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Identification by Laplace Transform in Nonlinear Panel or Time Series Models with Unobserved Stochastic Effects

Patrick GAGLIARDINI and Christian GOURIEROUX Montreal, November 21, 2014

• P. GAGLIARDINI and G. GOURIEROUX

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• 1. INTRODUCTION

This paper is about... ... identification in a general class of (semi-)parametric nonlinear models for time series or panel data with unobserved dynamic effects These models are characterized by an exponential affine specification for the nonlinear regression of data on lagged endogenous variables and unobserved dynamic effects

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• 1. INTRODUCTION

Exponential affine specifications are implied by standard distributional assumptions and are popular in asset pricing because of their analytical tractability The unobservable dynamic components capture time effects in panel data, stochastic volatilities and covolatilities of asset returns, systematic factors in risk analysis, etc. They complicate identification and estimation by standard methods such as Maximum Likelihood (ML)

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• 1. INTRODUCTION

The goal of the paper is to ... ... provide general procedures to obtain continuum sets of nonlinear moment restrictions for affine specifications based on the (conditional) Laplace transform Study when these sets of moment restrictions identify the nonlinear regression parameters, and the parametric or nonparametric specification for the distribution of dynamic effects The identification strategy naturally leads to Generalized Method of Moments (GMM) estimation

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OUTLINE

1

Introduction

2

The model: Examples and general specification

3

First-order moment restrictions: Nonlinear cross-differencing, moment restrictions from Laplace transform

4

Higher-order nonlinear moment restrictions

5

Application to linear and nonlinear latent factor models

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• 2. THE MODEL

2.1 Examples

Example 1: Count panel data with stochastic time effect A Poisson dynamic panel regression model: yi,t ∼ P(ft + αxi,t + cyi,t−1), i = 1, ..., n, t = 1, ..., T, (n and T large) with unobserved stochastic time effect ft Observed individual heterogeneity in the basic specification Extension to individual dynamic effects fi,t, e.g. “effort” processes for moral hazard in car insurance [Gourieroux, Jasiak (2004)]

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• 2. THE MODEL

2.1 Examples

Example 2: Linear factor model with lagged endogenous variables The vector yt of index returns for n markets yt = Bft + Cyt−1 + εt, t = 1, ..., T (T large) where the idiosyncratic shocks are εt ∼ IIN(0, Σ) and matrix Σ is diagonal The vector ft of K latent common factors represents systematic risks [see e.g. the Arbitrage Pricing Theory (APT), Ross (1976), Chamberlain, Rothschild (1983)] Lagged returns vector yt−1 accounts for contagion (e.g. spillover) effects across markets

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• 2. THE MODEL

2.1 Examples

Example 3: Systematic risk and contagion in time series count data Multivariate time series count data yt = (y1,t, ..., ym,t)′ Application to hedge fund survival: data are monthly liquidation counts in m management styles [Darolles, Gagliardini, Gourieroux (2011)] Conditionally on past counts and common factor path: yj,t ∼ P(aj + b′

jft + c′ jyt−1),

j = 1, ..., m, t = 1, ..., T, (T large) Systematic risk factor ft with loading matrix B = [b1, ..., bm]′ Contagion effects within and between groups through lagged counts and coefficients matrix C = [c1, ..., cm]′

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• 2. THE MODEL

2.1 Examples

Example 4: Conditionally Gaussian factor model with stochastic volatility in the factor yt = βh1/2

t

ηt + εt, t = 1, ..., T (T large) with (n, 1) loading vector β and bivariate latent factor ft = (ht, ηt)′ The factor stochastic volatility process (ht), the common shock ηt ∼ IIN(0, 1) and the idiosyncratic shocks vector εt ∼ IIN(0, Σ) are mutually independent Process (ht) is an Autoregressive Gamma (ARG) Markov process with noncentral gamma transition: E[exp(uht)|ht−1] = exp

• ρu

1 − cuht−1 − δ log(1 − cu)

• ,

where δ > 0 is the degree of freedom parameter, c > 0 is a scale parameter, and ρ ∈ (0, 1) is the first-order autocorrelation

