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UBC Theses and Dissertations

Segmented regression modelling with an application to German exchange rate data Susko, Edward Andrew 1992

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S E G M E N T E D R E G R E S S I O N M O D E L L I N G W I T H A N A P P L I C A T I O N T O G E R M A N E X C H A N G E R A T E D A T A by E D W A R D A N D R E W S U S K O B . A . , The Univers i ty of W i n d s o r , 1990 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E T H E F A C U L T Y O F G R A D U A T E S T U D I E S T H E D E P A R T M E N T O F S T A T I S T I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A November 1991 ©Edward A n d r e w Susko, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) A b s t r a c t Segmented regression models are the topic of this thesis. These are regression models i n which the mean response is thought to be linear i n the explanatory variables w i t h i n regions of a part i cu lar explanatory variable. A criterion for est imat ing the number of segments i n a segmented model is given and the consistency of this est imator is established under rather general conditions. There have been many studies on model ing and forecasting foreign exchange rates us- ing various models, notably the random walk model , the forward rate model , monetary models and vector autoregressions, see, for example, Meese and Rogoff (1983) and Ba i l l i e and M c M a h o n (1989). The general conclusions have been that most of the models can- not outperform the random walk model by a significant marg in . The observation that the dependence of the exchange rate on the key macroeconomic indicators is t ime varying , nonstat ionary and nonlinear leads to consideration of nonlinear models. In this thesis seg- mented models are f itted to G e r m a n exchange rate data using least squares and forecasting results obtained from these models are compared wi th forecasting results from widely used models i n exchange rate predict ion. The segmented models tend to perform better than models that have been established in the l i terature, notably, the random walk model . i i T a b l e o f C o n t e n t s Abs t rac t i i L i s t of Tables iv L i s t of Figures v Section 1: Exchange Rates 1-7 1.1 Introduction 1-3 1.2 Determinat ion of Exchange Rates 4-7 Section 2: D a t a Analys is 8-15 2.1 Introduction 8 2.2 Exp lanatory Variables 8-10 2.3 P lo t Suggestions 10-12 2.4 Results from Model F i t t i n g 13-16 2.5 Conclusions 17 Section 3: E s t i m a t i n g the Number of Segments 18-30 3.1 Introduction 18 3.2 A Cr i t e r i on for E s t i m a t i n g the Number of Segments 18-20 3.3 Consistent Es t imat i on of the Number of Segments 20-30 Bib l i ography 58-59 i i i L i s t o f T a b l e s Table 1 Forward Rate Models 31 Table 2 Segmented Mode l s -S tandard Pr inc ip le Components 32-34 Table 3 Segmented M o d e l s - Standard Pr inc ip le Components-Forecast ing Results 35 Table 4 Monetary Models 36 Table 5 Segmented M o d e l s - T s a y Pr inc ip le Components 37-38 Table 6 Segmented M o d e l s - Tsay Pr inc ip le Components -Forecast ing Results 39 iv L i s t o f F i g u r e s F igure 1 T i m e Series Plots-Interest Differentials 40 Figure 2 T i m e series Plots -Trade Balance Differences 41 Figure 3 T i m e series P lo ts -Money Supply Difference 42 Figure 4 T i m e Series Plots-Interest Differential P r i n c i p a l Components 43 Figure 5 T i m e series Plots -Trade Balance P r i n c i p a l Components 44 Figure 6 T i m e series P l o t s - M o n e y Supply P r i n c i p a l Components 45 Figure 7 Scatter Plots-Interest Differential P r i n c i p a l Component 1 46 Figure 8 Scatter Plots-Interest Differential P r i n c i p a l Component 2 47 Figure 9 Scatter Plots -Trade Balance P r i n c i p a l Component 1 48 Figure 10 Scatter Plots -Trade Balance Pr inc ip le Component 2 49 Figure 11 Scatter P lo ts -Money Supply P r i n c i p a l Component 1 50 Figure 12 Scatter P lo ts -Money Supply P r i n c i p a l Component 2 51 Figure 13 Three Dimensional P lots -Pr inc ip le Components 52 Figure 14 Three Dimensional Plots-Interest Differential P r i n c i p a l Component 1 53 Figure 15 L a g 1 P lot -Exchange Rates 54 Figure 16 Scatter P l o t - F o r w a r d Rates 55 Figure 17 Three Dimensional P lot (1971-1985) 56 Figure 18 T w o Dimensional Plots (1971-1985) 57 Section 1 Exchange Rates 1.1 Introduction Exchange rates are an important source of variat ion i n f inancial decisions. However, at- tempts to model and predict exchange rates beyond 1971 have i n general not performed better than that of a random walk , Meese and RogofF (1983). It is the purpose of at least part of this thesis to provide a model which helps to explain some the var iat ion contained i n exchange rates. F i x e d currency exchange rates have been the norm for many years (unt i l 1971 we had been more or less under a system of fixed exchange rates). They have also been one of the goals that the internat ional community has tr ied to achieve. W h e n exchange rates are fixed or reasonably predictable i t is believed that trading i n global markets is less tentative and that economic ac t iv i ty is more efficient. One of the problems w i t h t r y i n g to mainta in fixed exchange rates, however, is that government pol icy w i l l be restricted, sometimes to the detriment of domestic considerations. A s a result of this problem exchange rates have been allowed to float. P r i o r to the great depression the exchange rate market operated according to the gold standard. Under the gold standard exchange rates were fixed. Currencies were defined i n terms of go ld . A loss or gain i n domestic money supply corresponded directly w i t h a loss or gain i n gold. Th i s would then be accompanied by an appropriate change i n interest rates, G D P and prices. These changes would i n t u r n be accompanied by appropriate changes i n foreign investment and trade which would adjust the exchange rate to its appropriate price. A s mentioned above domestic pol icy was restricted by such a system. Furthermore the inf lation rate was dependent upon gold discoveries. The gold Standard collapsed dur ing the chaos of the great depression of the 1930's. In the intrests of economic efficiency the industr ia l ized nations met i n Bre t t on Woods, New Hampshire i n 1944 to set up a system which would fix exchange rates. The U n i t e d States pegged its currency to gold and the other nations pegged their currency to that of the Un i t ed 1 States. T h e International Monetary F u n d was set up to police the s i tuat ion . Once again one of the ma jor problems w i t h such a system was that countries were forced to adopt monetary policies that may not have been i n their best domestic interests. Furthermore , devaluations were permit ted for countries only after long balance of payment deficits. This made such devaluations easy to predict and speculators were hence able to increase the magnitude of the devaluation. Revaluations expected from balance of payment surplus nations were not readi ly received and so the Uni ted States was forced to accept a balance of payments deficit. A l l of these problems combined to cause the fal l of the Bre t t on Woods system i n 1971. In general exchange rate model ing attempts have been based on the theoretical re lat ion- ships between the exchange rate and known (or approximately known) indicators of exchange rate movements. One favoured method used i n exchange rate model ing is to look at the re la- t ionship between the spot rate and