1. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Searching for Nonlinearities in Real Exchange Rates By Yamin S. Ahmad † and Stuart Glosser ‡ † ‡ University of Wisconsin – Whitewater †† Corresponding Author: Dept of Economics, University of Wisconsin Whitewater, 800 W Main Street, Whitewater, WI 53190 Email: ahmady@uww.edu, Homepage: http://facstaff.uww.edu/ahmady/ Tel: + 1 (262) 472 5576 ‡‡ Dept of Economics, University of Wisconsin Whitewater, 800 W Main Street, Whitewater, WI 53190 Email: glossers@uww.edu, Tel: +1 (262) 472 5580

2. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 2 What Is This Paper About…? The question we address: How well are threshold models (in particular ESTAR models) able to capture the dynamics of real exchange rates given their recent success in the literature? Our Key Findings: The appropriateness of the ESTAR model depends on the country in question  Concern about power of Terasvirta’s LM test for STAR nonlinearity Underlying thresholds may not be within the exchange rate domain but rather within variables that pertain to macroeconomic policy  Overnight interest spread (Dollar-Sterling)  CPI spread (Dollar-Lira)

3. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 3 Related Literature Do bilateral real exchange rates follow a random walk?  Early literature focused on Unit Root tests and Cointegration  The PPP puzzle (Rogofff, 1996) Are bilateral real exchange rates nonlinear stationary?  Nonlinearities arise because of frictions that arise. Some examples are transactions costs (Michael, Nobay, and Peel, 1997) and noise traders (Killian and Taylor, 2003).  The presence of the frictions serves to create a wedge in price dynamics by creating “bands of inaction” where changes in the real exchange rate are merely random.  When the deviation of the real exchange rate from parity exceeds the threshold, goods arbitrage occurs and the real exchange rate reverts back towards the band.

4. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 4 Related Literature Threshold Autoregressive Models (TAR) The evolution of real exchange rates may exhibit different dynamic characteristics (regimes) depending on some underlying state as determined by the magnitude of a threshold variable. These regimes are characterized by past values of the threshold variable such as the real exchange rate or by macroeconomic variables related to the real exchange rate. These regimes are distinct from each other. Examples of papers that incorporate a TAR specification in examining exchange rates include, Obstfeld and Taylor (1997), Canjels, Prakash-Canjels and Taylor (2004), Obstfeld & Taylor (1997), Sarno, Chowdhury and Taylor (2004), Zussman (2003).

5. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 5 Related Literature Smooth Transition Threshold (STAR) Models The abrupt changes in regimes implied by the TAR model may be unrealistic. Smooth transition between regimes may be more realistic. Two variants of the the smooth transition autoregressive (STAR) model are the exponential STAR (ESTAR) and the logistic STAR (LSTAR). The ESTAR model have been successfully able to explain some observed characteristics of real exchange rates.  Examples of papers that incorporate the STAR and ESTAR specification include Michael, Nobay and Peel (1997), Taylor, Peel and Sarno (2001), Kilian and Taylor (2003), Sarno, Chowdhury and Taylor (2004), and Lothian and Taylor (2004).

6. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 6 Related Literature Threshold Models Other work has examined STAR models and found that the presence of “aberrant” observations can substantially distort the distributional properties of the LM type tests used to reject linearity (Escribano et al., 1998, van Dijk et al., 1999) The presence of additive outliers (due to data contamination, or because of aberrant observations), may lead researchers following a Teräsvirta(1994) type rule to incorrectly reject linearity and conclude nonlinear dynamics.

7. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Hence In This Paper… We examine the ability of nonlinear adjustment mechanisms to capture the dynamics of exchange rates. In particular, we investigate the ability of the ESTAR framework to explain the dynamics of five spot dollar real exchange rates. 7

8. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 8 The ESTAR Model Suppose y t represents the real exchange rate, or equivalently, deviations from PPP. The ESTAR model assumes symmetric adjustment because economic intuition suggests that both positive and negative deviations from PPP would have symmetric effects on the speed of adjustment (1)

9. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 9 The ESTAR Model The important parameters in this form are and *. It is easy to see from this specification that if there is any evidence of nonstationary behavior, either  or * will be greater than (or equal to) zero. However, if the outer regime is stationary and the inner regime nonstationary, then we must have *<0 and  + *<0. In other words, if deviations from PPP are small, then y t may follow a unit root process or may even be nonstationary. For large deviations from PPP, then the process for y t is mean reverting. Re-writing in deviation form: (2)

10. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 10 ESTAR Estimation Methodology Methodology 1.Pick the order of the autoregression, p.  Look at some type of information criteria, e.g. AIC, SBC  Examine the partial autocorrelation function (PACF) 2.Test null hypothesis of linearity (STAR tests) I.Pick a value for the delay parameter, d  Test null hypothesis of linearity by testing restrictions on coefficients of an auxiliary equation based on a third order Taylor expansion of the transition equation. II.Go back to step I and do for another value of d III.Pick the value of d that minimizes the p-value 3.If evidence points to nonlinear (STAR) process:  Choose between LSTAR and ESTAR transition functions based on results of STAR tests  Estimate parameters through nonlinear least squares

11. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 11 The spot dollar real exchange rates (rebased to 1973:1)

12. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 12 Augmented Dickey-Fuller and Phillips-Perron tests

13. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 13 Benchmark Linear Model Lag length selection based on PACF, AIC and SBC criteria. The values of the Ljung-Box Q statistics above show that the lag length selected for each country would suggest that the model is not underspecified.

14. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 14 Estimation of the linear AR(p) model :

15. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 15 Linearity Tests Teräsvirta’s (1994) Decision Rule: Min p- value at H 0,1 or H 0,3  Pick LSTAR H 0,2  Pick ESTAR

16. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 16 ESTAR Results (Unrestricted)

17. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Summary of ESTAR Results US-France  Evidence of stationary behavior in the outer regime and the dollar- franc real exchange rate appears to be nonlinear and globally stationary  Hypothesis that + *  0 is rejected at a 0.008 level of significance US-UK  Random walk type behavior in the inner regime  Parameters are significant in inner regime only!  Unable to reject the null hypothesis that either = *= 0 17

18. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Summary of ESTAR Results US-Italy  The hypothesis that + *  0 is rejected at 8% level of significance.  Skeptical about results since they are sensitive to the initial vector of parameters used to initiate the nonlinear least squares search.  Changing the starting value for the threshold parameter, c 0, yields a different ESTAR specification with a lower RSS.  However, parameter estimates appear to be counter-intuitive: is significantly less than zero and * is significantly greater than zero.  Problem: there is no underlying economic theory that allows us to discard the alternative specification that was estimated, apart from the idea that it runs contrary to the way we would expect real exchange rates to behave under this framework. 18

19. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 19 Key Findings From Testing For and Estimating an ESTAR Model 1.The STAR estimation methodology that we follow suggests the presence of nonlinearities in three of the five real exchange rates. 2.Only the dollar-franc rate appears to conform to the ESTAR specification as nonlinear stationary.

20. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 20 A Simulation Exercise… Simulation To investigate further, we conduct the following exercise: 1.Estimate a linear AR(p) model of the dollar-sterling real exchange rate, and keep coefficients 2.Using estimated coefficients, create pseudo-data that has the characteristics of the actual real exchange rate (i.e. mean, variance, and sample size), but where the underlying DGP is linear 3.For each trial, implement the ESTAR methodology to see if:  Linearity is accepted or rejected  Whether the methodology indicates (falsely) an ESTAR or LSTAR, given that it rejected linearity 4.Repeat for 5000 trials.

21. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Simulation Results 21 There are substantial distortions in the size of the test. Given a true null hypothesis that the data generating process is linear, the true linear null is rejected in favor the alternative of nonlinearity at a rate substantially higher than that of 5 percent.

22. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 22 Power of Terasvirta’s LM test? Perhaps the ESTAR model and the diagnostic tests suggested by Teräsvirta (1994) are falsely rejecting the linearity hypothesis. To investigate this possibility we analyze the standardized residuals of the linear model to look for outliers.

23. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 23 Significant Outliers Boxes represent outliers greater than 3 standard deviations.

24. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Small Sample in Outer Regime? If the influential observations depicted above represent valid observations in the “outer” regimes of a true STAR process:  the results would indicate we only have a few observations by which to make inferences about parameters in that regime. Lack of observations imply that any estimation procedure would be subject to small sample biases (Ahmad, 2008). 24

25. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 25 Does this mean that Escribano, Franses, and van Dijk (1998) were correct? “…The fact that a large part of these data points coincide for the different series strengthens the conclusion that they do not signal intrinsic nonlinearity in the process generating real exchange rates, but rather are caused by some aberrant exogenous events. Hence, the presence of transactions costs do not seem to imply nonlinear behavior of real exchange rates.” (P. 17)

26. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Removing the influential observations 26 According to the above table: Nonlinearity is rejected The results suggest an LSTAR model However, upon estimation, we are unable to reject + * ≥ 0.

27. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 An Alternative Nonlinear Specification: Open Loop TAR Models  Instead of using the real exchange rate as the threshold variable, we look to economic theory to determine the threshold.  According to the monetary model of exchange rates, exchange rates are determined by monetary policy considerations.  Therefore we consider:  Interest rate spreads (Dollar-Sterling)  The spread in price levels (Dollar-Lira) 27

28. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 TAR Methodology Tsay (1989): Use arranged auto regression and recursive estimation to test for Threshold Nonlinearity 28

29. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Dollar-Sterling TAR Model The spread between the federal funds rate and the U.K. overnight rate is used as the threshold variable. Results of TSAY’s Test: F( 4, 382 )=4.80047 P= 8.7e-04 Delay (d) = 1 month Threshold value of spread = 4.66% 29

30. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Dollar-Sterling TAR Model 30 X t is the previous month’s spread between the federal funds rate and the U.K. overnight rate.

31. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Dollar-Lira The spread between the log of the U.S. and Italian CPI is used as the threshold variable. Results of TSAY’s Test: F( 5, 298 )= 2.7 P=.021 Delay (d) = 4 months Threshold value of spread = 100.6 31

32. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Dollar-Lira 32 X t is the spread the log of the U.S. and Italian CPI lagged 4 months

33. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 33 Conclusions The underlying methodology that is used to detect nonlinearity and STAR type behavior, suggests that three of the five real exchange rate series may be modeled as an exponential STAR (ESTAR) process. However, we find that of the three real exchange rate series, only the dollar-franc conforms to an ESTAR specification The dollar-sterling and the dollar-lira do not conform to any STAR type specification, even when influential observations are removed.

34. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 Conclusions (cont.) We investigate the dollar-sterling and dollar-lira real exchange rates further and find that they are best modeled as open-loop TAR processes.  The underlying threshold does not appear to be within the exchange rate domain, but rather within variables that pertain to monetary policy: in the overnight interest rate spread for US-UK and in the CPI spread for US-Italy. We question whether the STAR framework adequately captures the nonlinear nature of real exchange rates because of both the power and the size of the test. Real exchange rates exhibit nonlinear dynamics. However, based on our findings, we suggest that we may need to need to look at macroeconomic theory to determine where threshold points may lie, rather than assuming that they lie within the domain of lagged real exchange rates. 34

35. Ahmad & Glosser, UW – Whitewater, Econometric Society Summer Meetings, 2008 The End 35