## 1. SNP Auxilliary Model

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---

### Overview and objectives
- Identify a class of jump-diffusion models that approximate short-term interest rates of emerging markets.
- Reconcile parameters of the term structure of interest rates with associated bond yields by estimating volatility and jump risk premia in highly volatile markets.
- Key empirical questions:
  - Are jumps in the term structure of interest rates statistically significant? — Most jumps in emerging markets coincide with financial crises.
  - To what extent are volatility and jump risk factors priced in emerging market bonds?
- Observations: frequent, large spikes in interest rates and bond yields interpreted as jumps; common crises referenced include the 1982 debt crisis, December 1994 Mexican Tequila effects, late-1997 Asian crisis, August 1998 Russian crisis, January 1999 Brazilian Sneeze, April 2000 NASDAQ rash.
- Estimation strategy: two-stage estimation (Benzoni (2002) extension):
  - Stage 1: SMM/EMM on daily short-term rates to estimate structural parameters using an SNP auxiliary model.
  - Stage 2: simulation on bond yields to estimate volatility and jump risk premia using continuous-time ATSM jump-diffusion bond pricing.

### Model specification and pricing kernel
- Model class: continuous-time affine jump-diffusion models with stochastic volatility and jump components in returns and volatility.
- Baseline: two-factor stochastic volatility (Andersen and Lund) with time-dependent volatility factor vt.
- Jumps: Poisson jumps in r and v with intensities λr and λv and jump sizes ξr and ξv; correlated Brownian motions and possible contemporaneous or independent jump arrivals.
- Market prices of risk:
  - Candidate pricing kernel assigns market prices of risk, yielding dynamics under martingale measure Q.
  - Volatility risk premia parameter: vη.
  - Compensating term for pure jump process expressed in the source as JJt tvηλ.

### Econometric approach
- First stage:
  - Use EMM (efficient method of moments) with an SNP density / quasi-MLE transition density auxiliary model to estimate structural parameters when latent volatility and jumps are present.
  - Long simulations compute expectations given candidate parameter vectors.
- Second stage:
  - Use continuous-time ATSM jump-diffusion bond pricing (discounted expected value under Q) and simulated method of moments to minimize bond yield errors and estimate risk premia parameters (vη, Jη).
  - Moment condition: E[PT(·) − P~(·)] = 0 where PT(·) is observed bond yield and P~(·) predicted price using the ATSM; error assumed stationary and ergodic with zero mean.

### Data sources and sample statistics
- Bond yields: Brady bond end-of-week yields from Datastream, May 1994 through December 2002, weekly frequency.
- Reported sample statistics (1994-2002 Brady Bond Price Statistics):
  - Argentina:
    - Mean 33.78
    - S.D. 0.21
    - Median 14.8
    - Skewness 5.4
    - Kurtosis 33.3
  - Brazil:
    - Mean 16.7
    - S.D. 4.9
    - Median 16.1
    - Skewness 1.8
    - Kurtosis 8.7

### Empirical results — model comparison and diagnostics
- General finding:
  - Diffusion-only models without stochastic volatility and unanticipated jumps cannot generate the non-normalities observed in interest rates; diffusion-only models fail to replicate fat tails.
  - Jump-diffusion models with stochastic volatility provide acceptable characterizations for emerging market short-term interest rates.

- A. SNP auxiliary model (selection and diagnostics)
  - SNP chosen by minimum BIC; semi-parametric GARCH auxiliary captures key features.
  - High-order AR terms required in mean equation (uL order 3); Garch(2,2) for conditional variance persistence.
  - Non-Gaussian innovations indicated by BIC reduction; heterogeneity in xK generally rejected.

- B. Simple diffusion benchmark (CIR: constant volatility, no jumps)
  - Estimated model: drt = kr(θ − rt) dt + σ dW.
  - Evidence of strong mean reversion (rk highly significant).
  - Goodness of fit: χ2 values in excess of 58.50[13] (Argentina) and 29.30[10] (Brazil) and rejected at the 95% level.
  - Constant variance σ approximately 7 percent for Argentina and Brazil, respectively (source statement).
  - Fails to account for leptokurtosis, ARCH/GARCH persistence, and important SNP score diagnostics.

