Blame the Discount Factor No Matter What the Fundamentals Are

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1 Blame the Discount Factor No Matter What the Fundamentals Are Anna Naszodi 1 Engel and West (2005) argue that the discount factor, provided it is high enough, can be blamed for the failure of the empirical literature to find significant relationship between the economic fundamentals and the exchange rate. In this paper I test whether the discount factor is of that magnitude. In order to obtain an estimates of the discount factor which is robust to the definition of the fundamental, I apply the following method. I treat the fundamental as being unobserved and filter it out by the Kalman-filter from survey data on exchange rate forecasts with multiple forecast horizons. The estimates for the discount factor is obtained by maximizing the filtering likelihood. The empirical results support the explanation of Engel and West for the disconnect puzzle. JEL: E44, F31, G10 Keywords: asset-pricing exchange rate model, disconnect puzzle, survey forecasts, Kalman-filter. 1 Introduction If the discount factor is in a certain interval then it can be blamed for the failure of the empirical literature to find significant relationship between the fundamentals and the nominal exchange rate, given that the process of the fundamentals is integrated of order one (I(1)). This result is obtained by Engel and West (2005). Sarno and Sojli (2009) estimate the discount factor and find that its magnitude can account for the feeble link between the exchange rate and the fundamentals along the arguments of Engel and West. The empirical method applied by Sarno and Sojli (2009) identifies the discount factor not only from the time series of the exchange rate and the survey data on the one-month-ahead exchange rate forecasts, but also from a set of macro 1 Magyar Nemzeti Bank (the Central Bank of Hungary), Budapest, Hungary naszodia@mnb.hu, anna.naszodi@gmail.com 1

2 fundamentals. 2 In this respect their estimates depends on how exactly the fundamental is defined. This paper proposes an alternative empirical strategy to estimate the discount factor with the same stylized asset-pricing exchange rate model used by Engel and West (2005) and Sarno and Sojli (2009). This alternative approach distinguishes itself from that of Sarno and Sojli by being robust to the definition of the fundamentals. The importance of this feature is supported by the following facts. First, there is no consensus among economists over what the macro fundamentals are. Second, some components of the commonly used fundamentals like the risk premium are unobservable. The main idea of the proposed approach is to use a richer data set on exchange rate forecasts which can take over the role of the fundamentals at identifying the discount factor. These richer data consist of survey forecasts with not only one forecast horizon, but more. I apply this approach to estimate the discount factors of four major exchange rates. Thereby, I test whether the magnitude of the estimates is consistent with the Engel and West explanation for the disconnect puzzle. 2 Exchange rate model I work with a conventional class of asset-pricing models in which the exchange rate is the discounted sum of the current and expected future fundamentals. 3 s t = (1 b) b j E t (f t+j ) with 0 < b < 1, (1) j=0 wheres t denotesthelogexchangerateattimet,whilef t+j isthefundamental at time t + j. E t (.) is the expectation operator, where the expectation is conditional on all the information available at time t. Parameter b is the discount factor, whose magnitude we are interested in. The reduced form of (1) can be rationalized by a number of structural models. For instance, Engel and West (2005) review some commonly used models, where b is either the semi-elasticity of money demand with respect 1 b 2 Sarno and Sojli (2009) define the fundamentals as being the weighted average of the foreign and domestic money supplies and outputs. As a robustness check, they use different weighting schemes. 3 See for instance Mark (2001), page 68. 2

3 b to the interest rate, or is the inverse of the relative weight of the exchange 1 b rate in the Taylor rule. Along these lines, estimates of the money demand or the Taylor rule can also be used to learn the magnitude of the discount factor b. While Engel and West (2005) chose the above approach, this paper provides estimates of the discount factor that is independent from any of the structural models and uses only the reduced form model of (1) which is common across a large set of structural models. The cost of generality, i.e., the cost of being independent from structural models, is that one has to make an assumption on the process of the fundamental. Here, the assumed process is f t = ρf t 1 +α(1 ρ)+ǫ f,t, where 0 < ρ 1, and ǫ f,t i.i.d N(0,σ 2 f). (2) Iftheautoregressiveparameter0 < ρ < 1thentheprocessofthefundamental isstationary, whileincaseofρ = 1itisintegratedoforderone. Inadditionto the magnitude of the discount factor, we are also interested in the estimates of the autoregressive parameter of the fundamental. The reason for our interestisthattheexplanationofengelandwest(2005)fortherandomwalk behavior of the exchange rate relies not only on the near unity of the discount factor, but also on the assumption that the process of the fundamental is integrated of order one. They show that all autocorrelations of ds will be very close to zero for b very close to one and ρ = 1. From the practical point of view of predicting the future exchange rate, a highly persistent stationary process of the fundamental also makes an interesting case. As it is argued by Engel and West (2005), there is continuity in the autocorrelations in the following sense: for b near one, the autocorrelations of ds will be near zero if the previous paragraph s condition that certain variables are I(1) is replaced with the condition that those variables are I(0) but with an autoregressive root very near one. For a given autoregressive root less than one, the autocorrelations will not converge to zero as b approaches one. But they will be very small for b very near one. It can be derived from Equations (1) and (2) that the expected k periodahead exchange rate is the following function of the fundamental f t and the discount factor b. 4 E t (s t+k ) = α (1 b)α ρk 1 ρb +(1 b) ρ k 1 ρb f t for all k > 0. (3) 4 See the derivation in the Appendix. 3

