Volatile and persistent real exchange rates without the contrivance of sticky prices

Volatile and persistent real exchange rates without the contrivance of sticky prices Michael J. Moore Queen’s University Belfast, Northern Ireland Maurice J. Roche National University of Ireland, Maynooth, Republic of Ireland The flexible-price two-country monetary model is extended to include a consumption externality with habit persistence. The model is simulated using the artificial economy methodology. It successfully explains (i) the high volatility of nominal and real exchange rates, (ii) the high correlation between real and nominal rates, and (iii) the persistence of real exchange rates. It offers a neo-classical explanation for the Meese-Rogoff exchange rate forecasting puzzle. Keywords: Artificial Economy; Real and Nominal Exchange Rates; Habit Persistence JEL classification: F3; F4; Michael J. Moore can be contacted at [email protected] Maurice J. Roche can be contacted at [email protected] If is there is an issue in international economics on which there is no consensus, it is why nominal exchange rates are so volatile. It is not hard to see why. In standard macroeconomic models1, the (log of) nominal exchange rate depends on the difference between (the logs of) home and foreign money and is proportional to the difference between (the logs of) home and foreign income. To mimic the volatility of nominal exchange rates, either money or income has to display variability that they obviously do not possess2. This has led a number of writers to despair of macroeconomics altogether. The most celebrated example is Flood and Rose (1999) who argue that forex market microstructure has more to contribute to our understanding of exchange rates than macroeconomics3. The apparently excessive volatility of nominal exchange rates is not an isolated puzzle in international economics. Real exchange rates exhibit similar volatility and to compound the mystery, real and nominal rates are closely correlated4. The traditional response to this twist is to invoke sticky prices, the most recent manifestation of which is the new open economy macroeconomics (for a survey, see Lane, 2001). Specifically, Chari, Kehoe and McGrattan (2001) show that with price stickiness for at least four quarters, very low levels of the intertemporal elasticity of substitution, and a high cross-country correlation in shocks to money growth, their international business cycle model produces simulated moments that broadly mimic the properties of real and nominal exchange rates. Using translog preferences, staggered contracts and pricing to market, Bergin and Feenstra (2001) also explain the same stylised facts. The problem with this is that not all economists are content with results that depend on the arbitrary assumption of pre-set prices. In this paper we simulate a very simple model that assumes perfectly flexible prices, complete risk sharing, plausible levels of the intertemporal elasticity of substitution and little or no cross-country correlation in shocks to money growth. The results also match the stylized facts5. Our modelling strategy is to extend Campbell and Cochrane (1999) 1 preferences to both a monetary and an international setting. With this specification there is an aggregate consumption externality (see for example Abel (1990) and Duesenberry (1949)) and utility is time-inseparable because of habit persistence. The utility function depends not only on the consumption of home and foreign goods but also on the surplus of consumption over an externally generated habit that is both volatile and persistent. This makes the marginal rate of substitution between home and foreign goods volatile enough to explain the variability in real exchange rates. The high volatility of nominal exchange rates follows since prices are pinned down by the modest volatility of the money stock. The high correlation between real and nominal exchange rates also follows since variations in both have their common source in the ‘surplus consumption ratio’. What of the persistence of real and nominal exchange rates? Chari, Kehoe and McGrattan (2001) argue “the persistence in the real exchange rate is essentially determined by persistence of consumption” and that habit persistence, while increasing persistence in consumption typically