- C. Stochastic volatility (SV) model (no jumps)
  - Allowing unobservable volatility (ρ = 0 for baseline) improves fit but is still strongly rejected.
  - Captures some asymmetry but fails to accommodate fat tails (insufficient excess kurtosis).
  - Does not fully fit ARCH and GARCH score diagnostics.

- D. Stochastic volatility with jumps (SVJ)
  - Add a Poisson jump process with Gaussian jump sizes and constant intensity λt = λ0.
  - Chi-square goodness-of-fit improves substantially: decreases to 20.34[8] and 20.88[7] for Argentina models and to 15.20[5] and 22.13[4] for Brazil models; some SVJ specifications cannot be rejected at conventional levels.
  - Jump parameters (J_μ, J_σ, 0_λ) are highly significant in SVJ specifications.
  - Jump frequency and sizes (source statements and table values):
    - "On average, we find that jumps occur (8,10) times a year in Argentina and Brazil, respectively."
    - Argentina (detailed from text): average 8.5 jumps per year; mean jump size 0.6±percent; standard deviation of jump size 0.4 percent.
    - Brazil (detailed from text): average 9.6 jumps per year; negative mean jump size (implying more negative than positive jumps).
  - Time-varying jump intensity specification (λt = λ0 + λ1 · volatility): λ0 and λ1 positive but λ1 estimates insignificant; linear jump intensity does not improve fit over constant intensity in these samples.

- E. Volatility and jump risk premia (reconciling short-term rates and bond yields)
  - Two pricing models considered:
    - Multifactor volatility model without jumps (SV).
    - Multifactor volatility model with jumps (SVJ).
  - Volatility risk premium (v_η):
    - Brazil SV: -0.4325 (0.0476)
    - Brazil SVJ: -0.1123 (0.0511)
    - Argentina SV: -0.3934 (0.0389)
    - Argentina SVJ: -0.0821 (0.0465)
    - Interpretation: vη is negative and significantly different from zero for the stochastic volatility model — variance risk is priced.
  - Jump risk premia (J_η) — sizes only; frequency risk premia assumed zero:
    - Argentina SVJ: 0.2561 (0.1176)
    - Brazil SVJ: 0.3397 (0.9766)
  - Relative magnitudes: jump risk premia are almost three times the volatility risk premia (source statement).

### Key table excerpts (selected EMM estimates and diagnostics)
- Table 2 — Argentina (sample October 1994 to December 1999), selected estimates (standard errors in brackets):
  - r_k:
    - CIR: 0.9814 (0.0910)
    - SV: 1.3243 (0.2430)
    - SVJ1: 0.8988 (0.0112)
    - SVJ2: 1.4555 (0.1350)
  - J_μ (SVJ models):
    - SVJ1: 0.3274 (0.0398)
    - SVJ2: -0.6026 (0.1043)
  - J_σ (SVJ models):
    - SVJ1: 0.2606 (0.0356)
    - SVJ2: 0.4090 (0.1322)
  - 0_λ (SVJ models):
    - SVJ1: 8.6321 (3.4522)
    - SVJ2: 8.5120 (2.3211)
  - dfχ:
    - CIR: 58.50[13]
    - SV: 21.58[10]
    - SVJ1: 20.34[8]
    - SVJ2: 20.88[7]

- Table 3 — Brazil (sample October 1994 to December 1999), selected estimates (standard errors in brackets):
  - r_k:
    - CIR: 0.8637 (0.0984)
    - SV: 1.0107 (0.1056)
    - SVJ1: 0.7240 (0.0602)
    - SVJ2: 0.7293 (0.1192)
  - J_μ (SVJ models):
    - SVJ1: -1.6966 (0.1004)
    - SVJ2: 0.2386 (0.5934)
  - J_σ (SVJ models):
    - SVJ1: 1.7405 (0.1145)
    - SVJ2: 1.3245 (0.2219)
  - 0_λ (SVJ models):
    - SVJ1: 9.5603 (2.4503)
    - SVJ2: 9.6321 (2.1146)
  - dfχ:
    - CIR: 29.30[10]
    - SV: 28.65[7]
    - SVJ1: 15.20[5]
    - SVJ2: 22.13[4]

- Table 4 — Volatility and Jump Risk Premia (standard errors in brackets):
  - v_η and J_η:
    - Brazil SV: v_η = -0.4325 (0.0476)
    - Brazil SVJ: v_η = -0.1123 (0.0511); J_η = 0.3397 (0.9766)
    - Argentina SV: v_η = -0.3934 (0.0389)
    - Argentina SVJ: v_η = -0.0821 (0.0465); J_η = 0.2561 (0.1176)

