Seasonal Unit Roots And HEGY Testing

Why Seasonal Unit Roots Matter

A time series can be persistent in ways that an ordinary zero-frequency unit-root test does not capture. For quarterly data, for example, values in the same quarter of successive years may remain strongly linked even after the overall level has no conventional random-walk trend. Such persistence can arise from a unit root at a seasonal frequency. A seasonal unit root affects how shocks persist and which differences may be needed to obtain a stationary series.

The HEGY framework—named for Hylleberg, Engle, Granger, and Yoo—tests for unit roots at particular frequencies. Its key idea is to factor the seasonal-difference polynomial, then express the candidate roots as separate coefficient restrictions in a regression. The test is not simply an ordinary Dickey–Fuller test repeated on several transformed series: its null distributions depend on the frequency, deterministic specification, and lag augmentation.

Definition: A seasonal unit root is a root on the unit circle of a time-series lag polynomial at a seasonal frequency. With \(s\) observations per year, the roots of the seasonal-difference polynomial \(1-L^s\) are \(z_k=\exp(2\pi i k/s)\), for \(k=0,\ldots,s-1\). The root \(z_0=1\) is the zero-frequency root; the other roots correspond to seasonal frequencies. Complex roots occur in conjugate pairs and represent two real restrictions.

Factor the Seasonal Polynomial

Write \(L\) for the lag operator, so \(Ly_t=y_{t-1}\). For quarterly observations, seasonal differencing uses \(1-L^4\). Its factorization over the real numbers is

$$ 1-L^4=(1-L)(1+L)(1+L^2). $$

The factor \(1-L\) gives the root \(1\), associated with zero frequency. The factor \(1+L\) gives the root \(-1\), associated with frequency one-half cycle per observation, or two cycles per year. Finally, \(1+L^2\) has roots \(i\) and \(-i\), a conjugate pair associated with frequency one-quarter cycle per observation, or one cycle per year. These roots do not mean the series must visibly repeat a fixed seasonal pattern. They represent stochastic persistence at the corresponding frequencies.

Worked Example: Factoring the Quarterly Difference

A quarterly series is transformed by \(\Delta_4 y_t=(1-L^4)y_t\). Identify its candidate unit-root frequencies.

Factor the lag polynomial:

$$ 1-L^4=(1-L)(1+L)(1+L^2). $$

The first factor has root \(L=1\), giving zero frequency. The second has root \(L=-1=\exp(\pi i)\), giving angular frequency \(\pi\) radians per observation, or one-half cycle per observation. The last factor has roots \(L=i\) and \(L=-i\), corresponding to angular frequencies \(\pi/2\) and \(-\pi/2\), or one-quarter cycle per observation and its conjugate.

Thus quarterly HEGY analysis distinguishes a zero-frequency root, a root at \(-1\), and a pair of seasonal roots at \(i\) and \(-i\). This factorization identifies the hypotheses to test; it does not establish that any of the roots is present in observed data.

For monthly data, the seasonal-difference polynomial is \(1-L^{12}\). Its roots are \(\exp(2\pi i k/12)\) for \(k=0,\ldots,11\): one root at \(1\), one at \(-1\), and five conjugate pairs. Each conjugate pair corresponds to a distinct seasonal frequency. A monthly HEGY specification therefore has more frequency-specific hypotheses than a quarterly one.

The HEGY Regression and Its Hypotheses

A useful way to state the testing logic is through a regression with \(s\) lagged levels and lagged seasonal differences:

\Delta_s y_t=d_t+\sum_{j=1}^{s}\beta_j y_{t-j} +\sum_{r=1}^{p}\phi_r\Delta_s y_{t-r}+\varepsilon_t, \qquad \Delta_s y_t=(1-L^s)y_t.

Here \(d_t\) collects the chosen deterministic terms, such as an intercept, trend, or seasonal indicators; \(p\) is the number of augmentation lags; and \(\varepsilon_t\) is the regression disturbance. HEGY implementations often use an equivalent transformed-regressor parameterization that makes the frequency-specific coefficients especially direct. The regression above makes the root restrictions explicit. Let \(B(z)=\sum_{j=1}^{s}\beta_j z^j\). At a root \(z\) of \(1-z^s\), the unit-root null requires \(B(z)=0\). The seasonal-difference factors vanish there, as do terms formed by multiplying them by lag polynomials.

For quarterly data, this gives the following restrictions:

\begin{aligned} z=1 &: \quad \beta_1+\beta_2+\beta_3+\beta_4=0,\\ z=-1 &: \quad -\beta_1+\beta_2-\beta_3+\beta_4=0,\\ z=i,\,-i &: \quad \beta_4-\beta_2=0,\quad \beta_1-\beta_3=0. \end{aligned}

The first two roots are real, so each has one restriction and is tested with a frequency-specific t-type statistic. The conjugate pair requires both the real and imaginary parts of \(B(i)\) to vanish, so it is tested jointly with an F-type statistic. These are HEGY statistics, not ordinary t or F tests with their usual reference distributions: their null distributions are nonstandard under unit-root asymptotics.

Testing principle: For a candidate root \(z\) of \(1-z^s\), the unit-root null imposes \(B(z)=0\). Test a real root with its single real restriction. For a nonreal conjugate pair, test the real and imaginary restrictions jointly. Use critical values or p-values appropriate to the HEGY specification, not the standard Student t or F distributions.

Why the Restrictions Identify the Frequencies

The restriction formula follows directly from the polynomial representation. Ignoring deterministic terms for the moment, move the lagged-level terms to the left side of the HEGY regression. Its autoregressive lag polynomial is

A(z)=(1-z^s)\left(1-\sum_{r=1}^{p}\phi_r z^r\right)-B(z).

