<figure><img src="image_1.png" /></figure>

HAL
open science

# Reference chronology of Business cycles for the UEMOA economies

Gohou Danon

► To cite this version:

Gohou Danon. Reference chronology of Business cycles for the UEMOA economies. 2020. hal-02877433v2

HAL Id: hal-02877433

https://hal.science/hal-02877433v2

Preprint submitted on 28 Jun 2020

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

Public Domain
Reference chronology of Business cycles for the
UEMOA economies.

Gohou Danon

University of Lorraine Nancy - France

gohou-ferdinand.danon@univ-lorraine.fr
Abstract

The importance of business cycle analysis justifies the existence of seve-
ral business cycle dating organizations for industrialized countries. Currently
there is no specific reference chronology for the purposes of diagnosis of the
cyclical behavior of the economy within the West African Economic and Mone-
tary Union (WAEMU). We use the classical NBER and a mixed multi-change
point approaches to describe the repetitive sequences that underlie business
cycles with reference to the economy of Cote d'Ivoire. We determine the refe-
rence chronology for business cycle on the basis of the Gross Domestic Product
(GDP) and a population of coincident indicators, which tracks fluctuations in
the ivorian economic activity. We sucefully anticipate business cycle upturns
and downturns of the leading economy in the WAEMU region. Moreover, we
add a reference chronology of the business cycle for the economy of Cote d'Ivoire
that can be used monitor the region's economy.

**Key words** : Business cycles, mixed multi-change point model, turning points, reference cycle
## 2.1 Introduction

Several business cycle dating organizations exist for industrialized countries, including the Organisation for Economic Co-operation and Development (OECD), the Economic Cycle Research Institute (ECRI) and the Centre for Economic and Policy Research (CEPR) for Europe and the National Bureau of Economic Research (NBER) for the United States. However, to the best of our knowledge, none of the West African Economic and Monetary Union (WAEMU<fnref n="3" />) countries has an official reference chronology of the economic cycle. From this fact, it seems necessary to create one, in order to measure the quality of the probabilities of recession which are the focus of chapter 3, when investigating a dynamic factor model with a Markov switching regime behaviour. The specification of the reference chronologies of economic cycles generally amounts to determining cyclical turning points. In this context, the literature observes two approaches. In the first approach, Burns and Mitchell (1946) initially identify turning points specific to each economic variable from a panel, then determine a turning point common to all the economic variables. Thanks to Bry-Boschan in 1971, the non parametric and popular Bry-Boschan algorithm, it is trivial to estimate the reference cycles chronology from either an aggregated series, for example GDP<fnref n="4" /> re-presenting global economic activity, or from several coincident time series as in Camacho, Gadea and Loscos (2018).

We implement both strategies for dating the turning points of a reference chronology within the WAEMU. To this end, we refer to the Bry-Boschan in 1971 BB algorithm, a non-parametric method allowing turning points to be dated for a single variable. The second method applies to a multivariate framework. It dates back to Burns and Mitchell (1946) and was further considered in the work of Harding et Pan-gan (2006). Indeed, the authors first identify the turning points of individual series, then look for a common date considered as a global turning point. Stock et Watson (2014) refer to a reverse path to that of Burns, by first dating the turning points of a selected population of variables and then determining average turning points along that population of points.

In this perspective, Camacho, Gadea and Loscos (2018) contribute twice to the

<footnote n="3">3. The West African Economic and Monetary Union (also known by its French acronym, UE-MOA) is a West African organization created on January 10, 1994, whose mission is to achieve the economic integration of member states with unified external tariffs. Members of the WAEMU are Benin, Burkina Faso, Côte D'Ivoire, Guinea-Bissau, Mali, Niger, Senegal, and Togo</footnote>

<footnote n="4">4. Gross Domestic Product</footnote>
literature as their turning point dating methodology simultaneously deals with the turning points from a set of economic variables while marking a point for inferences on the specific dates of regime change through a Markov chain process. This chapter examines both approaches to develop a reference cycle for Côte d'Ivoire's economy. Our final choice of the reference cycle depends on the quality of the results of these methods for this particular small open economy.

We consider this section to be a contribution to the literature, since this is, to our knowledge, the first reference chronology of business cycles in the CFA Franc zone. Section 2 is devoted to the methodology of the Bry-Boschan (BB algorithm) to dating the reference cycle. We present the data in section 3. The application of this univariate approach is presented in section 4 while section 5 presents the multivariate approach. Section 6 is dedicated to the application of the mixture multiple-points change model.

## 2.2 NBER type reference cycle : a BB algorithm approach

The persistent nature of cyclical imbalances fed the concept of economic cycle formalisation described by Burns and Mitchel in 1946. This essentially graphic orientation was automated through an algorithm by Bry-Boschan (1971), allowing the dating of the turning points with a rigorous description of a cycle' phases. Thus, the BB algorithm is a constrained filtering procedure of rules developed by Bry-Boschan (1971). In practice, the BB algorithm is used for detection of business cycles and operates directly on the data level (in contrast to the growth cycle) by selecting the local extrema of the time series in respect to key constraints on the length and amplitude of the expansion and recession phases. These constraints allow a coherent alternation of peaks and troughs belonging to the different phases of the cycles. The BB algorithm was modified by Harding and pagan (2002) to take into account quarterly data. This is the BBQ algorithm we implement in the next section : based on the gross domestic product series representing overall economic activity, this section explains the theory behind the analysis formulated by Bry-Boschan (1971) and Harding and pagan (2002). This non parametric specification allows us to define the turning points of a reference cycle for the economy of Cote d'Ivoire, representing 40 percent of the economy of the West African Economic and Monetary Union (WAEMU).
2.2.1 Procedure for calculating turning points

The location of every single turning point of the business cycle is the main focus
of the method of Bry et Boschan (1971). The BB algorithm is based on seasonally
adjusted monthly or quarterly time series. In practice, the BB algorithm is structured
around five successive steps, as follows :

— *First step : Identification of local extremums*

The BB algorithm identifies peaks and troughs over an interval of five (5) months. The adaptation made by Harding and pagan (2002), rather configures the local extremes on a periodicity of 2 quarters (or six months).