- Tables 5 and 6 — SNP score component estimates and t-ratios for Argentina and Brazil (selected entries preserved in source):
  - Argentina example entries:
    - Hermite a01: CIR 0.881 (0.89); SV 0.725 (0.73); SVJ -0.119 (-0.12); SVJ1 0.093 (0.09)
    - Hermite a02: CIR -6.530 (-1.99); SV 1.401 (0.43); SVJ -0.980 (-0.30); SVJ1 -5.227 (-1.59)
    - GARCH g1τ: CIR -18.367 (-2.38); SV -14.990 (-1.95); SVJ -8.013 (-1.04); SVJ1 -8.195 (-1.06)
  - Brazil example entries:
    - Hermite a01: CIR 0.506 (0.12); SV 0.678 (0.17); SVJ 1.035 (0.25); SVJ2 -0.319 (-0.078)
    - Hermite a02: CIR 1.918 (0.23); SV 2.093 (0.25); SVJ 5.330 (0.64); SVJ2 -2.604 (-0.313)
    - ARCH a1τ: CIR 11.606 (2.19); SV 10.693 (2.02); SVJ 9.668 (1.82); SVJ2 5.796 (1.093)

### Appendix I — SNP auxiliary model (specification and tuning)
- Transition density approximated by truncated Hermite expansion yielding a location-scale transform of an innovation tz.
- Innovation density: contains polynomial pzx in (x,z) of degree K multiplying multivariate normal φ.
- Semi-parametric specifications include Gaussian VAR, Gaussian ARCH, Semiparametric ARCH, Gaussian GARCH, Semiparametric GARCH, and nonlinear nonparametric forms as special cases depending on parameter settings.
- Tuning parameters (as in source):
  - zK degree of polynomial in z
  - xK degree of polynomial dependence on lags of tx
  - uL Lag Length of the VAR
  - rL Lag length of ARCH portion
  - gL Lag length of GARCH portion
  - pL Lag length of polynomial dependence on lags of tx
- Model selection: Schwarz Bayes Information criterion
  - BICsn = (−1/n) log( ) + (1/2)(p/n) log(n) as in the source (equation A.7); smaller BIC indicates better fit.

### Conclusions and extensions
- Main conclusions:
  - Diffusion-only models with stochastic volatility cannot fit non-normalities in short-term interest rates for emerging markets.
  - Augmenting with jump components yields robust specifications that explain leptokurtic behavior observed in volatile markets.
  - Strong evidence that investors attach substantial risk premia to frequent jumps in emerging market bond yields.
  - Where jumps are more common, jump risk premia are systematically higher than volatility risk premia.
- Potential extensions noted in the source:
  - Investigate contagious jumps from other markets and identify risk premia attached to jumps originating in other countries.

*Source: _wp07172 — 1. SNP Auxilliary Model (content unit extracted from the supplied PDF chapter).*

### 1. SNP Auxilliary Model ................................................................................................

### 1. SNP Auxilliary Model

### Overview
- This content unit corresponds to "1. SNP Auxilliary Model" and spans the indicated section of the source PDF.
- The section is followed by "References" and a set of Tables and a Figure listed in the source content.

### References
- A references section is present immediately after the SNP Auxilliary Model section.

### Tables (listed in source)
- Table 1: SNP Density Estimation
- Table 2: EMM Estimates of the Jump Diffusion Stochastic Model for Argentina
- Table 3: EMM Estimates of the Jump Diffusion Stochastic Model for Brazil
- Table 4: Volatility and Jump Risk Premia
- Table 5: Estimates and t-ratios of the Average SNP Score Components for Argentina
- Table 6: Estimates and t-ratios of the Average SNP Score Components for Brazil

### Figure
- A Figure is listed following the tables (title not specified in the provided content).

*Source: _wp07172 - 1. SNP Auxilliary Model ................................................................................................*

