A Bayesian Change Point Analysis of the USD/CLP Series in Chile from 2018 to 2020: Understanding the Impact of Social Protests and the COVID-19 Pandemic

Abstract

Exchange rates are determined by factors such as interest rates, political stability, confidence, the current account on balance of payments, government intervention, economic growth and relative inflation rates, among other variables. In October 2019, an increased climate of citizen discontent with current social policies resulted in a series of massive protests that ignited important political changes in Chile. This event along with the global COVID-19 pandemic were two major factors that affected the value of the US dollar and produced sudden changes in the typically stable USD/CLP (Chilean Peso) exchange rate. In this paper, we use a Bayesian approach to detect and locate change points in the currency exchange rate process in order to identify and relate these points with the important dates related to the events described above. The implemented method can successfully detect the onset of the social protests, the beginning of the COVID-19 pandemic in Chile and the economic reactivation in the US and Europe. In addition, we evaluate the performance of the proposed MCMC algorithms using a simulation study implemented in Python and R.

1. Introduction

In the analysis of time series, change points are typically defined as those time points where we observe sudden changes in the data. These changes are often produced by events that can be internal or external to the nature of the process, and therefore, in order to properly understand the impact and nature of such events, it is important to accurately identify the time points where changes occur in the time series of interest. For this reason, change point detection (CPD) methods have been applied in different fields, including speech recognition [1], climate change [2], medicine [3] and finance [4], to name a few.

Multiple algorithms have been proposed to identify change points, including binary segmentation, mixture models and some Bayesian methods; see [5]. In this work, we apply a semiparametric model that consists of two components: a parametric component that aims to detect “jumps” in the mean of the series and a second component that incorporates a functional additive effect to differentiate abrupt changes from medium-term changes, as described in [6]. These types of functional effects have been used, for example, in comparative genomic hybridization (CGH) data, where genetic and chromosomal alterations are detected through the wave pattern derived from the measurement instrument, and are modeled with smooth functions. A nonparametric approach to model these type of effects can be found in [6,7,8]. A few comprehensive surveys discussing CPD methods have appeared more recently in the literature. See for example [9,10,11,12,13,14,15].

In this work, we use a Bayesian approach to fit the model, which allows us to incorporate prior knowledge and obtain posterior probabilities that facilitate the interpretation of results and decisions related to the presence of change points and functional effects. Since Bayesian estimation is generally difficult, Markov Chain Monte Carlo (MCMC) methods are typically considered. In this context, some methods for the detection of multiple change points without functional effect have been proposed in [16,17,18]. Here, we propose an alternative MCMC algorithm and evaluate its performance using a simulation study implemented in R [19] and Python [20].

Finally, we apply the method to study the USD/CLP exchange rate, for a range of dates that included the social protests that occurred in Chile between October and November 2019 (triggered by social discontent) and the start of the COVID-19 pandemic, with September 2020 as the deadline.

The reminder of the paper is structured as follows: Section 2 describes the Bayesian methodology used to detect change points, the model formulation and estimation procedure. Section 3, shows the results of a simulation study implemented in Python and R, used to compare the performance of the proposed MCMC algorithm. In Section 4, we apply the methodology to identify change points in the USD/CLP series in the context of social protests and COVID-19. We conclude with a short discussion in Section 5.

2. Methodology

For the model formulation, we consider the segmentation approach introduced in [6], in which the different portions of the sequence are represented by its mean value and a functional component that aims to capture the effect that a certain phenomenon can have in the series. Specifically, given a sequence of observations $Y_1,\dots ,Y_n$, we assume

$$Y_t={\mu }_k+f\left(x_t\right)+{ϵ}_t,\mathrm{for}t\in [{\tau }_k,{\tau }_{k+1}],$$

(1)

where the ${\tau }_k$ and ${\tau }_{k+1}$ represent the change points at the end points of the segment and ${\mu }_k$ is the mean value within the segment. Here, $x_t$ denotes any covariate of interest and $f\left(x_t\right)$ corresponds to the unknown functional component evaluated at $x_t$. The error terms ${ϵ}_t$ are assumed to be white noise, that is, they are independent with ${ϵ}_t\sim \mathrm{Normal}(0,{\sigma }^2)$ for all t.