4 3 Empirical Implementation Our primary aim is to estimate the discount factor without using data on the fundamental, and without knowing the structural macro model. This Section shows one way of doing it. The basic idea is that survey forecasts with various forecast horizons can identify the discount factor together with the fundamental. A consistent estimator is the Kalman-filter that treats the fundamental as being unobserved. I assume that the survey forecasts are unbiased estimates of the expected future log exchange rates. z t,t+k = E t (s t+k )+ǫ t,k with ǫ t,k i.i.d N(0,σ 2 z,k), (4) where z t,t+k is the survey forecasts with the forecast horizon k. By substituting Equation (3) into (4), we obtain some of the observation equations with different forecast horizons k. z t,t+k = α (1 b)α ρk 1 ρb +(1 b) ρ k 1 ρb f t +ǫ t,k. (5) If we had forecasts for h number of different forecast horizons, then the number of observation equations would be h as well. However, in addition to the survey forecasts, we have data also on the spot exchange rate making the number of observation equations h + 1. Although the spot rate can be thought of as a special forecast with zero forecast horizon s t = z t,t, it is worth to treat it separately from the forecasts. In contrast to the expected future log exchange rate E t (s t+k ), the expected current log exchange rate E t (s t ), i.e., the log spot rate s t is observed without error. By using the identities s t = z t,t and ǫ t,0 = 0, Equation (5) gives s t = α (1 b)α 1 1 ρb +(1 b) 1 1 ρb f t. (6) Finally, Equation(2) corresponds to the transition equation in the state space model. In the filtering exercise, the initial value of the state variable f 0 is set to the log spot exchange rate of the beginning of the sample s 0. The initial state variance is set to 0.3, which reflects a high degree of uncertainty. The estimates of the discount factor and all the other parameters are obtained by maximizing the filtering likelihood. 4

5 The explanation of Engel and West (2005) for the disconnect puzzle relies on two conditions. First, the discount factor is high. Second, the process of the fundamental is I(1). Whether the discount factor is equal to unity, can be tested by the likelihood ratio test. LR = 2(logl r logl u ). (7) where logl r and logl u are the log likelihoods of the restricted and the unrestricted models respectively. Under the null hypothesis, H 0 : b = 1, the LR-statistics has a chi-square distribution with 1 degrees of freedom. The second condition is tested in three different ways. First, I apply the likelihood ratio test in order to see how likely the restriction of ρ = 1 is. Second, I test whether the time series of the filtered fundamentals have unit roots. For this purpose I use the augmented Dickey-Fuller (ADF) test, and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. 4 Survey Data and Empirical Results I use the survey data of the Consensus Economics on the forecasted 3-monthsahead, 1-year-ahead and 2-year-ahead exchange rates. 5 These forecasts are for the exchange rates of the euro(eur), the United Kingdom pound(gbp), the Japanese yen (JPY) against the US dollar (USD), and the Swiss franc (CHF) against the euro. The survey data are the mean of the forecasts of the individual survey participants presumably mirroring the consensus view of the professional forecasters. The frequency of the survey data is monthly. The sample is spanned by January 1999 and September The beginning of the sample is determined by the availability of the data, while the end of the sample is the last survey date preceding September 15, 2008, when Lehman Brothers has been filed for Chapter 11 bankruptcy protection. In September 2008, there was a salient structural break in different measures of uncertainty, like the implied volatilities of currency option prices and the cross-sectional standard deviation of the individual forecasts. As the jump in uncertainty would invalidate our assumption of having unchanged variance in the observation equations in a sample covering the period even after the 5 The reported forecasts are not the expected log exchange rates, but the expected exchange rates. I proxy the expected log exchange rates by the log of the reported expected exchange rates. An even more precise approximation would be based on adjusting by half of the variance. However, the difference between the two approximations is negligible. 5

