Time-Varying forward bias and the volatility of risk premium: a monetary explanation

TIME-VARYING FORWARD BIAS AND THE VOLATILITY OF RISK PREMIUM: A MONETARY EXPLANATION Juan Ángel Lafuente (*) Universidad Jaume I Unidad Predepartamental de Finanzas y Contabilidad Campus del Riu Sec, 12080, Castellón Jesús Ruiz (*) Instituto Complutense de Análisis Económico (ICAE) Universidad Complutense Campus de Somosaguas, 28223 Madrid ABSTRACT _________________________________________________________________________________ Forward exchange rate unbiassedness is rejected for international exchange markets. This paper proposes a stochastic general equilibrium model which generates substantial variability in the magnitude of predictable excess returns. Simulation exercises suggest that high persistency in the monetary policy produces greater bias in the estimated slope coefficient in the regression of the change in the logarithm of the spot exchange rate on the forward premium. Also, our model suggest that the nature of the transmission between monetary shocks can explain the excess return puzzle. Empirical evidence for the US-UK exchange rate according to our theoretical results is provided. _________________________________________________________________________________ RESUMEN _________________________________________________________________________________ La insesgadez del tipo de cambio forward es rechazada para los mercados cambiarios internacionales. Este trabajo propone un modelo de equilibrio general dinámico y estocástico que genera variabilidad suficiente en las magnitudes de los excesos de rendimientos predecibles. Los ejercicios de simulación realizados sugieren que una alta persistencia de la política monetaria produce un mayor sesgo en el coeficiente estimado de la pendiente de la regresión entre la primera diferencia del logaritmo del tipo de cambio spot sobre la prima forward. Además, nuestro modelo sugiere que la naturaleza de la transmisión entre shocks monetarios puede explicar la paradoja del exceso de rendimiento. Por último, proporcionamos evidencia empírica de acuerdo con nuestros resultados teóricos para el tipo de cambio entre EEUU y Reino Unido. _________________________________________________________________________________ (*) Los autores agradecen al profesor Alfonso Novales sus valiosos comentarios y sugerencias. Jesús Ruiz agradece también los comentarios y sugerencias proporcionados por los miembros del CREB así como los medios que han puesto a su disposición durante el tiempo que estuvo invitado como visitante (febrero y marzo de 2002). Agradecemos el apoyo financiero del Ministerio de Educación a través de las becas PB98-0831 y BEC2000-1388-C04-03. 1 I nt r oduct ion The puzzle of biasedness on forward exchange rat e refers that the estimated slope coe¢ cients in the regression of the change in the logarithm of the spot rate on the forward premium signi…cantly departs from one (see Zhu (2002) , Tauchen (2001), Baillie and Bollerslev (2000), Baillie and Ostenberg (2000) and M cCallum (1994), among many others). Such discrepancy from the underlying valuein theuncovered int er est rate parity impliesthat the forward rate is not an unbiased predictor of the future spot rat e, suggesting the possibility of unexploited pro…t opportunities. Potential explanations of this excess return puzzle generally are assigned to three kind of categor ies: a) the most popular is that such pattern arises as a consequence of a time-varying risk premia (see Fama (1984)); b) a second explanation relies on the nature of expectations. Under no rational expectations agents do not e¢ ciently use the available informat ion set, incurring in systematic forecasting errors over a signi…cant number of time periods ahead (see Froot and Frankel (1989)) ; and c) the peso problem, that is, market part icipants anticipate by rational learning process a future discrete shift in policy that is not performed within the sample period analyzed (see Lewis, (1995) ). Even though a substantial number of studies have addressed the ability of general equilibrium models related to t he Lucas (1982) model to explain the forward premium puzzle (see, for example, Hodrick (1989), M acklem (1991) , Canova and Marrinan (1993), Bekaert (1994)), they unsuccessfully explained the substantial variability that occur in the magnitude of predictable excess returns. Currently, there is no conclusive theory explaining the behavior of the bias of test for a risk premium in forward exchange rates, and it is yet regarded as one of the most impor tant unresolved puzzle in international …nance. In this paper we develop a theoretical general equilibrium model to explain short and long-run risk premium in forward markets for foreign exchange that, not only pr ovide additional insights about the potential explaining factor s of the for ward risk premium, but also reproduce the forward pr emium anomaly under rational expectations. T he model takes as benchmark the Dutton’s model (1993) which is based on the general equilibrium models of Lucas (1982). Our model extend the …rst one in three ways: a) we consider two forward (one and two periods) exchange rates as a hedge instruments for spot exchange rate a more realistic approach to real markets in where di¤er ent time to maturity can be traded. T his enriches the 2 analysis because of it would be possible to identify t he e¤ect of the time to maturity in the derivative contract on the forward market risk premia, b) it is considered the possibility that domestic and foreign consumptions goods will be complementary or substitutes. Therefore, the model allows to estimate the impact of the nature of consumpt ions goods. If, for example, dollars