Note that from this model formulation, we need to estimate the set of parameters $\theta =(\tau ,\mu ,K,f,{\sigma }^2)$, where $\tau =({\tau }_1,\dots ,{\tau }_K,{\tau }_{K+1})$ are the change points, $\mu =({\mu }_1,\dots ,{\mu }_K)$ correspond to the segments’ means and K is the (unknown) number of segments. To estimate f, we follow a nonparametric approach similar to the one described in [8] and assume

$$f\left(x_t\right)=\sum _{i=1}^M{\lambda }_j{\varphi }_j\left(x_t\right),$$

(2)

where ${\varphi }_j$ is a function chosen from a set of functions known as dictionary. This set can have as many functions as desired to improve the estimation of f. Therefore, if M denotes the number of functions that makes up the dictionary and ${\lambda }_j$ determines the coordinates of the $n\times M$ matrix of functions $F={\left[{\varphi }_j\left(x_t\right)\right]}_{t,j}$ obtained from the dictionary, we can write

$${(f\left(x_1\right),\dots ,f\left(x_n\right))}^{\prime }=\mathit{F}\mathit{\lambda }.$$

(3)

In general, the size M of the dictionary can be large and the matrix representation allows for a flexible framework to estimate functions with both smooth components and local irregularities. In practice, M varies between 200 and 300 functions due to its impact on computation time.

To identify the change points, we follow the variable selection approach proposed in [21]. We define $\mathit{X}$ to be the lower triangular matrix filled with 1’s

$$\mathit{X}=\left[\begin{array}{ccccc}{1}_{1,1}& 0& \dots & & 0\\ 1_{2,1}& 1_{2,2}& & & ⋮\\ 1_{3,1}& 1_{3,2}& \ddots & & \\ ⋮& ⋮& ⋮& \ddots & 0\\ 1_{n,1}& 1_{n,2}& \dots & 1_{n,n-1}& 1_{n,n}\end{array}\right],$$

(4)

and the $n\times 1$ vector $\mathit{\beta }$ to contain the values of the difference between the means of the segment $k+1$ and the previous segment k, where ${\mu }_0=0$. This way, we can rewrite Equation (3) in matrix form as

$$\mathit{Y}=\mathit{X}\mathit{\beta }+\mathit{F}\mathit{\lambda }+\mathit{ϵ}.$$

(5)

Note that the product $\mathit{X}\mathit{\beta }$ gives a vector in which the mean of each segment is repeated for all the coordinates corresponding to the segment, which allows us to determine both the positions of the change points and the mean value of each segment. As a result, the coordinates of the estimated vector $\widehat{\mathit{\beta }}$ correspond to estimates of the change points ${\widehat{\mu }}_k$. It follows that the vectors $\mathit{\beta }$ and $\mathit{\lambda }$ only have nonzero values either when change points occur or the functional component f is nonzero.

Bayesian Estimation

To estimate the parameters in Equation (5), we follow [22] and consider two instrumental variables, $\mathit{\gamma }=({\gamma }_1,\dots ,{\gamma }_n)$ and $\mathit{r}=(r_1,\dots ,r_M)$. These variables are associated with the parameters $\mathit{\beta }$ and $\mathit{\lambda }$, respectively, so that ${\gamma }_i=I_{({\beta }_i\ne 0)}$ for $i=1,\dots ,n$ and $r_j=I_{({\lambda }_j\ne 0)}$ for $j=1,\dots ,M$, where $I_{(·)}$ denotes the indicator function over the corresponding set. This way, we can learn whether the coefficients ${\beta }_i$ and ${\lambda }_j$ are present or not. If $d_{\gamma }$ represents the number of coefficients ${\beta }_i\ne 0$, and $d_r$ the number of coefficients ${\lambda }_i\ne 0$, we can consider subarrays ${\mathit{X}}_{\mathit{\gamma }}$, ${\mathit{\beta }}_{\mathit{\gamma }}$, ${\mathit{F}}_{\mathit{r}}$ and ${\mathit{\lambda }}_{\mathit{r}}$ so that we can establish the correspondence

$$\begin{array}{cc}\hfill \mathit{X}\mathit{\beta }& \iff {\mathit{X}}_{\mathit{\gamma }}{\mathit{\beta }}_{\mathit{\gamma }}\hfill \\ \hfill \mathit{F}\mathit{\lambda }& \iff {\mathit{F}}_{\mathit{r}}{\mathit{\lambda }}_{\mathit{r}},\hfill \end{array}$$

