Dynamic Panel Quantile Regression

Quick Reference

Class: panelbox.models.quantile.dynamic.DynamicQuantile Import: from panelbox.models.quantile import DynamicQuantile Stata equivalent: ivqreg y x1 x2 (L.y = L2.y L3.y), quantile(0.5) R equivalent: ivqr::ivqr() (for IV approach)

Overview

Dynamic Panel Quantile Regression extends standard panel quantile models to include lagged dependent variables. The model is:

\[Q_{y_{it}}(\tau | y_{i,t-1}, X_{it}, \alpha_i) = \rho(\tau) y_{i,t-1} + X_{it}'\beta(\tau) + \alpha_i\]

The key challenge is that the lagged dependent variable \(y_{i,t-1}\) is endogenous — it is correlated with the fixed effect \(\alpha_i\) and potentially with the error term. This endogeneity is present in both mean and quantile regression, but quantile regression lacks the within-transformation that partially addresses it in OLS.

PanelBox implements three approaches to handle this endogeneity:

• IV approach (Galvao, 2011): uses deeper lags as instruments
• Quantile Control Function (Powell, 2016): two-step control function approach
• GMM approach: Arellano-Bond type moment conditions adapted for quantile regression

The persistence parameter \(\rho(\tau)\) captures how past outcomes affect current outcomes at different parts of the distribution. If \(\rho(0.9) > \rho(0.1)\), high-outcome observations exhibit more persistence than low-outcome observations.

Quick Example

from panelbox.core.panel_data import PanelData
from panelbox.models.quantile import DynamicQuantile

panel_data = PanelData(data=df, entity_col="firm_id", time_col="year")

model = DynamicQuantile(
    data=panel_data,
    formula="investment ~ value + capital",
    tau=[0.25, 0.5, 0.75],
    lags=1,
    method="iv",
)
results = model.fit(iv_lags=2, bootstrap=True, n_boot=100)

When to Use

• Dynamic processes: outcomes persist over time (e.g., earnings, investment, GDP)
• State dependence: past outcomes causally affect current outcomes
• Quantile-specific persistence: \(\rho(\tau)\) varies across the distribution
• Short-run vs long-run effects: separate transitory from permanent impacts
• Heterogeneous dynamics: adjustment speed differs at different quantiles

Key Assumptions

• Balanced panel required: observations must be available for all entities at all time periods
• Valid instruments: deeper lags (\(y_{i,t-2}, y_{i,t-3}, \ldots\)) are uncorrelated with current errors
• Sufficient time periods: \(T\) must be large enough for the instruments to be relevant (\(T \geq 4\) minimum)
• Sequential exogeneity: \(E[\rho_\tau'(\varepsilon_{it}) | y_{i,t-1}, X_{it}, \alpha_i] = 0\)

Detailed Guide

The Endogeneity Problem

In a dynamic quantile model, \(y_{i,t-1}\) is correlated with \(\alpha_i\) by construction (past \(y\) depends on the fixed effect). The standard within-transformation does not eliminate this correlation in quantile regression because the check loss function is not linear.

Three solutions are available:

Method 1: Instrumental Variables (Galvao 2011)

Uses deeper lags \(y_{i,t-2}, y_{i,t-3}, \ldots\) as instruments for \(y_{i,t-1}\):

model = DynamicQuantile(
    data=panel_data,
    formula="y ~ x1 + x2",
    tau=0.5,
    lags=1,
    method="iv",     # Galvao (2011) IV approach
)
results = model.fit(
    iv_lags=2,       # use y_{t-2} and y_{t-3} as instruments
    bootstrap=True,
    n_boot=100,
)

The IV procedure:

1. First-difference to remove \(\alpha_i\): \(\Delta y_{it} = \rho \Delta y_{i,t-1} + \Delta X_{it}'\beta + \Delta \varepsilon_{it}\)
2. Use \(y_{i,t-2}, y_{i,t-3}\) as instruments for \(\Delta y_{i,t-1}\)
3. Estimate by instrumental variable quantile regression

Method 2: Quantile Control Function (Powell 2016)

A two-step approach that controls for endogeneity via a control function:

model = DynamicQuantile(
    data=panel_data,
    formula="y ~ x1 + x2",
    tau=0.5,
    lags=1,
    method="qcf",     # Powell (2016) control function
)
results = model.fit(bootstrap=True, n_boot=100)