are relatively risk, the uncertainty about the future spot exchange should a¤ect di¤erently on the forward risk pr emia under complement aries or substitutes consumption goods, and c) the weight of each, domestic and foreign, consumption good in the utility function is not necessary the same. Consequently, a broad set of scenarios can be simulated in order to explore for potential explanatory factors of the risk pr emium. The solut ion of the model involves to evaluate expectations of nonlinear expressions. Therefore, numerical solutions are provided. Under the assumption of rational expectations, our solution method allows to solve jointly for both prices and positions in one and two-periods ahead forward contr acts. This is an interesting extension relative to the Dutton‘s solution method. Simulation exercises are carried out with a variety of parameter values, revealing that, even when the econometric bias behind the regression of the change in the logarithm of the spot rate on the forward premium is taken into account, the model can reproduce the bias for forward exchange rate to pr edict the future evolution of spot rate. The model suggest what are the key factor generating high volatility for the risk premium is the persistence of the monetary policy. Under a relative high persistence the estimated slopes dr amatically decreases below one. M oreover , theoretical result s show that the time to maturity of forwar d contract is negatively correlated with the size of the slope coe¢ cient in the regression, that is, the estimated slopes corresponding to the long time to maturity cont ract are relatively lower. In accor dance with the theoretical simulations of the model, the paper reports empirical evidence for the US dollar -British pound, not only on the relationship between the correlation of the monetary shocks and the size of estimated slopes, but also on the linkage between the persistence of the monetary policy and the bias for forward exchange rate. The r est of the paper is organized as follows: sect ion 2 present the model. In section 3 simulations of risk pr emium are presented and theoretical r esults about the bias of forward premium are provided. Section 4 refers empirical evidence for the US-UK exchange rat e. Finally, section 5 summarizes and makes concluding remarks. 3 2 T he M odel There are two countries with its own currency and a single consumer. In each count ry the representative …rm r eceives an endowment of a single traded good. T he only tradable …nancial assets are the money forward periods exchange contracts. Also, there is no cont ingent claims markets, so all possibilities to reduce risk are concerning the forward exchange market, where two maturity contracts are available. The two consumers own titles to the …rms in their respective countries. The timing of the model can be summarized as follows: 1) at the beginning of each period, both … rms pay to the respective consumers in its count ry a dividend equal to all incomes achieved the previous period. Then, the consumer turns in its dividends for a new money, and t he old money becomes wor thless. This implies that all money will be spent; 2) after receiving the money supply, consumers liquidate t heir forward contracts traded in foreign exchange in the two previous periods, 3) consumers spend their money on the two goods. Domestic goods must be purchased with its own currency. All transactions take place at equilibr ium prices. 4) At the end of each period, consumers make forward contracts to delivery of currency in the next two periods. Endowments of goods and money supplies are stochastic, and its natural logarithm follow an autoregressive process with a Nor mal innovation. Let us to denote X t and M t for any good endowment or money supply, respectively: ln X t = ¹ X (1 ¡ ½X ) + ½X ln X t¡ 1 + »X ;t ; ³ »X ;t ~N 0; ¾2X ´ ³ ln X t¤ = ¹ X ¤ (1 ¡ ½X ¤ ) + ½X ¤ ln X t¡¤ 1 + »¤X ;t ; »X ¤ ;t ~N 0; ¾2X ¤ ln M t = ¹ M (1 ¡ ½M ) + ½M ln M t¡ 1 + »M ;t ; »M ;t ~N 0; ¾2M ln M t¤ = ¹ M ¤ (1 ¡ ½M ¤ ) + ½M ¤ ln M t¡¤ 1 + »¤M ;t ; ³ ³ , ´ ´ »M ¤ ;t ~N 0; ¾2M ¤ (1) , , ´ (2) (3) , (4) wheret he asterisk denotest he for eign country. Correlations between any four shocks (½M M ¤ ; ½X X ¤ ; ½M X ; ½M X ¤ ; ½M ¤ X ; ½M ¤ X ¤ ) are initially restricted to be zero. 2.1 T he Consum er ’s problem The utility function of the home consumer is a CES function: 4 1 [Á( CD;t ) ² + (1 ¡ Á) (CF ;t ) ² ]( 1¡ ° ) =² , (5) 1¡ ° where CD ;t and CF;t are the consumption levels of domestic and foreign goods at time t , ° is the coe¢ cient of r elative risk aversion, and 1¡1 ² is the elasticity of substitution, and Á is the weight for each consumption good. If ² approaches to zero consumption goods becomes more substitutes, whereas complementary arises when ² approaches to one. The parameter Á measures the weight of each consumption good in the utility function. T he optimization problem for the home consumer is: Ut = M ax E0 " 1 X 1 ¯ [Á(CD ;t ) ² + (1 ¡ Á) (CF;t ) ² ](1¡ 1¡ ° t= 0 t # ° ) =² (6) f CD;t ; CF;t g s:t: PD ;t CD;t + St PF;t CF ;t · Yt , S ¡ Ft¡ 1;1 S ¡ Ft¡ 2;2 Yt = M t + Tt¡ 1;1 t + Tt¡ 2;2 t , Ft¡ 1;1 Ft¡ 2;2 where PD;t and PF;t are the prices of domestic and foreign goods at time t, Yt is the total income in period t, St is the spot exchange rate, Ft¡ 1;1, Ft¡ 2;2 are the prices for the two maturity forward contracts available, Tt¡ 1;1 and Tt¡ 2;2 are the respective amount of its currency that the home country sold forward in the two previous periods. The money supply (M t ) plus the pro…ts on each forward currency trading in period t equals the t otal home income. A similar optimization problem can be pointed out for the foreign consumer , that is: M ax E0 ¤ f CD;t ; CF¤ ;t g