(6)

reducing the dimensionality of the problem and therefore simplifying the parametrization of the model to

$$\mathit{Y}={\mathit{X}}_{\mathit{\gamma }}{\mathit{\beta }}_{\mathit{\gamma }}+{\mathit{F}}_{\mathit{r}}{\mathit{\lambda }}_{\mathit{r}}+ϵ.$$

(7)

From here, the set of parameters that need to be estimated is ${\theta }^{\star }=({\mathit{\beta }}_{\mathit{\gamma }},\mathit{\gamma },{\mathit{\lambda }}_{\mathit{r}},\mathit{r},{\sigma }^2)$, which can be accomplished using a Bayesian approach, by drawing values from the posterior distribution $\pi ({\theta }^{\star }|y)$ associated with Equation (7).

From Bayes’s theorem, we have

$$\pi ({\theta }^{\star }|y)={\displaystyle \frac{L(y|{\theta }^{\star })\pi \left({\theta }^{\star }\right)}{\int L(y|{\theta }^{\star })\pi \left({\theta }^{\star }\right)d{\theta }^{\star }}},$$

(8)

where $L(y|{\theta }^{\star })$ is the likelihood of the sampling model and $\pi \left({\theta }^{\star }\right)$ is the prior distribution of the parameter ${\theta }^{\star }$. More specifically, we seek to determine the posterior joint distribution

$$\pi ({\mathit{\beta }}_{\mathit{\gamma }},\mathit{\gamma },{\mathit{\lambda }}_{\mathit{r}},\mathit{r},{\sigma }^2|\mathit{Y}).$$

(9)

In order to generate observations from the posterior distribution, we use Markov Chain Monte Carlo (MCMC) methods incorporating a Metropolis–Hastings (M-H) algorithm and a Gibbs sampler; see [23].

To complete the specification of the model, we assume:

(i)

${\gamma }_i\sim \mathrm{iid}\mathrm{Bernoulli}\left({\pi }_i\right)0\le {\pi }_i\le 1;i=2,\dots ,n;{\pi }_1=1$.

(ii)

$r_j\sim \mathrm{iid}\mathrm{Bernoulli}\left({\nu }_j\right)0\le {\nu }_j\le 1;j=1,\dots ,M$.

(iii)

${\sigma }^2$ follows a Jeffreys prior distribution.

(iv)

${\mathit{\beta }}_{\mathit{\gamma }}|\mathit{\gamma },{\sigma }^2$ follows a Zellner g-prior distribution.

$$\begin{array}{cc}\hfill {\mathit{\beta }}_{\mathit{\gamma }}|\mathit{\gamma },{\sigma }^2& \sim {\mathcal{N}}_{d_{\gamma }}(0,c_1{\sigma }^2{\left({\mathit{X}}_{\mathit{\gamma }}^{\prime }{\mathit{X}}_{\mathit{\gamma }}\right)}^{-1})\hfill \end{array}$$

(v)

${\mathit{\lambda }}_{\mathit{r}}|\mathit{r},{\sigma }^2$ follows a Zellner g-prior distribution.

$$\begin{array}{cc}\hfill {\mathit{\lambda }}_{\mathit{r}}|\mathit{r},{\sigma }^2& \sim {\mathcal{N}}_{d_r}(0,c_2{\sigma }^2{\left({\mathit{F}}_{\mathit{r}}^{\prime }{\mathit{F}}_{\mathit{r}}\right)}^{-1}).\hfill \end{array}$$

