Modeling the Structural Shifts in Real Exchange Rate with Cubic Spline Regression (CSR). Turkey

Transcription

1 International Journal of Business and Social Science Vol. 2 No Modeling the Structural Shifts in Real Exchange Rate with Cubic Spline Regression (CSR). Turkey Dr. Bahar BERBEROĞLU Anadolu University, Open Education Faculty. Eskisehir, Turkey E-ail: bdire@anadolu.edu.tr, Phone: /255 Prof. Dr. C. Necat BERBEROĞLU (Corresponding Author) Faculty of Econoic and Adinistrative Sciences Anadolu University Eskisehir, Turkey E-ail: nberbero@anadolu.edu.tr, Phone: /241 Abstract Econoic life is continuous but tie series presenting econoic data contain soe break points because of sharp changes and shifts because of iportant shocks such as econoic crises. In such cases any iportant structural shifts occur in soe paraeters of econoic odels. These shifts on econoic paraeters generally ay differ and occur with different tie lags. In this study it is aied to expose and analyze the occurrence of structural shifts and structural changes in Real Exchange Rate in Turkey during the period of We used cubic spline regression in odeling the structural changes in real exchange rate in Turkey. With this technique, several cubic spline regression odels of real exchange rate are constructed and discussed, and the ost significant one is chosen. Then in this study, we also tried to put forward how prediction su of squares statistic residuals can iprove the analysis in cubic spline odels. Keywords: Cubic Spline Regression, Structural Change, Real Exchange Rate, PRESS statistics. JEL Codes: C01, C10, C22, C51 1. INTRODUCTION Econoic tie series are usually affected by nuerous internal or external shocks and variables. Although the econoic life is continuous, soe events such as wars, econoic or political crises generally cause abrupt and sharp changes in the data collected in their periods. But ostly leaps or jups do not occur in these tie series, so they keep their continuity with soe structural shifts or structural changes. When a statistician and of course any other researcher such as an econoetrician sees soe break points, knots or structural shifts in a tie series in which he attepts to use in forecasting, or siply he has a priori reasons to suspect that iportant structural changes have occurred in that tie series, spline functions offer a way to use these prior inforation. Structural shifts are unexpected changes in the data pattern, but these changes do not always iply discontinuity in the data. Spline functions preserve the continuity of a tie series without creating jups and by this way these functions iprove forecasts when applied to data sets containing all kinds of structural changes. Spline functions offer a relatively siple technique to fit a function through a series of points. A spline function occurs when two or ore polynoial functions are connected end to end. The biggest advantage of spline functions is their flexibility in approxiation. Each portion of the function ay be structurally siilar or different. The spline function is less restrictive than iposing the sae structure on all subsets of the saple. With the use of regression analysis, trends are estiated to predict the future. For exaple, when iportant structural changes have occurred in alost all kinds of econoic tie series or data sets such as national incoe values, econoic growth rates, exports, iports, price indexes and interest rates, spline functions provide a very useful additional tool for tie series forecasting. Spline functions have been used priarily in statistics then in business and econoetrics. They were previously used by statisticians for testing structural shifts in various data sets; afterwards other researchers used splines in testing tie series of profits, interest rates, econoic growth rates, exports and iports which usually contain structural shifts (Marcus, 1987/1988). Whenever there is the suspicion of a structural change in a tie series, a spline function allows that a priori belief to be included in the odel. In the language of Bayesian statistics, spline functions can be said to offer a prior that structural change has occurred. 60

