Non-Linear Forecasting using a Taylor Rule Based Exchange Rate Model

Non-Linear Forecasting using a Taylor Rule Based Exchange Rate Model By Rudan Wang (Department of Economics, University of Bath) Bruce Morley* (Department of Economics, University of Bath) Michalis P. Stamatogiannis (Management School, University of Liverpool) Address for correspondence: Department of Economics, University of Bath, Bath, BA2 7AY, UK. Tel. +44 1225 386497, E-mail. Bm232@bath.ac.uk, Fax. +44 1225 383423. 1

Non-Linear Forecasting using a Taylor Rule Based Exchange Rate Model Abstract This research utilises a non-linear approach to modelling and forecasting the exchange rate, based on the recently developed Taylor rule model of exchange rate determination. The separate literatures on exchange rate models and the Taylor rule have already shown that the non-linear specification can outperform the equivalent linear one. In addition the Taylor rule based exchange rate model used here has been augmented with a wealth effect to reflect the increasing importance of the asset markets in monetary policy. Using a Smooth Transition Regression model (STR), the results offer strong evidence of non-linearity in the variables used and that the interest rate differential is the most appropriate transition variable. We conduct the conventional out-of-sample forecasting performance test, which indicates that the non-linear models outperform their linear equivalents as well as the random walk. Key Words: Exchange rate; Taylor rule; Forecasting; Non-linear J.E.L. E44, E47, F37, F41 2

1. Introduction The aim of this study is to determine whether a non-linear approach to the modelling and forecasting of the exchange rate, using a Taylor rule based model, outperforms an equivalent linear version. Since the start of the floating exchange rate era, a number of studies have attempted to explain exchange rate movements, although as suggested by Cheung et al. (2005), mostly with limited success. Explaining and predicting exchange rate movements is an important aspect of monetary policy, particularly as capital flows between international asset markets have increased. Recently, a new strand of the exchange rate literature has developed a series of models that combine interest rate reaction functions based on the Taylor rule and the exchange rate, which has produced more accurate forecasts (Molodtsova and Papell, 2009; Gloria, 2010). These models tend to reflect more realistically how monetary policy has been conducted or evaluated over the recent past and offers an alternative interpretation of exchange rate dynamics. Although these studies have provided evidence of short-term predictability (Molodtsova and Papell, 2009), the results can vary across countries and different time intervals. Further recent developments in the study of exchange rates and monetary policy in general have involved the use of non-linear estimation techniques, which have become prevalent in both the literature on Taylor rule models (Qin and Enders, 2008) and modelling of the exchange rate (Michael et al., 1997, De Grauwe and Grimaldi, 2005). As far as we know, there has as yet been no attempt to connect both strands of the literature and use nonlinear approaches in the Taylor rule type exchange rate models, particularly with respect to forecasting. The main contribution of this study is the estimation and forecasting of the Taylor rule based exchange rate model using non-linear techniques, having pre-tested for non-linearity in the variables. We have also incorporated wealth effects into the models to reflect the importance of asset markets to both the Tylor rule and exchange rate determination. A number of different 3

types of non-linear techniques have been used to model the exchange rate, in this study we use the Smooth Transition Regression (STR) models, which were originally applied to nonlinearities over the business cycle by Terasvirta and Anderson (1992). Over recent years, it has been applied successfully in many exchange rate studies including purchasing power parity (PPP) with Michael et al. (1997), monetary models and the uncovered interest parity (UIP) model with Baillie and Kilic, (2006). One feature of the STR methodology is that it allows the selection of a possible form of nonlinear behaviour without having to make prior assumptions about the structure of the nonlinearities. Moreover, compared to other alternative nonlinear models such as the Markov switching model and threshold autoregressive model, the STR family of models allow a smooth and gradual transition from one regime to another rather than sudden jumps between regimes. In addition the evidence suggests that the Markov switching model does not produce particularly good exchange rate forecasts (Engel, 1994) The theoretical basis for this study is the linear model of the exchange rate by Molodtsova and Papell (2009) and additionally it has been augmented by the inclusion of wealth effects, including both stock prices and house prices. As Case et al. (2005) suggest both have varying degrees of influence on the macro-economy. Other studies including Granger et. al. (2000) have analysed wealth effects in the form of stock prices and the exchange rate, finding a significant relationship. 1 Unlike most previous studies with the STR models, we have documented and compared results across different models with regard to a large number of macroeconomic transition variables including the real exchange rate, output gap, inflation difference, interest rate difference, wealth effect and a measure of exchange rate volatility. The 1 Wang et al. (2016) have demonstrated the robustness of the linear version of a wealth augmented Taylor rule based exchange rate model in terms of both the in-sample and out-of-sample performance relative to the original model. 4

