Tasmanian School of Business & Economics Economics & Finance Seminar Series 1 February 2016

Transcription

1 P A R X (PARX), US A A G C D K A R Gruppo Bancario Credito Valtellinese, University of Bologna, University College London and University of Copenhagen Tasmanian School of Business & Economics Economics & Finance Seminar Series 1 February 2016 Cavaliere (UniBO) PARX December / 29

2 Introduction Aim of the paper: Extension of Poisson autoregressive models (PAR) to exogenous variables (Q)ML estimation and asymptotic theory Use of "weak dependence" Modeling of US corporate defaults Cavaliere (UniBO) PARX December / 29

3 Introduction The PAR-X model Fokianos, Rahbek & Tjøstheim (2009, JASA) PAR model: y t F t 1 Poisson (λ t ) = P (λ t ) λ t measurable w.r.t. F t 1 := σ (y t 1, y t 2,..., y 1, λ 1 ), t = 1, 2,..., T PARX: Related literature: F t 1 := σ (y t 1, y t 2,..., y 1, x t 1,..., x t m, λ 1 ) x t R d x exogenous Realized GARCH (Hansen et al., 2012, JoAE) GARCH-X (Han&Park, 2008, JoE; Han&Kristensen, 2015, JBES) Cavaliere (UniBO) PARX December / 29

4 Data and Model US corporate default counts and their autocorrelations High temporal dependence in default counts. Existence of default clusters over time. Overdispersion of the distribution of default counts (average is 3.51, variance is 15.57). How do we model (and forecast) defaults so as to take into account these features? What are the determinants of defaults dynamics? How do we incorporate macro and finance factors in the model? Cavaliere (UniBO) PARX December / 29

5 Data and Model US corporate default counts Default clustering is related to the recent debate in the finance literature on contagion effects,and comovements in corporate solvency due to common macro and financial factors ("systematic risk"); see, e.g., Das et al. (2007) and Lando and Nielsen (2010). "[Is] time variation in... corporate defaults controlled by exogeneous factors with no feedback from actual defaults to these factors? Or can we statistically document "contagion" effects by which one firm s default increases the likelihood of other firms defaulting?" (Lando and Nielsen, 2010). Cavaliere (UniBO) PARX December / 29

6 Data and Model US corporate default counts Alternative explanations: Conditional independence: (y t x t,past) independent over time Contagion: (y t x t i ) depends on (y t i : i 1) Possible choice of the explanatory variables (x t ) Realized Vol BAA/10Y spread # downgrades Industrial production Leading index NBER recession indicator Cavaliere (UniBO) PARX December / 29

7 Data and Model Possible choice of the explanatory variables (x t ) Asymptotic analysis based on x t = g (x t 1, ε t ) Cavaliere (UniBO) PARX December / 29

8 Model and estimation PARX model - basic specification Conditionally on the past, y t N 0 has a Poisson distribution with time-varying intensity λ t : y t F t 1 = Poisson (λ t ). p q λ t = ω + α i y t i + β i λ t i + γ f (x t 1 ), i =1 i =1 where x t 1 contains relevant (observed) macro and finance factors. p i =1 α i > 0 captures possible contagion effects (through past counts). γ f (x t 1 ) captures macro/financial shocks to corporate solvency. q i =1 β i λ t i captures "long memory" Cavaliere (UniBO) PARX December / 29

9 Model and estimation Maximum-likelihood estimation We collect model parameters in θ = (α 1,..., α p, β 1,..., β q, γ) and write up conditional log-likelihood function of θ in terms of observations y 1,..., y T : L T (θ) = T l t (θ), l t (θ) : = y t log λ t (θ) λ t (θ), t=1 with λ t (θ) as defined earlier. (recall: if Y is P (λ), then P (Y = y) = exp ( λ) λ y /y!) The maximum likelihood estimator (MLE) is then defined as ˆθ T := arg max θ Θ L T (θ). For large sample theory it is essential that at the true value θ 0 L T (θ) T ( ) yt λt (θ) θ = 1 θ=θ0 t=1 λ t θ θ=θ0 is MDS w.r.t. F t 1. Cavaliere (UniBO) PARX December / 29