s:t: " 1 X # ³ ´ ² i (1¡ ° )=² 1 h ³ ¤ ´² ¤ ¯ Á CD;t + (1 ¡ Á) CF;t (7) 1 ¡ ° t= 0 t ¤ PD;t CD¤ ;t + St PF ;t CF;t · Yt ¤St , Yt¤ = M t¤ + Tt¡¤ 1;1 St ¡ Ft¡ 1;1 + Tt¡¤ Ft¡ 1;1St 5 2;2 St ¡ Ft¡ 2;2 . Ft¡ 2;2 St 2.1.1 Opt im al good choices. In any period t the home consumer chooses levels of CD ;t and CF ;t that maximize Ut subject to the level of total home income. First order conditions for choice of CD ;t and CF ;t ar e: [Á(CD;t ) ² + (1 ¡ Á) (CF ;t ) ² ] [Á(CD;t ) ² + (1 ¡ Á) (CF ;t ) ² ] 1¡ ° ² 1¡ ° ² ¡ 1 ¡1 (CD;t ) ² ¡ 1 ¡ ¸ t PD ;t = 0 , (8) ( CF;t ) ² ¡ 1 ¡ ¸ t St PF ;t = 0 , (9) Yt ¡ PD;t CD ;t ¡ PF ;t St CF ;t = 0 . (10) From 8 and 9 yields the following relationships: " CF;t = (1 ¡ Á) PD;t ÁPF;t St #¾ CD;t , (11) where ¾ = 1¡1 ² is the elasticity of substitution. Using 11 and the budget constr aint, t he demand function for the domestic good is the following: CD ;t = 1¡ ¾ PD;t + ³ Yt PD¡ ;t¾ ´ 1¡ Á ¾ (St PF;t ) 1¡ ¾ Á . (12) Substituting 12 into equation 11 the next demand function for t he foreign good arises: " CF ;t = (1 ¡ Á) PD ;t ÁPF;t St #¾ 1¡ ¾ PD;t + ³ Yt PD¡ ;t¾ ´ 1¡ Á ¾ (St PF;t ) 1¡ ¾ Á (13) Similar equation to (12) and (13) can be easily found for the foreign country: " CF¤ ;t ¤ CD;t = = (1 ¡ Á) PD ;t ÁPF;t St PD1¡;t ¾ + ³ #¾ CD¤ ;t Yt¤ St PD¡ ;t¾ ´ 1¡ Á ¾ (St PF ;t ) 1¡ ¾ Á ¤ Substituting 15 into 14 we obtain the analytical expression for CF;t . 6 (14) (15) 2.1.2 For war d Cont r act ing As well as the allocation of current r esources between the two goods, the home consumer choose in per iod t the levels of the one and two periods forward contracting, that is Tt;1 and Tt;2 . The Euler conditions are: " Et ¸ t+ 1¯ à " Et ¸ t+ 2¯ St+ 1 ¡ Ft;1 Ft;1 t+ 1 à St+ 2 ¡ Ft;2 Ft;2 t+ 2 ! # = 0 , (16) = 0 , (17) ! # where Et denotes the conditional expectation to the information set available in period t. From 16: Et [¸ t+ 1St+ 1] = E t [¸ t+ 1Ft;1 ] , and taking into account 8 yields: Ft;1 = h i @Ut+ 1 1 @CF ;t+ 1 PF ;t + 1 h i Ut + 1 1 Et @@ CF ;t + 1 PF ;t + 1 St + 1 Et . (18) Similar r earranging from 17 when taking into account 8 leads t o the following expression for the two-periods forward price: Ft;2 = h i @Ut+ 2 1 @CF ;t+ 2 PF ;t + 2 h i Ut + 2 1 Et @@ CF ;t + 2 PF ;t + 2 St + 2 Et . (19) Analogous expr essions to (18) and (19) can be obtained when the foreign consumer chooses in period t the levels of the one and two periods forward ¤ and T ¤ : contracting, that is Tt;1 t;2 · Ft;1 = Et · Et Et · Et ¸ @Ut¤+ 1 1 @CF¤ ;t + 1 PF ;t + 1 St + 1 · Ft;2 = ¤ @Ut+ 1 1 ¤ @CF ;t+ 1 PF ;t + 1 ¤ @Ut+ 1 2 ¤ @CF ;t+ 2 PF ;t + 2 , (20) ¸ . (21) ¸ @Ut¤+ 2 1 @CF¤ ;t + 2 PF ;t + 2 St + 2 7 ¸ 2.2 2.2.1 M arket -Clear ing Equilibr ium in t he Goods M ar ket . The world constraints on consumptions of the two traded goods in both count ries implies that the total endowment of the two goods must be equal the consumption of each good in the respective countries, that is: ¤ CD;t + CD;t = X D ;t , (22) ¤ CF;t + CF;t = X F;t . (23) Equilibrium prices of the two goods depend on t he home and foreign money suppliesas well as their total endowment in each countr y. Taking into account that a) money is worthless after each period and b) each country’s good only can be purchased with the country’s curr ency, the following cash-in-advance spending constraints must be hold: PD ;t X D;t = M t , (24) PF;t X F;t = M t¤ . (25) Since goods endowments X D;t and X F;t , and money supplies M t and M t¤ are exogenous, the two above equations determine pr ices of consumpt ion goods. The solution of the model requires the evaluation of expectations in equations 18 and 19, in where highly non-linear expressions appear. This avoids the possibility of an analytical solution. Appendix 1 provides detailed explanation about the solution method to obt ain simulated equilibrium in spot and forward exchange markets. It allows the joint search of all variables (prices and positions) concerning the forward market. In equilibrium, the following relationships between home and foreign derivative positions holds Tt ¡ l ;l = ¡ Tt¡¤ 3 l;l , l = 1; 2. (26) Simulat ion of for war d pr i ces and r isk pr emiums The equilibrium spot rates can be obtained as follows: using the budget ¤ = Y ¤S constr aints, PD ;t CD ;t + St PF;t CF;t = Yt and PD ;t CD¤ ;t + St PF;t CF;t t t 8 and equations (11) and (14) , we can solve analytically the spot exchange as a function of the exogenous stochastic variables X D;t ; X F ;t ; M t ; M t¤: à 1 ¡ Á X F;t St = Á X D;t 3.1 ! " Mt . M t¤ (27) D e…nit ion of R isk Pr em ium To avoid the implications of Siegel’s paradox we use the following de…nition of the r isk premium in the forward mar ket: r pt;t+ l = f t;l ¡ E t (st+ l ) , l = 1; 2. (28) where Et (¢) denotes the mathematical expectation conditioned on the set of all relevant infor mation at time t, st is the logarithm of thedomestic currency pr ice of foreign cur rency at time t and f t;l is the logarithm of the forward exchange rate with delivery at time t + l. 3.2 3.2.1 Par am et er scenar ios w her e t he for war d pr em ium anom aly ar ises Test ing t he unbiasedness hyp ot hesis The main objective of the paper is to analyze the parameter set that could repr oduce the forward premium bias. T he central hypothesis that we analyze in this paper is the Uncovered Interest Rate Parity (UIP) condition, which states that: Et (¢ st+ l ) = f t;l ¡ st = i t ¡ i ¤t; (29) where Et denotes the conditional expectation to the informat ion set available on time t; i t and i ¤t are the interest rates on domestic and foreign deposits, respectively, and ¢ denotes the …rst di¤er ence operator , that is, ¢ st+ l ´ st+ l ¡ st+ l¡ 1. To test for unbiasedness hypothesis, the lit er ature has widely focused on the following