2 The Special Issue on Conteporary Issues in Business Studies Centre for Prooting Ideas, USA A spline function joins a tie series of observations at points called join points, knots, or break points occurring in tie (Marcus, 1987/1988). Marcus (1987/1988) explained a quadratic spline with a single join point and estiated quadratic spline with using duy variables. Poirier (197) showed how cubic splines can be used to test the structural changes. In Poirier s study, spline theory and piecewise regression theory were integrated to provide a fraework in which structural change is viewed as occurring soothly. Specifically, structural change occurs at given points through jup discontinuities in the third derivative of a continuous piecewise cubic estiating function. In the study testing procedures were developed for detecting structural change. Poirier s study used cubic spline because of its actual piecewise nature which is justified fro a theoretical standpoint. Attention is focused on those piecewise regression odels when; (1) the piecewise nature reflects the occurrence of structural change, (2) the pieces are defined over adjacent intervals of the independent variable, () the overall odel is continuous with structural change revealing itself with jup discontinuities in the third derivative, and (4) the points of structural change are known (Poirier, 197). Poirier (197) proposed an excellent discussion of the basic theory of cubic regression splines. Arguing fro an econoic point of view, he developed the idea that structural change occurs in a sooth fashion so that splines for a natural tool for analyzing structural changes. He argued that knots should occur at a point in tie near the point of structural change. Poirier illustrated his point with an exaple based on Indianapolis 500 data. Buse and Li (1977) followed up Poirier s paper, arguing that cubic spline regression is a special case of restricted least squares and that the latter approach offered a richer enu of procedures (Wegan and Wright, 198). Gallant and Fuller (197) treated the join points (knots) as unknown paraeters and developed procedures for estiating their location. In the study of Buse A., and Li L., (1977) the standard cubic spline regression ethod was used and this ethod was shown as to be a special case of the restricted least squares (RLS) estiator. In that study the equivalence of the two procedures under a coon set of restrictions was proved. The greater flexibility of the restricted least squares estiator in ters of the nuber of restrictions and test of the hypotheses were also obviously shown in the study. Wold (1974) reflected a lot of experience with fitting regression splines. Based on his practical experience, Wold (1974) ade soe useful recoendations for knot points selection. These recoendations were based on the assuption of fitting a cubic spline, which is the ost popular case, and ay need soe odification for degree of the spline function, >. 1. Knot points should be located at data points. 2. A iniu of four or five observations should fall between knot points.. No ore than one extreu and one inflection point should fall between knots (because a cubic could not fit ore). 4. Extrea should be centered in intervals and inflection points should be located near knot points. A useful statistical approach for coparing the forecasting perforance of different odels is the (Predicted Residuals Su of Squares) PRESS statistics proposed by Allen (1974). PRESS statistics is based on the leaveone-out or jackknife technique. The idea is to fit the odel without the ith observation x i and use this fitted odel to predict the response ŷ i at x i. There is a well-known forula for coputing prediction su of squares (PRESS) residuals in a regression proble without having to refit the curve for each observation. Tarpey (2000a) showed that the sae basic results hold for fitting a regression function when the regression coefficients are subject to linear restrictions. 2. TURKISH REAL EXCHANGE RATE IN THE PERIOD OF The rapid econoic developent in Turkey which started in early 1960s with the ipleentation of planned industrialization and odernization odel based upon iport substitution strategy, has turned into a severe crisis as a result of the adjustent probles and the oil crisis of 1970s. In the beginning of crisis period the growth rate slowed down and then turned to negative, inflation began to increase, finding funds for iports becae harder and scarcity of goods eerged in the internal arkets, and Turkey had alost cae to an end econoically in The Stability Measures of the 24th of January, 1980 aied aintaining short-ter econoic stability, structural adjustent in the econoy, and changing long-ter accuulation regie. Even though IMF supported stability progras had been ipleented before 1980, with the ilitary intervention of the 12th of Septeber, 1980 the relations with IMF altered and, the 24th of January progra and its results changed in content and influences. The structural adjustent policies of the IMF and World Bank becae the ajor deterinants of Turkey s econoic policies in the early 1980s. 61