— *Second step : Respect of the alternation of peaks and throughs*

For successive peaks(troughs), the higher (lower) peak (trough) is selected.
Thus the necessary alternation of turning points is ensured.

— *Third step : Minimum duration of the cycle*

By applying the Spencer curve on the resulting series from step 2, we eliminate
cycles too short to be useful. Indeed, the minimum duration of 15 months for a
complete cycle comprises at least three reversal points, configured as follows :
peak-trough-peak or trough-peak-trough.

— *Fourth step : Minimum duration of a phase*

Taking into account information from the third step, any contraction (peak to peak) or expansion (peak to peak) must respect the minimum duration of 5 months required for a complete phase. We eliminate all turning points associated with a phase of less than 5 months.

— *Fifth step : Some rules of censorship*

1. If the first inflection point is a peak (trough), its value must necessarily be greater (lower) than the first point of the studied series. Otherwise, the phase is eliminated

2. If the last turning point is a peak (trough), its value must necessarily be greater (lower) than the first point of the studied series. If not, the phase is eliminated.

3. Finally, any phase whose peak value is less than that of the trough is eliminated.

Thus, the BB algorithm performs a selection of peaks and troughs, candidates to be
turning points, and then applies a series of operations to eliminate points that do
not meet the criteria characterising the cycles. The BB algorithm can be summarised
in table 1 as follows :

<table><caption>TABLE 1 – summary of the BB algorithm</caption><thead><tr><th></th><th>Business Cycle</th><th>Growth Cycle</th></tr></thead><tbody><tr><td>Pic</td><td>y<sub>t-k</sub> > y<sub>t</sub> < y<sub>t+k</sub></td><td>z<sub>t-k</sub> > z<sub>t</sub> < z<sub>t+k</sub></td></tr><tr><td>Creux</td><td>y<sub>t-k</sub> < y<sub>t</sub> > y<sub>t+k</sub></td><td>z<sub>t-k</sub> < z<sub>t</sub> > z<sub>t+k</sub></td></tr><tr><td>Durée minimale d'une phase</td><td>5 mois</td><td>5 mois</td></tr><tr><td>Durée minimale d'un cycle</td><td>15 mois</td><td>15 mois</td></tr></tbody></table>

## 2.3 Data

Business cycle forecasting exercises are based upon the use of actual and survey variables. In a non-parametric formulation including the BB algorithm, we consider monthly GDP as the benchmark measure of the overall state of an economy. Note that GDP<fnref n="5" /> at purchase price is calculated by the World Bank "*without deduction for the depreciation of manufactured assets or for the depletion and degradation of natural resources. This series has been linked to produce a coherent time series in order to counter for breaks in series over time due to changes in base years, source data and methodologies.*"

We also consider a set of variables correlated with the GDP, including industrial production index, household consumption of manufactured products, and import and export data. Indeed, the industrial production index, is arguably seen as one of the most important variables used to obtain quarterly GDP data. Moreover, other studies, such as Camacho, Bengoechea and Perez Quiros (2006) confirm that the manufacturing sector is the most affected by fluctuations in the economic cycles.

## 2.4 Application of the BB algorithm to Cote d'Ivoire

### 2.4.1 Reference chronology of the business cycle

#### 2.4.1.1 Data transformation and estimation method
Current practice in the process of dealing with economic cycles requires quarterly time series. Our goal being to set up a monthly chronology of business cycles, we convert the frequency of

<footnote n="5">5. Source: World Bank staff estimates based on World Bank national accounts data archives, OECD National Accounts Statistics and IMF World Economic Outlook database.</footnote>
the available annual GDP into monthly figures. Thus, we consider modern methods of temporal disaggregation, including Denton-Cholette, particularly the Dagum and Cholette (2006) method as an extension of the Denton (1971) and the Chow-Lin regression method explained in the work of Chow and Lin (1971). While the former method is a purely mathematical approach which is easy to perform and does not need any other data series, the economic value of the resulting series may be limited.

We convert the annual GDP into monthly series with an indicator instead, in a way that gives an economic value to the resulting series. Indeed, with an indicator, the “Denton-Cholette” method transfers the movement of the indicator to the resulting series. Thus, the monthly Industrial Production (IP) series serves as a movement indicator (Denton (1971)). We check for the obvious correlation between the indicator (IP) and the resulting GDP by performing the Chow-Lin regression (Chow and Lin (1971)).

It is not surprising that Sax and Steiner (2013) mention temporal disaggregation methods as widely used in official statistical institutions, including France, Italy and other European countries. For example, in the WAEMU region and particularly in Cote d'Ivoire, the National Institute of Statistics uses the Dagum and Cholette, 2006) method to compute the quarterly figures of the Gross Domestic Product. We use this method on the GDP data sourced from the World Bank for Cote d'Ivoire.

<figure><img src="image_2.png" /><figcaption>FIGURE 1 – Monthly seasonally adjusted GDP</figcaption></figure>
The dating of turning points through the Bry-Boschan algorithm, requires level series, rather than trend series. Consequently, the measurement of contractions is better seen as sequences of absolute declines in a series, rather than as periods of slow growth compared to a trend (Watson, M. (1994)).

Overall, the sum of the quarterly GDP series is consistent with the annual GDP series. This can be illustrated with the diagnostic of the Chow-Lin regression results :

<figure><img src="image_3.png" /><figcaption>FIGURE 2 – Chow-Lin Regression results</figcaption></figure>

The figure above shows that the coefficient of industrial production is different from zero. This suggests that the monthly GPD not only follows the movement of the industrial production series, as suggested by the Denton-Cholette method, but there is a significant correlation between these variables from the Chow-Lin perspective

