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

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SEGMENTED  REGRESSION  WITH A N APPLICATION TO GERMAN  MODELLING EXCHANGE  RATE  by EDWARD ANDREW  SUSKO  B . A . , T h e U n i v e r s i t y of W i n d s o r , 1990 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR T H EDEGREE OF MASTER OF SCIENCE  T H E F A C U L T Y OF G R A D U A T E STUDIES T H E D E P A R T M E N T O F STATISTICS  We accept this thesis as conforming to the required s t a n d a r d  T H E UNIVERSITY O F BRITISH C O L U M B I A November 1991 © E d w a r d A n d r e w Susko, 1991  DATA  In  presenting this  degree at the freely  thesis  in  University of  partial  fulfilment  of  this thesis for  department  or  by  his  or  scholarly purposes may be her  representatives.  permission.  Department The University of British Columbia Vancouver, Canada  for  an advanced  Library shall make  it  agree that permission for extensive  It  publication of this thesis for financial gain shall not  DE-6 (2/88)  requirements  British Columbia, I agree that the  available for reference and study. I further  copying of  the  is  granted  by the  understood  that  head of copying  my or  be allowed without my written  Abstract Segmented regression models are the topic of this thesis. These are regression models i n w h i c h the m e a n response is thought to be linear i n the e x p l a n a t o r y variables w i t h i n regions of a p a r t i c u l a r e x p l a n a t o r y variable. A criterion for e s t i m a t i n g the number of segments i n a segmented m o d e l is given a n d the consistency of this e s t i m a t o r is established under rather general conditions. T h e r e have been m a n y studies on m o d e l i n g and forecasting foreign exchange rates usi n g various models, n o t a b l y the r a n d o m walk m o d e l , the f o r w a r d rate m o d e l , m o n e t a r y models a n d vector autoregressions, see, for example, Meese a n d Rogoff (1983) a n d B a i l l i e a n d M c M a h o n (1989). T h e general conclusions have been that most of the models cannot o u t p e r f o r m the r a n d o m walk m o d e l by a significant m a r g i n .  T h e observation that  the dependence of the exchange rate on the key macroeconomic indicators is t i m e v a r y i n g , n o n s t a t i o n a r y a n d nonlinear leads to consideration of nonlinear models. In this thesis segmented models are fitted to G e r m a n exchange rate d a t a using least squares and forecasting results o b t a i n e d from these models are compared w i t h forecasting results from widely used models i n exchange rate p r e d i c t i o n . T h e segmented models tend to perform better t h a n models t h a t have been established i n the l i t e r a t u r e , notably, the r a n d o m walk m o d e l .  ii  Table of  Contents  Abstract  ii  L i s t of Tables  iv  L i s t of Figures  v  Section 1: E x c h a n g e Rates  1-7  1.1 I n t r o d u c t i o n  1-3  1.2 D e t e r m i n a t i o n of Exchange Rates  4-7  Section 2: D a t a A n a l y s i s  8-15  2.1 I n t r o d u c t i o n  8  2.2 E x p l a n a t o r y Variables  8-10  2.3 P l o t Suggestions  10-12  2.4 Results from M o d e l 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 N u m b e r of Segments  18-30  3.1 I n t r o d u c t i o n  18  3.2 A C r i t e r i o n for E s t i m a t i n g the N u m b e r of Segments  18-20  3.3 Consistent E s t i m a t i o n of the N u m b e r of Segments  20-30  Bibliography  58-59  iii  List of Tables  Table 1 F o r w a r d R a t e Models  31  Table 2 Segmented 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  32-34  Table 3 Segmented  Models-  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 Results  35  Table 4 M o n e t a r y M o d e l s  36  Table 5 Segmented M o d e l s - T s a y P r i n c i p l e Components  37-38  Table 6 Segmented  Models-  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 Results  iv  39  List of Figures  F i g u r e 1 T i m e Series Plots-Interest Differentials  40  F i g u r e 2 T i m e series P l o t s - T r a d e Balance Differences  41  F i g u r e 3 T i m e series P l o t s - M o n e y S u p p l y Difference  42  F i g u r e 4 T i m e Series Plots-Interest Differential P r i n c i p a l C o m p o n e n t s  43  F i g u r e 5 T i m e series P l o t s - T r a d e Balance P r i n c i p a l C o m p o n e n t s  44  F i g u r e 6 T i m e series P l o t s - M o n e y S u p p l y P r i n c i p a l C o m p o n e n t s  45  F i g u r e 7 Scatter Plots-Interest Differential P r i n c i p a l C o m p o n e n t 1  46  F i g u r e 8 Scatter Plots-Interest Differential P r i n c i p a l C o m p o n e n t 2  47  F i g u r e 9 Scatter P l o t s - T r a d e Balance P r i n c i p a l C o m p o n e n t 1  48  F i g u r e 10 Scatter P l o t s - T r a d e Balance P r i n c i p l e C o m p o n e n t 2  49  F i g u r e 11 Scatter P l o t s - M o n e y S u p p l y P r i n c i p a l C o m p o n e n t 1  50  F i g u r e 12 Scatter P l o t s - M o n e y Supply P r i n c i p a l C o m p o n e n t 2  51  F i g u r e 13 T h r e e D i m e n s i o n a l P l o t s - P r i n c i p l e Components  52  F i g u r e 14 T h r e e D i m e n s i o n a l Plots-Interest Differential P r i n c i p a l C o m p o n e n t 1  53  F i g u r e 15 L a g 1 P l o t - E x c h a n g e Rates  54  F i g u r e 16 Scatter P l o t - F o r w a r d Rates  55  F i g u r e 17 T h r e e D i m e n s i o n a l P l o t (1971-1985)  56  F i g u r e 18 T w o D i m e n s i o n a l P l o t s (1971-1985)  57  Section 1 Exchange  1.1  Rates  Introduction  E x c h a n g e rates are an i m p o r t a n t source of v a r i a t i o n i n financial decisions. However, attempts to m o d e l a n d predict exchange rates beyond 1971 have i n general not performed better t h a n that of a r a n d o m w a l k , Meese and RogofF (1983). It is the purpose of at least p a r t of this thesis t o provide a m o d e l w h i c h helps to e x p l a i n some the v a r i a t i o n contained i n exchange rates. F i x e d currency exchange rates have been the n o r m for m a n y years ( u n t i l 1971 we h a d been more or less under a system of fixed exchange rates). T h e y have also been one of the goals t h a t the i n t e r n a t i o n a l c o m m u n i t y has t r i e d to achieve. W h e n exchange rates are fixed or reasonably predictable i t is believed t h a t t r a d i n g i n g l o b a l markets is less tentative a n d t h a t economic a c t i v i t y is m o r e efficient.  O n e of the problems w i t h t r y i n g to m a i n t a i n fixed exchange rates,  however, is t h a t government p o l i c y w i l l be r e s t r i c t e d , sometimes to the detriment of domestic considerations. A s a result of this p r o b l e m exchange rates have been allowed to  float.  P r i o r to the great depression the exchange rate market operated according to the g o l d standard.  U n d e r the gold s t a n d a r d exchange rates were fixed.  Currencies were defined i n  terms of g o l d . A loss or g a i n i n domestic money s u p p l y corresponded d i r e c t l y w i t h a loss or gain i n g o l d .  T h i s w o u l d t h e n be accompanied by an a p p r o p r i a t e change i n interest rates,  G D P a n d prices.  These changes w o u l d i n t u r n be accompanied by a p p r o p r i a t e changes i n  foreign investment and trade w h i c h w o u l d adjust the exchange rate to its a p p r o p r i a t e price. A s m e n t i o n e d above domestic p o l i c y was restricted by such a system. F u r t h e r m o r e the i n f l a t i o n rate was dependent u p o n gold discoveries. T h e gold S t a n d a r d collapsed d u r i n g the chaos of the great depression of the 1930's. In the intrests of economic efficiency the i n d u s t r i a l i z e d nations met i n B r e t t o n W o o d s , N e w H a m p s h i r e i n 1944 to set up a system w h i c h w o u l d fix exchange rates. T h e U n i t e d States pegged i t s currency to gold a n d the other nations pegged their currency to that of the U n i t e d 1  States. T h e I n t e r n a t i o n a l M o n e t a r y F u n d was set up t o police the s i t u a t i o n . Once again one of the m a j o r problems w i t h such a system was that countries were forced to adopt m o n e t a r y policies t h a t m a y not have been i n their best domestic interests. F u r t h e r m o r e , devaluations were p e r m i t t e d for countries only after l o n g balance of payment deficits.  