Exchange Rate Dynamics and the Disconnect

In this paper we bring the survey of the contemporaneous research on the behaviour of exchange rate and so called exchange rate disconnect puzzle. We have introduced several directions in which research on this object could be developed. They are basically concerned with the monopolistically competition, which allows for exchange rate fluctuations from its long run equilibrium level, so that purchasing power parity condition doesn’t holds. The alternative approach incorporates heterogeneous agents’ setup.


Introduction
The exchange rate "disconnect puzzle" in a sense of Obstfeld and Rogoff (2000) is a missing link between exchange rates and the macroeconomic fundamentals that international macroeconomics theory suggests should be the driving force of the exchange rates. In particular, it seems that macroeconomic variables are not able to explain not only a current level of exchange rates but even their volatility. It should be mentioned that the "disconnect puzzle" in this meaning is a general term that includes also another well known puzzles: "forward bias puzzle" and "purchasing power parity puzzle".
In fact, however, the so called "exchange rate disconnect puzzle" refers to two different phenomena in the exchange rate literature: whereas the first refers to "exchange rate determination puzzle", i.e. the fact, that exchange rate can not be explained by the corresponding movement in macroeconomic fundamentals. 2 This puzzle was firstly established by Meese and Rogoff in 1983, when they have published their influential paper concerning out-of-sample forecasting accuracy of the standard macroeconomic models. They found that no so far known monetary model is able to outperform a naive random walk model of exchange rate. A second part of the puzzle 3 refers to extreme (or excessive) volatility of exchange rates with respect to other macroeconomic fundamentals. This second part concerns two concepts of extreme volatility of exchange rates: the excessive volatility under the given exchange rate regime (e.g. the fact that floating exchange rates are more volatile than the corresponding fundamentals) on the one hand, and the excessive volatility when moving from one exchange rate regime to another.
The rough insight into the problem of "exchange rate determination" puzzle gives us the following table about the variability of some key macroeconomic fundamentals in the CR. Those data brings us quite surprising insight: even when there were several different monetary policy regimes during the last decade in the CR, the variability of the growth of the important macroeconomic fundamentals differentials remains (roughly 2 The out of sample accuracy of traditional monetary models in Meese and Rogoff (1983) refers to this part of the "disconnect". Another disconnect literature could be found in Campbell (2006); Cheung -Chinn -Pascual (2005); Engel -West (2005); Flood -Rose (1999) and Sarno (2005). said) unchanged 4 , except for the pre-and post-crisis period, where, however, the money growth differential stays stable too. That fact can't be interpreted as the evidence for disconnect, but on the other hand, these findings seems to be quite surprising.
The aim of this paper is to summarize two directions of the research relating to this "puzzle", i.e. which aspects of these puzzles are those approaches able to solve, where are their limits, and where further research interest is most likely to turn.

Searching for Explanations of a "Disconnect Puzzle"
As this puzzle is undoubtedly controversial, several directions of possible explanations have revealed over time. One problem about the international macroeconomics is straightforward: the stylized fact revealed in empirical research is that the three key assumptions in standard monetary models do not hold: short term and sometimes even long run deviations of purchasing power parity, miserably performed uncovered interest parity and unstable money demand equations generally -this all together might be the cause of the "disconnect puzzle". But even if this would be the case, question why it is so still remains. Another open issue is whether the Dornbusch type overshooting 5 might be responsible for the "disconnect puzzle".

Models Based on Obstfeld and Rogoff (1995)
The first direction of the research tries to learn about disconnect from studying sticky price model based on Obstfeld and Rogoff´s (1995). As was mentioned in the introduction, in the case of exchange rate disconnect it seems, that there is only a limited feedback not only with price levels, but either with other current macroeconomics variables, such as money supply of GDP, so that "exchange rates have lives of their own" and traditional exchange rate models have failed when trying to forecast short to medium exchange rate movements even ex-post. In order to get closer to answers to those questions, it would be necessary to modify the baseline Obstfeld and Rogoff model in several directions. This sort of model is based on a dynamic general equilibrium (DGE) framework with well specified micro foundations, which allows for incorporating the assumptions of imperfect competition, incomplete asset markets and alternative pricing adjustments (as well as different types of nominal rigidities). The original model of Obstfeld and Rogoff does not allowed for the CPI-based deviations of purchasing power parity, as it assumes all goods to be tradables and complete pass-through of exchange rate changes to import prices (e.g. prices of tradables are set in buyer's currency). In the appendix to Similar to any other DGE model, we assume that there is a continuum of consumer-producers that produce nontradable goods and consume both, the nontradables and the tradables. Each representative household is consider to have 4 See Appendix for the graphical exposition of the variability of macroeconomic fundamentals. 5 Exchange rate overshooting phenomenon was firstly introduced by Dornbusch (1976). monopoly over production of one of the nontradables   where 0<<1 is the discount factor, T C stands for consumption of the traded good and N C is nontraded goods consumption index defined by Note that real balances enter utility function (eq. 1) directly and as a consequence of having general isoelastic function additively. Obstfeld and Rogoff (1995) showed, that the parameter   is the critical determinant of whether overshooting can occur or not.
where T P is the price of tradables. It is assumed that the law of one price holds only in tradables, so that we can write * T T P E P   ( we consider the foreign price level * T P as exogenous and constant). The nontraded goods price index N P is given by being the price of nontraded good z in money terms. As a consequence of given preferences it is assumed that the demand curve for nontraded goods is downward sloping and has the form A N C is aggregate per capita nontraded goods consumption, here taken as a given.
As bonds are denominated in tradables and r denotes the constant world net interest rate in tradables and   1 , the typical j's period budget constraint in money terms is defined as where per capita taxes  are denominated in tradables.
Since Obstfeld and Rogoff assume that Ricardian equivalence holds, they abstract from government spending and introduce balanced government budget in each period. The government budget constraint, in units of tradables, is therefore given by

