Time-varying Cointegration Models and Exchange Rate Predictability in Korea

Park, Soo Kyung; Park, Choel Beom Article Time-varying Cointegration Models and Exchange Rate Predictability in Korea KDI Journal of Economic Policy Provided in Cooperation with: Korea Development Institute (KDI), Sejong Suggested Citation: Park, Soo Kyung; Park, Choel Beom (2015) : Time-varying Cointegration Models and Exchange Rate Predictability in Korea, KDI Journal of Economic Policy, ISSN 2586-4130, Korea Development Institute (KDI), Sejong, Vol. 37, Iss. 4, pp. 1-20, https://doi.org/10.23895/kdijep.2015.37.4.1 This Version is available at: https://hdl.handle.net/10419/200777 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-sa/4.0/ KDI Journal of Economic Policy 2015, 37(4): 1 – 20 1 Time-varying Cointegration Models and Exchange Rate Predictability in Korea By S OOKYUNG P ARK AND C HEOLBEOM P ARK * We examine the validity of popular exchange rate models such as the purchasing power parity (PPP) hypothesis and the monetary model for Korean won/US dollar exchange rate. Various specification tests demonstrate that Korean data are more favorable for both models based on time-varying cointegration coefficients as compared to those based on constant cointegration coefficients. When the abilities to predict future exchange rates between those models based on timevarying cointegration coefficients are compared, an in-sample analysis shows that the time-varying PPP (monetary model) has better predictive power over horizons shorter (longer) than one year. Results from an out-of-sample analysis indicate that the time-varying PPP outperforms models based on constant cointegration coefficients when predicting future exchange rate changes in the long run. Key Word: Exchange rate, Monetary model, Predictability, Purchasing power parity, Time-varying cointegration JEL Code: F37, F41 I. Introduction nderstanding the movements of exchange rates is important, especially in Korea, where exports play an influential role in her growth. In spite of the importance of exchange rates, however, understanding the movements of exchange rates based on macroeconomic models has been a challenge to economists. Hence, the goal of this paper is to examine whether popular macroeconomic models such as the Purchasing Power Parity hypothesis (henceforth PPP) and/or the monetary model can explain fluctuations in the Korean won/US dollar exchange rate. More specifically, we investigate which model between the PPP hypothesis and the * Park: Research Professor, Department of Economics, Korea University, Anam-ro 145, Seongbuk-gu, Seoul, Korea 136-701 (e-mail: [email protected]); Park: (corresponding author) Professor, Department of Economics, Korea University, Anam-ro 145, Seongbuk-gu, Seoul, Korea 136-701 (e-mail: cbpark_kjs@ korea.ac.kr). * Received: 2015. 10. 14. * Referee Process Started: 2015. 10. 16. * Referee Reports Completed: 2015. 11. 23. U 2 KDI Journal of Economic Policy NOVEMBER 2015 monetary model provides a better tool for understanding exchange rate fluctuations in Korea. In order to achieve this goal, we relate the exchange rate to macroeconomic variables based on the concept of cointegration. The cointegration approach has been widely applied in the literature on exchange rates since its introduction by Engle and Granger (1987). However, the cointegration relationships presented in this study include the constant cointegration relationship, which has been examined in many studies, as well as the type of cointegration relationship which allows cointegration coefficients to vary gradually over time. There are reasons why we analyze the time-varying cointegration relationship between the exchange rate and macroeconomic variables in addition to the constant cointegration relationship between those variables. Many empirical studies have reported that the structure of the money market has changed over time.1 Cheung and Chinn (2001) also show that the macroeconomic variables and economic models utilized by foreign exchange traders to understand exchange rate movements shift over time. Bacchetta and van Wincoop (2013) demonstrate that the relationship between the exchange rate and macroeconomic variables varies over time when the pertinent structural parameters are unknown. Finally, Bierens and Martins (2010) and Park and Park (2013) provide evidence of time-varying cointegration relationships among variables under the PPP hypothesis and among variables under the monetary model, respectively. Most of the above-mentioned studies, however, focus on the US or other advanced economies.2 In addition to the reasons discussed above, considering the time-varying cointegration relationship is particularly relevant in Korea because her economic structure has changed over time. These structural changes include financial and economic reforms suggested by International Monetary Fund (IMF) during the Asian Financial Crisis of 1997 as well as various currency swap agreements during the Global Financial Crisis. Given that these reforms and agreements must result in gradual changes in the economic environment in which the exchange rate and other macroeconomic variables are determined, it is a meaningful exercise to extend the time-varying cointegration approach to the wondollar exchange rate in Korea under both the PPP hypothesis and the monetary model. The paper is organized as follows. Section II briefly presents both the PPP hypothesis and the monetary model. Section III provides a discussion of the data and the econometric methodology of the time-varying cointegration approach. Section IV reports that the PPP hypothesis and the monetary model based on the constant cointegration approach are limited in terms of being able to explain the movements of the exchange rate in Korea. Section V shows that Korean data are more favorable when used with the PPP and the monetary model based on the time-varying cointegration approach through various model specification tests. Section VI compares the capability to predict future exchange rate changes among the time-varying PPP, the time-varying monetary model, and a combination of the 1Studies such as Stock and Watson (1993) and Mulligan and Sala-i-Martin (2000) show that the money demand function has unstable coefficients. Clarida et al. (2000) and Kim and Nelson (2006) demonstrate that monetary policy rules have also shifted over time. 2As an exception, Kim and Jei (2013) and Kim et al. (2009) examine time-varying cointegration relationships among variables under the PPP hypothesis for Asian countries. VOL. 37 NO. 4 Time-varying Cointegration Models and Exchange Rate Predictability in Korea 3 two via in-sample and out-of-sample analyses. The in-sample analysis shows that the time-varying PPP shows better predictive power over horizons shorter than one year, whereas the time-varying monetary model outperforms when the horizons are longer than one year. The time-varying PPP shows better performance according to the results of the out-of-sample analysis. Concluding remarks are offered in Section VII. II. Theoretical Discussion: PPP and Monetary Model In this section, we present a brief discussion of the PPP hypothesis and the monetary model, and derive the time-varying cointegration version of these models. A. Purchasing Power Parity (PPP) The PPP hypothesis states that when goods are traded freely, the nominal exchange rate adjusts so that the goods must sell for the same price in both countries. This idea can be expressed by the following equation, (1) *, t t t P SA P  where t S is the nominal exchange rate, t P is the domestic price level (i.e., in Korea), and * t P is the foreign price level (i.e., in the US). A is a constant which captures trade barriers and difference in preferences between the two countries. When temporary deviations from the PPP relationship is allowed, Equation (1) can be re-expressed after taking logarithms, as follows: (2) * tttt sappε   where lower letters denote the logarithms of the corresponding capital letters, and t ε represents temporary deviations from the PPP. Although t s and * tt pp are known to be I(1) variables, * () ttt s pp must be stationary under the PPP, which implies a [1, -1] cointegration vector for the exchange rate and relative price. Many empirical studies have examined the validity of the PPP under constant cointegration coefficients, providing evidence that the PPP does not hold in most cases. However, this failure of the PPP may be due to the fact that the constant cointegration coefficient cannot reflect gradual changes in economies as opposed to a failure of the PPP in general. Hence, we also consider the following PPP relationship