T h i s made such  devaluations easy to predict a n d speculators were hence able to increase the magnitude of the d e v a l u a t i o n . R e v a l u a t i o n s expected f r o m balance of payment surplus nations were not r e a d i l y received a n d so the U n i t e d States was forced to accept a balance of payments deficit. A l l of these p r o b l e m s combined to cause the fall of the B r e t t o n W o o d s system i n 1971. In general exchange rate m o d e l i n g a t t e m p t s have been based o n the theoretical r e l a t i o n ships between the exchange rate a n d k n o w n (or a p p r o x i m a t e l y k n o w n ) indicators of exchange rate movements. One favoured m e t h o d used i n exchange rate m o d e l i n g is to look at the r e l a tionship between the spot rate and the f o r w a r d rate. T h e spot rate is the price of one currency i n terms of another currency. T h e f o r w a r d rate is the price at w h i c h one currency can be p u r chased i n terms of another currency at some prespecified t i m e i n the future. T h e u s u a l setup is: St+i = p + F + e +i, t  t  where St is the spot rate at t i m e t , F is the forward rate at t i m e t a n d p is a risk p r e m i u m . t  Often p is t a k e n to be equal to zero. A n o t h e r approach to exchange rate m o d e l i n g is to assume the a p p r o x i m a t e satisfaction of t h e o r e t i c a l money demand retationships. These type of models are called monetary models a n d are q u i t e c o m m o n i n the economic l i t e r a t u r e . See for instance B i l s o n (1978, 1979), Frenkel (1976), F r a n k e l (1979) a n d H o o p e r a n d M o r t o n (1982). T h e models t h a t f a l l under this category are u s u a l l y based o n the money d e m a n d e q u a t i o n . F o r a given c o u n t r y the aggregate supply of money is M / P where M is the n o m i n a l s u p p l y of money a n d P is the price level. T h e aggregate d e m a n d for money is of the f o r m m^Y,  where Y is real income a n d R is the p r e v a i l i n g  R,...),  interest r a t e . A widely used f o r m for m j is:  m  d  KY e- ,  =  a  pR  where K is a country specific constant a n d a , f3 are independent of the country we are t a l k i n g 2  about. So at e q u i l i b r i u m , M/P  =  KY e~ a  m  or, P =  M KY e-P ' a  R  Let q u a n t i t y * denote the relevant quantities for some other country. N o w the theory of p u r chasing power p a r i t y assumes t h a t the rate of exchange between two countries is d i r e c t l y r e l a t e d t o 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. L e t t i n g lower case l e t t e r = l n ( u p p e r case l e t t e r ) , we have s = ( m - m*) + {k* -k)  + a(y* - y) + (3{R -  R*).  For a further more detailed discussion of monetary models see B a i l l i e a n d M c M a h o n ( 1 9 8 9 ) . In the previous two paragraphs an a t t e m p t has been made t o i n t r o d u c e some of the prevalent models used i n forecasting a n d e m p i r i c a l research. A g a i n however these models have not been extremely successful i n p r e d i c t i o n or accurate m o d e l i n g . T h e r e are several problems w i t h the above approaches. T h e f o r w a r d rate models seem viable since they take i n t o account expectations concerning future market conditions a n d these expectations m a y be more i m p o r t a n t i n a forecasting sense t h a n the a c t u a l future state of affairs or predictions thereof. One p r o b l e m is that the f o r w a r d rates are constrained by the interest rate p a r i t y theory to be dependent p r i m a r i l y on the previous exchange rate a n d interest rates. T r a d e balance differentials a n d other economic i n d i c a t o r s of exchange rate movements are restricted i n t h e i r effect on the f o r w a r d rate. T o e x p l a i n , interest rate p a r i t y theory implies t h a t (1 + ^ - ( 1  ^/4)  +  ^/4)*"  where St is the rate of exchange for foreign currency i n terms of domestic currency, F is the t  corresponding f o r w a r d rate for one quarter ahead, Rd is the domestic a n n u a l interest rate a n d Rf is the foreign interest rate. T h u s i f as the f o r w a r d rate m o d e l i m p l i e s , S +i « F , then t  si t+  « s + ln(l t  + R /4) d  3  - /n(l +  R /4) f  t  N o t e the s i m i l a r i t y between this m o d e l a n d the m o n e t a r y models stated above. F o r R s m a l l a T a y l o r series a p p r o x i m a t i o n suggests t h a t ln(l + R) « R. Hence the m o d e l is a p p r o x i m a t e l y a special case of the above mentioned m o n e t a r y models w i t h some coefficients set equal to zero and a r a n d o m walk component. T h e m o n e t a r y models also have some difficulties. F o r one, there is no direct a t t e m p t t o include m o r e t h a n two countries i n the models.  It seems sensible t h a t the exchange rate is  determined by the e x p l a n a t o r y variables of more t h a n two nations. A n o t h e r p r o b l e m is t h a t as certain e x p l a n a t o r y variables approach i n o r d i n a t e l y h i g h or low values governments, central banks a n d foreign investors w i l l take note a n d take a c t i o n . T h e i m p l i c a t i o n is t h a t the values of parameters m a y not be stable across a l l regions of the e x p l a n a t o r y variables. It is p r i m a r i l y these last t w o criticisms t h a t the models to be suggested i n t e n d t o deal w i t h . Ideally, the more e x p l a n a t o r y variables we include the more i n f o r m a t i o n we g a i n . However due t o the l i m i t e d number of observations some type of trade-off must be made. B y t a k i n g p r i n c i p a l components w i t h respect to e x p l a n a t o r y variables across countries i t is hoped that some i n f o r m a t i o n can be gained about the relationships between the exchange rate a n d the e x p l a n a t o r y variables of several countries as opposed to just two.  B y considering a m o d e l that is segmented  with  respect to the values of a n e x p l a n a t o r y variable i t is hoped t h a t the p r o b l e m of i n s t a b i l i t y of the p a r a m e t e r values m a y be dealt w i t h .  1.2 D e t e r m i n a t i o n o f E x c h a n g e  Rates  T h e r e are several approaches to exchange rate d e t e r m i n a t i o n . One idea is t h a t of P u r c h a s ing P o w e r P a r i t y . U n d e r this theory the exchange rate between any two countries is believed to reflect the relative price levels of those two countries. A n i m p l i c a t i o n of this theory is that i n f l a t i o n rates are the m a j o r determinants of exchange rates. However, e m p i r i c a l l y such relationships have been a n y t h i n g but clear i n the 1970's a n d 1980's. S t i l l this theory was widely accepted pre 1971 for e x p l a i n i n g l o n g range behaviour of exchange rates. It w o u l d seem that this theory is useful i n e x p l a i n i n g exchange rate behaviour d u r i n g fixed exchange rate periods. A n o t h e r approach is to consider the exchange rate as an asset value. U n d e r this approach the exchange  rate one p e r i o d ahead is determined by the present exchange rate plus some 4  expected changes i n the exchange rate m a r k e t . T h e p r o b l e m then arises as to how to account for expected changes.  One t h i n g t o do is look at the f o r w a r d rates, as mentioned i n the  i n t r o d u c t i o n , this approach is useful i n t h a t i t takes expectations about future events i n t o account a n d often these expectations can have more relevance i n exchange rate prediction t h a n the a c t u a l occurrence of future events. These expectations w i l l be dependent u p o n m o n e t a r y sources, but i t should also be said that these expectations w i l l be dependent on other t h i n g s , such as news. A case i n p o i n t is the u p w a r d movements i n the exchange rates throughout m a n y of the i n d u s t r i a l i z e d nations ( n o t a b l y G e r m a n y ) i n 1980. It is hypothesized t h a t 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 R e a g a n could be expected to w i n the u p c o m i n g presidential election.  