2.4.1.2 **Empirical results from the BB algorithm.** As noted above, dating turning points through the Bry-Boschan algorithm, requires level series, rather than trend series. Consequently, the measurement of contractions is better seen as sequences of absolute declines in a series rather than as periods of slow growth compared to a trend (Watson, M. (1994)). The analysis of the GDP index shows the effectiveness of the Bry-Boschan algorithm in capturing the turning points of business cycles, rather than of growth cycles. Indeed, as expected, it appears that the BB algorithm does not consider falls of short duration followed by expansions, a process which is rather akin to acceleration cycles. The result from the R package 'tempdisagg' is as follows :
<table><caption>TABLE 2 – Chronology of economic contractions in Cote d'Ivoire</caption><thead><tr><th>Phase</th><th>Start</th><th>End</th><th>Duration</th></tr></thead><tbody><tr><td>Recession</td><td>1999 :12</td><td>2000 :12</td><td>12</td></tr><tr><td>Expansion</td><td>2000 :12</td><td>2002 :03</td><td>15</td></tr><tr><td>Recession</td><td>2002 :03</td><td>2002 :12</td><td>9</td></tr><tr><td>Expansion</td><td>2002 :12</td><td>2003 :11</td><td>11</td></tr><tr><td>Recession</td><td>2003 :11</td><td>2004 :11</td><td>12</td></tr><tr><td>Expansion</td><td>2004 :11</td><td>2005 :11</td><td>12</td></tr><tr><td>Recession</td><td>2005 :11</td><td>2007 :01</td><td>14</td></tr><tr><td>Expansion</td><td>2007 :01</td><td>2010 :10</td><td>45</td></tr><tr><td>Recession</td><td>2010 :10</td><td>2011 :04</td><td>6</td></tr></tbody></table>

Table 2 above indicates that the economy of Cote d'Ivoire experienced five (5) successive phases of contractions between December 2000 and April 2011. The average duration of recession was 10.6 months, while expansion periods were, on average, 20.75 months.

<figure><img src="image_4.png" /><figcaption>FIGURE 3 – Reference Chronology of Cote d'Ivoire Business Cycle</figcaption></figure>

### 2.4.2 Conclusion

The BB algorithm, applied to the seasonalised monthly GDP series, reveals that the economy of Cote d'Ivoire experienced five successive phases of contraction (see Table 2) between December 1999 and April 2011. The average duration of contraction was 10.6 months, while expansion periods averaged 20.75 months. Small periods of
with a negative value at $k = 7$ we conclude that the model with $k = 6$ is better. The model thus shows some differences with regards to the BB algorithm results. Although the MMC model did not manage to capture the trough of 2007, it was able, nevertheless, to detect a recession in 2008 and 2015, in contrast to the BB algorithm.

## 2 Conclusion

We construct an index of business cycle diffusion for the economy of Cote d'Ivoire. Such a dating process is aimed at assessing the extent of cyclical movements throughout a developing economy and particularly in the West African Economic and Monetary Union (WAEMU). This analysis of key turning points from a reference economic cycle is built upon several works. The Bry-Boschan algorithm and the Mixed Multi-change points (MMC) model ) identify business cycles turning points in a small open economy from the WAEMU. We find that the two approaches capture approximately the same number of cycles in Cote d'Ivoire between 1996 and 2018. Although the MMC model did not manage to capture the trough of 2007, it was able, nevertheless, to detect a recession in 2008 and 2015, in contrast to the BB algorithm. This is the first reference index which tracks fluctuations in the ivorian economic activity. This reference chronology is a key to monitor and analyse business cycles in the WAEMU economies.
points are set to represent the measure of the central tendency of the distribution of
the economic indicators. Hence, the authors ease the analysis with the possibility of
making statistical inferences about the estimated reference cycle.

In a natural extension of the population concept above, Camacho, Gadea and Loscos (2018) proposed an algorithm which minimizes the number of distributional hypotheses and considers estimators which simultaneously determine regime changes and their precise dates. Thus, the authors perform a finite mixed model approach which helps to classify observations to adjust for clustering, and to model unobserved heterogeneity. Indeed, considering finite mixed modelling, the observed data are assumed to belong to unobserved subpopulations and mixtures of probability densities, or regression models are used to model the outcome of interest. Finite mixed models have been used in many fields, including clustering of internet traffic (Jorgensen 2004) and perceived consumer risk (Wedel and DeSarbo 1993).

In this context, Camacho, Gadea and Loscos (2018) implemented several Monte Carlo simulations to reveal stylised facts associated with economic fluctuations from a set of disaggregated economic variables. The Markov-switching mixed model proposed by the authors addresses several questions, including the determination of reference cycle turning point dates with minimal distributional hypotheses, while assuming that each pair of turning point dates is the outcome of a combination of an undetermined number of individual bivariate normal distributions. The authors specified the overall reference cycle chronology as the means of these distributions. The regime change through the cycles involves a Markov chain with a measurable state space, where the probabilities of transition are not recurrent. In other words, the transition probability matrix is constrained such that we end up with multiple turning points, from which the reference cycle dates of peaks and troughs are selected through a finite mixed model.

We adopt this modelling framework as it seems suitable as an alternative to the
classic BB algorithm in the context of a multivariate environment with missing data
and ragged-edge data. To our knowledge, this is an important contribution to the
literature, since there is still no official reference chronology of business cycles for
WAEMU.
### 2.5.1 The mixture multiple-point change (MMC) model.

The mixed multiple-point change (MMC) model of Camacho, Gadea and Loscos (2020) is built on the population concepts of Stock and Watson (2014) with interesting innovations. Indeed, the idea of turning point clusters is not only maintained through the concept of the means of the separate normal distributions of regime change, but the authors contribute further since business cycle phases are obtained through estimation instead of being given. A second major innovation from the MMCM framework lies in the ability of the model to allow the determination of turning points by considering confidence intervals. In other words, the authors make it trivial to evaluate the reference cycle through statistical inference.

In the detail, given a period of time, K pairs of peaks and troughs dates are used to divide the entire reference business cycle into distinct phases of expansions and contractions. Following Camacho, Gadea and Loscos (2020), we note each pair of reference turning points dates as $\omega_k = (\omega_k^P, \omega_k^C)$ where P represents the peaks and C denotes the troughs. These pairs of turning points are calculated from a set of R coincident economic indicators. Thus, from each indicator r, specific pairs of turning points dates are collected in subsets of size $n_r$ where $r = 1, ..., R$. Hence from the BB algorithm (Bry-Boschan (1971)), one obtains $N = n_1+n_2+...+n_R$ individual pairs of turning points dates $\tau_j^r$ where $j = 1, ..., n_r$ and $r = 1, ..., R$. Therefore $\tau = (\tau_1, ..., \tau_N)$ represents the set of all turning points, with individual pairs $\tau_i = (\tau_i^P, \tau_i^C)$ and $i = 1, ..., N$.