### 1. SNP Sample Forecasts and Actual Data for Argentina and Brazil ......................................25

### 1. SNP Sample Forecasts and Actual Data for Argentina and Brazil ......................................25

### Introduction: purpose and key questions
- Objectives:
  - Identify a class of jump-diffusion models that approximate short-term interest rates of emerging markets.
  - Reconcile parameters of the term structure of interest rates with associated bond yields by estimating volatility and jump risk premia in highly volatile markets.
- Key empirical questions:
  - Are jumps in the term structure of interest rates statistically significant? — Most jumps in emerging markets coincide with financial crises.
  - To what extent are volatility and jump risk factors priced in emerging market bonds?
- Observations:
  - Frequent, large spikes in interest rates and bond yields interpreted as jumps.
  - Common crises referenced: 1982 debt crisis, December 1994 Mexican Tequila effects, late-1997 Asian crisis, August 1998 Russian crisis, January 1999 Brazilian Sneeze, April 2000 NASDAQ rash.
- Estimation strategy overview:
  - Two-stage estimation (Benzoni (2002) extension): first-stage SMM/EMM on daily short-term rates to estimate structural parameters; second-stage simulation on bond yields to estimate volatility and jump risk premia.

### Model specification and pricing kernel
- Model class: continuous-time affine jump-diffusion models with stochastic volatility and jump components in returns and volatility.
- Key model features preserved from the source:
  - Two-factor stochastic volatility baseline (Andersen and Lund) with time-dependent volatility factor vt.
  - Generalized specification allowing Poisson jumps in r and v with intensities λr and λv and jump sizes ξr and ξv.
  - Correlated Brownian motions and possibility of contemporaneous or independent jump arrivals.
- Market prices of risk:
  - Use of a candidate pricing kernel to assign market prices of risk, yielding dynamics under martingale measure Q.
  - Volatility risk premia parameter: vη.
  - Compensating term for pure jump process: JJt tvηλ (as expressed in the source).

### Econometric approach
- First stage:
  - Use EMM (efficient method of moments) with an auxiliary model (SNP density / quasi-MLE transition density approximation) to estimate structural parameters when latent volatility and jumps are present.
  - Long simulations are used to compute expectations given candidate parameter vectors.
- Second stage:
  - Use continuous-time ATSM jump-diffusion bond pricing (discounted expected value under Q) and simulated method of moments to minimize bond yield errors and estimate risk premia parameters (vη, Jη).
  - Moment condition: E[PT(·) − P~(·)] = 0 where PT(·) is observed bond yield and P~(·) predicted price using the ATSM; error assumed stationary and ergodic with zero mean.

### Data sources and sample statistics
- Bond yields:
  - Brady bond end-of-week yields from Datastream, May 1994 through December 2002, weekly frequency.
  - Large standard deviations, high kurtosis and skewness — indicating non-normality and crisis influence.
- Reported sample statistics (1994-2002 Brady Bond Price Statistics):
  - Argentina:
    - Mean 33.78
    - S.D. 0.21
    - Median 14.8
    - Skewness 5.4
    - Kurtosis 33.3
  - Brazil:
    - Mean 16.7
    - S.D. 4.9
    - Median 16.1
    - Skewness 1.8
    - Kurtosis 8.7

### Empirical results — model comparison and diagnostics
- General finding:
  - Models without stochastic volatility and unanticipated jumps cannot generate the non-normalities observed in interest rates; diffusion-only models fail to replicate fat tails.
  - Jump-diffusion models with stochastic volatility provide acceptable characterizations for emerging market short-term interest rates.

- A. SNP auxiliary model:
  - SNP chosen by minimum BIC; semi-parametric GARCH auxiliary captures key features.
  - High-order AR terms required in mean equation (uL order 3); Garch(2,2) for conditional variance persistence.
  - Non-Gaussian innovations indicated by BIC reduction with 0zK>; heterogeneity in xK generally rejected.

- B. Simple diffusion benchmark (CIR: constant volatility, no jumps):
  - Estimated model: drt = kr(θ − rt) dt + σ dW.
  - Evidence of strong mean reversion (rk highly significant).
  - Goodness of fit: 2χ values in excess of [58(13),29(10)] and rejected at the 95% level.
  - Constant variance σ approximately 7 percent for Argentina and Brazil, respectively.
  - Fails to account for leptokurtosis, ARCH/GARCH persistence, and important SNP score diagnostics.

- C. Stochastic volatility (SV) model (no jumps):
  - Allowing unobservable volatility (ρ = 0 for baseline) improves fit but is still strongly rejected.
  - Captures some asymmetry but fails to accommodate fat tails (insufficient excess kurtosis).
  - Does not fully fit ARCH and GARCH score diagnostics; adding more diffusion factors unlikely to resolve non-normalities.