Then, the posterior distribution of ${\theta }^{\star }$ can be written as

$$\pi \left(\mathit{\theta }|\mathit{Y}\right)={\displaystyle \frac{\pi \left(\mathit{Y}|{\mathit{\theta }}^{\star }\right)\pi ({\mathit{\beta }}_{\mathit{\gamma }}|\mathit{\gamma },{\sigma }^2)\pi ({\mathit{\lambda }}_{\mathit{r}}|\mathit{r},{\sigma }^2)\pi \left(\mathit{\gamma }\right)\pi \left(\mathit{r}\right)\pi \left({\sigma }^2\right)}{\pi \left(\mathit{Y}\right)}},$$

(10)

where the likelihood is of the form

$$\pi \left(\mathit{Y}|{\mathit{\theta }}^{\star }\right)\propto {\left({\displaystyle \frac{1}{2\pi {\sigma }^2}}\right)}^{\frac{n}{2}}exp\left(-{\displaystyle \frac{1}{2{\sigma }^2}}{\left(\mathit{Y}-{\mathit{X}}_{\mathit{\gamma }}{\mathit{\beta }}_{\mathit{\gamma }}-{\mathit{F}}_{\mathit{r}}{\mathit{\lambda }}_{\mathit{r}}\right)}^{\prime }\left(\mathit{Y}-{\mathit{X}}_{\mathit{\gamma }}{\mathit{\beta }}_{\mathit{\gamma }}-{\mathit{F}}_{\mathit{r}}{\mathit{\lambda }}_{\mathit{r}}\right)\right).$$

(11)

Finally, we estimate ${\theta }^{\star }$ using the following two-stage approach:

Stage 1:

The estimation of $\mathit{\gamma }$ and $\mathit{r}$ conditionally to $(\mathit{Y};{\widehat{\mathit{\beta }}}_{\mathit{\gamma }},{\widehat{\mathit{\lambda }}}_{\mathit{r}},{\widehat{\sigma }}^2)$ using the Metropolis–Hastings algorithm; see [6,24].

Stage 2:

The estimation of ${\mathit{\beta }}_{\mathit{\gamma }},{\mathit{\lambda }}_{\mathit{r}}$ and ${\sigma }^2$ given $\widehat{\mathit{\gamma }},\widehat{\mathit{r}}$ and $\mathit{Y}$ using the Gibbs sampler algorithm:

• ${\mathit{\beta }}_{\mathit{\gamma }}|{\mathit{\lambda }}_{\mathit{r}},{\sigma }^2,\widehat{\mathit{r}},\widehat{\mathit{\gamma }},\mathit{Y}\sim {\mathcal{N}}_{d_{\gamma }}\left({\displaystyle \frac{{\mathit{T}}_{\mathit{\gamma }}{\mathit{X}}_{\mathit{\gamma }}^{\prime }(\mathit{Y}-{\mathit{F}}_{\mathit{r}}{\mathit{\lambda }}_{\mathit{r}})}{{\sigma }^2}},{\mathit{T}}_{\mathit{\gamma }}\right)$
  ${\mathit{T}}_{\mathit{\gamma }}={\sigma }^2{\left[{\displaystyle \frac{1+c_1}{c_1}}{\mathit{X}}_{\mathit{\gamma }}^{\prime }{\mathit{X}}_{\mathit{\gamma }}\right]}^{-1}$
• ${\mathit{\lambda }}_{\mathit{r}}|{\mathit{\beta }}_{\mathit{\gamma }},{\sigma }^2,\widehat{\mathit{r}},\widehat{\mathit{\gamma }},\mathit{Y}\sim {\mathcal{N}}_{d_r}\left({\displaystyle \frac{{\mathit{W}}_{\mathit{r}}{\mathit{F}}_{\mathit{r}}^{\prime }(\mathit{Y}-{\mathit{X}}_{\mathit{\gamma }}{\mathit{\beta }}_{\mathit{\gamma }})}{{\sigma }^2}},{\mathit{W}}_{\mathit{r}}\right)$
  ${\mathit{W}}_{\mathit{r}}={\sigma }^2{\left[{\displaystyle \frac{1+c_2}{c_2}}{\mathit{F}}_{\mathit{r}}^{\prime }{\mathit{F}}_{\mathit{r}}\right]}^{-1}$
• ${\sigma }^2|{\mathit{\beta }}_{\mathit{\gamma }},{\mathit{\lambda }}_{\mathit{r}},\widehat{\mathit{r}},\widehat{\mathit{\gamma }},\mathit{Y}\sim IG\left(a,{\displaystyle \frac{b}{2}}\right)$
  $a={\displaystyle \frac{n}{2}}+{\displaystyle \frac{d_{\gamma }}{2}}+{\displaystyle \frac{d_r}{2}}$
  $b={\left(\mathit{Y}-{\mathit{X}}_{\mathit{\gamma }}{\mathit{\beta }}_{\mathit{\gamma }}-{\mathit{F}}_{\mathit{r}}{\mathit{\lambda }}_{\mathit{r}}\right)}^{\prime }\left(\mathit{Y}-{\mathit{X}}_{\mathit{\gamma }}{\mathit{\beta }}_{\mathit{\gamma }}-{\mathit{F}}_{\mathit{r}}{\mathit{\lambda }}_{\mathit{r}}\right)+{\mathit{\beta }}_{\mathit{\gamma }}^{\prime }\left({\displaystyle \frac{{\mathit{X}}_{\mathit{\gamma }}^{\prime }{\mathit{X}}_{\mathit{\gamma }}}{c_1}}\right){\mathit{\beta }}_{\mathit{\gamma }}+{\mathit{\lambda }}_{\mathit{r}}^{\prime }\left({\displaystyle \frac{{\mathit{F}}_{\mathit{r}}^{\prime }{\mathit{F}}_{\mathit{r}}}{c_2}}\right){\mathit{\lambda }}_{\mathit{r}}.$

There are many types of prior distributions that can be considered on the residual variance ${\sigma }^2$ and the vector parameters ${\mathit{\beta }}_{\mathit{\gamma }}$ and ${\mathit{\lambda }}_{\mathit{r}}$. We adopt the popular and convenient g-prior [25] for ${\mathit{\beta }}_{\mathit{\gamma }}$ and ${\mathit{\lambda }}_{\mathit{r}}$. It preserves correlation structure among predictors in its prior covariance and yields closed-form marginal likelihoods, which leads to huge computational savings by avoiding sampling in the parameter space, leading to the preference of these priors over many other conventional prior distributions [26]. For ${\sigma }^2$, we adopt a Jeffreys prior, which can be viewed as the limit of an inverse-gamma density. When combined with (iv) and (v), the prior (iii) results in an inverse-gamma posterior for ${\sigma }^2$ (see the last distribution of Stage 2) and as such it behaves as a conjugate prior. For information on the use of the g and Jeffreys priors in regression models, see [27,28] and references therein.

3. Simulation Study

We simulated 100 series, each with $n=100$ values in the scale of $[0,1]$, where the functional part of the model was determined according to

$$f\left(t\right)=0.3\times sin\left(2\pi \frac{t}{20}\right)+1.5I_{0.1}\left(t\right)-2I_{0.5}\left(t\right)+3I_{0.6}\left(t\right),$$

(12)

that is, a sinusoidal curve with jumps at $t=0.1$, $t=0.5$ and $t=0.6$.

The left panel of Figure 1 shows the functional part with the three peaks described above and the right panel depicts randomly simulated mean values for the four segments.

For each series, $K=4$ segments were simulated, and each of them took a random value to simulate its mean value ${\mu }_k$ of the set {0,1,2,3,4,5}, so that they did not repeat at adjacent intervals. The positions of the change points were randomly selected, where the minimum size in each segment was of five, and the position with respect to the peaks was at a distance of at least three. In each simulation, the value of the standard deviation $\sigma$ was constant. To evaluate the performance of the model at different noise levels, we considered the values $\sigma =0.1,0.5,1.0,1.5$.