based on the time-varying cointegration coefficient, (3) * (), ttttt saβpp ε   where βt is the cointegration coefficient which varies smoothly over time, 4 KDI Journal of Economic Policy NOVEMBER 2015 capturing the effect of gradual changes in the economic environment on the PPP relationship. B. Monetary Model The monetary model has been examined in many studies, and there has been some debate regarding whether it accurately explains fluctuations in exchange rates. For example, Mark (1995) and Chinn and Meese (1995) show that the monetary model has significant predictive power for exchange rate movements, whereas Kilian (1999) and Berkowitz and Giorgianni (2001) provide evidence against exchange rate predictability based on the monetary model. The traditional monetary model examined in these studies can be summarized by the following four equations: (4) * 1 () ( ) tt t t t t Es s δii π    (5) * () ttt sβpp (6) ttt tt mp i y v      (7) *** ** ttt tt mp i yv     where mt and yt are the logarithms of the money supply and real income level, respectively. it is the level of the nominal interest rate. t π is the deviation from uncovered interest parity (UIP) or the unobserved risk premium, while t v and * t v denote unobserved velocity shocks. We assume that t π, t vand * t v follow stationary processes.3 γ and  correspondingly represent the elasticity of the money-demand income and the semi-elasticity of the money-demand interest rate. Under the assumption of ( ) -*2 tt m Δmm~i.i.d.(0,σ) and ( ) *2 tt y Δyy~i.i.d.(0,σ-) , these four equations can be combined to express the following cointegration relationship between t s , ( ) * tt m-m and ( ) * tt y-y : (8)  ** 11 tttttt 22 11 λλγ smm-yy+u -λ- =- - λ  Where 1 βδ λ=δ+ β  , =+ 2 β λδ β f f,i ttt ti i uλuE λu                  12 1  , and 3These unobserved processes are assumed to follow nonstationary processes in Engel and West (2005), as they were not able to find evidence of cointegration between the exchange rate and observable fundamentals. Using the time-varying cointegration approach, however, we find cointegration evidence of these variables, as shown in the next section. As a result, we assume stationarity with regard to these unobserved components. VOL. 37 NO. 4 Time-varying Cointegration Models and Exchange Rate Predictability in Korea 5 =+ t* ttt π u(- v-v) δ f. Equation (8) implies the -λλγ ,, -λλ       11 22 1 11 cointegration vector for t s ,   * tt mm, and   * tt yy, and this constant cointegration vector becomes [1, -1, 1] under the assumption of βδγ   1 . As shown in Park and Park (2013), however, data in advanced economies are not favorable when used with the monetary model based on constant cointegration coefficients. Hence, when the underlying parameters (δ, β, ϕ, and γ) are allowed to vary over time, we can derive the following time-varying cointegration relationship: (9)     ** tttt tttt sαmm αyy u    12  where tt t ttt βδ λδ β  1  , tt t ttt β λδ β  2   , itt t tjti ij αλ E[( λ)λ]      1 11 2 1 10 and it t t t t j ti ti ij αγλE[( λ)γλ ]     1 21 2 1 10 . III. Data and Econometric Methodology A. Data This study ascertains the validity of the PPP and monetary model in Korea. As a result, empirical analyses require data for the exchange rate (Korean won per US dollar) and data for macroeconomic variables such as the price index, money supply and real income levels for Korea and the US. Because the frequency of the data utilized in the analyses is monthly, industrial production is used for real income in both countries.4 The M1 money stock and the Consumer Price Index (CPI) are used for the money supply and price-level variables.5 Data for Korea and the US are obtained from the websites of the Bank of Korea and the Federal Reserve Bank of St. Louis, respectively.6 The data of the money supply and real income levels are seasonally adjusted, and the sample period covers the period between January of 1980 and April of 2015. 4Due to data availability, the index for mining and manufacturing industrial products is used for the Korean real income data. 5Although some studies advocate the use of the Producer Price Index (PPI) to examine the PPP hypothesis, our results are not sensitive regarding whether CPI or PPI data are used in the analyses. Results based on the PPI are available upon request. 