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• 2. THE MODEL

2.1 Examples

Example 5: Multivariate stochastic volatility model Vector of returns yt for n assets yt = a +    Tr(B1Σt) . . . Tr(BnΣt)    + Σ1/2

t

εt, t = 1, ..., T (T large) where εt ∼ IIN(0, Idn) Stochastic process Σt of symmetric positive-definite (n, n) volatility-covolatility matrices: assume autoregressive Wishart dynamics [Bru (1991), Gourieroux (2006), Gourieroux, Jasiak, Sufana (2009)] Volatility risk premia parametrized by symmetric matrices B1, ..., Bn See Gagliardini, Gourieroux (2014) for identification and estimation

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• 2. THE MODEL

2.2 General specification

Endogenous variable yt, with dimension n Unobservable dynamic effect ft, with dimension K Observable covariate process xt Information sets: let yt−1 = (yt−1, yt−2, ...) and similarly for ft and xt Assumption A.1: Exponentially affine nonlinear regression model E[exp(u′yt)|yt−1, ft, xt] = exp

• a(u, xt, θ)′[Bft + Cyt−1 + d(xt, θ)] + b(u, xt, θ)
• where u is the multidimensional argument of the Laplace transform, θ, B, C

are parameters (possibly constrained) and a, b, d are known functions.

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• 2. THE MODEL

2.2 General specification

Assumption A.2: Exogeneity, strict stationarity and Markov property of unobservable dynamic effects and observable regressors The joint process (f ′

t , x′ t)′ is:

i) exogenous and Markov, that is, the conditional distribution of (f ′

t , x′ t)′ given

yt−1, ft−1 and xt−1 depends on ft−1, xt−1 only; ii) strictly stationary. The dynamics of the unobservable dynamic effect can be either specified parametrically, or let nonparametric

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• 2. THE MODEL

2.2 General specification

Nonlinear (semi-)affine state space specification as in Bates (2006), but suitable for the analysis of multivariate cross-sectional measurements (potentially of large dimension) The identification conditions in our moment-based setting are different from those in the likelihood-based framework of Bates (2006) The moment restrictions in this paper also differ from those underlying simulation based methods such as Simulated Method of Moments [SMM, McFadden (1989), Pakes, Pollard (1989)], Simulated Nonparametric Moments [SNM, Creel, Kristensen (2012)], etc. The (semi-)affine and (semi-)parametric framework in Assumption A.1 is more structural than the nonparametric setting in Hu, Shum (2012), Hu, Shiu (2013), allowing for more informative identifying restrictions

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.1 Nonlinear cross-differencing for panel data

Assume that variables yi,t and ft are one-dimensional, and individual histories (yi,t, xi,t), i varying, are independent conditional on factor path (ft). Then: E[exp(uyi,t)|yt−1, ft, xt] = exp{a(u, xi,t, θ)[ft + cyi,t−1 + d(xi,t, θ)] + b(u, xi,t, θ)} i.e. E

• exp{uyi,t − a(u, xi,t, θ)[cyi,t−1 + d(xi,t, θ)] − b(u, xi,t, θ)}|yt−1, ft, xt
• =

exp[a(u, xi,t, θ)ft], ∀i, u, ft (1) Assumption A.3*: The function u → a(u, xi,t, θ) is continuous and strictly monotonous w.r.t. argument u, for any given xi,t and θ. Then, we can define: u(v, xi,t, θ) as the solution of the equation a(u, xi,t, θ) = v, and replace argument u by u(v, xi,t, θ) in equation (1)

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.1 Nonlinear cross-differencing for panel data

We get: E

• exp{u(v, xi,t, θ)yi,t − v[cyi,t−1 + d(xi,t, θ)] − b[u(v, xi,t, θ), xi,t, θ]}|yt−1, ft, xt
• =

exp(vft) By differencing the equations for any pair of individuals i and j, with i = j, and computing conditional expectations given yi,t−1, yj,t−1, xt, we get First-order nonlinear moment restrictions with cross-differencing: E

• exp{u(v, xi,t, θ)yi,t − v[cyi,t−1 + d(xi,t, θ)] − b[u(v, xi,t, θ), xi,t, θ]}|yi,t−1, yj,t−1, xt
• = E
• exp{u(v, xj,t, θ)yj,t − v[cyj,t−1 + d(xj,t, θ)] − b[u(v, xj,t, θ), xj,t, θ]}|yj,t−1, yi,t−1, xt
• ,