In this context, Camacho, Gadea and Loscos (2020) make couple of assumptions for modelling the reference cycle turning points including:

1. the distribution of the clusters of the turning points across the reference cycle is heterogeneous while homogeneous within. To fix this, the authors conclude, on one hand, that the latter case implies that the groups of specific turning point dates cluster around different reference turning points, while the former statement suggests that $\tau$ has a different probability distribution around each reference turning point.

2. when considering these two facts together, the turning points at each phase $k$ are generated from a family of bivariate normal distributions with means $\omega_k$ and covariance matrices $\Sigma_k$.

3. specific turning point dates are generated from a mixture of K normal distri-
butions

4. to make the case for a Markov regime change process, the authors indicate that the reference cycle turning points are inferred from an unobserved state variable s taking values from 1 to K. These state variables are collected in S = (s₁, ..., sN). Hence, the process s follows a first-order Markov process implying that the probability of transitioning to the current state at time t depends only to the state at time t − 1 :

$$
\mathrm{Pr} (\tau_i = k | s_{i-1} = \ell, \dots, s_1 = \vartheta, \tau^{i-1}) = \mathrm{Pr} (s_i = k | s_{i-1} = \ell) = p_{\ell k} \quad (1)
$$

where $\tau^i = (\tau_1, \dots, \tau_i)$ represents a set of consecutive pairs of turning points dates. The process involves a $(K \times K)$ matrix $P$ with rows summing to 1.

5. for the turning point dates $\tau_i$ are displayed in an ascending order and the pairs of reference turning points dates $\omega_k$ do not overlap, $\omega_k^P < \omega_k^T < \omega_{k+1}^p$, for all $k = 1, \dots, K$, consequently, $s_i < s_{i+1}$.

In this context, it seems natural to considere a non-ergotic Markov chain. In other
words, we have a non-recurrent probability distribution, i.e. once the process reaches
a reference turning point date k for the first time, it remains there with probability
p<sub>kk</sub> until the next reference turning point date k + 1 with probability 1 − p<sub>kk</sub> for
k = 1, ..., K − 1. Thus the transition probability matrix P depending on probability
p<sub>kk</sub> :

$$
P = \begin{pmatrix}
p_{11} & 1-p_{11} & 0 & \cdots & 0 \\
0 & p_{22} & 1-p_{22} & \cdots & 0 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
0 & \cdots & 0 & p_{K-1K-1} & 1-p_{K-1K-1} \\
0 & \cdots & \cdots & 0 & 1
\end{pmatrix} \quad (2)
$$

where $p_{kk}$ are collected in the vector $\theta = (\theta_1, \theta_2, ..., \theta_K, \theta_P)$ with $\theta_k = (\omega_k, \sum_k)$ representing the parameters of the distribution for each pair $k$ and the following finite Markov mixture model captures the density of the distribution of turning points dates. :

$$
p(\tau_i | \theta, \tau^{i-1}) = \sum_{k=1}^{K} \text{prob}(S_i = k | \theta_k, \tau^{i-1}) p(\tau_i | \theta_k, \tau^{i-1}) \quad (3)
$$

where : $p(\tau_i|\theta, \tau^{i-1})$ is the normal density, $N(\omega_k, \sum_k)$ with different means and co-
variance matrices $prob(S_i = k|\theta_k, \tau^{i-1})$ is the probability of sampling with group k
with $0 \le prob(S_i = k|\theta_k, \tau^{i-1}) \le 1$ and $\sum_{k=1}^{K} prob(S_i = k|\theta_k, \tau^{i-1}) = 1$.

2.5.2 The Bayesian inference

The mixture multiple-point change (MMC) model above, requires a Bayesian in-
ference to maintain a complete posterior probability distribution on a set of the
coincident economic indicators. As such, we need to figure out unknown and unob-
servable quantities, given known facts. This is the purpose of a Bayesian inference,
which uses full-power probability theory to draw a conclusion and learn from data.
In the case of business cycle analysis, the unknown quantities we want to figure out
are the distribution of turning point dates.

In detail, we first use a prior probability distribution which represents what the
MMC knows before seeing the data. Second, we update the prior distribution with
data and, third, we get the posterior probability distribution that represents what
the MMC model knows after having seen the data.

In this context, we need to compute integrals such as the mean and the variance,
which are difficult to solve. This is where sampling algorithms based on Markov Chain
Monte Carlo techniques (MCMC) become a means of carrying out the inference.

MCMC sampling implies, that with *N* samples simulated from a distribution, it
is possible to calculate any statistic of a posterior distribution. To this end, Gibbs
sampling is one of the MCMC techniques best suited to our case. In other words, Gibbs
sampling makes it possible to generate posterior samples by scanning each variable
in order to sample its conditional distribution, the remaining variables being fixed at
their present values. Bayesian analysis, therefore, deduces the posterior distribution
of a parameter, then summarises the complete distribution. Thus, Gibbs' sampling
implies that the distribution of the parameters of the MMC model can be transformed
into quantity distributions.

2.5.3 The Gibbs sampling

In order to figure out the distributions of the turning point dates, we have to
estimate both the parameters of the distributions $\theta = (\theta_1, \theta_2, ..., \theta_K, \theta_P)$ and the infe-
rence about the state variable $S$. To this end, a Markov Chain Monte Carlo (MCMC)
method is implemented via a Gibbs sampler, which needs a preliminary classifi
cation.
Following Fruhwirth-Schnatter (2006), the Gibbs sampler can be set such that each pair, k, of reference turning point dates possesses a number of preliminary observations $N_k(S^{(0)})$ and its sample means $\omega_k^{(0)}$, with $S^{(0)} = (S_1^{(0)}, ..., S_N^{(0)})$ and $k = 1, ..., K$. In this context, the k-means clustering algorithm is effective in identifying groups of reference turning point dates and assigning the individual pairs of turning points to the correct group.