the forward rate. The spot rate is the price of one currency i n terms of another currency. The forward rate is the price at which one currency can be pur - chased i n terms of another currency at some prespecified t ime i n the future. The usual setup is : St+i = p + Ft + et+i, where St is the spot rate at t ime t , Ft is the forward rate at t ime t and p is a risk p r e m i u m . Often p is taken to be equal to zero. A n o t h e r approach to exchange rate model ing is to assume the approximate satisfaction of theoretical money demand retationships. These type of models are called monetary models and are quite common i n the economic l i terature . See for instance B i l s on (1978, 1979), Frenkel (1976), Franke l (1979) and Hooper and M o r t o n (1982). The models that fa l l under this category are usually based on the money demand equation. For a given country the aggregate supply of money is M / P where M is the nominal supply of money and P is the price level . The aggregate demand for money is of the form m^Y, R,...), where Y is real income and R is the prevai l ing interest rate. A widely used form for m j is : md = KYae-pR, where K is a country specific constant and a , f3 are independent of the country we are t a l k i n g 2 about. So at equ i l ibr ium, M/P = KYae~m or, M P = KYae-PR' Let quant i ty* denote the relevant quantities for some other country. Now the theory of pur - chasing power par i ty assumes that the rate of exchange between two countries is directly re lated to the relative price levels of those two countries. So S = P/P* = M/M* K y a e _ p R , where S is the exchange rate. Le t t ing lower case letter=ln(upper case letter) , we have s = ( m - m*) + {k* -k) + a(y* - y) + (3{R - R*). For a further more detailed discussion of monetary models see Ba i l l i e and McMahon(1989) . In the previous two paragraphs an attempt has been made to introduce some of the preva- lent models used i n forecasting and empir ical research. A g a i n however these models have not been extremely successful i n predict ion or accurate model ing. There are several problems w i t h the above approaches. The forward rate models seem viable since they take into account expec- tations concerning future market conditions and these expectations may be more important i n a forecasting sense than the actual future state of affairs or predictions thereof. One problem is that the forward rates are constrained by the interest rate par i ty theory to be dependent p r i - m a r i l y on the previous exchange rate and interest rates. Trade balance differentials and other economic indicators of exchange rate movements are restricted i n their effect on the forward rate. To expla in , interest rate par i ty theory implies that (1 + ^ / 4 ) ^ - ( 1 + ^ / 4 ) * " where St is the rate of exchange for foreign currency i n terms of domestic currency, Ft is the corresponding forward rate for one quarter ahead, Rd is the domestic annual interest rate and Rf is the foreign interest rate. Thus i f as the forward rate model impl ies , St+i « Ft, then st+i « st + ln(l + Rd/4) - /n( l + Rf/4) 3 Note the s imi lar i ty between this model and the monetary models stated above. For R smal l a Taylor series approximat ion suggests that ln(l + R) « R. Hence the model is approximately a special case of the above mentioned monetary models w i t h some coefficients set equal to zero and a random walk component. The monetary models also have some difficulties. For one, there is no direct attempt to include more than two countries i n the models. It seems sensible that the exchange rate is determined by the explanatory variables of more than two nations. Another problem is that as certain explanatory variables approach inordinately high or low values governments, central banks and foreign investors w i l l take note and take act ion. The impl i cat ion is that the values of parameters may not be stable across a l l regions of the explanatory variables. It is pr imar i ly these last two crit ic isms that the models to be suggested intend to deal w i t h . Ideally, the more explanatory variables we include the more information we gain . However due to the l i m i t e d number of observations some type of trade-off must be made. B y tak ing pr inc ipa l components w i t h respect to explanatory variables across countries i t is hoped that some in format ion can be gained about the relationships between the exchange rate and the explanatory variables of several countries as opposed to just two. B y considering a model that is segmented w i t h respect to the values of an explanatory variable i t is hoped that the problem of instabi l i ty of the parameter values may be dealt w i t h . 1.2 Determinat ion of Exchange Rates There are several approaches to exchange rate determinat ion. One idea is that of Purchas- ing Power P a r i t y . Under this theory the exchange rate between any two countries is believed to reflect the relative price levels of those two countries. A n impl i ca t i on of this theory is that inf lat ion rates are the major determinants of exchange rates. However, empir ica l ly such rela- tionships have been anything but clear i n the 1970's and 1980's. S t i l l this theory was widely accepted pre 1971 for explaining long range behaviour of exchange rates. It would seem that this theory is useful i n explaining exchange rate behaviour during fixed exchange rate periods. A no ther approach is to consider the exchange rate as an asset value. Under this approach the exchange rate one period ahead is determined by the present exchange rate plus some 4 expected changes i n the exchange rate market . The problem then arises as to how to account for expected changes. One th ing to do is look at the forward rates, as mentioned i n the in troduct ion , this approach is useful i n that i t takes expectations about future events into account and often these expectations can have more relevance i n exchange rate prediction than the actual occurrence of future events. These expectations w i l l be dependent upon monetary sources, but i t should also be said that these expectations w i l l be dependent on other things, such as news. A case i n point is the upward movements i n the exchange rates throughout many of the industr ia l ized nations (notably Germany) i n 1980. It is hypothesized that these changes were (Issard, 1983 ) at least p a r t l y a result of the news that R o n a l d Reagan could be expected to w i n the upcoming presidential election. W i t h his support for t ight monetary pol icy and his support for the st imulat ion of U . S . competetiveness i t was expected that the dollar would increase i n value. The point is that the exchange rate increased irrespective of whether U . S . monetary pol icy actual ly was t i ght , the determining factor was the expectation as opposed to the actual occurence of the event. It should be added that asset model aproaches are inter l inked in some sense w i t h monetary models. Expectat ions about future conditions w i l l be dependent upon news about po l i t i ca l s ituations and the l ike but they w i l l also be dependent on what the current or past state of affairs is l ike w i t h respect to key economic variables. Furthermore i t has been mentioned that at least theoretically the forward rate approach can be approximated by monetary approach w i t h constrained variables and a random walk component. The f inal approach I w i l l consider w i t h respect to exchange rate determinat ion, and the approach that forms the basis for my models, is that of monetary models. The idea is that the exchange rate is the price of one country's currency in terms of another country's currency and hence that the laws of supply and demand apply. Thus , this approach looks at economic variables which are considered important i n the determination of the supply of and demand for a country 's money by foreign interests. Some key factors i n such a determinat ion would be: 1. Trade Balance and Current Account - A country sells i ts exports i n i t ' s home currency. So ho ld ing a l l other variables constant an increase i n the demand for a country 's currency would correspond to an increase i n exports. 