6The web address for the Bank of Korea is http://ecos.bok.or.kr/. The web address for the Federal Reserve Bank of St. Louis is https://www.stlouisfed.org/. 6 KDI Journal of Economic Policy NOVEMBER 2015 B. Econometric Methodology Considering that the cointegration approach under constant cointegration is widely applied in empirical studies, we introduce briefly the time-varying cointegration approach as proposed by Park and Hahn (1999) in this subsection. Suppose that the following time-varying cointegration relationship holds between t s , 1t x , and 2t x , (10) tttttt sρρxρ x u    011 22 where t u denotes the cointegration residuals and 1t ρ and 2t ρ are the timevarying cointegration coefficients. Define smooth functions 1 ρ and 2 ρ on [0, 1] such that 11t t T     and 22t t T     , where  is the sample size. Under the assumption that 1 ρ and 2 ρ are smooth enough so that they can be approximated by a series of polynomials and/or trigonometric functions, Equation (10) can be written as follows: (11) tttttt t tt tt sρρxρxuρρ xρ x u TT            011 22 01 1 2 2 κκ ii t i i t κt ii tt ρθφxθφ x u TT                  12 11 2 2 01 2 11 '' κtκκtκκt ρχaχau   11 2 2 11 2 2 0 Where i φ 1 and i φ 2 are the corresponding series functions used to approximate ρ 1 and ρ 2 , 11 ' 11 1 κt1 1t tt χ,, x TT =                , 22 ' 22 2 t1 2t tt ,, x TT              , 11 111 1κ aθ,...,θ'     , 22 222 1κ aθ,...,θ'       , and 1 11 tt 1 ii 1t i1 tt uu x TT                    2 22 2ii2t i1 tt +x TT               . When t s, 1t x , and 2t x are nonstationary, canonical cointegration regression (CCR) offers better asymptotic results. Hence, Equation (11) under the CCR transformation becomes VOL. 37 NO. 4 Time-varying Cointegration Models and Exchange Rate Predictability in Korea 7 (12) 11 2 2 1' 1 2' 2 t0ktk ktkkt †††† usaa      Where 11 11 1 kt 1 k 1t †† tt ,..., x TT               , 22 22 2 kt 1 k 2t †† tt ,..., x TT             , and the superscript † denotes CCR transformed variables. Once the LS estimators for 1 1 κ a and 2 2 κ a in Equation (12) are obtained, ρ 1 and ρ 2 can then be approximated by  = 1 κ 11 ii i1 θφ ∑ and  = 2 κ 22 ii i1 θφ ∑ , respectively. Fourier Flexible Form (FFF) series functions, which include polynomials and trigonometric functions, are utilized to approximate ρ 1 and ρ 2 . IV. Assessment of Macroeconomic Models with Constant Cointegration Coefficients Before beginning any analysis to examine a cointegration relationship, we check whether variables under the PPP or monetary model are indeed nonstationary in Korea, as they are in other countries. For this purpose, we use the Augmented Dickey-Fuller (ADF) test and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test for both the level and the first difference of the exchange rate, relative money, relative income and relative price. As shown in Table 1, the unit root null hypothesis of the ADF test cannot be rejected for those variables in levels, while the unit root null hypothesis is rejected in the first difference of the variables. TABLE 1—UNIT ROOT TESTS FOR VARIABLES IN THE MONETARY MODEL AND PPP Variables ADF test KPSS test t s Level -2.780 (0.206) 0.177 First difference -6.090 (0.000) 0.043 () * tt m-m Level -0.439 (0.986) 0.553 First difference -5.602 (0.000) 0.089 ( ) * tt y-y Level -1.924 (0.640) 0.281 First difference -22.650 (0.000) 0.057 ( ) * tt p -p Level -1.717 (0.742) 0.410 First difference -4.720 (0.001) 0.086 Note: Numbers in parentheses are the MacKinnon (1996) one-sided p-values for the ADF test statistics. Each lag length is determined by the Akaike information criterion (AIC). The null hypothesis of the Kwiatkowski-Phillips-Schmidt-Shin test is of stationarity and the alternative is the presence of a unit root. The asymptotic critical value for the test statistics is from Kwiatkowski-Phillips-Schmidt-Shin (2002), and it is 0.146 at the 5% level. 