3 International Journal of Business and Social Science Vol. 2 No International trade played a inor role in the Turkish econoy before 1980, but grew rapidly after the 1980 refors. Especially exports and iports started to increase rapidly after 198 with the Governent Change. New governent ephasized foreign trade liberalization. After 198 econoic refors prooted foreign trade by reoving price controls, decreasing subsidies, reducing tariffs. In addition to rapid growth in both exports and iports, the refors brought a change in the structure of foreign trade, and the predoinant role of agricultural products cae to an end with the eergence of a greater ephasis on industrial products. Turkey signed a free trade agreeent with the European Free Trade Association (EFTA) in In 1992, Turkey and 10 other nations in the Black Sea region fored the Black Sea Econoic Cooperation Organization. Turkey becae a eber of the World Trade Organization (WTO) in 1995, and a year later signed a Custos Union Agreeent with the EU. Turkey also becae a eber of the Organization for Econoic Cooperation and the Organization of the Islaic Conference. Separately, Turkey has entered into free trade agreeents with Israel, and with several Central and Eastern European countries. During these developents in Turkey s international relations, Turkish Econoy experienced several econoic crises and ipleented several stabilization progras. Soe of these econoic crises eerged fro doestic political probles and governent changes and soe others arise fro wars and international crises and consequently all of these developents had great ipacts on Turkey s exports and iports. The ost iportant stabilization progras in period were, 5th of April 1994 Stabilization Progra which had ipacts in the sae year, and 21st of January 2001 Stabilization Progra whose ipacts were seen in The ajor effects of all these events were seen of course on the values of Real Exchange Rate. The changes in Turkey s Real Exchange Rate in this period can easily be seen in the Figure 1. Figure 1. Based on 1 USD+1.5 EUR basket, in the relative price calculations and consuer prices for Turkey are used. (1987. JAN. = 100) In the years of econoic crises and sharp depreciations of TL alost all of acroeconoic paraeters of Turkish Econoy were affected negatively at different degrees and with different tie lags. When the value of Real Exchange Rate is considered we assued that, structural changes occurred in data without any tie lag. So our hypothesis is that, the sharp depreciation of TL in the year of 1994 and the Econoic Crisis in 2001 were the ost iportant events of period in Turkish Econoy and structural changes in the value of Real Exchange Rate occurred in these years. For this reason, we applied cubic spline regression on these three paraeters and fored the knots t 1 =8 for 1994 and t 2 =15 for When Figure 1 is observed it could be understood that the odel fits four (4) conditions of Wold (1974) which was entioned before, and also it could be understood that cubic spline regression provides good odeling in such tie series. Since in this technique knots should be located at data points and five observations should fall between knot points, and the inflection points should fall between knots and should be located near knot points.. MODEL (CUBIC SPLINE REGRESSION (CSR) AND PRESS STATISTICS) Spline regressions are splines which are coputed according to a regression odel. The odel is generally as follows, y i = S x i + ε i, i = 1,2,, n (.1) Where S (x) is a piecewise polynoial spline with esh Δ and ε i are a white noise sequence. The proble then is to copute S (x) based on a least squares approach. Given the esh of knots, Δ={t 1 <t 2 < <t N }, Sith (1979) shows that a piecewise or segented polynoial ay be represented in the Heaviside function notation; that is, if we let u + =u if u>0 and u + =0 if u 0, then the general for of a segented polynoial regression odel, (.1), with N knots in the esh Δ having N+1 polynoial pieces each of degree ay be rewritten as follows (Wegan and Wright, 198): 62 y i = j=0 β 0j x t j + N k=1 j=0 β kj (x i t k ) + j + ε i (.2) In general, with N knots and the N+ 1 polynoial pieces each of degree ay be written a spline odel with no continuity restrictions as follows: S x = j=0 β 0j x t j + N k=1 j=0 β kj (x i t k ) + j. (.)