W i t h his support for tight monetary p o l i c y a n d  his support for the s t i m u l a t i o n of U . S . competetiveness i t was expected that the d o l l a r w o u l d increase i n value. T h e point is t h a t the exchange rate increased irrespective of whether U . S . m o n e t a r y p o l i c y a c t u a l l y was t i g h t , the d e t e r m i n i n g factor was the e x p e c t a t i o n as opposed t o the a c t u a l occurence of the event. It should be added that asset m o d e l aproaches are i n t e r l i n k e d i n some sense w i t h monetary models. E x p e c t a t i o n s about future conditions w i l l be dependent u p o n news about p o l i t i c a l s i t u a t i o n s a n d the like but they w i l l also be dependent on what the current or past state of affairs is l i k e w i t h respect to key economic variables. F u r t h e r m o r e i t has been mentioned that at least theoretically the forward rate a p p r o a c h can be a p p r o x i m a t e d by m o n e t a r y approach w i t h constrained variables and a r a n d o m w a l k component.  T h e final approach I w i l l consider w i t h respect to exchange rate d e t e r m i n a t i o n , and the approach t h a t forms the basis for m y models, is that of m o n e t a r y models.  T h e i d e a is t h a t  the exchange rate is the price of one country's currency i n terms of another country's currency and hence t h a t the laws of s u p p l y a n d d e m a n d apply. T h u s , this a p p r o a c h looks at economic variables w h i c h are considered i m p o r t a n t i n the d e t e r m i n a t i o n of the s u p p l y of a n d d e m a n d for a c o u n t r y ' s money by foreign interests. Some key factors i n such a d e t e r m i n a t i o n w o u l d be:  1. T r a d e B a l a n c e and C u r r e n t A c c o u n t - A country sells i t s exports i n i t ' s home currency. So h o l d i n g a l l other variables constant an increase i n the d e m a n d for a c o u n t r y ' s currency w o u l d correspond t o an increase i n e x p o r t s . 5  2. Foreign Investment - A n increase i n investment i n a c o u n t r y w o u l d be accompanied by an increase i n demand for that c o u n t r y ' s currency i n order to finance the investment.  3. Interest Rates - A n increase i n interest rates a t t r a c t s foreign investment.  4. M o n e y S u p p l y - A n increase i n a country's domestic money s u p p l y implies a n increase the money supplied to foreign investors.  5. Inflation rates - rising i n f l a t i o n rates make a country's currency less favourable to foreign investors.  Such factors are i m p o r t a n t i n the d e t e r m i n a t i o n of exchange rates but certain problems a n d ideas need to be kept i n m i n d when using these models. Changes i n such indicators w o u l d not u s u a l l y correspond to simultaneous changes i n money s u p p l y a n d demand this makes i t necessary t o investigate the use of e x p l a n a t o r y variables at different lags. V o l a t i l i t y m a y confuse relationships, large changes i n economic indicators m a y cause concern as to the h e a l t h of a n a t i o n ' s economy thus r e s u l t i n g i n the reverse of or at least t e m p e r i n g of the effect o n the exchange r a t e . M a n y of these variables are endogenous a n d thus again expected effects m a y be tempered.  A n o t h e r concern should be policy changes, they m a y change the relationships  between these variables a n d c e r t a i n l y affect expectations about these variables a n d exchange rates (the " L u c a s C r i t i q u e " , F r e n k e l 1983 ).  It is i m p o r t a n t to consider changes  between  countries w i t h respect to these variables. For instance, h o l d i n g a l l other variables constant a n 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 i n t o account that leads to the segmented regression models that I consider. I t r y to take the i m p o r t a n t determinants of s u p p l y a n d demand for foreign currency into account as e x p l a n a t o r y variables.  That is,  the variables mentioned above (1-5) a n d subsets of these variables are used to determine the segmented regression r e l a t i o n s h i p .  Since i n t e r n a t i o n a l trade is greatest between the larger  i n d u s t r i a l nations it is reasonable to restrict attention t o the exchange rate a n d e x p l a n a t o r y 6  variables corresponding to these n a t i o n s . T h u s the possible e x p l a n a t o r y variables were chosen to be the differentials of the i m p o r t a n t determinants between G - 7 n a t i o n s . Suppose we can assume t h a t government policies of the m a j o r t r a d i n g partners are more or less rational over the years a n d t h a t interest rates a n d money supplies digest market i n f o r m a t i o n relatively efficiently. T h e n since p o l i c y decisions w i l l often be determined or acompanied by h i - l o values of these variables i t seems reasonable to consider seperate regression segments w i t h respect to some segment e x p l a n a t o r y r a n d o m variable. F o r example governments are c e r t a i n t o act i n the case of large trade balance deficits or large interest rate differentials. W h a t emerges is a m o d e l where the exchange rate is dependent u p o n certain e x p l a n a t o r y variables a n d where this r e l a t i o n s h i p is different at extreme values of the e x p l a n a t o r y variables. T h e m o d e l i n g considered is inherently long r u n m o d e l i n g . 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 i m e l a g .  F u r t h e r m o r e , t h i s t i m e l a g may be of variable l e n g t h w i t h repect t o , say, m o n t h l y periods. Hence m o n t h l y models may not be appropriate i n that the t i m e l a g chosen as 'best' may not be constant w i t h respect to t i m e . If the t i m e periods concerned are increased to say q u a r t e r l y periods then the models should be more robust w i t h respect to this a s s u m p t i o n of constant t i m e l a g . F o r this reason i t was decided to concentrate o n l o n g r u n m o d e l i n g .  7  Section 2 Data Analysis  2.1  Introduction  In section 1 i t was mentioned t h a t p o l i c y considerations may have effects on the r e l a t i o n ships between exchange rates a n d certain economic i n d i c a t o r s . T h u s a segmented regression m o d e l c o u l d be a p p r o p r i a t e . 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. T h e general f o r m of a segmented t i m e series regression model can be stated as  Y = xj_ /? - + e , t  x  t  t  if x  td  e ( r , - _ i , n], i - 1 , . . . , / + 1,  where Y is the exchange rate at t i m e t , x< is a vector of e x p l a n a t o r y variables mentioned above t  at t i m e t — 1, e is an error t e r m , x d is one of the components of x (the segmentation variable), t  —oo =  To <  T\ <  t  •• • <  Ti+i  —  oo a n d {xt}  t  a n d {Q}  are independent series.  