Exchange Rate Forecasting: Results from a Threshold Autoregressive Model

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<ul><li><p>P1: SMA</p><p>Open economies review KL566-03-Pippenger January 27, 1998 17:23</p><p>Open economies review 9: 157170 (1998)c 1998 Kluwer Academic Publishers. Printed in The Netherlands.</p><p>Exchange Rate Forecasting: Resultsfrom a Threshold Autoregressive Model</p><p>MICHAEL K. PIPPENGER AND GREGORY E. GOERINGAssociate Professor, Department of Economics, School of Management, P. O. Box 756080,University of Alaska, Fairbanks AK 99775-6080, USA</p><p>Key words: exchange rates, threshold autoregression, forecasting</p><p>JEL Classication Number: F3, C5</p><p>Abstract</p><p>Structural models of exchange rate determination rarely forecast the exchange rate more accu-rately than a naive random walk model. Recent innovations in exchange rate modeling indicate thatchanges in the exchange rate may follow a self-exciting threshold autoregressive model (SETAR).We estimate a SETAR model for various monthly US dollar exchange rates and generate forecasts forthe estimated models. We nd: (1) nonlinearities in the data not uncovered by the standard nonlinear-ity tests and (2) that the SETAR model produces better forecasts than the naive random walk model.</p><p>Structural models of exchange rate determination typically produce exchangerate forecasts inferior to a naive random walk model.1 This nding leads manyto conclude that open economy macro-models of exchange rate determinationare at best incomplete. However, this result is perhaps not surprising when oneviews the exchange rate as an asset price.</p><p>As with many asset prices, exchange rates appear to contain signicantnonlinearities when higher frequency data are examined. In particular, dailyand weekly exchange rate data seem to contain conditional heteroskedasticity.However, many authors, such as Baillie and Bollerslev (1989), nd that nonlin-earities are typically not found when monthly or annual data are tested.</p><p>The standard non-linear model employed in exchange rate modeling is theAutoregressive Conditional Heteroskedasticity (ARCH) class model. A numberof pre-test procedures have been developed to test for ARCH processes. Inparticular, Engle's (1982) Lagrange Multiplier (LM) test can be used to detectnonlinearity and is typically used to test for ARCH. One criticism of the ARCHmethodology is that while there are theoretical implications of ARCH processes,(e.g., a time varying risk premium in asset pricing models), the occurrence ofARCH in economic data is, in general, not implied by theory.2</p><p>Another type of nonlinearity that may occur in economic data is the thresholdautoregressive process.3 Unlike ARCH and its variants, threshold processes are</p></li><li><p>P1: SMA</p><p>Open economies review KL566-03-Pippenger January 27, 1998 17:23</p><p>158 PIPPENGER AND GOERING</p><p>frequently implied by economic theory. In particular, an exchange rate targetzone would lead to a threshold type process for the nominal exchange rate(see Krugman, 1991). Similarly, exchange rate management such as leaningagainst the wind'' may lead to a threshold type process for changes in thenominal exchange rate.4 Also, transaction costs may lead to a threshold typeprocess for the real exchange rate.5</p><p>First, we nd that while ARCH nonlinearity is not detected in log changes inthe nominal exchange rate, threshold nonlinearity is found.6 We then build aself-exciting threshold autoregressive (SETAR) model and show that, in general,the SETAR model forecasts changes in the exchange rate superior to the naiverandom walk model.</p><p>Our paper is organized as follows. Section one motivates and discusses theuse of the specic SETAR model employed and outlines the identication andestimation procedures used in SETAR model building. Section two discussesthe data used, presents the estimated models, and shows that the ARCH non-linearity tests commonly used may fail to detect threshold nonlinearity. Sectionthree contains the forecasting properties of the estimated SETAR models andcompares those forecasts to the naive random walk model. Our conclusionsare contained in section four.</p><p>1. The SETAR model: Motivation, identication and estimation</p><p>The self-exciting threshold autoregressive (SETAR) model has the following gen-eral form:</p><p>xt D a j0 CkX</p><p>iD1a</p><p>ji xti C h j"t if r j1 &lt; xtd r j ; (1)</p><p>where r0 &lt; r1 &lt; r2 &lt; &lt; rl1 &lt; rl . The SETAR process in (1) is a piece-wise linear AR.k/ process and process switching depends upon the thresholdparameter r j and the value of xtd where d is the delay. In this general case,there are l regimes. This model is succinctly denoted as SETAR.lI k/.