4 The Special Issue on Conteporary Issues in Business Studies Centre for Prooting Ideas, USA j As in the special case just discussed, the presence of a β kj (x i t k ) + ter allows a discontinuity at t k in the jth (j) derivative of S, S (S (0) (j) =S ), and its absence forces the continuity of S at t k. Thus, S can be ade continuous at knot t k by oitting fro (.) the constant ter β k0 (x i t k ) 0 +, and S (j) can be ade continuous at j t k by oitting the ter β kj (x i t k ) +. If we wanted continuity in the jth derivative of S at a particular knot t k, we would probably also want continuity in all lower-order derivates at t k. This continuity is accoplished by oitting the ters β kn (x i t k ) n +, n=0,1,,j. Different continuity restrictions can be iposed at different knots siply by deleting the appropriate ters. The ore ters we delete, the worse the fit will be. Because the deletion of a + ter is equivalent to adding a continuity restriction, however, the curve will be soother. In order to deterine which continuity restrictions do not result in a significantly worse fit, we test in (.) whether the appropriate coefficients are zero. The soothest possible spline with polynoial pieces of degree with N knots is given by (Sith, 1979), S x = j=0 β 0j x t j + N k=1 β k (x t k ) + (.4) The degree of the spline function fro (.2),, depends on what is a realistic assessent of the nuber of derivatives available in the regression function. Obviously, this knowledge is frequently not available. Often the choice is siply =, which yields a cubic spline and which is the sallest yielding visual soothness. The choice =2 or =1 will yield, respectively, piecewise quadratics or piecewise lines (i.e.,quadratic and linear splines) (Wegan and Wright, 198). The cubic spline function is y i = j=0 β 0j x t j + N k=1 j=0 β kj (x i t k ) + j + ε i (.5) Forula (.5) ay be rewritten as follows: β 00 + β 01 x + β 02 x 2 + β 0 x, 0 < x t 1 E y i = β 10 + β 11 x t 1 + β 12 (x t 1 ) 2 + β 1 (x t 1 ), t 1 < x t 2 (.6) β 20 + β 21 x t 2 + β 22 (x t 2 ) 2 + β 2 (x t 2 ), t 2 < x Then as given by (Sith, 1979) soothest cubic spline with polynoial pieces of degree = with N knots will be, S x = j =0 β 0j x t j + N k=1 β k (x t k ) + (.7) Forula (.7) ay be rewritten as follows: β 00 + β 01 x + β 02 x 2 + β 0 x, 0 < x t 1 E y i = β 00 + β 01 x + β 02 x 2 + β 0 x + β 1 (x t 1 ), t 1 < x t 2 (.8) β 00 + β 01 x + β 02 x 2 + β 0 x + β 1 (x t 1 ) + β 2 (x t 2 ), t 2 < x Thus; E y i = β 00 + β 01 x + β 02 x 2 + β 0 x + β 1 (x t 1 ) + + β 2 (x t 2 ) + (.9) The ordinary least squares fit, the ith PRESS residual is defined as e i = y i - ŷ i, wherey i, is the predicted ith response fro a odel fitted with all the data except the ith observation. A siple way the copute the PRESS residuals without having to refit the odel for each of the observations is to note that; e (i) = y i y i 1 h ii, where h ii is the ith diagonal eleent of the hat atrix H = X(X X) 1 X (Allen 1974). For the restricted linear regression, there also exists a siple forula for coputing PRESS residuals. It can be expressed as the vector of predicted responses under the restricted odel by y = (H J)y, where J = X X X 1 R R X X 1 R 1 R(X X) 1 X. The PRESS residual for the leave-one-out analysis using the restricted least squares fit is e i = y i y i. 6

5 International Journal of Business and Social Science Vol. 2 No It can be expressed e i as e (i) = y i y i 1 h ii + j ii, Where j ii denote the ith diagonal eleent of the atrix J (Tarpey 2000b). In another work (Tarpey 2000a), Tarpey argued that PRESS statistic perfors better job for Restricted Least Squares. In both works (Tarpey, 2000a and 2000b) he supported this arguent by coparing PRESS statistics of restricted and unrestricted odels. 