5 2. Foreign Investment - A n increase i n investment i n a country would be accompanied by an increase i n demand for that country 's currency i n order to finance the investment. 3. Interest Rates - A n increase i n interest rates attracts foreign investment. 4. M o n e y Supply - A n increase i n a country's domestic money supply implies an increase the money supplied to foreign investors. 5. Inflation rates - r is ing inf lat ion rates make a country's currency less favourable to foreign investors. Such factors are important i n the determination of exchange rates but certain problems and ideas need to be kept i n m i n d when using these models. Changes i n such indicators would not usual ly correspond to simultaneous changes i n money supply and demand this makes i t necessary to investigate the use of explanatory variables at different lags. Vo la t i l i t y may confuse relationships, large changes i n economic indicators may cause concern as to the health of a nation 's economy thus result ing i n the reverse of or at least tempering of the effect on the exchange rate. M a n y of these variables are endogenous and thus again expected effects may be tempered. Another concern should be policy changes, they may change the relationships between these variables and certainly affect expectations about these variables and exchange rates (the " L u c a s C r i t i q u e " , Frenkel 1983 ). It is important to consider changes between countries w i t h respect to these variables. For instance, holding a l l other variables constant an increase i n one country's interest rates should have no effect on exchange rates i f the interest rates of a l l other nations increase at the same rate. It is this approach w i t h the above mentioned considerations taken into account that leads to the segmented regression models that I consider. I t ry to take the important determinants of supply and demand for foreign currency into account as explanatory variables. T h a t is , the variables mentioned above (1-5) and subsets of these variables are used to determine the segmented regression relat ionship. Since international trade is greatest between the larger industr ia l nations it is reasonable to restrict attention to the exchange rate and explanatory 6 variables corresponding to these nations. Thus the possible explanatory variables were chosen to be the differentials of the important determinants between G-7 nations. Suppose we can assume that government policies of the major t rad ing partners are more or less rational over the years and that interest rates and money supplies digest market in format ion relatively efficiently. T h e n since pol icy decisions w i l l often be determined or acompanied by hi - lo values of these variables i t seems reasonable to consider seperate regression segments w i t h respect to some segment explanatory random variable. For example governments are certain to act i n the case of large trade balance deficits or large interest rate differentials. W h a t emerges is a model where the exchange rate is dependent upon certain explanatory variables and where this relationship is different at extreme values of the explanatory variables. The model ing considered is inherently long run model ing. Changes i n these determinants and changes i n the exchange rates w i l l not be simultaneous, there w i l l be some t ime lag . Furthermore , this t ime lag may be of variable length w i t h repect to , say, monthly periods. Hence month ly models may not be appropriate i n that the t ime lag chosen as 'best' may not be constant w i t h respect to t ime. If the t ime periods concerned are increased to say quarterly periods then the models should be more robust w i t h respect to this assumption of constant t ime lag . For this reason it was decided to concentrate on long run model ing. 7 Section 2 D a t a Analys is 2.1 Introduction In section 1 i t was mentioned that pol icy considerations may have effects on the re lat ion- ships between exchange rates and certain economic indicators . Thus a segmented regression model could be appropriate. In this section the results of fitting various segmented models to the G e r m a n exchange rate movements post 1971 are discussed. The general form of a segmented t ime series regression model can be stated as Yt = xj_x/? t- + et, if xtd e ( r , - _ i , n], i - 1 , . . . , / + 1, where Yt is the exchange rate at t ime t , x< is a vector of explanatory variables mentioned above at t ime t — 1, et is an error t erm, xtd is one of the components of x t (the segmentation variable) , —oo = To < T\ < • • • < Ti+i — oo and {xt} and {Q} are independent series. 2.2 E x p l a n a t o r y Variables To use a segmented model i t was neccesary to make certain decisions about explanatory variables. Some important theoretical determinants of exchange rates were discussed i n section 1 and are l is ted below 1. Interest Rate Differentials 2. Trade Balance Differentials 3. M o n e y Supply Differentials 4. Inflation Rate Differentials Some things should be mentioned at this point . It is the differential that is important . In section 1 i t was pointed out, for instance, that i f a country increases its supply of money that this w i l l i n general make its currency less attract ive . Th i s w i l l not be true i f every other country also increases its money supply by an appropriate amount. Hence the importance of dif- ferentials. However i f differentials between a l l countries are considered there w i l l be hundreds of 8 explanatory variables. In view of this i t was decided to consider differentials between impor tant t rad ing partners and economic powers. In part i cu lar i t was decided to consider the differentials between Germany and the other G-7 nations. P r e l i m i n a r y simultaneous t ime series plots and scatter plots of the exchange rate vs. inf lation rates suggested that inflation rate differentials would not be very useful as explanatory variables and so i t was decided not to consider them. Further the effects of the inf lat ion rate differentials are expected to be imbedded i n the interest rate differentials. Thus the explanatory variables to be considered were interest rate differen- t ia ls , money supply differentials, and trade balance differentials a l l between Germany and the G-7 nations. It was important to consider these variables w i t h some t ime lag since it may be that the effect of an explanatory variable on the exchange rate w i l l not be felt immediately . Thus the explanatory variables were considered w i t h and without t ime lags. F i n a l l y some sort of s tandardizat ion , to be mentioned shortly, was necessary. In view of the fact that exchange rates were relatively fixed v i a the Bre ton Woods Monetary System u n t i l 1971 and that the explanatory variables are theoretically more valuable as l ong term predictor than as short term predictors, i t was decided to use quarterly da ta from 1971 through to the second quarter of 1990. T h e decision to use quarterly da ta as opposed to month ly da ta was par t ly because of some of the concerns mentioned i n the previous section and par t ly because of some pract i ca l problems related to the avai labi l i ty a n d / o r re l iabi l i ty of certain economic d a t a on a month ly basis. To repeat at least one of the concerns mentioned i n the previous section it can be expected that the lag used i n the explanatory variables is more l ike ly to be variable w i t h respect to t ime i n the monthly model than in the corresponding quarterly model . To avoid this problem without adding parameters a useful approach is to consider quarterly data . For each G-7 country the fol lowing data were obtained from the internat ional financial statistics published by the I M F (International Monetary Fund) . 