choice of the most appropriate transition variable is based on model specification and diagnostic testing. Moreover, we have identified the presence of any nonlinearity prior to estimation of the models and then selected the most appropriate STR specification using a linearity test. As with similar studies such as Molodtsova and Papell (2009), the main emphasis of this paper is on the out-of-sample forecasting performance of the nonlinear STR type exchange rate models, based on the approach of Clements and Smith (1999). However as with the exchange rate literature as a whole, the use of non-linear models for forecasting has produced mixed results, with the performance relative to the benchmarks varying across different samples as in Pavlidis et al. (2012) or the degree of non-linearity in the data as in Enders and Pascalau (2015). In this study the evaluation of the forecasting performance of the model is conducted against different benchmarks including the random walk model and linear Taylor rule based model. The results from estimation and forecasting of the nonlinear STR model provide evidence of the nonlinear relationship between the exchange rate and economic variables. Moreover, the STR models outperform both the random walk and the linear Taylor rule model in out-ofsample forecasting of the exchange rate. Following the introduction, section 2 introduces the STR model and the nonlinear Taylor rule exchange rate, it also outlines the specification, estimation and evaluation techniques used in this study. In section 3, we estimate the models and compare their in-sample specification and out-of-sample forecasting performance. The main conclusions are then drawn in the final section. 5

2. Methodology 2.1 The Modelling Framework The theoretical Taylor rule (Taylor, 1993) assumes that the nominal interest rate depends on changes in inflation and the output gap. Our starting point is the forward-looking Taylor rule (Clarida et al., 1998), where we assume the foreign country targets the exchange rate in its Taylor rule and the interest rate is assumed to adjust gradually towards its target level. In addition to this original specification, this study extends the model through the addition of a variable representing the effects of wealth on the baseline equation, as used in other studies such as Semmler and Zhang (2007). This modified Taylor rule is: i = μ + λπ + γy + δw + φq (1) i = (1 ρ)i + ρi + v (2) Where i is the target for the short-term nominal interest rate, π is the inflation rate, y is the output gap, or percent deviation of actual real GDP from an estimate of its potential level, w is the asset price, q is the real exchange rate, i is the actual observable interest rate, ρ denotes the degree of interest rate smoothing and v is the error term also known as the interest rate smoothing shock. Substituting (1) into (2) gives the following equation for the actual short-term interest rate: i = (1 ρ)(μ + λπ + γy + δw + φq ) + ρi + v (3) Taking the US as the domestic country and equation (3) as the interest rate reaction function for the foreign country; the monetary policy reaction function for the US is the same as in equation (3) with ϕ = 0. 6

To derive the Taylor rule based exchange rate equation, we follow the approach used by Molodtsova and Papell (2009). Letting ~ denote variables for the foreign country; the interest rate differential is constructed by subtracting the Taylor rule equation for the foreign country from that of the domestic country, the US. i ı = α + β π β π + β y β y + (β w β w ) β q + ρ i ρ ı + η (4) Where u and f are coefficients for the U.S. and foreign country respectively; α is a constant, β = λ(1 ρ), β = γ(1 ρ) and β = δ(1 ρ) for both countries, and β = ϕ(1 ρ) for the foreign country. Assuming Uncovered Interest Rate Parity (UIP) holds along with rational expectation: E( s ) = (i ı ) (5) Substituting (4) into (5) produces the following Taylor rule exchange rate equation: s = α + β π β π + β y β y + β w β w β q + ρ i ρ ı + η (6) Where s is the natural log of the U.S. nominal exchange rate, defined as the US dollar per unit of foreign currency, so that an increase in s represents a depreciation of the American dollar. A further homogenous model can also be estimated, where the two respective central banks are assumed to respond identically to changes in inflation, the output gap, the wealth effect and the 7

interest rate smoothing coefficients are equal. 2 This in effect restricts the foreign coefficient to being equal to the domestic one. So that β = β,β = β, β = β and ρ = ρ β. Therefore, the modified form of the Taylor rule model is as follows: s = α + β (π π ) + β (y y ) + β (w w ) β q + β (i ı ) + η (7) In this paper, we extend this linear specification by introducing a non-linear smooth transition regression model. There are a number of justifications for the presence of non-linearity in both monetary policy and the exchange rate. For monetary policy they include nonlinearities in the Phillips Curve (Nobay and Peel, 2000). The justification for non-linearity in the exchange rate include the effect of central bank intervention (Reitz et al., (2011)), speculative restrictions (Baillie and Kilic, 2006) and heterogeneous trading behaviour (Sarantis, 1999) This STR process is discussed in detail in (Teräsvirta 1994, Dijk, Teräsvirta et al. 2002) and applied to this setting, allows the transition to catch any smooth changes in our Taylor rule exchange rate model. The model assumes there are at least two regimes with different sets of coefficients and a transition variable which determines the movements across the regime. A standard two regime STR model has the following specification: y = φz + θz G(s ; γ, c) + u (8) 2 Estimation and forecasting have been conducted for both restricted and unrestricted models In general, we found the homogenous model generates better forecasts so report these results. 8