- D. Stochastic volatility with jumps (SVJ):
  - Add a Poisson jump process with Gaussian jump sizes and constant intensity λt = λ0.
  - Chi-square goodness-of-fit improves substantially: decreases to [20(8),15(5)] and cannot be rejected at the 1% level.
  - Jump parameters (μJ, σJ, λJ) are highly significant.
  - Jump frequency and sizes:
    - Source summary: "On average, we find that jumps occur (8,10) times a year in Argentina and Brazil, respectively."
    - Argentina (detailed): average 8.5 jumps per year; mean jump size 0.6±percent; standard deviation of jump size 0.4 percent.
    - Brazil (detailed): average 9.6 jumps per year; negative mean jump size (implying more negative than positive jumps).
  - SVJ accommodates linear aspects, leptokurtic behavior, and ARCH/GARCH dynamics (insignificant SNP score t‑ratios at 5% level).
  - Time-varying jump intensity specification (λt = λ0 + λ1 · volatility): λ0 and λ1 positive but λ1 estimates insignificant; linear jump intensity does not improve fit over constant intensity in these samples.

- E. Volatility and jump risk premia (reconciling short-term rates and bond yields)
  - Two models considered for pricing reconciliation:
    - Multifactor volatility model without jumps.
    - Multifactor volatility model with jumps.
  - Volatility risk premium:
    - Estimated vη is negative and significantly different from zero for the stochastic volatility model — variance risk is priced.
  - Jump risk premia (sizes only; frequency risk premia assumed zero):
    - Argentina jump risk premia: 0.256 basis points.
    - Brazil jump risk premia: 0.339 basis points.
  - Relative magnitudes:
    - Jump risk premia are almost three times the volatility risk premia (source statement).

### Conclusion: implications and extensions
- Main conclusions:
  - Diffusion-only models with stochastic volatility cannot fit non-normalities in short-term interest rates for emerging markets.
  - Augmenting with jump components yields robust specifications that explain leptokurtic behavior observed in volatile markets.
  - There is strong evidence that investors attach substantial risk premia to frequent jumps in emerging market bond yields.
  - Where jumps are more common, jump risk premia are systematically higher than volatility risk premia.
- Potential extensions noted in the source:
  - Investigate contagious jumps from other markets and identify risk premia attached to jumps originating in other countries.

_Italic: Source — content extracted from the PDF chapter "1. SNP Sample Forecasts and Actual Data for Argentina and Brazil" (pages summarized in the supplied content)._

### APPENDIX I. SNP AUXILLIARY MODEL

### APPENDIX I. SNP AUXILLIARY MODEL

### Model specification and transition density
- The auxiliary model follows Gallant and Tauchen’s method using the score function of an auxiliary model whose transition density is given by
  (,) (,) (,) fxy fyx fxy dx ρ ρ ρ = ∫  (A.1)
- Expanding (,)fxyρ in a Hermite series and deriving the transition density of the truncated expansion yields a transition density 1() Ktt fyx − that has the form of a location scale transform
  1t ttx yRzμ − =+  (A.2)
  of an innovation t z.
- The density function of the innovation t z is
  [] [] 2 2 (, ) () () (, ) K pzxz hzx pzxzdu φ φ = ∫ ,  (A.3)
  where (  ,   ) pzx is a polynomial in (  ,   ) zx of degree K and (  ) zφ denotes the multivariate normal density function with dimension M, mean vector zero and an identity variance-covariance matrix.
- The polynomial (  ,   ) pzx can be expressed in rectangular form
  00 (, ) x z K K pzx a x z βα βα αβ == ⎛⎞ = ⎜⎟ ⎝⎠ ∑∑ ,  (A.4)
  where α and β are multi-indexes of maximal degrees z K and x K, respectively.
- The function (  ,   ) pzx corrects for departures from Gaussianity.

