## _wp0478 - References

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---

### Introduction
- Empirical literature on contagion uses a range of methodologies: Sharpe (1964); Grubel and Fadner (1971); King and Wadhwani (1990); Engle, Ito, and Lin (1990); Bekaert and Hodrick (1992).
- Paper objective: provide a unifying framework based on a latent factor structure to highlight similarities and differences among approaches (Dungey and Martin (2001); Corsetti, Pericoli, and Sbracia (2001); Bekaert, Harvey, and Ng (2003)).
- Approaches compared:
  - Correlation analysis (Forbes and Rigobon (2002))
  - Vector autoregression (VAR) (Favero and Giavazzi (2002))
  - Probability models (Eichengreen, Rose, and Wyplosz (1995, 1996))
  - Co-exceedance approach (Bae, Karolyi, and Stulz (2003))
- Key insight: differences in contagion definitions are minor; under certain conditions approaches are equivalent and amount to tests of a slope dummy (parameter γ).

### II. A model of interdependence
- Baseline latent factor/trivariate factor model (non-crisis):
  - x_{i,t} = λ_i w_t + δ_i u_{i,t}, i = 1,2,3.
  - w_t ∼ (0,1); u_{i,t} ∼ (0,1), ∀i.
  - Independence: E[u_{i,t} u_{j,t}] = 0, ∀i ≠ j; E[u_{i,t} w_t] = 0, ∀i.
  - Covariances (non-crisis): E[x_{i,t} x_{j,t}] = λ_i λ_j, ∀i ≠ j.
  - Variances: E[x_{i,t}^2] = λ_i^2 + δ_i^2, ∀i.
- Special cases and identification:
  - λ_1 = λ_2 = λ_3 = 0 → independent asset markets.
  - Example identification: set λ_i λ_j = constant L for all i ≠ j (Mahieu and Schotman (1994)).

### III. Unanticipated-shock models of contagion
- Contagion definition: transmission of unanticipated local shocks across markets during crisis; test statistical significance of γ in crisis-period model.
- Crisis-period trivariate model (contagion from country 1 to 2):
  - y_{1,t} = λ_1 w_t + δ_1 u_{1,t}
  - y_{2,t} = λ_2 w_t + δ_2 u_{2,t} + γ u_{1,t}
  - y_{3,t} = λ_3 w_t + δ_3 u_{3,t}
- Bivariate implications:
  - Crisis covariance: E[y_{1,t} y_{2,t}] = λ_1 λ_2 + γ δ_1.
  - Change in covariance: E[y_{1,t} y_{2,t}] − E[x_{1,t} x_{2,t}] = γ δ_1.
  - Crisis variance for y_{2,t}: E[y_{2,t}^2] = λ_2^2 + δ_2^2 + γ^2.
  - Volatility decomposition (relative contributions): λ_2^2/(λ_2^2+δ_2^2+γ^2), δ_2^2/(λ_2^2+δ_2^2+γ^2), γ^2/(λ_2^2+δ_2^2+γ^2).
- Multivariate extension and identification:
  - General form: y_{i,t} = λ_i w_t + δ_i u_{i,t} + Σ_{j≠i} γ_{i,j} u_{j,t}.
  - For N=3: 6 contagion parameters γ_{i,j}; pre-crisis and crisis variance-covariance matrices provide 12 moments to identify 12 unknowns.
  - Joint test: compare unconstrained (q_u) and constrained (q_c) objective functions; under H0: q_c = 0 ∼ χ^2 with six degrees of freedom for the trivariate example.
- Structural breaks and excess moments:
  - Increased interdependence without contagion can arise if λ_i or δ_i change between subperiods.
  - Counting excess moments:
    - If idiosyncratic breaks allowed for all N: excess moments = N(N−1)/2.
    - If both global and idiosyncratic breaks allowed for all N: excess moments = N(N−3)/2.
    - Examples: N=3 with all breaks → just identified (no testable contagion channels); N=4 → 2 potential contagion channels; N=6 → 9 potential contagion channels.
- Using only crisis data:
  - Identification feasible when non-crisis moments unavailable if N large enough and limited contagion links entertained (example: N=4 gives 10 unique moments and allows two contagious links).
- Dynamics and heteroscedasticity:
  - Four modeling avenues: include lagged returns, lag in world factor w_t, autoregressive idiosyncratic u_{i,t}, or dynamics in variance (GARCH).
  - Example factor-GARCH specification:
    - w_t = e_t, e_t ∼ N(0, h_t)
    - h_t = (1 − α − β) + α e_{t−1}^2 + β h_{t−1}.
  - Conditional moments with contagion:
    - E_{t−1}[y_{2,t}^2] = λ_2^2 h_t + δ_2^2 + γ^2.
    - E_{t−1}[y_{1,t} y_{2,t}] = λ_1 λ_2 h_t + γ δ_1.
  - Factor-GARCH offers a parsimonious multivariate GARCH alternative to BEKK.

