflation rate forecasts to forecast the exchange rate. That is, IFUS,2008:3 = .00292 + .7001 x (.0221) = .01839 IFUK,2008:3 = .00713 + .4144 x (.0191) = .01505 sF2008:3 = IFUS,2008:3 - IFUK,2008:3 = .01839 - 01505 = .00334. SF2008:3 1.9914 USD/GBP x [1 + (.00334)] = 1.99802 USD/GBP. 2008:3 = SF2008:3-S2008:3 = 1.99802 – 1.7705 = 0.22752. That is, you predict, over the next quarter, an appreciation of the GBP. You can use this information to manage currency risk at your firm. For example, if, during the next quarter, the U.S. firm you work for expects to have GBP outflows, you can advise management to hedge. ¶ Example V.3: Out-of-sample Forecasting Exchange Rates with a Structural Ad-hoc Model Suppose a Malaysian firm is interested in forecasting the MYR/USD exchange rate. This Malaysian firm is an importer of U.S. goods. A consultant believes that monthly changes in the MYR/USD exchange rate are driven V.6 by the following econometric model (MYR = Malaysian Ringitt): 𝑠 / = α0 + α1 𝐼 , –𝐼 + α2 𝑦 –𝑦 +𝜀 (V.1) –𝐼 represents the inflation rate differential between Malaysia and the U.S., and where, 𝐼 represents the income growth rates differential between Malaysia and the U.S. The spot rate this 𝑦 –𝑦 month is St=3.1021 MYR/USD. Suppose equation (V.1) is estimated using 10 years of monthly data with ordinary least squares (OLS). We have the following excel output: SUMMARY OUTPUT Regression Statistics Multiple R R Square 0.092703 0.018594 Adjusted R Square Standard Error Observations -0.0087 0.051729 112 ANOVA df SS MS F 0.00126 0.00268 0.47242 Regression Residual 2 109 0.002528 0.291666 Total 111 0.294195 Coefficients Standard Error t Stat P-value Intercept 0.006934 0.005175 1.3399 0.18028 (IMYR – IUSD)t 0.215927 0.105824 2.04044 0.04131 (yMYR – yUSD)t 0.091592 0.051676 1.77243 0.07633 Significance F 0.624762 That is, the coefficient estimates are: a0 = 0.00693, a1 = 0.21593, and a2 = 0.09159. That is, the output from your OLS regression is: E[𝑠 / , ] = 0.00693 + 0.21593 𝐼 (1.34) (2.04) –𝐼 + 0.09159 𝑦 (1.77) –𝑦 , R2 = .0186. Let’s evaluate our ad-hoc model. The t-statistics (in parenthesis) for the two variables are all bigger than 1.65. Therefore, all the explanatory variables are statistically significant at the 10% level. This regression has an R2 –𝐼 and 𝑦 – 𝑦 explain less than two percent of the variability of equal to .0186. That is, 𝐼 changes in the MYR/USD exchange rate. This is not very high, but the t-stats give some hope for the model. The t-statistics (in parenthesis) for the two variables are all bigger than 1.65. Therefore, all the explanatory variables are statistically significant at the 10% level. The Malaysian firm decides to use this model to generate out-of-sample forecasts. V.7 Suppose the Malaysian firm has forecasts for next month for 𝐼 respectively. Then, –𝐼 and 𝑦 –𝑦 : 3% and 2%, sFMYR/USD,t+one-month = 0.00693 + 0.21593 * (0.03) + 0.09159 * (0.02) = .0152. The MYR is predicted to depreciate 1.52% against the USD next month. The spot rate this month is St=3.1021 MYR/USD, then, for next month, we predict: 𝑆 = 3.1021 MYR/USD (1 + .0152) = 3.1493 MYR/USD. Based on these results, the Malaysian firm, which imports goods from the U.S., decides to hedge its next month USD anticipated outflows. ¶  Some Practical Issues in Fundamental Forecasting There are several practical issues associated with any fundamental analysis forecasting, such as the forecasting model of equation (V.1): (1) Correct specification. That is, are we using the "right model?" (In econometrics jargon, "correct specification.") (2) Estimation of the model. This is not a trivial issue. For example, in equation (V.1) we need to estimate the model to get a0, a1, and a2. Bad estimates of a0, a1, and a2 will produce a bad forecast for sMYR/USD,t+one-month. This issue sometimes is related to (1). (3) Contemporaneous variables. In a model like equation (V.1), some of the explanatory variables are contemporaneous. We also need a model to forecast the contemporaneous variables. For example, in the equation (V.1) we need a model to forecast INTt and INCt. In econometrics jargon, this is called simultaneous equations models.  1.A.2 Fundamental Approach: Evidence In 1979 Richard Levich, from New York University, made a comparison of the forecasting performance of several models for the magazine Euromoney. Levich studied the forecasting performance of various services relative to the forward rate. The forward rate is a free, observable forecast. A good forecaster has to beat the forward rate. Let SFj,i be the forecasting service j's forecasted exchange rate for time t=i and Si the realized exchange rate at time t=i. Let Ft,i denote the forward rate for time T=i. Levich compared the mean absolute error (MAE) of the forecasting service and the forward market, defined as MAEFSj = { |𝑆 -𝑆 | + |𝑆 -𝑆 | +...