### Special cases and resulting approximations
- Homogeneous innovations:
  - If innovations t z are homogeneous, the distribution does not depend on 1t x − and t z density can be approximated by 2() [ ()] ()hzpzzφ→ where () pz is a polynomial of degree z K.
  - The resulting density function is 2 (;)[()] (  ,) Mx fyxpz n yρμ→∑ , where 1 () x z R yμ − = −. The leading term of the expansion is (,) Mx nyμ ∑, which is a Gaussian vector autoregression.
  - When z K is positive, the semi parametric VAR density can approximate a large class of densities.
- Gaussian ARCH / GARCH type specifications:
  - Letting R be a linear function of the absolute values of elements of vectors 1rtL r tLx yμ − − − through 2 1 t tx yμ − − − leads the variance-covariance matrix to become (A.5), where (   ) vech R denotes a vector of length (1) / 2MM + containing elements of the upper triangle of R, 0 ρ is a vector of length (1)/2MM+, (1) P through () r L P are (1)/2MM+ by M matrices.
  - Garch like specifications can also be derived from (A.6), where (1) G through () g L G are vectors of length (1) / 2MM+.
- Other models can be derived by assuming the leading term of the expansion (,) Mx nyμ ∑ follows a Gaussian ARCH by letting R depend on lagged transformed variables.

### Parametric forms and lag structures (as in source)
- Example expressions from the source (preserving original notation and ordering):
  - 111 ' ttt xxx RR −−− ∑ = 12 0()1 1 () r trtLi r L xitLix i vech RP yρμ −−−+ −− + = = + − ∑ 111 ' ttt xxx RR −−− ∑ = 122 0()1() 11 ()() () g r trtLitLi rg L L xitLixix ii vech RP ydiag Gvech Rρμ −−−+−−+ −− + == = + − + ∑∑
  - (,) Mx nyμ ∑ follows a Gaussian ARCH by letting R be a linear function of absolute values of elements of 1rtL r tLx yμ − − − through 2 1 t tx yμ − − −.

### Tuning parameters of the SNP auxiliary model
- z K degree of polynomial in z
- x K degree of polynomial dependence on lags of t x
- u L Lag Length of the VAR
- r L Lag length of ARCH portion
- g L Lag length of GARCH portion
- p L Lag length of polynomial dependence on lags of t x

### Model selection: Schwarz Bayes Information criterion
- The Schwarz Bayes Information criterion is computed as:
  ^ 1 ( ) log( ) 2 n p BICsn n ρ ρ = + ,  (A.7)
  where 1 1 1 ( ) log[ (; )] n nt t sfyx n ρρ − = = − ∑ and the term (1/ 2)(/ ) log( )pnn θ is the penalty to good fits obtained by excessive parameterizations.
- The smaller the BIC the better the fit.