where z is a vector of explanatory variables including a constant and possibly some lagged values of y. ϕ and θ represent parameter vectors for the linear and nonlinear parts of the model respectively. The error term u is assumed to be n.i.d. with zero mean and constant variance σ. G(s ; γ, c) is the transition function assumed to be continuous and bounded between 0 and 1. s is the transition variable, γ is the transition parameter, also known as the speed of transition, which determines how quickly the transition between regimes occurs and is restricted by γ > 0. c denotes the particular threshold level and corresponds to the value of the transition variable where the transition takes place. Both γ and c are estimated by the model and the transition variable s can be an element or a linear combination of z or a linear deterministic trend. There are two alternative functional forms of the transition function: - Logistic Function: - Exponential Function: G(s t ; γ, c) = 1 1 + exp [ γ(s t c)] (9) G(s t ; γ, c) = 1 exp γ(s t c) 2 (10) Equation (8) combined with transition function (9) jointly define the logistic STR (LSTR) model. Equation (8) with transition function (10) form an exponential STR (ESTR) model. Different functional forms of G(s ; γ, c) correspond to different types of exchange rate switching behaviour. For the logistic STR model, the transition function is a monotonically increasing function of s. Therefore, the LSTR models describe relationships that change relative to the level of the threshold variable. Given that G(s ; γ, c) is continuous and bounded between zero and one, the combined nonlinear coefficients ϕ + θ G(s ; γ, c) will change monotonically from ϕ to 9

(ϕ + θ) according to different values of s. When s c +, G(s ; γ, c) 1 and coefficients become ϕ + θ ; when s c, G(s ; γ, c) 0 and the coefficients become ϕ. In contrast to the logistic function, the exponential function is symmetric and U- shaped around c. It describes dynamic behavior which is the same for the high values of the transition variables as it is for low values. The transition function G(s ; γ, c) 1 both as s c and s c + and the coefficient in this approach becomes (ϕ + θ). In the case of s = c, G(s ; γ, c) = 0, the coefficients become ϕ. 2.2 The Modelling Strategy When testing for possible nonlinearity in the Taylor rule based exchange rate model, we use the procedure developed by Granger and Teräsvirta (1993) and Teräsvirta (1994). This modelling approach consists of three steps: specification, estimation and evaluation. This Taylor rule STR model takes the following form: s = φz + θz G(s ; γ, c) + ε (11) where G(.. ) acts as the transition function; z = ( π π, y y, q, i ı, w w ) is the vector of regressors in the above models. The vector ϕ = α, β, β, β, β, β and θ = α, β, β, β, β, β contain the parameters from the linear and nonlinear parts of the model. We have chosen six different transition variables in the nonlinear estimation. These 10

are the output gap, interest rate differential, inflation differential, wealth effect differential, real exchange rate and exchange rate volatility. 3 After selecting the predetermined transition variable, we follow the approach proposed by Teräsvirta (1994) in testing for the null of linearity against the alternative STR model. These tests are conducted through estimating the following auxiliary regression: s = δ z + δ z s + ε (12) Where z is the vector of variables in z without the constant; s is one of the elements of z. The null hypothesis is of linearity (H ): δ = δ = δ = 0; whilst the alternative hypothesis is at least one of δ 0, j = 1,2,3. As suggested by Teräsvirta (1994), F-versions of the LM test statistics are employed as these have better size properties than the χ -statistic. 4 Once the null hypothesis of linearity is rejected in favour of STR nonlinearity, we can choose the appropriate form of the transition function. The decision is based on testing the order of the polynomial in the auxiliary regression (12). Granger and Teräsvirta (1993) and Teräsvirta (1994) proposed the following sequence of null hypotheses: H : δ = 0 (13) H : δ = 0 given δ = 0 (14) 3 The nonlinearity in the Taylor rule exchange rate model may arise from either the Taylor rule part or the exchange rate. Therefore, the hypothesis of nonlinearity in the model can be tested simply by evaluating the setting in the functional form of the interest rate reaction function and exchange rate. The exchange rate volatility is used as a transition variable to study how the model is related to market risk by including a risk premium within the nonlinear part of the model. 4 In small or moderate sized samples, the χ -statistic may be heavily oversized (Dijk, D. v., et al. (2002)). 11

H : δ = 0 given δ = 0, δ = 0 (15) According to Teräsvirta (1994), the decision rules for choosing between LSTR and ESTR models are the following: We compare the significance level of the three F-tests, if the p-value of the test corresponding to H is the smallest among the three, then select the ESTAR model; otherwise a LSTAR model is chosen as the appropriate model. Both the LSTR and ESTR model are estimated using non-linear least squares (NLLS) with a grid search for the parameters γ and c with respect to the nonlinear optimization. For the γ estimation, we scale the transition function, by dividing it using the standard deviation of s (i.e. σ ) for the LSTR models and by the variance of s (i.e. σ ) in the case of ESTR. The transition function is standardised to make it easier to compare the estimates of the transition parameters across different equations. 5 The estimated STR models are then assessed using misspecification tests that have been specifically designed to evaluate the adequacy of a single equation STAR model, as suggested by Lin and Teräsvirta (1994). These are known as the LM tests of no error autocorrelation, the LM test of no remaining nonlinearity and the LM test of parameter constancy. Other commonly used tests in the STR literature include the Jarque-Berra test for the normal distribution of the errors and the LM test of no autoregressive conditional heteroscedasticity (ARCH) in the residual. 5 This is also recommended by Granger et al. (1993) and Teräsvirta (1994). They have argued that scaling the transition variable by its own standard deviation before empirical estimation not only speeds up the convergence but also improves the stability of the nonlinear least squares estimation algorithm. 12

