Exchange Rate Forecasting under Model Uncertainty

Transcription

1 Exchange Rate Forecasting under Model Uncertainty Ahmed S. Alzahrani University of Texas at Dallas September 7, 2012 Abstract We consider 100 economic models and 8 major exchange rates to examine their forecasting performance. We can t pin down any particular model which beats the random walk across all horizons and 8 currencies. We take this empirical result as evidence that all the economic models are mis-specified due to the exclusion of the common variables. We show theoretically that the common factor combined forecast provides minimum mean squared prediction error under model misspecification. And then we provide strong empirical results that the combined forecasts outperform random walk models across all currencies and over all horizons except for the first month horizon. Key Words: Forecast Combination, Common Factor Model, Nominal Exchange Rate, Long- Horizon Forecast, Exchange Rate Determination. 1

2 1 Introduction Forecasting a bilateral exchange rate is challenging. Many fundamental models have been proposed for the determination of exchange rates. The risk of out-of-sample forecasting of each of these models, compared to the risk of forecasting of the random walk model, measured by Theil s U-statistic, is sensitive to subsample periods, constraint specifications, estimation methods, and bilateral-exchange rates. The ranking of models according to their MSPEs is non-constant across bileteral exchange rates, time periods, specifications, and methods of estimation. In general, it is hard to judge whether fundamental economic models are actually helpful to forecast bilateral nominal exchange rates. Overall, there have been three improvements to the performance of out-of-sample forecasting. First, researchers have designed better models. From a naïve flexible price model to the recent Taylor rule (Mark and Sul, 2001), the determination of the exchange rate has been well developed. Second, better forecasting technique has been also developed. From a simple prediction regression to forecasting combination, econometricians have studied to improve forecasting performance. Third, greater information sets have been used in forecasting. From univariate simple time series regression to factor augmented forecasting (Greenaway-McGrevy, Mark, Sul and Wu, 2012 GMSW hereafter), researchers have understood complicated data much more deeply. However, even under such rapid developments, forecasting exchange rates is still questionable for two reasons. First, none of the methods always beats the random walk hypothesis for all subsamples, across all major exchange rates. Second, there have been no empirical studies that combine all three developed techniques at the same time. This paper aims to challenge this diffi cult task by using common factor combining forecasts. First, the combining forecast technique is the most recently developed forecasting method, and has been proven to be very effi cient (for example see Timmerman 2006, and Hansen 2007). Also the combining forecast literally uses all possible models to form the best possible forecast. Hence this method is free from any selection bias. Last, the common factor combining forecast utilizes cross sectional and time series information together. Hence this method nests the recent forecasting results by GMSW and Engel, Mark, and West (2012). Particularly, GMSW emphasizes the role of key exchange rates on the exchange rate determination. Three key exchange rates the Euro, Yen, and Swiss Franc are common factors to the other bilateral exchange rates. Hence, typical, two-country exchange rate models become mis-specified because they exclude the information of the three key exchange rates. We use broadly five generic models PPP, naïve flexible price, sticky price, UIP, and Taylor rule to generate 100 versions of detailed models for one bilateral exchange rate. For each model, we obtain forecasts so that we have 100 forecasts in our hands at each time horizon. From such rich forecasts, we extract the most significant common factors, and then use them as regressors 2