leads to less volatile consumption and, hence less volatile exchange rates. In our model this impediment to explaining persistence does not arise because utility also depends on the home and foreign surplus consumption ratios. These are exogenously driven by highly persistent stochastic processes. The closest contribution to the spirit of this paper is Finn (1999) who also uses a flexible price model to address some of the same issues. In her model, money is endogenous but there is a second source of real innovations in the form of shocks to the marginal efficiency of investment. She successfully explains the high correlation between real and nominal exchange rates as well as the persistence in both variables: however she is silent on their volatilities. Another anomaly is what Obstfeld and Rogoff (2001) label “the exchange-rate disconnect puzzle”. Meese and Rogoff (1983) showed that the forecasts from standard macroeconomic exchange rate models produce higher root mean squared errors than the 2 random walk model. In this paper we show that these results could be due to the fact that standard macroeconomic exchange rate models omit a key variable that is related to habit persistence. The plan of the paper is as follows. First, the model is developed. In section 2, we discuss the data. The model is calibrated in the third section. The results are presented in section 4. Finally, we make some concluding remarks. I. The model The basic structure of the model is the well-known Lucas (1982) two-country, two-good, two-money representative agent story. In this model the real exchange rate is equated with the intratemporal marginal rate of substitution between domestic and foreign goods and can be written as 2 1 /. /   it t it UC RUC (1) Utility is U, is the consumption of goods and services of country j by the household of country i at time t and j it C t R is the time t relative price of foreign to home goods expressed in a common currency. This is, of course, the real exchange rate6. It can be written as 2 / t1  tt t , R SP P where is the nominal exchange rate and is the price of country j goods in terms of country j money t Sj t P    22 11 // , //   it t t it t UC P SUC P (2) respectively. The exchange rate is measured as the home price of foreign currency at time t. It is obvious that persistent, volatile and highly correlated real and nominal exchange rates will primarily depend on the properties of the intratemporal marginal rate of substitution and not necessarily on the time series properties of consumption. 3 We will take the following simple model to illustrate the effects of introducing the habit externality on the time series properties of real and nominal exchange rates. Households in both countries7 are assumed to maximize the discounted expected value of lifetime utility. We will consider two cases. The first is the standard case where utility depends on consumptions only. The second case assumes that households also have habits in domestic and foreign goods. The utility functions are given as 1(1 ) 2(1 ) 12 00 () () (,) , 1,2, 11             tt it it it it tt CC UC C i (3) 11(1) 22(1) 1122 00 ()() (,) ,1 11                tt it it it it it it it it tt CH CH UC H C H i ,2, ij (4) where  is the discount factor, 1/ is the intertemporal elasticity of substitution, and is the subsistence consumption (or habit j it H 8) of goods and services of country j by the household of country i. Note that, if in equation 0, , j it H(4) collapses to equation (3), which represents standard addilog preferences with no habit. Habit persistence takes the form of an aggregate consumption externality i.e. ‘Keeping Up with the Jones’s’ effects along the lines of Duesenberry (1949) and Abel (1990). Recent work in the economics of happiness literature suggests that relative income is an important factor in individual’s levels of satisfaction; see for example Oswald and Clarke (1996) and Oswald (1997). We reparameterize the utility function in equation (4) in terms of j it X , the surplus consumption ratio of goods and services of country j by the household of country i , 1,2, 1,2.   