Why the Deterministic Terms Matter
A unit-root test asks whether a series has a stochastic trend, but the answer depends on what deterministic movement the model allows. A series fluctuating around zero, one fluctuating around a constant mean, and one fluctuating around a linear time trend call for different Dickey-Fuller specifications. The statistic’s reference distribution changes with that choice, so a correctly calculated statistic can still lead to an incorrect conclusion if it is compared with the wrong critical values.
The central coefficient is the same in all three versions: it measures how the previous level of the series enters its current change. The test concerns whether that coefficient is zero, which corresponds to a unit autoregressive root. What differs across specifications is the set of deterministic terms included alongside it.
From an AR(1) Model to the Dickey-Fuller Regression
Begin with a first-order autoregression that allows a deterministic component \(d_t\):
Here \(y_t\) is the observed series, \(\rho\) is its autoregressive coefficient, and \(\varepsilon_t\) is an innovation. For the basic Dickey-Fuller test, suppose the innovations have mean zero, constant finite variance, and no serial correlation; standard asymptotic results also require suitable regularity conditions on the initial observation. These conditions make the first-order regression an appropriate starting point. If the innovations are serially correlated, the basic test’s reference theory need not apply; the augmented Dickey-Fuller topic treats that issue.
Define the first difference by \(\Delta y_t=y_t-y_{t-1}\). Subtract \(y_{t-1}\) from both sides of the autoregression:
Writing \(\gamma=\rho-1\) gives the Dickey-Fuller regression. The unit-root null \(\rho=1\) is therefore the restriction \(\gamma=0\). A stationary alternative near the unit root has \(\rho<1\), or equivalently \(\gamma<0\).
The three common specifications set \(d_t\) to zero, to a constant, or to a constant plus a linear time trend. Under the alternative, these permit different stationary behaviors: fluctuations around zero, around a constant mean that may be zero or nonzero, or around a deterministic linear trend. These descriptions concern the stationary alternative; they do not mean the series must exhibit that particular mean or trend.
Three Specifications and Their Hypotheses
| Specification | Dickey-Fuller regression | Stationary alternative | Common statistic label |
|---|---|---|---|
| No constant or trend | \(\Delta y_t=\gamma y_{t-1}+\varepsilon_t\) | Stationary around zero | \(\tau\) |
| Constant | \(\Delta y_t=c+\gamma y_{t-1}+\varepsilon_t\) | Stationary around a constant mean, which may be zero or nonzero | \(\tau_\mu\) |
| Constant and trend | \(\Delta y_t=a+bt+\gamma y_{t-1}+\varepsilon_t\) | Stationary around a linear deterministic trend | \(\tau_\tau\) |
In every row, the coefficient test is \(H_0:\gamma=0\) against \(H_a:\gamma<0\). The deterministic terms change the model around that restriction. For example, under the constant specification and the null, the equation becomes \(\Delta y_t=c+\varepsilon_t\): a random walk with drift. Under the trend specification, the null becomes \(\Delta y_t=a+bt+\varepsilon_t\). Thus deterministic terms are not interchangeable decorations; they describe what systematic movement is allowed under the null as well as under the alternative.
The usual coefficient statistic is the estimated coefficient divided by its estimated standard error:
Although this has the algebraic form of a \(t\)-statistic, under the unit-root null it does not have the usual Student \(t\) distribution. The lagged level \(y_{t-1}\) is generated by a process with a unit root under \(H_0\), so the standard regression conditions that yield the usual \(t\) reference distribution do not apply. The limiting distributions are nonstandard and depend on which deterministic terms are included. The statistic labels in the table are a reminder to use the matching Dickey-Fuller distribution or critical values.
For orientation, the familiar large-sample 5% critical values are approximately \(-1.95\) for the no-constant case, \(-2.86\) for the constant case, and \(-3.41\) for the constant-plus-trend case. These are only reference approximations: finite-sample critical values can differ, and test tables or software must match the sample size and deterministic specification. A more negative statistic is needed to reject in the constant and trend cases than in the no-constant case.
Worked Example: Deriving and Fitting the No-Constant Regression
Suppose a hypothetical series has observed levels \(y_0=1,\ y_1=-1,\ y_2=-1,\ y_3=1,\ y_4=1\). For a no-constant Dickey-Fuller regression, the four lagged levels are \(1,-1,-1,1\), and the corresponding differences are \(-2,0,2,0\). The specification assumes no deterministic mean or trend and innovations meeting the basic conditions stated above.
The least-squares estimate with no intercept is the sum of products divided by the sum of squared lagged levels:
The fitted differences are \(-1,1,1,-1\), so the residuals are \(-1,-1,1,1\), and \(SSE=4\). With four observations and one estimated coefficient, the residual variance estimate is \(s^2=4/(4-1)=4/3\). Therefore,
This is the no-constant Dickey-Fuller statistic, not a conventional \(t\)-statistic with three degrees of freedom. Its inference requires the no-constant Dickey-Fuller reference distribution. The very small sample also cautions against treating an asymptotic critical value as a precise finite-sample cutoff.