+ 𝑆 V.8 -𝑆 | } * (1/n), MAEFM = { |Ft,1 - 𝑆 | + ||Ft+1,2 - 𝑆 | +...+ ||Ft+n-1,n - 𝑆 | } * (1/n). For the sample period 1977-1980 and for twelve major forecasting services, Levich compared the percentage of times the services did better than the forward market, that is MAEFS < MAEFM. The results were: Time Horizon % of Observations with MAEFS < MAEFM 1 month 28.8% 3 months 24.0% 6 months 28.8% 12 months 32.7% In terms of mean absolute error, on average the forecasting services underperformed the forward rate more than two-thirds of the time. Since observation of the forward rate is free and since, as mentioned above, the forward rate is not an unbiased forecast of the future spot rate, these results suggest that paying a forecasting service might not be a wise expense. However, Levich found that forecasting services have some ability to predict whether the future spot rate will be greater than or smaller than the current forward rate. For some investors the direction of the forecast maybe as important as the absolute or squared error. Example V.4: Suppose the current one-month forward rate is Ft,1-mo = .7335 USD/CAD and the forecast for the future spot rate is .7342 USD/CAD. Based on the forecasted CAD appreciation over the forward rate, Ms. Sternin, a U.S. speculator, decides to buy CAD forward. Suppose that the CAD appreciates to St+1-mo = .7390 USD/CAD. The direction of the forecast (relative to the forward rate) was right, the CAD appreciated over the forward rate. Then the (absolute) size of the forecast error in this case is: .7390 - .7342 = .0052 USD/CAD. Ms. Sternin would not be unhappy with the forecasting error, since she made a profit of: .7390 - .7335 = .055 USD/CAD. On the other hand, suppose that the CAD depreciates to St+1-mo = .7315 USD/CAD. Now, the direction of the forecast (relative to the forward rate) was wrong. The (absolute) size of the error was smaller: |.7315 - .7342| = .0027 USD/CAD. But, Ms. Sternin would have lost: .7315 - .7335 = -.0020 USD/CAD. Therefore, Ms. Sternin might prefer to have a forecast that is correct in direction, and might not care about the MAE. ¶ In 1980, Levich tested the number of times a profit is made from taking advantage of the direction forecasted by some forecasting services. He found that some forecasting services were able to V.9 correctly predict the direction more than 50 percent of the time. In 1982 and 1983, Levich repeated the same study and found that the forecasting services that performed well in 1980 were not the same as those that did well in the updated study. Therefore, it seems that it is very difficult to consistently predict the future spot rate relative to the forward rate.  Forecasting: “Que sera, sera… the future’s not ours to see” Forecasting exchange rates is very difficult. As mentioned above, forward rates are poor predictors of future spot rates. Forecasting services on average have a difficult time just beating the forward rate. The random walk tends to outperform the other foreign exchange rate models. Paraphrasing a popular American TV show from the fifties and early sixties, we can say: “Doris Day knows best.”  1.B Technical Approach The technical approach (TA) focuses on a smaller subset of the available data. In general, it is based on price information. The analysis is "technical" in the sense that it does not rely on a fundamental analysis of the underlying economic determinants of exchange rates or asset prices, but only on extrapolations of past price trends. Technical analysis looks for the repetition of specific price patterns. Technical analysis is an art, not a science. Computer models attempt to detect both major trends and critical, or turning, points. These turning points are used to generate trading signals: buy or sell signals. The most popular TA models are simple and rely on moving averages (MA), filters, or momentum indicators. 1.B.1 Technical Analysis Models 1.B.1.i MA Models The goal of a MA model is to smooth erratic daily swings of asset prices in order to signal major trends. A MA is simply an average of past prices. We will use the simple moving average (SMA). An SMA, 𝑆 , is the unweighted mean of the previous Q data points: 𝑆 = (𝑆 + 𝑆 + 𝑆 + ... + 𝑆 )/Q If we include the most recent past prices, then we calculate a short-run MA (SRMA). If we include a longer series of past prices, then we calculate a long-term MA (LRMA). The double MA system uses two 𝑆 ’s: a LRMA and a SRMA. A LRMA will always lag a SRMA because it gives a smaller weight to recent movements of exchange rates. In MA models, buy and sell signals are usually triggered when a SRMA of past rates crosses a LRMA. For example, if a currency is moving downward, its SRMA will be below its LRMA. When it starts rising again, it soon crosses