Application of Taylor rule fundamentals in forecasting exchange rates

Agyapong, Joseph Article Application of Taylor rule fundamentals in forecasting exchange rates Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Agyapong, Joseph (2021) : Application of Taylor rule fundamentals in forecasting exchange rates, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 9, Iss. 2, pp. 1-27, https://doi.org/10.3390/economies9020093 This Version is available at: https://hdl.handle.net/10419/257251 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. 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Application of Taylor Rule Fundamentals in Forecasting Exchange Rates. Economies 9: 93. https://doi.org/10.3390/ economies9020093 Academic Editor: Robert Czudaj Received: 20 May 2021 Accepted: 15 June 2021 Published: 21 June 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Faculty of Business, Economics and Social Sciences, Christian-Albrechts-University of Kiel, Olshausenstr. 40, D-24118 Kiel, Germany; [email protected] or [email protected]; Tel.: +49-15-252-862-132 Abstract: This paper examines the effectiveness of the Taylor rule in contemporary times by investigating the exchange rate forecastability of selected four Organisation for Economic Co-operation and Development (OECD) member countries vis-à-vis the U.S. It employs various Taylor rule models with a non-drift random walk using monthly data from 1995 to 2019. The efficacy of the model is demonstrated by analyzing the pre- and post-financial crisis periods for forecasting exchange rates. The out-of-sample forecast results reveal that the best performing model is the symmetric model with no interest rate smoothing, heterogeneous coefficients and a constant. In particular, the results show that for the pre-financial crisis period, the Taylor rule was effective. However, the post-financial crisis period shows that the Taylor rule is ineffective in forecasting exchange rates. In addition, the sensitivity analysis suggests that a small window size outperforms a larger window size. Keywords: Taylor rule fundamentals; exchange rate; out-of-sample; forecast; random walk; directional accuracy; financial crisis 1. Introduction Exchange rates have been a prime concern of the central banks, financial services firms and governments because they control the movements of the markets. They are also said to be a determinant of a country’s fundamentals. This makes it imperative to forecast exchange rates. Generally, one could ask, is there any benefit in accurately forecasting the exchange rates? Ideally, there is no intrinsic benefit to accurate forecasts; they are made to enhance the resulting decision making of policymakers (Hendry et al. 2019). One of the popular investigations into the exchange rate movements was made by Meese and Rogoff (1983). In their paper, they perform the out-of-sample exchange rates forecast during the post-Bretton Woods era. They found that the random walk model performs better with the exchange rate forecast than the economic fundamentals. This was the Meese–Rogoff puzzle. Subsequently, researchers have challenged the Meese and Rogoff findings. Mark (1995) uses the fundamental values to show the long-run predictability of the exchange rate. Clarida and Taylor (1997) use the interest rate differential to forecast spot exchange rates. Mark and Sul (2001) also find evidence of predictability for 13 out of 18 exchange rates using the monetary models. In 1993, John B. Taylor presented monetary policy rules that describe the interest rate decisions of the Federal Reserve’s Federal Open Market Committee (FOMC). In most literature, this has been named the Taylor rule. Taylor (1993) stipulates that the central bank regulates the short-run interest rate in response to changes in the inflation rate and the output gap (interest rate reaction function). This has become a monetary policy rule which the Federal Reserve (Fed) and other central banks have incorporated into their decision making (Taylor 2018). The Taylor rule principle is used in this study due to its effectiveness in monetary policy. It is superior to the traditional models as it combines the uncovered interest rate parity, the purchasing power parity and the other monetary variables for forecasting exchange rates. This makes it a more robust method for forecasting Economies 2021,9, 93. https://doi.org/10.3390/economies9020093 https://www.mdpi.com/journal/economies Economies 2021,9, 93 2 of 27 the exchange rates. The Taylor rule monetary policy operates well in countries that practice floating exchange rates with an inflation-targeting framework. Economists have derived two versions of the Taylor rule to forecast the exchange rate. These include Taylor rule differentials and Taylor rule fundamentals. Engel et al. (2008) developed the Taylor rule differentials model by subtracting the Taylor rule of the domestic country from that of the foreign country. Instead of using the estimated parameters, they apply the postulated parameters into the forecasting regression to perform the test. They perform out-of-sample predictability of the exchange rate and find that the Taylor rule differentials models perform better than the random walk in the long horizon compared to the short horizon. Other literature including Engel et al. (2009) provides supporting evidence of the Taylor rule models in predicting the exchange rate. The Taylor rule fundamentals model was first established by Molodtsova and Papell (2009). They deducted the Taylor rule of the foreign country from the domestic country and the variables contained in the Taylor rule equation are directly utilized to perform an out-of-sample prediction for the exchange rate. Rossi (2013) surveys the exchange rate forecast models in most literature and finds strong evidence in favor of the Taylor rule fundamentals by Molodtsova and Papell (2009). However, in reaction to the global financial crisis, the major central banks set short-term interest rates to a zero lower bound (ZLB) which renders the conventional monetary policies ineffective. This has led to a debate among thought leaders on the efficacy of the Taylor rule. The objective of this study is to check the effectiveness of the Taylor rule monetary policy in contemporary times by applying the Taylor rule fundamentals to forecast the exchange rates using current data and a new set of currency pairs. The research, therefore, contributes to the existing literature by investigating the usefulness of the Taylor rule-based exchange rate forecast in the pre- and post-crisis periods. More so, the study examines the sensitivity of the window size on the performance of the Taylor rule fundamentals. The research paper applies to four (4) OECD countries, namely, Norway, Chile, New Zealand and Mexico vis-à-vis the United States (U.S.). These countries adopt floating exchange rates with an inflation-targeting framework. The impetus for selecting these countries includes the fact that Norway is one of the long-standing trading partners of the United States. Norway invests about 35% of its government pension fund in the U.S. Bloomberg (2019) reported that Norway plans to increase its wealth fund by USD 100 billion in U.S. stocks 1 . This shows that their economies and the exchange rate could be affected by the Taylor rule policy. However, the Norwegian exchange rate has received less research. Chile and Mexico were selected for this research because they are among the seven largest economies in Latin America that are emerging economies and have a floating exchange rate and inflation target framework. These countries contribute to the research by depicting how the Taylor rule monetary policy affects the exchange rates of the emerging economies in Latin America. New Zealand is the first country to implement the inflation-targeting framework in the early 1990s. Therefore, the Reserve Bank of New Zealand would exhibit a greater experience of the Taylor rule policy. These countries have chronological market relations regarding trading with the U.S. Their economies rely very much on the importation and exportation of goods and services. Moreover, the Triennial Survey by the Bank for International Settlements (BIS) (2019) shows that the U.S. dollar is the most traded currency and the U.S. has been the center for international trading over the years. In 2019, the U.S. dollar contributed 88.3% of the total foreign exchange market volume. In 2019, the New Zealand dollar was ranked 11th among the global currencies trading, adding about 2.1% to the foreign exchange market volume. In the same currencies rankings, the Norwegian krone ranks 15th and the Mexican peso ranks 16th for contributing 1.8% and 1.7%, respectively. The U.S., Norway, Chile and New Zealand are noted as having “commodity currencies”2which influence the exchange rate changes (Chen et al. 2010). These features contribute to the choice of selecting the countries for this research. Economies 2021,9, 93 3 of 27 The domestic country considered in this study is the U.S. An out-of-sample forecast is performed in the short horizon for the Norwegian krone, Chilean peso, New Zealand dollar and Mexican peso exchange rates with the U.S. dollar. The benchmark model is the random walk. A linear model is used in this work since it is shown