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Lecture notes 260D revised 6804 Lecture 8 Risk topics Interest parity condition theory and tests Risk in model of sticky prices and monetary policy Possible student presentatitions Mark 85 if familiar with GMM estimation Possible dissertation topics Empirical tests and estimates of risk premium Part1 Risk premia and the interst rate parity conditions a Motivation May recall several conditions thought to hold in foreign exchange market in addition to PPP Uncovered interest rate parity Ifhave choice between assets in the two currencies for people to be willing to hold existing supply of both their returns must compensate people for an expected depreciation of the value of the currency l t lit l it Et em to tell allocation story 82 1 it or 62 1 Elem for later comparisons It Or if take logs and use approximations of Log1it it approx and difference of logs is percent change use tilde to show log of variable it 139 Et5m Et 139 expdepretiation Forward exchange rate efficiency If can buy contract this period for currency exchange next period the rate should be what people expect the spot rate to be next period fl E 8H4 Covered interest rate arit putting the two together 1 1 82 1 1 2 Plan Consider simple exible price model to show where these come from and how risk enters in Then consider some tests and how have risks have been incorporated into models of monetary policy b Model Description Number of assets households can hold home money nominal interestbearing assets each of two currencies which pay separate nominal interest rates Also can buy foreign currency asset along with a contract to convert returns back into home currency at a guaranteed rate Recall the covered interest rate parity story Call this asset F and rate of currency exchange set at period If for conversion in period tis l Household must decide allocation of savings between these nominal assets w t M Max Elg UCtt SI 13th 1139HBH e1zquot BH f11z391FH MH PtCt 3 eth etFt M FOCS Write in Bellman form and use lambda for lagrange multiplier on budget constraint FOCS on the three interestbearing assets and consumption 1 Home bonds Egilm l1 it 12 2 Foreign bonds F 8H1 1H1 l1 lit 2 822 3 Forward exChange 275 Mm if 11 2 ez t v 5 U Consumption lambda A P l Combine eguations and substitute for lambda 4 Covered interest rate parity combine 1 and 3 1 139 1 139 e This is same as previously does not involve expectations 5 Uncovered interest rate parity use 2 and l J Eta me 1 12 PM PM Is more complicated than simple version given before because involves expected marginal utilities In fact the left hand side term is same as in market efficiency condition 6 Market efficiency condition use 2 and 3 U39 U39 ZEEHMEM PM PM Also is different from simple version given before it involves expected marginal utilities as well Re ects a desire to smooth consumption Ifwant to compare other two equations to their previous counterparts it is easier to rewrite them somewhat Risk premium rewrite 6 to split up expectations U U fl E et1 00V 8244 5 7H4 El PM PM E 8H4 RP and Uncovered interest rate parity 1 q jaauum 11 Inteppret Both involve a risk premium which was not present in simple versions previously May require a higher return on an asset that is riskier This risk premium does not simply re ect the fact that you have to be rewarded for taking a position in a foreign currency This fallacy was debunked by Frenkel 79 JIE IfAmericans need to receive a premium to buy German assets this implies that Germans are receiving a negative premium when they buy American assets which is taking a position in a foreign currency from their perspective There is nothing illogical about a negative risk premium on holding foreign currency assets risk is a matter of how the value of the asset uctuates in relationship to things that the households value like consumption A premium is the reward for assuming risk that is not diversifiable Further if the risk premium is positive this implies the covariance term above is positive the covariance of the exchange rate and marginal utility is positive When marginal utility is high means consumption is low so indicates bad times Ife is high during this period it means that get many dollars for a yen so foreign asset pays off well relative to home asset So home asset is riskier undesirable in that it pays off worse during bad times Therefore foreign assets are a good hedge against consumption risk So home assets must pay a somewhat higher return because of the risk premium and the forward rate is somewhat higher Could also be negative correlation where home asset would be a better hedge Note that even if households are risk neutral U is constant we still have 1 1 f2 E et1 00V ez1 9HEz So the forward rate still is not equal to the expected future spot rate This extra term is often called a Jensen inequality term and just re ects the presence of convexity The risk premium sometimes is defined netting this out Forward Premium Sometimes encounter term forward premium This is defined as F P f ez representing how much extra need to pay to get foreign exchange for future than for today This implies that the forward premium may be viewed as the sum of the risk premium and the expected change in the spot exchange rate FP f2 ez fz E2ez1lE2ez1lezRB Ezlez1lez c Early