regression relating the change in the spot rate to the forwardspot spread: ¢ st+ l = ®l + ¯ l (f t;l ¡ st ) + ut+ l;l ; 9 (30) The estimat ion of equation (30) tries to test the ability of the forwardspot di¤erential to forecast the direction of change in spot rate. Regardless the sampling frequency, the UIP condition implies that ®l = 0 and ¯ l = 1. However, empirical evidence has widely reported on estimated slopes that turn out to be below than one or even negatives1. This …nding not only reject the UIP condition, but also is contradictory with either form of the expectations hypothesis. The analytical expression for the OLS estimation of ¯ l is: ¯ ol s = Cov (f t ;l ¡ st ; st+ l ¡ st ) , V ar (f t;l ¡ st ) (31) where V ar (¢) refers to variance, and Cov (¢) denotes the covariance. As pointed out in Engel (1996), if the estimator is consistent , under rational expectations it follows that: ³ ´ p lim ¯ ol s = 1 ¡ ¯ r p where ¯ r p = Cov ( E t (s t+ l )¡ st ; f t ;l ¡ E t ( st + l ) ) + V ar ( f t ;l ¡ E t ( st + l ) ) V ar ( f t;l ¡ st ) (32) . From t his expression it can be obser ved that low values of ¯ ols can be explained under rational expectations if Var (f t ;l ¡ E t (st+ l )) is enough large. The risk premium is widely considered themost likely source of thepuzzle, but taking into account the regression results reported in the literature the required volatility are far larger than most researchers would accept. One of the major task in the lit er ature concerns to explain why the risk premium has such a large variance. Our model provide some insights about t his issue. 3.2.2 T heor et ical R esult s In all numerical simulations the discount factor ¯ and t he relative risk aversion ° are constant and equal to 0:99 and 1:50, respectively2. We consider a variety of scenarios than can be summarized as follows: a) we focus the analysis on the e¤ects of the monetary policy (we leave further work the analysis of the e¤ects of real shocks on risk premia in for ward markets for 1A recent survey can be found in Engel (1996) . values inside t he int erval [0:90; 0:99] and [1:10; 5:00] for ¯ and ° , lead t o similar r esult s t o t hose report ed in t he paper. 2 Paramet er 10 foreign exchange). T herefore, only a uncertainty source is considered: monetary shocks. This way we consider either one or two shocks; b) we distinguish between situations in where there is no persistencein theshocks of both countries from other ones in where only the home country have persistence in the monetary shock 3. The nature of the int er action between monet ary policies is also examined. When two shocks are considered we allow for three possibilities: uncorrelated, positive and negatively correlated monetary shocks. The considered absolute value for t he correlation coe¢ cient between domestic and foreign shocks is 0.9. To summarize the theoretical results from estimating equat ion 30 using simulated spot and forward exchange rates with " = 14 , Table 1 reports the volatility of the forward premium and Figures 1 to 10 (Appendix 3) depict the estimated slopes as a function of the correlation between monetary shock and the persistence of the monetary policy when only a monetary shock is considered. Also we provide the con…dence intervals at the 5% signi…cance level based on the simulated distribut ion of slopes with one hundred of theor et ical observations. Several interesting questions emerge from this information set: 1. The estimated slopes are generally lower than one, a consistent …nding with expression 30. This means that ¯ r p > 0. This …nding has been documented in many empirical studies (see, for example, Bilson (1981) , Fama (1984), Bekaert and Hodrick (1993), Backus et al. (1993) and Mark et al. (1993)). 2. There is a negative relationship between the estimated slope coe¢ cient and the time to maturity. In the long-run the forward bias is greater than in shor t-r un, re‡ecting a higher uncertainty in the futures evolution of spot rates, 3. A relative higher persist ence in the monetar y policy pr oduces lower estimated value for the slope. This …nding is consistent with those reported in Baillie and Bollerslev (2000) . Those authors simulate forwar d premiums. according to a highly stylized UIP-FIGARCH model (Fractionally Integrated GARCH model), showing that a long memory 3 Under no corr elat ion bet ween monet ar y shocks t his sit uat ion can be int erpret ed as t he home count ry behaves as a leader since it can updat e t he forecast ing of money supply. The considered aut oregressive paramet er is 0.9. 4 Similar result s ar e found wit h " = 0, which are available fr om t he aut hors upon request . 11 in the forward premium produces wide disper sion in the slope coe¢ cients. Tauchen (2001) simulates the sampling distribution of t he slope coe¢ cient in equation (30), showing that such to be the case when spot rates are generated with a near to non-stationary AR(1) process. This is not surprising when equation (27) is observed. Under high persistence in the monetary policy of the domest ic countr y, spot rate is very autocorrelat ed, and consequently the forward premium should have high persistence. The negative relationship is clearer when. 4. More interestingly, our model suggest that under a relative high persist ence in t he domestic/ foreign monetary policy the volatility of the forward premium is greater. From table 1, it can be observed t hat the volatility under persistence is above …ve times the volatility that corresponds to the case where monetary policy forecast can not be updated using current information. 