### IV. Correlation and covariance analysis
- Forbes and Rigobon (2002) define contagion as increase in correlation during crisis and adjust for upward bias since correlations are positive functions of volatility.
- Forbes–Rigobon adjustment (bivariate):
  - Adjusted unconditional correlation: ν_y = ρ_y / { r1 + [ (σ_{y,1}^2 − σ_{x,1}^2) / σ_{x,1}^2 ] (1 − ρ_y^2) }.
  - Null hypothesis: H0: ν_y = ρ_x versus H1: ν_y > ρ_x.
  - Test statistics:
    - FR_1 = (ν̂_y − ρ̂_x) / sqrt(1/T_y + 1/T_x).
    - Fisher-transform statistic FR_2 as given in source equation (28).
- Regression (Chow-test) formulation:
  - Pre-crisis scaled regression: (x_{2,t}/σ_{x,2}) = α_1 (x_{1,t}/σ_{x,1}) + η_{x,t}, α_1 = ρ_x.
  - Crisis scaled regression: (y_{2,t}/σ_{x,2}) = β_1 (y_{1,t}/σ_{x,1}) + η_{y,t}, β_1 = ν_y.
  - Pooled regression with slope dummy: (z_{2,t}/σ_{x,2}) = β_1 (z_{1,t}/σ_{x,1}) + γ_1 (z_{1,t}/σ_{x,1}) d_t + η_t, where γ_1 = β_1 − α_1. Test H0: γ_1 = 0.
  - Regression approach invariant to scaling; small-sample standard errors differ from Forbes–Rigobon adjustment.
- Relationship to unanticipated-shock models:
  - Substituting u_{1,t} out of contagion model yields form equivalent to pooled regression with world factor as stacked source-country returns; Forbes–Rigobon and latent-shock tests are linked via γ.
- Multivariate extension:
  - For N=3, joint test α_{i,j} = β_{i,j}, ∀i≠j (six restrictions) via pooled system with slope dummies γ_{i,j}; implementable with LR/Wald/LM.
- DCC (determinant of change in covariance matrix) test (Rigobon):
  - DCC = ||Ω̂_y − Ω̂_x|| / σ̂_{DCC}; under H0 change = 0.
  - In bivariate factor setting numerator reduces to −γ̂^2 δ̂_2; DCC = 0 iff γ = 0.
  - Practical implementation often uses VAR residuals across total sample to form conditional covariances to mitigate simultaneity.
- Endogeneity and simultaneity:
  - Treating source country as exogenous (Forbes and Rigobon) tests one direction only; reversing exogeneity is inappropriate due to simultaneity bias.
  - Simultaneity bias small if correlations are small (Monte Carlo evidence).
  - Correct approach: estimate simultaneous equations (IV or structural estimators) and test γ_{i,j} on simultaneous-equation estimates.
  - Identification strategies:
    - Imposing α_i = β_i may preclude contagion detection.
    - Restricting no structural break in idiosyncratic variances can yield identification.
    - Expanding assets (N≥3) or including lags (VAR) provides additional moments/identification.
  - Instrument proposal: use stacked counterpart series s_i = [−x_{i,1},...,−x_{i,T_x}, y_{i,1},...,y_{i,T_y}]′ (Rigobon); akin to Wald (1940) dummies.