to be the most efficacious exchange rate forecastability in the literature (Rossi 2013). The forecast would be evaluated by using the mean squared forecast error. For the forecast comparison, Molodtsova and Papell (2009) state that the linear model is nested; therefore, Clark and West’s (2006,2007) model is used to perform the significant test 3 . In this paper, similar models and specifications by Molodtsova and Papell (2009) are used. It is important to stress that this study does not show which models beat the random walk but rather aims to show how accurately, significantly and reliably the Taylor rule fundamentals could forecast the exchange rate movements. Accuracy means that the forecasts are close to the values of the exchange rate. This follows a claim by Engel and West (2005) that the random walk performance is not a surprise but a result of rational expectations. This means that the exchange rate acts as a near-random walk and the random walk is not easy to beat (Diebold 2017). The study seeks to shed light on these research questions: Can the Taylor rule fundamentals models accurately forecast currency exchange rates? How significant are the Taylor rule fundamentals in forecasting the exchange rate during the global financial crisis and great recession? Has the Taylor rule been effective in describing the exchange rate changes after the financial crisis? Can the exchange rate directions be forecasted by the Taylor rule fundamentals? In this paper, it is observed that the Taylor rule fundamentals could effectively describe and forecast the exchange rates until the financial crisis. In contrast, the Taylor rule fundamentals have been insignificant in forecasting exchange rates in the post-financial crisis. In addition, the choice of window size selection affects the forecast outcome of the models. The study shows that the smaller window size (60 observations) influences the Taylor rule fundamentals models to forecast the exchange rate better than the larger window size (120 observations). The remainder of the study is structured as follows: Section 1.1 gives some literature reviews on the topic. Section 2provides a theoretical framework, and details of the Taylor rule fundamentals are essential to this study. Section 3describes the models and specifications for the forecast. Section 4discusses the empirical framework, which also contains the data. Section 5contains the main empirical test result, and Section 6provides some economic analysis of the results. Section 7concludes the study. 1.1. Literature Review Recent research studies in the exchange rate forecast have advanced our knowledge of the exchange rate movement in the market. Some of the literature explains how Taylor rule fundamentals are used to forecast the exchange rate in different countries and with diversified approaches. Molodtsova and Papell (2009) performed one month ahead of out-of-sample prediction of the exchange rate with the Taylor rule fundamentals for 12 OECD countries vis-à-vis the U.S. for the post-Bretton Woods period (from 1973 to 2006). Quasi-real-time data were used in their paper. Out of 16 specifications generated, they found a 5% level significant evidence of exchange rate forecast for 11 out of the 12 OECD Countries. Their strongest evidence results from the symmetric Taylor rule model with heterogeneous coefficients, interest rate smoothing and a constant. In addition, the paper finds strong evidence of exchange rate predictability with the Taylor rule fundamental models as compared to the conventional interest rates parity, purchasing power parity (PPP) and monetary models. In addition, Molodtsova et al. (2011) used real-time quarterly data to find proof of out-of-sample predictability of the USD/EUR exchange rate based on the Taylor rule fundamentals. Another research by Molodtsova and Papell (2012) finds evidence of USD/EUR exchange rate predictability with the Taylor rule fundamentals during the financial crisis and the great recession. Economies 2021,9, 93 4 of 27 Moreover, Ince (2014) applied real-time data to evaluate the out-of-sample forecast of the exchange rate with PPP and Taylor rule fundamentals using single-equation and panel methods. Using bootstrapped out-of-sample test statistics, Ince found that the Taylor rule fundamentals better forecast the exchange rate at the one-quarter-ahead. However, the Taylor rule fundamentals forecast performance is not improved with the panel estimation. Contrary to the Taylor rule fundamentals, the researcher found that the PPP model was better at forecasting the exchange rate in the longer horizon (16-quarter). Its forecast performance increases in the panel model relative to a single-equation estimation. Byrne et al. (2016) contributed to the study by forecasting the exchange rates using the Taylor rule fundamentals and inculcating Bayesian models of time-varying parameters. They incorporated the financial crisis into their work and found that the Taylor rule fundamentals have the power to predict the exchange rate. Ince et al. (2016) extended the work by Molodtsova and Papell (2009) and demonstrated short-run out-of-sample predictability of the exchange rate with the two versions of the Taylor rule model for eight exchange rates vis-à-vis the U.S. dollar. Their research found strong evidence of exchange rate predictability with the Taylor rule fundamental model as compared to the Taylor rule differential and much stronger proof than the traditional exchange rate predictors. Cheung et al. (2019) performed exchange rate prediction redux and found the Taylor rule fundamentals outperform the random walk when the models’ performances are measured with the mean squared prediction errors. However, they did not find statistically significant performance when the DMW test was conducted. In addition to the basic linear model, Caporale et al. (2018) investigated the Taylor rule in five emerging economies through an augmented rule including exchange rates and a nonlinear threshold specification, which was estimated by the generalized method of moments. They found an overall performance of the augmented nonlinear Taylor rule to describe the actions of monetary authorities in these five countries. Furthermore, Zhang and Hamori (2020) performed exchange rate prediction by combining modern machine learning methodologies (neural network models, random forest and support vector machine) with four fundamentals that include the Taylor rule models, uncovered interest rate, purchasing power parity and monetary model. Their root mean squared error and Diebold–Mariano test results prove that the fundamental models together with the machine learning perform better than the random walk. 2. Taylor Rule Fundamentals Researchers have discovered that macroeconomics policies that center on the price level (inflation) and real output directly perform better than other policies such as money supply targeting. In 1993, John B. Taylor proposed that for a flexible exchange rate regime, the central bank adjusts its short-term interest rate target in response to changes in the price level (inflation rate) and real output (output gap) from a target as given in Equation (1): it†=πt+θ(πt−πt†) + σyt+r†(1) where i t† is the target for the short-term nominal interest rate. πt and πt† are the inflation rate and target level of inflation, respectively 4 . y t is the output gap (percent deviation of actual real gross domestic product (GDP) from an estimate of its potential level) and r † is the equilibrium level of the real interest rate. The parameters θand σ are the weights representing the central bank’s reactions to the changes in the inflation rate and the output gap. Taylor assumes that inflation and output have the same weight of reaction (0.5 parameters each). Both the inflation target and the real interest rate are 2% at equilibrium. According to Taylor (1993), the short-term nominal interest rate would be raised by the Fed if the inflation rises over the target inflation level or the realized output is above the potential output and vice versa. Molodtsova and Papell (2009) proposed fundamentals on the account of the Taylor rule monetary policy. The Taylor rule fundamentals suggest that when two economies fix their interest rates based on the Taylor rule, their interests would influence the exchange Economies 2021,9, 93 5 of 27 rate through the concept of uncovered interest rate parity. Now, following the asymmetric model by Clarida et al. (1998), the real exchange rate is added to the Taylor rule for the foreign countries. The idea is that the Fed sets the target level of the exchange rate to make PPP hold. That is, the nominal interest rate rises or falls if the exchange rate depreciates or appreciates from the PPP. This is expressed in Equation (2) below: it†=πt+θ(πt−πt†) + σyt+r†+ϑzt(2) where ztis the real exchange rate and ϑis the coefficient. Also by the Clarida et al. (1998) smoothing model, Molodtsova and Papell (2009) assume in Equation (3) that the U.S. actual nominal interest rate adjusts to its target rate and lagged value. The lagged value is added since, in decision making, the central bank could not observe the ex-post-realized nominal interest rate. Hence, the lag value helps to account for delay adjustment. it=(1−γ)it†+γit−1+vt(3) where γ is the coefficient