3. Empirical results 3.1 Data The data is all quarterly and consists of the exchange rate returns measured in log-differences, and the Taylor rule based economic fundamentals for the United Stated, the United Kingdom, Sweden and Australia. Quarterly data was used as a result of the limited monthly data for GDP and house prices. These countries were selected as a result of the availability of the wealth data, especially for housing and also as a result of their strong housing markets. Due to data availability, the time period for these countries differs depending on the measure of the wealth effect. When stock prices represent the wealth effect, the data ranges from 1975Q1 to 2008Q4, whereas when house prices are used, the data is from 1980Q1 to 2008Q4, due to data availability. As a measure of wealth, stock prices have been chosen to represent the wealth effect for the UK and Australia 6. This is because they have particularly strong equity markets, whereas in Sweden they are of less importance compared to the banking system, therefore house prices were found to be the most appropriate wealth effect for Sweden. All variables except interest rates are in logarithms. The inflation rate is the annual inflation rate calculated using the CPI over the previous 4 quarters and real GDP is used to measure the level of output. As in other studies, the output gap is constructed as the percentage deviation of actual output from a Hodrick Prescott (HP) generated trend. The real foreign/u.s. exchange rate is calculated as the percentage deviation of the nominal exchange rate from the target defined by PPP (i.e., q = s (p p ), where p and p are natural logarithms for U.S. and 6 This is based on the linear estimation of the model, in which stock prices were found to be the most significant determinant for these two countries. Results available on request. 13

foreign price levels, respectively, as measured by respective CPI levels. Money market rates are used as a measure of short-term interest rates. The nominal exchange rate is defined as the U.S. dollar price of foreign currency and is taken as the end-of-month exchange rate. Studies by Molodtsova et al. (2008), among others, have highlighted the importance of realtime data for use in Taylor rule-based exchange rate forecasting. Real-time data are based on vintages of data that are available to researchers at each point in time (i.e., before data revisions are applied). As real-time data are only available for the U.S. among the countries studied, we followed Molodtsova and Papell (2009) and used quasi-real time data when measuring the output gap. In this case, the current vintage data were used, though trends at period t were calculated using observations 1 to t-1. Exchange rate volatility is measured using the conditional volatility series produced from a GARCH (1, 1) framework and has been widely used to proxy the risk premium. All the data was obtained from Thomson DataStream and the International Monetary Fund International Statistics (IMF). The quarterly closing prices of the main stock market indexes are used to represent equity prices in each country. House price indices are taken from Oxford Economics and measure the quarterly house price. 3.2 Nonlinear estimation results Table 1 contains the p-values of the LM tests, with the first column reporting the results of the tests for linearity against non-linearity based on the STR models with various transition variables 7 (i.e. H ). The following columns show the results of the model selection tests, which determine whether we use the LSTR or the ESTR approach (i.e. H, H, H ) and the 7 All variables were initially tested for stationarity using the Ng and Perron test, which suggested all were stationary at the 5% level of significance. 14

subsequent non-linear model specification. According to Teräsvirta (1994), since it is possible for the three hypotheses ( H, H, H ) to be simultaneously rejected, we have selected the one with the strongest level of rejection. The results provide strong evidence in favour of a nonlinear specification for the Taylor rule based exchange rate model. For each country, there is evidence of nonlinearity for more than one of the transition variables. The transition variable for which linearity is most strongly rejected does not necessarily imply that it is the best model overall, any variable with evidence of non-linearity can be included in the estimation and forecasting, then the decision regarding the best model can be made based on the model s evaluation and forecasting performance. Tables 2 to 8 present a summary of the results including a series of misspecification tests for the nonlinear exchange rate models for all three cases. The estimated beta coefficients for the models are not included in the tables due to space constraints and as the emphasis here is on out-of-sample forecasting performance and ensuring the models are well specified before using them to produce the forecasts 8. Dummy variables have been included in the models to overcome any problems of normality in the error term. 9 Before analysing the results, we have checked whether the nonlinearities are due to the presence of outliers using the method of Sarantis (1999). 10 The parameter estimates for the UK and Sweden are found to be similar before and after the dummies are added. Therefore, we conclude that nonlinearities in the UK 8 The estimates of the beta coefficients from the linear and non-linear parts of these models are available on request. 9 Details regarding the dummy variable are footnoted under each table. Results before the addition of the dummy variables are not presented here but are available upon request. 10 Sarantis (1999) checked for nonlinearities due to the presence of outliers by using dummies to filter them out. He argued that if all estimated parameters and diagnostic statistics are similar to the initial ones with residuals becoming normal, then we can conclude that the nonlinearity is not due to the effect of outliers. 15