### V. Models of asymmetries and nonlinearities
- Motivation: transmissions may be nonlinear or asymmetric during extreme returns.
- Four classes of approaches:
  A. Outliers (Favero and Giavazzi (2002))
  - Use VAR to identify unusual residuals; construct dummies for extreme residuals (e.g., d_{i,k,t} = 1 if |v_{i,t}| > 3 σ_i).
  - Structural model includes these dummies; contagion tested via significance of γ parameters on dummies.
  - Similar to Forbes–Rigobon: both effectively test slope dummies; differences in dummy construction and parameterization.
  - Simultaneity requires simultaneous-equation estimation; identification often via own lags with potential weak-instrument issues.
  B. Probability models — dichotomous classifications (Eichengreen, Rose, Wyplosz (1995, 1996); Kaminsky and Reinhart (2000))
  - Construct Exchange Market Pressure (EMP) and binary CRISIS indicator (example threshold: CRISIS if EMP > μ_EMP + 1.5 σ_EMP).
  - Probit framework: latent y^* driven by world factor and contagious dummy d_{1,t}; d_{2,t} = 1[y^* < f(EMP_{2,t})].
  - Test contagion via γ significance; choice of EMP threshold affects classification and results.
  - Probability change: ∂P_t/∂d_{1,t} = Φ(z_1) − Φ(z_0).
  - Least-squares on continuous equation can be consistent and more efficient than binary classification.
  C. Probability models — polychotomous classifications (Bae, Karolyi, Stulz (2003))
  - Tail focus via exceedances E_{i,t} = 1[y_{i,t} < THRESH] (THRESH example: −5 percent).
  - Model co-exceedance counts with multinomial logit:
    - P_{j,t} = exp(β_j x_{j,t}) / Σ_{k=0}^N exp(β_k x_{k,t}), j = 0,...,N.
  - Extends probit framework to multiple co-exceedance outcomes.
  D. One-sided asymmetries
  - Distinguish positive vs negative shocks, example:
    - y_{2,t} = λ_2 w_t + δ_2 u_{2,t} + γ^+ u_{1,t} d_t + γ^− u_{1,t} (1 − d_t), d_t = 1[u_{1,t} > 0].
  - Test symmetry H0: γ^+ = γ^−.
  - Evidence: Butler and Joaquin (2002) find correlations rise more in bear markets than bull markets.
  - Other approaches: event-based dummies for known dates; asymmetric GARCH in idiosyncratic shocks (Bekaert, Harvey, and Ng (2003)).

### VI. Additional methods
- Principal components and dynamic factor approaches:
  - Principal components via eigen-decomposition of covariance/correlation matrix (Calvo and Reinhart (1995); Kaminsky and Reinhart (2001)).
  - Constant variance-covariance assumption may be inappropriate across crisis periods; dynamic factors (Mody and Taylor (2003)) or extended latent factor model recommended.
- Multiple equilibria and mixture distributions:
  - Model with mixture distributions f(y_{i,t}) = φ f_1(y_{i,t}) + (1 − φ) f_2(y_{i,t}) (0 < φ < 1).
  - Hamilton (Markov-switching) representation with time-varying φ_t discussed; threshold dummy systems can generate multiple equilibria.
- Spillovers vs contagion:
  - Spillovers: transmissions via identifiable channels (trade links, financial flows, competitive devaluation); examples include Glick and Rose (1999); Dasgupta (2001); van Rijckeghem and Weder (2001).
  - Information flows and investor preference channels: calibration/simulation evidence (Calvo and Mendoza (2000); Chue (2002)); Rigobon (2002) example of rating upgrade changing regional commonality.
- Multiple asset classes:
  - Most literature focuses on single asset classes; joint study of currencies, equities, bonds is limited but present (Kaminsky and Reinhart (2001); Baig and Goldfajn (1999); Ellis and Lewis (2000); Dungey et al. (2002a,b, 2003)).
  - Cross-asset tools: common-factor models, Granger causality, bivariate GARCH, stochastic discount factor/no-arbitrage approaches (Flood and Rose (2003)).
  - Practical modeling challenges: time zones, data frequency, missing observations, volatility structures, dimensionality.