5. Also, the transmission of the monetary policy e¤ects between both count ries appears to be a signi…cant factor to explain departures from the UIP. Under a relat ive high persistence, the estimated slope show higher discrepancy with t he unitary value when monetary shocks are positively correlated. Indeed the maximum median anomaly for all simulations appears when shocks are positively cor related and the domestic monetary policy is very persistent. This a realist ic scenario for most of empirical studies that analyses the exchange rate between US and other country, which generally takes as a benchmark the Fed´ s monetary policy. Deviations from the UIP condition are negligible regardless the correlation between the monetary shocks only under no persistence in the monetary policy of both countr ies. But, what about the ability of t he model to generate bias for forward exchange rate?. To answer this question Figures 11 to 20 depict the sum of the asymptotic bias plus the median estimated slopes coe¢ cients and their corresponding con…dence intervals at the 5% signi…cance level, again using one hundr ed of theoretical observations. T hose graphs show two relevant aspects: 1. once we have …ltered the econometric bias a discrepancy with the unitary value remains, r evealing that the theoretical model can generate a bias for forward exchange rate. 12 2. the relative persistence in the monetary policy appears as the key factors behind the forward unbiasedness. It can be observed that the con…dence intervals are larger enough under a relative high persistence, suggesting a higher variability in the potential estimated slopes. In particular, when the two countries apply persistence in the monetary policy the con…dence intervals are much less informative. M oreover , under such scenario the corr elation between monetary shocks is an additional factor that explain the forward bias, revealing a higher median deviation fr om one when shocks are positively rather than negatively correlated (see Figures 19 and 20). Also in this case the con…dence intervals are less informative than under no In summary, our model suggest t hat the anomaly should appear when one country act as a leader when monetary policy is implemented and a high persistence is applied. Such is the case in most of empirical analysis that concerns the dollar exchange rate. In the next section we provide empirical evidence about this. 4 Em pi r ical evi dence. T he U S dollar -B r it ish pound ex chage r at e In this section we provide empirical evidence focusing not only on the relationship between the t ransmission of monetary shocks and slope coe¢ cients, but also on the link between the monetar y persistence and the bias for the US dollar-British puund forward exchange rate. The consider ed time to maturity is one month and the sample period covers from December, 1986 to November 2001. The model pr edict a negative relationship between the estimated slopes and the correlation between monetary shocks. Figure 21 show the XY plot of the r olling correlat ion between the M 1 cyclical components5 of US and UK and the rolling slopes using one month time to maturity US-UK forward exchange rate for the already referred sample period. The window size to compute the rolling statistics corresponds to …ve years. The US average rolling persistence in this period was 0.77. Clearly, and according with our theoretical results, a negative relationship ar ises. To quantitative account for this statement , we perform the following regression: 5 The Hodr ick-Prescot t …lt er is used t o det rend t he monet ary aggregat e. 13 ¯ t = ±0 + ±1½M M ¤ ;t + ul;t l = 1; 2 (33) where ¯ t denotes the actual rolling slope and ½M M ¤ ;t is the correlation coe¢ cient between the cyclical M1 components. The …tted line is: ¯^ l;t = 0:88(0:09) ¡ 2:39 (0:21) ½M M ¤ ;t where standard errors are in parentheses. The R-squared becomes 0.42. Also, the model suggests a negative relationship between the estimated slopes coe¢ cients and the persistence of the monetar y policy. Figure 22 depict the rolling persistence of the UK monetary policy and the corresponding rolling slopes using a …ve years moving window over the above referred sample. It can be observed that in three subsamples a negative relationship appears, suggesting that additional factors are a¤ecting the forward bias along the overall sample. 5 Sum mar y and concludi ng r emar k s In this paper we examine the bias of tests for a risk premium in forward exchange rat es which refers to signi…cant discrepancies with t he unitary value in the estimat ed slope coe¢ cients from regressions of the change in the logarithm of the spot rate on the forward premium. We perform a theoretical analysis by extending t he dynamic and stochastic general equilibrium model with goods endowment proposed in Dutton (1993). Our contribution is the intr oduction of a two-period forward contract in the derivative market . Also, a solution method under rational expectations is provided. Our main objective is to explore the e¤ects of the monetary policy and their interactions between the domest ic and for eign country on the behavior of the risk premium in order to explain the inconsistency wit h the UIP condition. Our simulations results suggest that a high persist ence in the domestic monetary policy produces gr eater volatility in the forward premium, and consequently the estimated slope coe¢ cients show greater deviations from one. M or eover, the nature of the t ransmission between monetary shocks is a potential explaining factor for excess return puzzle. Under persistence, the estimated slopes dramatically decrease below one when monet ary shocks are positively correlated. Finally, we …nd that the time to maturity of the derivative contract is positively related with the bias of risk premium in forward exchange