8 KDI Journal of Economic Policy NOVEMBER 2015 Furthermore, the stationarity null hypothesis of the KPSS test can be rejected for all of these variables in levels, whereas the stationarity null hypothesis is not rejected for any of the variables in the first difference. These results, shown in Table 1, strongly suggest that all of the variables under the PPP or the monetary model can be considered to be integrated of order one individually. A. The PPP hypothesis with constant cointegration coefficients In order to assess the capability of the PPP hypothesis to explain movements of the exchange rate in Korea, we initially examine whether = PPP * tttt zs-(p-p) is stationary. That is, we investigate the validity of the [1, -1] cointegration vector between t s and -* tt (p p ). Hence, the ADF test is conducted for P PP t z. As shown in Table 2, the unit root null hypothesis can be rejected marginally for P PP t z. This implies that there exists a long-run relationship between t s and -* tt (p p ) based on the [1, -1] cointegration vector. When we remove the restriction on the cointegration vector, however, the plausibility of the PPP hypothesis is altered. That is, we subsequently address whether there exists any other constant cointegration vector between  and -* tt (p p ) rather than [1, -1]. For this purpose, the Engle-Granger test is conducted. The estimated cointegration coefficients between  and (pt  pt *) in the first step of the Engle-Granger test are as follows: ( ) =+ + * tttt s7.991.19 -p ˆ pε (0.14) (0.15) where numbers in parentheses are standard errors.7 TABLE 2—ASSESSMENT OF THE PPP WITH CONSTANT COINTEGRATION COEFFICIENTS Test Statistic 5% Critical Value ADF test for * ttt s -(p - p ) -2.9231 -2.8683 The Engle-Granger Test -3.1870 -3.3654 The Johansen Cointegration Test With Trend Trace Statistic 16.3009 25.8721 Max Eigenvalue Statistic 11.8701 19.3870 Without Trend Trace Statistic 14.4693 15.4947 Max Eigenvalue Statistic 11.8701 14.2646 Note: For the Engle-Granger test, () * t01tt t s=β+βp-p +ε is run in the first step, after which the ADF test is conducted for the residuals in the first step. The critical value for the ADF statistic is from Phillips and Ouliaris (1990). For the Johansen test, when the ‘no cointegration’ null is tested, a linear deterministic trend in the data is allowed and the computation of the critical values is based on MacKinnon-Haug-Michelis (1999) p-values. Lag intervals are selected by the AIC. 7Following Stock and Watson (1993), we employ dynamic least squares to have optimal estimates of cointegration parameters. The estimated results for first-difference terms are not reported to conserve space, but are available upon request. VOL. 37 NO. 4 Time-varying Cointegration Models and Exchange Rate Predictability in Korea 15 (13) ++ =+ + TVC tk 0,k k t tkt,t s-s γγzw where ( ) = TVC * ttttt zs-β p -p under the PPP, and ( ) = TVC * tt1ttt zs-αm-m - ( ) * 2t t t αy-y under the monetary model. As the exchange rate moves toward longrun equilibrium over time,  should be negative. The results of the in-sample analysis are presented in Table 5. k γ (the slope coefficient in the predictive regression) always has correct signs and is significant regardless of forecast horizons or regardless of the underlying model used. The PPP model has impressive predictive power for horizons shorter than one year. Specifically, the PPP model can explain 31% of the variation in the exchange rate at a six-month horizon. As horizons become longer, however, the monetary model shows better forecasting ability. We also run the following regression to determine whether the combination of the PPP and the monetary model improves the predictive power: (14) st+k-st=γ0,k+γ1,kzt TVC,1+γ2,kzt TVC,2+wt+k, t where ( ) = TVC ,1 * ttttt zs-β p -p and ( ) ( ) = TVC ,2 * * tt1ttt2ttt zs-αm-m -αy-y . As shown in the last three-columns of Table 5, 1,k γ is significant over shorter horizons, while 2,k γ is significant over longer horizons. However, the combination of the two models does not improve the predictive ability much over the PPP for short horizons and/or over the monetary model for long horizons. This implies that the time-varying PPP relationship is the dominant reason for the timevarying monetary model among four structural equations (Equations (4) – (7)) under the monetary model. TABLE 5—PREDICTIVE REGRESSIONS: IN-SAMPLE ANALYSIS Forecasting Horizon (k) Time-varying PPP Time-varying Monetary Model Time-varying PPP + Time-varying Monetary Model γk (T-statistics) R  2 γk (T-statistics) R  2 γ1,k (T-statistics) γ2,k (T-statistics) R  2 1 -0.2757 (-6.1937) 0.1006 -0.24 (-5.3517) 0.0988 -0.1712 (-3.4517) -0.1444 (-2.9295) 0.1199 6 -1.2317 (-6.8911) 0.3113 -0.8523 (-5.5743) 0.1967 -1.0305 (-3.8621) -0.2775 (-1.3152) 0.3215 12 -1.2195 (-7.6954) 0.1589 -1.2816 (-5.1457) 0.2226 -0.4897 (-2.2506) -1.0099 (-3.4466) 0.2348 24 -0.9074 (-5.6222) 0.0692 -0.9462 (-4.4534) 0.0865 -0.373 (-1.5247) -0.7385 (-2.6095) 0.0884 36 -0.8034 (-5.0077) 0.0704 -0.7703 (-5.4014) 0.0746 -0.4183 (-1.9073) -0.5307 (-2.7486) 0.0762 48 -0.9496 (-4.3616) 0.0885 -0.9224 (-5.817) 0.0935 -0.4826 (-1.733) -0.6403 (-3.0603) 0.0953 Note: Numbers in parentheses are t-statistics based on Newey-West standard errors. 