of lag interest rate. Putting Equation (2) into (3) gives the interest rate reaction function of the U.S. it=(1−γ) [πt+θπt−πt†+σyt+r†+ϑzti+γit−1+vt(4) where ϑ = 0 for the U.S. if the real exchange rate approaches equilibrium. Molodtsova and Papell (2009) derive the Taylor rule fundamentals-based forecasting equation by subtracting the interest rate reaction function of the foreign country from the U.S. This results in an interest rate differential function represented in Equation (5). it−it∗=α+αππt−αππt∗+αyyt−αyyt∗+γit−1−γit−1∗−αzzt∗+vt(5) where * denotes foreign variables, the constants are: απ = (1 −γ )(1 + θ ), αy = σ (1 −γ ), αz = ϑ(1 −γ), and α= (1 −γ)(θπ* + r*), and vtis the shock term. The observation from Equation (5) is that, if the inflation rate rises over the target in the U.S. economy, the Fed responds to it by increasing the interest rate. It is worth noting that the monetary model of exchange rate implies an opposite relationship between interest rates and exchange rate, with higher domestic interest rate leading to an exchange rate depreciation. If uncovered interest rate parity (UIRP) holds, Dornbusch (1976) proposes that overshooting causes the U.S. dollar (USD) to later depreciate. It is empirically proven in most literature (Chinn and Quayyum 2012) that UIRP does not hold in the short run; hence, following Gourinchas and Tornell (2004), Molodtsova and Papell (2009) shows that the interest rate increment leads to a continuous rise in the USD. According to the Taylor rule (1993), the appreciation of the USD causes the inflation rate in foreign countries to rise. Applying the symmetric model, the foreign central banks respond by increasing the foreign interest rate. Investors begin to move their capital from the U.S. to foreign countries, since there would be higher returns on foreign investment. The demand for the USD diminishes, the exchange rate immediately appreciates up to the point where the interest rate differential equals the expected depreciation, and the dollar starts to depreciate (forward premium). Another reaction from the Taylor rule (1993) is that if the U.S. output gap increases, the Fed raises the Federal funds rate by αy , causing the USD to appreciate. By contrast, if the foreign country’s output gap increases and follows the Taylor rule, its central bank raises its interest rate, causing the USD depreciation. Moreover, the foreign central bank raises its interest rate when it observes a fall in its real exchange rate. This leads to a fall in the demand for the USD and immediate or forecasted depreciation. If the countries practice the smoothing model, a higher lagged interest rate increases current and expected future interest rates, which leads to an immediate and sustained USD appreciation. However, a higher lagged foreign interest rate causes a current or expected fall in the U.S. interest rate, Economies 2021,9, 93 6 of 27 and the USD is predicted to depreciate. From the rational expectations and the predictions explained above, it is observed that interest rate shocks that cause the central banks to respond to interest rate adjustment also have an impact on the exchange rate. Combining the analyses from Equation (5), the Taylor-rule-based exchange rate forecasting equation is developed as: ∆st+1=β−βππt+β∗ππt∗−β∗yyt∗−βit−1+β∗it−1∗+β∗zzt∗+vt(6) where s t is the log of the U.S. dollar nominal exchange rate taken as the domestic price of foreign currency and ∆st+1 is the change in the nominal exchange rate. βi represent the parameters of the forecasting equation. 3. Model Description Rossi (2013) explains how successful the linear equation model has been in forecasting the exchange rate. Therefore, a single-equation linear model as represented in Equation (6) is analyzed in this research. The same specifications proposed by Molodtsova and Papell (2009) would be used in this paper. Firstly, as proposed in Taylor (1993), there is a symmetric model ( βz * = 0) if the Fed and the foreign central banks follow the same rule to set the nominal interest rate based on current inflation, inflation gap (actual–target inflation), the output gap (actual–potential GDP) and equilibrium real interest rate. If the foreign central bank adds the real exchange rate to its Taylor rule ( βz * 6= 0), it is described as an asymmetric model (Clarida et al. 1998). Secondly, smoothing is considered, which is the interest rate expressed on its lag variable ( βi6= 0, βi * 6= 0). Contrary, without interest rate lag it is termed as no smoothing ( βi = 0, βi * = 0). The third model used in Molodtsova and Papell (2009) is homogeneous. This occurs when the domestic and foreign central banks have the same parameter in their Taylor rule fundamental variables ( βπ = βπ *, βy = βy *, βi = βi *). However, if their response parameters are not the same, the heterogeneous model would be constructed for it ( βπ 6=βπ *, βy6=βy *, βi6=βi *). Constant ( β6= 0) and no constant ( β = 0) are constructed as the fourth model. If the domestic and foreign central banks do not have the same target inflation rates and equilibrium real interest rates, a constant is added to the right-hand side of the equation and vice versa. The specifications by Molodtsova and Papell (2009) are modified to construct 16 models for this research as below: Model 1: Symmetric, Smoothing, Homogeneous Coefficients and a Constant { βπt−πt * yt−yt* it−1−it−1*} Model 2: Symmetric, Smoothing, Homogeneous Coefficients and no Constant { πt−πt * yt−yt* it−1−it−1*} Model 3: Symmetric, Smoothing, Heterogeneous Coefficients and a Constant { βπtπt * y t yt* it−1it−1*} Model 4: Symmetric, Smoothing, Heterogeneous Coefficients and no Constant { πtπt * y t yt* it−1it−1*} Model 5: Symmetric, no Smoothing, Homogeneous Coefficients and a Constant { βπt− πt* yt−yt*} Model 6: Symmetric, no Smoothing, Homogeneous Coefficients and no Constant { πt−πt * yt−yt*} Model 7: Symmetric, no Smoothing, Heterogeneous Coefficients and a Constant { βπtπt * ytyt*} Model 8: Symmetric, no Smoothing, Heterogeneous Coefficients and no Constant { πtπt * ytyt*} Model 9: Asymmetric, Smoothing, Homogeneous Coefficients and a Constant { βπt−πt * yt−yt* it−1−it−1* zt*} Model 10: Asymmetric, Smoothing, Homogeneous Coefficients and no constant { πt−πt * yt−yt* it−1−it−1* zt*} Model 11: Asymmetric, Smoothing, Heterogeneous Coefficients and a constant { βπtπt * y t yt* it−1it−1* zt*} Economies 2021,9, 93 7 of 27 Model 12: Asymmetric, Smoothing, Heterogeneous Coefficients and no Constant { πtπt * y t yt* it−1it−1* zt*} Model 13: Asymmetric, no Smoothing, Homogeneous Coefficients and constant { βπt−πt * yt−yt* zt*} Model 14: Asymmetric, no Smoothing, Homogeneous Coefficients and no Constant { πt− πt* yt−yt* zt*} Model 15: Asymmetric, no Smoothing, Heterogeneous Coefficients and Constant { βπtπt * ytyt* zt*} Model 16: Asymmetric, no Smoothing, Heterogeneous Coefficients and no Constant { πtπt * ytyt* zt*} 4. Empirical Framework 4.1. Benchmark Model and Window Sensitivity Selection The choice of benchmark and window size usually has an impact on the forecast results. After Meese and Rogoff (1983), it has been widely debated in most studies that the exchange rate follows a random walk. This implies that the exchange rate has a minimal chance of forecasting. There are two forms of random walk models discussed by Rossi (2013). These include a random walk without drift: ∆st+1 = 0. This is a martingale difference, which means that the current exchange rate steps from the previous exchange rate observation. Another form of a random walk considered in the literature is a random walk with drift. This is shown as ∆st+1 = δt , where δt is a drift term included in the random walk. The drift can be thought of as determining a trend in the exchange rate. The exchange rate forecast surveyed by Rossi (2013) affirms that random walk without drift as a benchmark performs better than with drift. Therefore, in this paper, the random walk without drift is used as the benchmark. Inoue and Rossi (2012) have shown that rolling windows of small size are more helpful to check predictive power. Although larger window size reduces the effect of outliers, Elliott and Timmermann (2016) explain that larger window size sometimes includes past data which are not important for current prediction. Hendry et al. (2019) add that a smaller window size excludes irrelevant information that might cause forecast failure. In this study, the empirical analysis is performed with a fixed-length rolling window with a 60 window size for the estimation. 