rates. The UIP condition only holds in the absence of persistence 14 when monetary shocks are uncorrelat ed. or negatively correlated. However , this is an unlikely scenario for most of developed economies. The paper provides empir ical evidence for the US dollar-British pound exchange rate. In accordancewith our theoretical results, a negative relationship between the forward bias and the UK monetary persistence is observed along three di¤erent subsamples from December, 1986 to November 2001. Moreover, a negative relationship between the forward bias and the cor relation between monet ary shocks arises during the overall sample, where a high persistence in the US monetary policy is detected. While the focus of this paper is the e¤ect of the monetary policy, a similar analysis can be made taking into account t he presence of both monetary and real shocks. We leave further work under such scenarios for further research. R efer ences [1] Backus, D., Gregory A. and C. Telmer (1993), Accounting for forward rates in markets for foreign currency, Jour nal of Finance 48, 1887-1908. [2] Baillie, R.T . and T . Bollerslev (2000), T he forward premium anomaly is not as bad as you think, Jour nal of International Money and Finance 19, 471-488. [3] Baillie, R.T . and W. P. Ostenberg, Deviations from daily uncovered int er est rate parity and t he role of intervention, Jour nal of International Financial Markets, Institutions and Money 10, 363-379. [4] Bilson, J. (1981), The ” speculative e¢ ciency” hypothesis, Jour nal of Business 54, 435-452. 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Frankel (1989), Forward discount bias: is it an exchange risk premium, The Quater ly Jour nal of Economics 104, 139161. [12] Hodr ick, R.J. ( 1989), Risk, uncertainty and exchange rates, Journal of Monetary Economics 23, 433-459. [13] Lewis, K .K. (1995). Puzzles in international …nancial market s. In: Gr ossman, G. and K . Rogao¤ ( Eds.). Handbook of International Economics, vol. 3. North Holland, Amsterdam, pp. 1913-1971. [14] Lucas, Robert (1982), Interest rates and currency prices in a two-count ry world, Journal of Monetary Economics 10, 335-59. [15] Macklem, R.T . (1991), Forward exchange rates in art i…cial economies, Jour nal of International Money and Finance 10, 365-391. [16] Mark, N., Wu Y. and W. Hai (1993) , Understanding spot and forward exchange rate regressions. Ohio State University, Columbus, OH. [17] McCallum, B.T. (1994), A reconsiderat ion of the uncovered interest rate parity relationship, Journal of Monetar y Economics 33, 105-391. [18] Tauchen, G. (2001), The bias of test s for a risk premium in forward exchange rates, Journal of Empir ical Finance 8, 695-704. [19] Zhu, Z. (2002), T ime-varying forward bias and the expected excess return, Jour nal of Inter national Financial Markets, Institutions and Money 12, 119-137. 16 Appendix 1. Solution Method This appendix contains the step that we use in the solution method. As we pointed out in Section 3, the problem concerning the home and foreign consumer is highly non-linear, not allowing to achieve an analytical solution. Therefore a numerical approach must be used. After providing numerical values for the structural parameters involved in the theoretical economy, that is, f ¯ ; ° ; Á; "; ¾X ; ¾X ¤ ; ¾M ; ¾M ¤ ; ¹ X ; ¹ X ¤ ; ¹ M ; ¹ M ¤ ; ½X ; ½X ¤ ; ½M ; ½M ¤ g, the next stages are: 1. We obtain one hundred realizations for the st ochastic variables X D ;t ; X F;t ; M t ; M t¤ in each time period t = 1; :::100. 2. One hundred realizations of bot h home and foreign prices of the consumption goods are computed according to equations (24) and (25), in each time period. Let us to denot e this numer ical set as f (PD;t;i ; PF ;t;i ) ; i; t = 1; :::100g, where i and t denote the realization and the time period, respectively. 3. Similar numerical set to the previous one for PD and PF is computed for thespot exchange rate using equation (27), that is, f St;i i ; t = 1; :::100g. Computation of theforward pricesand derivative positions for the one and two period ahead traded contract s [Ft;1; Ft;2 ; Tt;1Tt;2]. From equations (11) and (14), substituting into equations (18) and (19) the following expressions can be obtained: " Et Ft;1 = " Et = "¡ 1 CD;t+ 1 "¡ 1 CD;t+ 1 ³ ³ ÁCD" ;t+ 1 + (1 ¡ ´ Á)CF" ;t+ 1 ´ " " ÁCD;t+ 1 + (1 ¡ Á)C F;t+ 1 E t [WD ;t+ 1] , Et [WD;t+ 1 =St+ 1] 1¡ ° " 1¡ ° " ¡ 1 # 1 PD ;t + 1 ¡1 # 1 PD ;t + 1 St + 1 (34) 17 " Et Ft;2 = " "¡ 1 CD;t+ 2 Et ³ "¡ 1 CD;t+ 2 ³ ÁCD" ;t+ 2 ´ Á)CF" ;t+ 2 + (1 ¡ ´ " " ÁCD;t+ 2 + (1 ¡ Á)C F;t+ 2 1¡ ° " # ¡ 1 1 PD ;t + 2 1¡ ° " ¡1 # 1 PD ;t + 2 St + 2 E t [WD ;t+ 2] , Et [WD;t+ 2 =St+ 2] = " ¤ "¡ 1 CF;t+ 1 Et F t;1 = " CF¤ ;t+"¡11 Et ³ ³ (35) ¤ " ÁCD;t+ 1 ¤ " ÁCD;t+ 1 ´ + (1 ¡ + (1 ¡ 1¡ ° " ¤ " Á)CF;t+ 1 ¤ " Á)CF;t+ 1 ´ # ¡ 1 1 PF ;t+ 1¡ ° " ¡ 1 1 # 1 PF ;t + 1 St+ 1 Et [WF;t+ 1] , E t [WF ;t+ 1=St+ 1 ] = " (36) ³ ´ 1¡ ° " ¤ "¡ 1 ¤ " ¤ " E t CF;t+ 2 ÁC D;t+ 2 + (1 ¡ Á)C F;t+ 2 F t;2 = " Et CF¤ ;t+"¡21 ³ ¤ " ÁCD;t+ 2 + (1 ¡ ¤ " Á)CF;t+ 2 ´ 1¡ ° " ¡ 1 # 1 PF ;t+ ¡ 1 2 # 1 PF ;t + 2 St+ 2 Et [WF;t+ 2] . (37) E t [WF ;t+ 2=St+ 2 ] We solve jointly Ft ;1, Ft;2, Tt;1 and Tt;2 by searching values that satisfy the following approximations of the equations (34) to (37): = " P N "¡ 1 i= 1 CD ;t+ 1;i Ft;1 = " P N "¡ 1 i = 1 CD ;t+ 1;i ³ ³ ÁCD" ;t+ 1;i ÁCD" ;t+ 1;i + (1 ¡ + (1 ¡ " Á)CF;t + 1;i Á)CF" ;t+ 1;i ´ ´ 1¡ ° " # ¡ 1 1 PD ;t + 1;i 1¡ ° " ¡ 1 # 1 PD ;t + 1;i St+ 1;i P N = P N i = 1 [WD;t+ 1] i=1 P N [WD ;t+ 1=St + 1 ] " "¡ 1 i = 1 C D;t+ 2;i Ft;2 = P N " "¡ 1 i = 1 CD ;t+ 2;i = ³ ³ (38) ÁCD" ;t+ 2 " ÁCD;t+ 2;i P N P N , i = 1 [WD;t+ 2;i ] i = 1 [WD ;t+ 2;i =St+ 2;i ] + (1 ¡ + (1 ¡ , Á)CF" ;t+ 2;i " Á)CF;t+ 2;i ´ ´ 1¡ ° " 1¡ ° " # ¡1 1 PD ;t + 2;i ¡ 1 PD ;t+ # 1 