## Model II-Impulse-Response Functions

## Source details

**Canonical URL:** [Model II-Impulse-Response Functions](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11113.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11113.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11113.pdf.json)

---

### Data, sample, and estimation setup
- Sample period: January 2004–October 2010 (82 months).
- Yield dataset: 984 monthly observations of zero-coupon yields for 12 maturities: 1, 2, 3, 4, 6, 9, 12, 18, 24, 30, 36, and 48 months.
- Instruments used to construct zero-coupon yields: LTN and NTN-F bonds (LTN up to 24 months; interpolation and bootstrapping used for longer maturities).
- Macroeconomic variables (monthly, same sample): four inflation measures (including IPCA and implied inflation from 1-year and 3-year inflation-linked bonds), real activity measures (real economic activity relative to potential and industrial production), risk perception measures (Brazilian CDS spread and Bovespa index), and the policy rate (Selic).
- PCA used to summarize each of the first three macroeconomic groups into one series each, producing four macro variables labeled “inflation,” “economic activity,” “risk perception,” and “policy rate.”
- Modeling approach: three-factor Nelson–Siegel (level, slope, curvature) for yields; factors follow a VAR(1) and are extended with macro factors in a 6-variable VAR(1) (yield factors + three macro factors + policy/risk variable).
- Bootstrapping (non-parametric Monte Carlo tailored for state-space models) used to improve standard error and impulse response inference.

### Stylized facts of the Brazilian yield curve (Jan 2004–Oct. 2010)
- The average yield curve is upward sloping and concave.
- Yield-curve shapes over time: upward sloping, downward sloping, humped, and inverted humped.
- Yield dynamics are persistent; spread dynamics are less persistent.
- The short end of the yield curve is more volatile than the long end.
- Factor properties:
  - Level: highly persistent; high variation relative to its mean.
  - Slope: less persistent than any single yield; highly variable relative to its mean.
  - Curvature: least persistent; largest variability relative to its mean.
- Correlations:
  - Level is not highly correlated with slope and curvature.
  - Slope and curvature show highly negative pairwise correlation (correlation between slope and curvature = -0.51).
- Average monthly changes in yields at all maturities are negative (average decline in yield curve level over the sample).

### Key descriptive numbers for yields (selected)
- 1-month yield: Mean 13.558, Std. Dev. 3.289, Minimum 8.636, Maximum 19.990, ρ(1) 0.988, ρ(12) 0.617, ρ(24) 0.520.
- 48-month yield (interpreted as level): Mean 14.446, Std. Dev. 2.607, Minimum 10.144, Maximum 22.576, ρ(1) 0.922, ρ(12) 0.526, ρ(24) 0.312.
- Slope (empirical proxy): Mean 0.888, Std. Dev. 1.991, Minimum -3.146, Maximum 6.650, ρ(1) 0.849, ρ(12) 0.099, ρ(24) 0.225.
- Curvature (empirical proxy): Mean -0.306, Std. Dev. 1.364, Minimum -2.797, Maximum 2.844, ρ(1) 0.911, ρ(12) -0.495, ρ(24) 0.149.

### Yield-only Nelson–Siegel model: fit and estimated factor properties
- Model: three-factor Nelson–Siegel (level, slope, curvature) fitted to monthly cross-sections; decay parameter ς estimated along with factors.
- Fit quality: average fitted curve and average actual yield curve are very close; residuals exhibit negligible mean errors (basis points) and small standard deviations.
- Estimated factors (Yield-only model, Jan. 2004–Oct. 2010):
  - Level: Mean 14.965, Std. Dev. 2.819, Minimum 9.639, Maximum 26.064, ρ(1) 0.860, ρ(3) 0.639, ρ(6) 0.411.
  - Slope: Mean 1.440, Std. Dev. 3.139, Minimum -5.347, Maximum 9.128, ρ(1) 0.873, ρ(3) 0.651, ρ(6) 0.354.
  - Curvature: Mean -1.138, Std. Dev. 4.331, Minimum -10.031, Maximum 8.787, ρ(1) 0.906, ρ(3) 0.629, ρ(6) 0.089.
- Correlations between estimated factors and empirical proxies: level 0.89, slope 0.98, curvature 0.99.