2.2 E x p l a n a t o r y V a r i a b l e s  T o use a segmented m o d e l i t was neccesary to make certain decisions about e x p l a n a t o r y variables. Some i m p o r t a n t theoretical determinants of exchange rates were discussed i n section 1 a n d are l i s t e d below 1. Interest R a t e Differentials 2. T r a d e B a l a n c e Differentials 3. M o n e y S u p p l y Differentials 4. Inflation R a t e Differentials Some things should be mentioned at this p o i n t . It is the differential t h a t is i m p o r t a n t . In section 1 i t was p o i n t e d out, for instance, that i f a country increases its s u p p l y of money t h a t this w i l l i n general make its currency less a t t r a c t i v e . T h i s w i l l not be true i f every other country also increases its money supply by an a p p r o p r i a t e a m o u n t . Hence the i m p o r t a n c e of differentials. However i f differentials between a l l countries are considered there w i l l be hundreds of 8  e x p l a n a t o r y variables. In view of this i t was decided to consider differentials between i m p o r t a n t t r a d i n g partners and economic powers. In p a r t i c u l a r i t was decided to consider the differentials between G e r m a n y a n d the other G - 7 nations. P r e l i m i n a r y simultaneous t i m e series plots a n d scatter plots of the exchange rate vs. i n f l a t i o n rates suggested t h a t inflation rate differentials w o u l d not be very useful as e x p l a n a t o r y variables a n d so i t was decided not to consider t h e m . F u r t h e r the effects of the i n f l a t i o n rate differentials are expected to be imbedded i n the interest rate differentials. T h u s the e x p l a n a t o r y variables t o be considered were interest rate different i a l s , money s u p p l y differentials, and trade balance differentials a l l between G e r m a n y a n d the G - 7 nations. It was i m p o r t a n t to consider these variables w i t h some t i m e l a g since it m a y be t h a t the effect of a n e x p l a n a t o r y variable o n the exchange rate w i l l not be felt i m m e d i a t e l y . T h u s the e x p l a n a t o r y variables were considered w i t h and w i t h o u t t i m e lags. F i n a l l y some sort of s t a n d a r d i z a t i o n , to be mentioned shortly, was necessary. In view of the fact t h a t exchange rates were relatively fixed v i a the B r e t o n W o o d s M o n e t a r y System u n t i l 1971 and that the e x p l a n a t o r y variables are theoretically more valuable as l o n g t e r m predictor t h a n as short t e r m predictors, i t was decided t o use q u a r t e r l y d a t a f r o m 1971 t h r o u g h to t h e second quarter of 1990.  T h e decision to use quarterly d a t a as opposed t o  m o n t h l y d a t a was p a r t l y because of some of the concerns mentioned i n the previous section a n d p a r t l y because of some p r a c t i c a l problems related to the a v a i l a b i l i t y a n d / o r r e l i a b i l i t y of c e r t a i n economic d a t a on a m o n t h l y basis. T o repeat at least one of the concerns mentioned i n the previous section i t can be expected that the l a g used i n the e x p l a n a t o r y variables is more l i k e l y to be variable w i t h respect to t i m e i n the m o n t h l y m o d e l t h a n i n the corresponding q u a r t e r l y model.  T o a v o i d this p r o b l e m w i t h o u t a d d i n g parameters a useful approach is t o consider  q u a r t e r l y d a t a . F o r each G - 7 country the following d a t a were obtained f r o m the i n t e r n a t i o n a l financial  statistics p u b l i s h e d by the I M F (International M o n e t a r y F u n d ) .  1.  E x c h a n g e Rates - These were end of p e r i o d spot rates.  T h e y were expressed  as  M a r k s / U . S . Dollar. 2.  M o n e y 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 r c u l a t i o n . For each country money supply was given i n that country's currency. 9  3. Interest Rates - T h i s was the discount r a t e / B a n k rate for the country of interest. 4. T r a d e B a l a n c e - T h i s was calculated as merchandise i m p o r t s minus merchandise e x p o r t s . T h i s q u a n t i t y was given i n U . S . D o l l a r s . 5. C . P . I - C o n s u m e r P r i c e Index. T h i s q u a n t i t y was used as the i n d e x for i n f l a t i o n of a p a r t i c u l a r country. Several things should be mentioned w i t h respect to the d a t a .  Some s t a n d a r d i z a t i o n is  necessary. In p a r t i c u l a r one wants t o consider real values. In order to do so i t was necessary to adjust for the rate of i n f l a t i o n . T h u s to get real money supply the n o m i n a l money s u p p l y was m u l t i p l i e d by (lOO/CPI try)coun  m u l t i p l i e d by (100/CPIus)by CPIus/CPlGermany),  T o get real trade balance n o m i n a l trade balance  was  T o get real b i l a t e r a l exchange rates, exchange rates were m u l t i p l i e d 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 o b t a i n money supply differentials. E a c h country's money s u p p l y was s t a n d a r d i z e d , i.e.  (quantity-mean(quantity)/s.e.(quantity)).  N o w even i n the presently described s i t u a t i o n where only differentials between G - 7 nations are considered there are s t i l l 3 x 6 = 1 8 e x p l a n a t o r y variables. Since there are 78 observations this w o u l d have been too m a n y e x p l a n a t o r y variables for a reasonable segmented m o d e l . T h u s it was decided to consider as possible e x p l a n a t o r y variables the p r i n c i p a l components for each differential w h i c h explained most of the w i t h i n differential v a r i a b i l i t y . F o r 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, G e r m a n y - F r a n c e interest  rate d i f f e r e n t i a l , . . . , G e r m a n y - U . S . interest rate differential). A n o t h e r possible approach was to consider the p r i n c i p a l components as defined i n T s a y (1990). These p r i n c i p a l components  are  o p t i m a l i n a predictive sense. M o d e l s using these p r i n c i p a l components were explored.  2.3 P l o t Suggestions  T h e next step i n the analysis was to consider the plots of the various e x p l a n a t o r y variables vs. the exchange rates. O f course the e x p l a n a t o r y variables are related to each other a n d hence two or even three d i m e n s i o n a l plots w i l l not give the complete p i c t u r e . T h e 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 line. T h e corresponding scale is given on the left h a n d side of the p l o t . E a c h plot also gives the sample p a t h of some e x p l a n a t o r y variable w i t h a broken line.  T h e corresponding scale being given o n the right h a n d side of  the p l o t . N o t i c e the v o l a t i l i t y of the exchange rate p a r t i c u l a r l y post 1980. T h e p e r i o d i n the m i d '80's is p a r t i c u l a r l y v o l i t i l e . T h i s ' h i l l ' is not seen i n m a n y of the other t i m e series plots a l t h o u g h there is some i n d i c a t i o n of i t i n for instance the t i m e series plot of s t a n d a r d i z e d money s u p p l y differential w i t h respect to G e r m a n y a n d the U . S . . A l s o , notice the t i m e series plots of the trade balance differentials of the G - 7 nations given o n page 40. These plots hint at the interdependence of the exchange rates a n d the chosen e x p l a n a t o r y variables. T h e plots of the t r a d e balance differentials often exhibit ' h i l l s ' a n d 'valleys' after t h a t same behavior is observed i n the exchange rates.  T h e next t y p e of plots t o be observed are the two dimensional scatter plots given on pages 45-50. These plots are l a b e l e d l a g plots i n reference t o the fact t h a t they plot the e x p l a n a t o r y variable l i s t e d at the top of the page vs.  exchange rates at six different t i m e lags.  These  plots must be observed w i t h c a u t i o n of course since the e x p l a n a t o r y variables are certain to be i n t e r r e l a t e d . F o r instance money supply differentials are definitely going to be dependent u p o n prevailing interest rates to a certain extent.  T h e plots are not altogether encouraging.  For  example the relationships between money supply differentials, t r a d e balance differentials a n d exchange rates are not p a r t i c u l a r l y evident i n the corresponding scatter plots of the exchange rates vs.  the first p r i n c i p a l components.  