### VII. Conclusions
- Unifying result: diverse empirical contagion tests can be reinterpreted as tests on a common parameter γ within a latent factor framework.
- Five focal methods cast as γ-tests:
  - Latent factor models (Dungey et al.; Corsetti, Pericoli, Sbracia; Bekaert, Harvey, Ng).
  - Correlation adjustment (Forbes and Rigobon).
  - VAR/outlier identification (Favero and Giavazzi).
  - Probability/probit approaches (Eichengreen, Rose, Wyplosz).
  - Tail/co-exceedance multinomial approaches (Bae, Karolyi, Stulz).
- Open empirical issues highlighted:
  - Time zones, data frequency, missing observations, endogenous crisis definitions (addressed in companion work Dungey et al. (2004)).
  - Small-sample properties of tests and comparative performance across crises remain to be explored.

*IMF Working Paper content unit _wp0478 - References*

### References..............................................................................................................

### _wp0478 - References..............................................................................................................

### Introduction
- Empirical literature on contagion uses a range of methodologies (Sharpe (1964); Grubel and Fadner (1971); King and Wadhwani (1990); Engle, Ito, and Lin (1990); Bekaert and Hodrick (1992)).
- Paper objective: provide a unifying framework based on a latent factor structure to highlight similarities and differences among approaches (Dungey and Martin (2001); Corsetti, Pericoli, and Sbracia (2001); Bekaert, Harvey, and Ng (2003)).
- Approaches compared:
  - Correlation analysis (Forbes and Rigobon (2002))
  - Vector autoregression (VAR) (Favero and Giavazzi (2002))
  - Probability models (Eichengreen, Rose, and Wyplosz (1995, 1996))
  - Co-exceedance approach (Bae, Karolyi, and Stulz (2003))
- Key insight: differences in contagion definitions are minor; under certain conditions approaches are equivalent and amount to tests of a slope dummy (parameter γ).

### II. A model of interdependence
- Baseline latent factor/trivariate factor model (non-crisis):
  - Asset returns vector: {x1,t, x2,t, x3,t}.
  - Model: x_{i,t} = λ_i w_t + δ_i u_{i,t}, i = 1,2,3.
  - World factor: w_t ∼ (0,1).
  - Idiosyncratic factors: u_{i,t} ∼ (0,1), ∀i.
  - Independence assumptions: E[u_{i,t} u_{j,t}] = 0, ∀i ≠ j; E[u_{i,t} w_t] = 0, ∀i.
  - Covariances during non-crisis: E[x_{i,t} x_{j,t}] = λ_i λ_j, ∀i ≠ j.
  - Variances: E[x_{i,t}^2] = λ_i^2 + δ_i^2, ∀i.
- Special cases:
  - λ_1 = λ_2 = λ_3 = 0 → independent asset markets.
  - Identification strategy example: λ_i λ_j set to constant L for all i ≠ j (Mahieu and Schotman (1994)).