4.2. Data Description The countries under study include Norway, Chile, New Zealand and Mexico vis-à- vis the United States of America. The currencies include U.S. dollar (USD), Norwegian krone (NOK), Chilean peso (CLP), New Zealand dollar (NZD) and Mexican peso (MXN). Considering the indirect quotation of the exchange rate data, the USD is used as the base currency in this study. Monthly data of each country from 1995M1 to 2019M12 are applied to the estimation and forecasting of the exchange rate and includes a key financial phenomenon such as the 2008 global financial crisis which affected the foreign exchange movement. The raw data used include the foreign exchange rates (S t ), interest rate (i t ), income (output) (y t ) and prices (p t ) 5 . The consumer price index (CPI) is used to measure the price level in the economy. The federal funds rate is used as a short-term interest for the U.S. The money market rates are used as the short-run interest rate for Norway, New Zealand and Mexico. The deposit rates are used as the short-run interest rate for Chile since there were no available money market rate data. The industrial production (IP) index is used to replace countries’ national income because GDP data are not consistently published. From 1995, enormous fluctuations in the exchange rates have been experienced in the countries. The dot-com boom between 2000 and 2001 led to economic growth in the U.S. As a result, the NOK, CLP, NZD and MXN currencies depreciated against the USD. Norway introduced inflation targeting in 2001 after the NOK depreciated highly in 2000 and the USD/NOK exchange rate reached 9.65. During the 2007–2008 financial crisis, the USD Economies 2021,9, 93 8 of 27 depreciated, and the NOK, CLP, NZD and MXN appreciated. This caused their exchange rates against the USD to fall. For instance, in 2008, the USD/NOK declined to about 4.94. The exchange rates immediately increased in 2009 when the USD appreciated against the other currencies. However, commodity currencies such as NOK, CLP and NZD quickly appreciated due to a boom in commodities prices in 2009. The depreciation of the NOK, CLP, NZD and MXN against the USD was observed in 2019. The MXN has especially been on an incessant path of depreciation against the USD due to loss of productivity in Mexico comparative to the U.S. (see Figure 1for details). Economies 2021, 9, x FOR PEER REVIEW 8 of 28 income (output) (yt) and prices (pt)5. The consumer price index (CPI) is used to measure the price level in the economy. The federal funds rate is used as a short-term interest for the U.S. The money market rates are used as the short-run interest rate for Norway, New Zealand and Mexico. The deposit rates are used as the short-run interest rate for Chile since there were no available money market rate data. The industrial production (IP) index is used to replace countries’ national income because GDP data are not consistently published. From 1995, enormous fluctuations in the exchange rates have been experienced in the countries. The dot-com boom between 2000 and 2001 led to economic growth in the U.S. As a result, the NOK, CLP, NZD and MXN currencies depreciated against the USD. Norway introduced inflation targeting in 2001 after the NOK depreciated highly in 2000 and the USD/NOK exchange rate reached 9.65. During the 2007–2008 financial crisis, the USD depreciated, and the NOK, CLP, NZD and MXN appreciated. This caused their exchange rates against the USD to fall. For instance, in 2008, the USD/NOK declined to about 4.94. The exchange rates immediately increased in 2009 when the USD appreciated against the other currencies. However, commodity currencies such as NOK, CLP and NZD quickly appreciated due to a boom in commodities prices in 2009. The depreciation of the NOK, CLP, NZD and MXN against the USD was observed in 2019. The MXN has especially been on an incessant path of depreciation against the USD due to loss of productivity in Mexico comparative to the U.S. (see Figure 1 for details). Figure 1. Foreign exchange rates. The output gap (yt) in this paper is measured as the percentage deviation of actual output from a Hodrick and Prescott (1997) (HP) generated trend6. This is because there is no standard description of potential GDP used in the central banks’ interest rate reaction function. There are other alternatives measures of the output gap for example percentage deviations of actual output from a linear time trend or a quadratic time trend. However, the HP trend is proven to be a more accurate measure than the other two measures (Ince and Papell 2013). All variables except interest rates are in logarithms. The raw data are combined to construct data for the forecast models7. Figure 1. Foreign exchange rates. The output gap (y t ) in this paper is measured as the percentage deviation of actual output from a Hodrick and Prescott (1997) (HP) generated trend 6 . This is because there is no standard description of potential GDP used in the central banks’ interest rate reaction function. There are other alternatives measures of the output gap for example percentage deviations of actual output from a linear time trend or a quadratic time trend. However, the HP trend is proven to be a more accurate measure than the other two measures (Ince and Papell 2013). All variables except interest rates are in logarithms. The raw data are combined to construct data for the forecast models7. 4.3. Estimation and Out-of-Sample Forecasting From one to three months-ahead out-of-sample forecast for USD/NOK, USD/CLP, USD/NZD and USD/MXN exchange rates are generated. The reason for the multistep is to check how the models forecast the exchange rates as the forecast horizon increases. The 16 models in Section 3are estimated by the ordinary least squared (OLS) using rolling windows (Molodtsova and Papell 2009). In the time-series data, periods 1995M1 to 1999M12 are used for the estimation and the remaining for the out-sample forecast. Thus, the first 60 observations of the time-series data are used to perform the first-month out-of-sample forecast in observation 61. The first data point is dropped, and observation 61 is added in the estimation sample and estimates the model over to forecast observation 62. The process is continued to extract the forecast error vector. A similar procedure is done for the 2 and 3 months-ahead out-of-sample forecast. 4.4. Forecast Assessment Approach There are different loss functions used in evaluating the out-of-sample forecast. These include mean squared error (MSE), mean absolute error (MAE) and root mean squared Economies 2021,9, 93 15 of 27 the economic performance of the Taylor rule. In this study, the sample data are adjusted to cover the financial crisis and the great recession and realized evidence of exchange rate forecastability with the Taylor rule fundamentals models 19 . The interest rate data show that until the 2008 financial crisis, the central banks adjusted their interest rates to control inflation. Hence, the monetary policy became very active as the central banks followed the Taylor rule descriptions. However, in the post-financial crisis period, the Taylor rule fundamentals could not forecast the exchange rate better than the random walk. The Fed lowered the interest rate to zero lower bound. Norway’s interest rate also decreased close to the lower bound. In 2019, New Zealand lowered the interest rate to 1%. Mexico experienced a 3% interest rate from 2013 to 2015, and it increased to 4.5% in 2019. Chile’s interest rate declined to 0.5% in 2009, increased after 2010 to 5% and then decreased after 2012 to 1.75% (check Figure 2) . From the empirical analysis, the Taylor rule lost its efficacy in forecasting the exchange rates after the financial crisis because the interest rates hit the zero lower bound. Economies 2021, 9, x FOR PEER REVIEW 15 of 28 The strongest evidence comes from the models with heterogeneous coefficients, which is consistent with the result of Molodtsova and Papell (2009). The most performing model based on the empirical result analysis is model 7, which incorporates symmetric with no interest rate smoothing and heterogeneous coefficients with a constant. This implies that the inflation rate and output gap influence the changes in the exchange rates. The heterogeneous coefficient means that the Fed and the foreign central banks respond differently to change in the inflation rate and the output gap. The constant shows that the central banks do not have the same target inflation rates and equilibrium real interest rates. In addition, the symmetric model explains that the Fed and the foreign central banks follow the same Taylor rule model. When the real exchange rate is added to the models (asymmetric), the performance was again boosted. This shows that the central banks react to the adjustment of PPP, which influences the exchange rate movements. The financial crisis causes a structural break in the sample data. Therefore, the coefficients might not be constant over time, and the model could favor the short-run period. Nikolsko-Rzhevskyy et al. (2014) test for multiple structural changes to examine the economic performance of the Taylor rule. In this study, the sample data are adjusted to cover the financial crisis and the great recession and realized evidence of exchange rate forecastability with the Taylor rule fundamentals models19. The interest rate data show that until the 2008 financial crisis, the central banks adjusted their interest rates to control inflation. Hence, the monetary policy became very active as the central banks followed the Taylor rule descriptions. However, in the post-financial crisis period, the Taylor rule fundamentals could not forecast the exchange rate better than the random walk. The Fed lowered the interest rate to zero lower bound. Norway’s interest rate also decreased close to the lower bound. In 2019, New Zealand lowered the interest rate to 1%. Mexico experienced a 3% interest rate from 2013 to 2015, and it increased to 4.5% in 2019. Chile’s interest rate declined to 0.5% in 2009, increased