Time Series Models for the Colombian TRM Exchange Rate Luis Giraldo-Alvarado, Javier Riascos-Ochoa Department of Basic Sciences and Modeling, Universidad Jorge Tadeo Lozano, Bogotá, Colombia Abstract The US Dollar and the Colombian Peso currency exchange rate (Tasa Representativa del Mercado, TRM) is a financial series characterized by periods of high volatility. In this paper it was applied three classes of time series: the ARMA, ARMA-GARCH and Markov Switching (MS) models to represent the log- returns of the TRM between 2013-01-03 and 2020-07-02. The best models among several fitted models for each class were determined based on the Akaike and the Bayesian information criteria (AIC and BIC, respectively) and one-step forecasts in the period 2020-07-03 to 2020-07-31. The ARMA-GARCH model allowed a more precise description of the conditional variance than the ARMA model. Furthermore, the MS model defined 3 regimes each one with its own AR process. The regime with highest variability showed sporadic occurrences, and can be associated to three important events at global scale: the oil crisis in 2014, the US-China trade war in 2018, and the COVID-19 pandemics in march 2020. The results demonstrate the robustness of the models for forecasting in one-step or longer time windows. Keywords Exchange rate, volatility, ARMA, ARMA-GARCH, Markov Switching, Forecast, COVID-19. 1. Introduction The value of the US Dollar and its market value in Colombia, known as the TRM, has been in the general interest of the government, shareholders and other economic agents as it is the reference currency for conducting international transactions in the country. The presence of periods of high volatility is characteristic of the TRM series since its conversion to a floating exchange rate in September 1999 [7]. Therefore, it has been essential to have robust tools to describe its behavior and to forecast its future values and volatility. Several studies based on time series models have been conducted to analyze and predict the TRM. In [9], Hernandez and Mesa evaluated the impact of the intervention of the Central Bank on factors that affected the average response and variance of the TRM. Lega in 2007 found that the volatility has high persistence and amplitude, specially in periods of devaluation [10]. Cepeda and Casas in 2008 used a GARCH model to describe the variance of the TRM, finding that the GARCH(1,2) best explained the performance of stock prices in the period of evaluation [11]. In 2017, González applied latent regime models such as Markov Switching (MS), ARIMA ICAIW 2020: Workshops at the Third International Conference on Applied Informatics 2020, October 29–31, 2020, Ota, Nigeria " luism.giraldoa@utadeo.edu.co (L. Giraldo-Alvarado); javier.riascos@utadeo.edu.co (J. Riascos-Ochoa)  0000-0001-6680-8510 (J. Riascos-Ochoa) © 2020 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings http://ceur-ws.org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) and Random walk models, finding that the MS model performed better in a forecast horizon of 25 days and ARIMA in one-step forecasting [14]. Modeling currencies exchange rates from other Latin-american countries has also been of interest in recent years. Ortiz et al. fitted the US Dollar and Mexican Peso (MXN-USD) currency exchange rate with two classes of GARCH models with different distributions of the residuals and compared their performances [12]. Espinosa Gonzáles et al. implemented a