Impulse Response Functions¶

Quick Reference

Class: panelbox.var.irf.IRFResult Method: PanelVARResult.irf() Import: from panelbox.var import PanelVAR (IRFs accessed via results) Stata equivalent: irf create, irf graph R equivalent: vars::irf()

Overview¶

Impulse Response Functions (IRFs) trace the dynamic effect of a one-time shock to one variable on all variables in the system over time. They are the primary tool for understanding how shocks propagate through a Panel VAR system.

An IRF answers the question: If variable \(j\) receives a one-standard-deviation shock at time \(t=0\), how does variable \(i\) respond at horizons \(h = 0, 1, 2, \ldots, H\)?

Formally, the IRF at horizon \(h\) is defined through the Moving Average (MA) representation of the VAR:

\[ Y_t = \sum_{s=0}^{\infty} \Phi_s \varepsilon_{t-s} \]

where \(\Phi_s\) are the MA coefficient matrices computed recursively from the VAR coefficient matrices:

\[ \Phi_0 = I_K, \quad \Phi_h = \sum_{l=1}^{\min(h, p)} A_l \Phi_{h-l} \]

PanelBox supports two identification methods: Cholesky (orthogonalized) and Generalized (Pesaran-Shin), with bootstrap confidence intervals.

Quick Example¶

from panelbox.var import PanelVARData, PanelVAR

# Estimate model
var_data = PanelVARData(df, endog_vars=["gdp", "inflation", "rate"],
                         entity_col="country", time_col="year", lags=2)
model = PanelVAR(data=var_data)
results = model.fit(cov_type="clustered")

# Compute IRFs with bootstrap confidence intervals
irf = results.irf(
    periods=20,
    method="cholesky",
    ci_method="bootstrap",
    n_bootstrap=500,
    ci_level=0.95,
)

# Access specific response
response = irf["gdp", "inflation"]  # GDP response to inflation shock

# Plot all IRFs
irf.plot()

When to Use¶

Analyzing shock transmission: How does a monetary policy shock affect output and inflation?
Dynamic multiplier analysis: What is the cumulative effect of a fiscal shock on GDP?
Structural analysis: Identifying the causal chain of macroeconomic shocks
Policy evaluation: Comparing the dynamic effects of different interventions

Key Assumptions

Stability: The VAR must be stable (results.is_stable() == True) for IRFs to converge to zero
Identification: Cholesky IRFs depend on variable ordering; Generalized IRFs are order-invariant but shocks are not orthogonal
Homogeneity: The same impulse responses apply to all entities (pooled coefficient assumption)

Detailed Guide¶

Identification Methods¶

The raw VAR residuals are typically correlated across equations. To isolate the effect of a shock to a single variable, the residuals must be orthogonalized.

Cholesky (Recursive)Generalized (Pesaran-Shin)

The Cholesky decomposition of \(\Sigma = PP'\) produces a lower-triangular matrix \(P\). Orthogonalized IRFs are:

\[ \Phi_0^{\text{orth}} = P, \quad \Phi_h^{\text{orth}} = \sum_{l=1}^{\min(h,p)} A_l \Phi_{h-l}^{\text{orth}} \]

Ordering matters: Variables listed first are treated as more "exogenous" -- they respond only to their own shocks contemporaneously, while later variables respond to shocks from earlier variables as well.

# Default ordering (order of endog_vars)
irf = results.irf(periods=20, method="cholesky")

# Custom ordering: rate is most exogenous, gdp is most endogenous
irf = results.irf(
    periods=20,
    method="cholesky",
    order=["rate", "inflation", "gdp"],
)

Choosing Variable Ordering

Place variables that respond more slowly (or are set by policy) first. Common macroeconomic orderings:

["output", "inflation", "interest_rate"] -- Output-first (Christiano, Eichenbaum & Evans)
["interest_rate", "inflation", "output"] -- Policy-first

Generalized IRFs (Pesaran & Shin, 1998) are invariant to variable ordering. The GIRF for a shock to variable \(j\) is:

\[ \text{GIRF}_j(h) = \frac{1}{\sqrt{\sigma_{jj}}} \Phi_h \Sigma e_j \]

where \(\sigma_{jj}\) is the variance of \(\varepsilon_j\) and \(e_j\) is the \(j\)-th unit vector.

irf = results.irf(periods=20, method="generalized")

Trade-off: Generalized IRFs avoid the arbitrary ordering problem but the shocks are not orthogonal -- they reflect the actual correlation structure of the residuals.

Computing IRFs¶

irf = results.irf(
    periods=20,                       # Forecast horizons
    method="cholesky",                # "cholesky" or "generalized"
    shock_size="one_std",             # "one_std" or numerical value
    cumulative=False,                 # True for accumulated responses
    order=None,                       # Custom ordering (Cholesky only)
    ci_method="bootstrap",            # None or "bootstrap"
    n_bootstrap=500,                  # Bootstrap replications
    ci_level=0.95,                    # Confidence level
    bootstrap_ci_method="percentile", # "percentile" or "bias_corrected"
    n_jobs=-1,                        # Parallel jobs (-1 = all cores)
    seed=42,                          # Reproducibility
    verbose=True,                     # Show progress bar
)

Parameters:

Parameter	Type	Default	Description
`periods`	`int`	`20`	Number of horizons
`method`	`str`	`"cholesky"`	`"cholesky"` or `"generalized"`
`shock_size`	`str` or `float`	`"one_std"`	Shock magnitude
`cumulative`	`bool`	`False`	Accumulated responses (\(\Psi_h = \sum_{s=0}^{h} \Phi_s\))
`order`	`list[str]`	`None`	Variable ordering (Cholesky only)
`ci_method`	`str`	`None`	`None` or `"bootstrap"`
`n_bootstrap`	`int`	`500`	Number of bootstrap replications
`ci_level`	`float`	`0.95`	Confidence level for CIs
`bootstrap_ci_method`	`str`	`"percentile"`	`"percentile"` or `"bias_corrected"`
`n_jobs`	`int`	`-1`	Parallel jobs for bootstrap
`seed`	`int`	`None`	Random seed for reproducibility

Accessing Results¶

The IRFResult object provides flexible access to the computed IRFs.