Tests of Interest parity and market efficiency Framework The earliest tests of forward market efficiency just regressed the future spot rate on the forward rate in logs et1 a0 alfl 82 Tested if 6100 and 611 Null hypothesis is that the forward rate provides an unbiased forecast of the future spot exchange rate Note the equation tested here replaces the expected value of em in the original condition with the actual future value So the regressions imposes rational expectations with perfect foresight This is done because we do not have direct data on people s expectations So keep in mind this assumption could in part explain rejections of the equation Some researchers Frankel have collected survey data on traders exchange rate expectations and tried using this in the tests without much success Another problem with the equation tested above is that the exchange rate is probably nonstationary This implies that the estimate of a1 is biased toward 10 This is a reason not to trust early tests many of which claimed to support the market efficiency hypothesis To deal with the nonstationarity issue most researches subsequently have tested the equation with e1 subtracted on each side 8H1 eza0 a1fz ezgz Sometimes this forward market efficiency equation is tested in a different form Since covered interest parity seems to hold well we can replace the right hand side term in the regression with the interest rate differential The regression then becomes a test of uncovered interest parity em ez a0 6116 i gz These researchers realized a risk premium might be present but they tended to assume it was not timevarying If the risk premium was constant it would appear in the an term making it deviate from the hypothesized value of zero But it should not affect the at term So the researchers tended to focus on the hypothesis that 6111 Results 1 Backus Gregog and Telmer 1 1993 JOF monthly data on onemonth forward rates and the corresponding future spot rates for CN dollar French franc mark yen and pound all relative to the dollar from July l974April 1990 In all cases they find estimate of a1 is significantly less than one and may be negative 2 Froot 1 1990 J EPerspectives Summarize the literature of 75 papers on the subject The average estimate of al over 75 papers is 088 only a few nd a1gt0 and none nd a1gt1 3 Mcallum 1 1994 JME nds estimate is 74 using yen mark and pound rates against the dollar on monthly data A dissenting nding 4 Ma 1eld and Mu h 1992 Economics Letters estimate a version of the equation above simultaneously for 3 currencies and allow for a common timevarying intercept term they can no longer reject an estimate of 611 5 Flood and Rose 1 1994 NBERWP estimate over different time periods and suggest that nding of estimate a1lt0 applies only exible exchange rate periods Though even in xed rate periods the estimate still is less than 1 Conclusions In general papers reject the hypothesis that 6111 and most nd that it is actually negative This means that a country s currency is expected to appreciate in future periods when its interest rate is high Rather than offsetting a high interest rate future appreciation makes it even more profitable to buy a currency How can this be consistent with market equilibrium How can we explain this nding An explanation for the strange estimates of a1 Timevaging risk premia Ifthe risk premium varies over time we could write the equation from before 8H1 eza0 a1fz ezRPz 52 Where RR and et together are the error term in the regression So the error term includes a component that may be correlated with regressors and this would bias the estimate of all To solve this problem some researchers have tried to use ARCH models to model the risk premium Domowitrz and Hakkio 1995 HE They propose a separate regression for the risk premium itself as a function of various things like the interest rate differential But results were not encouraging It appears that the risk premium may well be timevarying but it is not a simple function of the interest rate differential It moves in a more complicated fashion For example recall the results we saw in Eichenbaum and Evans QJE The forward premium there appeared to move conditionally on monetary policy shocks d GMM tests Several tests in late 1980s and 1990s have estimated the nonlinear Euler equations above by GMM methods The idea is to test the conditions in a form where the risk premium is still built in General Method rearranging condition 6 U E ez1 f2 0 21 The term inside brackets may be regarded as an excess return on forward exchange and the condition states that this excess return should be zero This implies that any excess return that exists should not be predictable using any information know in period I Can test this by taking a vector of various instruments 2 know at t and requiring U39E Ezez1 fz P72 22 0 21 This forms a set of restrictions that can be used to estimate parameters in the utility function risk aversion and test the equation The GMM estimator is the value which minimizes the quadratic form J g Wg Where g is the sample counterpart of the set of orthogonality conditions above and W is a weighting matrix of the orthogonality conditions Can use J to evaluate the validity of the overidentifying restrictions When multiplied by the number of regression observations J is assymptotically chisquare