However, the plot of the exchange rates vs.  the  first p r i n c i p a l component of the interest rate differentials does suggest a r e l a t i o n s h i p , i n fact i t appears f r o m the plots t h a t a segmented approach w i t h respect t o this e x p l a n a t o r y variable m a y be the way to go. F o r a given e x p l a n a t o r y p r i n c i p a l component notice the relative homogeneity of the plots of the exchange rates vs. e x p l a n a t o r y p r i n c i p a l components across different t i m e lags. In v i e w of this apparent s i m i l a r i t y w i t h respect to lags i t was decided that i t m a y not be neccesary t o fit a lot of models w i t h m a n y different l a g combinations. E x p e r i m e n t a t i o n w i t h a few lags, to be mentioned below, confirmed that this was indeed the s i t u a t i o n . A l s o , notice t h a t the scatter plots of the exchange rates vs. the interest differentials seem to hint at the existence of at least two segments. T h e plot of the exchange rate vs. the exchange rate lagged 11  one quarter is given on page 53. C l e a r l y there is a very good linear r e l a t i o n s h i p . Hence i t was decided to include the exchange rate lagged one quarter i n a l l further m o d e l i n g a t t e m p t s . T h e plot of the f o r w a r d rate vs the spot rate one period forward is given on page 54. A g a i n there is a clear linear relationship a l t h o u g h the v a r i a b i l i t y is somewhat greater t h a n that i n the p l o t 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 m a y not be sufficient i n e x p l o r i n g the possible relationships i n the d a t a . O n pages 51-52 several three d i m e n s i o n a l plots are g i v e n . O n page 51 three d i m e n s i o n a l plots of each of the first e x p l a n a t o r y p r i n c i p a l components w i t h the exchange rates lagged one p e r i o d and at present are given.  T h e l a t t e r five plots do not  give m u c h suggestion to possible relationships along the d i r e c t i o n of the e x p l a n a t o r y variable but the plot concerning the p r i n c i p a l component for the interest differentials does suggest a possible segmented relationship w i t h possibly three segments. T h e plot o n page 52 rotates the three d i m e n s i o n a l plot of the first p r i n c i p a l component for the interest rate differentials.  As  i n the two d i m e n s i o n a l plot a segmented relationship appears possible a n d i t appears t h a t a segmented m o d e l w i t h possibly two or three segments m a y be a p p r o p r i a t e . In table 3, some forecasting results are quoted. One s t u n n i n g 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 a n d M o n e y S u p p l y P C I as an a d d i t i o n a l segmentation variable. T h e r e is quite a large dicrepency between  the  m o d e l w i t h two segments and the m o d e l w i t h three segments. T h e m o d e l w i t h three segments performs s u b s t a n t i a l l y better t h a n the m o d e l w i t h two segments. T h u s a n i m m e d i a t e question is to what extent the number of segments could have been predicted v i a e x p l o r a t o r y plots. One such plot is given on page 55. Y — t  fiiYt-i,  i = 1. ..I)  versus lag-1 interest rate differential  and money s u p p l y differential was p l o t t e d , where 1 was any plausible n u m b e r of segments and $i the estimated coefficients.  T h e plots were t h e n r o t a t e d to get images f r o m different angles.  T h e plot shown is the one corresponding to / = 3, a n d i t does appear i n this plot that three segments are a p p r o p r i a t e .  12  2.4 R e s u l t s f r o m M o d e l F i t t i n g  In this subsection the f i t t i n g of various models is discussed. T h e general f o r m of the models can be stated as  Y  t  = x't.j/?,- + e , t  if x d G (r<_i, T ], i = 1 , . . . , / + 1, t  {  where Yt is the exchange rate at t i m e t , x is a vector of e x p l a n a t o r y variables at t i m e t — 1, e  t  t  is a n error t e r m , x d is one of the components of x* (the segmentation variable) a n d —oo = To < t  T\ < • • • < Ti  +i  = oo. N o t e that the special case where / = 0, To = — o o , T i = oo is the f a m i l i a r  linear regression setup.  T h e s i t u a t i o n where I > 0 is t h a t of segmented regression, namely  a l i n e a r regression setup is assumed on 'segments' of a p a r t i c u l a r e x p l a n a t o r y variable. T h e types of models t h a t were fitted can be categorized. T h e first type is the segmented models w i t h s t a n d a r d p r i n c i p a l components. These models are segmented models w i t h various combinations of the e x p l a n a t o r y variables mentioned i n section 2.2 each m o d e l h a v i n g one e x p l a n a t o r y variable d e t e r m i n i n g the segmentation. T h e second type of models is the segmented models w i t h T s a y t y p e p r i n c i p a l components. These models are the same i n spirit as the previous t y p e of models but the p r i n c i p a l components adopted were taken according to T s a y (1990) i n order to m a x i m i z e the l a g 1 a u t o c o r r e l a t i o n of a linear c o m b i n a t i o n of the variables of interest. T h e r e are two other types of models, these are c o m p e t i n g models i n the sense t h a t they are the favoured models i n the economic l i t e r a t u r e . These models have been described i n the first section where the general problems i n exchange rate d e t e r m i n a t i o n were discussed. T h e first of the two w o u l d be m o n e t a r y models. T h e f o r m a n d theoretical background b e h i n d these models was discussed i n the first section. F o r previous analysis of these models using m o n t h l y d a t a pre 1980's see Meese and Rogoff (1983). T h e other type of competing m o d e l is the f o r w a r d rate m o d e l . A g a i n , some of the m e r i t s a n d pitfalls of using this type of model have been discussed i n section 1. Segmented regression is a f o r m of nonlinear regression a n d c r i t e r i o n for testing m o d e l assumptions a n d parameter values are not well developed a n d w o u l d involve assumptions about independent i d e n t i c a l l y d i s t r i b u t e d errors. A s a p r a c t i c a l issue the first t h i n g we are compelled to consider is the a p p r o p r i a t e number of segments. the next section as i t pertains to large samples. 13  T h i s issue is discussed i n more detail i n  In the present s i t u a t i o n we do not have a  large sample. F o r s m a l l samples this issue has not been resolved a n d often the best p o l i c y is t o experiment w i t h different numbers of segments perhaps as suggested i n the plots. F u r t h e r m o r e s m a l l samples place restrictions on the a b i l i t y to estimate large numbers of parameters a n d thus some further retrictions must be placed u p o n the number of segments.  However c r i t e r i a  are needed t o adress the issue of m o d e l appropriateness. T h e p r i m a r y c r i t e r i a considered were mean squared error ( M S E ) , a n d the s u m of out of sample one step ahead forecasting errors for a five year p e r i o d ( S S F E ) f r o m the second quarter of 1985 to the second quarter of 1990. T h e quantities f r o m the r a n d o m walk m o d e l were used as the y a r d s t i c k to measure  performance.  Some r e s i d u a l analysis was done as w e l l . Scatter plots of the residuals acf plots a n d t i m e series plots of the residuals were e x a m i n e d . T h e results for the segmented models w i t h s t a n d a r d p r i n c i p a l components  are l i s t e d o n  pages 31-34. A variety of combinations of e x p l a n a t o r y variables, segmentation variables a n d lags were experimented w i t h . In most cases the r e d u c t i o n i n mse over the r a n d o m walk was s i m i l a r for models h a v i n g the same e x p l a n a t o r y a n d segmentation variables but w i t h different lags. T h i s tends t o agree w i t h the plot suggestions.  It does not appear t h a t choosing the correct  l a g for the e x p l a n a t o r y variables is of utmost i m p o r t a n c e .  T h i s m a y be a result of the fact  that q u a r t e r l y as opposed to m o n t h l y d a t a were used. T h e models i n w h i c h the segmentation variable was the p r i n c i p a l component from the interest differentials h a d significant reductions i n M S E over those models w h i c h used a different segmentation v a r i a b l e . A g a i n this was i n d i c a t e d to a certain extent by the plots.  F r o m the results corresponding to M S E i t was decided t o  use the models w h i c h h a d the p r i n c i p a l component f r o m the interest rate differentials as the segmentation variable to evaluate forecasting ability. A s is seen on page 34 the r e d u c t i o n i n one step ahead s u m of squared forecasting error ( S S F E ) over the r a n d o m walk m o d e l for a five year p e r i o d (from the second quarter of 1985 to the second quarter of 1990) was as large as 42 percent i n the case where the p r i n c i p a l components from the interest differentials a n d money supply differentials were the e x p l a n a t o r y variables. T h i s result is quite impressive i n view of the fact t h a t i t has been widely accepted that large reductions i n forecasting a b i l i t y over the r a n d o m w a l k w i l l not usually be o b t a i n e d . T h e results for the segmented models using T s a y type p r i n c i p a l components 14  are listed  on pages 36-38.  