4. ESTIMATION RESULTS: REAL EXCHANGE RATE CSR MODELS In our work, we produced four different odels fro the sae data set by ipleenting cubic spline regression fro piecewise polynoial structure to the soothest structure and showed the produced odels in Table 1. Model III was not statistically significant but the others were. So we tested the significance of the coefficients of Models I. II and IV. Table 1: Coefficients of constructed CSR odels for Real Exchange Rate We will present all of the odels which we produced and suarized in Table 1 in graphics one by one. In these graphics one can see, how the curve transfors to the soothest shape fro piecewise for. As it can be seen, Model I is in piecewise for and Model IV is so sooth. While foring cubic spline regression odels, knots appeared in the years of structural changes. For this reason when we estiate the equation (.6), our data set knots appeared at t 1 =8 and t 2 =15. So the equation could be written as below, β 00 + β 01 x + β 02 x 2 + β 0 x, 0 < x 8 E y i = β 10 + β 11 x 8 + β 12 (x 8) 2 + β 1 (x 8),8 < x 15 β 20 + β 21 x 15 + β 22 (x 15) 2 + β 2 (x 15), 15 < x Figure 2. CSR odel (Model I) for Turkey s Real Exchange Rate in In the Figure 2 which presents Model I, it is shown that cubic spline curve is indeed a discontinuous function at the knots as in Sith s (1979) work. Also since Model I is a discrete function, constants of β 00, β 10 and β 20 are shown separately. If we wish to fit polynoial pieces of different degrees, odel (.) can be odified and ultiple regression techniques applied as usual. In soe cases, the odification ay necessarily be substantial. The siplest type of odification occurs when only polynoial pieces of degrees n and n-1 are used in the odel. We ust then ipose the condition in odel (.) that soe paraeters are zero and certain pairs of paraeters su to zero. If polynoial pieces of degrees other than n and n-1 are desired, the odifications to (.) ust be involved ore (Sith, 1979). We begin by using (.) to write the odel for a three-piece cubic (largest-degree piece) continuous spline: E y i = β 00 + β 01 x + β 02 x 2 + β 0 x + β 11 x β 12 x β1 x β21 x β 22 x β2 x 15 + Figure. CSR odel (Model II) for Turkey s Real Exchange Rate in The absence of the β 10 and β 20 ters guarantees that the function is continuous, and the presence of higher-order ters allow for possible discontinuities in the derivatives. Our third continuous spline is; E y i = β 00 + β 01 x + β 02 x 2 + β 0 x + β 12 (x 8) + 2 +β 1 (x 8) + + β 22 (x 15) + 2 +β 2 (x 15) + Figure 4. CSR odel (Model III) for Turkey s Real Exchange Rate in In this odel in addition to the ters β 10 and β 20, two other ters β 11 and β 21 do not exist because of the 0 restriction on the. Testing the significance of the odel also eans testing the significance of the restrictions at the sae tie. Although Model III was not significant, in order to show how the curve is getting soother we presented its graphics in Figure 4. Our fourth continuous spline is; E y i = β 00 + β 01 x + β 02 x 2 + β 0 x + β 1 (x 8) + + β 2 (x 15) + Figure 5. CSR odel (Model IV) for Turkey s Real Exchange Rate in

6 The Special Issue on Conteporary Issues in Business Studies Centre for Prooting Ideas, USA In Model IV, besides the ters β 10, β 20, β 11, β 21 we used the 0 restriction for β 12 and β 22 ters. Because of the significance of the odel which can be seen easily in Table 1, all of these restrictions are also significant. Furtherore this odel is the soothest one aong all cubic splines and which is produced according to the forula (.), so without any hesitation we choose Model IV as the best one. If the fourth continuous spline is observed carefully it can be seen that, two extree points exist in this spline, the year 1991 is the axiu and 1995 is the iniu. As Wold (1974) entioned, two extrees ust exist in a cubic function and only one be placed between knots. Furtherore, in our function the year 199 is the inflection point, and as the year 1994 is the first knot, the inflection point is placed near the knot. Table 2: Coparison of CSR odel (Model IV) and OLS odels with residuals su of squares In order to test the significance of the CSR odel constructed in Table 2 and to obtain a positive F test statistical value, the RSS value of CSR odel should be greater than the RSS of OLS odel. Model IV is significant at 0.05 levels. But according to Tarpey s approach the PRESS statistics of CSR odel should be saller than OLS odel in order the restrictions which are deterined according to the knots should be significant. Here it can be seen that the PRESS statistic of our CSR odel Model IV is saller than the PRESS statistic of OLS odel. 5. CONCLUSIONS The econoic issues and probles experienced in a country cause big changes particularly