Mark 1 1985 JME This method first done by Mark Data Work with four currencies simultaneously Can dollar DM Neth guilder UK pound all relative to the US dollar Monthly data 7383 Instruments For instruments want variables which might reasonably be of importance to agents in solving their forecasting problem Since agents are concerned about predicting the future return in terms of the consumption good the current consumption is used as an instrument Also current and past movements in the exchange rates are likely to provide information on expected returns Use two versions of each instrument Consumption nondurables with services and without Exchange rates 1 Current and past value for realized pro ts from foreign exchange speculation on currencies gap between forward rate and actual spot rate one month later Idea is that future speculative profits might be partly predictable from past profits Current and past forward premia on the currencies gap between forward rate and actual current spot rate The idea is that the forward premium can be expressed as the sum of the expected rate of change of the exchange rate and the risk premium and so it contains information about expected speculative profits N V Choose to use range of lags for the instruments Note assume a CRRA utility so need to estimate intertemporal elasticity l gamma Results Estimates of risk aversion are large intertemporal elasticities small Needed to explain large risk premium Instrument list 1 using speculative profits Cannot reject model for any of the lags considered for either definition of consumption Tmz39n is the statistic distributed chisquare the minimized value of distance function J times the sample size Instrument list 2 using forward premium Can reject for case of zero lags only current values of the instruments but not for cases including lags Conclude Reject the model only in a couple cases when forward premium is used as an instrument So a mixed result Forward premium has some predictive power There is a risk premium at work but to degree the model is not rejected this premium seems to obey the general properties the consumptionbased theory says it should have Other papers Hodrick 1 1989 J ME more recent data and more countries seven Uses quarterly data from 73III 871V Also estimates large values of gamma in most cases Not reject Modjtahedi 1 1991 JIE Again similar to Mark 1985 but look at other maturities 13 and 6 months Reject the model a stronger rejection than Mark 1985 Backus Gregog and Telmer 1 1993 JOF Try with other utility functions than CRRA 7 habit persistence Reject the model Overall conclusion from the GMM literature Very mixed results Often the consumptionbased model does not perform very well Generally consumption is not variable enough to explain the high risk premium unless it uses a suspiciously high value of risk aversion for utility Further the returns from speculation the foreign exchange market seem to be predictable contrary to the theory based on past values of the forward premium Idea for future research Other model variations have proved useful in explaining similar domestic puzzles like equity premium puzzle Not know of anyone apply these yet Part2 Theoretical papers with risk and stickV prices a Obstfeld and Rogoff 20011998 WP Motivation The previous models of monetary policy we considered either had no uncertainty such as Obstfeld iRogoff J PE 1995 or they imposed certainty equivalence by linearizing the model But we know uncertainty can be important for agents behavior Concern over exchange rate risk is a main motivation for establishing fixed exchange rate regimes Interested in guestions 1 How does a model that includes sticky prices help us explain the forward premium puzzle 2 What are the welfare implications of exchange rate variability in such a model Model The specification is generally similar to that in Obstfeld and Rogoff 1995 JPE Two countries Not necessarily same size Home is size n and foreign is ln Home preferences separable in consumption real money balances and leisure which is a function of output Interest elasticity of money is 1 8i and interest elasticitv of quot 1 p so p is the coefficient of relative risk aversion Perfect insurance between home and foreign households see next page This implies the following risk sharing condition U U e P P A useful simplif 39ng assumption using Cobb Douglas preferences may skip this in lecture if covered earlier CobbDouglas speci cation The consumption aggregate here is careful to assume a unitary elasticity of substitution between the home goods aggregate and the foreign goods aggregate Home consumption of home goods per capita CH is CES aggregate of differentiated varieties with elasticity of substitution 0 Same for CF home consumption of foreign goods per capita Total aggregate home consumption is CobbDouglas aggregate of these two C C I C fquot Where n is also the share of home consumption as well as country size Implication for real income Implies that home consumer will split total consumption expenditure between the groups of goods in constant ratio CHniC and CF1 niC PH PF Price index is CobbDouglas as well And assume LOP holds so here PPP will hold as well PH 2P P ePi So can write foreign demand for home good as quot P C nP C n C PH PH Insert these into the marketclearing condition for