after 2010 to 5% and then decreased after 2012 to 1.75% (check Figure 2). From the empirical analysis, the Taylor rule lost its efficacy in forecasting the exchange rates after the financial crisis because the interest rates hit the zero lower bound. Figure 2. Nominal interest rates. Taylor (2015) explained during the 2015 IMF conference that the Fed has not adhered to the prescription of the Taylor rule. This rendered the monetary policy as being passive. Most central banks set short-term interest rates to a zero lower bound (ZLB) in response to the financial crisis and have adopted quantitative easing or large-scale asset purchases (balance sheets) to pursue their policies. For the Taylor rule fundamentals to be more descriptive, the study suggests that the central banks control the interest rate lower bound. Following Ben S. Bernanke’s (2015) presentation at the 2015 IMF conference, the Fed could also devise a monetary policy that would increase the inflation target, targeting the price level and targeting the output gap. If these variables are controlled by the central banks, the models would forecast the exchange rate better in the post-financial crisis period. In addition, Bernanke (2015) suggests mix monetary and proactive fiscal policies as they would ideally control the zero lower bound interest rate. 0 20 40 60 80 1995m1 2000m1 2005m1 2010m1 2015m1 2020m1 monthly I_US I_NOR I_CHI I_NZD I_MEX Figure 2. Nominal interest rates. Taylor (2015) explained during the 2015 IMF conference that the Fed has not adhered to the prescription of the Taylor rule. This rendered the monetary policy as being passive. Most central banks set short-term interest rates to a zero lower bound (ZLB) in response to the financial crisis and have adopted quantitative easing or large-scale asset purchases (balance sheets) to pursue their policies. For the Taylor rule fundamentals to be more descriptive, the study suggests that the central banks control the interest rate lower bound. Following Ben S. Bernanke’s (2015) presentation at the 2015 IMF conference, the Fed could also devise a monetary policy that would increase the inflation target, targeting the price level and targeting the output gap. If these variables are controlled by the central banks, the models would forecast the exchange rate better in the post-financial crisis period. In addition, Bernanke (2015) suggests mix monetary and proactive fiscal policies as they would ideally control the zero lower bound interest rate. The empirical results in this paper also depict that policymakers could not accurately apply the Taylor rule fundamentals to forecast the USD/NOK, USD/CLP, USD/NZD and USD/MXN in the 2 and 3 months ahead. The model is significant in only the one month-ahead out-of-sample forecast. Furthermore, central banks and asset managers or investors are usually interested in knowing the directional signs of the exchange rate in the market, as it equips them to efficiently and strategically decide either to sell or buy a security. The exchange rate directional accuracy was tested by applying the Pesaran and Timmermann (1992) test. The results show that the Taylor rule fundamentals models could accurately forecast the directional change of the USD/NZD exchange rate. This means that, other things equal, investing in New Zealand is more profitable compared to the other three countries since investors could forecast the exchange rate changes. The success of the directional accuracy of the New Zealand exchange rate change has a minimum of 50.47%. Except after the financial crisis where the Taylor rule has not been effective, Mexico’s exchange rate has a potential for directional accuracy considering the PT test results. However, the PT test Economies 2021,9, 93 16 of 27 results for Norway and Chile depict difficulties in evaluating the directional exchange rate changes in these two economies. 7. Conclusions In this research, the out-of-sample forecast is used to examine the application of the Taylor rule fundamentals in forecasting the exchange rates. To this end, an inference can be made that for the whole sample data, the Taylor rule fundamentals significantly forecast the exchange rates at the 1 month ahead using the Clark and West (2006,2007) test. However, at the 2 and 3 months-ahead forecast horizon, weak evidence of exchange rates forecastability is realized with the Taylor rule fundamentals models against the random walk in the four countries. The best-performing model is symmetric with no interest rate smoothing and heterogeneous coefficients with a constant. Taylor rule fundamentals models perform better in forecasting the exchange rates until the financial crisis. The Taylor rules’ performance is observed to be insignificant in the post-financial crisis. This could be attributed to interest rates approaching the zero lower bound. Moreover, the study showed that the models’ performance is sensitive to changes in the window size. The models perform better with small window size than with larger window size. New Zealand was the best-performing country and Chile the worst. It means that Chile slightly follows the Taylor rule for its monetary policy (Moura and Carvalho 2010;Moura 2010). For directional accuracy, the PT test demonstrates results in favor of the USD/NZD and USD/MXN exchange rates. By and large, this study provides evidence of exchange rate forecastability with the Taylor rule fundamentals using the CW test statistics. However, the Taylor rule fundamentals do not put the estimated coefficients on the Taylor rule variables. Rather, the models only examine out-of-sample forecastability (Ince et al. 2016). Therefore, further investigation into the connection between the Fed using the Taylor rule and the out-of-sample exchange rate forecastability is needed. Given the experience that the Taylor rule has not been effective since after the 2008 financial crisis, there is also a need for a broader approach such as the use of a balance sheet of the central banks. Funding: This research received no external funding. Data Availability Statement: The data presented in this study are available publicly. All sources are cited in the study. Acknowledgments: The author appreciates Stefan Reitz for his indelible supervision and immeasurable support. Conflicts of Interest: The author declares no conflict of interest. Appendix A. Clark and West (CW) Test The Clark and West (2006,2007) test uses simulations to show the existence of linear forecastability in a given series, contrary to the null hypothesis that the series follows a martingale sequence or difference (also known as a random walk). They compare the out-of-sample MSFE of the random walk and the alternative of linear forecastability. Model 1 is the parsimonious model (null model of random walk). Model 2 is the nested large model (alternative model). Given a linear regression as yt=βXt’+ et(A1) where y t is a dependent variable whose interest we want to predict (expected nominal exchange rate); X t’ is a vector of variables; and e t is the error term. Clark and West state under the null hypothesis that β = 0 and under the alternative hypothesis that β6= 0. They assume that under both hypotheses a martingale difference exists. This gives the conditional expectations of the errors being zero: Et−1et≡E(et|Xt, et−1, Xt−1, et−2, . . . ) = 0 Let ˆ y1t,t+τdenotes the forecasts of model 1 at period tof yt+τ. Economies 2021,9, 93 17 of 27 Where ˆ y2t,t+τ denotes the forecasts of model 2 at period tof yt+τ . τ is the forecast horizon, and yt+τis our actual value which is used as ∆st+1 in our analysis. Model 1 errors equal (yt+τ−ˆ y1t,t+τ), and model 2 errors equal (yt+τ−ˆ y2t,t+τ) MSPE1=P−1∑T t=T−P+1(yt+τ−ˆ y1t,t+τ)2(A2) MSPE2=P−1∑T t=T−P+1(yt+τ−ˆ y2t,t+τ)2(A3) Clark and West (2006,2007) solve the nested problem associated with the DMW test by introducing an adjustment term (adj). adj =P−1∑T t=T−P+1(ˆ y1t,t+τ−ˆ y2t,t+τ)2(A4) This gives the differences between the MSFE of the alternative hypothesis and the new adjustment term as (MSFE 2− adj). Hence, Clark and West test the null hypothesis of equal MSFE. H0: MSFE1=(MSFE2−adj)(A5) H1: MSFE1>(MSFE2−adj)(A6) After some computations: ˆ ft+τ=yt+τ−ˆ y1t,t+τ)2−yt+τ−ˆ y2t,t+τ)2−(ˆ y1t,t+τ−ˆ y2t,t+τ)2](A7) The mean then becomes: f=P−1∑T t=T−P+1ˆ ft+τ(A8) Clark and West (2006) states that if the test is against a random walk, the forecast of model 1 is just a constant value of zero, such that ˆ y1t,t+τ equals zero. Therefore, the MSFE 1 becomes the sample mean squared of the actual value ( ∆ s t+1 ). Thus, MSFE 1 = P −1 ∑T t=T−P+1yt+τ2. After some computation, ˆ ft+1=2(yt+1)(ˆ y2t+1)(A9) We then arrive at the test statistics: √P f /[sample variance of ˆ ft+1−f]1/2 For the one month-ahead forecast, the normal OLS standard error could be applied, since the forecast errors are white noise. However, Clark and West (2006,2007) state that as the forecast horizon increases, there would be overlapping in the data in forecasting τ—steps ahead. Therefore, MSPE1= (P−τ+1)−1∑T−τ+1 t=T−P+1(yt+τ,τ)2(A10) MSPE2= (P−τ+1)−1∑T t=T−P+1(yt+τ,τ−ˆ y2t,t+τ)2(A11) According to Clark and West, the time series follows a moving average (τ−1). That means there would be a serial correlation in the residuals. To solve this problem, Clark and West propose regressing a