### Yield–macro VAR results: Model specifications and estimation
- Model I: three yield factors + inflation principal component + economic activity principal component + policy rate (Selic).
- Model II: three yield factors + inflation PC + economic activity PC + risk perception PC (replacing policy rate).
- Estimation constraints: each yield–macro VAR(1) contains six variables; only 82 monthly observations lead to potential small-sample issues.
- Inference mitigation: bootstrapping used to obtain bootstrapped standard errors and impulse response bounds.
- Estimated factors in both yield–macro models closely match yield-only estimates:
  - Level: Mean 14.965, Std. Dev. 2.819, ρ(1) 0.860.
  - Slope: Mean 1.441, Std. Dev. 3.140, ρ(1) 0.873.
  - Curvature: Mean -1.137, Std. Dev. 4.331, ρ(1) 0.906.
- Measurement errors (residuals) remain small and similar to the yield-only model.

### Key parameter and significance findings
- Model I:
  - Total of 11 coefficients significant (4 diagonal and 7 off-diagonal).
  - All diagonal coefficients of the macro block (A_ss) are significant.
  - Diagonal curvature in the yield block (A_yy) is significant.
  - Macroeconomic factors generally more persistent than yield curve factors (except curvature whose persistence exceeds the policy rate’s).
  - Policy rate (Selic) is influenced positively by the level and curvature yield factors and by inflation and its own lag.
  - Policy rate did not appear to significantly influence next-period inflation or economic activity.
  - Inflation significantly affects the level and slope of the yield curve (A_sm block).
- Model II:
  - Total of 14 coefficients significant (all 6 diagonal elements and 8 off-diagonal).
  - Macroeconomic factors again more persistent than yield curve factors.
  - No lagged yield factor significantly affects current macro factors (A_sy not significant).
  - Inflation has a positive and significant effect on next-period curvature.
  - Economic activity significantly and positively affects next-period level and slope.
  - Risk perception (RA) significantly affects all three yield factors: increases in risk aversion increase next-period level and slope more than proportionally and strongly reduce curvature.
  - In A_yy, the level significantly affects the curvature positively and the slope negatively.
  - Notable magnitude: Model II diagonal risk-aversion coefficient reported greater than 1 (VAR remains stationary as eigenvalues < 1).

### Counts and notable parameter magnitudes (selected)
- Model I: 11 significant coefficients total.
- Model II: 14 significant coefficients total.

### Impulse response function findings (bootstrapped)
- Four impulse-response groups analyzed for each model: (i) macro to macro shocks; (ii) macro to yield shocks; (iii) yield to macro shocks; (iv) yield to yield shocks.

- Model I (policy rate)
  - Macro→Macro: standard small-macro-model behavior; high persistence for macro variables; policy rate responds positively to inflation shocks.
  - Macro→Yield: generally small and mostly insignificant; notable temporary negative response of policy rate to a slope shock and positive response to a curvature shock.
  - Yield→Macro: generally unimportant/insignificant.
  - Yield→Yield: yield factors significantly influence their own dynamics; level and slope respond negatively to a curvature shock; shocks to level produce a positive slope response and negative curvature response; slope surprises reduce the level factor.

- Model II (risk perception)
  - Macro→Macro: persistence in macro variables, especially strong for risk aversion; increases in risk perception associated with reductions in inflation and increases in economic activity (difficult to fully rationalize).
  - Macro→Yield: shocks to risk perception have statistically significant temporary effects on all three yield factors; for the period analyzed, increases in risk perception were associated with decreases in level and slope and increases in curvature (the decrease in level and slope is noted as difficult to rationalize).
  - Yield→Macro: generally unimportant/insignificant; temporary positive effect of a curvature shock on economic activity.
  - Yield→Yield: broadly similar to Model I; persistence of curvature and level shocks declines in Model II while slope shock persistence increases.

### Summary implications and interpretation
- Macroeconomic variables exhibit higher persistence than yield curve factors over the sample.
- Bilateral feedback between the yield curve and macroeconomic variables is limited and appears slightly stronger from macro variables to the yield curve than vice versa.
- Changes in perceptions about Brazil’s risk (risk aversion) strongly influence yield curve dynamics.
- The policy rate during the period was informed by both financial factors (level and curvature) and inflation, but the policy rate itself did not show a significant direct effect on next-period inflation or economic activity in these VAR(1) specifications.
- Shocks to yield curve factors have meaningful effects primarily on other yield factors (internal yield dynamics) rather than on macro variables.

