Cross-Sectionally Augmented Unit-Root Tests

Why Augment a Panel Unit-Root Test?

Panel unit-root tests that assume cross-sectional independence can be misleading when units share unobserved shocks. For example, national economies may respond to global demand, or firms may face a common change in credit conditions. If these shocks persist over time, the units’ outcomes can move together even after observed covariates are included. Treating that dependence as absent can distort the size of a unit-root test: its rejection rate under a true null may differ from the nominal significance level.

Cross-sectionally augmented Dickey–Fuller tests (CADF) address this problem by adding cross-sectional averages of the outcome and its changes to each unit’s regression. The cross-sectional augmented IPS statistic (CIPS) then averages the resulting individual CADF t-statistics. These procedures do not estimate every common factor explicitly. Instead, under suitable factor and loading conditions, the averages serve as proxies for common components.

Model, Hypotheses, and Conditions

Definition: Let $y_{it}$ denote the outcome for unit $i=1,\ldots,N$ at time $t=1,\ldots,T$. A common-factor representation is $$ y_{it}=d_{it}+\boldsymbol{\lambda}_i'\boldsymbol{f}_t+e_{it}, $$ where $d_{it}$ contains deterministic components, $\boldsymbol{f}_t$ is a vector of unobserved common factors, $\boldsymbol{\lambda}_i$ is the unit-specific loading vector, and $e_{it}$ is an idiosyncratic component. Cross-sectional dependence arises when the common factors affect multiple units. In a panel unit-root test, the null is that all units have a unit root; the alternative allows a nonzero fraction of units to be stationary.

A unit root means that shocks can have persistent effects on the level of a series. For an AR(1) without additional lags, $\Delta y_{it}=b_i y_{i,t-1}+\varepsilon_{it}$ corresponds to an autoregressive coefficient $1+b_i$. The unit-root null is $b_i=0$. Mean reversion in this AR(1) requires $|1+b_i|<1$, equivalently $-2<b_i<0$; $b_i<0$ alone is not sufficient. In the augmented regressions below, the coefficient on the lagged level is the unit-root parameter being tested, while lagged differences account for short-run dynamics.

The cross-sectional augmentation is useful when common factors generate dependence and the cross-sectional averages contain information about those factors. A standard justification requires sufficiently many units, factor loadings whose cross-sectional averages span the relevant common-factor space, and an adequately specified dynamic regression. In particular, the average loadings must not cancel in a way that makes the averages uninformative about a factor. The method is not a guarantee against every form of dependence: if the factors are poorly represented by the averages, or important dynamics are omitted, the test may not have its intended size.

For each unit, the test includes an intercept, and it may also include a deterministic time trend when that specification is appropriate. The deterministic specification, lag order, and reference critical values must match. The precise null and alternative are commonly stated as:

Hypotheses: $H_0$: every panel unit has a unit root. $H_a$: a nonzero fraction of units is stationary, with the remaining units permitted to have unit roots. The CIPS statistic is evaluated in the lower tail: sufficiently negative values count against $H_0$.

From an ADF Regression to CADF

An individual augmented Dickey–Fuller regression relates $\Delta y_{it}$ to the lagged level $y_{i,t-1}$, lagged first differences, and any selected deterministic terms. Its weakness in this setting is that it leaves common shocks out of the regression. CADF adds observed cross-sectional averages as proxies for those omitted common components.

Define the cross-sectional averages at time $t$ by $\bar y_t=N^{-1}\sum_{i=1}^N y_{it}$ and $\Delta\bar y_t=N^{-1}\sum_{i=1}^N\Delta y_{it}$. With $p_i$ lagged differences, the intercept-only CADF regression for unit $i$ is

\Delta y_{it} =a_i+b_i y_{i,t-1}+c_i\bar y_{t-1} +\sum_{j=1}^{p_i}d_{ij}\Delta y_{i,t-j} +\sum_{j=0}^{p_i}e_{ij}\Delta\bar y_{t-j} +u_{it}.

The lagged level $y_{i,t-1}$ tests the unit-root null. The cross-sectional lagged level $\bar y_{t-1}$ and changes $\Delta\bar y_{t-j}$ augment the regression to control for common components and their dynamics. The contemporaneous cross-sectional change, with $j=0$, is included in the usual specification. If the chosen deterministic specification includes a trend, a unit-specific trend is added. The regression is estimated separately for each unit, and the CADF statistic for unit $i$ is the t-statistic for $b_i$:

CADF_i=\frac{\widehat b_i}{SE(\widehat b_i)}.

This ratio is calculated from the estimated CADF regression. Its null distribution is not generally the ordinary Student t distribution: the unit-root null is nonstandard, and cross-sectional augmentation affects the reference distribution. Therefore, the t-ratio is a test statistic, not a license to use ordinary regression t critical values.

Worked Example: Constructing Cross-Sectional Averages

Suppose three hypothetical units have outcomes $y_{1,t-1}=10$, $y_{2,t-1}=14$, and $y_{3,t-1}=12$. Their current outcomes are $10.5$, $13.6$, and $12.4$, respectively. Calculate the lagged-level and differenced cross-sectional averages used in a CADF regression.

The lagged-level average is $$ \bar y_{t-1}=\frac{10+14+12}{3}=\frac{36}{3}=12. $$ The individual changes are $0.5$, $-0.4$, and $0.4$. Thus the average change is $$ \Delta\bar y_t=\frac{0.5-0.4+0.4}{3}=\frac{0.5}{3}=0.1667. $$ This agrees with calculating the average current outcome, $12.1667$, and subtracting the lagged average, $12$. These averages are regressors common to the unit-specific CADF regressions at time $t$; the values here are hypothetical and illustrate their construction, not a test result.

