# 2 inferring heterogeneous recessions in this section

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2 Inferring Heterogeneous Recessions In this section, we introduce a new class of dynamic factor models, in which the common factor follows Markov-switching dynamics that are subject to time-varying means. The proposed model summarizes the information contained in a set of real activity indicators into a common factor that accounts for heterogeneous recessions, and that can be interpreted as an index that proxies the business cycle dynamics of a given economy. For convenience of exposition, we first focus on describing the dynamics of the latent common factor, and then, we proceed to detail how such a latent factor is extracted from the observed data. We assume that the common factor, f t , follows nonlinear dynamics that are flexible enough to accommodate the realization of recessions of different magnitudes, f t = μ 0 (1 - s t ) + μ 1 s t + s t x t + e f,t , e f,t ∼ N (0 , σ 2 f ) i.i.d., (1) where s t ∈ { 0 , 1 } is a latent discrete variable that equals 0 when the economy is in a ‘normal’ episode, and takes the value of 1 when the economy faces an ‘abnormal’ episode. The variable s t is assumed to follow a two-state Markov chain defined by transition probabilities: Pr( s t = j | s t - 1 = i, s t - 2 = h, ... ) = Pr( s t = j | s t - 1 = i ) = p ij . (2) Notice that since there are two states, these probabilities can be summarized by the chance of remaining in a normal state, p , and the chance of remaining in an abnormal state, q . The variable x t in Equation ( 1 ) is defined as another unobserved process that evolves over time as follows: x t = s t x t - 1 + (1 - s t ) v t , v t ∼ N (0 , σ 2 v ) i.i.d. (3) ECB Working Paper Series No 2381 / March 2020 6
This law of motion implies that during normal times, when s t = 0, x t is a white noise which has no impact on the common factor f t . However, during an abnormal episode, when s t = 1, the value of x t remains fixed and is passed to the common factor. Hence, the common factor has the same constant mean μ 0 in normal times, but each abnormal episode is unique in the sense that the common factor during such episode would have a mean μ 1 adjusted by the value of x t , which, in turn, is unique for each episode. Obviously, this value x t is calculated from the observed data, that determine the magnitud that better fit each recession period. The novelty of the proposed nonlinear factor model is in the dynamics of the common factor, which in our case is a function of a random variable that fluctuates around a state-dependent mean to account for heterogeneous recessions. The mean of the common factor is at the same constant level in each period when the economy is in the normal state. However, when the economy switches to an abnormal state, the prior mean is drawn from a random distribution, learns from the data, and remains the same for the entire duration of the abnormal episode, until the economy reverts back to a normal episode. In other words, all normal episodes come with the same mean of the common factor, whereas each abnormal episode comes with its own unique common-factor mean.

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• Fall '19
• Economics, Recession, Late-2000s recession, GWI

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