T h e results are similar to the results obtained using the s t a n d a r d p r i n c i p a l  components. In fact plots of the p r i n c i p a l components suggested that the p r i n c i p a l components o b t a i n e d i n this fashion were very s i m i l a r to the p r i n c i p a l components obtained the s t a n d a r d way. T h e results using s t a n d a r d p r i n c i p a l components tended to be somewhat better t h a n the results using T s a y type p r i n c i p a l components but it seems clear from the s i m i l a r i t y of the results t h a t the appropriateness of the p r i n c i p a l components taken was not that much of an issue. T h e results f r o m f i t t i n g f o r w a r d rate models are given o n page 30. T h e forecasts came f r o m the models St+i = Ft +  e, t  and S  =p  t+1  + F + €. t  t  T h e scatter plot of of the exchange rate vs. the f o r w a r d rate i n conjunction w i t h the interest rate p a r i t y theory suggested that i t w o u l d be appropriate to ignore the risk p r e m i u m p. analyses were performed w i t h a n d w i t h o u t the risk p r e m i u m . W h e n St,  So  were expressed  F  t  i n n o m i n a l terms there was a s m a l l r e d u c t i o n i n forcasting error over the r a n d o m w a l k . A s mentioned before large gains cannot reasonably be expected f r o m these types of forecasts since F « St due to arbitrage concerns. W h e n models w i t h adjustments for i n f l a t i o n were i n c l u d e d t  there was no r e d u c t i o n i n forecasting errror over the r a n d o m walk m o d e l . T h e results of f i t t i n g the m o n e t a r y models are l i s t e d on page 35. Some discussion of these models was given i n the first section. F o r a more complete discussion of these models see Meese and Rogoff (1983). T h e u n d e r l y i n g equation for these models is  s = a + a (m 0  x  - m*) + a (y 2  - y*) + a (r 3  s  - r *) s  + a (7r - 7r*) + a TB 4  e  5  + a TB* 6  + u,  where s is the l o g a r i t h m of the price of dollars i n t e r m of foreign currency, m - m * the l o g a r i t h m of foreign to U . S . money supply, y - y * the l o g a r i t h m of foreign to U . S . real income, r  s  - r*  the interest rate differential, a n d w — 7r* is the expected i n f l a t i o n differential. T B a n d T B * e  represent the foreign a n d U . S . c u m u l a t i v e trade balances. T h e respective b a n k rates were taken as the short t e r m interest rates. T h e expected inflation rate was taken to be the i n f l a t i o n rate of the previous p e r i o d . In d e t a i l , ir = (CPI e  t  - CPI -\)ICPI -\. t  15  t  F r o m this general f o r m u l a t i o n  corne the three models t h a t were a n a l y z e d . T h e flexible price ( F r e n k e l - B i l s o n ) m o n e t a r y m o d e l in which a  4  a  5  = as = ae = 0. T h e s t i c k y price ( D o r n b u s c h - F r a n k e l ) m o n e t a r y m o d e l i n w h i c h  = as = 0 a n d the s t i c k y price ( H o o p e r - M o r t o n ) asset m o d e l i n w h i c h none of the coefficients  are zero.  See B i l s o n (1978, 1979), Frenkel (1976), D o r n b u s c h (1976), F r a n k e l (1979,  for further discussion of these models.  1981)  These models were fit w i t h no l a g i n the e x p l a n a t o r y  variables. T h e results were p o o r , none of the models fared better i n terms of M S E or S S F E i n c o m p a r i s o n to the r a n d o m walk m o d e l . T h e other type of m o d e l w i t h w h i c h comparisons were made were vector autoregression ( V A R ) type models as described i n Meese a n d Rogoff (1983). These models performed considerably better. T h e y can be described by  St = O-nSt-l + di2St-2 + . . .ai St-n n  where X -j t  + B X -\ a  t  + B X -2 i2  t  + • • -B Xtin  n  + V-it,  is the vector of the e x p l a n a t o r y variables i n the e q u a t i o n above lagged j periods.  T h e results are l i s t e d on page 35.  T h e best result was f r o m the D o r n b u s c h - F r a n k e l m o d e l  w i t h two lags. T h i s m o d e l achieved a r e d u c t i o n of 2.26 percent over the r a n d o m w a l k m o d e l i n S S F E . N o n e of the other models reported a r e d u c t i o n over the r a n d o m w a l k m o d e l .  It  should be mentioned t h a t these results should be interpreted w i t h c a u t i o n when c o m p a r i n g to the segmented models.  Since the results quoted for the segmented models were i n terms of  the exchange rate as opposed to the l o g a r i t h m of the exchange r a t e the sums of squares were calculated i n the same fashion for the m o n e t a r y models. T h i s tended to give more d r a m a t i c results t h a n w o u l d have been o b t a i n e d i f the sums of squares h a d been c o m p u t e d d i r e c t l y using the l o g a r i t h m i c values. F o r instance the above mentioned 2.26 percent r e d u c t i o n w o u l d be a r e d u c t i o n of 12.26 percent when the In t r a n s f o r m a t i o n was not t a k e n i n c o m p u t i n g S S F E . S t i l l the results are s t r i k i n g i n contrast to the superior results o b t a i n e d f r o m the segmented models. A question t h a t arises i s , to w h a t extent can the nonlinear s r t u c t u r e noticed i n the complete d a t a set be seen i n t r u n c a t e d versions of the data?  One w o u l d hope, for instance, t h a t the  number of segments w o u l d r e m a i n constant. In order to a t t e m p t t o answer this question several plots have been i n c l u d e d w h i c h give a v i s u a l comparison of the d a t a p r i o r to 1985 w i t h the f u l l d a t a set. These are given on pages 56 a n d 57.  16  2.5  Conclusions  T h e m o n e t a r y models i n general d i d poorly. T h e r e was an exception as mentioned above but even this m o d e l s t i l l performed p o o r l y i n comparison to the segmented m o d e l a n d the best reductions were comparable to the best reductions a decade ago as reported i n Meese a n d Rogoff (1983). T h e f o r w a r d models gave slight reductions i n S S F E over the r a n d o m walk models. T h i s can be expected since the interest rate p a r i t y theory implies that F  t  « St- Hence large gains  over a r a n d o m w a l k m o d e l should never be expected. T h e segmented models do seem to do better t h a n the r a n d o m walk m o d e l .  Certainly  more testing u s i n g the exchange rates of other n a t i o n s , f o l l o w i n g up w i t h further testing on the G e r m a n exchange rates of the future a n d other variations on the theme are neccesary to make any statments w i t h a degree of certainty. T h e m a i n reason that this type of m o d e l i n g was i n i t i a t e d was the concern about the effect of p o l i c y decisions, speculation a n d central bank actions i n times when economic indicators are t a k i n g on extreme or u n u s u a l values. It appears that this m a y be a v a l i d concern. T h e segmented models w i t h the p r i n c i p a l component  from  the interest differentials as the segmentation variable performed significantly better t h a n m a n y other models.  It does appear that there is some segmentation w i t h respect to this variable.  Perhaps this can be a t t r i b u t e d to market efficiency i n some m a n n e r . W h e n differentials are large c e n t r a l banks a n d speculators w i l l be forced to pay a t t e n t i o n but when the differentials are s m a l l i t m a y be that o n l y the more i n f o r m e d , aware a n d less risk averse p a r t i p a n t s act.  17  Section 3  E s t i m a t i n g the N u m b e r o f Segments  3.1  Introduction  Segmented models m a y be useful i n m a n y s i t u a t i o n s . F o r example Y e h et al.