</p><p>The motivation for applying the SETAR model to log changes in the nominalexchange rate is straight forward. When a central bank follows a foreign ex-change market intervention rule based on the magnitude of previous changesin the nominal exchange rate (e.g., leaning against the wind''), the time se-ries process for the exchange rate will switch when intervention occurs. Thesimplest intervention rule would be a two sided rule where intervention occurswhen the change in the exchange rate is considered by the central bank as toolarge'' in absolute value. Hence, the specic SETAR model we examine is theSETAR(3; 1):7</p><p>xt D a10 C a11 xt1 C "t if 1 &lt; xt1 r1xt D a20 C a21 xt1 C "t if r1 &lt; xt1 r2 (2)xt D a30 C a31 xt1 C "t if r2 &lt; xt1 C1;</p></li><li><p>P1: SMA</p><p>Open economies review KL566-03-Pippenger January 27, 1998 17:23</p><p>EXCHANGE RATE FORECASTING 159</p><p>where r1 &lt; r2 and xt represents changes in the log exchange rate. Thus, as (2)indicates, we have chosen a three regime model with homoskedastic errors.Also, "t is assumed to be Gaussian. When the percentage change in the ex-change rate is larger than r2 or smaller than r1, the central bank intervenes inthe foreign exchange market in the next time period causing the process for theexchange rate to shift. Under a simple intervention rule, as long as the percent-age change in the exchange rate is between r1 and r2 the central bank does notintervene in the foreign exchange market.8</p><p>If the threshold parameters are known prior to model building, estimationis quite simple. First, re-order the data according to xt1 and segment thedata according to the threshold parameters r1 and r2. Finally, estimate thecoefcients for each regime via OLS.9 However, if the threshold parametersare not known, a number of techniques have been developed to identify theseparameters.10</p><p>The rst technique we discuss is the recursive regression technique of Tsay(1989). The data series of n paired observations .xt ; xt1/ are obviously orderedaccording to time. Step 1 of the procedure re-orders the n pairs of obser-vations according to the size of xt1. Denote these new paired observationsX D .x ; x1/. Next a recursive regression is estimated of the form:</p><p>x D b0 C b1x1 C ! ; (3)</p><p>where no longer represents time but rather the th paired observations ofthe reordered series. Next plot the t-ratio of the recursive estimates of theautoregressive coefcient against x1. Under ideal conditions, if the modelis linear the recursive t-ratio should monotonically increase in absolute valuewhen plotted with respect to x1. Under a two threshold SETAR model therecursive t-ratio should initially monotonically increase in absolute value untilthe process switches. At the point where the process switches, the recursivet-ratio should change direction and this turning point will indicate the regionwhere the rst threshold parameter should lie. After this turning point, the t- ratioshould monotonically increase in absolute value until the next process switchoccurs. Once again the t-ratio should change direction and this turning pointshould indicate the region in which the second threshold parameter should lie.Simulation experiments indicate that this procedure is fairly effective at ndingthe threshold parameters in simulated data. However, one problem with thistechnique is that if the rst threshold occurs near the beginning of the re-ordereddata set, this procedure fails to provide a clear indication of the location of therst threshold.</p><p>Another technique for identifying threshold processes uses the Akaike In-formation Criterion (AIC) as suggested by Tong (1983). Consider the possiblethresholds r1 and r</p><p>2 . Using these threshold parameters and the re-ordered data</p><p>.x ; x1/ estimate the regression parameters for each of the three regimes given</p><p>by r1 and r2 . Next calculate the AIC under each of the three regimes denoted</p><p>as AIC1.r1 /; AIC2.r1 ; r2 /; and AIC3.r</p><p>2 /. Let AIC.r</p><p>1 ; r2 /DAIC1.r1 /CAIC2.r1 ; r2 /</p></li><li><p>P1: SMA</p><p>Open economies review KL566-03-Pippenger January 27, 1998 17:23</p><p>160 PIPPENGER AND GOERING</p><p>CAIC3.r2 /. Then, for all values for r1 and r2 choose as the threshold parametersthe values of r1 and r2 which minimizes AIC.r1 ; r</p><p>2 /.</p><p>11</p><p>We use a combination of the two techniques outlined above. First we employthe Tsay (1989) recursive t-stat technique to identify the likely regions where thethreshold parameters lie and then over these regions nd the values for r1 andr2 which minimizes AIC.r1 ; r</p><p>2 /.</p><p>2. Data and estimates</p><p>We use monthly end of period US dollar exchange rates for Austria, Belgium,Canada, Denmark, France, Germany, Ireland, Italy, Japan, Netherlands, Norway,Switzerland, and the UK from the International Monetary Fund's InternationalFinancial Statistics on CD-ROM. Exchange rates are denominated in units offoreign currency per US dollar. The period under examination is from March1979 to December 1991. The data are transformed by taking the natural logand rst differencing.</p><p>Monthly data are used for two reasons. First, this data frequency simpliesthe choice of a delay parameter since exchange rate intervention will occur inless than one month. If daily or even weekly data are used, choice of the delayparameter may not be so clear cut thus adding complication to the identicationprocedure. The other reason monthly data are used is that higher frequency datamay contain other types of nonlinearity in addition to that attributable to thresh-olds. In particular, ARCH may also occur in higher frequency data thus com-plicating the model building process. Furthermore, monthly data allows us toexamine the ability of the standard ARCH tests to detect threshold nonlinearity.