in econoic tie series. They are usually affected by various internal or external events. Such cases as wars, econoic or politic crises generally cause abrupt and sharp changes in the data collected in their periods but these tie series usually keep their continuity with soe structural shifts or structural changes. In this study our attept is to shed light on the question that whether Real Exchange Rate which are in interaction, change in the years of econoic crises. In the years of econoic crises and sharp depreciation of TL alost all of the econoic paraeters of Turkish Econoy were affected negatively but the effects on any paraeters seen usually at different degrees and tie lags. When we consider the values of Real Exchange Rate, our assuption was that the effects were siilar and the structural changes occurred in data without any tie lag. So we proposed that, sharp depreciation of TL in 1994 and the 2001 Econoic Crises were the ost iportant events of period in Real Exchange Rate. For this reason we applied cubic spline regression on these three paraeters and fored the knots t 1 =8 for 1994 and t 2 =15 for In our study we used cubic spline regression odel in order to expose how the econoic policies which were ipleented to proote Turkish foreign trade caused structural changes in related data. With this effort we realized that the devaluation in 1994 and the big econoic crisis experienced in 2001 caused structural breaks in Real Exchange Rates of Turkey. These structural breaks can only be exained with real data (without any transforing or soothing) by eploying Cubic Spline Regression. According to spline regression theory, spline regression is a preferred ethod in interpolation. With this ethod, one can easily reach iniu residual su of squares and achieve this result by using the real econoic data without changing their nature. If an econoist cares and takes this advantage into account, he/she will odel the breaks in tie series with splines and observe the deep ipacts of crises better. In our interpolation process, we are quite enlightening the question What should be in econoic sense. In period 2001 crisis started because of a political proble without any expectations in governental and econoic areas. But 1994 crisis was foreseen, so governent ipleented 5 th April Stabilization Progra before it. It is so interesting that, in our statistically significant odel the inflection point is the year 199 and by the turn of diinishing to an increase the change in the concavity of our spline occurs in this year. When observed carefully, it can be seen that the increase continues both before and after the year 2001 and also the structural break in 2001 crisis seen briefly. We don t have theoretical proofs to offer but, when we reeber that 2001 crisis was unforeseen and evaluate our odel in this sense, we can easily suggest that our odel fits the structure of such econoic case and data. When we exaine the Real Exchange Rates of Turkey, it can be seen that the rate was 1.8 in 199 and declined to 101. in This decline ight be accepted as a requested result of devaluation in the sae year. But also in 2001 Crisis which started suddenly in February, although an econoic stability progra based on exchange rate was ipleented in 1999, siilar changes occurred and Real Exchange Rate declined to 12.1 in 2001 although it was 152 in In conclusion it can be said that, our arguent on spline regression is supported with the results of this work. 65

7 International Journal of Business and Social Science Vol. 2 No Fro the perspective of spline theory, cubic spline regression provides a good odel in interpolation when there are significant structural breaks in tie series. Although 2008 Crisis had caused another structural break in the values of real exchange rate values in 2009, we could not include that year, because in this ethod in order to include and analyze this new structural break we should need to odel a tie series which also covers the year As entioned above there ust be a period of at least 4 or 5 years between the knot values. Furtherore, again in the perspective of regression analysis the estiation of the values of 2009 and 2010 with extrapolation ethod will be useless. Because as known, 2008 Crisis created a structural