How the Choice Changes the Interpretation
A constant belongs in the regression when the model should allow a nonzero drift under the unit-root null and a nonzero constant mean under a stationary alternative. A trend belongs when a linear deterministic trend is part of the model being tested. Omitting a needed term can distort the test’s behavior; including unnecessary deterministic terms can reduce its ability to detect a unit root’s absence. The choice should be based on the substantive model and the behavior the specification is meant to allow, not selected simply because one version produces a more favorable statistic.
The coefficient estimate itself also needs care. In the constant specification, if \(\gamma\ne 0\) and the process is stationary, the implied mean is \(-c/\gamma\). This can be zero when \(c=0\) or nonzero when \(c\ne0\); having a constant in the regression does not require a nonzero mean. For the constant-plus-trend specification, the deterministic terms allow a linear trend under the stationary alternative.
A Dickey-Fuller rejection is evidence against the unit-root null under the specified model. It is not proof that the series has every property of a correctly specified stationary process. Conversely, failure to reject does not establish that a unit root is true. The test can have limited power, especially when a stationary process is highly persistent, as in the near-unit-root behavior discussed in earlier material.
Worked Example: A Constant-Specification Statistic
Suppose a hypothetical constant-only regression, fitted to a short series, reports \(\hat{\gamma}=-0.70\) and \(SE(\hat{\gamma})=0.48\). Assume the series’ deterministic behavior is appropriately represented by a constant and that the innovation conditions for the basic test are reasonable.
The coefficient statistic is
The statistic is negative, but it should be compared with the constant-specification Dickey-Fuller distribution, not with a conventional Student \(t\) distribution. For context, it is not as negative as the approximate large-sample 5% constant-case critical value of \(-2.86\), so that asymptotic comparison would not reject \(H_0\) at 5%. Because the sample is short, an appropriate finite-sample critical value is preferable before making an inferential decision.
Worked Example: A Constant-and-Trend Test
Suppose a hypothetical regression with a constant and linear trend, fitted to 120 observations, reports \(\hat{\gamma}=-0.12\) and \(SE(\hat{\gamma})=0.05\). Assume that including a trend is substantively justified and that the innovation conditions are adequate for the basic Dickey-Fuller reference theory.
Calculate the statistic:
The approximate large-sample 5% critical value for the constant-and-trend specification is \(-3.41\). Since \(-2.40>-3.41\), this comparison does not reject \(H_0:\gamma=0\) at the 5% level. The conclusion is that this test has not supplied sufficient evidence against a unit root, allowing for the specified trend; it is not a conclusion that a unit root has been proved. A critical value for the constant-only model would not be valid for this regression.
Why the Reference Distribution Must Match
Under a unit root, the lagged level grows with the accumulated innovations rather than behaving like an ordinary stationary regressor. Its large-sample behavior is described by Brownian-motion functionals. Adding a constant or time trend changes how the regression accounts for deterministic components and changes those functionals. That is why the three coefficient statistics have different limiting distributions, even though each tests the same algebraic restriction \(\gamma=0\).
The practical test sequence is therefore straightforward, but the specification decision comes first:
Choose no constant, a constant, or a constant and trend to represent the deterministic behavior allowed by the model.
Regress \(\Delta y_t\) on \(y_{t-1}\) and precisely those deterministic terms.
Calculate \(\hat{\gamma}/SE(\hat{\gamma})\) and use a left-tailed rejection rule.
Compare with the Dickey-Fuller distribution for the same deterministic specification and, when available, the relevant finite sample.
Common Mistake: Using an Ordinary \(t\) Table
A common error is to see \(\hat{\gamma}/SE(\hat{\gamma})\), call it a \(t\)-statistic, and compare it with a Student \(t\) critical value using the regression’s residual degrees of freedom. That reasoning overlooks the unit-root null: under \(H_0\), the lagged level is nonstationary, and the statistic has a nonstandard Dickey-Fuller distribution. The correction is to use the Dickey-Fuller reference distribution that matches the deterministic terms. A second, closely related error is to use the constant-only critical value for a regression that includes a trend; that also mismatches the statistic and its reference distribution.
Check Your Understanding
Answer each question by identifying the specification and the relevant unit-root inference.
- Starting from \(y_t=c+\rho y_{t-1}+\varepsilon_t\), derive the constant-specification Dickey-Fuller regression and express its unit-root null in terms of \(\gamma\).
- In the constant-only regression, can the stationary alternative have a zero mean? Explain using the relationship between the intercept, \(\gamma\), and the stationary mean.
- Why is \(\hat{\gamma}/SE(\hat{\gamma})\) not compared with an ordinary Student \(t\) distribution under the unit-root null?
- A regression includes a constant and time trend. Which Dickey-Fuller statistic and reference distribution should be used, and what is the left-tailed alternative?
- What does a failure to reject the unit-root null establish, and what does it not establish?