and Sweden s exchange rate are not due to the effect of any outliers. However, the parameter estimates and diagnostic statistics change substantially for Australia 11. As a result, the linearity test has been rerun using the model with dummy variables to control for any nonlinear transition caused by the outliers. These results are presented in Table 6. After adjusting for outliers, the Australian exchange rate performs in a linear fashion apart from when interest rate differences are used as the transition variable. Based on these results, we have concluded that for the UK, the Logistic STR model with interest rate differences as the transition variable produces a well specified model, which outperforms the other models. In particular, the model has passed all the diagnostic tests, the parameters seem to be constant over time and there is no evidence of any remaining nonlinearity of the STR-type. For Sweden, the different criteria are consistent in selecting models with the interest rate and wealth effect as the transition variable, producing the best fitted models and passing all misspecification tests. For the others, when exchange rate volatility is used as the transition variable, the diagnostic and adequacy tests suggest there is evidence of parameter inconsistency in the way that the parameter may change monotonically over time. When the inflation difference is used, some evidence of nonlinearity exists although it is not very strong. The Australian exchange rate with interest rates as the transition variable are well specified after being tested for normality, autocorrelation, ARCH effects, and constancy of coefficients. Moreover, there is no evidence of any remaining nonlinearity of the STR-type. 11 A potential reason for the difference between the Australian results and the other currencies is that as Chen and Rogoff (2006) suggests, it is heavily influenced by the commodity markets. 16

3.3 Modelling the Transition Figures 1-3 show plots of the transition functions over time during our sample period. The change in the parameters, which depend on different transition variables, can be viewed as an indicator of the overall economic conditions or the monetary policy stance. For the UK, in the case where the interest rate difference or real exchange rate are the transition variable, the transition starting points are almost identical, with the first transition at the beginning of 1979. This is the occasion when the authorities objective of nominal exchange rate stability began to conflict with their monetary targeting, with the monetary target at risk of being overshot and unsustainable. The fluctuation in the interest rate within regimes, approximately between 1979 and 1988, together with a stable exchange rate corresponded to the period when the UK tried to stabilize the exchange rate through its interest rate policy. From these figures, it is interesting to note that the transition functions between 1989 and 1994 are very close to unity and after 1994, they are more often close to zero. This is similar to the results of Baillie and Kiliç (2006). Baillie and Kiliç (2006) argued this can been explained by the fact that the US has lower interest rates than the UK, and US dollars were quoted at a premium during this period. Another notable finding is that in general larger changes in parameters occurred more frequently prior to 1992. After this date, the UK adopted inflation targeting with no obvious changes in monetary policy. This coincides in our figure with the parameters of the nonlinear Taylor rule exchange rate generally changing less often after 1992. 12 For Sweden, the large change in parameters and the frequent shift of transition functions ends around 1994. After 1994, the transition functions stay in the lower regime most of the time and 12 The fact that there are lower parameter changes in the nonlinear Taylor rule after 1992 has also been found by Brüggemann and Riedel (2011). 17

parameter values change less frequently. This pattern reflects Swedish economic policy at the time, which experienced substantial interventions in the foreign exchange markets during the fixed rate regime period (i.e. before 1990). Following the banking crisis, there was a policy realignment within the Swedish economy in early 1990. And then, finally, the economy stabilised with no evident changes in monetary policy after 1995. It is also noticeable that between the periods 1990 to 1994, the transition functions attained values mostly in the upper regime with values close to unity, this reflects the severe banking and subsequent financial and economic crises experienced by Sweden in the early 90s. Figure 3 is the plot of the estimated transition function over time for Australia. Similar to the plots of the UK and Sweden, frequent shifts between regimes and large changes in parameters have occurred mostly before Australia adopted the floating exchange rate system at the end of 1983. Overall, the nonlinear specification improves upon the linear one by explaining some of the variation in the exchange rate related to the extreme peaks of various transition variables. Figures of the transition functions over time indicate that transition between regimes with large changes in parameters occurred most frequently prior to the introduction of the floating exchange rate system and inflation targeting. This is because the nonlinear Taylor type exchange rate relationships are based on Taylor rule interest rate models which are mainly used in studying the change and setting of monetary policy. As monetary policy has not greatly changed after the introduction of the floating system, so the same is the case with the exchange rate. 3.4 Out-of-sample forecasting Finally, we have evaluated the non-linear STR models in terms of their out-of-sample forecasting performance, including all the transition variables which produced evidence of non- 18