MS to describe the exchange rate of the Peruvian Real (PEN-USD) finding that the best description of the series was given by a three regime model [6]. Later, Rodríguez et al. characterized the volatility of daily stock markets returns in Argentina, Brazil, Chile, Mexico and Peru, by applying extended linear models contrasting with traditional time series models [13]. Meneses and Alvarado im- plemented back propagation artificial neural networks to describe and predict in one-step and longer time windows the MXN-USD exchange rate [15]. However, there is a need to implement more robust models for the TRM that provide better forecasts for longer time windows. This article applies ARMA, GARCH, and MS models for the daily value of the TRM by fitting in a longer time window than the preceding works for the TRM, specifically from January 3, 2013 to July 2, 2020. This period is characterized by several events of high volatility and abrupt changes in the mean level. 2. Time series models In order to apply the time series models explained in this section, the data series must have a constant zero mean. Therefore, in the following it was assumed that 𝑦𝑡 is the log-returns of the TRM series: 𝑦𝑡 = 𝑙𝑛(𝑇 𝑅𝑀𝑡 ) − 𝑙𝑛(𝑇 𝑅𝑀𝑡−1 ). (1) Section 2 shows the statistical tests that assure that 𝑦𝑡 satisfies these minimal stationary conditions. 2.1. Autoregressive moving average models 𝐴𝑅𝑀𝐴(𝑝, 𝑞) The process 𝐴𝑅𝑀𝐴(𝑝, 𝑞) is constituted by the sum of an autoregressive process 𝐴𝑅(𝑝) and a moving average process 𝑀𝐴(𝑞) [1]: 𝑝 𝑞 𝑦𝑡 = 𝜙0 + ∑ 𝜙𝑖 𝑦𝑡−𝑖 + ∑ 𝜃𝑗 𝜖𝑡−𝑗 + 𝜖𝑡 , (2) 𝑖=1 𝑗=1 where 𝜖𝑡 is a white noise process at time 𝑡 with zero mean and variance 𝜎 2 , the AR com- ponents are the contribution of the previous values of 𝑦𝑡 in the 𝑝 previous times, and the MA components are the contribution of the white noise until 𝑞 periods in the past. Under certain conditions of the parameters {𝜙𝑖 } and {𝜃𝑗 }, ARMA processes are stationary, i.e., their mean and variance are constant in time. 147 2.2. Generalized autoregressive model of conditional heteroscedasticity 𝐺𝐴𝑅𝐶𝐻 (𝑝, 𝑞) One of the main problems in time series modeling is that the conditional variance 𝜎𝑡|𝑡−1 2 of the series upon its past values is not stationary in time (i.e., the series is heterocedastic). By defining the white noise 𝜖𝑡 with a conditional variance as an 𝐴𝑅𝑀𝐴(𝑝, 𝑞), it is obtained a GARCH model. Specifically: 𝜖𝑡 = 𝜎𝑡|𝑡−1 𝜐𝑡 , (3) 2 𝜎𝑡|𝑡−1 = 2 𝜔 + 𝛼1 𝜖𝑡−1 2 + ... + 𝛼𝑝 𝜖𝑡−𝑝 2 + 𝛽1 𝜎𝑡−1|𝑡−2 2 + ... + 𝛽𝑞 𝜎𝑡−𝑞|𝑡−𝑞−1 , (4) where 𝜐𝑡 is a white noise process with zero mean and variance equal to one, and {𝛼𝑖 }, {𝛽𝑗 }, 𝜔 are parameters that need to be estimated from the data. For stability, stationarity and positive values in the series, it is necessary that 𝛼𝑖 , 𝛽𝑗 ≥ 0, ∑𝑖=1 𝛼𝑖 + ∑𝑗=1 𝛽𝑗 < 1, and 𝜔 > 0 [2, 3, 4]. 𝑝 𝑞 2.3. Regime change models: Markov Switching Many financial series show abrupt changes caused by macroeconomic factors, government policies, or other circumstances [8]. This is materialized in the series through the appearance of sudden changes in their usual behavior, generating breaks in the mean and variance of the process. For instance, a relative long period of low volatility can abruptly change to a high volatility one, which in some cases can last for long periods, generating clusters in the series. This clearly implies a problem when trying to apply the assumptions of stationary. One of the approaches is to assume that the time series parameters depend on an external environment that