1. Exchange Rates - These were end of period spot rates. T h e y were expressed as M a r k s / U . S . Do l la r . 2. Money supply - T h i s was taken to be M l money. It was calculated as demand de- posits plus currency i n c i rcu lat ion . For each country money supply was given i n that country's currency. 9 3. Interest Rates - Th i s was the discount r a t e / B a n k rate for the country of interest. 4. Trade Balance - Th i s was calculated as merchandise imports minus merchandise exports. Th is quanti ty was given i n U . S . Dol lars . 5. C . P . I - Consumer Price Index. T h i s quanti ty was used as the index for inf lat ion of a part icular country. Several things should be mentioned w i t h respect to the data . Some standardizat ion is necessary. In part i cular one wants to consider real values. In order to do so i t was necessary to adjust for the rate of inf lat ion. Thus to get real money supply the nominal money supply was mul t ip l i ed by (lOO/CPIcountry)- To get real trade balance nominal trade balance was mul t ip l i ed by (100/CPIus)- To get real b i lateral exchange rates, exchange rates were mul t ip l i ed by CPIus/CPlGermany), since the exchange rate can be regarded as the price i n Deutsche M a r k s for one U . S . D o l l a r . Since money supplies were given i n home currencies a further adjustment was necessary to obta in money supply differentials. E a c h country's money supply was standardized, i.e. (quantity-mean(quantity) /s .e . (quantity) ) . Now even i n the presently described s i tuat ion where only differentials between G-7 nations are considered there are s t i l l 3x6=18 explanatory variables. Since there are 78 observations this would have been too many explanatory variables for a reasonable segmented model . Thus it was decided to consider as possible explanatory variables the pr inc ipa l components for each differential which explained most of the w i t h i n differential variabi l i ty . For example the p r i n - cipal components from ( G e r m a n y - C a n a d a interest rate differential, Germany-France interest rate di f ferential , . . . ,Germany-U.S. interest rate differential). Another possible approach was to consider the pr inc ipa l components as defined i n Tsay (1990). These pr inc ipa l components are opt imal i n a predictive sense. Models using these pr inc ipa l components were explored. 2.3 Plot Suggestions The next step i n the analysis was to consider the plots of the various explanatory variables vs. the exchange rates. O f course the explanatory variables are related to each other and hence two or even three dimensional plots w i l l not give the complete picture . The plots are shown on pages 39-54. T i m e series plots are given on pages 39-44. T h r o u g h each of the plots the real 10 G e r m a n exchange rate movements are traced w i t h a solid l ine. The corresponding scale is given on the left hand side of the plot . Each plot also gives the sample pa th of some explanatory variable w i t h a broken l ine. The corresponding scale being given on the right hand side of the plot . Not ice the vo lat i l i ty of the exchange rate part i cu lar ly post 1980. The period i n the m i d '80's is par t i cu lar ly vol i t i le . Th i s ' h i l l ' is not seen i n many of the other t ime series plots a l though there is some ind icat ion of i t i n for instance the t ime series plot of standardized money supply differential w i t h respect to Germany and the U . S . . A l s o , notice the t ime series plots of the trade balance differentials of the G-7 nations given on page 40. These plots hint at the interdependence of the exchange rates and the chosen explanatory variables. The plots of the trade balance differentials often exhibit ' h i l l s ' and 'valleys' after that same behavior is observed i n the exchange rates. The next type of plots to be observed are the two dimensional scatter plots given on pages 45-50. These plots are labeled lag plots i n reference to the fact that they plot the explanatory variable l is ted at the top of the page vs. exchange rates at six different t ime lags. These plots must be observed w i t h caution of course since the explanatory variables are certain to be interrelated. For instance money supply differentials are definitely going to be dependent upon prevail ing interest rates to a certain extent. The plots are not altogether encouraging. For example the relationships between money supply differentials, trade balance differentials and exchange rates are not par t i cu lar ly evident i n the corresponding scatter plots of the exchange rates vs. the first pr inc ipa l components. However, the plot of the exchange rates vs. the first pr inc ipa l component of the interest rate differentials does suggest a relationship, i n fact i t appears f rom the plots that a segmented approach w i t h respect to this explanatory variable may be the way to go. For a given explanatory pr inc ipal component notice the relative homogeneity of the plots of the exchange rates vs. explanatory pr inc ipa l components across different t ime lags. In view of this apparent s imi lar i ty w i t h respect to lags i t was decided that i t may not be neccesary to fit a lot of models w i t h many different lag combinations. Exper imentat ion w i t h a few lags, to be mentioned below, confirmed that this was indeed the s i tuat ion. A l s o , notice that the scatter plots of the exchange rates vs. the interest differentials seem to hint at the existence of at least two segments. The plot of the exchange rate vs. the exchange rate lagged 11 one quarter is given on page 53. C lear ly there is a very good l inear relationship. Hence i t was decided to include the exchange rate lagged one quarter i n a l l further model ing attempts. T h e plot of the forward rate vs the spot rate one period forward is given on page 54. A g a i n there is a clear l inear relationship although the var iabi l i ty is somewhat greater than that i n the plot of the exchange rate vs. the exchange rate lagged one quarter. S t i l l the plot does lend credence to the forward rate models. It has been mentioned that two dimensional plots may not be sufficient i n exploring the possible relationships i n the data . O n pages 51-52 several three dimensional plots are given. O n page 51 three dimensional plots of each of the first explanatory pr inc ipa l components w i t h the exchange rates lagged one period and at present are given. The lat ter five plots do not give much suggestion to possible relationships along the direction of the explanatory variable but the plot concerning the pr inc ipa l component for the interest differentials does suggest a possible segmented relationship w i t h possibly