∀v, ∀i, j, i = j ⇒ A continuum of nonlinear moment restrictions for any pair of individuals

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.1 Nonlinear cross-differencing for panel data

Proposition 1: For a nonlinear panel data model with unobserved dynamic effects, under Assumptions A.1-A.3*, generically, the first-order nonlinear moment restrictions identify: i) the regression parameters B, C, θ by nonlinear cross-differencing. Then ii) the unconditional distribution of the dynamic effect ft is nonparametrically identified A nonlinear cross-sectional extension of the quasi-differencing approach usually applied to panel data with individual fixed effects and based on the first-order moments only [see Mullahy (1997), Wooldridge (1997), (1999)]

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.1 Nonlinear cross-differencing for panel data

Example 1: Count panel data with stochastic time effect (cont.) From the Laplace transform of the Poisson distribution: E

• exp{uyit + (1 − exp u)(αxi,t + cyi,t−1)}|yi,t−1, yj,t−1, xt
• =

E

• exp{uyj,t + (1 − exp u)(αxj,t + cyj,t−1)}|yi,t−1, yj,t−1, xt
• , ∀i = j, ∀u

By considering the first-order expansion w.r.t. argument u at u = 0, the equations become: E[yi,t−αxi,t−cyi,t−1|yi,t−1, yj,t−1, xt] = E[yj,t−αxj,t−cyj,t−1|yi,t−1, yj,t−1, xt], ∀i = j, which are the analogue of the moment restrictions considered in Windmeijer (2000), Blundell et al. (2002)

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.2 (Non-)parametric identification of the marginal distribution of the dynamic effect

In the framework of Assumptions A.1 and A.2 we have: E[exp{u′yt − a(u, xt, θ)′[Cyt−1 + d(xt, θ)] − b(u, xt, θ)}|yt−1, ft, xt] = exp{a(u, xt, θ)′Bft} (2) Assumption A.3: There exist a change of argument from u ∈ Rn to v = (v′

1, v′ 2)′ ∈ Rn−K × RK such that u(v, xt, θ, B), say, satisfies:

a[u(v, xt, θ, B), xt, θ]′B = v′

1,

∀v, B, θ, xt. We get the continuum of nonlinear first-order moment restrictions for the marginal distribution of (ft) (conditional on xt): E[exp{u(v, xt, θ, B)′yt − a[u(v, xt, θ, B), xt, θ]′[Cyt−1 + d(xt, θ)] − b[u(v, xt, θ, B), xt, θ]}|xt] = E[exp(v′

1ft)|xt],

∀v

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.2 (Non-)parametric identification of the marginal distribution of the dynamic effect

Proposition 2: Under Assumptions A.1-A.3, generically, the first-order nonlinear moment restrictions identify: i) The unconditional distribution of process (ft), once the parameters B, C, θ are known; ii) The regression parameters B, C, θ and the parameters characterized by the marginal of (ft), when the distribution of (ft) is specified parametrically. The parameters characterizing the dynamics of (ft) are not identifiable from the first-order moment restrictions

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.3 Examples: Factor models

Example 3: Linear model with common factor and contagion (cont.)    yt = Bft + Cyt−1 + εt, εt ∼ IIN(0, Σ) ft = Φft−1 + ηt, ηt ∼ IIN(0, Id − ΦΦ′) under the identification constraint that matrix B′B is diagonal First-order nonlinear moment restrictions: E0[exp(u′(yt − Cyt−1))] = exp 1 2u′(Σ + BB′)u

• ,

∀u ⇔ V0(yt − Cyt−1) = Σ + BB′ ⇔ Γ0(0) − Γ0(1)C′ − CΓ0(1)′ + CΓ0(0)C′ = Σ + BB′, (3) where Γ0(0) = V0(yt) and Γ0(1) = Cov0(yt, yt−1) Equation (3) is a linear combination of the first two Yule-Walker equations for (yt)

