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## Advantages of the MS-DFM

Among the major advantages of MS-DFM over the other methods of business cycle turn-ing point identification are timeliness (the MS-DFM estimates are available as soon as the dataset is updated), informativeness (contrary to non-parametric methods, MS-DFM is capable to distinguish the switch in mean from the switch in variance) and transparency (it is a fully replicable econometric technique and not an expert opinion). The primary output of the MS-DFM is, therefore, the estimate of the current state of the cycle, i.e. its nowcast (detection). Due to its high performance, it has been used in the several pa-pers analyzing various economies, such as Darbha (2001) for India, Mills and P. (2003) for the UK, Watanabe (2003) for Japan, Bandholz and Funke (2003) for Poland and

Chapter 1. General Introduction 8

Hungary, Bai and Ng (2013) Germany, Chauvet and Senyuz (2012) and Kim and Yoo (1995) for the US, Darné and Ferrara (2011) for France. The literature is very dynamic, and many extensions to the baseline model exist. Among them, of particular interest is the bi-factor MS-DFM, introduced by Chauvet (1998), which studies the comovement of the financial and the business cycles in order to identify whether the former causes the latter, the fact which can further be used for the construction of early warning indicators and prediction of recessions in the business cycles.

**Brief literature review**

The Dynamic Factor Model with Markov Switching (MS-DFM) was first suggested by Diebold and Rudebusch (1996)2. This paper relies on the seminal paper by Hamilton (1989) which applies a univariate Markov-Switching model to business cycle analysis. It was then formalized for the multivariate case by Kim (1994) and by Kim and Yoo (1995) and used afterwards by Chauvet (1998), Kim and Nelson (1998), Kaufmann (2000). While the original model assumes switches in mean, other types of non-linearity were proposed by Kholodilin (2002a), Kholodilin (2002b), Dolega (2007), Bessec and Bouab-dallah (2015) where the slope of factors or exogenous variables is state dependent; or by Chauvet (1998), Chauvet (1999), Kholodilin (2002a), Kholodilin (2002b), Kholodilin and Yao (2004), Billio et al. (2007) where the variance of the idiosyncratic component is state dependent; and lastly by Chauvet and Potter (1998) and Carvalho and Lopes (2007) where the authors allow for structural breaks in factor loadings.

According to Google Scholar, about 15 new papers on Markov-Switching Dynamic Factor models applied to the business cycle analysis appear every year.3 Current research in this area is led is several directions. Besides rapidly growing number of empirical applications, the most important are

• MS-DFM with mixed frequencies, allowing to consider even larger datasets (see Camacho and Martinez-Martin (2015) for the latest paper) and other improvements of the DFM part of the model (see Hindrayanto et al. (2016)).

• multivariate MS-DFM studying the interaction between the cycles (see Leiva-Leon (2017), Koopman et al. (2016)).

• highly dimensional Markov-switching models and their estimation methods (see Droumaguet et al. (2017), for example).

• time-varying MS-DFM (see Bazzi et al. (2017)).

Of course, this list is by no means exhaustive, but it still helps to better understand the contribution of this thesis to the literature.

**The model and the estimation methods**

The general framework for Markov switching factor models has been first proposed by Kim (1994) and was then used by Kim and Yoo (1995) to study the US business cycle. In the present paper, we take the same kind of specification as in Kim and Yoo (1995), and we assume that the growth rate cycle of the economic activity has only two regimes (or states), associated with its low and high levels. The economic activity itself is rep-resented by an unobservable factor, which summarizes the common dynamics of several observable variables. It is assumed that the switch between regimes happens instanta-neously, without any transition period (as is considered, for example, by STAR family models). This assumption can be motivated by the fact that the transition period before deep crises is normally short enough to be omitted. For example, the growth rate of French GDP fell from 0.5% in the first quarter of 2008 to -0.51% in the second quarter of the same year, and further down to -1.59% in the first quarter of 2009.