### Authors’ distilled conclusions
- The three-factor Nelson–Siegel model fits Brazilian zero-coupon yield data well (Jan 2004–Oct. 2010) and reproduces stylized facts of the yield curve via the level, slope, and curvature factors.
- Expanding the model to include macroeconomic variables (two six-variable VAR(1) specifications) yields economically meaningful links between yield factors and macro fundamentals, subject to small-sample estimation constraints that were mitigated via bootstrapping.
- Key empirical conclusions for the sample period:
  - Macro variables are more persistent than yield factors.
  - Macro variables show mostly insignificant responses to yield-curve shocks.
  - Risk-perception shocks strongly influence yield-curve dynamics.
  - Policy-rate setting incorporated financial factors and inflation, yet the policy rate did not have a clear significant direct effect on next-period inflation or economic activity in these models.
  - Bilateral feedback is limited and slightly stronger from macro to yield curve than the other way around.

*Source: Authors’ estimates and analysis reported in "Model II-Impulse-Response Functions" (content unit from _wp11113 - 6. Model II-Impulse-Response Functions, Jan. 2004–Oct. 2010).*

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

### _wp11113 - References .............................................................................................................

### Appendix
- Appendix I. Constructing the Government Zero Coupon Yield Curve Database for Brazil ...32

### Tables
- 1. General Characteristics of Federal Government Bonds Used for Computing Zero-Coupon Yields .........................................................................................................................................7
- 2. Descriptive Statistics, Yield Curves: Jan. 2004–Oct. 2010 .................................................10
- 3. Descriptive Statistics, Yield Spreads: Jan. 2004–Oct. 2010 ................................................11
- 4. Yield-Only Model, Residuals: Jan. 2004–Oct. 2010 ...........................................................12
- 5. Yield-Only Model, Estimated Factors: Jan. 2004–Oct. 2010 ..............................................13
- 6. Yield-Macro Model I, Estimated Factors: Jan. 2004–Oct. 2010 .........................................16
- 7. Yield-Macro Model II, Estimated Factors: Jan. 2004–Oct. 2010 ........................................17
- 8. Yield-Macro Model I, Residuals: Jan. 2004–Oct. 2010 ......................................................17
- 9. Yield-Macro Model II, Residuals: Jan. 2004–Oct. 2010 .....................................................17
- 10. Yield Macro Model, Model I Parameter Estimates (Bootstrapped standard errors in parentheses and t-statistics in brackets) ...................................................................................18
- 11. Yield Macro Model, Model II Parameter Estimates (Bootstrapped standard errors in parentheses and t-statistics in brackets) ...................................................................................21

### Figures
- 1. Observed Yield Curves ..........................................................................................................9
- 2. Observed and Estimated Average Yield Curve ...................................................................12
- 3. Estimates of the Level, Slope, and Curvature in The Yields-Only Model ..........................13
- 4. Model-Based Level, Slope, and Curvature vs. Data-Based Level, Slope, and Curvature ...14
- 5. Model I-Impulse-Response Functions .................................................................................23

*Source: _wp11113 - References .............................................................................................................*

### 6. Model II-Impulse-Response Functions ................................................................................2

### Model II-Impulse-Response Functions

### Data, sample, and estimation setup
- Sample period: January 2004–October 2010 (82 months).
- Yield dataset: 984 monthly observations of zero-coupon yields for 12 maturities: 1, 2, 3, 4, 6, 9, 12, 18, 24, 30, 36, and 48 months.
- Instruments used to construct zero-coupon yields: LTN and NTN-F bonds (LTN up to 24 months; interpolation and bootstrapping used for longer maturities).
- Macroeconomic variables (monthly, same sample): four inflation measures (including IPCA and implied inflation from 1-year and 3-year inflation-linked bonds), real activity measures (real economic activity relative to potential and industrial production), risk perception measures (Brazilian CDS spread and Bovespa index), and the policy rate (Selic).
- Principal Component Analysis (PCA) used to summarize each of the first three macroeconomic groups into one series each, producing four macro variables labeled “inflation,” “economic activity,” “risk perception,” and “policy rate.”
- Modeling approach: three-factor Nelson–Siegel (level, slope, curvature) for yields; factors follow a VAR(1) and are extended with macro factors in a 6-variable VAR(1) (yield factors + three macro factors + policy/risk variable). Bootstrapping used to improve standard error and impulse response inference.