2;i St + 2;i (39) 18 " P N F t;1 = ¤ "¡ 1 i = 1 C F;t+ 1;i " P N ¤ "¡ 1 i = 1 CF;t+ 1;i = " ¤ "¡ 1 i= 1 CF;t+ 2;i P N ³ ³ P N ¤ " ÁCD;t+ 2;i i = 1 [WF;t + 2;i ] i = 1 [WF ;t+ 2;i =St+ 2;i ] # ¡ 1 1 PF ;t + 1¡ ° " ¡1 PF ;t+ 1;i # 1 1;i St + 1;i (40) ¤ " ÁCD;t + 2;i P N ´ 1¡ ° " , i = 1 [WF ;t+ 1;i =St+ 1;i ] " + (1 ¡ ´ Á)CF¤ ;t+" 1 ¤ " ¤ " ÁCD;t+ 1;i + (1 ¡ Á)C F;t+ 1;i i = 1 [WF;t + 1;i ] P N ¤ "¡ 1 i = 1 CF;t+ 2;i = " ÁCD¤ ;t+ 1;i P N P N F t;2 = ³ ³ + (1 ¡ + (1 ¡ ¤ " Á) CF;t+ 2;i ¤ " Á)CF;t+ 2;i ´ ´ 1¡ ° " 1¡ ° " # ¡ 1 1 PF ;t + 2;i ¡1 PF ;t+ # 1 2;i St + 2;i . (41) Taking into account that under rat ional expectations Et [Wt+ 1 ] = ª 1 at + Et¡ 1 [Wt+ 1], where at is a white noise, the expression of the two period forwar d price in t ¡ 1 is: ³ P N Ft¡ 1;2 ´ P N ¡ ª D ;1 WD;t ¡ i= 1 [WD ;t;i ] ³ ´ , (42) = PN P N ~ [W =S ] ¡ ª W =S ¡ [W =S ] D;t+ 1;i t+ 1;i D ;1 D;t t D;t;i t ;i i= 1 i= 1 i= 1 [WD ;t+ 1;i ] or equivalently for the foreign consumer: ³ P N ´ P N i = 1 [WF;t+ 1;i ] ¡ ª F ;1 WF;t ¡ i= 1 [WF ;t;i ] ³ ´ . (43) Ft¡ 1;2 = P N P N ~ [W =S ] ¡ ª W =S ¡ [W =S ] F ;t+ 1;i t+ 1;i F ;1 F;t t F ;t;i t;i i= 1 i=1 Next, we proceed as follows: i )We posit initial conditions for the paramet ers { ª (0) ~ (0) ( 0) ~ ( 0) D;1ª D ;1ª F ;1ª F ;1} . ii ) Also, we need an init ial vector. Let us to denote it by f F0;1; F¡ 1;2; T0;1 ; T¡ 1;1g. Then, one hundred realizations of CD ;1;i ; CF ;1;i ; CD¤ ;1;i ; CF¤;1;i ; Y1;i ; Y1;i¤ in t = 1 trough equations (11) , (14) and the following expressions: à Y1;i = M 1;i ¤ Y1;i = ¤ M 1;i ! à ! S1;i ¡ F0;1 S1;i ¡ F ¡ 1;2 + T0;1 + T¡ 1;2 ; F0;1 F¡ 1;2 à ! à ! S1;i ¡ F0;1 S1;i ¡ F¡ 1;2 ¡ T0;1 ¡ T¡ 1;2 ; F 0;1 S1;i F¡ 1;2S1;i 19 CD ;1;i = CD¤ ;1;i = ¾ PD¡ ;1;i + ¾ PD¡ ;1;i + ³ ¾ Y1;i PD¡ ;1;i ´ 1¡ Á ¾ (S1;i PF ;1;i ) 1¡ ¾ Á ¡ ¾ Y1;i¤ S1;i PF;1;i ³ ´¾ 1¡ Á (S1;i PF ;1;i ) 1¡ ¾ Á ¤ iii ) With the pr evious data set, f CD;1;i ; CF ;1;i ; CD;1;i ; CF¤;1;i ; Y1;i ; Y1;i¤ g100 i = 1 , we iterate using the Gauss-Newton algorithm in the system concerning equations ( 38), (39), (42) and (43). After achieving the …xed point in the space (F1;1; F0;2; T1;1; T0;2 ) and evaluating in t = 1 with the variables f F1;1; F0;2 ; T1;1; T0;2 g the corresponding expressions, it is possible to compute values for ¤ CD ;1; CF;1 ; CD¤ ;1; CF;1 ; Y1; Y1¤, independently of the realization values. iv) T he steps ii ) and iii ) are repeated recursively for each time period, allowing to obtain the numerical solutions for the remainder of the sample size, ¤ that is, f CD ;t ; CF;t ; CD;t ; CF¤ ;t ; Yt ; Yt¤ g100 t= 2 : However , this solution depends on (0) ~ (0) (0) ~ (0) the initial condit ion { ª D;1 ª D;1 ª F;1ª F;1} . To …lter this e¤ect, we estimate an autoregressive pr ocess for t he expr essions of WD ;t ; (WD ;t =St ); WF ;t ; (WF ;t =St ) that can be computed with the simulated series of the previous solution. We use …ve lags in the AR speci…cat ion, a robust structur e in or der to forecast theprevious expressions. With the …tted autoregressive processes, estimation 0 (0) ~ (0) ~ (0) of ª s are recovered to evaluate the discrepancy with (ª (0) D;1 ; ª D;1 ; ª F;1 ; ª F;1 ) ¡ 6 using the euclidean norm. The used convergence criterion is 10 : When ¤ ; C ¤ ; Y ; Y ¤} 100 is the … the norm is lower, { CD ;t ; CF ;t ; CD;t nal numerical t F;t t t= 1 solution, whereas the norm is higher we back t o step i) to iterate with the ~ ( 0) ( 0) ~ ( 0) new initial condition for the vector f ª (0) D ;1ª D ;1ª F ;1ª F ;1g. 20 Appendix 2. St atist ical Tables. Table 1. Risk premium volatility ¾2M = 0:005; ½M = ½M ¤ = ½X = ½X ¤ = 0; ¾2M ¤ = ¾2X = ¾2X ¤ = 0 regression with l = 1 regression with l = 2 Á = 0.9 Á = 0.1 Á = 0.9 Á = 0.1 1 [V ar (f t;l ¡ E (st+ l ))] 2 0.0017 0.0017 0.0012 0.0012 ¾2M = 0:005; ½M = 0:9; ½M ¤ = ½X = ½X ¤ = 0; ¾2M ¤ = ¾2X = ¾2X ¤ = 0 regression with l = 1 regression with l = 2 Á = 0.9 Á = 0.1 Á = 0.9 Á = 0.1 1 2 [V ar (f t;l ¡ E (st+ l ))] 0.0092 0.0092 0.0168 0.0169 2 2 ¾M = ¾M ¤ = 0:005; ½M M ¤ = 0; ½M = ½M ¤ = ½X = ½X ¤ = 0; ¾2X = ¾2X ¤ = 0 regression with l = 1 regression with l = 2 Á = 0.9 Á = 0.1 Á = 0.9 Á = 0.1 1 [V ar (f t;l ¡ E (st+ l ))] 2 0.0019 0.0019 0.0023 0.0022 2 2 ¾M = ¾M ¤ = 0:005; ½M M ¤ = 0:9; ½M = ½M ¤ = ½X = ½X ¤ = 0; ¾2X = ¾2X ¤ = 0 regression with l = 1 regression with l = 2 Á = 0.9 Á = 0.1 Á = 0.9 Á = 0.1 1 [V ar (f t;l ¡ E (st+ l ))] 2 0.0008 0.0008 0.0006 0.0006 2 2 ¾M = ¾M ¤ = 0:005; ½M M ¤ = ¡ 0:9; ½M = ½M ¤ = ½X = ½X ¤ = 0; ¾2X = ¾2X ¤ = 0 regression with l = 1 regression with l = 2 Á = 0.9 Á = 0.1 Á = 0.9 Á = 0.1 1 [V ar (f t;l ¡ E (st+ l ))] 2 0.0020 0.0020 0.0020 0.0020 ¾2M = ¾2M ¤ = 0:005; ½M M ¤ = 0; ½M = 0:9; ½M ¤ = ½X = ½X ¤ = 0; ¾2X = ¾2X ¤ = 0 regression with l = 1 regression with l = 2 Á = 0.9 Á = 0.1 Á = 0.9 Á = 0.1 1 2 [V ar (f t;l ¡ E (st+ l ))] 0.0074 0.0074 0.0153 0.0154 2 2 ¾M = ¾M ¤ = 0:005; ½M M ¤ = 0:9; ½M = 0:9; ½M ¤ = ½X = ½X ¤ = 0; ¾2X = ¾2X ¤ = 0 regression with l = 1 regression with l = 2 Á = 0.9 Á = 0.1 Á = 0.9 Á = 0.1 1 [V ar (f t;l ¡ E (st+ l ))] 2 0.0094 0.0094 0.0187 0.0188 2 2 ¾M = ¾M ¤ = 0:005; ½M M ¤ = ¡ 0:9; ½M = 0:9; ½M ¤ = ½X = ½X ¤ = 0; ¾2X = ¾2X ¤ = 0 regression with l = 1 regression with l = 2 Á = 0.9 Á = 0.1 Á = 0.9 Á = 0.1 1 [V ar (f t;l ¡ E (st+ l ))] 2 0.0070 0.0069 0.0127 0.0109 Not e: Á measures t he degree of substit ut ability or complement ary. 21 Appendix 3. Figures Median Estimated slopes One-period ahead forward premium One monetary shock Dashed lines are the bands for the 95% confidence interval 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country 0.7 0.8 0.9 Figure 1 Median estimated slopes Two-periods ahead forward premium One monetary shock Dashed lines are the bands for the 95% confidence interval 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 