(1983) discuss the the i d e a of an 'anaerobic t h r e s h o l d ' . It is hypothesized t h a t i f a person has his w o r k l o a d steadily increased t h r o u g h some f o r m of exercise there comes a point where the muscles cannot get enough oxygen a n d w h a t were anaerobic m e t a b o l i c processes become aerobic  processes.  T h i s point is referred to as the 'anaerobic t h r e s h o l d ' . In this s i t u a t i o n two segments are what is suggested by the subject oriented theory. So i t w o u l d be n a t u r a l for the modeler t o fit a m o d e l w i t h t w o segments. However i n some s i t u a t i o n s i t m a y be suspected t h a t a segmented m o d e l should be a d o p t e d but the a p p r o p r i a t e n u m b e r of segments m a y not be k n o w n . F o r instance i n the exchange rate p r o b l e m i t is suspected t h a t a segmented m o d e l is a p p r o p r i a t e due to p o l i c y changes.  It is n o t , however, clear how m a n y segments w i l l be necessary  beforehand.  One i m m e d i a t e a p p r o a c h to this p r o b l e m is to g r a p h i c a l l y a t t e m p t to d e t e r m i n e how m a n y thresholds seem to be a p p r o p r i a t e . T h i s is w o r t h w h i l e as a first step and i n the case of a single e x p l a n a t o r y variable but m a y not be a p p r o p r i a t e i n the m u l t i v a r i a t e case. In the m u l t i v a r i a t e case the i n t e r r e l a t i o n s h i p s of the e x p l a n a t o r y variables m a y confuse such an a p p r o a c h . F u r t h e r this a p p r o a c h lacks o b j e c t i v i t y , some sort of a u t o m a t e d rule is desired. In this section I discuss a consistent procedure for i d e n t i f y i n g the n u m b e r of segments.  3.2 A C r i t e r i o n for E s t i m a t i n g the N u m b e r of Segments  C o n s i d e r the f o l l o w i n g segmented l i n e a r regression m o d e l .  = x' Pi + 6 , if x  Y  t  where e  t  —  t  5^o° 1^**1 <  ipiZt-i,  independent of {x }, t  x  t  td  f  = ( 1 , x \,..., t  00  >  x p)' t  € ( r f _ i , T j ] , i = 1 , . . . , / + 1, ? i t h the {z } w  i i d , mean zero a n d variance a  2  t  a n d —oo =  assume t h a t there exists 6 > 3 / 2 , k > 0 9 |V>;| < k/i  s  18  TQ  <  T\  <  • • • <  TJ  + 1  and  = oo. F u r t h e r we  V i . N o t e that this i m p l i e s t h a t {e } t  is a  s t a t i o n a r y ergodic process. C o n s i d e r the following regression setup.  R e c a l l t h a t the least squares estimate of j3 is given by  and the s u m of squared error is given by  where H  n  = X (X' X ) n  n  n  X' .  1  n  T h e s i t u a t i o n 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 m a t r i x A, a n d 1(.) is the i n d i c a t o r  function,  T h e n , i n terms of true parameters, our model can be r e w r i t t e n i n the vector f o r m ,  T h e e s t i m a t i o n 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 n u m b e r of thresholds r i , . . . , T j o . T h i s is done by m i n i m i z i n g the modified S c h w a r z ' c r i t e r i o n  U n d e r some regularity conditions essential to the i d e n t i f i a b i l i t y of the regression parameters, we s h a l l see below t h a t the o r d i n a r y least squares estimates $ • w i l l be unique w i t h p r o b a b i l i t y  C o n s i d e r the segmented linear regression m o d e l discussed i n the previous section. m i n i m i z e MIC(l)  Let /  . T o identify the number of thresholds /, a n d hence the number of segments  consistently, assume:  U n d e r Condition  1, the design m a t r i x X (a,j3)  every open i n t e r v a l  has f u l l c o l u m n rank a.s. as n —• oo for  n  i n the s m a l l neighborhood of t h e true thresholds r,,i  w h i c h Xd has positive p r o b a b i l i t y density. A n d X (a,/3)  = 1 , . . . , / for  w i l l have full c o l u m n r a n k for large  n  n a n d for every open i n t e r v a l (a,(3) i n the s m a l l n e i g h b o r h o o d of rf,i  = 1 , . . . , / ° for w h i c h  2 is satisfied. So /?,• w i l l be unique w i t h p r o b a b i l i t y tending to 1 as n —• oo, for  Condition  i = 1 , . . . , /, p r o v i d e d t h a t I converges t o /, the true n u m b e r of thresholds i n p r o b a b i l i t y . W h e n the segmented regression m o d e l reduces t o the segmented p o l y n o m i a l regressions or f u n c t i o n a l segmented regressions as discussed by Feder (1975), then a s i m i l a r c o n d i t i o n to either Condition  1 or Condition  2 is essential for i d e n t i f y i n g the segmented m o d e l parameters.  For details, see Feder (1975). In p a r t i c u l a r , for the segmented p o l y n o m i a l regression m o d e l , C o n d i t i o n 1 is a u t o m a t i c a l l y satisfied i f the key covariate Xd has positive density w i t h i n a s m a l l neighborhood of each of the thresholds. In a d d i t i o n , we need to place some r e s t r i c t i o n o n the d i s t r i b u t i o n of the i i d errors A n d this is the so-called local exponential to be locally exponentially  boundedness condition.  {z }. t  A r a n d o m variable Z is said  bounded i f there exist constants CQ a n d To i n (0,oo) such t h a t < e ° " , V |u| < T .  E(e )  c  uZ  2  (3.1)  0  M a n y of the c o m m o n l y used d i s t r i b u t i o n s such as the n o r m a l , the s y m m e t r i z e d Poisson and exponential d i s t r i b u t i o n s have such a property. In reality, the sample size n is always finite a n d hence the number of thresholds t h a t can be effectively identified is always bounded.  So we w i l l assume throughout t h a t there always  exists an upper b o u n d L of the true n u m m b e r of thresholds. A n o t h e r s i m p l i f i c a t i o n we g a i n i n the nonlinear m i n i m i z a t i o n of S ( T I , ..., T\ <  r/) is o b t a i n e d by l i m i t i n g the possible values of  < Ti t o the finite discrete set, {xid, •..,x d}.  ...  T h i s r e s t r i c t i o n induces no loss of  n  generality. Theorem  1  Suppose {z } t  variance  a. 2  Consider  the segmented linear  are iid with a locally  exponentially  regression  21  independent  having  of  l. n  mean zero and  I, that I < L for some specified upper  1 or 2 is satisfied.  as n —y oo.  n  bounded distribution  Assume for the true number of thresholds,  bound L > 0 and that one of Conditions  model with X  Then I converges to I in  probability  The proof of the theorem will be given after a series of related lemmas.  Proof  It follows from M a r k o v ' s i n e q u a l i t y t h a t for 0 < t  F r o m the previous l e m m a there exists M i such that  0  < T satisfying \t a.i\ < T for a l l 0  Since A ( k ) a n d B ( k ) are independent we get that  F i n a l l y , to conclude the proof,we note t h a t  Lemma 2  Consider  either Condition  the segmented regression  1 or Condition  bounded and are independent  2. Assume  of X . n  model with the design  that the iid errors {zt}  matrix  X  are locally  t  n  Since H (x d,xu) n  s  we have t h a t  is i d e m p o t e n t , i t can be decomposed as H (x d,  is o r t h o g o n a l a n d A = diag(l, following results:  locally  of{z }.  Conditioning on X ,  Proof  exponentially  Then  where po is the true order of the model and To is the constant associated with the nential boundedness  satisfying  n  n  s  Xtd) = WAW,  • • •, 1,0, • • •,0). N o t i n g t h a t rank(A)  > rank(AB)  where W we get the  as n —• oo, where c is the constant i n L e m m a 1.  F i n a l l y , by a p p e a l i n g to the d o m i n a t e d  convergence theorem we o b t a i n the desired result. Lemma 3 Condition  Proof  Consider  the segmented regression model with the design matrix X  1 or Condition  n  2. Assume  that the iid errors {zt}  satisfying  are locally exponentially  either bounded  It suffices to show the case when 1 = 1. Since the proof under either Condition  C o n d i t i o n 2 is essentially the same, we shall proceed by verifying the l e m m a under  1 or  Condition  It t h e n follows from the L a w of L a r g e N u m b e r s for s t a t i o n a r y ergodic stochastic processes  Therefore,  S i m i l a r l y , i t can be shown t h a t  It remains t o show that  T h u s proceeding as before, using the l a w of large numbers we get that  w h i c h gives the result. L e m m a 3.4  Consider the segmented regression model (2.1) with the design matrix X  ing either Condition  n  1 or Condition  bounded and are independent  2. Assume  of X . n  that the iid errors {zt} are locally  satisfy-  exponentially  Let 1° denote the true I. Let ( i f , . . . , T;°) denote the true  Hence i t suffices t o show t h a t for each  T h e n i t follows f r o m the previous lemmas a n d the law of large  In c o n c l u d i n g the theoretical section of this paper some m e n t i o n should be made as to the need for further research i n t o the s t a t i s t i c a l properties of these models.  