home goods nCH l nC nY nlnPC 1 nPC J nPHY Do same for demands for foreign goods and market clearing condition l n nPC1 nPCJ l nPFY Implies no borrowing or current account Together it is clear that the two conditions above imply that PHY PFY This implies that countries will have constant and equal shares of world real income regardless of shocks The intuition is that relative prices adjust to insure against shocks if a supply shock lowers the quantity of home output the relative scarcity will drive up the relative price of home goods in proportion Cole and Obstfeld 1991 demonstrated this property and used it to argue that international trade in assets would then be redundant The equilibrium acts like a complete assets market economy without the need for trade in assets Given isoelastic preferences over total consumption this means that countries will consume their incomes each period and the current account will always be zero Ifone looks at the intertemporal Euler equation we see that both countries want to keep consumption level smooth as shocks make their incomes vary over time Given that the incomes in both countries move exactly together the solution will imply that the consumption levels in both countries will move together So the solution implies C CPFI and CC P P The fact that there is no borrowing and hence no current account or wealth reallocation greatly simplifies the solution This is a widely used device which we will see employed in several subsequent papers in this literature Implications for Risk Sharing The allocation here will mimic exactly a complete markets equilibrium In particular it will satisfy the risk sharing condition characteristic of complete asset markets U ePquot for all periods and all states of nature U C P Under purchasing power parity this simply states that the marginal utilities of consumption should always be equal across countries Us U And since we showed above that C C is always satis ed we know here that the marginal utilizes will be equal also Note this conclusion will not be valid under some alternative specification We will see both ofthese cases in subsequent papers 1 Local currency pricing If the law of one price is violated then cannot substitute home for foreign prices in deriving the real income condition above 2 Presence of nontraded goods Suppose we specify consumption as including nontraded home goods C C D7ny where traded goods are a CobbDouglas aggregate of home and foreign The law of one price does not apply to the nontraded part of consumption so the real income condition above implies that C CTquot for traded goods alone instead of C C for overall consumption Firm behavior Main idea Exchange rate risk can affect pricesetting behavior of the rms Firms have same production function as before and set prices one period ahead of time They will accommodate the level of demand that prevails given the change in the actual aggregate price level and exchange rate But set price so that in expected value it will equalize the marginal utilitv of marginal revenue and the marginal disutility of extra effort lost leisure Boils down to where theta is elasticity of demand 52 EC 0 1 E0PH Analogous condition for foreign pricing involving P In past we took linear approximation to this pricesetting rule But here not Assume shocks lognormally distributed Take log of expected value which introduces a Jensen variance term and break expected value of product into product of expected values which introduces a covariance term in logs Use to express terms of trade PH 7 EHe 41 270022 2052 Interpret Suppose covariance term is positive Says that currency is week at same time when consumption level high Both events tend to raise demand for home good and make disutility of leisure higher than expected So high covariance raises volatility of this term Response of firm set price higher than otherwise would because want higher marginal revenue to hedge against the risk of this variance in marginal disutility of effort Note on variance term if home country is small nlt05 then variance of exchange rate makes it raise its price higher as well This is because exchange rate changes affect it more if it is small because larger fraction of the demand it faces is coming from abroad Explaining the forward premium puzzle Main idea Want to explain why a fall in the interest rate predicts a currency depreciation rather than appreciation as required by interest rate parity lI21 i1 Eiez1lez sz i i i For i to fall and still have an expected depreciation could be explained if there were a large fall in the risk premium To solve the puzzle we need a reason why there would be such a fall in the risk premium Recall a general expression for the risk premium from earlier in the lecture U39 U39 RP Ezez1 PM Ez 21 21 U U Under the risk sharing condition P 5 e P and the fact that marginal utility is c where l p is the intertemporal elasticity p 7p RPtEt C 21 El C 21 P 21 B 1 Now we use a trick to rewrite this Ifa variable X is log normally distributed then we can write logEXElogXvarlogX Ifwe assume that the shocks of the model are log normally distributed then the consumption and price levels will be also So we can rewrite the log of the risk premium as lower case letters represent logs overbars indicate steady state values IOgRB 3Pp5c l 36am 1 p2 var c 1 2pm pm 