its LRMA, generating a buy foreign currency signal. V.10 Sell FC LRMA Buy FC SRM time Buy FC signal: When SRMA crosses LRMA from below. Sell FC signal: When SRMA crosses LRMA from above. Example V.5: Generating trading signals for the (USD/GBP) using the Double MA model. We generate a SRMA using 30 days of information (red line) We generate a LRMA using 150 days of information (green line). Every time there is a crossing, the double MA model generates a trading signal. The double MA model generates many trading signals, as indicated by the crossings between the SRMA (red line) and the LRMA (green line). For example, there is a sell GBP signal in late 2007. By April 2009, the model generates a buy GBP signal. ¶ 1.B.1.ii Filter Models V.11 This is probably the most popular TA model. It is based on the finding that asset prices show significant small autocorrelations. If price increases tend to be followed by increases and price decreases tend to be followed by decreases, trading signals can be used to profit from this autocorrelation. The key of the system relies on determining when exchange rates start to show significant changes, as opposed to irrelevant noisy changes. Filter methods generate buy signals when an exchange rate rises X percent (the filter) above its most recent trough, and sell signals when it falls X percent below the previous peak. Again, the idea is to smooth (filter) daily fluctuations in order to detect lasting trends. The filter size, X, is typically between 0.5% and 2.0%. Example V.6: Determination of Trading signals with a filter model. Let the filter, X, be 1% => X= 1%. First, we need to determine a peak or a through. Then, we generate trading signals. St = 1.47114 CHF/USD Trough Peak Sell USD Peak = 1.486 CHF/USD (X = CHF .01486)  When St crosses 1.47114 CHF/USD, Sell USD Trough = 1.349 CHF/USD (X = CHF .01349)  When St crosses 1.36249 CHF/USD, Buy USD. ¶ Note that there is a trade-off between the size of the filter and transaction costs. Low filter values, say 0.5%, generate more trades than a large filter, say 2%. Thus, low filters are more expensive than large filters. Large filters, however, can miss the beginning of trends and then be less profitable. 1.B.1.iii Momentum Models Momentum models determine the strength of an asset by examining the change in velocity of the movements of asset prices. If an asset price climbs at significant increasing speed, a buy signal is issued. The trader needs to determine what constitutes “significant” increasing speed. 1.B.1.iv Newer Models V.12 These models monitor the derivative (slope) of a time series graph. Signals are generated when the slope varies significantly. There is a great deal of discretionary judgement in these models. Signals are sensitive to alterations in the filters used, the period length used to compute MA models and the method used to compute rates of change in momentum. These facts can make two TA practitioners using the same model, but different parameters, to generate different signals. To solve this problem, there are several newer TA methods that use more complicated mathematical formulas to determine when to buy/sell, without the subjectivity of selecting a parameter. Clements (2010, Technical Analysis in FX Markets) describes four of these methods: Relative strength indicator (RSI), Exponentially weighted moving average (EWMA), Moving average convergence divergence (MACD) and (iv) Rate of change (ROC). 1.B.2 Technical Approach: Evidence In Chapter III we presented evidence that for out-of-sample forecasts, the random walk model seems to be as good as any other model. Therefore, it should not be a surprise that many economists have a strong negative view of technical analysis forecasts. The usual argument against technical analysis is based on the concept of market efficiency (discussed in the Review Chapter). The efficiency argument says that there cannot be reliable price patterns in exchange rates. If there were reliable price patterns, traders would use them, and, then, by using them, traders would destroy them. The marketplace, however, is full of newsletters and consultants selling technical analysis forecasts and predictions. Recall that the random walk model makes the exchange rate independent of its past history. Empirical studies that test the efficiency of exchange rate markets usually employ simple linear regression analysis. John F. O. Bilson (1990) points out that linear comparisons are meaningless, since technical analysts rely on nonlinearities. For example, Bilson estimates the following model: y = b1 X1 + b2 X2 + b3 X3 + u, where y = (St+1 - Ft,1)/Ft,1, X1 = (Ft,1 - St)/Ft,1, X2 = (St - Ft-3,1)/Ft-3,1, X3 = X2 exp(|X2|/std(X2)), realized ex-post returns on the currency, a yield indicator (interest rate differential), a