16 KDI Journal of Economic Policy NOVEMBER 2015 B. Out-of-sample Analysis We also compare the out-of-sample performance of both models to that of the random walk without drift, which has been the benchmark for assessing the out-ofsample performance of exchange rate models in many studies since Meese and Rogoff (1983). In the out-of-sample analysis, time-varying cointegration errors are constructed using data up to January of 2005, and then the predictive regression in Equation (13) is run to estimate , and . Using the estimated values of , and  and the last observation of cointegration errors, forecasts are made for future changes in the exchange rate. We repeat these steps while keeping the window size constant. When comparing the out-of-sample performance of the time-varying PPP or the time-varying monetary model to that of a random walk model, we employ the Clark and West (2007) test statistic. The null hypothesis is that two competing forecasting models have an equal mean-squared prediction error. We construct the Clark-West test statistic so that it has a significantly positive sign if the regression model with time-varying cointegration errors exhibits superior predictive power in relation to the random walk model. Table 6 reports the test results. Unlike the impressive results in the in-sample analysis, the time-varying PPP and the time-varying monetary model outperform the random walk model at the 10% level only when the forecast horizon reaches 48 months. Even if the time-varying PPP and the time-varying monetary model are combined, the out-of-sample performance does not improve at all. This deterioration of the performance of both time-varying models in the out-of-sample analysis, however, may not result from the nature of those models. Instead, it may be related to the loss of power resulting from the smaller sample size for the estimation in the out-of-sample analysis, as emphasized by Inoue and Kilian (2004) and Bacchetta et al. (2010). This issue should be further investigated with more observations in the future. We also compare the out-of-sample performances of time-varying models with those of the counterparts based on constant cointegration models. The results are reported in Table 7. Consistent with Table 6, the Clark-West test statistic is designed to have a significantly positive sign if the regression model with timeTABLE 6—PREDICTIVE REGRESSIONS: OUT-OF-SAMPLE ANALYSIS: COMPARISON WITH RANDOM WALK Forecasting Horizon (k) Time-varying Monetary Model vs. Random Walk Time-varying PPP vs. Random Walk Time-varying Monetary Model + Time-varying PPP vs. Random Walk 1 -0.7897 (0.7852) 0.1875 (0.4256) -0.6504 (0.7423) 6 -0.2459 (0.5971) -2.7513 (0.997) -2.0785 (0.9812) 12 -0.7704 (0.7795) -0.9674 (0.8333) -0.7397 (0.7703) 24 -2.5407 (0.9945) -0.5072 (0.694) -2.4493 (0.9928) 36 0.2753 (0.3915) 1.2295 (0.1094) -0.2793 (0.61) 48 1.8133 (0.0349) 1.6084 (0.0539) 1.786 (0.0371) Note: Numbers in parentheses are p-values. VOL. 37 NO. 4 Time-varying Cointegration Models and Exchange Rate Predictability in Korea 17 TABLE 7— PREDICTIVE REGRESSIONS: OUT-OF-SAMPLE ANALYSIS: COMPARISON BETWEEN TIME-VARYING COINTEGRATION MODEL AND CONSTANT COINTEGRATION MODEL Forecasting Horizon (k) Time-varying Monetary Model vs. Constant Coefficient Monetary Model Time-varying PPP vs. Constant Coefficient PPP Time-varying Monetary Model + Time-varying PPP vs. Constant Coefficient Monetary Model + Constant Coefficient PPP 1 -2.4145 (0.9921) -2.7930 (0.9974) -0.6604 (0.7455) 6 -2.132 (0.9835) -2.7896 (0.9974) -1.6917 (0.9546) 12 0.082 (0.4673) -1.9335 (0.9734) 0.1603 (0.4363) 24 1.2522 (0.1053) 1.7249 (0.0423) 0.1027 (0.4591) 36 1.7973 (0.0361) 2.8229 (0.0024) 1.3525 (0.0881) 48 2.4844 (0.0065) 3.6397 (0.0001) 3.0417 (0.0012) Note: Numbers in parentheses are p-values. TABLE 8—PREDICTIVE REGRESSIONS: OUT-OF-SAMPLE ANALYSIS: COMPARISON BETWEEN TIME-VARYING MODELS Forecasting Horizon (k) Time-varying PPP