The model is thus decomposed into two equations, the first one defining the factor model, and the second one describing the Markov switching autoregressive model which is assumed for the common factor. More precisely, in the first equation, each series of the information set is decomposed into the sum of a common component (the common factor loads on each of the observable series with a specific weight) and an idiosyncratic component: yt = γft + zt, (3.1). where yt is a N × 1 vector of economic indicators, ft is a univariate common factor, zt is a N × 1 vector of idiosyncratic components, which is uncorrelated with ft at all leads and lags, γ is a N × 1 vector. In this equation all series are supposed to be stationary, so that some of the components of yt may be the first diﬀerences of an initial non-stationary economic indicator.

**One-step estimation method**

In this section, we recall the estimation method which has been introduced by Kim (1994) and Kim and Yoo (1995): it is a one-step method, but it can be employed only for a small set of observable series. Using the state-space representation of the model, which is given by equations (7) and (8), the Kalman filter can be written condition-ally on the realizations of the state variable at time t and t − 1. If Xt(|j,it−)1 denotes the predicted value of the variable Xt conditional on the information available up to t − 1 and on the realizations St = j and St−1 = i, the Kalman filter formulas are the following: Prediction step: α(j,i) = T α(i) + µ(j), (3.9).

### Data, reference dating and quality indicators

For the purpose of comparison one would like to run the two estimation methods on the same dataset. However due to the diﬀerent requirements on the number of series in the database for each method (large dataset for the two-step method in order to get the consistency of the PCA factor estimate, small dataset for the one-step method to obtain the convergence of the algorithm), we are unable to perform this kind of analysis. We therefore use a separate dataset for each method.

The database for the two-step procedure is constructed following Stock and Watson (2014) for the US and Bessec and Doz (2014) for France. It contains 151 monthly se-ries spanning the period May 1993 – March 2014.7 The data cover information on the production sector, financial sector, employment, households, banking system, business surveys, international trade, monetary indicators, major world economic indicators, and other indicators.

For the one-step method it is crucial to select series properly. The series must be an indicator of the economic cycle and should be available in monthly frequency. We choose 25 series out of the 151 series of the database for the two-step method and to use them in combinations of four, overall C254 = 12650 combinations. The strategy of trying all possible combinations of 4 out of 25 may seem too bulky and inelegant, however we deliberately avoided any data selection technique in order to minimize its possible im-pact on the output of the one-step results. The selection was made on the basis of the existing literature on the one-step method applied to business cycle analysis. To the four classical indicators for business cycle dating of the US economy (total personal income, total manufacturing and trade sales, number of employees on nonagricultural payrolls, total industrial production index) we added series used in Kholodilin (2006) (French stock market index CAC40, interest rates on the 3 months and 12 months government bonds, imports and exports), selected series of business surveys proved to be useful by Bessec and Doz (2014), the components of the OECD Composite Leading Indicator, as well as several series characterizing the dynamics of the major trade partners (Germany, USA, Asia). Since almost all of these series have been already used in the analysis of the French business cycle (and the others are likely to comove with it), we suppose that the common component of each combination can be considered as an estimate of the business cycle.

All series are seasonally adjusted, tested for the presence of unit roots and transformed to stationarity if necessary, then centralized and normalized. Detailed lists of series for both methods are given in Tables A.1 and A.2 of Appendix A.

#### Diﬀerent set of series for diﬀerent crises

As for OECD detected recessions, it is important to keep in mind that none of them (at least the five recessions we consider here) had the same origins as any of the other, so it is possible that the determinants of economic activity evolved with time, and so it is likely that the common factor of a particular set of series does not reflect the Great recession as well as it reflected the crisis of 1992-1993. However, in order to construct a universal instrument, it is preferable to find series that would capture the recession in all cases, if possible. For this purpose we compare the quality indicators of 72 sets of variables for each crisis separately. Table A.4 in Appendix B summarizes the information on the best combinations by crisis. Here FPS shows the proportion of months of each crisis incorrectly determined as expansion, i.e. the lower FPS, the better a crisis is captured.

We can see that:

• the combination consisting of volume of total retail trade, unemployment rate, trade balance and order books in the building industry, with the highest loadings on unemployment, is the best to detect the first and the last crisis and captures the second crisis well, too.

• the combination consisting of new passenger cars sales and registration, retail trade orders intentions, export, confidence indicator in services, with the highest loadings on retail trade and Confidence Index in services, is leading in case of the second, the third and the fourth crises, being significantly superior to the other combinations for the third and the fourth recessions.