linearity. This is done by comparing each nonlinear specification s forecasting performance with those of the equivalent linear model, as well as with the random walk, which is the standard benchmark when forecasting exchange rates. Forecasting with nonlinear models is more complicated than with the linear versions (Teräsvirta 2006). To obtain the out-of-sample forecasts, we follow the approach of Clements and Smith (1999) by applying a simulationbased procedure (i.e. a Monte Carlo method). 13 Firstly, all models were re-estimated up to 1999Q4 and these estimates were used to generate a set of recursive forecasts for 2000Q1 to 2008Q4. Each out-of-sample forecast is constructed using all the data up to the forecasting period. So, in total, we obtain 36 forecasts for each model. As noted in Clements and Smith (1999), we can write the nonlinear forecasting equation as: s = g(x ; θ) + ε (16) Assuming the error terms in this nonlinear model are Gaussian, we define the optimal forecast for s at time t is being equal to the conditional mean: s = E[ s Ω] = E[g(X ; θ) Ω] (17) where Ω is a set of past information. For a given set of X, we simulate a realization for s (h 1) by the Monte Carlo forecast formula: s = 1 N g X, ; θ (18) where each n value of ε in X, is a random draw from its standard normal distribution of residuals from the estimated model. We repeat the process 10000 times, this will give 10000 13 Each forecast is obtained as the average over 10,000 replications. 19

simulated realizations for s. The forecast of s given X t is the mean of the 10000 simulated realizations. The Law of Large Numbers guarantees that the sample means converge to the true forecasts. In the comparison exercise, we assess the out-of-sample forecasting performance in terms of the traditional mean squared forecast error (MSPE) criterion and Theil s U ratio. Models with a smaller MSPE and Theil s U ratio produce a better forecasting performance. Furthermore, in order to assess whether the nonlinear model forecasts have superior predictive ability over the linear ones, we perform the Clark and West (2007) (henceforth CW) test. All the test procedures are almost identical to the ones used in the linear model. With respect to the predictability tests with nonlinear models, Liu et al. (2010) have compared the power and size properties of the Diebold and Mariano (1995) and West (1996) (DMW) test with the CW test in nonlinear models. Their results indicate the CW test is appropriately sized and has good power when applied to nonlinear models. Table 8 presents the out-of-sample forecasting results, using the transition variables which provided evidence of non-linearity. In terms of the MSPEs and Theil s U ratios, they are less than one for every specification estimated. Therefore, we can conclude that for all countries studied, the non-linear STR model provides more accurate forecasts than the equivalent linear model and random walk. The last two columns show the CW statistics with respect to the two benchmarks. The values showing a rejection of the null hypothesis of equal forecasting accuracy are indicated with asterisks. When the benchmark is the random walk, we notice that the STR specifications outperform the random walk model in almost all cases at the 5% significant level. This result for the UK/US exchange rate is similar to Pavlidis et al. (2012), who find the non-linear model outperforms the equivalent linear model. The only exception is 20

Sweden s exchange rate when the interest rate difference acts as the transition variable, but it still yields superior forecasts at the 10% level. When comparing the forecasting performance of the STR specifications with the linear models, we find that for every specification where there is evidence of nonlinearity, the non-linear model outperforms the linear model in all cases in terms of forecasting accuracy at the 10% significance level and in approximately two thirds of the case at the 5% significance level. It therefore appears that the out-of-sample forecasting performance of the STR specifications is more accurate than both the linear specifications and the random walk. Figure 4 to Figure 6 provides a graphical representation of how well the forecasting from our nonlinear model tracks the actual exchange rate change over time. The lower and upper dotted lines represent the 95% prediction intervals of the forecast. According to this figure, the nonlinear STR model was able to produce reliable forecasts of the exchange rate changes up to the last quarter of 2008. Most actual changes in the exchange rate within the forecasting period lie within 95% of the predictive intervals. However, the reliability of our nonlinear forecasts breaks down by the end of 2008. However, given the unprecedented world economic turmoil unleashed by the events in the U.S. financial system during September 2008, this is not surprising. 4. Conclusion This study has analysed the forecasting performance of the non-linear Taylor rule based exchange rate model, complementing recent research which has found that the linear version of this model outperforms a random walk. Using quarterly data on dollar-sterling, dollar- Swedish krona and dollar-australian dollar exchange rates over the period 1975 to 2008, we found strong evidence of nonlinearities in the exchange rate with respect to several macroeconomic determinants, suggesting the Taylor rule exchange rate models can be improved by consideration of regime changes. 21

However the form of the non-linearity and the transition process has been found to vary across the countries studied, reflecting the differences in these economies. In general, the interest rate differential has been found to be an important source of nonlinearities in exchange rates for all the countries we studied. For both the UK and Australia, the Logistic STR model with interest rate differences as the transition variable produced a well specified model, which prevails over other nonlinear models. Whereas for Sweden s exchange rate, the estimation results based on the wealth effect as the transition variable generally give the best specification. The out-ofsample forecasting using the driftless random walk and linear Taylor rule as the benchmark models provides evidence that in general the predictive performance of the STR models improves upon its equivalent linear specification for all the countries studied. On the basis of these forecasting results, the evidence suggests the non-linear approach is the most effective way to model the exchange rate based on the Taylor rule methodology. Given the importance of predicting and explaining exchange rate movements for monetary policy, the main implication of the study is that using a non-linear approach to modelling and forecasting produces better outcomes than the more usual linear approaches. In addition the inclusion of wealth effects within the model adds to the previous evidence showing that possibly as a result of increased capital movements between international financial markets, asset prices have an important effect on exchange rates. Future research in this area could consider alternative non-linear approaches to modelling and a wider selection of potential transition variables. 22

