IPS Test (Im-Pesaran-Shin)¶

Quick Reference

Class: panelbox.validation.unit_root.ips.IPSTest H₀: All panels contain unit roots (\(\rho_i = 0\) for all \(i\)) H₁: Some panels are stationary (\(\rho_i < 0\) for some \(i\)) Stata equivalent: xtunitroot ips variable, lags(aic p) R equivalent: plm::purtest(x, test="ips")

What It Tests¶

The Im-Pesaran-Shin (IPS) test examines whether panels contain unit roots, allowing each entity to have a different autoregressive parameter \(\rho_i\). Unlike the LLC test which assumes a common \(\rho\), the IPS test permits heterogeneous adjustment speeds across entities.

If the test rejects H₀, there is evidence that at least some panels are stationary (not necessarily all). This makes the IPS test more flexible and realistic for panels with heterogeneous entities.

Quick Example¶

from panelbox.datasets import load_grunfeld
from panelbox.validation.unit_root.ips import IPSTest

data = load_grunfeld()

# IPS test with constant
ips = IPSTest(data, "invest", "firm", "year", trend="c")
result = ips.run()
print(result)

Output:

======================================================================
Im-Pesaran-Shin Panel Unit Root Test
======================================================================
W-statistic:       -2.8534
t-bar statistic:   -2.1240
P-value:           0.0022
Lags:              1
Observations:      180
Cross-sections:    10
Deterministics:    Constant

H0: All panels contain unit roots
H1: Some panels are stationary

Conclusion: Reject H0: Evidence that some panels are stationary
======================================================================

Interpretation¶

p-value	Decision	Meaning
\(p < 0.01\)	Strong rejection of H₀	Strong evidence that some panels are stationary
\(0.01 \leq p < 0.05\)	Rejection of H₀	Evidence of stationarity in some panels
\(0.05 \leq p < 0.10\)	Borderline	Weak evidence; investigate individual entities
\(p \geq 0.10\)	Fail to reject H₀	Consistent with unit root in all panels

Examine Individual Statistics

After rejecting H₀, inspect result.individual_stats to see which entities drive the rejection. Entities with more negative t-statistics are more likely to be stationary.

for entity, t_stat in result.individual_stats.items():
    status = "likely stationary" if t_stat < -2.0 else "likely unit root"
    print(f"Entity {entity}: t = {t_stat:.3f} ({status})")

Mathematical Details¶

Model¶

Each entity has its own ADF regression with a potentially different \(\rho_i\):

\[\Delta y_{it} = \rho_i y_{i,t-1} + \sum_{j=1}^{p_i} \theta_{ij} \Delta y_{i,t-j} + \alpha_i + \delta_i t + \varepsilon_{it}\]

where \(\rho_i\) is heterogeneous across entities.

Procedure¶

Individual ADF tests: For each entity \(i\), run an ADF test and obtain the individual t-statistic \(t_i\) for testing \(\rho_i = 0\).
Average t-statistic: Compute the cross-sectional average:

\[\bar{t} = \frac{1}{N} \sum_{i=1}^{N} t_i\]
Standardize: The W-statistic is:

\[W = \frac{\sqrt{N}\left(\bar{t} - E[\bar{t}]\right)}{\sqrt{\text{Var}[\bar{t}]}} \xrightarrow{d} N(0,1)\]

where \(E[\bar{t}]\) and \(\text{Var}[\bar{t}]\) are the expected value and variance of the individual ADF t-statistics under H₀, tabulated in Im, Pesaran & Shin (2003).

Test Statistic¶

The W-statistic follows a standard normal distribution under H₀:

\[W \xrightarrow{d} N(0, 1) \quad \text{as } N, T \to \infty\]

Configuration Options¶

IPSTest(
    data,                   # pd.DataFrame: Panel data in long format
    variable,               # str: Variable to test for unit root
    entity_col,             # str: Entity identifier column
    time_col,               # str: Time identifier column
    lags=None,              # int, dict, or None: Lags (None = auto per entity)
    trend='c',              # str: 'n' (none), 'c' (constant), 'ct' (constant + trend)
)

Lag Specification¶

The IPS test supports flexible lag selection:

Automatic (recommended)Fixed lagsPer-entity lags

# AIC-based lag selection per entity
ips = IPSTest(data, "invest", "firm", "year", lags=None)
result = ips.run()

# Same lag length for all entities
ips = IPSTest(data, "invest", "firm", "year", lags=2)
result = ips.run()

# Different lags per entity via dictionary
lags_dict = {"firm_1": 1, "firm_2": 2, "firm_3": 1}
ips = IPSTest(data, "invest", "firm", "year", lags=lags_dict)
result = ips.run()

Result Object: `IPSTestResult`¶

Attribute	Type	Description
`statistic`	`float`	W-statistic (standardized \(\bar{t}\))
`t_bar`	`float`	Average of individual ADF t-statistics
`pvalue`	`float`	P-value from standard normal
`lags`	`int` or `list`	Lags used (int if uniform, list if varying)
`n_obs`	`int`	Total number of observations
`n_entities`	`int`	Number of cross-sectional units
`individual_stats`	`dict`	Individual ADF t-statistics per entity
`test_type`	`str`	`"IPS"`
`deterministics`	`str`	Description of trend specification
`conclusion`	`str`	Test conclusion at 5% level

When to Use¶

Use IPS when:

Entities have heterogeneous dynamics (different \(\rho_i\) are plausible)
You want to know whether some (not necessarily all) panels are stationary
The panel is (approximately) balanced

Examples of appropriate settings:

GDP across countries with different economic structures
Stock prices of firms in different industries
Any panel where entities may have different adjustment speeds

Comparing LLC and IPS¶

from panelbox.validation.unit_root.llc import LLCTest
from panelbox.validation.unit_root.ips import IPSTest

data = load_grunfeld()

# LLC: common rho, H1 = ALL stationary
llc = LLCTest(data, "invest", "firm", "year", trend="c")
llc_result = llc.run()

# IPS: heterogeneous rho_i, H1 = SOME stationary
ips = IPSTest(data, "invest", "firm", "year", trend="c")
ips_result = ips.run()

print(f"LLC:  stat={llc_result.statistic:.4f}, p={llc_result.pvalue:.4f}")
print(f"IPS:  W={ips_result.statistic:.4f}, t_bar={ips_result.t_bar:.4f}, p={ips_result.pvalue:.4f}")

Aspect	LLC	IPS
\(\rho\)	Common	Heterogeneous \(\rho_i\)
H₁	All stationary	Some stationary
Power	Higher if common \(\rho\) holds	Higher if \(\rho_i\) vary
Balance	Required	Preferred

Common Pitfalls¶

Cross-Sectional Independence

Like LLC, the IPS test assumes cross-sectional independence. Correlated errors across entities (e.g., due to common shocks) can cause size distortions. Consider using the Fisher test or checking for cross-sectional dependence with the Pesaran CD test.

Small N

The IPS test requires a moderate number of entities for the central limit theorem approximation to work. With very few entities (N < 5), the standardized W-statistic may not be well-approximated by the normal distribution.

Partial Stationarity

When IPS rejects H₀, it only indicates that some panels are stationary, not which ones or how many. Examine result.individual_stats to identify which entities drive the rejection.

References¶

Im, K. S., Pesaran, M. H., & Shin, Y. (2003). "Testing for unit roots in heterogeneous panels." Journal of Econometrics, 115(1), 53-74.
Baltagi, B. H. (2021). Econometric Analysis of Panel Data (6^th ed.). Springer. Chapter 12.