Optimality Conditions
By choosing the optimal level of consumption, money and bond holdings and the output of nontraded goods, the individuals solves their intertemporal maximization problem. In particular, first order conditions are found by maximizing eq. (1) with respect to constraints given by eq. (5) and (6). After doing some algebraic calculations, we get The term i stands for nominal interest rate and is defined by the Fisher identity (8) is the so called Euler equation resolving optimal intertemporal allocation of tradables, whilst eq. (9) governs the optimal intratremporal levels of consumption of nontradables and tradables. Equation (10) is the money market equilibrium condition, where the money demand is the positive function of consumption and negative function of the interest rate. Equation (11) is the optimal labour-leisure choice where the marginal disutility of producing one extra unit of a nontraded goods is equal the marginal utility of consuming one.
Following Obstfeld and Rogoff (1995) 6 , we derive the symmetric steady state in which all exogenous variables are constant and the initial level stock of net foreign assets is zero. Then the model is solved for log-linearized deviations of initial steady state. One can derive, that the equation for the long run equilibrium level of exchange rate, money supply and price level is (the variables with overbars/hats stands for loglinear deviation of initial steady state).
As long as law of one price holds only for tradables in small open economy manner, the price of traded goods changes in proportion to the exchange rate. be derived that in this case the demand for real balances will rise by   M This means that when 1   , the demand for real balances will rise by less than the supply and the price of tradables (the exchange rate) would have to rise further to reach equilibrium. Therefore the exchange rate overshoots its long run equilibrium level.
There is now a huge literature extending the OR model, usually referred to as the New Open Economy Macroeconomics. While Obstfeld and Rogoff (1995) assume complete pass through of exchange rates to prices because they assume that nominal prices are set in the currency of the producer, a lot of their successors have extended their framework to the local currency pricing (LCP) case 7 . The assumption of LCP has become one of the strongest competing explanation of the "disconnect puzzle", because this explanation of excessive exchange rate volatility concerns the behaviour of tradables´ prices. Under LCP firms set prices in their own currency for sale to households located in their country, but set a price in foreign currency for sales to foreign households. They believe that within this framework it is possible to study and learn much more about the nature of exchange rate volatility when moving from one exchange rate regime to another. For example Duarte and Stockman (2005) came out with a modified version of Obstfeld and Rogoff (1995), when assuming the concept of goods market segmentation, local-currency pricing to market, incomplete asset markets and autonomous changes in risk premium that have been incorporated into the model. However, the common feature of the models of this kind is (as similar to classic Dornbusch model) the so called "indeterminacy" of the steady state. This means that the time path of exchange rate dynamics will depend on the initial conditions. The fact that in model of that kind the exchange rate is indeterminate made the potential policy implications, which arise from that kind of models quite questionable.