1 l 3an p p2 varc1 2pm p 4 Assuming a symmetric steady state and perfect insurance which makes consumption levels move together across countries in expectation this simpli es to 1 Z Z Z 2 RP 3 0w 0p p0 pt 05p Where 0 Varzpz1 and 0 COV2cz1pz1 The first bracketed term above is just the Jensen inequality term The second bracketed term represents how a home nominal asset can act as a hedge against consumption risk Ifthe covariance between consumption and the price level is positive this means when consumption is low the price level is also low and so the real value of home nominal assets is high Since home nominal assets pay off best in bad states of the world they are a good hedge This implies a low or even negative risk premium on home assets Now consider why the risk premium might fall that is when the covariance between consumption and prices rises This will be in cases when monetary policy is driving most of the uctuations Solve for the risk premium as a function of the variance in home monetarv shocks and assume for now that there are no foreign monetarv shocks RP a 1 was 1 where 039 is the volatility of shocks to money Recall that is the interest elasticity of money demand So provided money demand is not too elastic a rise in the volatility of money supply can lower the risk premium and this could explain the forward premium puzzle The intuition is that monetary shocks tend to make the covariance of consumption positive in the sticky price model When money supply rises only part of this translates into a rise in price and to the degree that price doesn t rise there is a rise in demand which raises production and consumption We have already seen that the comovement in consumption and prices makes home nominal assets an attractive hedge for home consumers against the risk imposed by these shocks Conclusion This paper offers one potential explanation for the forward premium bias and why the forward premium appears to move systematically with monetary policy When moneta policy becomes especially active this is when home nominal assets become an especially attractive hedge against the consumntion volatilitv created by volatile monetary shocks and this would lower the risk premium Welfare implications As in their JPE 95 paper the authors plug the equilibrium solutions into the utility function to evaluate welfare implications 2 ElulEClip Y2 Qexpl p Maz af 1 p 7 1 p E where Q w E Hp 201 p 0 So high variability in consumption or the exchange rate lowers utility Both are bad Why do volatile exchange rates worsen utility The reason is that exchange rate risk makes the rms want to produce less for trade and domestic consumption as we saw above Recall in OR 1995 that the main welfare problem was that monopolistically competitive rms restrict production below the socially optimal level This is made even worse here An unanticipated currency depreciation means a rm s goods are cheaper for foreign consumers so demand rises This forces the rm to produce more than its pro t maximizing level To hedge against this risk the monopolistically competitive rms charge an even higher markup above marginal costs This restricts output and consumption even further below the socially optimal level The 39 39 is exchange rate variabilitv hurts welfare because currech uctuations cause rms to produce and trade too little So a xed exchange rate regime can improve welfare by eliminating this risk and boosting production and consumption This conclusion stands in contrast to that of Helpman 1981 which concluded that there was no difference between xed and exible exchange rates This simple conclusion changes under a model with various market imperfections b Bacchetta and van Wincoo AER 2000 Motivation This paper shows that the welfare conclusion of Obstfeld and Rogoff 1998 is not general but depends on a particular case Depending on the type of utility function and depending on the nature of monetary policy a exible exchange rate regime that permits exchange rate variability may not hurt welfare so much The paper also explores the longstanding presumption that exchange rate variability hurts welfare by inhibiting international trade Model The model in most essential respects resembles the twocountry nonlinear sticky price model of Obstfeld and Rogoff 1998 above There are some important differences however 1 The model here assumes pricing to market so that firms can set different prices in each of the national markets 2 A general utility function is assumed UCL without imposing a particular functional form up front This differs from the utility function in ORl998 which was additively separable in consumption and leisure There are also some minor differences which do not much affect the results 3 Money is introduced by cash in advance constraints rather than in the utilty function This implies a simple money market equilibrium condition Y M where Y is total nominal income 4 Each country is equal in size so the share n 12 here 5 A simple linear production function each good is produce with one unit of labor 6 There is only one period Notation For the sake of consistency and clarity Iwill here use the same parameter names as in OR 1998 above even thought these differ from the parameter names in the actual BvW 2000 paper So 0 is again the demand elasticity facing each of the monopolistically