*Source: APPENDIX I. SNP AUXILLIARY MODEL (from the supplied PDF content)*

### References

### _wp07172 - References

### References
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- Bansal, R., A.R. Gallant, R. Hussey and G. Tauchen, 1993, Computational Aspects of Nonparametric Simulation, In D.A. Belsley (ed.) Computational Techniques for Econometrics and Economic Analysis, pp. 3-22.
- Bansal, R., A.R. Gallant, R. Hussey and G. Tauchen, 1995, Non parametric Estimation of Structural Models for High Frequency Currency Market Data, Journal of Econometrics, Vol. 66, pp. 251-287.
- Benzoni, L., 2002, Pricing Options under Stochastic Volatility: An Empirical Investigation, Working Paper, Finance Department-Calrson School of Management.
- Brennan, M.J. and E.S. Schwartz, 1980, Analyzing Convertible Bonds, Journal of Financial and Quantitative Analysis, Vol. 15, pp. 907-929.
- Chacko, G. and L.M. Viceira, 1999, Spectral GMM Estimation of Continuous-Time Processes, Working Paper, Harvard University.
- Chacko, G. and S. Das, 2002, Pricing Interest Rate Derivatives: A General Approach, Review of Financial Studies, Vol. 15, pp. 195-241.
- Chan K.C., G.A. Karolyi, F.A. Longstaff, and A.B. Sanders, 1982, An Empirical Comparison of Alternative Models of the Short Term Interest Rate, Journal of Finance, Vol. 47, pp. 1202-1227.
- Chernov, M. and E. Ghysels, 2000, A Study Towards a Unified Approach to the Joint Estimation of Objective and Risk Neutral Measures for the Purpose of Options Valuation, The Journal of Financial Economics, Vol. 56, pp. 407-458.
- Conley, T.G., L.P. Hansen, E.G.J. Luttmer and J. Scheinkman 1997, Short term Interest Rates as Subordinated Diffusions, The Review of Financial Studies, Vol. 10, pp. 525-577.
- Cox, J.C., J.E. Ingersoll, and S.A. Ross, 1985, A Theory of the Term Structure of Interest Rates, Econometrica Vol. 53, pp. 385-407.
- Das, S., 2001, The Surprise Element: Jumps in Interest Rates, Journal of Econometrics, Vol. 106, pp. 27-65.
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- Eraker, B., 2000, Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices, Working Paper, University of Chicago.
- Gallant, A.R. and G. Tauchen, 1989, Semi nonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications, Econometrica, Vol. 57, pp. 1091-1120.
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- Johannes, M., 2001, The Statistical and Economic Role of Jumps in Continuous-Time Interest Rate Models, Working Paper, Columbia University.
- Johannes, M., 1999, A Nonparametric Approach to Jumps in Interest Rates, Working Paper, Columbia University.
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- Merton, R.C., 1973, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, Vol. 4, pp. 141-183.
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### Table 2 — EMM Estimates of the Jump Diffusion Stochastic Model for Argentina (sample period October 1994 to December 1999)
- Models reported: CIR, SV, SVJ1, SVJ2. Standard errors in brackets.
- r_k:
  - CIR: 0.9814 (0.0910)
  - SV: 1.3243 (0.2430)
  - SVJ1: 0.8988 (0.0112)
  - SVJ2: 1.4555 (0.1350)
- r_θ:
  - CIR: 6.2780 (0.5257)
  - SV: 6.2191 (0.4352)
  - SVJ1: 6.1406 (0.5263)
  - SVJ2: 6.4192 (0.3213)
- v_k (SV/SVJ models):
  - SVJ1: 0.1139 (0.0403)
  - SVJ2: 0.5235 (0.0551)
  - SV: 0.5051 (0.0135)
- v_θ (SV/SVJ models):
  - SVJ1: 0.6466 (0.6341)
  - SVJ2: 0.4805 (0.2337)
  - SV: 0.3870 (0.2134)
- r_σ:
  - SV: 0.0735 (0.0103)
- v_σ (SV/SVJ models):
  - SVJ1: 0.0855 (0.0220)
  - SVJ2: 0.0679 (0.0321)
  - SV: 0.0967 (0.0631)
- J_μ (SVJ models):
  - SVJ1: 0.3274 (0.0398)
  - SVJ2: -0.6026 (0.1043)
- J_σ (SVJ models):
  - SVJ1: 0.2606 (0.0356)
  - SVJ2: 0.4090 (0.1322)
- 0_λ (SVJ models):
  - SVJ1: 8.6321 (3.4522)
  - SVJ2: 8.5120 (2.3211)
- 1_λ:
  - SVJ2: 2.4955 (1.9321)
- dfχ:
  - CIR: 58.50[13]
  - SV: 21.58[10]
  - SVJ1: 20.34[8]
  - SVJ2: 20.88[7]

### Table 3 — EMM Estimates of the Jump Diffusion Stochastic Model for Brazil (sample period October 1994 to December 1999)
- Models reported: CIR, SV, SVJ1, SVJ2. Standard errors in brackets.
- r_k:
  - CIR: 0.8637 (0.0984)
  - SV: 1.0107 (0.1056)
  - SVJ1: 0.7240 (0.0602)
  - SVJ2: 0.7293 (0.1192)
- r_θ:
  - CIR: 6.1370 (0.6541)
  - SV: 5.7419 (0.4981)
  - SVJ1: 8.7012 (0.3822)
  - SVJ2: 7.7512 (0.4954)
- v_k (SV/SVJ models):
  - SVJ1: 1.3328 (0.1089)
  - SVJ2: 1.2797 (0.0606)
  - SV: 1.1958 (0.0628)
- v_θ (SV/SVJ models):
  - SVJ1: 0.8305 (0.2412)
  - SVJ2: 0.8321 (0.2476)
  - SV: 0.3471 (0.0293)
- r_σ:
  - SV: 0.2228 (0.1239)
- v_σ (SV/SVJ models):
  - SVJ1: 0.3443 (2.0342)
  - SVJ2: 0.7212 (1.2156)
  - SV: 0.0636 (0.0519)
- J_μ (SVJ models):
  - SVJ1: -1.6966 (0.1004)
  - SVJ2: 0.2386 (0.5934)
- J_σ (SVJ models):
  - SVJ1: 1.7405 (0.1145)
  - SVJ2: 1.3245 (0.2219)
- 0_λ (SVJ models):
  - SVJ1: 9.5603 (2.4503)
  - SVJ2: 9.6321 (2.1146)
- 1_λ:
  - SVJ2: 0.2240 (1.4521)
- dfχ:
  - CIR: 29.30[10]
  - SV: 28.65[7]
  - SVJ1: 15.20[5]
  - SVJ2: 22.13[4]