From this specification, Camacho, Gadea et Loscos (2020) approximate the distribution of the parameters through a prior distribution of simulated values through $m = 1, ..., M_0, M_{0+1}, ..., M_0 + M$ steps where $M_0$ is not taken into account in obtaining the best possible posterior from the Gibbs sampler. Note that the MCMC can be achieved within two steps.

### 2.5.3.1 First step: Sampling parameters $\theta^{(m)}$ based on classification $S^{(m-1)}$

In finite mixture modeling, the parameters $\theta_1, ..., \theta_K$ are set to be independent of the $\theta_P$. Following Camacho, Gadea and Loscos (2020), this implies that:

$$p(\theta) = p(\theta_1, ..., \theta_K) p(\theta_P). \quad (4)$$

As in Chib (1998), the transition probabilities $p_{kk}$ are assumed to follow Beta distributions and to be independent both a priori ($p_{kk} \sim \text{Beta}(e_{k1}, e_{k2})$) and a posteriori ($p_{kk} | S \sim \text{Beta}(e_{k1}(S), e_{k2}(S))$) with:

$$e_{k1}(S) = e_{k1} + N_{kk}(S), \quad (5)$$

$$e_{k2}(S) = e_{k2} + N_{kk+1}(S), \quad (6)$$

for $k = 1, ..., K-1$ and $N_{ij}(S)$ standing for the number of transitions from state $i$ to state $j$. Given the assumed transition probability matrix $P$ in (2), $N_{kk+1}(S) = 1$. Accordingly, the vector parameters $\theta_P^{(m)}$ can be sampled from posterior distribution $p(\theta_P | S^{(m-1)})$.

We recall that the reference turning point dates $\omega_k$ and their covariance matrices $\Sigma_k$ are independent. The integrals of such a multivariate distribution can be difficult to calculate. Fortunately, the pattern of a Wishart distribution can approximate this kind of distribution through a conjugate prior for multivariate normal sampling. Thus considering the same conjugate prior for each covariance matrix $\Sigma_k \sim W(c_{0k}, C_{0k}^{-1})$ with fixed bivariate means $\omega_k$ of each cluster of turning
point dates, Camacho, Gadea and Loscos (2020) argue that the posterior density is : $\sum_k^{-1} |\omega_k, S, \tau \sim W(c_k(S), C_k(S)^{-1})$ where

$$c_k(S) = c_{0k} + N_k(S) \quad (7)$$

$$C_k(S) = C_{0k} + \sum_{i:s_i=k} (\tau_i - \mu_k)(\tau_i - \mu_k)' \quad (8)$$

Hence, it is easier to sample $\sum_k^{-1(m)}$ given $S^{(m-1)}$ et $\mu_k^{(m-1)}$. Indeed, using a conjugate prior makes it possible to remedy the impossibility of obtaining, for example, a posterior distribution of the variance in the case where we had a beta distribution on the variance of a normal distribution.

The same logic holds while maintaining the covariance $(\sum_k)$ of each group of turning point dates. Thus, under the prior for each mean $\omega_k \sim N(b_{0k}, B_{0k})$, the posterior density can be written as $\omega_k | \sum_k, S, \tau \sim N(b_k(S), B_k(S))$, such that :

$$B_k(S) = \left( B_{0k}^{-1} + N_k(S) \sum_k^{-1} \right)^{-1} \quad (9)$$

$$b_k(S) = B_k(S) \left( B_{0k}^{-1} b_{0k} + \sum_k^{-1} \sum_{i:s_i=k} \tau_i \right) \quad (10)$$

Exactly as in the previous case, sampling $\omega_k^{(m)}$ becomes naturally straightforward given $S^{(m-1)}$ and $\sum_k^{-1(m)}$.

In implementing the Gibbs sampler, Camacho, Gadea and Loscos (2020) set the priors $e_{k1} = 6$ and $e_{k2} = 0.1$ as in Chib (1998) while noninformative priors $b_{0k} = 0$, $c_{0k} = 0$, $B_{0k} = 1000I_2$, and $C_{0k} = I_2$ are used.

### 2.5.3.2 Second step : Sampling the classification $S^{(m)}$ given the parameters $\theta^{(m)}$.
Once the conditional posterior distribution is known through the parameters $\theta^{(m)}$, sampling the entire path $S$ becomes the main focus in the Gibbs process. To this end, the first-order Markov chain properties help Camacho, Gadea and Loscos (2020) to note the procedure as :

$$\Pr(S|\theta, \tau^N) \prod_{i=1}^{N-1} \Pr(s_i|s_{i+1}, \theta, \tau^i), \text{ where } \tau^i = (\tau_1, \dots, \tau_i) \quad (11)$$
The above procedure is based on the filtered probability of each state Pr ($s_i = k|\theta, \tau^i$). Such a filter is detailed in Hamilton (1989). Thus, a two steps recursive procedure takes an initial probability value Pr ($s_0 = k|\theta, \tau^0$) = 0 and consists of one-step ahead prediction which is updated in a second step.

### 2.5.3.3 One-step ahead prediction.
Given $i = 1, ..., N$, the one-step ahead prediction is computed in Camacho, Gadea and Loscos (2020) as:

$$ \mathrm{Pr} \left( s_i = k | \theta, \tau^{i-1} = \sum_{l=1}^{K} plk \mathrm{Pr} \left( s_{i-1} = l | \theta, \tau^{i-1} \right) \right) \quad (12) $$

### 2.5.3.4 Updated filtered probability
The above filtered probability is updated as below once a consecutive turning point is added

$$ \mathrm{Pr} (s_i = k | \theta, \tau^i) = \frac{p(\tau_i | s_i = k, \theta, \tau^{i-1}) \mathrm{Pr} (s_i = k | \theta, \tau^{i-1})}{\mathrm{Pr} (\tau_i | \theta, \tau^{i-1})}, \quad (13) $$

where $\mathrm{Pr}(\tau_i|\theta, \tau^{i-1}) = \sum_{k=1}^{K} p(\tau_i|s_i=k, \theta, \tau^{i-1}) \mathrm{Pr}(s_i=k|\theta, \tau^{i-1})$.