0.1 0.2 0.3 0.4 0.5 persistence in the domestic country Figure 2 22 0.6 0.7 0.8 0.9 1.4 Median estimated slopes One-period ahead forward premium Two monetary shocks. No persistency in the foreign country Dashed lines are the bands for the 95% confidence interval 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country 0.7 0.8 0.9 Figure 3 Median estimated slopes Two-periods ahead forward premium Two monetary shocks. No persistency in the foreign country Dashed lines are the bands for the 95% confidence interval 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country Figure 4 23 0.7 0.8 0.9 Median estimated slopes One-period ahead forward premium Two monetary shocks. High persist. in the foreign country Dashed lines are the bands for the 95% confidence interval 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country 0.7 0.8 0.9 Figure 5 Median estimated slopes Two-periods ahead forward premium Two monetary shocks. High persist. in the foreign country Dashed lines are the bands for the 95% confidence interval 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country Figure 6 24 0.7 0.8 0.9 Median estimated slopes One-period ahead forward premium Two monetary shocks. No persistency in both countries Dashed lines are the bands for the 95% confidence interval 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 Correlation between monetary shocks 0.3 0.5 0.7 0.9 Figure 7 Median estimated slopes Two-periods ahead forward premium Two monetary shocks. No persistency in both countries Dashed lines are the bands for the 95% confidence interval 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 Correlation between monetary shocks Figure 8 25 0.3 0.5 0.7 0.9 Median estimated slopes One period ahead forward premium Two monetary shocks. High persist. in the domestic country Dashed lines are the bands for the 95% confidence interval 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 Correlation between monetary shocks 0.3 0.5 0.7 0.9 Figure 9 Median estimated slopes Two- periods ahead forward premium Two monetary shocks. High persist. in the domestic country Dashed lines are the bands for the 95% confidence interval 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 Correlation between monetary shocks Figure 10 26 0.3 0.5 0.7 0.9 Median estimated slopes + asymptotic bias One-period ahead forward premium One monetary shock Dashed lines are the bands for the 95% confidence interval 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country 0.7 0.8 0.7 0.8 0.9 Figure 11 Median estimated slopes + asymptotic bias Two-periods ahead forward premium One monetary shock Dashed lines are the bands for the 95% confidence interval 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country Figure 12 27 0.9 Median estimated slopes + asymptotic bias One-period ahead forward premium Two monetary shocks. No persistency in the foreign country Dashed lines are the bands for the 95% confidence interval 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country 0.7 0.8 0.9 0.7 0.8 0.9 Figure 13 Median estimated slopes + asymptotic bias Two-periods ahead forward premium Two monetary shocks. No persistency in the foreign country Dashed lines are the bands for the 95% confidence interval 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country Figure 14 28 Median estimated slopes + asymptotic bias One-period ahead forward premium Two monetary shocks. High persist. in the foreign country Dashed lines are the bands for the 95% confidence interval 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country 0.7 0.8 0.9 0.7 0.8 0.9 Figure 15 Median estimated slopes + asymptotic bias Two-periods ahead forward premium Two monetary shocks. High persist. in the foreign country Dashed lines are the bands for the 95% confidence interval 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 persistence in the domestic country Figure 16 29 Median estimated slopes + asymptotic bias One-period ahead forward premium Two monetary shocks. No persistency in both countries Dashed lines are the bands for the 95% confidence interval 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 Correlation between monetary shocks 0.3 0.5 0.7 0.9 0.5 0.7 0.9 Figure 17 Median estimated slopes + asymptotic bias Two-periods ahead forward premium Two monetary shocks. No persistency in both countries Dashed lines are the bands for the 95% confidence interval 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 Correlation between monetary shocks 0.3 Figure 18 30 Median estimated slopes + asymptotic bias One period ahead forward premium Two monetary shocks. High persist. in the domestic country Dashed lines are the bands for the 95% confidence interval 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 Correlation between monetary shocks 0.3 0.5 0.7 0.9 0.5 0.7 0.9 Figure 19 Median estimated slopes + asymptotic bias Two-periods ahead forward premium Two monetary shocks. High persist. in the domestic country Dashed lines are the bands for the 95% confidence interval 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 Correlation between monetary shocks 0.3 Figure 20 31 a t e s l 5 years rolling slopes and rolling correlation Sample: January 1982 - December 2001 1 month time to maturity forward rate 1.5 p 1.0 0.5 e 0.0 E -0.5 -1.0 -1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 negative correlation 0.30 0.8 o r correlation between M1 cyclical components l r Figure 21 1.5 1.0 negative correlation negative correlation o Rolling slopes and persistence with 5 years moving window Sample: 1982, January to December 2001 1 month time to maturity UK-USA forward 0.25 0.20 0.5 0.15 0.10 s 0.0 0.05 -0.5 0.00 -1.0 c -0.05 -1.5 -0.10 rolling slopes persist_uk Figure 22 32