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Table 1 linearity tests Transition variable H 0 H 01 H 02 H 03 Type of model UK π t π t 0.2102 0.7110 0.7228 0.0172 Linear y t y t 0.0118* 0.1847 0.0310* 0.0425 ESTR i t 1 t 1 0.0283* 0.2822 0.0853 0.0336* LSTR w t (s) w t (s) 0.0601** 0.2656 0.0351* 0.2463 ESTR Sweden q t 0.0755** 0.0473* 0.5596 0.0965 LSTR volatility 0.0062* 0.0695 0.0008 0.8983 ESTR π t π t 0.0090* 0.0142* 0.1912 0.0526 LSTR y t y t 0.2050 0.0753 0.7366 0.2502 Linear i t 1 t 1 0.0380* 0.0105* 0.2584 0.3091 LSTR w t (h) w t (h) 0.0134* 0.0735 0.0046* 0.4984 ESTR q t 0.3666 0.5038 0.0889 0.7752 Linear volatility 0.0006* 0.0030 0.0009* 0.8954 LSTR Australia π t π t 0.2967 0.6039 0.1481 0.0518 Linear y t y t 0.0040* 0.0154* 0.0867 0.0359 LSTR i t 1 t 1 0.0032* 0.0029* 0.1698 0.0547 LSTR w t (s) w t (s) 0.6889 0.5514 0.5853 0.5904 Linear q t 0.0002* 0.0188 0.4393 0.0000* LSTR volatility 0.3034 0.0797 0.9328 0.2712 Linear Note: The table shows the p-values for the Teräsvirta (1994) linearity test, for which the null hypothesis of linearity is tested against the alternative of the STR model; the* and ** implies rejection of the null hypothesis at 5% and 10% significance levels, respectively. If the p-value of the linearity test H0 is less than significance level, the null is rejected, then we proceed to choose the one for which the p-value of the test is minimized among H01, H02 and H03 and use this to determine the best nonlinear model specification to implement. 26

Table 2 Estimates of STR models UK (1) (2) (3) (4) (5) Sample 75Q1:08Q4 75Q1:08Q4 75Q1:08Q4 75Q1:08Q4 75Q1:08Q4 Model ESTR LSTR ESTR LSTR ESTR Transition variable (s t ) Model parameters y t y t i t 1 t 1 w t w t (s) q t volatility γ 1.091 10.547 2.596 16.253 0.455 γ σ s (or σ 2 s ) 4164.052 4.120 516.118 117.63 980462.30 c -0.013* (0.003) Summary statistics 2.322* (0.355) -0.016* (0.007) 0.620* (0.000) 0.003* (0.000) R 2 0.223 0.250 0.234 0.246 0.295 adj. R 2 0.137 0.161 0.154 0.163 0.218 SSR ratio 0.891 0.857 0.878 0.864 0.807 σ 0.048 0.047 0.047 0.046 0.045 Log likelihood 222.072 224.470 223.066 224.050 228.535 Note: (a). Models are estimated by NLLS. γ is the speed of transition between regimes. γ/σs is the scaled speed for comparison across models. c is the threshold value for particular transition variable. The estimated standard errors are given in parentheses. * and ** denote significant at the 5% and 10% level, respectively. (b). In the case of interest rate smoothing as the transition variable, the result presented here has corrected the nonnormality by adding a dummy variable for 1988Q1. This represents the time that the UK government tried to stabilize the exchange rate by intervening in the foreign exchange market and changing interest rates. (c). adj.r2 is the adjusted R2.SSRratio denote sum of squared residuals ratio between the STR model and the linear specification. A lower ratio (i.e. less than one) indicates a better fit for the nonlinear model and vice versa. σ is the standard errors of regression. We prefer the regression model possessing higher values of adj.r2, and log likelihood, but lower values of SSRratio and σ. 27

Table 3 Diagnostic tests for STR models - UK (1) (2) (3) (4) (5) Sample 75Q1:08Q4 75Q1:08Q4 75Q1:08Q4 75Q1:08Q4 75Q1:08Q4 Model ESTR LSTR ESTR LSTR ESTR Transition variable (s t ) y t y t i t 1 t 1 w t w t (s) q t volatility Residual Tests JB 0.858 0.238 0.142 0.438 0.249 ARCH-LM(1) 0.945 0.802 0.461 0.985 0.799 LM(1) 0.812 0.563 0.066** 0.441 0.283 LM(4) 0.007* 0.147 0.013* 0.049* 0.018* Remaining Nonlinearity π t π t 0.814 0.564 0.868 0.736 0.868 y t y t 0.982 0.810 0.644 0.654 0.644 i t 1 t 1 0.858 0.916 0.793 0.367 0.793 w t w t (s) 0.436 0.204 0.519 0.229 0.519 q t 0.465 0.556 0.680 0.975 0.679 volatility 0.391 0.474 0.201 0.479 0.201 Parameter Constancy H 1 0.847 0.568 0.699 0.966 0.698 H 2 0.459 0.894 0.593 0.590 0.208 H 3 0.994 0.992 0.969 0.973 0.861 Note: numbers in this table are p-values. * and ** represent rejection of the null at the 5% and 10% significance levels, respectively. JB denotes the Jarque-Bera test for the null of normality of residuals. LM (1) and LM (4) denote LM tests for the null of no first and forth order serial correlation. ARCH-LM (1) denotes the null of no first order residual heteroskedasticity. For parameter constancy, rejection of either one of the null H1, H2 and H3 will lead a conclusion favouring parameter non-constancy, otherwise the parameters are time-invariant. 28

