Why Panel Unit-Root Tests Differ
A panel combines observations across units and time, but there is no single panel unit-root test with a single alternative. Some tests pool evidence around a common adjustment parameter; others retain more heterogeneity or combine separate unit-level tests. Consequently, two tests applied to the same panel can answer different questions even when both use the phrase “unit-root test.”
This tutorial compares the Levin–Lin–Chu (LLC), Im–Pesaran–Shin (IPS), and Fisher-type approaches under cross-sectional independence. Each requires a suitable treatment of serial dependence, deterministic terms, and the test statistic’s null distribution. The independence assumption is especially important: shared shocks can make unit-level evidence dependent and invalidate reference distributions that presume independent units.
A Common Regression Framework
For unit \(i=1,\ldots,N\) and time \(t=1,\ldots,T\), a common starting point is an augmented Dickey–Fuller regression:
Here, \(y_{it}\) is the observed series, \(d_{it}\) contains the selected deterministic terms, \(p_i\) is the number of lagged differences, and \(\varepsilon_{it}\) is the remaining innovation. Common deterministic specifications are no intercept or trend, an intercept, or an intercept and linear trend. A specification should match the question and the plausible deterministic behavior of the series; changing it changes the test’s calibration and can change its power.
The lagged differences are included to model short-run serial correlation. Appropriate lag selection and the test-specific corrections are part of the procedure, not optional details. For the comparisons here, cross-sectional independence means that the unit-specific processes (and hence their test statistics under the null) are independent across \(i\), conditional on the specified deterministic terms. The units may have serial dependence over time; the augmentation is intended to account for it. A common shock affecting multiple units violates the key cross-sectional assumption.
Three Tests, Three Alternatives
| Test | Null hypothesis | Alternative hypothesis | Main source of pooling |
|---|---|---|---|
| LLC | All units have a unit root: \(\rho_i=0\) for every \(i\) | A common adjustment parameter is negative: \(\rho_i=\rho<0\) for all \(i\) | Common persistence parameter; nuisance terms and short-run dynamics may differ |
| IPS | All units have a unit root: \(\rho_i=0\) for every \(i\) | A nonzero fraction of units has \(\rho_i<0\) | Average of unit-specific ADF statistics |
| Fisher-type | All unit-level tests have unit-root nulls | At least one unit is stationary; power is especially relevant when a nonzero fraction is stationary | Combines unit-level p-values |
These hypotheses describe distinct scientific claims. LLC is most restrictive under its alternative because it imposes a common autoregressive adjustment parameter. IPS permits heterogeneous adjustment and is designed to detect a stationary fraction. Fisher-type tests also allow heterogeneity, but a rejection only says that the combined evidence is inconsistent with all units having unit roots; by itself, it does not reveal which units drive that evidence.
Levin–Lin–Chu: A Common Adjustment Parameter
LLC tests the joint null \(H_0:\rho_i=0\) for every unit against the common-parameter alternative \(H_a:\rho_i=\rho<0\) for every unit. The deterministic coefficients and short-run lag coefficients can be unit-specific, while the persistence parameter is restricted to be common. Thus, LLC gains precision by pooling information under a stronger alternative than the heterogeneous tests.
The LLC procedure does not simply compare an unadjusted pooled regression coefficient with a conventional \(t\) distribution. Its adjusted statistic accounts for nuisance quantities, including short-run serial correlation and deterministic components. With the appropriate specification and asymptotic conditions, the adjusted statistic is calibrated to a standard normal reference distribution, with rejection in the lower tail. The finite-sample behavior can still depend on \(N\), \(T\), lag choices, and the data-generating process.
Worked Example: Interpreting an LLC Result
Suppose a hypothetical panel has \(N=24\) units and \(T=18\) observations per unit. The analysis includes a unit-specific intercept, allows unit-specific lagged-difference terms, and imposes a common \(\rho\). Assume the units are cross-sectionally independent and the selected lags adequately address serial correlation. An appropriately calculated LLC adjusted statistic is \(-2.31\).
For a lower-tail standard normal calibration, the p-value is \(P(Z\leq -2.31)\approx 0.0104\), where \(Z\sim N(0,1)\). At \(\alpha=0.05\), \(0.0104<0.05\), so the test rejects the joint unit-root null in favor of the common negative-adjustment alternative.
The conclusion is conditional on the common-parameter alternative and the test assumptions. It is not evidence that every unit has been individually shown to be stationary. Nor can the statistic be recovered from the panel dimensions alone: it depends on the data, regression estimates, and LLC adjustments.
Im–Pesaran–Shin: Averaging Unit-Specific Evidence
IPS estimates a separate augmented unit-root regression for each unit and averages the resulting unit-specific \(t\)-statistics for testing \(\rho_i=0\). Its null is \(H_0:\rho_i=0\) for every \(i\). Its heterogeneous alternative allows a nonzero fraction of units to have negative adjustment parameters, while other units may remain nonstationary. This is less restrictive than requiring the same \(\rho\) for all units.