3 to forecast the out-of-sample horizons for the bilateral exchange rate. This method is called the common factor combining forecast in the forecasting literature. The estimated common factors from the forecasts must include the information of the three key exchange rates and other potential common information, which may be missed in the GMSW setting. The forecasting performance of the common factor combining method is outstanding and significantly better than the conventional random walk forecasts with/without drift for all eight bilateral exchange rates. The remainder of this paper is as follows. The next section develops the intuitive explanation of why the common factor combining method works well, especially under model misspecification from excluding common variables. In this section, we lay out the models used in the paper and present the forecasting equations. In Section 3, we explain the forecast combination methods and provide detailed information on how to construct common factors to forecasts. We also report empirical findings. Section 4 concludes. At the end,we report graphically the density forecast for all models in "Fan-Chart" figures, for each exchange rate. The density of the out-of-sample forecasts of all models illustrate clearly model uncertainty and emphasizing our conclusion We also find the same conclusion when we exclude each model, one at a time. From these exercises, we confirm that all individual models are misspecified. 2 Exchange Models and Forecasting under Model Misspecification Let the log nominal exchange rate between a foreign country and the US be s t. We consider the following five models for the determination of the nominal exchange rate. The first three models are determinants for the log level nominal exchange rate given by PPP : s t = f (p t, p t ) + u p,t Flexible : s t = f (m t, m t, y t, yt ) + u f,t (1) Sticky : s t = f (m t, m t, y t, yt, i t, i t ) + u s,t where m t, p t, i t and y t are the log money supply, price, nominal interest rate, and income for the domestic country and the superscript * stands for the foreign country. Since s t is nonstationary, actual forecasting is usually based on the following error correction model s t+k s t = b k u j,t + ɛ j,t+k, for j = p, f, s. (2) where u j,t are assumed to be I (0) and treated as the equilibrium error. Note that we didn t express the constant term for notational simplicity. However in actual out-of-sample forecasts, we include the constant term in (2). Hence even though the first three models are determinants for the log level nominal exchange rate, via error correction representation, all the models can be re-written as the differenced model. 3

4 The next two models are determinants for the log differenced nominal exchange rate given by UIP : s t+k s t = f (i t, i t ) + ɛ u,t+k (3) Taylor : s t+k s t = f ( π t, π t, ỹ t, ỹt ) + ɛ τ,t+k (4) where π t and ỹ t are the inflation gap and the output gap for the domestic country, and the superscripted variables are for the foreign country. Inflation and output gaps are computed from p and y, respectively, using an HP-filter. It is important to note that none of the models can nest all the other models. The largest model is the sticky price model, which includes six variables in total. However, from the error correction model in (2), which is based on the sticky price model, it is not possible to derive either the UIP or the Taylor rule forecasting regressions. This implies that if one of the models were correct, then the rest of the models would be seriously misspecified. Also, as GMSW argue, none of the models include the key exchange rates as extra regressors. Hence, if the domestic currency is influenced by the key exchange rates, then all models become misspecified and the regression errors include some missing variables, namely, the key exchange rates. Now, we introduce a 1 K vector of the additional variables, F t, into each model. These variables could be the three key depreciation rates as GMSW suggest, or could be other key variables which economic researchers haven t considered before. Let s assume the true model is given by s t+k s t = X t β + ɛ t+k, for ɛ t+k = F t δ + v t+k, (5) where X t is a 1 m vector of all the relevant variables. Of course, the true model and missing variables are unknown. Here we allow that E(X lt F st ) 0 for some l and s, where l and s are the lth and sth elements of X t and F t, respectively. Consider a set of selected regressors Z it, where Z it may not be equal to X t for all i, or may be the same as X t for a particular i. Then the ith forecasting regression is given by s t+k s t = α k + Z it β ik + ε it+k, where ε it+k = Z it β ik + X t β + F t δ + v t+k = Z it β ik + w t + v t+k where w t = X t β + F t δ. Due to the endogeneity between Z it and ε it+k, the OLS estimator ˆβ ik becomes biased and inconsistent. We are not able to evaluate models based on such inconsistent point estimates. More precisely ˆβ ik = β ik + ( Z iz i ) 1 Z i ε i = ( Z iz i ) 1 Z i [w + v]. 4