competitive producers And 6 is exchange rate not S as in paper Some ke E uilibrium conditions here In this highly symmetric setup the equlibrium nominal exchange rate is determined simply by the ratio of national money supplies MAJ The nominal wage is determined by the ratio of marginal utlities of consumption and leisure U WPE Firms set prices at home p H and abroad p2 as generalized markups over marginal costs W where the markup depends on the demand elasticity 91 0 EUM pH EUM a EUMJ pH EUM So the price setting behavior depends on the comovement of wages with money in the home market and the comovement of wages with the exchange rate in setting the price for the foreign market The intuition is similar to that in OR 1998 These prices are important for the results that follow immediately below Consider the effects of change in money supplies Solve for consumption Cand leisure l C MP with prices sticky and l 1 7 05MP 7 05MP So under this symmetric equilibrium a shock to either domestic or foreign money supply affects demand for the foreign good in the same way and lowers leisure But it is only a domestic monetary shock that affects domestic consumption here Implications for trade volume To gauge the effects on trade volume define this as imports plus exports as a ratio to GDP Using the demand functions and symmetry it turns out that this is simply a function of the relative prices changed across the two national markets Exports Im ports 1 GDP 1212pHp1 9 This is convenient because it means that prices give direct results about trade volume For example under fixed exchange rate regime where MM always we know that p H p and our measure of trade volume equals Trade one But under exible exchange rates if p H lt p trade is lower than one This happens when the foreign market is riskier than the domestic market so that firms charge a higher price there which reduces foreign demand for home goods In this case a exible exchange rate regime reduces trade On the other hand if there were some reason why p H gt p2 where the foreign market is less risky than the domestic one then trade could be increased by the exible exchange rate regime Which price is higher depends on the nature of the utility function in particular on whether consumption and leisure are separable or interact in the utility Looking at the price setting equations above we see that prices are different depending on whether the two numerators EUM and E UM differ from each other First under the case of separable utility where U does not depend on consumption we see that the two numerators are the same This is because as shown above shocks to M and M affect leisure the same way So in this case a oating exchange rate regime which permits different monetary policies does not affect the volume of trade at all Second suppose that leisure and consumption are substitutes Then the fact that a rise in M raises consumption while M does not means that U rises less for Mshocks than for M shocks and domestic sales are less risky for the firm This means there is less trade under a oating exchange rate regime Third suppose that leisure and consumption are complements Then the fact that a rise in M raises consumption while M does not means that U rises even more for Mshocks than for M shocks and domestic sales are riskier In this case there is more trade under a oating regime The general conclusion is that oating exchange rates can either help or hurt trade volume depending on the nature of the utility function The usual presumption that variable exchange rates inhibits trade is not true in general See figure 1 for an illustration Implications for Welfare As in previous papers the mainfactors determining the welfare are l the variance of consumption 2 the variance of leisure 3 the covariance of consumption and leisure if utility is nonsneparable Ifwe wish to compare the welfare of fixed versus oating the first element does not matter Since CMP consumption variance is the same under both regimes Considering the second element the variance of leisure will be higher under a fixed regime since I 1 7 05MP 7 05MP Under a fixed regime M and M always move together making leisure move a lot whereas under a oating regime they move independently So this element makes exible exchange rates more desirable depending on how risk averse people are in their utility for leisure Considering now the third element consumption and leisure are more negatively correlated under a xed exchange rate regime because a rise in M makes C and leisure move in opposite directions while a shock in M alone only moves leisure So if consumption and leisure are substitutes it is a good thing for them to move in opposite directions and this makes a xed regime desirable And if they are complements it is a bad thing so this makes a exible exchange rate regime more desirable The third element above suggests that welfare is higher for whichever regime promotes more trade But this is not true in general since the second element above suggests that a exible exchange rate regime can be welfare superior in cases where consumption and leisure are weak substitutes even though there is less trade in this case than under a xed regime For a xed exchange rate regime to be superior in welfare it requires that consumption and leisure be fairly strong substitutes See gure 2 20