In order to sample the entire path $S$ as indicated in equation 11, Camacho, Gadea and Loscos (2020) begin sampling the last observation $s_N$ which coincides with the last filtered probability $\mathrm{Pr}(s_N|\theta, \tau^N)$.

In the detail, the authors set out couple of assumptions including:

- a random number $u$ with a uniform distribution between 0 and 1

- $w$ the number of times that $\sum_{k=1}^{k^*} \mathrm{Pr}(S_N = k|\theta, \tau^N) < u$, with $k^* = 1, ..., K$

- $S_N$ is sampled as $1+W$.

Thus it becomes relatively easier to sample $s_i$, $i = N-1, N-2, ..., 1$ as the result of comparing the random numbers generated with the conditional distribution $\mathrm{Pr}(s_i|s_{i+1}, \theta, \tau^i)$. In this context, assuming $s_{i+1} = l$, the conditional distribution of $s_i = j$ can be set as:

$$ \mathrm{Pr} (s_i = j | s_{i+1} = l, \theta, \tau^i) = \frac{p_{jl} \mathrm{Pr} (s_i = j | \theta, \tau^{i-1})}{\sum_{k=1}^{K} p_{kl} \mathrm{Pr} (s_i = k | \theta, \tau^{i-1})} \quad (14) $$

To minimize a possible identification problem due to a relatively large number of different modes $K!$ resulting from the posterior mixture model, the authors indicate an “indentifiability constraint” which suggests that the segmentation of the
contractions followed by expansion are not considered (voir Figure 2). This confirms
the characteristics of the BB algorithm in terms of measuring exclusively the business
cycle relative to growth and acceleration cycles. The following parametric method tries
to achieve the same goal, while using a population of economic indicators relatively
correlated to the GDP.

2.5 Reference cycles Chronology : a mixture multiple-point change (MMC) model approach

The Bry-Boschan algorithm and the Markov-switching (MS) model are widely considered for dating business cycles. However, the univariate analysis suggested by these approaches is seen by many researchers, including Chen, Mills Flasaquier (2009) as irrelevant for various reasons. First of all, the authors argue that there are no variables that can, taken alone, adequately represent the aggregate economic activity. Second, a univariate framework is obsolete when considering the co-movement of many economic variables as an essential factor in economic cycles.

This section is dedicated to a multivariate environment, an approach investigated
through several works, including Burns and Mitchell (1946) , Harding and Pagan
(2006) and Stock and Watson (2010, 2014) . These major contributions to the litera-
ture assume that the chronology of the reference cycle can be captured from clusters
of turning point dates from a set of coincident economic indicators. In this context
Harding et Pagan (2006) go beyond the visual inspection of Burns and Mitchell (1946)
and propose a non-parametric algorithm that extracts reference cycles from clusters
of turning point dates, then averages them. Thus, the authors replicate the chronology
of the reference cycles from several coincident economic indicators, as established by
the NBER. The framework of clusters naturally opens ways to achieve a substantial
improvement.

In 2010, Stock and Watson came up with a similar idea; however, the estimates
in this case are based on the known occurrence of a phase change through the bu-
siness cycles span. The authors first average turning points, before estimating the
parameters based on ordinary least squares. Despite the limitations of this method
Camacho, Gadea and Loscos (2018) saw the estimation procedure as interesting, since
it embeds missing data and ragged time series. Moreover, Stock and Watson (2014) ,
even brought in an innovative idea by suggesting a population concept where turning
what the values of these unknown quantities of could be. Bayesian inference allows,
in theory, first to obtain a posterior distribution of the parameters of interest and
then to generate samples from the posterior distribution. Finally, Bayesian inference
allows discrete formulas to be applied to the samples of the posterior distribution in
order to summarize the statistics of the parameters. Thus, any statistic that describes
a sample of data can describe a sample of a posterior distribution.

2.6.1 The data

We use a panel of coincident monthly economic indicators to illustrate a chronology of the reference cycle of the Cote d'Ivoire economy. Some indicators have missing observations, while others are only available for a short period of time. The data cover information on the production sector, the financial sector, international trade, the main indicators of the world economy and other indicators. In the absence of available series for survey balances, we limit ourselves to three groups of indicators, namely :

1. real indicators : the industrial production index, imports and exports,

2. nominal indicators : cocoa price indices, crude oil prices, BRVM (Bourse Régionale des Valeurs Mobilières) stock exchange index,

3. indicators reflecting the international environment : the exchange rate of the CFA franc against the US dollar,

4. indicators characterizing the dynamics of the economies of the Netherlands and Germany and Nigeria.

Indeed, these series are relatively correlated with the industrial production index, which we consider in this study to be the reference series. The variables, which are mostly monthly, were seasonally adjusted through the X12-ARIMA method for the log series. The quarterly GDP series was transformed into a monthly series using a linear interpolation technique. The BRVM stock market index, which is published daily, was also transformed into a monthly series, using the last observation of the month.

We perform an analysis of stationarity on all variables taken in logarithm by applying several unit root tests, including the ADF, Phillips Perron and KPSS tests. The results are summarized in Table 3. Any non-stationary variables were differentiated to obtain stationary variables, and then all variables were centred and reduced.
## 2.6.2 The Monte Carlo Simulation

Different Monte Carlo simulation experiments can be used to determine the exact number of separate turnaround-point date groups. With each turning point in the group that suits it, the Monte Carlo simulation thus makes it possible to estimate the chronology of the reference cycle for the economy of Cote d'Ivoire.

Seven scenarios can be used to simulate the dates of specific turning points from 1000 mixtures of three bivariate normal distributions.

### 1. Scénario 1

We generate, for each mixture, specific dates around three pairs of reference cycle dates that are generally the means of the Gaussian distributions corresponding to the average characteristics of the reference cycle highlighted by the BB algorithm. Thus the first reference peak is fixed at $\mu_1^P$ in March 2003. We establish at 15 months the distance between each reference peak and the corresponding reference trough and at 60 months the distance between the reference trough and the next reference peak.