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.3 Examples: Factor models

Special case I: Φ = 0 and C = 0, i.e. static factor and no contagion The first-order identification restriction becomes: Γ0(0) = Σ + BB′ ⇒ Σ = Σ0, B = B0 (4) [see e.g. Anderson, Rubin (1956), Lawley, Maxwell (1971)] Special case II: Φ = 0, i.e. static factor but possibly contagion The order condition for identification from restriction (3) n(n + 1) 2 + K(K − 1) 2 ≥ n(K + n + 1) is not satisfied for n ≥ K ≥ 1 Parameter (B0, C0, Σ0) is not point identifiable from the first-order nonlinear moment restrictions, i.e. the set E0 = {(B, C, Σ) : solution of the first-order restriction (3) } is not a singleton

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.3 Examples: Factor models

However, the first-order nonlinear moment restrictions (3) can be rewritten as Σ0 + B0B′

0 + (C − C0)Γ0(0)(C − C0)′ = Σ + BB′

⇒ Parameter (B0, C0, Σ0) is set identifiable under (4) since: Σ0 + B0B′

0 = min{Σ + BB′ : (Σ, B, C) ∈ E0},

where the minimum is with respect to the ordering on symmetric matrices Set identification holds even if the order condition fails!

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.3 Examples: Factor models

General case: dynamic factor and contagion The parameters are neither point identifiable, nor set identifiable, from the first-order moment restrictions Higher-order moment restrictions are needed: next section! Darolles, Dubecq, Gourieroux (2014) show that the loadings matrix B is identifiable under a full-rank condition for a multivariate partial autocovariance

• f order 2 of the observable process

Their identification strategy relies on the fact that the linear combinations of the components of yt, which are immune to the unobserved factor ft, are uncorrelated with yt−2, yt−3, ..., conditional on yt−1

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.3 Examples: Factor models

Example 4: Conditionally Gaussian factor model with stochastic volatility in the factor (cont.)    yt = βh1/2

t

ηt + εt, εt ∼ IIN(0, Σ), ht ∼ ARG(δ, ρ, c), ηt ∼ IIN(0, 1), under the identification constraint c = 1 − ρ, which implies ht ∼ γ(δ) Variance-covariance matrix Σ is not necessarily diagonal The first-order nonlinear moment restrictions are: E0[exp(u′yt)] = exp u′Σu 2 − δ log

• 1 − (u′β)2

2

• ,

∀u

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• 3. FIRST-ORDER NL MOMENT RESTRICTIONS

3.3 Examples: Factor models

Parameters β, Σ and δ are identifiable since: u′Σu 2 − δ log

• 1 − (u′β)2

2

• = u′Σ0u

2 − δ0 log

• 1 − (u′β0)2

2

• ,

∀u ⇔ Σ = Σ0, β = β0, δ = δ0 The autocorrelation ρ of the SV process is not identifiable from the first-order nonlinear moment restrictions Parameter identification is simpler in this nonlinear framework compared to the Gaussian linear factor model!

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.1 Second-order nonlinear moment restrictions

Let us define ψt(u, C, θ) = a(u, xt, θ)′[Cyt−1 + d(xt, θ)] + b(u, xt, θ). Then: E

• exp{u′yt + ˜

u′yt−1 − ψt(u, C, θ) − ψt−1(˜ u, C, θ)}|yt−2, ft, xt

• =

exp{a(u, xt, θ)′Bft + a(˜ u, xt−1, θ)′Bft−1}, ∀ u, ˜ u By the change of variable in Assumption A.3, and integrating out the unobservable effects (ft, ft−1), we get a continuum of second-order nonlinear moment restrictions: E

• exp{u′

tyt + ˜

u′

tyt−1 − ψt(ut, C, θ) − ψt−1(˜

ut, C, θ)}|xt

• = E
• exp{v′

1ft + ˜

v′

1ft−1}|xt

• ,

for all admissible v, ˜ v, where ut = u(v, xt, θ, B) and ˜ ut = u(˜ v, xt, θ, B)

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.1 Second-order nonlinear moment restrictions

Proposition 3: Under Assumptions A.1-A.3, the first- and second-order nonlinear moment restrictions allow for identifying (generically): i) The complete model, if the distribution of Markov process (ft) is specified parametrically. ii) The stationary and transition distribution of Markov process (ft), if the regression parameters are known and the distribution of process (ft) is let unspecified.