trend indicator, where Ft,1 is the one-month exchange rate forecast.. The third variable accounts for the nonlinear pattern of serial correlation by discounting large trend values. Bilson collected New York interbank spot and one-month forward rates, from April 1975 to April 1991, on the three major currencies -GBP, DEM, and JPY. The estimates of the above equation are (t-statistics in parentheses): V.13 Ŷ= -1.0041 X1 (3.03) .0939 X2 (2.66) + .6009 X3. (5.10) All the coefficients are significantly different from zero at the 1, which shows that past information has some predictive power. Bilson later examined the forecast ability of the above model. Let Ŷ denote the fitted value from the regression estimated above. He regressed y against Ŷ, that is, yt = a + b Ŷt + t. The regression results, for the three major currencies, are given below: Currency a b R2 GBP -.1071 (0.41) 1.0948 (3.26) .052, DEM .2436 (0.41) 1.2637 (3.64) .065, JPY .5911 (0.41) 1.4089 (4.23) .085. The expected returns estimated by Bilson account for between 5% and 8% of the monthly variation in the ex-post realized returns on the currency. It should be noted, however, that the R2's are small. Therefore, one should be very careful in the interpretation of the above results. Bilson argues that these results demonstrate that the expected returns are a statistically significant indicator of actual returns. The relation between past trends and the present might be non-linear. His model predictions are entirely based on past price information. He argues in favor of technical analysis and non-linear filters. Many practitioners claim that technical analysis has value since it can be useful to forecast exchange rates, especially in the short term. They claim that, using technical analysis forecasts, profitable trading strategies can be constructed. In general, in-sample results tend to be good –i.e., profitable– for TA strategies in the FX markets. For example, an old paper by Richard Sweeney (1986), published in the Journal of Finance, investigates the power of technical forecasting. He finds that simple filter rules were able to generate excess returns during the period 1973-1980. For example, Sweeney estimated that a 1% filter rule generated a return of 2.8% while a buy-and-hold strategy generated a 1.6% return. But, the out of sample performance of TA models is weak. LeBaron (1999) speculates that the apparent success of TA in the FX market is influenced by the periods where there is CB intervention. There is also some evidence that, even in-sample, the performance of TA models has declined over time. Ohlson (2004) finds that the profitability of TA strategies in the FX market have significantly declined over time, with about zero profits by the 1990s. In a 2007 survey, Park and Irwin (2007) published in the Journal of Economic Surveys, review the TA V.14 recent literature in different markets. They report that out of 92 modern academic papers, 58 found that TA strategies are profitable. Park and Irwin point out, however, many problems with most studies: data snooping, ex-post selection of trading rules, difficulties in the estimation of risk and transaction costs. II. Looking Ahead: Exchange Rate Volatility We have seen that exchange rates are very difficult to forecast. This difficulty makes it difficult to estimate the future value assets and liabilities denominated in foreign currency. In finance, uncertainty about the value of securities creates risk. Many financial models associate the variance of an asset with an asset's riskiness. Investors require higher rates of return for riskier assets. The volatility of an asset is also important in other fields of finance. For example, option pricing formulas use as a vital input an estimate for the variance. For many years, researchers assumed that the variance of asset returns was constant. The assumption of a constant variance is called homoscedasticity. This assumption is a common assumption in econometrics. For example, in the derivations of Appendix III we state that the variance of the error term is constant. Homoscedasticity is a convenient assumption. It simplifies the estimation of time series models. Variances of asset returns, however, tend to be time-varying. That is, the returns of financial assets are heteroscedastic over time -i.e., non-constant variance. For example, Figure V.2 plots on the first panel the weekly CHF/USD exchange rate and on the second panel the CHF/USD percent changes. There are periods where exchange rates are quite volatile, while during other periods exchange rates do not move very much. For example, consider the volatile 1978-1980 period with the more quiet 2006-2007 period. It is easily observed that the assumption of a constant variance does not hold for the CHF/USD exchange rate. V.15