The CADF and CIPS Statistics

Once each unit’s CADF regression has been estimated, the CIPS statistic is the arithmetic mean of its $N$ CADF t-statistics:

CIPS=\frac{1}{N}\sum_{i=1}^{N}CADF_i.

A more negative average indicates stronger evidence against the joint unit-root null, but the decision depends on the appropriate CIPS critical value. Those critical values depend on the deterministic terms, the panel dimensions, and the test specification; they are obtained from suitable published tables or a justified simulation procedure, not from a standard normal or Student t table. The test’s lower-tail direction matters: a statistic that is numerically larger (less negative) is not stronger evidence against $H_0$.

Worked Example: Calculating One CADF Statistic

For a hypothetical unit, suppose an intercept-only CADF regression includes the required cross-sectional averages and selected lagged differences. The estimated coefficient on $y_{i,t-1}$ is $\widehat b_i=-0.180$, and its regression standard error is $SE(\widehat b_i)=0.0900$. Calculate the CADF statistic.

By definition, $$ CADF_i=\frac{\widehat b_i}{SE(\widehat b_i)} =\frac{-0.180}{0.0900}=-2.00. $$ The statistic is $-2.00$. Its negative sign is consistent with a fitted pull away from the unit-root null, but it does not by itself establish rejection. A decision requires the relevant nonstandard CADF or CIPS reference distribution; comparing $-2.00$ with a conventional Student t cutoff would not be justified. The assumed regression specification and standard error are supplied as hypothetical regression output, so no raw observations are being used to claim a real result.

Worked Example: Calculating and Interpreting CIPS

Suppose four hypothetical units produce CADF statistics $-2.10$, $-3.40$, $-1.70$, and $-2.80$. For this illustration, assume the appropriate 5% lower-tail CIPS critical value for the specified deterministic terms, lag choices, and panel dimensions is $-2.35$. Calculate the statistic and state the decision.

The statistic is $$ CIPS=\frac{-2.10-3.40-1.70-2.80}{4} =\frac{-10.00}{4}=-2.500. $$ Because $-2.500<-2.35$, the statistic falls in the 5% rejection region under the stipulated reference value. In this hypothetical example, reject $H_0$ at the 5% level and conclude that the panel provides evidence against the null that all units have a unit root. The alternative allows a nonzero fraction of units to be stationary; rejection does not establish that every unit is stationary. The critical value here is explicitly assumed for illustration and must not be reused for a different panel or specification.

Why the Averages Can Help: A Justification

The intuition for augmentation follows from the common-factor representation. Taking the cross-sectional average gives

\bar y_t=\bar d_t+ \left(\frac{1}{N}\sum_{i=1}^{N}\boldsymbol{\lambda}_i\right)'\boldsymbol{f}_t +\bar e_t.

If the average loadings preserve information about the common factors and the idiosyncratic average becomes small as $N$ grows, then $\bar y_t$ contains a useful proxy for the factor component. Differencing the same representation yields

\Delta\bar y_t=\Delta\bar d_t+ \left(\frac{1}{N}\sum_{i=1}^{N}\boldsymbol{\lambda}_i\right)'\Delta\boldsymbol{f}_t +\Delta\bar e_t.

Thus the average level and its changes can track both persistent common movements and their short-run evolution. Adding them to the unit-level ADF regression can absorb dependence induced by the shared factors, conditional on the proxy and dynamic assumptions. This is a justification for the construction, not a claim that averaging always recovers every factor: for example, loadings that cancel across units can make the average uninformative. CADF and CIPS inference should be treated as specification-dependent, with critical values chosen for the actual design.

Specification and Common Pitfalls

Several choices determine whether the test answers the intended question. A trend should be included when the maintained deterministic specification calls for one, not added mechanically. The lag order should be adequate to capture short-run dynamics; too few lags can leave serial dependence in the regression errors, while unnecessary lags consume degrees of freedom, especially when $T$ is limited. Cross-sectional averages must be calculated from the same panel and time periods used in the regressions, with any missing-data treatment made explicit.

Using ordinary t critical values. The CADF t-ratio looks like a familiar coefficient t-statistic, but the unit-root null and panel dependence yield a nonstandard reference distribution. Use matching CADF or CIPS critical values.
Reading rejection as universal stationarity. CIPS rejection supports the alternative that a nonzero fraction of units is stationary; it does not show that all units are stationary.
Assuming any cross-sectional average captures every factor. Averages work as proxies only under suitable loading and factor conditions. If average loadings cancel, or factors remain inadequately represented, the augmentation may fail.
Equating a negative level coefficient with mean reversion in every AR(1). For $\Delta y_{it}=b_i y_{i,t-1}+\varepsilon_{it}$, the AR coefficient is $1+b_i$, so stationarity requires $-2<b_i<0$, not merely $b_i<0$.

Key takeaway: CADF adds cross-sectional averages to each unit’s unit-root regression; CIPS averages the resulting CADF t-statistics. The approach targets common-factor dependence, but valid interpretation depends on the factor-proxy conditions, regression specification, and matching nonstandard critical values.

Check Your Understanding

Use these questions to check your understanding of cross-sectionally augmented panel unit-root tests.

In the common-factor representation, what feature of cross-sectional averages makes them potentially useful proxies for unobserved common components?
Which coefficient is tested in a CADF regression, and how is its t-statistic calculated?
Four units have CADF statistics $-1.5$, $-2.5$, $-3.0$, and $-2.0$. Calculate CIPS.
Why is a conventional Student t critical value not generally appropriate for deciding whether to reject using CIPS?
For an AR(1) written as $\Delta y_{it}=b_i y_{i,t-1}+\varepsilon_{it}$, state the unit-root value and the full range of $b_i$ values corresponding to stationarity.
What does rejection of the CIPS null establish, and what does it not establish?