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.2 Third-order nonlinear moment restrictions

Third-order nonlinear moment restrictions from the joint Laplace transform of yt, yt−1, yt−2 conditional on yt−3, ft, xt Allow to identify the conditional transition densities of the unobservable effect g1(ft|ft−1; B, C, θ) at horizon 1, and g2(ft|ft−2; B, C, θ) at horizon 2, for given regression parameter values B, C, θ Kolmogorov relationship implies an infinite number of restrictions: g2(ft|ft−2; B, C, θ) =

• g1(ft|ft−1; B, C, θ)g1(ft−1|ft−2, B, C, θ)dft−1,

∀ ft, ft−2, B, C, θ, which can be used to identify the regression parameters Proposition 4: Under Assumptions A.1-A.3, the first-, second- and third-order nonlinear moment restrictions (generically) identify the regression parameters and the unspecified transition of Markov process (ft).

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.3 Examples: Factor models

Example 3: Linear model with common factor and contagion (cont.) The nonlinear moment restrictions at order 1, 2 and 3 are : Γ0(0) + CΓ0(0)C′ − Γ0(1)C′ − CΓ0(1)′ = BB′ + Σ, for order 1, Γ0(1) + CΓ0(1)C′ − Γ0(2)C′ − CΓ0(0) = BΦB′, to be added for order 2, Γ0(2) + CΓ0(2)C′ − Γ0(3)C′ − CΓ0(1) = BΦ2B′, to be added for order 3, where Γ0(h) = Cov0(yt, yt−h), h = 0, 1, 2, 3

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.3 Examples: Factor models

Let us focus on second-order moment restrictions If symmetric matrix Σ is unrestricted, the order condition for identification is not satisfied If Σ is assumed diagonal, the order condition is satisfied if K is sufficiently small In both cases, the Gaussian linear model with common unobservable factor and contagion is not identifiable Thus, the order condition is neither sufficient, nor necessary, for identification

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.3 Examples: Factor models

Example 4: Conditionally Gaussian factor model with stochastic volatility in the factor (cont.) The second-order nonlinear moment restrictions are E0[exp(u′yt + ˜ u′yt−1)] = exp 1 2u′Σu + 1 2˜ u′Σ˜ u

• E
• exp

(u′β)2 2 ht + (˜ u′β)2 2 ht−1

• ,

for all u, ˜

• u. Let us assume β0,1 = 0, and consider argument vectors

u = ( √ 2v/β0,1, 0, . . . , 0)′ and ˜ u = ( √ 2˜ v/β0,1, 0, . . . , 0)′. We get: E0

• exp

√ 2v β0,1 y1,t + √ 2˜ v β0,1 y1,t−1

• = E [exp (v(ht + λ0) + ˜

v(ht−1 + λ0))] , ∀v,˜ v, where β0,1 and λ0 = σ2

0,11/β2 0,1 are identified from the first-order restrictions

The ARG dynamics of (ht) is identifiable from the second-order restrictions

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.4 Semi-nonparametric identification of the linear model with contagion and (non Gaussian) common factor

Consider the semiparametric model with common factor and contagion yt = B0ft + C0yt−1 + εt where the unobservable processes (εt) and (ft) are mutually independent and (εt) is a n-dim. stationary strong white noise with unconditional Laplace transform E0[exp(u′εt)] = exp[ψ0

ε(u)]

(ft) is a K-dim. stationary first-order Markov process with joint Laplace transform E0[exp(u′ft + w′ft−1)] = exp[ψ0

f (u, w)]

Parameter (B0, C0, ψ0

ε, ψ0 f ) has both finite- and infinite-dimensional

components

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.4 Semi-nonparametric identification of the linear model with contagion and (non Gaussian) common factor

i) Identification of K, B0 and C0 Consider the identifiable function: h0(u, ˜ u, C) := log E0 [exp (u′(yt − Cyt−1) + ˜ u′(yt−1 − Cyt−2))] It is such that: h0(u, ˜ u, C0) = ψ0

ε(u) + ψ0 ε(˜

u) + ψ0

f (B′ 0u, B′ 0˜

u) Main idea: cross-terms in u, ˜ u are informative on common factor dynamics and factor loadings! The matrix of second-order cross partial derivatives of function h0(., ., C0) ∂2h0(u, ˜ u, C0) ∂u∂˜ u′ = B0 ∂2ψ0

f (B′ 0u, B′ 0˜

u) ∂v∂w′ B′ has reduced rank ≤ K and its null space contains the orthogonal complement