evolves following a markovian process, for example a discrete Markov chain. In this context, the state of the Markov chain 𝑠𝑡 is called the regime, takes values in a countable set (𝑠𝑡 = 1, 2, … , 𝑛), and the transition between regimes 𝑘 and 𝑙 in one-step has probability 𝑃𝑘𝑙 which depends only on the present state (the markovian property): 𝑃𝑘𝑙 = 𝑃 (𝑠𝑡+1 = 𝑙|𝑠𝑡 = 𝑘) = 𝑃 (𝑠𝑡+1 = 𝑙|𝑠𝑡 = 𝑘, 𝑠𝑡−1 = 𝑘1 , …) . (5) A Markov Switching (𝑀𝑆) model for 𝑦𝑡 with 𝑛 regimes and 𝐴𝑅(𝑝) processes, denoted as 𝑀𝑆(𝑛)𝐴𝑅(𝑝), has the form: 𝑝 𝑦𝑡 = 𝜙0,𝑠𝑡 + ∑ 𝜙𝑖,𝑠𝑡 𝑦𝑡−𝑖 + 𝜖𝑡 . (6) 𝑖=1 That is, the AR process followed by 𝑦𝑡 has parameters 𝜙𝑖,𝑠𝑡 and white noise with variance 𝜎𝑠2𝑡 that depends on the regime 𝑠𝑡 at time 𝑡 [5]. When fitting an 𝑀𝑆(𝑛)𝐴𝑅(𝑝) to a time series, the AR parameters 𝜙𝑖,𝑠𝑡 for each regime 𝑠𝑡 = 1, … , 𝑛 and the one-step transition probabilities 𝑃𝑘𝑙 of the Markov chain must be determined. 3. Modeling of the TRM series This section presents a descriptive analysis of the series of the TRM and its log-returns, a methodology for the fitting and selection of the models, and the results of the fitted ARMA, GARCH, and MS models to the log-return series. 148 Figure 1: (a) TRM series and (b) log-return of the TRM series from 2013-01-02 to 2020-07-02. 3.1. The TRM and log-return series It is clear from Figure 1(a) that the TRM series is not stationary. In fact, the Dickey-Fuller test suggests that the hypothesis of an stationary series must be rejected with a p-value of 0.9093. With the log-return series 𝑦𝑡 defined by eq. (1) the hypothesis of a constant mean of this series cannot be rejected, according to the Dickey-Fuller test with a p-value of 0.01 (see Figure 1(b)). Therefore, it is justified to apply the models introduced in section 2 to this series. It is important to note in Figure 1(a) that the mean of the TRM series has at least two periods of regular increase, from 2015-01 to 2016-01 and from 2018-06 to 2020-07. Besides, Figure 1(b) shows clusters of volatility in the period 2015-01 to 2016-01, from 2018-01 to 2018-06 and the period between 2020-03 an 2020-07. It is well known that three exogenous events influenced these abrupt changes in the TRM structure: the oil crisis 2014-10 that came with a quick fall of the hydrocarbons prices [16], the US-China trade war 2018-07 with duties imposed for both sides, and the global expansion of COVID-19 in 2020 [18]. The autocorrelograms of 𝑦𝑡 in Figure 2 suggest that the series could be an 𝐴𝑅(1) process as the 𝑃𝐴𝐶𝐹 and 𝐴𝐶𝐹 present sharp falls after lags 1 and 2, respectively. After these lags the autocorrelations seem to be not significantly different to 0. 149 Figure 2: Autocorrelograms of the Log-returns series 𝑦𝑡 . Table 1 𝐵𝐼 𝐶, 𝐴𝐼 𝐶 and 𝑀𝐴𝐸 values of the best five ARMA models for 𝑦𝑡 . Model BIC AIC ARMA(1,0) -12689.42 -12678.39 ARMA(0,1) -12689.23 -12678.2 ARMA(2,0) -12687.69 -12671.15 ARMA(0,2) -12687.68 -12671.13 ARMA(1,1) -12687.68 -12671.13 Table 2 ARMA(1,0) model parameters and log-likelihood values. Model 𝜙1 𝜎 Log-likelihood ARMA(1,0) 0.165 0.00766 6347 3.2. Methodology for the fitting and selection of the time series models It was used the R software for the fitting, selection and forecast of the time series models. An iterative methodology was implemented in order to select the best ARMA, ARMA-GARCH and MS-AR models. First, several number of models for each class were adjusted via maximum- likelihood estimation and the BIC and AIC indexes were calculated. Then, with the best five models according to the lowest values of BIC and AIC, one-step forecasts were generated at a horizon of 21 days. Finally, the errors MAPE, MAE and RMSE were calculated and the model with the lowest MAPE is chosen. This methodology assures the simplicity and precision of the selected model. 