10 Model and estimation Asymptotic analysis of MLE Log likelihood: L T (θ) = T t=1 y t log λ t (θ) λ t (θ) Consider the following reformulation of the model λ t = ω + αy t 1 + βλ t 1 + γf (x t 1 ) (simple PARX(1,1)) y t = N t (λ t ) where N t ( ) is a Poisson process of unit intensity (iid over time) x t = g (x t 1, ε t ) (Markov chain) Suppose that {ε t, N t } is i.i.d. α + β < 1 f (x) f ( x) K x x and E g (x, ε t ) g ( x, ε t ) 2 a x x 2 ( a < 1) Employing the techniques of Doukhan and Winterberger (2008), we show: {(y t, x t )} stationary This result provides us with a LLN essential to show n(ˆθ θ 0 ) d ( ) N 0, I 1 (θ 0 ), I (θ) = E [ 2 ] l t (θ) θ θ. Cavaliere (UniBO) PARX December / 29

11 Model and estimation Finite sample simulations λ t = ω + αy t 1 + βλ t 1 + γ exp (x t ), x t = (1/2)x t 1 + ε t Cavaliere (UniBO) PARX December / 29

12 Model and estimation Finite sample simulations λ t = ω + αy t 1 + βλ t 1 + γ exp (x t ), x t = (1/2)x t 1 + ε t Cavaliere (UniBO) PARX December / 29

13 Model and estimation Finite sample simulations λ t = ω + αy t 1 + βλ t 1 + γ exp (x t ), x t ARFIMA (0, 1/4, 0) Cavaliere (UniBO) PARX December / 29

14 Model and estimation Finite sample simulations λ t = ω + αy t 1 + βλ t 1 + γ exp (x t ), x t ARFIMA (0, 1/4, 0) Cavaliere (UniBO) PARX December / 29

15 On asymptotic theory Model and estimation Recall L T (θ) θ = θ=θ0 is MDS with respect to F t 1. Consider initially no λ t 1 and no "X" y t is a Markov chain on {0, 1, 2,...} T ( yt 1 t=1 λ t ) λt (θ) θ y t F t 1 P (λ t ), λ t = ω + αy t 1 θ=θ0 If 0 α < 1: geometrically ergodic (e.g. Jensen-Rahbek, 2007, ET), i.e. P n ( x) π ( ) ρ n g (x), ρ < 1 ( ( ) is the total variation norm) Proof: use the drift function (Grundwald,..., Tweedie, 1997; Nummelin, 1998) V (Y ) = 1 + Y 2 : E (V (Y t ) Y t 1 = y) = 1 + λ t (1 + λ t ) = c + α 2 V (y) Cavaliere (UniBO) PARX December / 29

16 Model and estimation On asymptotic theory Next, add λ t 1 y t F t 1 P (λ t ), λ t = ω + βλ t 1 + αy t 1 We have that y t is not longer a Markov chain, although (y t, λ t ) : Markov chain λ t : Markov chain Approach (Meitz and Saikkonen, 2008, ET; Carrasco and Chen, 2002, ET): Use N t (s), 0 s <, N t ( ) unit intensity Poisson process (integer valued!) Then λ t = ω + βλ t 1 + αn t 1 (λ t 1 ) R + Cannot establish φ-irriducibility: φ(a) > 0, 0 P (A λ) > 0 only open sets (γ, δ) can be reached (irrationals = A: φ (A) > 0 but A not open) Cavaliere (UniBO) PARX December / 29

17 Model and estimation On asymptotic theory Two approaches: Fokianos et al. (2009, JASA): λ t = ω + αy t 1 + βλ t 1 λ t = ω + αy t 1 + βλ t 1 + δi ( y t 1 = 1) η t, η t iid U[0, 1] (perturbation) Establish: (1) (y t, λ t ) geometrically ergodic (2) the difference (y t, λ t ) (y t, λ t ) is small Doukhan - Winterberger (2008) concept of "weak dependence" used to show LLN and CLT Cavaliere (UniBO) PARX December / 29