Method 3: GMM Approach

Uses Arellano-Bond type moment conditions:

model = DynamicQuantile(
    data=panel_data,
    formula="y ~ x1 + x2",
    tau=0.5,
    lags=1,
    method="gmm",
)
results = model.fit(iv_lags=2)

Estimation with Bootstrap Inference

Bootstrap is recommended for dynamic quantile models because analytical standard errors are complex:

results = model.fit(
    iv_lags=2,         # instrument depth
    bootstrap=True,    # bootstrap inference
    n_boot=100,        # bootstrap replications
    verbose=True,      # print progress
)

Interpreting Results

# Access results for each quantile
for tau in model.tau:
    r = results.results[tau]
    print(f"tau={tau:.2f}:")
    print(f"  Lag coefficient (rho): {r.params[0]:.4f}")
    print(f"  Other coefficients: {r.params[1:]}")
    if hasattr(r, "se_boot"):
        print(f"  Bootstrap SEs: {r.se_boot}")

Interpreting \(\rho(\tau)\):

• \(\rho(\tau) > 0\): positive persistence at quantile \(\tau\)
• \(\rho(\tau) \approx 1\): near-unit root behavior (strong persistence)
• \(\rho(\tau)\) increasing in \(\tau\): high-outcome observations are more persistent
• \(\rho(\tau)\) decreasing in \(\tau\): mean reversion is stronger at the top

Choosing the Number of Instrument Lags

More instrument lags increase efficiency but may weaken relevance:

# Compare different instrument depths
for iv_depth in [1, 2, 3]:
    model = DynamicQuantile(data=panel_data, formula="y ~ x1 + x2",
                             tau=0.5, lags=1, method="iv")
    results = model.fit(iv_lags=iv_depth)
    print(f"iv_lags={iv_depth}: rho = {results.results[0.5].params[0]:.4f}")

Rule of Thumb

Start with iv_lags=2 (instruments \(y_{i,t-2}\) and \(y_{i,t-3}\)). If results are unstable, try iv_lags=1 for fewer but stronger instruments.

Configuration Options

Parameter	Type	Default	Description
data	PanelData	required	Panel data object
formula	str	None	Model formula "y ~ x1 + x2"
tau	float/array	0.5	Quantile level(s) in \((0, 1)\)
lags	int	1	Number of dependent variable lags
method	str	"iv"	Estimation: "iv", "qcf", "gmm"

Fit Parameters

Parameter	Type	Default	Description
iv_lags	int	2	Additional lags used as instruments
bootstrap	bool	False	Use bootstrap for inference
n_boot	int	100	Number of bootstrap replications
verbose	bool	False	Print estimation progress

Diagnostics

Checking Instrument Validity

# The lag coefficient should be stable across instrument specifications
for iv_depth in [1, 2, 3, 4]:
    model = DynamicQuantile(data=panel_data, formula="y ~ x1",
                             tau=0.5, lags=1, method="iv")
    r = model.fit(iv_lags=iv_depth)
    print(f"iv_lags={iv_depth}: rho={r.results[0.5].params[0]:.4f}")
# Large variation suggests instrument problems

Quantile Process

# Estimate across many quantiles to trace persistence
tau_grid = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]
model = DynamicQuantile(data=panel_data, formula="y ~ x1",
                         tau=tau_grid, lags=1, method="iv")
results = model.fit(iv_lags=2, bootstrap=True, n_boot=100)

# Examine how persistence varies across quantiles
rho_values = [results.results[tau].params[0] for tau in tau_grid]

Tutorials

Tutorial	Description	Link
Dynamic Quantile	IV estimation with lagged dependent variables

See Also

• Pooled Quantile Regression — static quantile model without lags
• Fixed Effects Quantile Regression — static model with FE
• Diagnostics — bootstrap inference and model comparison
• Difference GMM — Arellano-Bond for mean regression

References

• Galvao, A. F. (2011). Quantile regression for dynamic panel data with fixed effects. Journal of Econometrics, 164(1), 142-157.
• Powell, D. (2016). Quantile treatment effects in the presence of covariates. RAND Working Paper.
• Arellano, M., & Bonhomme, S. (2016). Nonlinear panel data estimation via quantile regressions. The Econometrics Journal, 19(3), C61-C94.
• Galvao, A. F., & Kato, K. (2016). Smoothed quantile regression for panel data. Journal of Econometrics, 193(1), 92-112.