150 Figure 3: Autocorrelograms of residuals of the ARMA(1,0) model for 𝑦𝑡 . Figure 4: One-step ARMA(1,0) forecast. 3.3. ARMA models for the log-return series Following the previous methodology, 900 ARMA(p,q) models (with different orders 𝑝, 𝑞) were evaluated. The five models with the lowest 𝐵𝐼 𝐶 and 𝐴𝐼 𝐶 values are presented in Table 1. The model with the best forecast is 𝐴𝑅𝑀𝐴(1, 0) (i.e., 𝐴𝑅(1)); the parameters are shown in Table 2. Observe that 𝜙1 is less than 1, which assures stationary, and 𝜙0 is fixed to zero to assure a zero mean for the log-return series. The value for 𝜙1 shows that the log-returns of the TRM at 𝑡 + 1 are related positively with the returns at 𝑡 in around 16%. Regarding independence of the residuals, the 𝐿𝑗𝑢𝑛𝑔 − 𝑏𝑜𝑥 test shows that the residuals of the model are uncorrelated, which also can be verified graphically in the autocorrelogram of the series of the residuals (Figure 3). Finally, the one-step forecast was performed with the function forecast in R. The projected 151 Table 3 𝐵𝐼 𝐶 and 𝐴𝐼 𝐶 values of the best five ARMA-GARCH models for 𝑦𝑡 . Model BIC AIC ARMA(1,0) GARCH(1,1) -7.1915 -7.1765 ARMA(0,1) GARCH(1,1) -7.1898 -7.1748 ARMA(1,1) GARCH(1,1) -7.1904 -7.1724 ARMA(0,2) GARCH(1,1) -7.1902 -7.1722 ARMA(2,0) GARCH(1,1) -7.1901 -7.1721 Table 4 ARMA(1,0)GARCH(1,1) model parameters values. ARMA 𝜙0 𝜙1 𝜐𝑡 ∼ (1,0) 0.18e-3 0.1924 Normal GARCH 𝜔 𝛼1 𝛽1 (1,1) 5.560e-5 0.1018 0.8951 values show a fairly good fit that hardly departs from the actual value of the series in most periods. It is notable that in times of sudden changes, the level and trend correction took few steps to perform (Figure 4). 3.4. ARMA-GARCH models for the log-return series Suppose an ARMA(𝑝, 𝑞) model for 𝑦𝑡 . If the white noise process 𝜖𝑡 follows the conditions for a GARCH (𝑝 ′ , 𝑞 ′ ) process (section 2.2), then 𝑦𝑡 follows an ARMA(𝑝, 𝑞)GARCH (𝑝 ′ , 𝑞 ′ ) model. Around 3700 combinations of ARMA-GARCH models were evaluated by varying the orders 𝑝, 𝑞, 𝑝 ′ , 𝑞 ′ , and their parameters were estimated with the rugarch package in R. The model with the best performance was ARMA(1, 0)GARCH (1, 1) (Table 3). Recall that the ARMA(1,0) model for the mean was also the best ARMA model in section 3.3, although with slightly dif- ferent parameters. However, they also fulfill the stationary conditions as well as the fitted parameters of the GARCH : 𝜙1 < 1, 𝛼1 > 0, 𝛽1 > 0, 𝛼1 + 𝛽1 ≤ 1 and 𝜔 > 0 (Table 4). Finally, the Ljung-box test indicates that the residuals from the model fit are not correlated, which also can be evidenced in the autocorrelograms of the series of the residuals in Figure 5. With respect to the one-step forecast of the selected model, the projected value shows a fairly good approximation that hardly departs from the actual value of the series. The confidence in- tervals, calculated based on the one-step predicted conditional variance from the GARCH(1,1) model, contain the real values of the series, although they are tighter compared with the con- fident intervals of the ARIMA model of section 3.3. Hence, the ARMA(1,0)GARCH(1,1) gives a better representation of the variability of the TRM series. 