For the algorithm, we considered a dictionary of over 120 functions, including Haar functions [29] of the form $t\mapsto 2^{7/2}\varphi \left({\displaystyle \frac{2^{7}t}{n}}-k\right),k=0,\dots ,2^7-1$, where

$$\varphi \left(t\right)=\left\{\begin{array}{cc}1,\hfill & \hfill \mathrm{if}0\le t\le 1\\ 0,\hfill & \hfill \mathrm{otherwise}.\end{array}\right.$$

and Fourier functions of the form $t\mapsto sin\left(2\pi j{\displaystyle \frac{t}{n}}\right)$ and $t\mapsto cos\left(2\pi j{\displaystyle \frac{t}{n}}\right)$, with $j=1,\dots ,10$, and polynomial functions of the form t, $t^2$ and constant.

The hyperparameters $c_1$ and $c_2$ were set to 50. In each iteration, it was proposed to change two change points and two in the selection of functions. To start the algorithm, we started with three segments and functions from the dictionary. The threshold for the cutoff probability in the decision to consider or not as a change point was set to $0.5$, as for the selection of the dictionary functions.

To evaluate the performance of the algorithm in the simulated series, the root mean square error $\mathrm{RMSE}\left(\mu \right)$ was calculated for the detected change points:

$$\mathrm{RMSE}\left(\mu \right)=\sqrt{\left(\frac{1}{n}\sum _{t=1}^n{(\mu \left(t\right)-\widehat{\mu }\left(t\right))}^2\right)},\mathrm{with}\mu \left(t\right)={\mu }_k.$$

(13)

For the functional part, we obtained the RMSE of the distance between the true function and the estimated one:

$$\mathrm{RMSE}\left(f\right)=\sqrt{\left(\frac{1}{n}\sum _{t=1}^n{(f\left(t\right)-\widehat{f}\left(t\right))}^2\right)}.$$

(14)

Figure 2 shows a simulated series of the 100 generated for each variance. Simulated change points were included in each image as red lines and fitted curve as segmented lines. Figure A1 and Figure A2 of Appendix A show two additional simulated series for each noise level used ($\sigma$ in {$0.1,0.5,1.0,1.5$}) with the adjusted model.

The average RMSE values for the change points and the functions are shown in Figure 3. As expected, we observe how the error increases as the series has greater variability, which occurs in both cases.

To analyze how far our estimates are from the true simulated values for the 100 series, standard deviations were averaged for each level of $\sigma$ used. Here, it is noteworthy that for $\sigma$ in $\{0.1,0.5\}$ we had $\sigma$ reasonable average estimates of $0.17$ and $0.35$, respectively; however, for $\sigma \ge 1.0$ the estimates were too high, with averages of $0.79$ and $1.23$, respectively (see Figure 4).

Metropolis–Hastings Performance

The results for the simulations of the 100 series for each variance value were run on a Desktop Macbook Pro with a $2.5$ GHz Dual-Core Intel Core i5 processor, 12 GB of 1333 MHz DDR3 main memory, with 256 KB and 3 MB cache in L2 by core and L3, respectively. For each simulation, the execution times for the implementation of the Metropolis–Hastings algorithm were measured.

The execution time (in seconds) for the implementation in Python applied to the simulated series showed a decrease with an average time of $13.8$ s vs. $45.6$ in R for the simulated series with $\sigma =0.1$ (see the first two boxplots from left to right in Figure 5). In addition, using the Wilcoxon test for differences of means with $t_R-t_{Python}$ in independent samples at $95\%$ confidence, we obtained an interval of $[31.98;32.80]$ seconds; that is, the execution in R took between $31.98$ and $32.80$ s longer than in Python. In Figure 5, it can be seen that the times in Python for all levels of simulated noise were less than the times in R, where the smallest difference was reached for the simulated series with $\sigma =1.5$, obtaining a difference of means for $t_R-t_{Python}$ using Wilcoxon’s test at $95\%$ confidence, where R was between $26.54$ and $27.01$ s slower than Python. The difference evidenced that when we have a lower dispersion of the series, obtaining higher execution times in both R and Python (although this was more noticeable in R) can be explained because when the sparse matrices have less variance they will have more nonzero values; therefore, matrix multiplication will be more computationally expensive than when there are more zeros in them.