### III. Unanticipated-shock models of contagion
- Contagion definition used: transmission of unanticipated local shocks across markets during crisis.
- Crisis-period model (contagion from country 1 to country 2):
  - y_{1,t} = λ_1 w_t + δ_1 u_{1,t}
  - y_{2,t} = λ_2 w_t + δ_2 u_{2,t} + γ u_{1,t}
  - y_{3,t} = λ_3 w_t + δ_3 u_{3,t}
- Empirical aim: test statistical significance of γ (test γ = 0).
- Bivariate implications:
  - Crisis covariance: E[y_{1,t} y_{2,t}] = λ_1 λ_2 + γ δ_1.
  - Change in covariance: E[y_{1,t} y_{2,t}] − E[x_{1,t} x_{2,t}] = γ δ_1.
  - Crisis variance for y_{2,t}: E[y_{2,t}^2] = λ_2^2 + δ_2^2 + γ^2.
  - Volatility decomposition (relative contributions): λ_2^2/(λ_2^2+δ_2^2+γ^2), δ_2^2/(λ_2^2+δ_2^2+γ^2), γ^2/(λ_2^2+δ_2^2+γ^2).
- Multivariate extension:
  - General form: y_{i,t} = λ_i w_t + δ_i u_{i,t} + Σ_{j≠i} γ_{i,j} u_{j,t}.
  - For N=3 there are 6 contagion parameters γ_{i,j}; identification requires combining moments from pre-crisis and crisis variance-covariance matrices (total 12 moments to identify 12 unknowns).
  - Joint test via comparing unconstrained (q_u) and constrained (q_c) objective functions; under H0: q_c = 0 ∼ χ^2 with six degrees of freedom (for the trivariate example).
- Structural breaks:
  - Increased interdependence without contagion can arise if λ_i or δ_i change between subperiods.
  - Allowing structural breaks increases parameter counts; counting excess moments:
    - If idiosyncratic breaks allowed for all N: excess moments = N(N−1)/2.
    - If both global and idiosyncratic breaks allowed for all N: excess moments = N(N−3)/2.
    - Examples: N=3 with all breaks → just identified (no testable contagion channels); N=4 → 2 potential contagion channels; N=6 → 9 potential contagion channels.
- Using only crisis data:
  - Possible when non-crisis moments are unavailable; identification feasible if N large enough and limited contagion links entertained (example: N=4 gives 10 unique moments and allows two contagious links).
- Dynamics and heteroscedasticity:
  - Four avenues: include lagged returns (spillovers), lag in world factor w_t, autoregressive idiosyncratic u_{i,t}, or dynamics in variance (GARCH).
  - GARCH world factor specification example:
    - w_t = e_t, e_t ∼ N(0, h_t)
    - h_t = (1 − α − β) + α e_{t−1}^2 + β h_{t−1}.
  - Conditional variance and covariance in crisis include contagion terms: E_{t−1}[y_{2,t}^2] = λ_2^2 h_t + δ_2^2 + γ^2; E_{t−1}[y_{1,t} y_{2,t}] = λ_1 λ_2 h_t + γ δ_1.
  - Factor-GARCH provides parsimonious multivariate GARCH alternative to BEKK.