152 Figure 5: Autocorrelograms of residuals of the ARMA(1,0)GARCH(1,1) model for 𝑦𝑡 . Figure 6: One-step ARMA(1,0)GARCH(1,1) model forecast for 21 days. 3.5. Markov Switching models for the log-return series 𝑀𝑆(𝑛)𝐴𝑅(𝑝) models were adjusted using the MSwM package. The five models with the lowest 𝐵𝐼 𝐶 and 𝐴𝐼 𝐶 are shown in Table 5 from 63 models evaluated. The 𝑀𝑆(3)𝐴𝑅(0) was selected as it performed better in the forecast, that is, the best model consists in three regimes each one driving an 𝐴𝑅(0) process. Table 6 presents the values of the intercepts 𝜙0,𝑠𝑡 and the standard deviation of the white noise 𝜎𝑠𝑡 for each regime 𝑠𝑡 . Note that regime 2 has the higher volatility of the three regimes, 153 Table 5 Selection criteria of the best five MS-AR models. Model BIC AIC MS(3)AR(0) -13191.16 -13127.99 MS(3)AR(1) -13190.13 -13125.46 MS(2)AR(1) -13164.99 -13116.86 MS(4)AR(0) -13187.41 -13105.70 MS(2)AR(2) -13154.44 -13102.81 Table 6 AR(0) model parameters for the three regimes 𝑠𝑡 = 1, 2, 3 and transition probability matrix for 𝑠𝑡 . Regime 𝜙0,𝑠𝑡 𝜎𝑠𝑡 Transition probabilities 𝑠𝑡 1 2 3 1 0.0005 0.006 0.97 0.011 0.023 2 -0.0007 0.0114 0.008 0.98 0.00174e-6 3 0 0.0035 0.0177 0.002e-6 0.97 while regime 3 has the lowest by around half of the regime 1. Also, the coefficient 𝜙0 for regime 2 is the highest (in absolute terms) among the other regimes. This suggests that this regime predominates in the periods with rupture in the mean and with higher variance of the TRM. Moreover, Table 6 also shows the transition probability matrix of the Markov chain 𝑠𝑡 of the regimes. It is possible to observe that once in any regime 𝑖 = 1, 2, 3, 𝑠𝑡 will stay there for around 98% of the times. However, the probability of transition from regime 1 to regime 2 is very low (0.011) and almost zero from regime 3 to regime 2. This reflects the fact that periods of high volatility of the TRM series are rare but can persist for long periods. And on the other hand, there exists a major probability of transitions between regime 1 and regime 3, and vice versa. This represents the normal dynamic of the TRM with long periods of low-mid variability through time. The smoothed probabilities show excellent segmentation of the periods where the log-returns of the TRM series show marked fluctuations (Figure 7). It is observed that regime 2 periods start approximately at the same time when three critical worldwide situations initiated: the oil crisis, starting at 2015-01, the US-China trade war starting at 2018-06 and the COVID-19 out- break, which reached Colombia in 2020-03. These periods showed the fastest depreciation and volatility of the TRM in the last decade. The one-step forecast of the selected model in Figure 8 shows acceptable results in the de- scription of the average response, however the confidence intervals are wide. Figure 8 also shows the smoothed probabilities of each regime in the period 2020-05 to 2020-07. It is clear that the model develops a gradual transition from regime 2 (of high volatility) to regime 3 (of low-mid volatility), and that regime 1 loses preponderance. Regime 3 probability increases to almost 95% after an eight-day forecast. Finally, the residuals obtained from the model fit show 154 Figure 7: Smoothed probabilities of model MS(3)AR(1). Gray zones represent the smoothed probability of each regime. no significant correlations after the first lag, which is verifiable from Figure 9. 