three segments. T h e plot on page 52 rotates the three dimensional plot of the first pr inc ipa l component for the interest rate differentials. A s i n the two dimensional plot a segmented relationship appears possible and it appears that a segmented model w i t h possibly two or three segments may be appropriate . In table 3, some forecasting results are quoted. One stunning aspect of this table is w i t h respect to the models w i t h Interest Differential P C I as segmentation variable and Money Supply P C I as an addi t ional segmentation variable. There is quite a large dicrepency between the model w i t h two segments and the model w i t h three segments. T h e model w i t h three segments performs substantial ly better than the model w i t h two segments. Thus an immediate question is to what extent the number of segments could have been predicted v i a exploratory plots. One such plot is given on page 55. Yt — fiiYt-i, i = 1. ..I) versus lag-1 interest rate differential and money supply differential was p lot ted , where 1 was any plausible number of segments and $i the estimated coefficients. The plots were then rotated to get images from different angles. The plot shown is the one corresponding to / = 3, and it does appear i n this plot that three segments are appropriate. 12 2.4 Results from M o d e l F i t t i n g In this subsection the f i t t ing of various models is discussed. The general form of the models can be stated as Yt = x't.j/?,- + et, if xtd G (r<_i, T{], i = 1 , . . . , / + 1, where Yt is the exchange rate at t ime t , x t is a vector of explanatory variables at t ime t — 1, et is an error t e r m , xtd is one of the components of x* (the segmentation variable) and —oo = To < T\ < • • • < Ti+i = oo. Note that the special case where / = 0, To = — oo,Ti = oo is the famil iar l inear regression setup. T h e s ituation where I > 0 is that of segmented regression, namely a l inear regression setup is assumed on 'segments' of a part i cular explanatory variable. The types of models that were fitted can be categorized. The first type is the segmented models w i t h standard pr inc ipa l components. These models are segmented models w i t h various combinations of the explanatory variables mentioned i n section 2.2 each model having one explanatory variable determining the segmentation. The second type of models is the segmented models w i t h Tsay type pr inc ipa l components. These models are the same i n spirit as the previous type of models but the pr inc ipa l components adopted were taken according to Tsay (1990) i n order to maximize the lag 1 autocorrelation of a l inear combination of the variables of interest. There are two other types of models, these are competing models i n the sense that they are the favoured models i n the economic l i terature . These models have been described i n the first section where the general problems i n exchange rate determination were discussed. The first of the two would be monetary models. The form and theoretical background behind these models was discussed i n the first section. For previous analysis of these models using monthly data pre 1980's see Meese and Rogoff (1983). The other type of competing model is the forward rate model . A g a i n , some of the merits and pitfalls of using this type of model have been discussed i n section 1. Segmented regression is a form of nonlinear regression and cr iterion for testing model assumptions and parameter values are not well developed and would involve assumptions about independent identical ly d istr ibuted errors. A s a pract ical issue the first thing we are compelled to consider is the appropriate number of segments. Th i s issue is discussed i n more detail i n the next section as i t pertains to large samples. In the present s i tuat ion we do not have a 13 large sample. For smal l samples this issue has not been resolved and often the best pol icy is to experiment w i t h different numbers of segments perhaps as suggested i n the plots. Furthermore smal l samples place restrictions on the ab i l i ty to estimate large numbers of parameters and thus some further retrictions must be placed upon the number of segments. However cr i ter ia are needed to adress the issue of model appropriateness. The p r i m a r y cr i ter ia considered were mean squared error ( M S E ) , and the sum of out of sample one step ahead forecasting errors for a five year period ( S S F E ) from the second quarter of 1985 to the second quarter of 1990. T h e quantities from the random walk model were used as the yardstick to measure performance. Some residual analysis was done as wel l . Scatter plots of the residuals acf plots and t ime series plots of the residuals were examined. The results for the segmented models w i t h standard pr inc ipa l components are l isted on pages 31-34. A variety of combinations of explanatory variables, segmentation variables and lags were experimented w i t h . In most cases the reduction i n mse over the random walk was s imi lar for models having the same explanatory and segmentation variables but w i t h different lags. T h i s tends to agree w i t h the plot suggestions. It does not appear that choosing the correct lag for the explanatory variables is of utmost importance. Th i s may be a result of the fact that quarterly as opposed to month ly data were used. The models i n which the segmentation variable was the pr inc ipal component from the interest differentials had significant reductions i n M S E over those models which used a different segmentation variable. A g a i n this was indicated to a certain extent by the plots. F r o m the results corresponding to M S E it was decided to use the models which had the pr inc ipa l component from the interest rate differentials as the segmentation variable to evaluate forecasting abil ity. A s is seen on page 34 the reduction i n one step ahead sum of squared forecasting error ( S S F E ) over the random walk model for a five year period (from the second quarter of 1985 to the second quarter of 1990) was as large as 42 percent i n the case where the pr inc ipa l components from the interest differentials and money supply differentials were the explanatory variables. T h i s result is quite impressive i n view of the fact that i t has been widely accepted that large reductions i n forecasting ab i l i ty over the random walk w i l l not usually be obtained. The results for the segmented models using Tsay type pr inc ipa l components are l isted 14 on pages 36-38. The results are s imilar to the results obtained using the standard pr inc ipa l components. In fact plots of the pr inc ipal components suggested that the pr inc ipal components obtained i n this fashion were very s imi lar to the pr inc ipa l components obtained the standard way. The results using standard pr inc ipa l components tended to be somewhat better than the results using Tsay type pr inc ipal components but it seems clear from the s imi lar i ty of the results that the appropriateness of the pr inc ipal components taken was not that much of an issue. T h e results from f i t t ing forward rate models are given on page 30. T h e forecasts came from the models St+i = Ft + et, and St+1 =p + Ft + €t. T h e scatter plot of of the exchange rate vs. the forward rate i n conjunction w i t h the interest rate par i ty theory suggested that i t would be appropriate to ignore the risk premium p. So analyses were performed w i t h and without the risk premium. W h e n St, Ft were expressed i n nomina l terms there was a smal l reduction i n forcasting