# Access specific IRF: response of variable i to shock in variable j
response = irf["gdp", "inflation"]       # np.ndarray of shape (periods+1,)

# Convert to DataFrame
df_all = irf.to_dataframe()                        # All IRFs
df_gdp_shock = irf.to_dataframe(impulse="gdp")     # All responses to GDP shock
df_infl_resp = irf.to_dataframe(response="inflation")  # How inflation responds to all shocks
df_pair = irf.to_dataframe(impulse="gdp", response="inflation")  # Single pair

# Summary at selected horizons
print(irf.summary(horizons=[0, 1, 5, 10, 20]))

IRFResult Attributes:

Attribute	Type	Description
`irf_matrix`	`np.ndarray`	Shape `(periods+1, K, K)`. `irf_matrix[h, i, j]` = response of var \(i\) to shock in var \(j\) at horizon \(h\)
`var_names`	`list[str]`	Variable names
`periods`	`int`	Number of horizons
`method`	`str`	Identification method used
`cumulative`	`bool`	Whether cumulative
`ci_lower`	`np.ndarray` or `None`	Lower CI bounds `(periods+1, K, K)`
`ci_upper`	`np.ndarray` or `None`	Upper CI bounds `(periods+1, K, K)`
`ci_level`	`float` or `None`	Confidence level
`bootstrap_dist`	`np.ndarray` or `None`	Full bootstrap distribution `(n_bootstrap, periods+1, K, K)`

Bootstrap Confidence Intervals¶

Bootstrap CIs account for both parameter uncertainty and estimation error. The procedure:

Resample residuals with replacement
Reconstruct data using the estimated VAR coefficients and resampled residuals
Re-estimate the VAR on the bootstrap sample
Compute IRFs from the bootstrap estimates
Repeat \(B\) times and compute percentile intervals

# Standard percentile bootstrap
irf = results.irf(
    periods=20,
    method="cholesky",
    ci_method="bootstrap",
    n_bootstrap=1000,
    ci_level=0.95,
    bootstrap_ci_method="percentile",
    seed=42,
)

# Bias-corrected percentile (Hall 1992) -- more accurate in small samples
irf_bc = results.irf(
    periods=20,
    method="cholesky",
    ci_method="bootstrap",
    n_bootstrap=1000,
    bootstrap_ci_method="bias_corrected",
    seed=42,
)

Cumulative IRFs¶

Cumulative IRFs show the accumulated response over time:

\[ \Psi_h = \sum_{s=0}^{h} \Phi_s \]

Useful when variables are in growth rates (differences) and you want to see the level effect.

irf_cum = results.irf(periods=20, cumulative=True, ci_method="bootstrap")
irf_cum.plot()

Visualization¶

# Plot all IRFs in a grid
irf.plot(backend="matplotlib")

# Plot responses to a specific shock
irf.plot(impulse="gdp", backend="plotly")

# Plot how a specific variable responds to all shocks
irf.plot(response="inflation")

# Customize
irf.plot(
    impulse="gdp",
    response="inflation",
    ci=True,                    # Show confidence intervals
    backend="plotly",           # Interactive plot
    theme="academic",           # Visual theme
)

Interpretation Guidelines¶

Significant effects: An IRF is statistically significant at horizon \(h\) if the confidence interval at \(h\) excludes zero.

Pattern	Interpretation
Positive, decaying	Positive shock has temporary expansionary effect
Negative, decaying	Positive shock has temporary contractionary effect
Oscillating, converging	Shock triggers cyclical adjustment
Persistent (non-zero at long horizons)	Possible non-stationarity or near-unit root
Zero throughout	No dynamic effect -- variables are independent

Common Pitfalls

Ordering sensitivity: Always report which ordering you used for Cholesky IRFs, and check robustness with alternative orderings
Wide CIs: Very wide confidence intervals suggest insufficient data or a weakly identified system. Consider reducing the number of variables or lags
Unstable system: If results.is_stable() == False, IRFs will diverge -- this is a fundamental problem, not a display issue

Configuration Options¶

Method Comparison¶

Feature	Cholesky	Generalized
Ordering-dependent	Yes	No
Shocks orthogonal	Yes	No
Structural interpretation	With correct ordering	Limited
Best for	Theory-guided analysis	Exploratory analysis

Tutorials¶

Tutorial	Description	Link
IRF Analysis	Computing and interpreting IRFs

References¶

Luetkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer-Verlag.
Pesaran, H. H., & Shin, Y. (1998). Generalized impulse response analysis in linear multivariate models. Economics Letters, 58(1), 17-29.
Sims, C. A. (1980). Macroeconomics and reality. Econometrica, 48(1), 1-48.
Kilian, L. (1998). Small-sample confidence intervals for impulse response functions. Review of Economics and Statistics, 80(2), 218-230.
Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer Science & Business Media.