If \(z\) is a root of \(1-z^s\), then the first term is zero, leaving \(A(z)=-B(z)\). Therefore \(A(z)=0\), meaning the full lag polynomial has a root at \(z\), exactly when \(B(z)=0\). The augmentation terms do not change these restrictions because they multiply \(1-z^s\), which vanishes at every candidate seasonal root.

For a quarterly complex root, evaluate \(B\) at \(i\):

B(i)=\beta_1i-\beta_2-\beta_3i+\beta_4 =(\beta_4-\beta_2)+i(\beta_1-\beta_3).

A complex number is zero only if both its real and imaginary parts are zero. This gives the two restrictions for the \(i,-i\) pair. It also explains why a single t-test cannot establish or rule out that pair.

Worked Example: Calculating Quarterly Root Restrictions

Suppose a hypothetical quarterly HEGY regression estimates \((\hat\beta_1,\hat\beta_2,\hat\beta_3,\hat\beta_4)=(0.10,-0.04,0.10,-0.04)\). Calculate the restrictions evaluated at each candidate root.

At \(z=1\), the restriction expression is \(B(1)\):

$$ B(1)=0.10-0.04+0.10-0.04=0.12. $$

At \(z=-1\):

$$ B(-1)=-0.10-0.04-0.10-0.04=-0.28. $$

For the complex pair, calculate the real and imaginary restrictions:

\hat\beta_4-\hat\beta_2=-0.04-(-0.04)=0, \qquad \hat\beta_1-\hat\beta_3=0.10-0.10=0.

The estimated coefficients satisfy the two complex-pair restrictions exactly, but not the point restrictions for \(1\) or \(-1\). This arithmetic alone is not a statistical test: sampling uncertainty matters. In particular, it cannot establish a seasonal unit root merely because the estimated restrictions happen to equal zero.

Specification and Interpretation

The deterministic terms must reflect the question and the behavior allowed under the alternative. An intercept or trend can change the null distribution of the HEGY statistics. Seasonal indicators can represent fixed, deterministic differences among quarters or months; they are not substitutes for testing stochastic seasonal roots. A series may have deterministic seasonal means without seasonal unit roots, or seasonal unit roots without a simple fixed seasonal pattern.

Lag augmentation serves a different purpose: it helps make the regression disturbance sufficiently free of serial correlation for the test approximation to be credible. Too few lags can leave serial correlation and distort inference; adding many unnecessary lags can reduce power. The residual diagnostics and lag choice should therefore be reported alongside the deterministic specification and the test results. Critical values should match those choices and the sampling frequency.

A rejection is evidence against the particular root restriction being tested, not proof that the series is stationary. For example, rejecting the zero-frequency root does not rule out a root at \(-1\) or at a seasonal conjugate pair. Conversely, failure to reject is not proof that the root is present; the test may have limited power, especially in a short sample. Conclusions should identify the frequency and state whether the test rejects its unit-root null.

Worked Example: Interpreting Hypothetical HEGY Results

A hypothetical quarterly analysis uses an intercept and seasonal indicators, includes enough augmentation lags to leave no material residual serial correlation, and reports HEGY-calibrated p-values of 0.018 for the zero-frequency root, 0.31 for the \(-1\) root, and 0.007 for the \(i,-i\) pair. Use a 5% significance level.

For the zero-frequency test, \(0.018<0.05\), so reject the null of a root at \(1\). The result is evidence against a zero-frequency unit root under this specification.

For the \(-1\) test, \(0.31>0.05\), so do not reject the null of a root at \(-1\). This is insufficient evidence to rule out that root; it is not proof that the root is present.

For the joint test of the \(i,-i\) pair, \(0.007<0.05\), so reject the joint null that both real restrictions hold. The test provides evidence against that seasonal unit-root pair. The three results concern different frequencies; neither of the rejections determines the outcome of the other test or establishes that the series as a whole is stationary.

Common Mistakes and Practical Consequences

Using ordinary t or F critical values. HEGY unit-root statistics have nonstandard null distributions. Use critical values or p-values calibrated for the HEGY test, deterministic terms, and frequency.
Testing a complex pair with only one restriction. A pair such as \(i,-i\) requires both real and imaginary restrictions. Use the joint test.
Treating seasonal indicators as seasonal unit-root tests. Indicators capture fixed seasonal means; HEGY tests stochastic persistence at seasonal frequencies.
Concluding stationarity from one rejection. Each test addresses a specific root. Other roots, as well as the remaining stability conditions, still matter.
Reading non-rejection as proof. A large p-value means the data do not provide sufficient evidence to reject that root null at the chosen level; it does not confirm the null.

These distinctions matter when choosing transformations and interpreting long-run behavior. Seasonal differencing may remove a seasonal root, but unnecessary differencing can discard information and introduce avoidable dependence. Frequency-specific tests help target the relevant persistence rather than automatically applying seasonal differences to every series.

Check Your Understanding

Answer each question by identifying the relevant root or restriction.

Factor \(1-L^4\) over the real numbers and identify the frequency associated with each factor.
Why does a quarterly root at \(i\) require two real restrictions rather than one?
For quarterly coefficients \((\beta_1,\beta_2,\beta_3,\beta_4)\), write the restriction for the root at \(-1\).
A HEGY test for a seasonal pair has p-value 0.08. What is the decision at a 5% level, and what does that decision not establish?
Explain why seasonal indicators and seasonal unit-root tests address different features of a series.