### Stylized facts of the Brazilian yield curve (Jan 2004–Oct 2010)
- The average yield curve is upward sloping and concave.
- The yield curve assumes a variety of shapes through time: upward sloping, downward sloping, humped, and inverted humped.
- Yield dynamics are persistent; spread dynamics are less persistent.
- The short end of the yield curve is more volatile than the long end.
- The level factor is highly persistent and exhibits high variation relative to its mean compared to individual yields and other factors.
- The slope is less persistent than any single yield but highly variable relative to its mean.
- The curvature is the least persistent factor and displays the largest variability relative to its mean.
- Level is not highly correlated with slope and curvature, while slope and curvature show highly negative pairwise correlation (correlation between slope and curvature = -0.51).
- Average monthly changes in yields at all maturities are negative (average decline in yield curve level over the sample).

Key descriptive numbers for yields (selected)
- 1-month yield: Mean 13.558, Std. Dev. 3.289, Minimum 8.636, Maximum 19.990, ρ(1) 0.988, ρ(12) 0.617, ρ(24) 0.520.
- 48-month yield (interpreted as level): Mean 14.446, Std. Dev. 2.607, Minimum 10.144, Maximum 22.576, ρ(1) 0.922, ρ(12) 0.526, ρ(24) 0.312.
- Slope (empirical proxy): Mean 0.888, Std. Dev. 1.991, Minimum -3.146, Maximum 6.650, ρ(1) 0.849, ρ(12) 0.099, ρ(24) 0.225.
- Curvature (empirical proxy): Mean -0.306, Std. Dev. 1.364, Minimum -2.797, Maximum 2.844, ρ(1) 0.911, ρ(12) -0.495, ρ(24) 0.149.

### Yield-only Nelson–Siegel model: fit and estimated factor properties
- Three-factor Nelson–Siegel (level, slope, curvature) fitted to monthly cross-sections; decay parameter ς estimated along with factors.
- Good fit: average fitted curve and average actual yield curve are very close; residuals exhibit negligible mean errors (basis points) and small standard deviations.
- Estimated factor summary (Yield-only model, Jan. 2004–Oct. 2010):
  - Level: Mean 14.965, Std. Dev. 2.819, Minimum 9.639, Maximum 26.064, ρ(1) 0.860, ρ(3) 0.639, ρ(6) 0.411.
  - Slope: Mean 1.440, Std. Dev. 3.139, Minimum -5.347, Maximum 9.128, ρ(1) 0.873, ρ(3) 0.651, ρ(6) 0.354.
  - Curvature: Mean -1.138, Std. Dev. 4.331, Minimum -10.031, Maximum 8.787, ρ(1) 0.906, ρ(3) 0.629, ρ(6) 0.089.
- Correlations between estimated factors and empirical proxies: level 0.89, slope 0.98, curvature 0.99 (i.e., strong match to level/slope/curvature interpretations).

### Yield–macro VAR results: Model I (policy rate) and Model II (risk perception)
Model specifications
- Model I: three yield factors + inflation principal component + economic activity principal component + policy rate (Selic).
- Model II: three yield factors + inflation PC + economic activity PC + risk perception PC (replacing policy rate).

Estimation constraints and inference
- Each yield–macro VAR(1) contains six variables; estimation involves many parameters relative to available observations (only 82 monthly observations for yields), producing potential small-sample issues.
- To mitigate inference problems, bootstrapping (non-parametric Monte Carlo tailored for state-space models) was used to obtain bootstrapped standard errors and impulse response bounds.

Estimated factor statistics in yield–macro models
- Estimated factors in both yield–macro models closely match yield-only estimates (identical descriptive statistics reported for Models I and II):
  - Level: Mean 14.965, Std. Dev. 2.819, ρ(1) 0.860.
  - Slope: Mean 1.441, Std. Dev. 3.140, ρ(1) 0.873.
  - Curvature: Mean -1.137, Std. Dev. 4.331, ρ(1) 0.906.
- Measurement errors (residuals) remain small and similar to the yield-only model.