error over the random walk. A s mentioned before large gains cannot reasonably be expected from these types of forecasts since Ft « St due to arbitrage concerns. W h e n models w i t h adjustments for inf lat ion were included there was no reduction i n forecasting errror over the random walk model . The results of f i t t ing the monetary models are l isted on page 35. Some discussion of these models was given i n the first section. For a more complete discussion of these models see Meese and Rogoff (1983). The under ly ing equation for these models is s = a0 + ax(m - m*) + a2(y - y*) + a3(rs - rs*) + a 4 (7r e - 7r*) + a5TB + a6TB* + u, where s is the l ogar i thm of the price of dollars i n term of foreign currency, m - m * the logar i thm of foreign to U . S . money supply, y -y* the logar i thm of foreign to U . S . real income, rs - r* the interest rate differential, and we — 7r* is the expected inf lat ion differential. T B and T B * represent the foreign and U . S . cumulative trade balances. T h e respective bank rates were taken as the short term interest rates. The expected inflation rate was taken to be the inf lation rate of the previous per iod . In detai l , ire = (CPIt - CPIt-\)ICPIt-\. F r o m this general formulation 15 corne the three models that were analyzed. The flexible price (Frenkel-Bi lson) monetary model i n which a4 = as = ae = 0. The st icky price (Dornbusch-Frankel) monetary model i n which a5 = as = 0 and the st icky price (Hooper -Morton) asset model i n which none of the coefficients are zero. See B i l s on (1978, 1979), Frenkel (1976), Dornbusch (1976), Frankel (1979, 1981) for further discussion of these models. These models were fit w i t h no lag i n the explanatory variables. T h e results were poor, none of the models fared better i n terms of M S E or S S F E i n comparison to the random walk model . The other type of model w i t h which comparisons were made were vector autoregression ( V A R ) type models as described i n Meese and Rogoff (1983). These models performed considerably better. They can be described by St = O-nSt-l + di2St-2 + . . .ainSt-n + BaXt-\ + Bi2Xt-2 + • • -BinXt-n + V-it, where Xt-j is the vector of the explanatory variables i n the equation above lagged j periods. T h e results are l isted on page 35. The best result was from the Dornbusch-Frankel model w i t h two lags. T h i s model achieved a reduction of 2.26 percent over the random walk model i n S S F E . None of the other models reported a reduction over the random walk model . It should be mentioned that these results should be interpreted w i t h caution when comparing to the segmented models. Since the results quoted for the segmented models were i n terms of the exchange rate as opposed to the logar i thm of the exchange rate the sums of squares were calculated i n the same fashion for the monetary models. T h i s tended to give more dramat ic results than would have been obtained i f the sums of squares had been computed directly using the logar i thmic values. For instance the above mentioned 2.26 percent reduction would be a reduction of 12.26 percent when the In transformation was not taken i n computing S S F E . S t i l l the results are s t r ik ing i n contrast to the superior results obtained from the segmented models. A question that arises is , to what extent can the nonlinear srtucture noticed i n the complete da ta set be seen i n truncated versions of the data? One would hope, for instance, that the number of segments would remain constant. In order to attempt to answer this question several plots have been included which give a visual comparison of the da ta pr ior to 1985 w i t h the fu l l data set. These are given on pages 56 and 57. 16 2.5 Conclusions The monetary models i n general d id poorly. There was an exception as mentioned above but even this model s t i l l performed poorly i n comparison to the segmented model and the best reductions were comparable to the best reductions a decade ago as reported i n Meese and Rogoff (1983). The forward models gave slight reductions i n S S F E over the random walk models. Th i s can be expected since the interest rate par i ty theory implies that Ft « St- Hence large gains over a random walk model should never be expected. T h e segmented models do seem to do better than the random walk model . Certa in ly more testing using the exchange rates of other nations, fo l lowing up w i t h further testing on the G e r m a n exchange rates of the future and other variations on the theme are neccesary to make any statments w i t h a degree of certainty. The m a i n reason that this type of model ing was in i t ia ted was the concern about the effect of pol icy decisions, speculation and central bank actions i n times when economic indicators are tak ing on extreme or unusual values. It appears that this may be a val id concern. The segmented models w i t h the pr inc ipa l component from the interest differentials as the segmentation variable performed significantly better than many other models. It does appear that there is some segmentation w i t h respect to this variable. Perhaps this can be at tr ibuted to market efficiency i n some manner. W h e n differentials are large central banks and speculators w i l l be forced to pay attent ion but when the differentials are smal l i t may be that only the more informed, aware and less risk averse part ipants act. 17 Section 3 Est imat ing the N u m b e r of Segments 3.1 Introduction Segmented models may be useful i n many situations. For example Yeh et al.(1983) discuss the the idea of an 'anaerobic threshold ' . It is hypothesized that i f a person has his workload steadily increased through some form of exercise there comes a point where the muscles cannot get enough oxygen and what were anaerobic metabolic processes become aerobic processes. Th i s point is referred to as the 'anaerobic threshold ' . In this s ituation two segments are what is suggested by the subject oriented theory. So i t would be natura l for the modeler to fit a model w i t h two segments. However i n some situations i t may be suspected that a segmented model should be adopted but the appropriate number of segments may not be known. For instance i n the exchange rate problem it is suspected that a segmented model is appropriate due to pol icy changes. It is not , however, clear how many segments w i l l be necessary beforehand. One immediate approach to this problem is to graphical ly attempt to determine how many thresholds seem to be appropriate. Th is is worthwhile as a first step and i n the case of a single explanatory variable but may not be appropriate i n the mult ivariate case. In the mult ivariate case the interrelationships of the explanatory variables may confuse such an approach. Further this approach lacks object ivity , some sort of automated rule is desired. In this section I discuss a consistent procedure for identi fying the number of segments. 