### IV. Correlation and covariance analysis
- Forbes and Rigobon (2002): define contagion as increase in correlation during crisis; adjust for upward bias since correlations are positive functions of volatility.
- Forbes–Rigobon adjustment (bivariate):
  - Adjusted unconditional correlation: ν_y = ρ_y / { r1 + [ (σ_{y,1}^2 − σ_{x,1}^2) / σ_{x,1}^2 ] (1 − ρ_y^2) } (equation (24) in source).
  - Null hypothesis: H0: ν_y = ρ_x versus H1: ν_y > ρ_x.
  - Test statistics:
    - FR_1 = (ν̂_y − ρ̂_x) / sqrt(1/T_y + 1/T_x).
    - Fisher-transform statistic FR_2 (see equation (28) for exact form).
- Regression (Chow-test) formulation:
  - Pre-crisis scaled regression: (x_{2,t}/σ_{x,2}) = α_1 (x_{1,t}/σ_{x,1}) + η_{x,t}, α_1 = ρ_x.
  - Crisis scaled regression: (y_{2,t}/σ_{x,2}) = β_1 (y_{1,t}/σ_{x,1}) + η_{y,t}, β_1 = ν_y.
  - Pooled regression with slope dummy: (z_{2,t}/σ_{x,2}) = β_1 (z_{1,t}/σ_{x,1}) + γ_1 (z_{1,t}/σ_{x,1}) d_t + η_t, where γ_1 = β_1 − α_1. Test H0: γ_1 = 0.
  - This regression approach is invariant to scaling; standard errors differ in small samples compared with Forbes–Rigobon adjustment.
- Relationship to unanticipated-shock models:
  - Substituting u_{1,t} out of contagion model yields form equivalent to pooled regression with world factor as stacked source-country returns; thus Forbes–Rigobon and latent-shock tests are linked via γ.
- Multivariate extension:
  - For N=3, pre-crisis and crisis systems of scaled equations with parameters α_{i,j} and β_{i,j}; joint test α_{i,j} = β_{i,j}, ∀i≠j (six restrictions).
  - Implementation via pooled system with slope dummies γ_{i,j}; test H0: γ_{i,j} = 0, ∀i≠j using LR/Wald/LM.
- DCC (determinant of change in covariance matrix) test (Rigobon):
  - DCC = ||Ω̂_y − Ω̂_x|| / σ̂_{DCC}; under H0 change = 0.
  - In bivariate factor setting, numerator reduces to −γ̂^2 δ̂_2, and DCC = 0 iff γ = 0.
  - Practical implementation often uses VAR residuals across total sample to form conditional covariances (to mitigate simultaneity).
- Endogeneity issues:
  - Forbes and Rigobon treat source country as exogenous and test in one direction only; reversing exogeneity is inappropriate due to simultaneity bias.
  - Simultaneity bias small if correlations are small (Monte Carlo evidence).
  - Correct approach: estimate simultaneous equations (IV or structural estimators) and test γ_{i,j} on simultaneous-equation estimates.
  - Identification strategies:
    - Imposing α_i = β_i (as in some applications) may preclude contagion detection.
    - Restricting no structural break in idiosyncratic variances can yield identification with available moments.
    - Expanding assets (N≥3) or including lags (VAR approach) provides additional moments/identification.
  - Instruments proposal: use stacked counterpart series s_i = [−x_{i,1},...,−x_{i,T_x}, y_{i,1},...,y_{i,T_y}]′ (Rigobon); similar to Wald (1940) dummies.

### V. Models of asymmetries and nonlinearities
- Motivation: transmissions may be nonlinear or asymmetric during extreme returns.
- Four classes:
  A. Outliers (Favero and Giavazzi (2002)):
    - Use a VAR to identify unusual residuals; construct dummies for extreme residuals (e.g., d_{i,k,t} = 1 if |v_{i,t}| > 3 σ_i).
    - Structural model includes these dummies; contagion tested via significance of dummies (γ parameters).
    - Similarity to Forbes–Rigobon: both effectively test slope dummies; differences: Forbes–Rigobon uses period-long higher-volatility dummy, Favero–Giavazzi uses multiple short-lived outlier dummies and assigns separate parameters.
    - Simultaneity requires simultaneous-equation estimation; identification often via own lags—may produce weak-instrument issues if autocorrelation in returns is weak.
  B. Probability models — dichotomous classifications (Eichengreen, Rose, Wyplosz (1995, 1996); Kaminsky and Reinhart (2000)):
    - Construct Exchange Market Pressure (EMP) index and binary CRISIS indicator based on thresholds (e.g., CRISIS if EMP > μ_EMP + 1.5 σ_EMP in source).
    - Specify probit model: latent y^* driven by world factor and contagious dummy d_{1,t}; dependent binary d_{2,t} = 1[y^* < f(EMP_{2,t})].
    - Test contagion via significance of γ in the probit. Choice of EMP threshold affects classification and results.
    - Probabilities of crisis changes can be computed ∂P_t/∂d_{1,t} = Φ(z_1) − Φ(z_0) with z_1, z_0 as evaluated.
    - Least-squares on the continuous equation (equation (38) form) can deliver consistent and more efficient estimates than binary classification.
  C. Probability models — polychotomous classifications (Bae, Karolyi, Stulz (2003)):
    - Focus on tails using exceedances E_{i,t} = 1[y_{i,t} < THRESH] (THRESH = −5 percent example).
    - Define co-exceedances (simultaneous exceedances) and model counts with multinomial logit:
      P_{j,t} = exp(β_j x_{j,t}) / Σ_{k=0}^N exp(β_k x_{k,t}), j = 0,...,N.
    - Extends probit framework to multiple co-exceedance outcomes; alternative thresholds studied by Baur and Schulze (2002).
  D. One-sided asymmetries:
    - Distinguish positive (good) vs negative (bad) news. Example model:
      y_{2,t} = λ_2 w_t + δ_2 u_{2,t} + γ^+ u_{1,t} d_t + γ^− u_{1,t} (1 − d_t), d_t = 1[u_{1,t} > 0].
    - Test symmetry H0: γ^+ = γ^−.
    - Butler and Joaquin (2002): correlations rise much more in bear markets than bull markets.
    - Other approaches use event-based dummies for known dates (Baig and Goldfajn (2000); Ellis and Lewis (2000); Kaminsky and Schmukler (1999)).
    - Bekaert, Harvey, and Ng (2003) incorporate asymmetric GARCH in idiosyncratic shocks.