jj jit it it j it CH X i j C (5) When =, : this is the worst possible state. By contrast, as rises, the surplus consumption ratio converges on unity. We closely follow Campbell and Cochrane (1999) by assuming that the log of the surplus consumption ratios evolve as follows j it Cj it H0 j it Xj it C 4   11 (1 ) ( ) , 1,2,      jjjjj tttt xxxxvj (6) where <1, is the habit persistence parameter, j x is the steady state value for the logarithm of the surplus consumption ratio for good j and is the shock to consumption growth in country j. The function j t v ( ) j t x describes the sensitivity of the future log surplus consumption ratio to endowment innovations. It depends non-linearly on the current log surplus consumption ratio. The form of the sensitivity function( ) j t x is max max 2 max 12( ) () 1for 0for 1( ) where . 2        jj jj t tt j jj t j jj xx  j x xx X x x X xx (7) j X is the steady state value of the surplus consumption ratio for good j and is defined as . 1     j j v X (8) We define as the standard deviation of the innovation to the consumption of the jth good. There are a couple of advantages to specifying the habit along the lines of equations j v (6)-(8). Firstly, the habit is predetermined at the steady state. This means that it takes time for the consumption externality to affect an individual agents habit. The second advantage avoids a possible difficulty with the first. The habit is not predetermined outside of the steady state but if it were, a sufficiently low realization of consumption would mean that habit exceeded current consumption. The arguments of the utility functions in equation (4) become negative. Our habit specification prevents this by ensuring that the habit moves nonnegatively with consumption everywhere. These two features are illustrated in detail in Campbell and Cochrane (1999). Most importantly, for the problems that we are addressing, 5 the form of the external habit in equations (6)-(8) guarantees that the intratemporal marginal rate of substitution in (1) and (2) is both volatile and persistent. The rest of the model is as follows. The agent in the goods market faces the following cash-in-advance constraint (9) ,1,2,1 jjj it t it M PC i j ,2, where j it M is the amount of money of country j held by the household of country i for transactions in the goods market at time t. At the end of period t (or the beginning of period t+1), the domestic households holding of domestic currency 111 11 1 ,  ttt 1 t M PC B (10) is made up of proceeds from the sale of the endowment and the redemption of nominal discount bonds, j it B . The domestic household's holding of foreign currency is 2 11 1 .  t 2 t M B (11) Analogously the foreign households holding of foreign currency is 222 21 2 ,  ttt 2 t M PC B (12) and of domestic currency is 1 21 2 .  t 1 t M B (13) The only role for the government is to have a central bank that engages in open market operations. In each period the central bank of each country changes the money stock by issuing one-period discount bonds. The bonds are redeemed at the end of period t (or the beginning of period t+1). Equilibrium in the goods market is given by (14) 12 ,1  jjj ttt CCC j,2. ,2, Equilibrium in the money market given by: (15) 12 ,1  jjj ttt MMM j 6 Each household maximizes Eq. (4) subject9 to Eqs. (5)-(15). Like Lucas (1982) we assume that there is perfect international risk pooling in equilibrium. Recent work by Brandt, Cochrane and Santa-Clara (2001) suggests that international risk sharing is very high. With perfect risk sharing the equilibrium consumption of each good equals half of the current endowment i.e. , where is endowment of the ith country at time t. The solution is conventional and the expression for the real exchange rate can be found readily from equation 0.5 i jt t Ci Y, 1,2 i t Yi (1) and the utility function (4) 22 11 () . ()   tt t tt CX RCX   (16) Recalling that we use lowercase letters to indicate the log of a variable, we can write (17) 12 12 ()( ttt t ryy xx    ) t t m and (18) 12 12 1 2 ( 1)( ) ( ) ( ).      