5 Without loss of generality, we may let the jth element of the vector Z it be z jit = φ j w t + z o jit so that Z it β ik = φ i w t + z o it, for z o it = Z o itβ ik. Note that and k j is the dimension of Z it. k i φ i = φ j β jik, Zit o = j=1 Then it is straightforward to show that Z itˆβik = ˆφ i w t + ẑ o it, [ ] z1it, o..., zk o j it, which implies that the forecasts follow an approximate common factor structure. More interestingly, the number of common factors becomes one. Note that the MSPE for the ith model is given by s l+k ŝ il+k = Z ilˆβik + w l + v l+k = (1 ˆφ i ) w l + v l+k EP 1 P l=1 ( (s l+k ŝ il+k ) 2 = E 1 ˆφ ) 2 i EP 1 P l=1 w 2 l + EP 1 P l=1 v 2 l+k = ξ 2 i σ 2 w + σ 2 v = MSP E i Since we always rewrite ) (h ˆφ i w t = (ˆφi h 1 ) w t = ˆφ i wt so that we have to normalize the unknown common factor w t. Due to this normalization, the PC estimate of w t must be rescaled by running the following regression. s t+k s t = λ k ŵ t + v t+k ; t = T o + 1,..., T k. Then we have EP 1 P l=1 (s l+k ŝ l+k ) 2 = σ 2 v + = MSP E pc where is included since ŵ t is used in the actual forecast. So this is an approximation error. It is ( ) well known that the approximation error is O p T 1/2 as long as the total number of forecasting models is larger than the total number of the time periods used for estimating w t. o 5

6 Hence as long as ξ 2 i σ 2 w, MSPE pc must be smaller than or equivalent to MSPE i for all i. Simply we may let MSPE min = min [MSP E i ]. Then MSP E min MSP E pc as T o, P. When T o and P are not large, MSPE pc may not be smaller than MSPE min but the difference between the two would not be statistically significant as long as F t exists or all of the models are misspecified. On the contrary, suppose that F t does not exist, but one of the structural models, say model m, is the true model; then, MSPE m would equal MSPE min when Z it = X t. Also, this evidence should be consistent over the choice of subsamples, and across the horizon, k. 3 Empirical Models and Data Data is obtained from IHS Global Insight, International Monetary Fund (IMF), International Financial Statistics (IFS). All of the variables are measured monthly from to 2009:12. The variables are nominal exchange rates measured as the period average of the market rate, (s), consumer prices measured as the national CPI for all consumers, (p), an index of total industry production (seasonally adjusted), (y), an index of broad money M 2 (seasonally adjusted), (m), and nominal 3-month-interbank interest rates, (i). Variables for the USA are denoted by stars. The eight major exchange rates we considered are: the EURO, Pound, Japanese Yen, Norwegian Krone, Swedish Krona, Canadian Dollar, Danish Krone, and Korean Won against the US dollar. The total number of time series available at the time of this study was 132 observations for each variable. We use one third of the total sample, from to , to generate monthly forecasts for the next 44 months, from to , for each of the 100 individual models and for each of the 12 horizons. We consider only up to the 12-month horizon, due to the lack of degrees of freedom. For the 12-month horizon forecast of , we use the lagged sample from to in order to estimate the forecasting regressions. Since we have to use 12-month differences, we are losing 12 additional observations. The maximum number of explanatory variables, including a constant, is 8, which leaves us with only 12 degrees of freedom for the first 12-month horizon forecast. We use a recursive estimation to increase the degrees of freedom for the later forecasts. From the 44 forecasting errors of each model, we estimate the dominant common factor for each exchange rate. We considered more but we found that there was little difference. The actual outof-sample forecasting, for the last 44 months, starts from and ends at , which is the most volatile time period. All exchange rates have the same high peak in 2009 so that forecasting exchange rates during this time period becomes challenging. 6