### Table 4 — Volatility and Jump Risk Premia
- Parameters reported for Brazil and Argentina under SV and SVJ specifications.
- v_η:
  - Brazil SV: -0.4325 (0.0476)
  - Brazil SVJ: -0.1123 (0.0511)
  - Argentina SV: -0.3934 (0.0389)
  - Argentina SVJ: -0.0821 (0.0465)
- J_η:
  - Brazil SVJ: 0.3397 (0.9766)
  - Argentina SVJ: 0.2561 (0.1176)

### Table 5 — Estimates and t-ratios of the Average SNP Score Components for Argentina
- Reports Hermite coefficients, AR coefficients, ARCH coefficients, GARCH coefficients for models CIR, SV, SVJ, SVJ1 with t-ratios in parentheses.
- Selected entries (Hermite a01, a02; AR ψ2, ψ3; GARCH g1τ, g2τ) for illustration:
  - Hermite a01:
    - CIR: 0.881 (0.89)
    - SV: 0.725 (0.73)
    - SVJ: -0.119 (-0.12)
    - SVJ1: 0.093 (0.09)
  - Hermite a02:
    - CIR: -6.530 (-1.99)
    - SV: 1.401 (0.43)
    - SVJ: -0.980 (-0.30)
    - SVJ1: -5.227 (-1.59)
  - AR ψ2:
    - CIR: 21.720 (1.75)
    - SV: 14.859 (1.20)
    - SVJ: 17.974 (1.45)
    - SVJ1: 21.087 (1.70)
  - GARCH g1τ:
    - CIR: -18.367 (-2.38)
    - SV: -14.990 (-1.95)
    - SVJ: -8.013 (-1.04)
    - SVJ1: -8.195 (-1.06)
  - GARCH g2τ:
    - CIR: -17.131 (-2.27)
    - SV: -14.342 (-1.90)
    - SVJ: -7.402 (-0.98)
    - SVJ1: -7.731 (-1.02)

### Table 6 — Estimates and t-ratios of the Average SNP Score Components for Brazil
- Reports Hermite coefficients, AR coefficients, ARCH coefficients, GARCH coefficients for models CIR, SV, SVJ, SVJ2 with t-ratios in parentheses.
- Selected entries (Hermite a01, a02; AR ψ0; ARCH a0τ, a1τ) for illustration:
  - Hermite a01:
    - CIR: 0.506 (0.12)
    - SV: 0.678 (0.17)
    - SVJ: 1.035 (0.25)
    - SVJ2: -0.319 (-0.078)
  - Hermite a02:
    - CIR: 1.918 (0.23)
    - SV: 2.093 (0.25)
    - SVJ: 5.330 (0.64)
    - SVJ2: -2.604 (-0.313)
  - AR ψ0:
    - CIR: 23.150 (0.58)
    - SV: 20.822 (0.52)
    - SVJ: 24.315 (0.60)
    - SVJ2: 14.756 (0.367)
  - ARCH a0τ:
    - CIR: 60.255 (0.46)
    - SV: 52.628 (0.40)
    - SVJ: 20.146 (0.15)
    - SVJ2: -81.817 (-0.621)
  - ARCH a1τ:
    - CIR: 11.606 (2.19)
    - SV: 10.693 (2.02)
    - SVJ: 9.668 (1.82)
    - SVJ2: 5.796 (1.093)

### Additional materials in the unit
- Panel A: SNP density estimation notes and table headings (reports structure of estimated densities and objective function values including AIC, HQ, BIC).
- Panel B: Parameter setting table mapping parameter configurations to density function types (e.g., iid Gaussian, Gaussian VAR, Semiparametric VAR, Gaussian ARCH, Semiparametric ARCH, Gaussian GARCH, Semiparametric GARCH, Nonlinear Nonparametric).
- Figure 1: SNP Sample Forecasts and Actual Data for Argentina and Brazil (figure referenced).

*Source: _wp07172 - References*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07172.pdf_