18 Model and estimation On asymptotic theory Fokianos et al (2009) y t = N t (λ t ), λ t = ω + βλ t 1 + αn t 1 (λ t 1 ) yt = N t (λt ), λt = ω + βλt 1 + αn ( t 1 λ ) t 1 + ε δ t Lemma: 1 L T (θ) T θ = 1 T θ=θ0 Proof: 1 Using LT (θ) = T t=1 1 LT (θ) T θ = 1 θ=θ0 T ( ) yt λt (θ) 1 t=1 λ t θ d N (0, Ω 0 ) ( ( )) yt log λt λt + log f εt δ, establish T 2 Show that Ω δ 0 Ω 0 as δ 0 3 Show that T ( y t t=1 λt ( 1 L lim lim supp T (θ) 1 δ 0 T T θ ) λ 1 t (θ) ( ) θ d N 0, Ω0 δ L T (θ) ) T θ > δ = 0 (Brockwell and Davis, 1991, prop ) Cavaliere (UniBO) PARX December / 29

19 On weak dependence Model and estimation Recall: y t = N t (λ t ), λ t = ω + αy t 1 + βλ t 1 + γf (x t 1 ) x t = g (x t 1, ε t ) [exogenous] Construct the representation Z t := y t λ t x t = F Z t 1, ε t, N t ( ) }{{} "innovations" Doukhan and Winterberger (2008): if E F (z, ε, N) F ( z, ε, N) c z z ( c < 1 contraction ), then stationarity, ergodicity & LLN/CLT apply Cavaliere (UniBO) PARX December / 29

20 Empirical analysis of default counts PARX analysis of US corporate default counts Choice of macro and financial (X) factors: Realized volatility (RV ) of the S&P 500 Baa Moody s rated to 10-year Treasury spread (SP) Number of Moody s rating downgrades (DG ) NBER recession indicator (NBER) Negative and positive part of year-to-year change in Industrial Production Index (IP ( ) and IP (+) ) Negative and positive part of Leading Index released by the Fed (LI ( ) and LI (+) ) PARX(2,1)-X; α i 0, β 0 0, γ i 0 Cavaliere (UniBO) PARX December / 29

21 Empirical analysis of default counts Estimated models (t-stat s in parentheses) Cavaliere (UniBO) PARX December / 29

22 In-sample fit Empirical analysis of default counts Cavaliere (UniBO) PARX December / 29

23 In-sample fit Empirical analysis of default counts Cavaliere (UniBO) PARX December / 29

24 Forecasting Forecasting of default probabilities ) y t+h F t = Poisson (λ t+h t, where λ t+h t solves the following recursion, max{p,q} λ t+k t = ω + {α i + β i } λ t+k i t + γ x t+k 1 t, k = 1,..., h, i =1 with "initial value" p q λ t+1 t = ω + α i y t+1 i + β i λ t+1 i + γ x t. i =1 i =1 Cavaliere (UniBO) PARX December / 29

25 Forecasting Forecasting of default probabilities Recursive estimation Forecast interval: [Lt α/2, Ut α/2 ],where Pr(y t L α/2 t F t 1 ) = α/2; Pr(y t U α/2 t F t 1 ) = 1 α/2 Cavaliere (UniBO) PARX December / 29

26 Structural Breaks Pseudo-out-of-sample Forecasting: PARX(2,1) with RV & LI- Cavaliere (UniBO) PARX December / 29

27 Structural Breaks Structural breaks and subsample estimates Strong evidence of structural breaks at: 1998 and 2007 (beginning of last two financial crises). We re-do model selection and estimation for , , and In-sample fit and forecasting performance improve. ω α 1 α 2 β RV LI ( ) PAR(2,0) t-stats (7.04) (5.29) (8.32) PARX(1,1) t-stats 0.10 (2.04) - (7.31) (8.65) PARX(0,1) t-stats (0.00) - - (6.81) (2.17) (2.38) Cavaliere (UniBO) PARX December / 29

28 Structural Breaks Interpretation of results : Macro factors irrelevant, strong contagion effects (α 1 + α 2 = 0.65) : RV very strong explanator of defaults, weak contagion effect (α 1 + α 2 = 0.09) : RV and LI ( ) very strong explanators, no contagion effect (α 1 + α 2 = 0.00). This goes against much of the analysis of contagion effects and systemic risk found in the literature. The above results are based on the assumption that there is no feedback effect from # defaults to macro/finance factors. If there are feedback effects, then the picture changes. (to be continued...) Cavaliere (UniBO) PARX December / 29

29 Conclusions Conclusions PAR-X: dynamic properties of US corporate default counts Financial and macro variables explanatory power Contagion effects change over time Full theory (properties of y t, x t ; properties of ˆθ T ) Extension: multivariate modeling Cavaliere (UniBO) PARX December / 29