For each of the three reference dates, a sample of 300 specific dates of bivariate Gaussian distributions is established, whose means $\mu_k$ represent the reference dates and whose covariance matrices $\Sigma_k$, are the mean of the covariance matrices estimated when the MMC model is applied to all the indicators. In this scenario, the covariance is equal to zero.

For each of the 1000 simulations, 2500 prints are generated from the subsequent distributions. However, the first 500 simulations are ignored if they satisfy the initial conditions.

### 2. Scénario 2

As in Scenario 1, we simulate a reference cycle that is relatively longer than normal.

### 3. Scénario 3

This scenario allows us to simulate a double-dip recession cycle. In other words, this time we take into account possible periods of recession followed by brief expansions. This case appears in 2010 in Cote d'Ivoire, whose reference cycle contains a brief 7-month expansion. The 3 following scenarios take into account the low quality of the coincident indicators
4. *Scénario 4*

Here, we capture the fact that the reference turning points are difficult to identify due to the very high dispersion of date groups of specific turning points. Consequently, this scenario is a case of a data generation process whose variances are twice as large as the mean of the estimated variances of the empirical application.

5. *Scénario 5*

We deal with incomplete series with missing data at the beginning of the sample. This involves calculating inferences from a smaller set of observations. In this case, we generate only 50 observations for the first cluster.

6. *Scénario 6*

In this scenario, specific dates are obtained from a small set of indicators. The total number of specific dates simulated is, therefore, 50 in this case.

7. *Scénario 7*

The final scenario is outside the theoretical framework where we are able to capture overall economic activity from certain indicators. However, the leading or lagging trends of some indicators can lead to asymmetry around groups of turning points. These differences in symmetries can be studied by simulating the specific turning points from a distribution spread to the right (scenario 7a) and a distribution spread to the left (scenario 7b), using a mixture of the two distributions. The first is a normal distribution with the true mean parameters and the second is a normal distribution that shifts the mean parameters by -3 months in scenario 7a and +3 months in scenario 7b. The proportion of the two distributions is 90% and 10% respectively.

2.6.3 Empirical results

This section, therefore allows us to evaluate the ability of the MMC model to accurately infer, relative to the reference results provided by the BB algorithm, the dates of economic downturn in Cote d'Ivoire.

2.6.3.1 Identification of individual chronologies
We identify the inherited regime-switching dates for each economic indicator by applying the Bry-Boschan (1971) algorithm. As described in section 3.2, the BB algorithm isolates the local
minima and maxima of each of the ten indicators, considering length and amplitude
constraints on expansions and contractions. In this context, we apply the formulation
of the BB algorithm of Camacho, Gadea and Loscos (2020), which is based on the
work of Watson (1994). We record the result in figure 2 below.

<figure><img src="image_5.png" /><figcaption>FIGURE 4 – Distribution of groups of turning-points dates</figcaption></figure>

Indeed, by visualizing the density of the core of the distribution of specific peaks
and troughs, we can see a distribution of several groups of turning points around the
dates of regime change observed in the reference cycle obtained in section 3.3.

**2.6.3.2 Determination of reference dates** This section presents the formal determination of the date groups of turning points using the Monte Carlo simulation described above. Thus Camacho, Gadea and Loscos (2020) evaluate the performance of the MMC model by combining all the simulated pairs of change point dates into specific groups. Indeed, for each of the 1000 simulations for each of the seven scenarios, the authors estimate a set of models $M_k$ for each group $K$ (for $k = 1, ..., 7$) of cyclical turnaround points, based on the technique for calculating groups of turning point dates described in section 3.4.4
<table><caption>TABLE 3 – Selection of the number of cycles</caption><thead><tr><th>k</th><th>Log Vrais</th><th>AIC</th><th>BIC</th><th>Entropy</th><th>BIC - Entropy</th><th>Facteur de Bayes K-1/K</th></tr></thead><tbody><tr><td>1</td><td>-356.64</td><td>723.28</td><td>730.76</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>-127.39</td><td>276.78</td><td>293.24</td><td>0.61</td><td>294.47</td><td>437.52</td></tr><tr><td>3</td><td>-111.97</td><td>253.89</td><td>279.33</td><td>0.37</td><td>280.06</td><td>13.91</td></tr><tr><td>4</td><td>-10994</td><td>269.93</td><td>304.35</td><td>0.78</td><td>305.91</td><td>25.02</td></tr><tr><td>5</td><td>100.09</td><td>259.17</td><td>301.57</td><td>0.57</td><td>302.71</td><td>2.78</td></tr><tr><td>6</td><td>-94.63</td><td>258.25</td><td>300.63</td><td>0.01</td><td>300.63</td><td>10.06</td></tr><tr><td>7</td><td>-88.56</td><td>258.13</td><td>300.49</td><td>0.28</td><td>300.77</td><td>-8.85</td></tr></tbody></table>

Observation of the estimates given in Table 3 above shows an increase (−356.64 à −88) in the log likelihood in parallel with an equally uninterrupted increase in the number of *K* (1 to 7). According to our selection criteria, the minimum values AIC (258.25) and BIC (300.63) correspond to the smallest value of the entropy (0.01) and are related to *k* = 6. We therefore conclude that the whole period under consideration declines 6 reference cycles. This result is confirmed by the sequence of Bayes factors which compares two models with *k* − 1 and *k* different numbers of groups of turning point dates. Indeed, with a negative value at *k* = 7 we conclude that the model with *k* = 6 is better.