7 In order to generate multiple forecasts from each individual model, we used alternative specifications ranging from fully restricted to fully unrestricted parametrization of each of the five structural models we considered in (1), (3), and (4). To save space, we take the example of PPP fundamentals to illustrate the alternative specifications and how to generate the forecasts. The equilibrium error u jt for j = 1, 2, 3, 4, 5, 6 is defined as û p,1t = s t (p t p t ), û p,2t = s t â 1 (p t p t ), û p,3t = s t â 1 p t â 2 p t, û p,4t = s t â 0 (p t p t ), û p,5t = s t â 0 â 1 (p t p t ), û p,6t = s t â 0 â 1 p t + â 2 p t. and then forecasts are generated from the following six forecasting regressions ŝ i,t+k = s t + ˆβ i1 u p,jt + ɛ i,t+k for i = 1, 2, 3 (6) ŝ i,t+k = s t + β 0i + ˆβ i1 u p,jt + ɛ i,t+k for i = 4, 5, 6. (7) Note that the forecasting regressions in (7) for the cases of i = 1, 2, 3 become equivalent to those in (6) for the cases of i = 4, 5, 6. Hence for the flexible price and sticky price models, we have 20 and 66 forecasting regressions, respectively. For the UIP and Taylor rule models, we have 3 and 5 forecasting regression, respectively. Hence, the total number of forecasts becomes 100. From these forecasts, we extract the first, largest, common factor w t, by using principal components estimation. Table 1 reports the model number that minimizes MSPE for each horizon, the minimum Theil s U-statistics against a random walk without drift, U 2, and the ratio between U 2 and Theil s U- statistics against a random walk with drift, U 1. The first capital initial stands for a model group; and next numeric number stands for a particular model specification. For example, S63 stands for the sticky price model with the particular 63th specification. Here we don t provide detailed model specifications for two reasons. First we want to save space; and second we will show later that distinguishing model specifications is not at all meaningful. Evidently the relative Theil s ratios, U 2 /U 1, are larger than unity for most case except for the Japanese Yen. This implies that the random walk without drift is more diffi cult to beat. Hence, we use the random walk without drift as a benchmark model. Since the best model is chosen to minimize MSPE for each horizon over 100 models, it is not surprising to see that U 2 is smaller than unity. In fact, as the horizon increases, U 2 seems to converge to zero. We use Clark and West (2006) s asymptotic test to examine whether the minimum model beats the random walk. We reject the equal forecastability at the 5% level for all horizons except for the first horizon. For the Euro dollar case, there seems to be no dominant model across all of the horizons. Surprisingly, the flexible price models dominate the sticky price models. F10 and F16 seem to be one of best models. F10 is s t = a + b [m t m t y t + yt ] + u t and its counterpart, ECM, is given 7

8 by s t+k = α k,10 + β k,10 u t + ɛ t+k. F16 is s t = a + b (m t m t ) c (y t yt ) + u t and its ECM is given by s t+k = α k16 + β k16 u t + ɛ t+k. Here, the point estimates are not reported since the signs of these values are changing over time and across horizons. Interestingly, for the Japanese Yen (F17), Canadian Dollar (P2), Norwegian Krone (S31) and Swedish Krona (S39), there seems to be a single economic model which minimizes MSPE across all horizons. However, this evidence is illusive. Table 2 reports the best models when the forecasts end at Obviously the model numbers and groups are changing dramatically. For the Euro, the best group becomes the sticky price models; and model number changes to 66. For the Japanese Yen, F17 is no longer the best model. And for other exchange rates, all of the best models are changing over the choice of forecasting periods. In fact, such empirical findings coincided with those found by other researchers. Hence, we conclude that all models are misspecified since there is no single model that minimizes MSPE across all horizons and over every subsample. Next, we examine the out-of-sample forecasting performance of the common factor combined forecasts. Table 3 presents the minimum Theil s U-statistics from the best model (U 2 ), and that of the common factor combined forecasts (U c ). Also we report Theil s U statistics with combined forecasts excluding the Taylor rule models, denoted by U t. We use Diebold and Mariano statistics to evaluate the equal predictability of the best model to the combined forecasts; but we don t report the p-values here since we are not able to reject the null of the equal predictability for all horizons and all exchange rates. In fact, Theil s U-statistics of the best model are very similar to those of the combined forecasts. In general, as the forecast horizon increases, Theil s U-statistics of the combined forecasts seem to decrease more than those of the best model. However, as we discussed just before, they are not statistically different from each other. We also exclude each model group one by one, and re-calculate Theil s U-statistics. For example, U t shows Theil s U-statistics of the combined forecasts when Taylor models are excluded. If the Taylor rule group of models includes a true model, then excluding the Taylor rule group leads to lowering Theil s U-statistics of the combined forecasts, as the most important forecasts are excluded. However, evidently we can t find any significant difference between U c and U t. In fact, the difference between the two cannot be found until 4th digit. We also find the same conclusion when we exclude each model, one at a time. From these exercises, we confirm that all individual models are misspecified. 4 Concluding Remarks GMSW (2012) shows that three empirical factors Euro/dollar, Swiss franc/dollar and yen/dollar explain a large proportion of the exchange rate variation over time and have significant out-ofsample predictive power. However, including three empirical factors in the forecasting regression 8