The raw average is not generally compared with a standard normal distribution. For the chosen deterministic specification, sample size, and relevant lag setup, the null mean and variance of the unit-level statistic are used to standardize the average. Under the required asymptotic conditions, the standardized statistic has a standard normal reference distribution. The details of that calibration matter: using a null mean or variance for a different deterministic specification can give misleading inference.
Worked Example: Calculating an IPS Standardized Statistic
Consider four hypothetical cross-sectionally independent units. Their unit-level ADF \(t\)-statistics are \(-3.2\), \(-2.8\), \(-2.1\), and \(-1.7\). Suppose the appropriate null calibration for the chosen deterministic terms gives \(E(t_i)=-1.50\) and \(\operatorname{Var}(t_i)=0.80\). The average statistic is
The standardized IPS statistic is
Using the standard normal lower tail gives a p-value of approximately \(0.0168\). At the 5% level, this rejects the all-unit-root null, conditional on the calibration and independence assumptions. It does not establish that all four units are stationary: IPS allows a stationary fraction, and the average statistic does not identify the contributing units.
Fisher-Type Tests: Combining Unit-Level P-Values
A Fisher-type test first obtains a valid unit-root test p-value \(p_i\) for each unit using a chosen deterministic specification and suitable treatment of serial dependence. It then combines the p-values using
Under the joint null, if the unit-level p-values are independent and continuously uniform on \((0,1)\), then \(-2\ln(p_i)\) has a chi-square distribution with 2 degrees of freedom, and their sum has a chi-square distribution with \(2N\) degrees of freedom. The combined statistic is therefore compared with that reference distribution. This calibration requires both valid individual p-values and cross-sectional independence; uniform marginal p-values alone do not guarantee a chi-square sum when the p-values are dependent.
Worked Example: Combining Four Unit-Level P-Values
Suppose four hypothetical unit-level tests produce p-values \(0.01\), \(0.04\), \(0.20\), and \(0.50\). Assume each test is correctly calibrated under the unit-root null and the units are independent. The product is
Thus,
The upper-tail probability is approximately \(0.0094\), so the combined test rejects at the 5% level. The result is evidence against the hypothesis that every unit has a unit root. It does not, on its own, identify a particular stationary unit or show that most units are stationary. The small p-value from one unit contributes strongly, but the combined statistic uses all four p-values.
Why Fisher’s Reference Distribution Holds
The Fisher calibration follows directly from the distribution of a transformed uniform p-value. If \(p_i\) is uniform under its null, then for \(x\geq 0\),
This is the cumulative distribution function of a chi-square random variable with 2 degrees of freedom. If the \(p_i\) are also independent, the transformed quantities are independent, and the sum of \(N\) such chi-square variables has \(2N\) degrees of freedom. This derivation also shows why cross-sectional independence is central: dependence between unit-level p-values breaks the sum-of-independent-variables argument.
Choosing and Interpreting the Test
The choice should follow the substantive alternative, not just which procedure returns the smallest p-value. LLC is suited to a common adjustment parameter; IPS allows heterogeneous adjustment and uses an average; Fisher combines individual p-values and can reject when only some units provide strong evidence. The deterministic terms should be chosen and stated explicitly for each unit-level regression or pooled procedure.
A rejection is not a proof of stationarity in every unit, and a failure to reject is not proof that every unit has a unit root. Both conclusions are limited by power, sample size, specification, and assumptions. In particular, if shared shocks or other cross-sectional dependence are present, the independence-based reference distributions may be unreliable. A test designed for independent units should not be treated as automatically valid merely because it was calculated on panel data.
Common Mistakes
- Treating a negative \(\rho_i\) as sufficient proof of stationarity. It is not: stability of the remaining short-run dynamics is also required. Without it, the process can be explosive despite a negative coefficient.
- Describing LLC as fully heterogeneous. LLC imposes a common adjustment parameter under its alternative. IPS and Fisher-type methods accommodate more heterogeneity in the persistence evidence.
- Interpreting a panel rejection as a finding about every unit. IPS and Fisher-type rejections can arise when only a fraction—or even one unit in the Fisher alternative—contributes stationary evidence.
- Using the chi-square Fisher reference distribution when unit-level tests are dependent. Its standard derivation requires independent uniform null p-values, not merely individually valid p-values.
- Leaving deterministic terms unstated. Intercepts and trends affect both the model being tested and its null calibration. Report the specification rather than presenting a test name alone.
Check Your Understanding
Answer each question using the assumptions and alternatives stated above.
- What restriction does LLC impose under its alternative that IPS does not?
- For which procedure is a nonzero fraction of stationary units the central heterogeneous alternative: LLC, IPS, or both?
- Why does the standard Fisher chi-square reference distribution require cross-sectional independence?
- Suppose an IPS test rejects its all-unit-root null. What does the rejection establish, and what does it not establish about individual units?
- Why is \(\rho_i<0\) alone insufficient to conclude that unit \(i\)'s series is stationary?