6 structural break, I have decided to exclude observations after September In addition to the survey forecasts, I use the spot exchange rates on the days of the surveys. These data are from the European Central Bank. 6 The Kalman-filter is used for each of the 4 exchange rates separately. The length of the time series of all variables, i.e., the exchange rate and the forecasts with 3 different forecast horizons, is 116. Therefore, 116 times 4 observations help filter one single time series of the unobserved factor and to estimate 7 parameters in case of each of the exchange rate. Table 1 shows the estimates of the discount factor, the AR parameter of the fundamental together with the rest of the parameters. In economic sense, the point estimates of the discount factors are close to one for all the four exchange rates. However, the likelihood ratio test makes us to reject the null of b = 1. The same can be said about the AR parameters of the fundamentals: the estimates of ρ s are very close to unity for all the four currency pairs, but statistically they are significantly smaller than one. Table 2 presents some further statistics about the processes of the fundamentals. The null hypothesis of the ADF-test that the filtered fundamentals have unit roots cannot be rejected for any of the exchange rates. In accordance with this result, stationarity can be rejected based on the KPSS-test for three exchange rates out of the four. The only exception is the yen, where the KPSS test-statistics is not significant. Although the unit root tests of the filtered fundamentals contradict to the likelihood tests most of the times, the processes of the fundamentals are proved to be highly persistent by both of the approaches. To sum up the results, both the discount factors and the autoregressive parameters of the fundamentals are very close to unity, but certain statistical tests suggests that they are smaller than unity for the four exchange rates examined. 5 Conclusion The near random walk behavior of the exchange rate, and also the weak link between the nominal exchange rate and some economic fundamentals are well documented by the empirical literature. These disappointing findings have questioned the relevance of the conventional exchange rate models for a long time until the breakthrough work of Engel and West (2005). They reconciled 6 6

7 the empirics with the theory by demonstrating that under certain conditions the findings are consistent with the models. These conditions are (i) at least one of the factors driving the exchange rate follows an I(1) process, and (ii) the discount factor is relatively large, e.g., close to unity. The primary aim of this paper is to test the second condition by providing estimates of the discount factor. I use an estimator that is robust to the underlying macro model specifying the fundamentals. The motivation for developing such an estimator is twofold. First, the menu of the theoretically appealing fundamentals is large. Second, some components of the fundamentals are unobservable. The main finding is that no matter what the fundamentals are, the discount factor is high. Thereby, the test gives robust support to the resolution of the exchange rate disconnect puzzle proposed by Engel and West. 7

8 Literature Cited Engel, Charles, and Kenneth D. West. (2005). Exchange Rates and Fundamentals. Journal of Political Economy 113, Mark, Nelson C. (2001). International Macroeconomics and Finance: Theory and Econometric Methods. Oxford: Blackwell. Sarno, Lucio, and Elvira Sojli. (2009). The Feeble Link between Exchange Rates and Fundamentals: Can We Blame the Discount Factor? Journal of Money, Credit and Banking 41,

9 Tables Table 1: The Estimates of the Parameters b ρ α σ 2 f σ 2 z,k=3m σ2 z,k=1y σ 2 z,k=2y CHF/EUR LR EUR/USD LR GBP/USD LR JPY/USD LR Notes: *** indicate that we can reject the hypothesis at 1% significance level. The hypothesis of column b is H 0 :b = 1. The hypothesis of column ρ is H 0 :ρ = 1. 9

10 Table 2: Unit Root Tests of the Filtered Fundamentals ADF-test KPSS-test p-value CHF/EUR EUR/USD GBP/USD JPY/USD Notes:,, indicate significance at 10%, 5%, and 1% respectively. At the Augmented Dickey-Fuller (ADF) test the hypothesis is that there is unit root in the process of the fundamental, while at the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test the hypothesis is that the process of the fundamental is stationary. 10

11 Appendix In this Appendix I show that under the assumed process of the fundamental f t of (2), the k period-ahead expected log exchange rate E t (s t+k ) is given by Equation (3) in the reduced form asset-pricing model of (1). First, E t (f t+k ) is expressed as a function of f t from Equation (2). Second, by substituting (8) into (1), we obtain E t (f t+k ) = α(1 ρ k )+ρ k f t. (8) s t = (1 b) b j[ ] α(1 ρ j )+ρ j f t. (9) By using j=0 bj = 1 (0 < b < 1) and 1 b j=0 bj ρ j = 1 1 bρ Equation (9) simplifies to j=0 (0 < ρ 1), s t = α b(1 ρ) 1 bρ + 1 b 1 bρ f t. (10) From (10), we get [ E t (s t+k ) = E t α b(1 ρ) 1 bρ + 1 b ] 1 bρ f t+k. (11) Finally, we obtain Equation (3) by substituting Equation (8) into (11). 11