9 does not always improve the predictive power. Meanwhile, this paper shows the existence of unknown empirical factors by using the recently developed combined forecasts. We show that the combined forecasts minimize the asymptotic MSPE when economic models exclude some important common factors. Also, we find that the combined forecasts always beat those of random walk models for all exchange rates and all horizons, except for the first month horizon. Moreover, the out-ofsample forecast performance of combined models are as good as those of the best individual model that minimizes MSPE across all models for each horizon. From this exercise, we verify that the common factor combined forecast is the best econometric method to predict future exchange rates. Also, more importantly, we provide strong empirical evidence that the common factors are missing in all previous exchange rate models. However we didn t show explicitly whether the missing common factors are the only three empirical factors or not. In addition, we didn t investigate which economic models with three empirical factors perform best in terms of in-sample and out-of-sample predictive power. We leave these two important tasks for future research. References [1] Bai, Jushan and Ng, Serena, (2002), "Determine the Number of Factors in Approximate Factor Models", Econometrica 70, [2] Bates, J.M., Granger, C.W.J., (1969), "The Combination of Forecasts". Operations Research Quarterly 20: [3] Clemen, R.T., (1989), "Combining Forecasts: A Review and Annotated Bibliography". International Journal of Forecasting 5: [4] Diebold, Francis X. and Mariano, Roberto S., (1995), Comparing Predictive Accuracy, Journal of Business & Economic Statistics, Vol. 13, No. 3 (Jul., 1995), pp [5] Engel, C., J. Wang, and J. Wu, (2010),"Long-Horizon Forecasts of Asset Prices when the Discount Factor is Close to Unity", Working Paper. [6] Greenaway-McGrevy, Mark, Sul and Wu, (2012), "Exchange Rates as Exchange Rate Common Factors", Working Paper. [7] Engel, C., N. Mark, and K. West, (2007), "Exchange Rate Models are Not as Bad as You Think", in NBER Macroeconomics Annual, 2007, D. Acemoglu, K. Rogoff and M.Woodford (eds.), Chicago: University of Chicago Press. 9

10 [8] Engel, C., N. Mark, and K. West, (2009), Factor Model Forecasts of Exchange Rates, Working Paper. [9] Greenaway-McGrevy, Mark, Sul and Wu, (2012), "Exchange Rates as Exchange Rate Common Factors", Working Paper. [10] Hansen, Bruce E.(2007), "Least Squares Model Averaging", Econometrica", 75, [11] Hubrich, Kirstin and Kenneth D. West, (2009), "Forecast Comparisons for Small Nested Model Sets", ECB Working Paper No [12] Inoue, Atsushi and Lutz Kilian, (2004), "In-Sample or Out-of-Sample Tests of Predictability: Which One Should We Use?", Econometric Reviews 23(4), [13] Mark, N.C., Sul, D., (2001), "Nominal Exchange Rates and Monetary Fundamentals: Evidence from a Small Post-Bretton Woods Panel", Journal of International Economics 53, [14] Mark, Nelson A., and Donggyu Sul, (2001), "Nominal Exchange Rates and Monetary Fundamentals: Evidence from a Small Post-Bretton Woods Sample", Journal of International Economics 53, [15] Meese, Richard A., and Kenneth Rogoff, (1983a), "Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?", Journal of International Economics 14, [16] Meese, Richard A., and Kenneth Rogoff, (1983b), "The Out-of-Sample Failure of Empirical Exchange Rate Models: Sampling Error or Misspecification", in J. A. Frenkel, (ed.) Exchange Rates and International Macroeconomics (Chicago: University of Chicago Press). [17] Rapach, David E., and Mark E. Wohar, (2004), "Testing the Monetary Model of Exchange Rate Determination: A Closer Look at Panels", Journal of International Money and Finance, 23(6), [18] Stock, James H. and Mark W. Watson, (2006), "Forecasting with Many Predictors", in Handbook of Economic Forecasting, Vol 1, G. Elliott, C.W.J. Granger and A. Timmermann (eds), Amsterdam: Elsevier. [19] Timmermann, Allan, (2006), "Forecast Combinations",. In: Elliott, G., Granger, C.W.J., and Timmermann, A. Handbook of Economic Forecasting, Elsevier. [20] Verdelhan, Adrien, (2011). The Share of Systematic Variation in Bilateral Exchange Rates, manuscript, MIT. 10