### 2.6.3.3 Model Estimation and Accuracy of the MMC Model

From the penalties introduced in the model selection, Table 4 below reports the values of the log likelihood criteria (−95,59), AIC (261.18), BIC (313.56), when *k*=6 which characterises the best MMC model.
<table><caption>TABLE 4 – Estimation of the MMC model</caption><thead><tr><th colspan="2">Bry-Boschan model</th><th colspan="2">MMC model</th></tr><tr><th>Pic</th><th>Creux</th><th>Pic</th><th>Creux</th></tr></thead><tbody><tr><td>1999 :06</td><td>2000 :12</td><td>2000 : 02<br>(1996 : 08 – 2001 : 09)</td><td>2000 : 11<br>(1998 : 12 – 2002 : 4)</td></tr><tr><td>2002 :03</td><td>2002 :12</td><td>2002 : 02<br>(2000 : 09 – 2003 : 10)</td><td>2003 : 02<br>(2001 : 06 – 2004 : 12)</td></tr><tr><td>2004 :06</td><td>2004 :11</td><td>2004 : 09<br>(2002 : 07 – 2006 : 06)</td><td>2005 : 05<br>(2003 : 10 – 2007 : 04)</td></tr><tr><td>2006 :01</td><td>2007 :01</td><td>-</td><td>-</td></tr><tr><td>2007 :08</td><td>2008 :12</td><td>2008 : 01<br>(2006 : 05 – 2009 : 04)</td><td>2009 – 04<br>(2008 : 10 – 2007 : 04)</td></tr><tr><td>2010 :03</td><td>2011 :04</td><td>2010 : 09<br>(2010 : 03 – 2011 : 12)</td><td>2011 : 08<br>(2011 : 04 – 2013 : 06)</td></tr><tr><td>-</td><td></td><td>2015 : 01<br>(2014 : 05 – 2015 : 12)</td><td>2015 : 06<br>(2015 : 02 – 2016 : 10)</td></tr></tbody></table>

<table><caption>FIGURE 5 – MMC based Reference Chronology of Cote d'Ivoire Business Cycle</caption><thead><tr><th>Log Vrais.</th><th>AIC</th><th>BIC</th><th>Entropy</th></tr></thead><tbody><tr><td>-95.59</td><td>261.18</td><td>313.56</td><td>0.27</td></tr></tbody></table>

<figure><img src="image_6.png" /></figure>

The Mixture Multi-point Change (MMC) Model reveals six (6) complete business cycles. This result is confirmed by the Bayes factor sequence which compares two models with *k* − 1 and *k* different numbers of groups of turning point dates. Indeed,
with a negative value at $k = 7$ we conclude that the model with $k = 6$ is better. The model thus shows some differences with regards to the BB algorithm results. Although the MMC model did not manage to capture the trough of 2007, it was able, nevertheless, to detect a recession in 2008 and 2015, in contrast to the BB algorithm.

# 2 Conclusion

Weconstructed anindex of business cycle diffusion for the economy of Cote d'Ivoire. Such a dating process is aimed at assessing the extent of cyclical movements throughout a developing economy and particularly in the West African Economic and Monetary Union (WAEMU). This analysis of key turning points from a reference economic cycle is built upon several works. The Bry-Boschan algorithm and the Mixed Multi-change points (MMC) model identify business cycles turning points in a small open economy from the WAEMU. We find that the two approaches capture approximately the same number of cycles in Cote d'Ivoire between 1996 and 2018. Although the MMC model did not manage to capture the trough of 2007, it was able, nevertheless, to detect a recession in 2008 and 2015, in contrast to the BB algorithm. This is the first reference index which tracks fluctuations in the ivorian economic activity. This reference chronology is a key to monitor and analyse business cycles in the WAEMU economies.
Références

[1] HB98b Burns, Arthur F., et Wesley C. Mitchell (1946), Measuring Business Cycles, New York : NBER.

[2] Bry, G., and Boschan, C. (1971), Cyclical Analysis of Time Series : Selected Procedures and Computer Programs, NBER, New York.

[3] Camacho, M., Bengoechea, P., and G., Perez Quiros (2006). A useful tool for forecasting the Euro-area business cycle phases. International Journal of Forecasting 22 : 735-749.

[4] Camacho, M., Gadea, M., and Gomez-Loscas, A. (2020). A new approach to dating the reference cycle.

[5] Chow G. C. and An-loh Lin (1971). "Best Linear Unbiased Interpolation, Distribution and Extrapolation of Time Series by Related Series". Review of Economic and Statistics, Vol 53, November, PP372-375

[6] Chib, S. 1998. Estimation and comparison of multiple change-point models. Journal of Econometrics 86. 221-241.

[7] Dagum, E. B., & Cholette, P. A. (2006). Benchmarking, temporal distribution, and reconciliation methods for time series (Vol. 186). Springer Science & Business Media.

[8] Denton F. (1971), "Adjustment of monthly or quarterly series to annual totals : an approach based on quadratic minimization", Journal of the American statistical association.

[9] Fruhwirth-Schnatter, S. 2006. Finite Mixture and Markov Switching models. Springer Series in Statistics. New York, NY : Springer.

[10] Hamilton, J.D. (1989), "A New Approach to the Economic Analysis of Non-stationary Time Series and the Business Cycle", Econometrica, 57, 357-384.

[11] Harding, D., and Pagan, A.R., (2002), "Dissecting the cycle : A methodological investigation, Journal of Monetary Economics 49, pp 365-381.
[12] Harding, D., and Pagan, A.R., (2006). Measurement of business cycles. [pdf] Minerva. https://minerva-access.unimelb.edu.au/bitstream/handle/11343/34176/66612_00001789_01_966.pdf?sequence=1.

[13] Ndiaye F. (2014). La Methode de Denton et les comptes trimestriels”, STATECO N*107, 2012-2013.

[14] Jorgensen, M. 2004. Using multinomial mixture models to cluster Internet traffic. Australian & New Zealand Journal of Statistics 46 : 205–218.

[15] Sax, C., & Steiner, P. (2013). Temporal disaggregation of time series.

[16] Stock, J. H., & Watson, M. W. (2010). Modeling inflation after the crisis (No. w16488). National Bureau of Economic Research.

[17] Stock, J and Watson, M. (2014). Estimating turning points using large data stes. Journal of Econometrics, 178 :368-381.

[18] Chen Xiaoshan, Mills Terence C. Evaluation de la synchronisation des cycles économiques dans la zone euro. In : Revue française d'économie, volume 24, n°4, 2010. pp. 165-187.

[19] Wedel, M., and W. S. DeSarbo. 1993. A latent class binomial logit methodology for the analysis of paired comparison choice data : An application reinvestigating the determinants of perceived risk. Decision Sciences 24 : 1157–1170.

[20] Watson, M. (1994). Business Cycle Durations and Postwar Stabilization of the U.S. Economy. American Economic Review, 84 : 24-46