• f the range of matrix B0, for all u, ˜

u

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.4 Semi-nonparametric identification of the linear model with contagion and (non Gaussian) common factor

Assumption A.4: i) ∂2h0(u, ˜ u, C) ∂u∂˜ u′ has reduced rank ∀(u, ˜ u) iff C = C0. ii) Processes (ft) and (εt) are not Gaussian, and process (ft) is not i.i.d. iii) ∃(v∗, w∗) : ∂2ψ0

f (v∗, w∗)

∂v∂w′ has full rank. Proposition 5: Under Assumption A.4, the dimension K of the factor space, the range of matrix B0, and the contagion matrix C0 are identifiable. Under Assumption A.4 identification is possible even if the order condition fails!

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.4 Semi-nonparametric identification of the linear model with contagion and (non Gaussian) common factor

ii) Identification of ψ0

ε and ψ0 f

Assumption A.5: The factor process (ft) is Compound Autoregressive (CaR): E0[exp(u′ft)|ft−1] = exp[a0(u)′ft−1 + b0(u)] Functional parameters: a0 and marginal log-Laplace transform ϕ0

f = ψ0 f (·, 0)

Normalize (ft) such that the upper (K, K) block of matrix B0 is IdK, and consider the identifiable function τ0(u, v; h) := log E0[exp(u′˜ yt + v′˜ yt−h)] E0[exp(u′˜ yt)]E0[exp(v′˜ yt−h)], where ˜ yt denotes the (K, 1) vector process of the first K elements of yt − C0yt−1. Then: τ0(u, v; h) = ϕ0

f [a◦h 0 (u) + v] − ϕ0 f [a◦h 0 (u)] − ϕ0 f (v),

∀u, v, ∀h ≥ 1, (6) where a◦h

0 denotes function a0 compounded h times with itself

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• 4. HIGHER-ORDER NL MOMENT RESTRICTIONS

4.4 Semi-nonparametric identification of the linear model with contagion and (non Gaussian) common factor

From (6) it follows τ0(u, v; 2) = τ0[a0(u), v; 1], ∀u, v and ∂τ0 ∂v (u, 0; 1) = ∂ϕ0

f

∂u [a0(u)], ∀u Proposition 6: Suppose that Assumptions A.4-A.5 hold and function a0 : D0 → R0 is one-to-one, where domain D0 is a neighbourhood of 0 in CK, and R0 ⊂ D0. Then: i) Function a0(u) is identifiable for u ∈ D0, ii) Function ϕ0

f (u) is identifiable for u ∈ R0, and

iii) Function ψε(u) is identifiable for B′

0u ∈ R0.

• P. GAGLIARDINI and G. GOURIEROUX

Identification by Laplace Transform November 21, 2014 36 / 38

CONCLUDING REMARKS

This paper studies identification in nonlinear panel or time series models with unobservable dynamic effects under the assumption of an exponential affine specification Nonlinear moment restrictions are obtained from the conditional Laplace transform of the endogenous observable variables yt, yt−1, ... by integrating

• ut the unobservable effects

Second-order nonlinear moment restrictions generically identify all model parameters, under a parametric specification for the individual effects dynamics Third-order nonlinear moment restrictions are generically sufficient for nonparametric identification of the individual effects dynamics

• P. GAGLIARDINI and G. GOURIEROUX

Identification by Laplace Transform November 21, 2014 37 / 38

CONCLUDING REMARKS

Even when the order condition fails, identification can be possible by exploiting “corner”, or reduced-rank, properties of the true parameter values Identification may be simpler in a nonlinear framework, with non Gaussian factors and innovations, and with dynamic factors The proposed identification strategies suggest parametric estimation with GMM [see e.g. Singleton (2001), Jiang, Knight (2002), Chacko, Viceira (2003), Carrasco, Chernov, Florens, Ghysels (2007)] and nonparametric estimation by plug-in methods

• P. GAGLIARDINI and G. GOURIEROUX

Identification by Laplace Transform November 21, 2014 38 / 38