11 [21] West, Kenneth D., (1996), "Asymptotic Inference About Predictive Ability", Econometrica 64, [22] West, Kenneth D., (2006), "Forecast Evaluation", in Handbook of Economic Forecasting, Vol. 1, G. Elliott, C.W.J. Granger and A. Timmerman (eds), Amsterdam: Elsevier. 11

12 Table 1: Theils U-Statistics: Random Walk with Drift (U 1 ) and without Drift (U 2 ) Euro UK Pound JAP Yen Can Dollar (CAND) k Best U 2 U 2 /U 1 Best U 2 U 2 /U 1 Best U 2 U 2 /U 1 Best U 2 U 2 /U 1 1 F S T S F S T S F T F S F T F S F T F P F S F P F S F P F S F P F T F P S T F P F T F P S T F P

13 Table 1: Continue NOR Krone (NOK) SEW Krona (SEK) DEN Krone (SKK) KOR Won (KRW) k Best U 2 U 2 /U 1 Best U 2 U 2 /U 1 Best U 2 U 2 /U 1 Best U 2 U 2 /U

14 Table 2: Unstable Individual Forecasts First Sample: , Second Sample: Euro Pound Yen CAN D k F16 T99 S70 T100 T97 T93 S67 T93 2 F16 S66 S30 T100 T99 S61 S49 S29 3 F10 S66 T96 T100 F17 S62 S45 S37 4 F16 S66 T96 T100 F17 S62 S45 S35 5 F10 S66 T96 T100 F17 S62 P2 P2 6 F10 S66 S63 S70 F17 S62 P2 P2 7 F16 S66 S63 S63 F17 F11 P2 P2 8 F16 S66 S63 S63 F17 F11 P2 P2 9 F16 S66 T96 S63 F17 F11 P2 P2 10 S46 S66 T96 T96 F17 F11 P2 S35 11 F18 S66 T96 T96 F17 F17 P2 S35 12 S30 S66 T96 T96 F17 F17 P2 S35 NOK SEK DKK KRW 1 S31 T100 S51 T100 T100 F7 S68 T93 2 S31 S51 S51 S51 S27 T100 S47 T97 3 S31 S51 S52 T100 S27 S27 S47 S47 4 S31 S51 S39 S69 S41 S27 S63 S48 5 S31 S51 S39 S69 S41 S53 S33 S48 6 S31 S31 S39 T95 S41 S53 S33 S55 7 S31 S31 S39 S53 S53 S53 S33 S63 8 S31 S31 S39 S53 S53 S53 T96 S63 9 S31 S31 S39 S39 S53 S53 T96 S63 10 S31 S31 S39 S39 S53 S53 T96 S63 11 S31 S31 S39 S63 S53 S53 T96 S63 12 S31 S31 S39 S63 S31 S53 T96 T96 14

15 Table 3: Comparison among Best (U 2 ), CF Combined Forecast (U c ) with all models and Combined Forecast (U t ) without Taylor models Euro Pound Yen Can D k U 2 U c U t U 2 U c U t U 2 U c U t U 2 U c U t NOK SEK DKK KRW

16 16

17 17

18 18

19 19

20 Data Description (Source: IHS Global Insight) Variable Exchange Rate Name s All exchange rates Exchange Rate, Market Rate, Period Average p EUR Consumer Prices, Harmonized CPI All but EUR Consumer Prices, CPI National, All Consumers m All exchange rates Broad money (M2) SA / Index publication base SA y DKK Production of Total Industry SA / Index Publication Base SA All but DKK Production, Industrial Production, S.A i CAD Interest Rates, Central Bank Policy Rate KRW Interest Rate, 3-Month Certificates of Deposit All but CAD and KRW Interest Rate, 3-Month Interbank Note: USA variables are matched with the available variables for each exchange rate. 20