vs. Time-varying Monetary Model Time-varying PPP vs. Time-varying Monetary Model + Time-varying PPP P PP M RMSE RMSE Diebold-Mariano statistic + PPP M PPP RMSE RMSE Diebold-Mariano Statistic 1 0.9478 1.2387 (0.1077) 0.9743 0.8406 (0.2003) 6 1.039 -0.7739 (0.7805) 1.0131 -0.9156 (0.8201) 12 0.9232 1.8169 (0.0346) 0.9181 2.8012 (0.0025) 24 0.9246 1.5419 (0.0615) 0.94 2.3414 (0.0096) 36 0.8960 1.9954 (0.023) 0.8681 2.3003 (0.0107) 48 0.9835 0.2582 (0.3981) 0.9781 0.6961 (0.2432) Note: Numbers in parentheses are p-values. varying cointegration errors exhibits superior predictive power to the corresponding constant cointegration model. The constant cointegration approach shows significantly better out-of-sample performances in short horizons than the time-varying cointegration approach, regardless of underlying macroeconomic models. As the forecast horizon increases, however, the time-varying models show better out-of-sample forecast ability regardless of underlying macroeconomic models. The superiority of the predictability from the time-varying model over the counterparts from the constant cointegration model becomes significant at horizons longer than one or two years. Finally, we compare the out-of-sample performances among the time-varying PPP, the time-varying monetary model, and the combination of the two. The first two columns of Table 8 show horse race results between the time-varying PPP and 18 KDI Journal of Economic Policy NOVEMBER 2015 the time-varying monetary model. The time-varying PPP outperforms the timevarying monetary model at all horizons except for the six-month horizon, and the gap in the forecast performance is significant at 12–36 month horizons according to the Diebold-Mariano test. Similarly, the time-varying PPP always shows better outof-sample performance against the combination of the two models except with a six-month horizon, as shown in the last two columns of Table 8. Again, the superior performance of the time-varying PPP relative to the combined model is significant at 12–36 month horizons according to the Diebold-Mariano test. Although the time-varying PPP shows the best out-of-sample performance, the results should be interpreted with caution, as argued by Inoue and Kilian (2004) and Bacchetta et al. (2010). VII. Discussion This paper shows that when cointegration coefficients are allowed to vary over time, both the PPP and the monetary model can pass various specification tests, implying that macroeconomic variables based on those models are tightly linked with the exchange rate in Korea. When the abilities to predict future exchange rates between those models based on time-varying cointegration coefficients are compared, the in-sample analysis shows that the time-varying PPP (monetary model) shows better predictive power with horizons shorter (longer) than one year. The results of the out-of-sample analysis indicate that the time-varying PPP performs better when used to predict future changes in the exchange rate. In addition to these findings, the movements of time-varying coefficients appear to have some signaling power for the Korean economy. The time-varying cointegration coefficient based on the PPP increased around the periods of the Asian currency crisis and the global financial crisis, which may reflect the drastic depreciation of the Korean currency around the time of those crises. The seemingly upward-sloping trend in the time-varying coefficients based on the PPP in Figure 1 suggests a depreciation of the real exchange rate resulting from the slowdown of the growth in the Korean economy.10 The time-varying coefficients based on the monetary model in Figures 2 and 3 also behaved abnormally around these two crises. The coefficients of the relative money (for the relative income) are usually positive (negative) as the theoretical model predicts, but they became negative (positive) around the two crises, implying a drastic depreciation of the Korean currency as compared with fundamentals at those times. REFERENCES Andrews, D. K. 1991. “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.” Econometrica 59: 817–858. Bacchetta, P., E. van Wincoop. 2013. “On the Unstable Relationship between Exchange Rates and Macroeconomic Fundamentals.” Journal of International Economics 91: 18–26. 