tttttt syyxxm In the standard case where there are no habits in the utility function: the j t x terms are not present. In Chari, Kehoe and McGrattan (2001) the log-linearized real and nominal exchange rates depend on both aggregate income and money differentials in a similar way to (17) and (18) and it is obvious why they need a large value for . The curvature parameter10 needs to be above unity so as to generate positive correlations between real and nominal exchange rates and needs to be large to generate the required volatility. We do not need this restriction because it is the presence of the log surplus consumption ratio differential in both (17) and (18) that secures this correlation in our model. In addition Chari, Kehoe and McGrattan (2001) need to assume price stickiness for at least four quarters in order to generate persistence in the real exchange rate. If households have Campbell and Cochrane (1999) habits in domestic and foreign goods neither large  nor price stickiness is required to 7 generate persistent, volatile and highly correlated real and nominal exchange rates. The j t x terms generate volatility, persistence and high correlation. Since the classic paper by Meese and Rogoff (1983) economists have found it difficult to explain why the random walk exchange rate model outperforms many other models in short horizon forecasting. They calculated the root mean squared forecast error (RMSE) for the monetary model (RMSEmod) and for the random walk model (RMSErw) and computed RMSEmod /RMSErw. The relative RMSE tends to be greater than unity at most forecast horizons. To investigate this we conduct two types of experiment. We follow Mark (1995) and Faust, Rogers and Wright (2001) and estimate the following exchange rate equation at forecast horizon k (19) 2 ~(0,)   tk t k k t t t e ss ze eiid , . t s where zt is the log deviation of the exchange rate from fundamentals predicted by the “monetary model” and is given by (20) 12 1 2 (1)( )( )     ttttt zyymm In these studies point estimates of  k are positive and tend to increase with the horizon k. Mark (1995) and Faust, Rogers and Wright (2001) compute recursive out of sample forecasting exercises. They leave the last forty observations of a sample of size T for evaluation. They estimate (19) with T-40 observations and produce forecasts for horizons 1, 4, 8, 12 and 16 quarters. Then they add one observation to the end of the estimation sample and repeat the forecasting exercise. This results in 40 k=1 quarter forecasts, 37 k=4 quarter forecasts, 33 k=8 quarter forecasts, 29 k=12 quarter forecasts and 25 k=16 quarter forecasts. Using real time data Faust, Rogers and Wright (2001) show that the relative RMSE was greater than unity and increasing with the forecast horizon for three out the four US dollar exchange rates studied11. For ease of comparison, an example of their results is reproduced as Table 2. 8 analysis and shows that the results of Table 6 are not peculiar to the selected values for the exogenous forcing processes. The reason why linear exchange forecasting models such as Eq. (19) are so poor is that the differential in the log surplus consumption ratio is omitted. Though this series is stationary it is both highly autoregressive and volatile. It is non-linearly related to fundamentals: though the model, of course, has the properties of long-run mean reversion and long-horizon predictability by construction, the short-run deviations of the nominal exchange rate from fundamentals are very volatile. No linear forecasting equation such as (19) or (21) is capable of capturing this subtlety. Our habit model lends support to the work of Taylor and Peel (2000), Clarida, Sarno, Taylor, and Valente (2001), Kilian and Taylor (2001) and Taylor (2001) all of which emphasize the importance of non-linearities in nominal exchange rate and purchasing power parity modelling. V. Conclusions The point of this paper is simple. Using a flexible price model with a simple twist, it is perfectly possible to explain many of the puzzles associated with purchasing power parity and nominal exchange rates under floating exchange rates. The volatility and persistence of both real and nominal exchange rates along with their correlation can be mimicked so long as preferences are subject to an aggregate consumption externality. The model, proposed here, still has limitations. It only describes an exchange economy. Ljungqvist and Uhlig (2000) have pointed out that there are problems in expanding the Campbell and Cochrane (1999) framework to a production economy. ‘Consumption bunching’ rather than consumption smoothing becomes welfare optimal. However, this only arises if the habit is internalised: our habit is strictly an externality. It would be useful to extend our model to include production in order examine related puzzles that only arise in 15 that context. These include the high correlation between real exchange rates and international output ratios, (See Finn 1999). Finally the model, as it stands, cannot explain why real exchange rates are more volatile under floating than fixed exchange rates. Notwithstanding these reservations, this paper demonstrates that the despair of Flood and Rose (1999) is premature and the contrivance of sticky prices may not be necessary. 