</p><p>The time period was chosen to coincide with the advent of the EuropeanMonetary System (EMS). During all of this period Belgium, Denmark, France,Germany, Ireland, Italy, and the Netherlands participated in the EMS throughoutthe sample period. Hence, at least for these countries, US dollar exchange ratesshould follow a consistent process over the time period under examination.</p><p>The plots of the recursive t-ratios for Canada and Japan failed to indicatethe presence of thresholds. However, for the remaining countries the recursivet-ratios indicate threshold nonlinearity and the likely regions for two thresholds.For example, gure 1 below depicts the recursive t-ratios of the estimated AR(1)coefcient for Germany.12</p><p>Figure 1 illustrates the recursive t-ratios for Germany plotted against laggedchanges in the exchange rate. The gure illustrates the instability of the t-ratio over the initial part of the data which makes the identication of the rstthreshold difcult using only the recursive t-ratio technique. This instability is at-tributable to the small number of observation used to estimate the initial t-ratios.However, the region for the upper threshold is clearly indicated as around 0.03.</p><p>The general pattern shown by gure 1 is found for all the remaining coun-tries. Over these regions we conducted a grid search and used as thresholdparameters the values which minimized the AIC as discussed above.</p></li><li><p>P1: SMA</p><p>Open economies review KL566-03-Pippenger January 27, 1998 17:23</p><p>EXCHANGE RATE FORECASTING 161</p><p>Table 1. Preliminary tests of monthly US dollar exchange rates.</p><p>Box-Ljung McLeod-Li Kurtosistest test LM-test test</p><p>Austria 23.2 2.55 0.249 1.31</p><p>Belgium 24.2 1.66 0.245 1.46</p><p>Denmark 27.0 1.11 0.004 0.21</p><p>France 23.2 3.53 1.31 1.75</p><p>Germany 22.3 1.94 0.192 1.21</p><p>Ireland 21.5 1.81 0.449 1.03</p><p>Italy 28.3 1.74 0.0002 0.76</p><p>Netherlands 25.3 1.15 0.002 1.09</p><p>Norway 27.6 2.08 0.589 3.83</p><p>Switzerland 21.7 1.47 0.075 0.28</p><p>UK 17.3 7.03 1.37 1.83</p><p>Note: The Box-Ljung test is distributed 2(24)10% critical value: 33.2.The McLeod-Li test is distributed 2(4)10% critical value: 7.78 LM-testis distributed 2(1)10% critical value: 2.71. The Kurtosis test statisticis distributed standard normal. Data are in log changes.</p><p>Figure 1. Recursive T-statisticsGermany.</p><p>Table 1 contains the results of the standard nonlinearity tests for log changesin the exchange rate for the various countries as well as the estimated kurtosis.13</p><p>These tests, in effect, test the adequacy of the standard random walk modelfor the log exchange rate.14 Also, the Box-Ljung Q-test for white noise wasapplied to log changes in nominal exchange rates (see Ljung and Box, 1978).The McLeod-Li Q-test for nonlinearity and the Engle's Lagrange Multiplier (LM)</p></li><li><p>P1: SMA</p><p>Open economies review KL566-03-Pippenger January 27, 1998 17:23</p><p>162 PIPPENGER AND GOERING</p><p>test for ARCH were conducted and the data was tested for excess kurtosis.The McLeod-Li tests applies the Box-Ljung Q-test to the random walk modeltesting for up to 4th order autocorrelation in the squared residuals. The LM-testtests for ARCH(1) errors in the random walk model.15</p><p>The log rst difference of the nominal exchange rate appears to follow awhite noise process in every case. The kurtosis test indicates excess kurtosisonly for Norway and possibly for France and the UK. More importantly, neitherthe McLeod-Li nor the LM-test indicates the existence of ARCH type nonlin-earity. This conrms the ndings of previous studies that low frequency datatypically does not seem to contain ARCH. For example, Baillie and Bollerslev(1989) only nd ARCH in high frequency exchange rate data. However, whilethese tests may have power in detecting ARCH type nonlinearity, as shown byGoering and Pippenger (1994) these tests may have low power under this typeof threshold alternative. Thus, although monthly exchange rate data fails to in-dicate ARCH type nonlinearity, the recursive t-ratios strongly indicate thresholdnonlinearity. For example, although the t-ratios of Germany in gure 1 indicatethe presence of thresholds, the McLeod-Li and LM-tests for Germany do notdetect this threshold nonlinearity. This suggests that if researchers only usestandard ARCH tests, they may mistakenly conclude nonlinearity does not playa signicant role in exchange rate determination when low frequency data areexamined.</p><p>Table 2 contains the estimated threshold parameters as well as nonlinearityand excess kurtosis tests of the residuals from the estimated threshold model.</p><p>Using the Box-Ljung Q-test, autocorrelation was not detected for any of themodels at the 10% signicance level.16 Also, Table 2 indicates that ARCH typeor similar nonlinearity is not detected in the residuals of the threshold models.</p><p>Table 2. Threshold estimates and nonlinearity tests.</p><p>Upper Lower Box- McLeod-Li Kurtosisthreshold threshold Ljung test test LM-test test</p><p>Austria 0.01966 0.02969 25.2 0.827 0.24...</p></li></ul>


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