10According to the Balassa-Samuelson hypothesis, a country with an increase in productivity relative to other countries experiences a rise in its price level and real appreciation. VOL. 37 NO. 4 Time-varying Cointegration Models and Exchange Rate Predictability in Korea 19 Bacchetta, P., E. van Wincoop, and T. Beutler. 2010. “Can Parameter Instability Explain the Meese-Rogoff Puzzle?” NBER International Seminar on Macroeconomics 2009: 125–173. Berkowitz, J., L. Giorgianni. 2001. “Long Horizon Exchange Rate Predictability?” Review of Economics and Statistics 83: 81–91. Bierens, H., L. Martins. 2010. “Time-Varying Cointegration.” Econometric Theory 26: 1453– 1490. Cheung, Y., M. D. Chinn. 2001. “Currency Traders and Exchange Rate Dynamics: A Survey of the US Market.” Journal of International Money and Finance 20: 439–471. Chinn, M. D., R. A. Meese. 1995. “Banking on Currency Forecasts: How Predictable is Change in Money?” Journal of International Economics 38: 161–178. Clarida, R., J. Gal´ı, and M. Gertler. 2000. “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory.” Quarterly Journal of Economics 115: 147–180. Clark, T. E., K. D. West. 2007. “Approximately Normal Tests for Equal Predictive Accuracy in Nested Models.” Journal of Econometrics 138: 291–311. Engel, C., K. D. West. 2005. “Exchange Rates and Fundamentals.” Journal of Political Economy 113: 485–517. Engle, R. F., C. W. J. Granger. 1987. “Co-Integration and Error Correction: Representation, Estimation and Testing.” Econometrica 55: 251–276. Inoue, A., L. Kilian. 2004. “In-sample or out-of-sample tests of predictability: Which one should we use?” Econometric Reviews 23: 371–402. Johansen, S. 1988. “Statistical Analysis of Cointegrating Vectors.” Journal of Economic Dynamics and Control 12: 231–254. Johansen, S., K. Juselius. 1990. “Maximum Likelihood Estimation and Inference on Cointegration – With Applications to the Demand for money.” Oxford Bulletin of Economics and Statistics 62: 169–210. Kilian, L. 1999. “Exchange Rates and Monetary Fundamentals: What Do We Learn from LongHorizon Regressions?” Journal of Applied Econometrics 14: 491–510. Kim, B. H., H. K. Kim, and K. Y. Oh. 2009. “The Purchasing Power Parity of Southeast Asian Currencies: A Time-varying Coefficient Approach.” Economic Modelling 26: 96–106. Kim, H. G., S. Y. Jei. 2013. “Empirical Test for Purchasing Power Parity Using a Time-varying Parameter Model: Japan and Korea cases.” Applied Economics Letters 20: 525–529. Kim, C., C. Nelson. 2006. “Estimation of a Forward-Looking Monetary Policy Rule: A TimeVarying Parameter Model using Ex-Post Data.” Journal of Monetary Economics 53: 1949– 1966. Kwiatkowski, D., P. C. Phillips. P. Schmidt., and Y. Shin. 1992. “Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root: How Sure are We that Economic Time Series Have a Unit Root?” Journal of Econometrics 54: 159–178. MacKinnon, J. G. 1996. “Numerical Distribution Functions for Unit Root and Cointegration Tests.” Journal of Applied Econometrics 11: 601–618. MacKinnon, J. G., A. A. Haug, and L. Michelis. 1999. “Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration.” Journal of Applied Econometrics 14: 563–577. N. C. Mark. 1995. “Exchange Rates and Fundamentals: Evidence on Long-horizon Predictability.” The American Economic Review 85: 201–218. Meese, R. A., K. Rogoff. 1983. “Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?” Journal of International Economics 14: 3–24. Mulligan, C. B., X. Sala-i-Martin. 2000. “Extensive Margins and the Demand for Money at Low Interest Rates.” Journal of Political Economy 108: 961–991. Park, C., S. Park. 2013. “Exchange Rate Predictability and a Monetary Model with Timevarying Cointegration Coefficients.” Journal of International Money and Finance 37: 394– 410. Park, J. Y. 1992. “Canonical Cointegration Regressions.” Econometrica 60: 119–143. Park, J. Y., S. B. Hahn. 1999. “Cointegrating Regression with Time Varying Coefficients.” Econometric Theory 15: 664–703. Phillips, P. C. B., S. Ouliaris. 1990. “Asymptotic Properties of Residual Based Tests for Cointegration.” Econometrica 58: 165–193. 20 KDI Journal of Economic Policy NOVEMBER 2015 Shin, Y. 1994. “A Residual-based Test of the Null of Cointegration Against the Aalternative of No Cointegration.” Econometric Theory 10: 91–115. Stock, J., M. Watson. 1993. “A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems.” Econometrica 61: 783–820.