break in the tie series and the value of real exchange declined and becae in In the study we deonstrated how Cubic Spline Regression (CSR) becoes a very useful tool in the existence of structural shifts or changes in tie series, we also showed that that prediction su of squares residuals (PRESS) statistics iprove the analysis in cubic spline odels. Buse and Li (1977) anyway showed the equality of Restricted Least Squares and Cubic Spline Regression. The Real Exchange Rate is a well known subject in econoics and Real Exchange Rate is analyzed with various odels, but if great structural shifts occur in Real Exchange Rate paraeters in the cases of big econoic crises and devaluations cubic spline odels which describe the break points as knots can do better job in estiations. The odels in this study will be ore useful in understanding the real effects of econoic crisis in a country and they will be helpful in realizing the size and length of crisis experienced in reality. REFERENCES Allen, D. M., (1974), The Relationship Between Variable Selection and Data Augentation and a Method for Prediction, Technoetrics, 16, No.1, Buse, A., Li, L., (1977), Cubic Splines as a Special Case of Restricted Least Squares, Journal of the Aerican Statistical Association, 72, No.57, (Access Date: ) Gallant, A. R., Fuller W. A., (197), Fitting Segented Polynoial Regression Models Whose Join Points Have to Be Estiated, Journal of the Aerican Statistical Association, 68, No.41, Marcus, R. D., ( ), How To Deal With Structural Changes In The Data, The Journal of Business Forecasting Methods & Systes, 6, 4, Poirier, D.J. (197), Piecewise Regression Using Cubic Splines, Journal of the Aerican Statistical Association, 68, No.4, Sith, P. L., (1979), Splines As a Useful and Convenient Statistical Tool, The Aerican Statistician, Vol., No. 2 (May, 1979), pp Wegan, E.J., Wright I.W.,(198), Splines in Statistics, Journal of the Aerican Statistical Association, 78, No.82, Wold, Svante, (1974), Spline Functions in Data Analysis, Technoetrics, 16, No.1, 1-11 Tarpey, T.,(2000a), A Note on the Prediction Su of Squares Statistic for Restricted Least Squares, The Aerican Statistician, 54, No.2, Tarpey, T., (2000b), Spline Bottles, The Aerican Statistician, 54, No.2,

8 The Special Issue on Conteporary Issues in Business Studies Centre for Prooting Ideas, USA Figures: Observed Real Exchange Rate For Turkey ( ) Observed Exchange Rate Figure 1. Based on 1 USD+1.5 EUR basket, in the relative price calculations and consuer prices for Turkey are used. (1987. JAN. = 100) Model I Figure 2. CSR odel (Model I) for Turkey s Real Exchange Rate in Model II Observed Exchange Rate Exchange Rate(CSR) Figure. CSR odel (Model II) for Turkey s Real Exchange Rate in

9 International Journal of Business and Social Science Vol. 2 No Model III Observed Exchange Rate Exchange Rate (CSR) Figure 4. CSR odel (Model III) for Turkey s Real Exchange Rate in Model IV Observed Exchange Rate Exchange Rate (CSR) Figure 5. CSR odel (Model IV) for Turkey s Real Exchange Rate in

10 Real Exchange Rate 15<x 8<x 15 0<x 8 The Special Issue on Conteporary Issues in Business Studies Centre for Prooting Ideas, USA Tables: Table 1: Coefficients of constructed CSR odels for Real Exchange Rate Model I Model II Model III Model IV Coefficients β 00 β 11 β 12 β 1 β 10 β 11 β 12 β 1 β 20 β 21 β 22 β 2 Estiated Coefficients (and s.e.) * (1.764) (12.445) * (.124) * (0.229) (17.056) (17.81) * (5.211) (0.444).497 (7.159) * (8.567) * (.475) * (0.8) Partial t Estiated Coefficients (and s.e.) * (14.06) (12.7) * (.177) * (0.22) ** (12.07) * (2.901) 0.04 (0.49) * (6.514) * (2.27) * (0.218) Partial t Estiated Coefficients (and s.e.) Partial t Estiated Coefficients (and s.e.) * (17.1) 2. * (10.17) * (1.677) * (0.08) * (0.092) * (0.01) F 1.15 *** 0.9 *** *** Note: * Coefficients are significant at 95% levels, ** Coefficients are significant at 90% levels, *** CSR Models are significant at 95% levels. Model was not significant in F test, so we didn t need to copute t values. Partial t Tablo 2: Coparison of CSR odel (Model IV) and OLS odels with residuals su of squares Residuals su of squares of Constructed CSR Models and OLS Models RSS e (i) PRESS e ~ ( i) CSR Model (Model IV) 2185,96 420,006 OLS Model 1854, ,508 69