3.2 A C r i t e r i o n for Est imat ing the N u m b e r of Segments Consider the fol lowing segmented l inear regression model . Yt = x'tPi + 6 f , if xtd € ( r f _ i , T j ] , i = 1 , . . . , / + 1, where et — ipiZt-i, 5^o° 1̂ **1 < 0 0 > ? w i t h the {zt} i i d , mean zero and variance a2 and independent of {xt}, x t = (1 , xt\,..., xtp)' and —oo = TQ < T\ < • • • < T J + 1 = oo. Further we assume that there exists 6 > 3 /2 , k > 0 9 |V>;| < k/is V i . Note that this implies that {et} is a 18 stat ionary ergodic process. Consider the fol lowing regression setup. Reca l l that the least squares estimate of j3 is given by and the sum of squared error is given by where Hn = Xn(X'nXn) 1X'n. T h e s i tuat ion here i n the segmented regression case is completely analogous. where i n general A w i l l denote a generalized inverse for any matr ix A, and 1(.) is the indicator funct ion, T h e n , i n terms of true parameters, our model can be rewritten i n the vector form, T h e est imation of a l l the parameters is done p r i m a r i l y i n two steps. F i r s t we estimate / , the number of thresholds r i , . . . , T j o . Th i s is done by m i n i m i z i n g the modified Schwarz ' cr iterion Under some regularity conditions essential to the identi f iabi l i ty of the regression parameters, we shal l see below that the ordinary least squares estimates $ • w i l l be unique w i t h probabi l i ty Consider the segmented l inear regression model discussed i n the previous section. Let / min imize MIC(l) . To identify the number of thresholds /, and hence the number of segments consistently, assume: Under Condition 1, the design m a t r i x Xn(a,j3) has f u l l co lumn rank a.s. as n —• oo for every open interval i n the smal l neighborhood of the true thresholds r,,i = 1 , . . . , / for which Xd has positive probabi l i ty density. A n d Xn(a,/3) w i l l have ful l co lumn rank for large n and for every open interval (a,(3) i n the smal l neighborhood of rf,i = 1 , . . . , / ° for which Condition 2 is satisfied. So /?,• w i l l be unique w i t h probab i l i ty tending to 1 as n —• oo, for i = 1 , . . . , /, provided that I converges to / , the true number of thresholds i n probabi l i ty . W h e n the segmented regression model reduces to the segmented po lynomia l regressions or functional segmented regressions as discussed by Feder (1975), then a s imi lar condit ion to either Condition 1 or Condition 2 is essential for ident i fy ing the segmented model parameters. For details, see Feder (1975). In part i cu lar , for the segmented po lynomia l regression model , Cond i t i on 1 is automat ica l ly satisfied i f the key covariate Xd has positive density w i t h i n a smal l neighborhood of each of the thresholds. In add i t i on , we need to place some restrict ion on the d is tr ibut ion of the i i d errors {zt}. A n d this is the so-called local exponential boundedness condition. A random variable Z is said to be locally exponentially bounded i f there exist constants CQ and To i n (0,oo) such that E(euZ) < e c ° " 2 , V |u| < T0. (3.1) M a n y of the commonly used distr ibutions such as the n o r m a l , the symmetrized Poisson and exponential d istr ibut ions have such a property. In reality, the sample size n is always finite and hence the number of thresholds that can be effectively identified is always bounded. So we w i l l assume throughout that there always exists an upper bound L of the true nummber of thresholds. Another s impli f icat ion we gain i n the nonlinear m i n i m i z a t i o n of S ( T I , ..., r/) is obtained by l i m i t i n g the possible values of T\ < ... < Ti to the finite discrete set, {xid, •..,xnd}. Th i s restrict ion induces no loss of generality. T h e o r e m 1 Consider the segmented linear regression model with Xn independent of ln. Suppose {zt} are iid with a locally exponentially bounded distribution having mean zero and variance a2. Assume for the true number of thresholds, I, that I < L for some specified upper bound L > 0 and that one of Conditions 1 or 2 is satisfied. Then I converges to I in probability as n —y oo. 21 The proof of the theorem will be given after a series of related lemmas. P r o o f It follows from M a r k o v ' s inequality that for 0 < t0 < T satisfying \t0a.i\ < T for a l l F r o m the previous l e m m a there exists M i such that Since A ( k ) and B ( k ) are independent we get that F i n a l l y , to conclude the proof,we note that L e m m a 2 Consider the segmented regression model with the design matrix Xn satisfying either Condition 1 or Condition 2. Assume that the iid errors {zt} are locally exponentially bounded and are independent of Xn. Then where po is the true order of the model and To is the constant associated with the locally nential boundedness of{zt}. P r o o f Cond i t i on ing on Xn, we have that Since Hn(xsd,xu) is idempotent , i t can be decomposed as Hn(xsd, Xtd) = WAW, where W is orthogonal and A = diag(l, • • •, 1,0, • • •,0). Not ing that rank(A) > rank(AB) we get the fol lowing results: as n —• oo, where c is the constant i n L e m m a 1. F i n a l l y , by appealing to the dominated convergence theorem we obtain the desired result. L e m m a 3 Consider the segmented regression model with the design matrix Xn satisfying either Condition 1 or Condition 2. Assume that the iid errors {zt} are locally exponentially bounded P r o o f It suffices to show the case when 1 = 1. Since the proof under either Condition 1 or Cond i t i on 2 is essentially the same, we shall proceed by verifying the l e m m a under Condition It then follows from the L a w of Large Numbers for stat ionary ergodic stochastic processes Simi lar ly , i t can be shown that Therefore, It remains to show that Thus proceeding as before, using the law of large numbers we get that which gives the result. L e m m a 3.4 Consider the segmented regression model (2.1) with the design matrix Xn satisfy- ing either Condition 1 or Condition 2. Assume that the iid errors {zt} are locally exponentially bounded and are independent of Xn. Let 1° denote the true I. Let ( i f , . . . , T;°) denote the true Hence it suffices to show that for each T h e n it follows from the previous lemmas and the law of large 30 In concluding the theoretical section of this paper some mention should be made as to the need for further research into the statist ical properties of these models. For instance i t would be useful to have l ike l ihood based tests for testing the existence of thresholds. A l so of interest would be the asymptot ic distributions of the estimated parameter values under m i l d stat ionarity assumptions. T a b l e 1 F o r w a r d R a t e M o d e l s Random Walk Mode l Forward Rate Mode l Unadjusted sse 1.67 1.66 per cent reduction 1.04 five year mfse 0.45 0.42 per cent reduction 6.67 Adjusted for Inflation sse 1.29 1.31 five year mfse 0.44 0.44 W i t h Intercept Unadjusted sse 1.67 1.65 per cent reduction 1.14 five year mfse 0.45 0.43 per cent reduction 4.44 Adjusted for Inflation sse 1.29 1.30 five year mfse 0.44 0.47 31 T a b l e 2 S e g m e n t e d M o d e l s - S t a n d a r d P r i n c i p l e C o m p o n e n t s Percent reduction in root mse model description from random walk model Interest Differential -Pr inc iple Component l ( L a g 1) Two Segments 16.69 Three Segments 23.61 Two Segments (random walk coeff.=l) 6.75 - Principle Component l ( L a g 2) Two Segments 15.20 - Principle Component l ( L a g 3) Two Segments 9.85 Trade Balance Differential -Pr inc iple Component l ( L a g 1) Two Segments 6.28 Two Segments (random walk coeff.=l) 2.46 - P r i n c i p l e Component l ( L a g 2) Two Segments 4.92 - P r i n c i p l e Component l ( L a g 3) Two Segments 6.90 Money Supply Differential -Pr inc iple Component l ( L a g 1) Two Segments 13.46 Two Segments (random walk coefF.