• although good during certain periods of the timeline, unfortunately none of these sets of variables could be used as a ’core’ set due to their relatively poor performance on the expansion periods and non-detected crises.

The set of data contained in these two combinations appears to be suﬃcient to identify all five crises with a special role given to unemployment, retail trade orders intentions and confidence index in services.

**Table of contents :**

**1 General Introduction **

1.1 Business Cycle

1.1.1 History

1.1.2 Definition

1.1.3 Existing explanations

1.1.4 Dating of the turning points

1.2 Markov-Switching Dynamic Factor Model

1.2.1 General assumptions

1.2.2 Advantages of the MS-DFM

1.2.3 Brief literature review

1.3 Contribution

**2 Introduction Générale **

2.1 Cycle économique

2.1.1 Histoire

2.1.2 Définition

2.1.3 Les explications existantes

2.1.4 La datation des points de retournement

2.2 Le modèle MS-DFM (Markov-Switching Dynamic Factor Model)

2.2.1 Hypothèses générales

2.2.2 Avantages du modèle MS-DFM

2.2.3 Littérature

2.3 Contribution

**3 Dating Business Cycle Turning Points for the French Economy: anMS-DFM approach **

3.1 Introduction

3.2 The model and the estimation methods

3.2.1 The model

3.2.2 One-step estimation method

3.2.3 Two-step estimation method

3.3 Data, reference dating and quality indicators

3.3.1 The dataset

3.3.2 Reference dating

3.3.3 Measures of quality

3.4 Estimation results

3.4.1 One-step method

3.4.2 Two-step method

3.4.3 Comparison: one-step vs two-step

3.5 Conclusion

**4 On the consistency of the two-step estimates of the MS-DFM: a Monte Carlo study **

4.1 Introduction

4.2 Markov-Switching Dynamic Factor Model

4.3 Two-step estimation method

4.4 Monte Carlo simulations

4.4.1 Experimental setup

4.4.1.1 The DGP

4.4.1.2 The parameters of control

4.4.1.3 Estimation

4.4.1.4 Measures of quality

4.4.2 Simulation results

4.4.2.1 The impact of the first step

4.4.2.2 Consistency and small-sample performance of the twostep estimates ˆ( ˆ f)

4.4.3 Identification of states

4.4.4 Analysis of t-statistics

4.4.5 Other scenarios

4.5 Conclusion

**5 Dynamical Interaction Between Financial and Business Cycles **

5.1 Introduction

5.2 The DI-MS-FM

5.2.1 The general presentation

5.2.2 Construction of RFt and FFt

5.2.3 The interaction mechanism

5.2.4 Granger causality

5.2.5 Extension: policy analysis

5.3 Estimation and Forecasting

5.3.1 Maximum Likelihood Estimation

5.3.2 Forecasting

5.3.3 In-sample and out-of-sample performance

5.3.3.1 In-sample performance

5.3.3.2 Out-of-sample performance

5.4 Interaction between financial and business cycles in the US

5.4.1 Data description

5.4.2 Characteristics of cycles and identified interaction regimes

5.4.3 Identifying the periods of recession, financial downturn and high interdependence between the cycles

5.4.4 Robustness check

5.4.5 Transition probabilities and smoothed probabilities of future states

5.5 Conclusion

**A Appendix to “Dating business cycle turning points for the French economy: an MS-DFM approach” **

A.1 Datasets

A.2 One-step estimation results

A.3 Estimation results for one-step and two-step methods

**B Appendix to “On the consistency of the two-step estimates of the MSDFM: A Monte-Carlo study” **

B.1 Variance of the factor

B.2 Two-step estimates distribution

B.3 Results on unfiltered data

B.4 ML estimates obtained with the observable factor ft

B.5 Properties of empirical distributions of t-statistics corresponding to the two-step estimates

B.6 Simulations with various scenarios

**C Appendix to “Dynamical Interaction between Financial and Business Cycles” **

C.1 Composition of factors RFt and FFt

C.2 Dynamics of RFt and FFt

C.3 Robustness check

C.3.1 Alternative dataset

C.3.2 Estimation on subsets

**Bibliography**