16 References Abel, Andrew. “Asset Prices Under Habit Formation and Catching Up with the Joneses.” A.E.R., Papers and Proceedings 80 (May 1990): 38-42. Bergin, Paul R., and Feenstra, Robert C. “Pricing-to-Market, Staggered Contracts and Real Exchange Rate Persistence.” J. Internat. Econ. 54, (August 2001): 333-359. Brandt, Michael W., Cochrane, John H. and Santa-Clara, Pedro “International Risk Sharing is Better Than You Think (or Exchange Rates Are Much Too Smooth).” Working Paper no. 8404. Cambridge, Mass.: NBER, (July 2001). 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Taylor, Mark P. and Peel, David A. “Nonlinear Adjustment, Long Run Equilibrium and Exchange Rate Fundamentals.” Journal of International Money and Finance, 19, (2000): 3353. 20 TABLE 1 Properties of exchange rates and consumer price indices in the data Standard deviations AR(1) coefficients Crosscorrelations Price Ratio Nominal Exchange Rate Real Exchange Rate Price Ratio Nominal Exchange Rate Real Exchange Rate Real and Nominal Exchange Rates Austria 1.61 8.60 8.44 0.89 0.77 0.76 0.98 Belgium 2.13 9.47 8.95 0.95 0.81 0.78 0.97 Denmark 1.24 8.50 8.45 0.69 0.78 0.77 0.99 Finland 1.90 8.49 7.85 0.89 0.84 0.83 0.98 France 1.24 8.96 8.42 0.91 0.81 0.78 0.99 Germany 1.50 8.84 8.60 0.90 0.78 0.76 0.99 Greece 2.75 7.08 7.01 0.67 0.75 0.70 0.94 Ireland 2.30 8.74 7.68 0.86 0.81 0.73 0.97 Italy 1.67 8.91 8.22 0.85 0.81 0.78 0.98 Luxembourg 2.32 9.47 8.77 0.95 0.81 0.77 0.97 Netherlands 1.70 8.80 8.61 0.90 0.78 0.76 0.98 Portugal 3.86 8.63 7.22 0.82 0.81 0.68 0.90 Spain 2.30 9.03 8.69 0.89 0.84 0.82 0.97 Sweden 1.76 8.64 8.16 0.80 0.80 0.78 0.98 United Kingdom 2.54 8.45 8.06 0.85 0.81 0.78 0.98 EU 1.37 8.19 7.65 0.91 0.81 0.78 0.99 Note to the table: the statistics are based on logged and H-P filtered quarterly data for the period 1973:1-1998:4. The statistics for the European Union are trade-weighted aggregates of all countries in the table with the exception of Denmark and Greece. 21 TABLE 2 Relative RMSE of the monetary model in out-of-sample forecasting Forecast Horizon k 1 4 8 12 16 Canada 0.99 0.81 0.63 0.65 0.55 Germany 1.01 1.10 1.87 2.52 2.90 Japan 1.00 0.98 1.01 1.17 1.40 Switzerland 1.00 0.93 1.02 1.01 1.13 Note to the table: These results are taken from Table 1 in Faust, Rogers and Wright (2001). The entries in the table show the ratio of the out-of-sample RMSE from the monetary model to that of the driftless random walk model using real-time data. The exchange rates are versus the U.S. dollar. 22 TABLE 3 Baseline parameterization Consumption growth Money growth AR(1) coefficient 0.00 0.35 Standard deviation of shock 0.56% 0.60% Curvature of the utility function  0.10 Persistence of the log surplus-consumption ratio  0.97 23 TABLE 4 Properties of exchange rates and consumer price indices in the simulated models Standard Deviations AR(1) coefficients Crosscorrelations Price Ratio Nominal Exchange Rate Real Exchange Rate Price Ratio Nominal Exchange Rate Real Exchange Rate Real and Nominal Exchange Rates Data 1.37 8.19 7.65 0.91 0.81 0.78 0.99 Habit 1.74 (0.00) 7.74 (0.002) 7.88 (0.002) 0.78 (0.02) 0.65 (0.004) 0.62 (0.004) 0.91 (0.003) Standard 1.69 (0.000) 0.001 (0.000) 0.78 (0.002) 0.69 (0.002) -0.51 (0.005) CKM 5.46 (0.75) 7.86 (0.80) 7.77 (0.72) 0.93 (0.02) 0.69 (0.08) 0.62 (0.08) 0.76 (0.06) Note to the table: ‘Data’ refers to the USA/EU stylized facts in the last row of Table 1. Each series is logged and detrended using the Hodrick-Prescott filter. The means of each detrended series in the habit and standard models based on 1000 simulations are reported with the associated standard error in parenthesis. The means of each series in the Chari, Kehoe and McGrattan (2001) (CKM) benchmark economy model based on 100 simulations are reported with the associated standard error in parenthesis in the last row. 24