Table 4 Estimates of STR models Sweden (1) (2) (3) (4) Sample 80Q1:08Q4 80Q1:08Q4 80Q1:08Q4 80Q1:08Q4 Model LSTR LSTR ESTR LSTR Transition variable (s t ) π t π t i t 1 t 1 w t w t (h) volatility Model parameters γ 1.757 27.316 0.096 2.885 γ σ s (or σ 2 s ) 73.224 6.568 47.977 4331.120 c 0.005* (0.000) Summary statistics 2.882* (0.000) 0.017* (0.000) 0.005* (0.000) R 2 0.412 0.451 0.463 0.463 adj. R 2 0.321 0.353 0.367 0.367 SSR ratio 0.6391 0.5956 0.5820 0.5820 σ 0.049 0.048 0.047 0.047 Log likelihood 188.678 192.615 193.847 193.838 Note: (a). in the original models, we found the Jarque-Bera normality test has been rejected for all specifications. Therefore, the two dummy variables added to the models are 1993Q3 and 2008Q4. 2008Q4 represents the financial crisis. 1993Q3 is around the time that the Governing Board of the Riksbank Sweden announced it would apply inflation targets and the Swedish economy had begun to recover. (b). Models are estimated by NLLS. γ is the speed of transition between regimes. γ/σs is the scaled speed for comparison across models. c is the threshold value for particular transition variable. The estimated standard errors are given in parentheses. * and ** denote significant at the 5% and 10% level, respectively. (c). adj.r2 is the adjusted R2.SSRratio denote sum of squared residuals ratio between the STR model and the linear specification. A lower ratio (i.e. less than one) indicates a better fit for the nonlinear model and vice versa. σ is the standard errors of regression. We prefer the regression model possessing higher values of adj.r2, and log likelihood, but lower values of SSRratio and σ. 29

Table 5 Diagnostic tests for STR models - Sweden (1) (2) (3) (4) Sample 80Q1:08Q4 80Q1:08Q4 80Q1:08Q4 80Q1:08Q4 Model LSTR LSTR ESTR LSTR Transition variable (s t ) π t π t i t 1 t 1 w t w t (h) volatility Residual Tests JB 0.143 0.102 0.187 0.273 ARCH-LM(1) 0.259 0.826 0.676 0.690 LM(1) 0.026* 0.551 0.334 0.497 LM(4) 0.188 0.763 0.594 0.695 Remaining Nonlinearity π t π t 0.994 0.154 0.458 0.593 y t y t 0.707 0.792 0.873 0.340 i t 1 t 1 0.073** 0.925 0.860 0.974 w t w t (h) 0.967 0.929 0.999 0.892 q t 0.495 0.302 0.962 0.532 volatility 0.693 0.810 0.811 0.916 Parameter Constancy H 1 0.475 0.169 0.848 0.018* H 2 0.501 0.452 0.905 0.777 H 3 0.999 0.965 0.997 0.999 Note: Values in this table are p-values. * and ** represent rejection of the null at the 5% and 10% significance levels, respectively. JB denotes the Jarque-Bera test for the null of normality of residuals. LM (1) and LM (4) denote LM tests for the null of no first and forth order serial correlation. ARCH-LM (1) denotes the null of no first order residual heteroskedasticity. For parameter constancy, rejection of either one of the null H1, H2 and H3 will lead a conclusion favouring parameter non-constancy, otherwise the parameters are time-invariant. 30

Table 6 Linearity test (after adjustment) - Australia Transition variable H 0 H 01 H 02 H 03 Type of model π t π t 0.1671 0.5251 0.9532 0.0051 Linear y t y t 0.6697 0.7711 0.3310 0.7468 Linear i t 1 t 1 0.0178* 0.0046* 0.8667 0.0086 LSTR w t (s) w t (s) 0.1554 0.0710 0.3478 0.1298 Linear q t 0.1654 0.6119 0.1710 0.0755 Linear volatility 0.1561 0.0058 0.7650 0.6237 Linear Note: The table shows p-values of the linearity test after introducing dummy variables in the models for which the null hypothesis of linearity is tested against the alternative of the STR model; The dummy variables added to the model are 2008Q4, 1986Q3 and 1985Q2. the* and ** implies rejection of the null hypothesis at the 5% and 10% significance level, respectively. If the p-value of the linearity test H0 is less than the significance level, the null is rejected, then we proceed to choose the one for which the p-value of the test is minimized among H01, H02 and H03 and determine for which the best nonlinear model specification is going to be implemented. 31