=l) 5.98 - P r i n c i p l e Component l ( L a g 2) Two Segments 15.00 32 Percent reduction in root mse model description from random walk model Interest Diferential -Pr inc ipa l Component 1 (lagl) and Trade Balance Differential -Pr inc ipa l Component 1 ( lagl) I .D. segmentation variable - T w o Segments 16.38 -Three Segments 22.77 T . B . segmentation variable - T w o Segments 11.45 Interest Diferential -Pr inc ipa l Component 1 ( lagl) and Money Supply Diferential -Pr inc ipa l Component 1 (lagl) I .D. segmentation variable - T w o Segments 20.76 -Three Segments 26.03 T . B . segmentation variable - T w o Segments 13.37 Money Supply Diferential -Pr inc ipa l Component 1 (lagl) and Trade Balance Diferential -Pr inc ipa l Component 1 ( lagl) T . B . segmentation variable - T w o Segments 7.03 M . S . segmentation variable - T w o Segments 12.66 33 Percent reduction in root mse model description from random walk model Interest Diferential -Pr inc ipa l Component 1 (lagl) and Money Supply Diferential -Pr inc ipa l Component 1 (lagl) and Trade Balance Diferential -Pr inc ipa l Component 1 (lagl) I .D. segmentation variable - T w o Segments 21.28 -Three Segments 27.56 T . B . segmentation variable - T w o Segments 11.67 -Three Segments 12.55 34 T a b l e 3 S e g m e n t e d M o d e l s — S t a n d a r d P r i n c i p l e C o m p o n e n t s F o r e c a s t i n g R e s u l t s Model Description Sum of Squared Forecasting Error (SSFE) S S F E (Random Walk) Percent Reduction in S S F E Interest Differential Two Segments Three Segments 0.299 0.368 0.437 0.437 31.56 15.63 Interest Differential and Trade Balance Differential Two Segments Three Segments 0.424 0.490 0.437 0.437 2.92 Interest Differential and Money Supply Differential Two Segments Three Segments 0.636 0.252 0.437 0.437 42.33 Interest Differential and Money Supply Differential and Trade Balance Differential Two Segments Three Segments 0.630 0.507 0.437 0.437 35 T a b l e 4 M o n e t a r y M o d e l s percent reduction percent reduction (without In Mode l root M S E root M S E S S F E S S F E transf.) Random Walk 0.063 0.0908 Frenkel-Bilson 0.2219 0.7780 Dornbusch-Frankel 0.192 0.8594 Hooper-Morton 0.194 2.6440 Vector Autoregressions Frenkel-Bilson -2 lags 0.0598 5.06 0.1003 0.86 Frenkel-Bilson -3 lags 0.061 3.15 0.1244 Dornbusch-Frankel -2 lags 0.0583 7.44 0.0887 2.26 12.36 Dornbusch-Frankel -3 lags 0.0604 4.11 0.1154 Hooper-Morton -2 lags 0.0545 13.48 0.1025 36 T a b l e 5 S e g m e n t e d M o d e l s — T s a y P r i n c i p l e C o m p o n e n t s Percent reduction in root mse model description from random walk model Interest Differential -Pr inc iple Component l ( L a g 1) Two Segments 15.83 Three Segments 15.53 Trade Balance Differential -Pr inc iple Component l ( L a g 1) Two Segments 5.46 Three Segments 13.23 Money Supply Differential -Principle Component l ( L a g 1) Two Segments 12.90 Three Segments 15.33 Interest Diferential -Pr inc ipa l Component 1 (lagl) and Trade Balance Differential -Pr inc ipa l Component 1 ( lagl) I .D. segmentation variable - T w o Segments 15.03 -Three Segments 15.78 T . B . segmentation variable - T w o Segments 9.14 -Three Segments 15.20 37 Percent reduction in root mse model description from random walk model Interest Diferential -Pr inc ipa l Component 1 ( lagl) and Money Supply Diferential -Pr inc ipa l Component 1 ( lagl) I .D. segmentation variable - T w o Segments 16.39 -Three Segments 16.43 M . S . segmentation variable - T w o Segments 14.72 -Three Segments 15.88 Money Supply Diferential -Pr inc ipa l Component 1 ( lagl) and Trade Balance Diferential -Pr inc ipa l Component 1 ( lagl) T . B . segmentation variable - T w o Segments 8.71 -Three Segments 13.14 M . S . segmentation variable - T w o Segments 14.34 -Three Segments 17.72 Interest Diferential -Pr inc ipa l Component 1 ( lagl) and Money Supply Diferential -Pr inc ipa l Component 1 ( lagl) and Trade Balance Diferential -Pr inc ipa l Component 1 ( lagl) I .D. segmentation variable - T w o Segments 15.68 -Three Segments 16.55 38 T a b l e 6 S e g m e n t e d M o d e l s - T s a y P r i n c i p l e C o m p o n e n t s F o r e c a s t i n g R e s u l t s Model Description Sum of Squared Forecasting Error (SSFE) S S F E (Random Walk) Percent Reduction in S S F E Interest Differential Two Segments Three Segments 0.329 0.360 0.437 0.437 24.65 17.65 Interest Differential and Money Supply Differential Two Segments Three Segments 0.331 0.378 0.437 0.437 24.15 13.38 Interest Differential and Trade Balance Differential Two Segments Three Segments 0.378 0.416 0.437 0.437 13.52 4.69 Interest Differential and Money Supply Differential and Trade Balance Differential Two Segments Three Segments 0.403 0.443 0.437 0.437 7.65 39 40  42 43 44  Exchange Rate 2.0 2.5 Exchange Rale 20 2.5 Exchange Rate 20 2.5 d 3 <D 5T ta II H 3' Exchange Rate 2.0 2.5 46 Exchange Rate 2.0 2.5 =1 3 5T ta n u Exchange Rate 2.0 2.5 Exchange Rate 2.0 2.5 3 o Exchange Rate 2.0 2.5 Exchange Rate 2.0 2.5 47 48 49 50 51 52 53 1.5 exchange rate(t) 2.0 2.5 3.0 i . * 54 2.0 exchange rate(t) 2.5 3.0 3.5 55 56 57 Bibl iography Baillie, R. and McMahon, P. (1989), The foreign exchange market. Theory and econometric evidence (Cambridge University Press). Bilson, J . F . O . (1978), Rational expectations and exchange rates, in J .A . Frenkel and H . G . Johnson, The Economics of Exchange Rates: Selected Studies, 75-76. Bilson, J . F . O . (1979), The deutsche mark/dollar rate - A monetary analysis, in: Karl Brun- ner and Allan H. Meltzer, eds., Policies for employment, prices and exchange rates, Carnegie - Rochester Conference 11 (North Holland Publishing Company, Amsterdam). Box, G .E .P . and Tiao, G . C . (1977) A canonical analysis of multiple time series, Biometrika, 64, 2, 355-365. Dornbusch, R. (1976), Exchange rate expectations and monetary policy, Journal of Inter- national Economics, 6, 231-244. Feder, P.I. (1975a) On asymptotic distribution theory in segmented regression problems- identified case, Annals of Statistics, 3, 49-83. Feder, P.I. (1975b) The log likelihood ratio in segmented regression, Annals of Statistics, 3, 84-97. Frankel, J . A . (1979), On the mark: theory of floating exchange rates based on real interest rate differentials, American Economic Review, 69, 610-622. Frenkel J .A . (1976), A monetary approach to the exchange rate: doctrinal aspects and empirical evidence., Scandinavian Journal of Economics, 78 , 200-224. Frenkel, J .A . (1983), An introduction to exchange rates and international macroeconomics, in: J .A . Frenkel ed., Exchange rates and international macroeconomics, 1-18, University of Chicago Press. Hooper, P. and Morton, J . E . (1982), Fluctuations in the dollar: A model of nominal and real exchange rate determination, Journal of International Money and Finance, 1, 39-56. 58 Issard, P. (1983), An accounting framework and some issues for modeling how exchange rates respond to the news, in: J A . Frenkel ed., Exchange rates and international macroeco- nomics, 19-56, University of Chicago Press. Kouri, P.J.K (1976), The exchange rate and balance of payments in the short run and in The long run, a monetary approach, Scandinavian Journal of Economics, 7 8 , 280-304. Liu, J . , Wu, S., Zidek, J .V . (1991), On Segmented Multivariate Regressions, Dept. of Statistics Technical Report No. 109, University of British Columbia. Meese, R.A. and Rogoff, K. (1983), Empirical exchange rate models of the seventies: Do they fit out of sample? Journal of International Economics, 14, 3-24. Serfling, R .J . (1981), Approximation theorems of mathematical statistics, Wiley, New York. Tsay, R.S. (1990), Correlation transformation and components of stock prices, unpublished manuscript. Yeh, M.P. , Gardner, R . M . , Adams, T . D . , Yanowitz, F . G . , and Crapo, R.O. (1983), "anaer- obic threshold':Problems of determination and validation. J A p p l . Physiol. Respirit. Envioron. Excercise Physiol., 55, 1178-1186. 59

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