4. Application to the USD/CLP Dataset

The USD/CLP series under study considers the dates included between January 2018 and September 2020 (available in http://www.bcentral.cl, accessed on 23 January 2021) with daily frequency, with a total of $n=655$ records. This series is shown in Figure 6, highlighting the important dates that refer to the onset of the social crisis and COVID-19 pandemic events.

For the implementation of the model, we considered a dictionary with $M=248$ functions, including Haar functions [29], Fourier functions, second-degree polynomials and B-Splines [30]. The order of the dictionary is detailed in

Table 1. Details of the 248-function dictionary applied to the dollar/Chilean peso series.

Index	Function
1	$Constant=1$
2	Haar function in $t=1$
⋮	⋮
65	Haar function in $t=64$
66	$sin\left(2\pi \times 1\times {\displaystyle \frac{t}{100}}\right)$
67	$cos\left(2\pi \times 1\times {\displaystyle \frac{t}{100}}\right)$
⋮	⋮
90	$sin\left(2\pi \times 13\times {\displaystyle \frac{t}{100}}\right)$
91	$cos\left(2\pi \times 13\times {\displaystyle \frac{t}{100}}\right)$
92	t
93	$t^2$
94	B-Splines order 1, with 1 node
⋮	⋮
124	B-Splines order 1, with 30 nodes
125	B-Splines order 2, with 1 node
⋮	⋮
154	B-Splines order 2, with 31 nodes
155	B-Splines order 3, with 1 node
⋮	⋮
186	B-Splines order 3, with 32 nodes
187	B-Splines order 4, with 1 node
⋮	⋮
248	B-Splines order 4, with 62 nodes

. In

Table 2. Constants used for model execution.

Constants	Value
Iteration number	160,000
First simulated values (burn-in)	40,000
$c_1$ and $c_2$	50
Number of initial segments	5
Number of initial dictionary functions	5
Number of proposed change points in each iteration	1
Number of dictionary functions proposed in each iteration	1

, we summarize the constants used to analyze the USD/CLP dataset.

The methodology considered first the detection of change points without prior knowledge, expressed with a uniform probability of $0.001$ for each point in the series and also using different values of prior probabilities (${\pi }_i$) to suggest that 14 October 2019 (the onset of the social protests in Chile) and 3 March 2020 (when the first COVID-19 case was detected in Chile) should be identified as change points.

4.1. USD/CLP without Prior Knowledge

Figure 7 shows results obtained using the same prior probability $\pi \left(\mathit{\gamma }\right)=0.001$ for all the points of the series, where $\pi \left(\mathit{r}\right)=0.01$ for all M functions, and considering a threshold of $0.6$ and $0.5$ for the posterior probabilities of the selection of the change points and functions of the dictionary.

From Figure 7, we considered five selected change points and 12 functions. The change points were on the following dates:

3 May 2018:

The curve shows an increase in the dollar value that differs from its previous trend. We must mention that during March 2018, the president of the United States Donald Trump announced the imposition of tariffs for USD 50 mil to products of Chinese origin, under the assumption of “unfair trade practices” and “theft of intellectual property”. After this announcement, China informed that tariffs would be applied to 128 products from the United States. Given these facts, we associate this change point with a consequence of the trade war started between China and the United States.

1 August 2019:

This date could be related by the beginning of the economic war between the USA and China since President Trump unveiled the 10% tariff plan on 1 August 2018, blaming China for not following through on promises to buy more American agricultural products.

13 November 2019:

The abrupt rise in the value of the dollar is clearly appreciated on this date. We know that on 18 October 2019 the social protests began in Chile, with the dollar reaching an observed value of $713.23$ Chilean pesos, and then on 29 November 2019 reaching a historical maximum of $828.25$ pesos. One day before the date of this change point was detected, Tuesday 12 November 2019, there was a national strike in Chile, being the day with the greatest violence registered. This day of great violence motivated the agreement for social peace and the new constitution on 15 November 2019, which is focused on the creation of a new Magna Carta, to replace the one of 1980. Consequently, this change point can be explained given the particular situation that occurred in Chile in this period.