3.6. Forecast comparison When the results of the best ARMA, ARMA-GARCH, and MS models are compared, it was found that the Markov Switching model achieves the best one-step adjustment, getting lower error indicators (4% less) and 𝐵𝐼 𝐶 index as shown in Table 7. That is, by making the mean and variance depending on different regimes, it was possible a better forecast of the TRM series in the short term. When evaluated in a longer time window, in this case a horizon of 21 days, the forecast obtained with the ARMA model shows a better performance, although similar to the 𝐴𝑅𝑀𝐴 − 𝐺𝐴𝑅𝐶𝐻 model (see Table 7). Then, for this situation an ARMA model is a simpler and a more easy-to-implement representation, but the 𝐴𝑅𝑀𝐴 − 𝐺𝐴𝑅𝐶𝐻 model provides a more accurate representation of the variance. 155 Figure 8: One-step MS(3)AR(1) model forecast for 21 days. Figure 9: Analysis series of residuals MS(3)AR(1) model. In the case of the modeling of the series over long periods of time, the fact of being able to characterize several regimes to represent specific realities of the financial series, has a positive effect when obtaining an approximation for the average value of the series. In the same way and as in applying the forecast to one step, there is a clear advantage in having the possibility of modeling the variance through time, which is at the same time a more realistic representation of the behavior of the TRM. 156 Table 7 Forecast comparison for the best models 𝑀𝑆(3)𝐴𝑅(1), 𝐴𝑅𝑀𝐴(1, 0), 𝐴𝑅𝑀𝐴(1, 0) − 𝐺𝐴𝑅𝐶𝐻 (1, 1). Model/Forecast MAPE MAE RMSE BIC MS AR / One-Step 0.5186 18.991 23.8121 -13127.99 ARMA / One-Step 0.5375 19.7004 24.2352 -12678.39 ARMA-GARCH / One-Step 0.5389 19.7529 24.264 -7.1765 MS AR / Horiz. 2.989 108.88 115.048 -13127.99 ARMA / Horiz. 2.738 99.72 106.96 -12678.39 ARMA-GARCH / Horiz. 2.944 107.24 113.44 -7.1765 4. Conclusions The TRM is one of the most important economic items for the Colombian economy. This finan- cial series is influenced by various factors originated by national and international political and economic environment, adding stochastic behaviors to its structure and high volatility clusters, making its forecast quite difficult to carry out. In this paper the TRM series was studied over a longer period of time (from 2013-01 to 2020-07) compared with previous works, with the aim to propose a more robust time series model. For this, an iterative methodology was proposed for the fitting, selection and forecast of the most adequate models from three classes of time series: ARMA, ARMA-GARCH and Markov Switching (MS). The latter two with the property that they can model the non-stationary behavior of the TRM variance. A MS model with 3 regimes and 𝐴𝑅(0) processes achieved the best fit over the ARMA(1,0) GARCH(1,1) and the ARMA(1,0) models, showing closer values to the real series for a one-step prediction horizon. It is noteworthy that the best MS model for the PEN-USD exchange rate in [6] had also three regimes, one of them representing the high volatility periods and the other two expressing the mid-low volatility periods in a "normal" economy context. In the present work, regime 2 represented these periods of high volatility, which in the time window at con- sideration started approximately simultaneously with three critical events at global scale: the oil crisis (2015-01), the US-China trade war (2018-06), and the COVID-19 pandemics in Colom- bia (2020-03). Additionally, it is worthy to note the improvement in accuracy for the testing window compared with the MS model in [14]. 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