Models Build Upon Non-rational Behaviour of Economic Agents
Another group of researchers is trying to answer the question of excessive volatility in resignation to rationality of agents´ behaviour in exchange rate modelling and explore models with heterogeneous agents' structure starting from the microstructure of the foreign exchange market. This approach represents quite huge departure from traditional macroeconomic approach relaxing its key underlying assumptions.
We can mention Devereux and Engel (2002) among others, who believes that there are swings in expectations about future values of the exchange rates that stand behind so volatile exchange rates, which are (in his view) unrelated to prices and interest differentials. There are several issues that support his view: the fact that exchange rate expectations are inconsistent and seem to fail in forecasting future values, the failure of UIP that could be due to the non-rational behaviour of economic agents, failing of efficient market hypothesis. Last but not least, issues arising from switching from one exchange rate regime or monetary policy framework to another, which is generally followed by the changing of the way how expectations in the economy is made.
Under fixed exchange rate regime the government is obliged to stabilize the exchange rate at some predefined value. Countries moving from fixed to floating exchange rates face a remarkable change in the behaviour of real exchange rates 8 . They are much more volatile under floating exchange rate regimes, where expectations are not tied down by the threat of potential central bank intervention. However under the assumption that expectations are based on fundamentals, the relationship between fundamentals and exchange rates should be the same if there is no corresponding change in underlying fundamentals.
One non-rational direction of resolving exchange rate determination puzzle is the introduction of irrational behaviour of one group of agents in the foreign exchange market. The representative "non-rational" approach is for example the model with interaction of "chartists and fundamentalists", where agents have different beliefs about the future changes of exchange rates. The simple "chartists-fundamentalists" model could have for example following structure: Fundamentalists´ excess demand depends on the deviations between the current exchange rate and its fundamental value 9 where is the coefficient for the sensitivity to deviations of f s and t s .
The chartists´ excess demand will generally depend on the expectations over future depreciation of the exchange rate, i.e. on the expected future change of the exchange rate. 10 where is again the sensitivity parameter.
The depreciation of the exchange rate (the change of the exchange rate) is the outcome of the overall excess demand, as well as interaction between the agents: Under the assumption that chartists adjust their expectations adaptively to actual change in exchange rate, i.e. that  , where  is the speed of adjustment of adaptively made expectations. Their behaviour may be like this due to lack of experience, which leads to their irrational behaviour.
The non-rational behaviour of agents is the consequence of chartists´ adaptive expectations on one hand and the neglecting of the existence of chartists by fundamentalist on the other hand.
The previous equations constitute following system of two differential equations: 8 See Appendix for the evidence in the case of the CR. 9 This value might be given by PPP value as an example; it is the fundamental value that should be known to the fundamentalists. The chartist and fundamentalists approach was introduced by Frankel and Froot (1986) 10 In continuous time setting the difference between expected and the actual exchange rate can be replaced by the expected differential of exchange rate with respect to time.
Under  As we can see, this kind of model is able to bear high volatility of exchange rates, as well as the well known Dornbusch-type of overshooting with mean reversion. As for example De Grauwe and M. Grimaldi (2005) have pointed out, models of exchange rate that incorporate heterogeneous beliefs of agents (and therefore nonlinearities in the exchange rate dynamics) are able to explain several puzzles, including the "disconnections" (in all their meanings). They distinguish two agent types: i.e. "chartists" with a positive feedback rule and "fundamentalists" with a negative one. The nonlinearities came into the model through the transaction costs and nonlinear adjustment in the goods market. They have shown that the disconnection of actual exchange rate from its long run fundamental value is the natural consequence of nonlinearities in the model. Moreover, the nonlinearity of the model can produce endogenous regime shifts (structural changes) such that the link between the exchange rate and its fundamentals could be broken even when there is no change in the policy regime.
Despite of the promising nature of these kinds of models, their practical usage is quite questionable. As Duarte and Stockman (2005) have pointed out, one should not resign to find explanations of the "disconnect puzzle" within the traditional class of models. The reason is, that you can always build up a model, which is able to explain the data, but there is a question whether the model is able to provide reasonable policy implications.

Conclusions
The determination of the real and nominal exchange rates is still dominant problem among the monetary economics, especially due to the existence of quite large number of puzzles that the contemporaneous research has been unable to resolve. One of them, the "disconnect puzzle", states that the enormous exchange rate volatility seems to have no corresponding counterpart in the variability of macroeconomic fundamentals. In last few years, researchers are trying to resolve this puzzle together with another "price puzzle" in the international economics, the purchasing power parity puzzle that could be in some sense considered to be a special form of the "disconnect puzzle".
We have shown two possible ways of extending contemporaneous research to resolve the exchange rate disconnect. One based on extending Obstfeld and Rogoff (1995) approach, questioning two competing approaches with nontradables and local currency pricing. The microstructure approach based on heterogeneous agents´ expectations has brought another explanation to the exchange rate "disconnect" issue. Another issue that haven't been discussed is the consideration of other type of nonlinear models that are able to capture in some sense the role of deviations from the purchasing power parity. Those models have therefore potential to mimic the real exchange rate behaviour.
In any cases it is worth to say, however, that the role of macroeconomic fundamentals in explaining exchange rate behaviour is undoubtedly controversial and open issue in international macroeconomics.

Exchange Rate Dynamics and the Disconnect Abstract
In this paper we bring the survey of the contemporaneous research on the behaviour of exchange rate and so called exchange rate disconnect puzzle. We have introduced several directions in which research on this object could be developed. They are basically concerned with the monopolistically competition, which allows for exchange rate fluctuations from its long run equilibrium level, so that purchasing power parity condition doesn't holds. The alternative approach incorporates heterogeneous agents' setup.
Key words: exchange rate determination; exchange rate disconnect; overshooting; volatility.