### VI. Additional methods
- Principal components and dynamic factor approaches:
  - Principal components via eigen-decomposition of covariance/correlation matrix (Calvo and Reinhart (1995); Kaminsky and Reinhart (2001)).
  - Assumption of constant variance-covariance may be inappropriate across crisis periods; use dynamic factors (Mody and Taylor (2003)) or the extended latent factor model.
- Multiple equilibria and mixture distributions:
  - Contagion theories often yield multiple equilibria; model with mixture distributions f(y_{i,t}) = φ f_1(y_{i,t}) + (1 − φ) f_2(y_{i,t}) (0 < φ < 1).
  - Hamilton (Markov-switching) representation with time-varying φ_t discussed (Jeanne and Masson (2000); Masson (1999c)).
  - Threshold dummy systems (e.g., Peseran and Pick (2003); Favero and Giavazzi style) can generate multiple equilibria.
- Spillovers vs contagion:
  - Spillovers: transmissions via identifiable channels (trade links, financial flows, competitive devaluation); examples: Glick and Rose (1999); Dasgupta (2001); van Rijckeghem and Weder (2001).
  - Information flows and investor preference channels: calibration/simulation evidence (Calvo and Mendoza (2000); Chue (2002)); Rigobon (2002) example of rating upgrade changing regional commonality.
- Multiple asset classes:
  - Most literature focuses on single asset classes across countries; joint study of currencies, equities, bonds is limited but present (Kaminsky and Reinhart (2001); Baig and Goldfajn (1999); Ellis and Lewis (2000); Dungey et al. (2002a,b, 2003)).
  - Cross-asset modeling tools: common-factor models, Granger causality, bivariate GARCH, stochastic discount factor/no-arbitrage approaches (Flood and Rose (2003)).
  - Practical modeling challenges: time zones, data frequency, missing observations, volatility structures, and dimensionality.

### VII. Conclusions
- Unifying result: diverse empirical contagion tests can be reinterpreted as tests on a common parameter γ within a latent factor framework.
- Five focal methods cast as γ-tests:
  - Latent factor models (Dungey et al.; Corsetti, Pericoli, Sbracia; Bekaert, Harvey, Ng).
  - Correlation adjustment (Forbes and Rigobon).
  - VAR/outlier identification (Favero and Giavazzi).
  - Probability/probit approaches (Eichengreen, Rose, Wyplosz).
  - Tail/co-exceedance multinomial approaches (Bae, Karolyi, Stulz).
- Open empirical issues highlighted:
  - Time zones, data frequency, missing observations, endogenous crisis definitions (addressed in companion work Dungey et al. (2004)).
  - Small-sample properties of tests and comparative performance across crises remain to be explored.

*Italicized source attribution: IMF Working Paper content unit _wp0478 - References*

### References

### References

### Bibliography (alphabetical)

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*Italicized source attribution line provided separately by the pipeline.*

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