Key parameter and significance findings
- Model I:
  - Total of 11 coefficients significant (4 diagonal and 7 off-diagonal).
  - All diagonal coefficients of the macro block (A_ss) are significant.
  - Diagonal curvature in the yield block (A_yy) is significant.
  - Macroeconomic factors are generally more persistent than yield curve factors (except curvature whose persistence exceeds the policy rate’s).
  - The policy rate (Selic) is influenced positively by the level and curvature yield factors and by inflation and its own lag; the policy rate did not appear to significantly influence next-period inflation or economic activity.
  - Inflation significantly affects the level and slope of the yield curve (A_sm block).
- Model II:
  - Total of 14 coefficients significant (all 6 diagonal elements and 8 off-diagonal).
  - Macroeconomic factors again more persistent than yield curve factors.
  - No lagged yield factor significantly affects current macro factors (A_sy not significant).
  - Inflation has a positive and significant effect on next-period curvature.
  - Economic activity significantly and positively affects next-period level and slope.
  - Risk perception (RA) significantly affects all three yield factors: for the sample period, increases in risk aversion increase next-period level and slope more than proportionally and strongly reduce curvature.
  - In A_yy, the level significantly affects the curvature positively and the slope negatively.

Counts and notable parameter magnitudes (selected)
- Model I: 11 significant coefficients total.
- Model II: 14 significant coefficients total.
- Model II diagonal risk-aversion coefficient reported greater than 1 (noting VAR remains stationary as eigenvalues < 1).

### Impulse response function findings (bootstrapped)
Four groups of impulse responses analyzed for each model: (i) macro to macro shocks; (ii) macro to yield shocks; (iii) yield to macro shocks; (iv) yield to yield shocks.

Model I (policy rate)
- Macroeconomic responses to macro shocks: standard small-macro-model behavior; high persistence for macro variables; policy rate responds positively to inflation shocks.
- Macroeconomic responses to yield shocks: generally unimportant/insignificant; notable temporary negative response of policy rate to a slope shock and positive response to a curvature shock.
- Yield responses to macro shocks: generally small and mostly insignificant; notable positive association between the level factor and expected inflation.
- Yield responses to yield shocks: yield factors significantly influence their own dynamics; level and slope respond negatively to a curvature shock; shocks to level produce a positive slope response and negative curvature response; slope surprises reduce the level factor.

Model II (risk perception)
- Macroeconomic responses to macro shocks: persistence in macro variables, especially strong for risk aversion; increases in risk perception associated with reductions in inflation and increases in economic activity (difficult to fully rationalize).
- Macroeconomic responses to yield shocks: generally unimportant/insignificant; temporary positive effect of a curvature shock on economic activity.
- Yield responses to macro shocks: shocks to risk perception have statistically significant temporary effects on all three yield factors; for the period analyzed, increases in risk perception were associated with decreases in level and slope and increases in curvature (the decrease in level and slope is noted as difficult to rationalize).
- Yield responses to yield shocks: broadly similar to Model I; persistence of curvature and level shocks declines in Model II while slope shock persistence increases.

Summary implications and interpretation
- Macroeconomic variables exhibit higher persistence than yield curve factors over the sample.
- Bilateral feedback between the yield curve and macroeconomic variables is limited and appears slightly stronger from macro variables to the yield curve than vice versa.
- Changes in perceptions about Brazil’s risk (risk aversion) strongly influence yield curve dynamics.
- The policy rate during the period was informed by both financial factors (level and curvature) and inflation, but the policy rate itself did not show a significant direct effect on next-period inflation or economic activity in these VAR(1) specifications.
- Shocks to yield curve factors have meaningful effects primarily on other yield factors (internal yield dynamics) rather than on macro variables.

### Conclusions (authors’ distilled conclusions)
- The three-factor Nelson–Siegel model fits Brazilian zero-coupon yield data well (Jan 2004–Oct 2010) and reproduces stylized facts of the yield curve via the level, slope, and curvature factors.
- Expanding the model to include macroeconomic variables (two six-variable VAR(1) specifications) yields economically meaningful links between yield factors and macro fundamentals, subject to small-sample estimation constraints that were mitigated via bootstrapping.
- Key empirical conclusions for the sample period:
  - Macro variables are more persistent than yield factors.
  - Macro variables show mostly insignificant responses to yield-curve shocks.
  - Risk-perception shocks strongly influence yield-curve dynamics.
  - Policy-rate setting incorporated financial factors and inflation, yet the policy rate did not have a clear significant direct effect on next-period inflation or economic activity in these models.
  - Bilateral feedback is limited and slightly stronger from macro to yield curve than the other way around.

*Source: Authors’ estimates and analysis reported in "Model II-Impulse-Response Functions" (content unit from _wp11113 - 6. Model II-Impulse-Response Functions, Jan. 2004–Oct. 2010).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11113.pdf_