2 June 2020:

At this point on the curve, we see a sharp fall in the dollar both in Chile and globally. In the Chilean case, it is mainly due to the fact that during June 2020 copper presented a significant rise due to the reopening of the US and European economies.

18 June 2020:

This date could be associated with the reopening of the economy in the US and Europe after going through the peak period of the COVID-19 pandemic.

4.2. Changing Prior Probabilities

To explore the prior effect, we considered two dates: 18 October 2019, the start of the protest in Chile, and 11 March 2020, when the World Health Organization (WHO) declared the COVID-19 pandemic. For each of these dates, the prior $\pi \left(\mathit{\gamma }\right)=0.3$ was set. Figure 8 shows the results.

The change points detected corresponded to the following dates:

2 May 2018

We detected again this point, which is associated with the trade war between the USA and China.

12 November 2019

As in the case without prior knowledge, the model detected this change point, which was expected both visually and by known facts, as it corresponds to the social crisis. This time, the date on which the change point was detected is closer to the date set with prior knowledge than to the date set without prior knowledge.

3 June 2020

At this point, it can be seen how the model fits well to the fall in the value of the dollar, which did not occur without applying prior knowledge. This date coincides with the reopening of the economy in the US and Europe after going through the peak period of the COVID-19 pandemic.

18 June 2020

This date could be associated with the reopening of the economy in the US and Europe after going through the peak period of the COVID-19 pandemic.

8 July 2020

This date could be associated with the reopening of the economy in the US and Europe after going through the peak period of the COVID-19 pandemic.

It is important to note that two dates selected with higher probabilities, with and without prior knowledge, are related to the social crisis in Chile, 12 October 2019, and the COVID-19 pandemic, 3 June 2020.

The model which considers prior knowledge yielded an estimated variance ${\sigma }^2=0.0331$ less than that without prior knowledge on the proposed dates.

For the functional part, the model selected one Fourier function, $cos\left(2\pi \times {\displaystyle \frac{13t}{655}}\right)$, one Haar function, the function $t\mapsto t$, and finally four B-splines of second order and three B-splines of third order. These results are consistent in that models with splines are widely used to approximate these types of functions. In [31], the authors showed a model with splines to adjust the interest rate structure.

5. Discussion

The main concern of the change point problem is the complexity of building a flexible and effective model to estimate the unknown number and locations of break points in time series. In this paper, we have proposed to apply a flexible methodology in order to detect change points in the USD/CLP series in the context of social crisis and COVID-19.

For the dollar/Chilean peso time series, the model managed to detect the change points corresponding to the social protests and the start of the COVID-19 pandemic in Chile. It also suggested change points for the year 2018, dates that represented a certain beginning of the trade war between the US and China as it entered phase 2 with new tariff measures. The change points in the period of June–July 2020 coincided with the reopening of the economy in the US and Europe, after going through the first peak of COVID-19 infections.

The applied model was well adjusted for time series with small variances $\le 0.5$, which is disclosed in the simulation study. As the variance grew, we had greater errors in the detection of change points, and this was reflected in the absence of the detection of these.

For the model, we used MCMC methods, which require long processing times. By implementing these in the Python programming language, a decrease in the average execution time of between $31.98$ and $32.80$ seconds was obtained with respect to R for the Metropolis–Hastings algorithm in simulated series with $\sigma =0.1$. We recommend using Python for iterative models over R.

The advantage of the Bayesian model used is that it allows us to introduce prior knowledge, which helped direct the model in the appropriate case. Another advantage is that we have the flexibility to decide the cutoff threshold to be applied, both to define what a change point will be or which dictionary function will be present.

The results suggest that the events that affected the large economies of the US and Europe affected the dollar/Chilean peso exchange rate, as the trade war between the US and China and the start of the COVID-19 pandemic led to the devaluation of the Chilean peso with respect to the dollar. When economic reactivation occurred in the US and Europe, there was also a drop in the dollar value. On the other hand, the social outburst in Chile seems to have caused a sharp rise in the value of the dollar, due to higher country risk.