Newey–West robust variance estimator (Newey and West 1987) on ˆ f. This results in ˆ gt= 2yt(ˆ y2t, t+τ). The sample mean becomes g. Consistent sample variance, ˆ V= (P−2τ+2)−1∑T−τ+1 t=T−P+τ(ˆ gt−g)2(A12) Decision rule: Economies 2021,9, 93 18 of 27 Clark and West’s statistics follow a one-sided test that captures only the upper tail. This implies that if the test statistic is greater than +1.282 (0.10 test) or +1.645 (0.05 test), we reject the null hypothesis of a random walk model. Appendix B. Stationarity Test (Augmented Dickey–Fuller Test) Table A1. Unit root test with ADF. Norway Chile New Zealand Mexico U.S M T-Stat M T-Stat M T-Stat M T-Stat M T-Stat ∆st+1 2−11.046 *** 2 −11.944 *** 2 −10.044 *** 2 −13.660 *** - - it−12−1.394 2 −3.188 ** 2 −1.536 2 −3.533 *** 2 −1.616 πt3−5.497 *** 3 −3.348 * 3 −3.240 * 3 −3.503 ** 3 −5.073 *** yt3−5.635 *** 3 −12.589 *** 3 −10.567 *** 3 −4.561 *** 3 −2.957 zt2−1.713 2 −1.830 2 −1.546 2 −2.568 - - it−1−it−1* 1 −1.972 * 2 −4.402 *** Drift −2.007 ** 2 −3.910 *** - - πt−πt* 3 −4.306 *** 2 −3.347 ** 3 −3.567 ** 1 −2.534 ** - - yt−yt* 3 −6.109 *** 3 −13.307 *** 3 −10.118 *** 3 −5.091 *** - - zt(D) 2 −11.240 *** 2 −12.579 *** 2 −10.727 *** 2 −11.807 *** - - it−1(D) 2 −8.737 *** 2 −14.847 *** 2 −8.104 *** 2 −13.135 *** 2 −5.980 *** This table displays the stationarity results for the variables found in Equation (6). The augmented Dickey–Fuller is used to test the null hypothesis of a unit root (left-sided hypothesis). The decision test is that if the test statistic (z t ) is less than the critical value at either 1%, 5% and 10% significant level, and the null hypothesis is rejected. ***, ** and * mean the variable is stationary at 1%, 5% and 10% significant levels, respectively. The columns with M represent the model used for the stationarity test. Testing with model 1 shows that no constant, and no trend is added to the regression equation. Constant but no trend in the regression is explained by model 2. This implies that the fundamental variables are expected to move to equilibrium in the long run. Model 3 shows that we run the ADF test with both constant and trend since some of the variables such as prices and industrial production can change over time. z t (D) and i t−1 (D) represent the first difference of the real exchange rate and the lag interest rate, respectively. Appendix C. Out-of-Sample Forecast with 60 Window Size Table A2. One month-ahead forecasts using Taylor rule fundamentals with 60 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.8982 0.8192 0.8529 −0.3793 0.9868 3.1196 *** 0.9324 1.3117 * 2 0.9244 0.9753 0.8904 −0.6690 0.9163 0.7625 0.9221 0.0089 3 0.8834 2.5139 *** 0.7563 0.5004 0.8964 3.5655 *** 0.7891 0.9487 4 0.8329 1.3197 * 0.7602 −0.3995 0.8504 1.4779 * 0.7849 −1.0651 5 0.9247 0.6476 0.9434 −0.4662 0.9507 0.8869 0.9273 0.0639 6 0.9649 0.9933 0.9617 −0.5418 0.9779 1.3359 * 0.9491 −0.3468 7 0.9637 3.2694 *** 0.8496 0.0465 0.9321 2.1055 ** 0.9134 1.3351 * 8 0.8715 0.6963 0.8526 −0.9668 0.9105 1.4001 * 0.8935 −0.9495 9 0.8257 −0.8020 0.8435 −0.3542 0.9288 2.1614 ** 0.9085 0.9310 10 0.8993 0.8944 0.8546 −0.3696 0.9207 2.2608 ** 0.9298 1.2328 11 0.8444 1.6622 ** 0.7340 −0.3737 0.8345 2.6262 *** 0.7408 0.4761 12 0.8981 2.6203 *** 0.7598 0.5393 0.8375 2.3610 *** 0.7781 1.0582 13 0.8781 −0.2905 0.8826 −1.2108 0.8767 −0.9117 0.9167 0.0532 14 0.9255 0.6613 0.9429 −0.4715 0.9517 0.9092 0.9265 0.0717 15 0.9065 2.0943 ** 0.8003 −1.1445 0.8441 0.9267 0.8821 0.3895 16 0.9656 3.2811 *** 0.8505 0.0575 0.8948 1.9619 ** 0.9161 1.3074 * Table A2 presents the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West statistics for a 1 month-ahead out-of-sample forecast for the full sample from 1995 M1 to 2019M11. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The OLS estimation is performed using a rolling regression with 60 window size from 1995 M1 to 1999 M12 and the remaining sample for the out-of-sample forecast. For Norway, Chile and Mexico, the number of observations (T + 1) is 299, and the number of forecasts (P) is 239. However, due to the unavailability of data for New Zealand, (T + 1) = 274 and P = 214. R.MSFE above 1 indicates that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model. ***, ** and * show that the random walk is rejected at 1%, 5% and 10% significance level, respectively. Economies 2021,9, 93 19 of 27 Table A3. Taylor rule fundamentals forecast until the financial crisis with 60 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.9130 0.8512 0.8389 −0.0752 1.0111 2.3596 *** 1.0087 1.9035 ** 2 1.0129 1.8783 ** 0.8536 −0.5864 0.9819 1.9450 ** 1.0355 2.0877 ** 3 0.9449 1.6752 ** 0.7362 0.1460 0.9616 2.0835 ** 0.9746 1.6280 * 4 0.8462 1.0371 0.6939 −0.1092 0.9352 1.8031 ** 0.9606 0.8050 5 0.9354 0.7297 0.9147 −0.4414 1.0144 2.2626 ** 0.9956 0.8469 6 1.0066 1.7160 ** 0.9398 −0.6915 1.0465 2.8568 *** 1.0390 1.5825 * 7 0.9861 2.3374 *** 0.7856 −1.2723 1.0178 1.8237 ** 1.0109 1.8679 ** 8 0.9010 1.1180 0.7763 −0.9972 0.9571 2.0061 ** 0.9705 0.6645 9 0.8530 −0.6033 0.9342 −0.0923 0.9407 1.4517 * 1.0245 1.1560 10 0.9139 0.8741 0.8406 −0.0605 0.9853 2.1143 ** 1.0100 1.9186 ** 11 0.9389 1.1637 0.7254 −1.1851 0.8903 1.2233 0.9328 0.6491 12 0.9793 1.8889 ** 0.7406 0.1703 0.8890 1.4653 * 0.9842 1.7114 ** 13 0.9159 0.0625 0.8477 −1.5401 0.9282 0.4731 0.9871 0.3743 14 0.9361 0.7205 0.9142 −0.4438 1.0163 2.2849 ** 0.9961 0.8554 15 0.9638 1.4798 * 0.7407 −2.4105 0.9787 1.3009 * 1.0178 1.0772 16 0.9928 2.3920 *** 0.7864 −1.2923 0.9434 1.4581 * 1.0200 1.8680 ** This table presents the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West (CW) statistics for a 1 month-ahead out-of-sample forecast for the period until the financial crisis and the great recession. The sample covers from 1995 M1 to 2008 M12. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The OLS estimation is performed using a rolling regression with 60 window size from 1995 M1 to 1999 M12 and the remaining sample for the out-of-sample forecast. For Norway, Chile, New Zealand and Mexico, the number of observations (T + 1) is 168, and the number of forecasts (P) is 108. R.MSFE above 1 indicates that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model. ***, ** and * show that the random walk is rejected at 1%, 5% and 10% significance level, respectively. Table A4. Taylor rule fundamentals forecast in the post-financial crisis with 60 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.8985 −0.0223 0.8684 −0.6357 0.9087 1.6652 ** 0.9811 1.4600 * 2 0.8756 −0.9647 1.0034 0.9318 0.9215 −1.9769 0.8728 −1.0589 3 0.7832 0.5816 0.7355 −0.8177 1.1478 2.8820 *** 0.7222 0.4526 4 0.8367 0.0172 0.8238 −0.719 0.8205 0.6305 0.6978 −1.7000 5 0.9562 0.5537 0.9890 0.6090 0.9158 −1.6638 0.9187 −0.1533 6 0.9629 −0.9979 1.0016 0.9312 0.9553 −0.7282 0.9290 −1.083 7 0.8754 −0.1025 0.9388 −0.3689 0.8219 −1.4401 0.9228 0.7682 8 0.9059 0.4585 0.9470 −0.2228 0.9272 −0.0414 0.8678 −2.6233 9 0.7626 −0.5391 0.8774 −0.6870 0.7730 0.6323 0.8473 0.2281 10 0.9004 0.1153 0.8720 −0.6895 0.8014 −0.1290 0.9733 1.3052 * 11 0.7272 −0.3102 0.7325 -0.5015 0.9318 1.8273 ** 0.6055 0.1523 12 0.7873 0.6003 0.7391 −0.8850 0.8374 1.2645 0.6898 0.4658 13 0.8285 −0.0407 0.8794 −0.2763 0.7878 −2.2377 0.8658 −0.3735 14 0.9530 0.5297 0.9879 0.5823 0.9245 −1.4380 0.9172 −0.1245 15 0.8084 −0.1346 0.8172 −0.5581 0.6991 −2.2012 0.8026 −0.5160 16 0.8670 −0.1669 0.9365 −0.4199 0.8463 −1.1448 0.9185 0.6142 This table reports the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West statistics for a 1 month-ahead out-of-sample forecast for the period after the financial crisis. The sample runs from 2009 M1 to 2019M11. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The OLS estimation is performed using a rolling regression with 60 window size from 2009 M1 to 2013 M12 and the remaining sample for the out-of-sample forecast. For Norway, Chile and Mexico, the number of observations (T + 1) is 131, and the number of forecasts (P) is 71. However, due to the unavailability of data for New Zealand, (T + 1) = 106 and P = 46. R.MSFE above 1 indicates that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model. ***, ** and * show that the random walk is rejected at 1%, 5% and 10% significance level, respectively. Economies 2021,9, 93 20 of 27 Table A5. Two months-ahead forecasts using Taylor rule fundamentals with 60 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.8095 −0.9226 0.7973 −1.0027 0.9152 1.7413 ** 0.8888 0.5754 2 0.8692 −0.0550 0.8420 −1.2095 0.8583 −0.2475 0.8922 −0.6344 3 0.7399 1.0431 0.6466 −0.1134 0.7361 1.9249 ** 0.6468 −0.8145 4 0.7056 −0.9817 0.6342 −1.3666 0.7275 0.0013 0.7016 −2.4359 5 0.8601 −0.5850 0.9169 −1.0181 0.8989 −0.1538 0.8972 −0.6623 6 0.9354 0.3380 0.9394 −1.1774 0.9388 0.4637 0.9300 −0.7510 7 0.8710 1.7787 ** 0.7573 −1.0310 0.8217 0.6960 0.8389 −0.3220 8 0.7893 −1.0326 0.7677 −1.4836 0.7995 0.0367 0.8543 −1.7698 9 0.7268 −2.5201 0.7641 −0.4412 0.8270 0.4461 0.8469 0.1555 10 0.8095 −0.8510 0.7998 −0.9923 0.8392 0.9762 0.8875 0.5153 11 0.6732 0.2553 0.5868 −0.4163 0.6428 1.0762 0.5635 −1.2967 12 0.7526 1.2942 * 0.6519 −0.0551 0.6598 0.5047 0.6255 −0.7125 13 0.7938 −1.7760 0.8391 −1.2078 0.8034 −1.8932 0.8677 −0.8523 14 0.8615 −0.5556 0.9164 −1.0272 0.9000 −0.1150 0.8968 −0.6373 15 0.7857 1.0447 0.7010 −1.3586 0.7087 −0.3285 0.7921 −1.4813 16 0.8728 1.8382 ** 0.7587 −1.0201 0.7782 0.4674 0.8404 −0.3776 This table presents the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West statistics for a 2 months-ahead out-of-sample forecast for the full sample from 1995 M1 to 2019M11. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The estimation is performed using a rolling regression with 60 window size from 1995 M1 to 1999 M12 and the remaining sample for the out-of-sample forecast. The serial correlation is checked using the Newey–West estimator with lag 4. For Norway, Chile and Mexico, the number of observations (T + 1) is 299, and the number of forecasts (P) is 239. However, due to the unavailability of data for New Zealand, (T + 1) = 274 and P = 214. R.MSFE above 1 indicates that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model. ** and * show that the random walk is rejected at 5% and 10% significance level, respectively. Table A6. Three months-ahead forecasts using Taylor rule fundamentals with 60 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.7936 −1.0581 0.7825 −0.9523 0.8749 1.2196 0.9050 1.2010 2 0.8558 −0.3015 0.8308 −1.1224 0.8181 −0.8687 0.9050 −0.2514 3 0.7061 0.6939 0.5964 −0.0274 0.6735 1.9382 ** 0.6213 0.1006 4 0.6741 −1.2309 0.5927 −1.3343 0.6825 −0.3268 0.7119 −1.6080 5 0.8400 −0.8240 0.9112 −1.1327 0.8695 −0.6967 0.9062 −0.3429 6 0.9278 0.2569 0.9366 −1.2529 0.9185 0.0806 0.9372 −0.4941 7 0.8426 1.4340 * 0.7316 −1.1514 0.7888 0.3984 0.8457 0.1092 8 0.7664 −1.1463 0.7419 −1.6913 0.7639 −0.1768 0.8774 −1.2281 9 0.7011 −2.3364 0.7799 0.3491 0.7788 −0.2020 0.8672 1.2043 10 0.7932 −0.9875 0.7876 −0.9218 0.7909 0.3491 0.9043 1.1360 11 0.6040 0.0915 0.5238 −0.3922 0.5386 0.7354 0.5170 −0.1280 12 0.7165 0.9625 0.6044 0.0806 0.5860 −0.0114 0.5910 0.1893 13 0.7667 −1.7693 0.8304 −1.1784 0.7675 −2.0489 0.8869 0.1127 14 0.8420 −0.7888 0.9103 −1.1429 0.8705 −0.6441 0.9065 −0.3069 15 0.7315 0.7575 0.6639 −1.1044 0.6324 −0.7536 0.8108 −0.1792 16 0.8440 1.5029 * 0.7320 −1.1299 0.7409 0.1582 0.8487 0.1066 This table presents the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West statistics for a 3 months-ahead out-of-sample forecast for the full sample from 1995 M1 to 2019M11. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The estimation is performed using a rolling regression with 60 window size from 1995 M1 to 1999 M12 and the remaining sample for the out-of-sample forecast. The serial correlation is checked using the Newey–West estimator with lag 4. For Norway, Chile and Mexico, the number of observations (T + 1) is 299, and the number of forecasts (P) is 239. However, due to the unavailability of data for New Zealand, (T + 1) = 274 and P = 214. R.MSFE above 1 indicate that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model. ** and * show that the random walk is rejected at 5% and 10% significance level, respectively. Economies 2021,9, 93 21 of 27 Table A7. Directional accuracy test using Taylor rule fundamentals with 60 window size. Model Norway Chile New Zealand Mexico PT p-Value Directional Accuracy PT p-Value Directional Accuracy PT p-Value Directional Accuracy PT p-Value Directional Accuracy 1 0.3523 51.05% 0.4806 50.21% 0.0020 59.81% *** 0.0255 55.65% ** 2 0.7992 47.28% 0.4813 50.21% 0.1592 53.27% 0.4249 50.63% 3 0.6699 48.54% 0.4226 50.63% 0.0002 62.15% *** 0.0080 57.74% *** 4 0.1386 53.56% 0.5817 49.37% 0.1364 53.74% 0.3751 51.05% 5 0.1384 5356% 0.5282 49.79% 0.1782 53.27% 0.5926 49.37% 6 0.4239 50.63% 0.4753 50.21% 0.0341 56.07% ** 0.8404 46.86% 7 0.1827 52.72% 0.2380 52.30% 0.0728 55.14% * 0.1356 53.56% 8 0.2877 51.88% 0.6927 48.54% 0.0499 55.61% ** 0.8918 46.03% 9 0.8882 46.03% 0.9455 45.19% 0.1439 53.74% 0.5375 49.79% 10 0.4021 50.63% 0.4803 50.21% 0.0730 55.14% * 0.0487 54.81% ** 11 0.7288 48.12% 0.5223 49.79% 0.0014 60.28% *** 0.7269 48.12% 12 0.5153 49.79% 0.3247 51.46% 0.0012 60.28% *** 0.0080 57.74% *** 13 0.2355 52.30% 0.9329 45.19% 0.4155 50.47% 0.5948 49.37 14 0.0889 54.39% * 0.5277 49.79% 0.1732 53.27% 0.5382 49.79% 15 0.3105 51.46% 0.7996 47.28% 0.0096 57.94% *** 0.5783 49.37% 16 0.1521 53.14% 0.1655 53.14% 0.0056 58.41% *** 0.2001 52.72% The table reports the directional accuracy which explains the percentage change of the exchange rates that were accurately forecasted with the Taylor rule fundamentals. The directional accuracy is tested after performing the 1 month-ahead out-of-sample forecast for the full sample from 1995M1 to 2019M11. The window size used is 60. PT-test is used in this table. The null hypothesis is that the actual and forecasted exchange rate values are independently distributed. The columns are directional accuracy and the p-values of the PT-test. ***, ** and * indicate that the null hypothesis is rejected at 99%, 95% and 90% confidence interval, respectively. Table A8. Directional accuracy test until the financial crisis with 60 window size. Model Norway Chile New Zealand Mexico PT p-Value Directional Accuracy PT p-Value Directional Accuracy PT p-Value Directional Accuracy PT p-Value Directional Accuracy 1 0.2261 53.70% 0.2268 53.70% 0.0060 62.04% *** 0.0214 57.41% ** 2 0.8498 45.37% 0.5888 49.07% 0.0271 59.26% ** 0.0873 56.48% * 3 0.5767 49.07% 0.2268 53.70% 0.0167 60.19% ** 0.0319 58.33% ** 4 0.2252 53.70% 0.3683 51.85% 0.1263 55.56% 0.2725 52.78% 5 0.1263 55.56% 0.4387 50.93% 0.0169 60.19% ** 0.4637 50.00% 6 0.1713 54.63% 0.5014 50.00% 0.0004 65.74% *** 0.7102 47.22% 7 0.0857 56.48% * 0.6683 48.15% 0.0414 58.33% ** 0.1626 54.63% 8 0.2268 53.70% 0.4434 50.93% 0.0056 62.04% *** 0.8898 44.44% 9 0.8303 45.37% 0.9375 43.52% 0.0947 55.56% * 0.4556 50.00% 10 0.2261 53.70% 0.2268 53.70% 0.0580 57.41% * 0.0214 57.41% ** 11 0.3828 50.93% 0.5856 49.07% 0.1160 55.56% 0.9162 43.52% 12 0.3576 51.85% 0.2269 53.70% 0.0417 58.33% ** 0.0319 58.33% ** 13 0.2269 53.70% 0.9701 41.67% 0.1204 55.56% 0.7094 47.22% 14 0.0624 57.41% * 0.4387 50.93% 0.0418 58.33% ** 0.5499 49.07% 15 0.2916 52.78% 0.9522 42.59% 0.0019 63.89% *** 0.7748 46.30% 16 0.0873 56.48% * 0.5922 49.07% 0.0091 61.11% *** 0.2134 53.70% This table presents the directional accuracy which explains the percentage change of the exchange rates that were accurately forecasted with the Taylor rule fundamentals. The directional accuracy is tested after performing the 1 month-ahead out-of-sample forecast for the sample until the financial crisis from 1995M1 to 2008M12. The window size used is 60. PT-test is used in this table. The null hypothesis is that the actual and forecasted exchange rate values are independently distributed. The columns are directional accuracy and the p-values of the PT-test. ***, ** and * indicate that the null hypothesis is rejected at 99%, 95% and 90% confidence interval, respectively. Economies 2021,9, 93 22 of 27 Table A9. Directional accuracy test in the post-financial crisis with 60 window size. Model Norway Chile New Zealand Mexico PT p-Value Directional Accuracy PT p-Value Directional Accuracy PT p-Value Directional Accuracy PT p-Value Directional Accuracy 1 0.4667 49.30% 0.6811 47.89% 0.0173 63.04% ** 0.0639 60.56% * 2 0.5559 47.89% 0.3563 53.52% 0.5000 50.00% 0.9813 40.85% 3 0.7539 46.48% 0.8370 43.66% 0.0159 65.22% ** 0.5356 52.11% 4 0.4946 50.70% 0.6275 46.48% 0.1303 56.52% 0.9347 42.25% 5 0.2478 56.34% 0.8019 47.89% 0.5732 47.83% 0.7506 47.89% 6 0.8584 45.07% 0.3248 53.52% 0.7525 45.65% 0.9813 40.85% 7 0.9011 43.66% 0.2360 54.93% 0.4408 47.83% 0.3092 53.52% 8 0.8219 46.48% 0.4079 52.11% 0.1930 54.35% 0.9041 43.66% 9 0.6879 45.07% 0.6098 47.89% 0.1629 52.17% 0.2738 54.93% 10 0.5559 47.89% 0.6630 47.89% 0.1930 54.35% 0.1567 57.75% 11 0.9168 42.25% 0.2145 50.70% 0.0031 67.39% *** 0.7506 47.89% 12 0.7539 46.48% 0.7768 45.07% 0.0510 60.57% * 0.3304 54.93% 13 0.2740 54.93% 0.3227 47.89% 0.6032 47.83% 0.7315 49.30% 14 0.2478 56.34% 0.8589 46.48% 0.3668 52.17% 0.5964 50.70% 15 0.5808 49.30% 0.3227 47.89% 0.6096 43.48% 0.4702 50.70% 16 0.8442 45.07% 0.2360 54.93% 0.2022 52.17% 0.2887 53.52% This table presents the directional accuracy which explains the percentage change of the exchange rates that were accurately forecasted with the Taylor rule fundamentals. The directional accuracy is tested after performing the 1 month-ahead out-of-sample forecast for the sample after the financial crisis from 2009M1 to 2019M11. The window size used is 60. PT-test is used in this table. The null hypothesis is that the actual and forecasted exchange rate values are independently distributed. The columns are directional accuracy and the p-values of the PT-test. ***, ** and * indicate that the null hypothesis is rejected at 99%, 95% and 90% confidence interval, respectively. Appendix D. Out-of-Sample Forecast with 120 Window Size Table A10. One month-ahead forecasts using Taylor rule fundamentals with 120 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.9063 −2.0031 0.9592 −0.2488 0.8771 0.3452 0.9543 0.0799 2 0.9444 −1.8616 0.9742 −0.1443 0.9334 −0.2917 0.9562 −0.3261 3 0.9779 2.1202 ** 0.9303 0.8035 0.8851 1.5198 * 0.8611 −0.5517 4 0.9463 1.6581 ** 0.9124 −0.2047 0.9175 1.0503 0.9053 0.0158 5 0.9294 −2.0619 0.9740 −0.3223 0.9476 0.0666 0.9709 0.2446 6 0.9396 −2.3887 0.9838 −0.0265 0.9611 0.2315 0.9655 −1.2765 7 0.9906 2.2043 ** 0.9476 0.5955 0.9115 1.5477 * 0.8872 −1.8763 8 0.9533 0.2775 0.9624 −0.1645 0.9382 0.3376 0.9457 −0.5228 9 0.9031 −2.1569 0.9453 −0.0644 0.8608 −0.1151 0.9122 −1.2996 10 0.9082 −1.9923 0.9590 −0.2339 0.8742 −0.2544 0.9523 0.0339 11 0.9378 1.9324 ** 0.9175 0.7506 0.8527 1.0948 0.8450 −0.7142 12 0.9784 2.0693 ** 0.9307 0.8029 0.8682 0.6805 0.8576 −0.6301 13 0.8877 −2.8950 0.9661 −0.2035 0.9276 −0.3192 0.9403 −0.8985 14 0.9295 −2.0453 0.9740 −0.3240 0.9399 −0.0676 0.9681 0.2120 15 0.9473 1.5808 * 0.9288 0.1818 0.8782 1.0077 0.8608 −2.2444 16 0.9948 2.0611 ** 0.9492 0.6005 0.8837 1.0024 0.8807 −1.9585 Note: This table presents the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West statistics for a 1 month-ahead out-of-sample forecast for the full sample from 1995 M1 to 2019M11. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The OLS estimation is performed using a rolling regression with 120 window size from 1995 M1 to 2004 M12 and the remaining sample for the out-of-sample forecast. This is done to check the window sensitivity effect on the forecast. For Norway, Chile and Mexico, the number of observations (T + 1) is 299, and the number of forecasts (P) is 179. However, due to the unavailability of data for New Zealand, (T + 1) = 274 and P = 154. R.MSFE above 1 indicate that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model. ** and * show that the random walk is rejected at 5% and 10% significance level, respectively. Economies 2021,9, 93 23 of 27 Table A11. Taylor rule fundamentals forecast until the financial crisis with 120 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.8852 −2.0073 0.9418 −0.4307 0.8785 −1.1306 0.9697 0.6728 2 0.9099 −1.9923 0.9509 −0.3719 0.8878 −0.9423 0.9735 0.0772 3 1.0900 1.3703 * 0.8952 −0.0084 0.8971 −0.0339 0.9965 1.3638 * 4 0.9816 0.7211 0.8286 −0.7747 0.9041 −0.0054 1.0485 1.7695 ** 5 0.9075 −1.7657 0.9428 −0.4945 0.9344 −0.3158 1.0263 1.4759 * 6 0.9231 −1.8502 0.9483 −0.3994 0.9731 0.3959 1.0166 1.7695 ** 7 1.1572 1.5436 * 0.9330 −0.2035 0.9471 0.3600 0.9512 −0.4345 8 0.9177 −0.8924 0.9009 −0.8425 0.8529 −1.2831 1.0251 1.4005 * 9 0.8898 −1.4926 0.9539 −0.4171 0.8592 −1.5545 0.9652 0.3590 10 0.8869 −2.0436 0.9425 −0.4210 0.8643 −1.4193 0.9688 0.6423 11 1.0983 1.0720 0.8996 0.0564 0.8881 −0.0162 0.9934 0.7555 12 1.1164 1.3685 * 0.8980 0.0147 0.9047 0.1239 0.9980 1.3315 * 13 0.9074 −1.4510 0.9572 −0.3681 0.9190 −0.3573 1.0176 0.8172 14 0.9084 −1.7872 0.9430 −0.4890 0.9214 −0.4453 1.0257 1.4806 * 15 1.1416 1.3313 * 0.9545 −0.0076 0.8691 −0.2929 0.9377 −1.1007 16 1.1768 1.5382 * 0.9381 −0.1377 0.8642 −1.0485 0.9536 −0.4134 This table presents the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West (CW) statistics for a 1 month-ahead out-of-sample forecast for the period until the financial crisis and the great recession. The sample covers from 1995 M1 to 2008 M12. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The OLS estimation is performed using a rolling regression with 120 window size from 1995 M1 to 2004 M12 and the remaining sample for the out-of-sample forecast. For Norway, Chile, New Zealand and Mexico, the number of observations (T + 1) is 168, and the number of forecasts (P) is 48. R.MSFE above 1 indicates that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model. ** and * show that the random walk is rejected at 5% and 10% significance level, respectively. Table A12. Taylor rule fundamentals forecast in the post-financial crisis with 120 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.9873 0.2859 0.9888 0.1923 0.4821 −0.1519 1.0832 1.3836 * 2 0.8243 −1.3386 1.0631 2.0984 ** 1.0249 0.9143 0.7852 −0.7178 3 0.5715 −1.3473 1.0213 0.9981 0.9444 0.5387 0.9334 0.2996 4 0.5528 −1.3412 0.9931 0.2134 0.5873 −0.4984 0.6344 −1.4861 5 0.9873 0.2171 1.0684 2.3011 ** 1.0186 0.5244 0.7718 −0.6564 6 0.8874 −1.3408 1.0661 2.4418 *** 1.0263 0.9407 0.9806 −0.6138 7 0.9461 0.5983 1.0595 1.9601 ** 0.9223 −0.2711 0.7155 −0.8690 8 0.9755 0.7698 1.0576 1.7816 ** 0.9179 −0.3024 0.9627 −0.1872 9 1.0379 0.9937 0.9962 0.2962 0.4420 −0.1482 1.0545 1.0591 10 0.9887 0.3092 0.9869 0.1491 0.6625 −0.0935 1.0875 1.5015 * 11 0.4934 −1.2789 1.2106 2.5037 *** 1.1149 1.2577 0.7497 0.6434 12 0.5717 −1.3399 1.0245 1.0625 0.6338 −0.2372 0.9027 −0.2176 13 1.0379 0.8520 0.9898 0.1858 0.9805 0.2122 0.7788 −0.7543 14 0.9930 0.3341 1.0670 2.2501 ** 1.0239 0.6777 0.7513 −0.6540 15 0.8756 −0.4526 1.0151 0.5896 0.8187 −0.4810 0.7079 −0.8501 16 0.9267 0.3508 1.0578 1.9536 ** 0.9112 −0.3509 0.6714 −0.7914 This table reports the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West statistics for a 1 month-ahead out-of-sample forecast for the period after the financial crisis. The sample runs from 2009M1 to 2019M11. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The OLS estimation is performed using a rolling regression with 120 window size from 2009M1 to 2018M12 and the remaining sample for the out-of-sample forecast. For Norway, Chile and Mexico, the number of observations (T + 1) is 131, and the number of forecasts (P) is 11. However, due to the unavailability of data for New Zealand, 96 window size is used for the rolling regression. (T + 1) = 106 and P = 10. R.MSFE above 1 indicate that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model. ***, ** and * show that the random walk is rejected at 1%, 5% and 10% significance level, respectively. Economies 2021,9, 93 24 of 27 Table A13. Two months-ahead forecasts using Taylor rule fundamentals with 120 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.8803 −2.2716 0.9534 −0.3441 0.8392 −0.3273 0.9403 −0.3125 2 0.9276 −1.7871 0.9638 −0.4639 0.9014 −0.8334 0.9444 −0.6587 3 0.9232 1.2349 0.9038 0.4537 0.8160 0.3999 0.8258 −1.2834 4 0.8937 0.6735 0.8862 −0.4857 0.8492 0.0195 0.8785 −0.4377 5 0.9094 −2.2157 0.9604 −0.7320 0.9218 −0.5795 0.9646 0.0180 6 0.9289 −2.4494 0.9700 −0.4971 0.9404 −0.3609 0.9634 −1.3256 7 0.9497 1.4161 * 0.9193 −0.0666 0.8652 0.6480 0.8583 −2.3034 8 0.9189 −0.6074 0.9376 −0.7488 0.9009 −0.4585 0.9295 −0.9545 9 0.8589 −2.8561 0.9394 −0.0351 0.8209 −0.6857 0.8787 −1.7814 10 0.8831 −2.2473 0.9532 −0.3229 0.8373 −0.8153 0.9382 −0.3561 11 0.8721 1.1844 0.3881 0.2842 0.7758 0.0431 0.7959 −1.4531 12 0.9219 1.2620 0.9038 0.4536 0.7941 −0.4536 0.8212 −1.3982 13 0.8470 −3.1445 0.9565 −0.3710 0.8993 −0.9476 0.9168 −1.3938 14 0.9105 −2.2087 0.9604 −0.7318 0.9122 −0.7071 0.9619 −0.0051 15 0.8885 0.6817 0.8919 −0.4111 0.8279 0.2137 0.8143 −2.4786 16 0.9502 1.2829 * 0.9211 −0.0472 0.8426 0.2793 0.8506 −2.3818 This table presents the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West statistics for a 2 months-ahead out-of-sample forecast for the full sample from 1995 M1 to 2019M11. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The estimation is performed using a rolling regression with 120 window size from 1995 M1 to 2004M12 and the remaining sample for the out-of-sample forecast. The serial correlation is checked using the Newey–West estimator with lag 4. For Norway, Chile and Mexico, the number of observations (T + 1) is 299, and the number of forecasts (P) is 179. However, due to the unavailability of data for New Zealand, (T + 1) = 274 and P = 154. R.MSFE above 1 indicate that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model. * shows that the random walk is rejected at 10% significance level, respectively. Table A14. Three months-ahead forecasts using Taylor rule fundamentals with 120 window size. Model Norway Chile New Zealand Mexico R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat R.MSFE T-Stat 1 0.8710 −2.4126 0.9616 −0.1370 0.8200 −0.6012 0.9443 −0.1471 2 0.9191 −1.9359 0.9733 −0.1804 0.8871 −1.0439 0.9480 −0.5165 3 0.8933 0.9542 0.8895 0.2959 0.7854 0.1078 0.8317 −0.8595 4 0.8675 0.2673 0.8793 −0.4550 0.8154 −0.2684 0.8855 −0.0686 5 0.9016 −2.4215 0.9645 −0.6893 0.9148 −0.8014 0.9712 0.2162 6 0.9244 −2.6400 0.9730 −0.4653 0.9351 −0.5786 0.9704 −0.8319 7 0.9252 1.2043 0.9144 −0.1315 0.8545 0.5759 0.8683 −2.0687 8 0.9047 −0.8939 0.9407 −0.5960 0.8999 −0.5462 0.9421 −0.5791 9 0.8379 −2.8230 0.9433 0.1731 0.8003 −0.8976 0.8864 −1.2824 10 0.8744 −2.3761 0.9614 −0.1168 0.8195 −1.0777 0.9424 −0.1968 11 0.8282 0.8312 0.8540 0.1699 0.7347 −0.3572 0.8020 −0.8008 12 0.8896 1.0079 0.8887 0.2882 0.7600 −0.8445 0.8275 −0.9585 13 0.8258 −3.0366 0.9561 −0.3502 0.8912 −1.0845 0.9307 −0.7334 14 0.9033 −2.4166 0.9644 −0.6922 0.9044 −0.9198 0.9689 0.1927 15 0.8485 0.3265 0.8777 −0.4391 0.8160 0.1595 0.8260 −2.0376 16 0.9224 1.0422 0.9155 −0.1209 0.8400 0.3093 0.8607 −2.1412 This table presents the relative mean squared forecast error (R.MSFE) and the test statistics of Clark and West statistics for a 3 months-ahead out-of-sample forecast for the full sample from 1995 M1 to 2019M11. The random walk without drift is used as the null hypothesis where the alternative hypothesis is a linear model with the Taylor rule fundamentals. The estimation is performed using a rolling regression with 120 window size from 1995 M1 to 2004 M12 and the remaining sample for the out-of-sample forecast. The serial correlation is checked using the Newey–West estimator with lag 4. For Norway, Chile and Mexico, the number of observations (T + 1) is 299, and the number of forecasts (P) is 179. However, due to the unavailability of data for New Zealand, (T + 1) = 274 and P = 154. R.MSFE above 1 indicates that the alternative model outperforms the random walk. CW is a standard normal with a one-sided test, and it tests the significance and accuracy of the alternative model.