F o r instance i t  w o u l d be useful to have l i k e l i h o o d based tests for t e s t i n g the existence of thresholds. A l s o of interest w o u l d be the a s y m p t o t i c d i s t r i b u t i o n s of the estimated parameter values under m i l d s t a t i o n a r i t y assumptions.  30  Table 1 Forward Rate Models  Unadjusted sse per cent reduction five year mfse per cent reduction A d j u s t e d for Inflation sse five year mfse W i t h Intercept Unadjusted sse per cent reduction five year mfse per cent reduction Adjusted for Inflation sse five year mfse  Random Walk Model  Forward Rate M o d e l  1.67  1.66 1.04 0.42 6.67  0.45  1.29 0.44  1.31 0.44  1.67  1.65 1.14 0.43 4.44  0.45  1.29 0.44  1.30 0.47  31  Table 2 Segmented Models-Standard Principle Components  Percent reduction i n root mse from random walk model  model description Interest Differential - P r i n c i p l e Component l ( L a g 1) T w o Segments Three Segments T w o Segments (random walk coeff.=l) - P r i n c i p l e Component l ( L a g 2) T w o Segments - P r i n c i p l e Component l ( L a g 3) T w o Segments Trade Balance Differential - P r i n c i p l e Component l ( L a g 1) T w o Segments T w o Segments (random walk coeff.=l) - P r i n c i p l e Component l ( L a g 2) T w o Segments - P r i n c i p l e Component l ( L a g 3) T w o Segments M o n e y Supply Differential - P r i n c i p l e Component l ( L a g 1) T w o Segments T w o Segments (random walk coefF.=l) - P r i n c i p l e Component l ( L a g 2) T w o Segments  32  16.69 23.61 6.75 15.20 9.85  6.28 2.46 4.92 6.90  13.46 5.98 15.00  Percent reduction i n root mse from r a n d o m walk model  model description Interest Diferential - P r i n c i p a l Component 1 (lagl) and Trade Balance Differential - P r i n c i p a l Component 1 ( l a g l ) I.D. segmentation variable - T w o Segments - T h r e e Segments T . B . segmentation variable - T w o Segments Interest Diferential - P r i n c i p a l Component 1 (lagl) and M o n e y Supply Diferential - P r i n c i p a l Component 1 (lagl) I.D. segmentation variable - T w o Segments - T h r e e Segments T . B . segmentation variable - T w o Segments M o n e y Supply Diferential - P r i n c i p a l Component 1 (lagl) and Trade Balance Diferential - P r i n c i p a l Component 1 (lagl) T . B . segmentation variable - T w o Segments M . S . segmentation variable - T w o Segments  33  16.38 22.77 11.45  20.76 26.03 13.37  7.03 12.66  Percent reduction i n root mse from random walk model  model description Interest Diferential - P r i n c i p a l Component 1 (lagl) and M o n e y Supply Diferential - P r i n c i p a l Component 1 (lagl) and Trade Balance Diferential - P r i n c i p a l Component 1 (lagl) I . D . segmentation variable - T w o Segments - T h r e e Segments T . B . segmentation variable - T w o Segments - T h r e e Segments  34  21.28 27.56 11.67 12.55  Table 3 Segmented Models—Standard Principle Components Forecasting Results  M o d e l Description  S u m of Squared Forecasting E r r o r (SSFE)  SSFE (Random W a l k )  Percent Reduction in S S F E  0.299 0.368  0.437 0.437  31.56 15.63  2.92  Interest Differential T w o Segments Three Segments Interest Differential and Trade Balance Differential T w o Segments Three Segments Interest Differential and M o n e y Supply Differential T w o Segments Three Segments  0.424 0.490  0.437 0.437  0.636 0.252  0.437 0.437  Interest Differential and M o n e y Supply Differential and Trade Balance Differential T w o Segments Three Segments  0.630 0.507  0.437 0.437  35  42.33  Table 4 Monetary  Model Random Walk Frenkel-Bilson Dornbusch-Frankel Hooper-Morton Vector Autoregressions Frenkel-Bilson -2 lags Frenkel-Bilson -3 lags Dornbusch-Frankel -2 lags Dornbusch-Frankel -3 lags Hooper-Morton -2 lags  root M S E  Models  percent reduction root M S E  0.063 0.2219 0.192 0.194  SSFE  percent reduction SSFE  (without In transf.)  0.0908 0.7780 0.8594 2.6440  0.0598  5.06  0.1003  0.061  3.15  0.1244  0.0583  7.44  0.0887  0.0604  4.11  0.1154  0.0545  13.48  0.1025  36  0.86  2.26  12.36  Table 5 Segmented Models—Tsay Principle Components  Percent reduction i n root mse from random walk model  model description Interest Differential - P r i n c i p l e Component l ( L a g 1) T w o Segments Three Segments  15.83 15.53  Trade Balance Differential - P r i n c i p l e Component l ( L a g 1) T w o Segments Three Segments M o n e y S u p p l y Differential - P r i n c i p l e Component l ( L a g 1) T w o Segments Three Segments  5.46 13.23  12.90 15.33  Interest Diferential - P r i n c i p a l Component 1 (lagl) and Trade Balance Differential - P r i n c i p a l Component 1 (lagl) I.D. segmentation variable - T w o Segments - T h r e e Segments T . B . segmentation variable - T w o Segments - T h r e e Segments  15.03 15.78 9.14 15.20  37  Percent reduction i n root mse from random walk model  model description Interest Diferential - P r i n c i p a l Component 1 ( l a g l ) M o n e y Supply Diferential - P r i n c i p a l Component 1 (lagl) I.D. segmentation variable - T w o Segments - T h r e e Segments M . S . segmentation variable - T w o Segments - T h r e e Segments M o n e y S u p p l y Diferential - P r i n c i p a l Component 1 (lagl) Trade Balance Diferential - P r i n c i p a l Component 1 (lagl) T . B . segmentation variable - T w o Segments - T h r e e Segments M . S . segmentation variable - T w o Segments - T h r e e Segments Interest Diferential - P r i n c i p a l Component 1 (lagl) M o n e y Supply Diferential - P r i n c i p a l Component 1 (lagl) Trade Balance Diferential - P r i n c i p a l Component 1 (lagl) I.D. segmentation variable - T w o Segments - T h r e e Segments  and  16.39 16.43 14.72 15.88 and  8.71 13.14 14.34 17.72 and and  15.68 16.55  38  Table 6 Segmented M o d e l s - T s a y Principle Components Forecasting Results  M o d e l Description  S u m of Squared Forecasting Error (SSFE)  SSFE (Random Walk)  Percent R e d u c t i o n in S S F E  0.329 0.360  0.437 0.437  24.65 17.65  Interest Differential T w o Segments Three Segments Interest Differential and M o n e y Supply Differential T w o Segments Three Segments Interest Differential and Trade Balance Differential T w o Segments Three Segments  0.331 0.378  0.437 0.437  24.15 13.38  0.378 0.416  0.437 0.437  13.52 4.69  Interest Differential and M o n e y Supply Differential and Trade Balance Differential T w o Segments Three Segments  0.403 0.443  0.437 0.437  7.65  39  40  42  43  44  Exchange Rate 2.0  2.5  d  3  <D  5T  ta II  Exchange Rale 20  2.5  H  3'  Exchange Rate 20  Exchange Rate  2.5  2.0  46  2.5  Exchange Rate 2.0  2.5  =1 3 5T  ta  un  Exchange Rate 2.0  Exchange Rate  2.5  2.0  2.5  3  o  Exchange Rate 2.0  Exchange Rate  2.5  2.0  47  2.5  48  49  50  51  52  53  exchange rate(t) 1.5  2.0  2.5  3.0 i  .  *  54  exchange rate(t) 2.0  2.5  55  3.0  3.5  56  57  Bibliography  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 Brunner 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 International Economics, 6, 231-244. Feder, P.I. (1975a) On asymptotic distribution theory in segmented regression problemsidentified 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, 7 8 , 200-224. Frenkel, J . A . (1983), A n 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), A n accounting framework and some issues for modeling how exchange rates respond to the news, i n : J A . Frenkel ed., Exchange rates and international macroeconomics, 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 . , W u , S., Zidek, J . V . (1991), O n 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), "anaerobic threshold':Problems of determination and validation. J A p p l . Physiol. Respirit. Envioron. Excercise Physiol., 55, 1178-1186.  59  

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