Augmented Dickey-Fuller Lag Selection And Diagnostics

Why the Augmented Regression Needs Lagged Differences

The Dickey–Fuller test asks whether a series has a unit root, but its simplest regression can leave serial correlation in the errors when the series has richer short-run dynamics. That matters: the usual Dickey–Fuller reference distributions presume an appropriately specified disturbance process. The augmented Dickey–Fuller (ADF) regression adds lagged changes to represent short-run dependence while retaining the lagged level that carries the unit-root test.

The number of lagged differences is a specification choice, not the unit-root hypothesis itself. Too few may leave correlated residuals; too many use up degrees of freedom and can weaken the test in a finite sample. Lag selection therefore combines a fit-versus-complexity criterion with direct checks of the fitted residuals.

Definition: For a series $y_t$, an ADF regression with $p$ lagged differences has the form $$ \Delta y_t = d_t + \gamma y_{t-1} + \sum_{j=1}^{p}\psi_j\Delta y_{t-j}+u_t. $$ Here $\Delta y_t=y_t-y_{t-1}$, $d_t$ denotes the chosen deterministic terms (such as an intercept or a time trend), and $u_t$ is the regression disturbance. The unit-root null is $H_0:\gamma=0$; the usual one-sided alternative is $H_a:\gamma<0$. The lag order $p$ counts the lagged differences, not the lagged level.

The deterministic specification must be chosen for the question and series at hand, then held fixed while comparing lag orders. The material on Dickey–Fuller tests under alternative deterministic terms explains why the intercept and trend choices affect the test. AIC or BIC comparisons across models with different deterministic terms would mix two specification decisions and would not isolate the choice of $p$.

Why Lagged Differences Absorb Short-Run Dependence

A short derivation shows how the augmentation arises. Suppose, for illustration, that $y_t$ follows an autoregression of order two:

y_t=a_1y_{t-1}+a_2y_{t-2}+\varepsilon_t.

Subtract $y_{t-1}$ from both sides and use $y_{t-2}=y_{t-1}-\Delta y_{t-1}$. Then

\begin{aligned} \Delta y_t &=(a_1+a_2-1)y_{t-1}-a_2\Delta y_{t-1}+\varepsilon_t\\ &=\gamma y_{t-1}+\psi_1\Delta y_{t-1}+\varepsilon_t, \end{aligned}

where $\gamma=a_1+a_2-1$ and $\psi_1=-a_2$. A unit root in this example corresponds to $a_1+a_2=1$, or $\gamma=0$. The lagged difference represents short-run dynamics that would otherwise be left out of the regression disturbance. Higher-order dynamics lead to additional lagged differences. This algebra explains their role; it does not guarantee that any chosen finite $p$ is sufficient for a particular data set.

In applications, the intended disturbance $u_t$ should be approximately serially uncorrelated for the chosen lag order and deterministic specification. The test statistic for $\gamma$ is not an ordinary $t$-statistic with a standard normal or Student $t$ reference distribution under the unit-root null. Use Dickey–Fuller critical values appropriate to the deterministic terms and test setup. Lag selection does not change that principle.

Information Criteria for Choosing the Lag Order

Information criteria balance in-sample fit against the number of estimated regression coefficients. For a candidate order $p$ fitted on the same $n$ observations, let $RSS_p$ be its residual sum of squares and $k_p$ its number of regression coefficients. A common regression form is

AIC(p)=n\ln(RSS_p/n)+2k_p,\qquad BIC(p)=n\ln(RSS_p/n)+k_p\ln n.

The Hannan–Quinn criterion uses a penalty between the usual AIC and BIC penalties for large samples:

HQ(p)=n\ln(RSS_p/n)+2k_p\ln(\ln n).

In these expressions, smaller scores are preferred. For comparisons within a fixed deterministic specification, $k_p$ can be counted as the regression coefficients (including the intercept if present, the lagged-level coefficient, and the lagged-difference coefficients). Some software includes the estimated error variance in the parameter count. That adds a common penalty across candidate lag orders, so it does not change their ranking when the sample and likelihood convention are otherwise the same. Always check the software's convention before comparing reported score values.

These criteria are not interchangeable. AIC usually penalizes added coefficients less heavily than BIC as $n$ grows, and can choose a longer lag. BIC's larger large-sample penalty often yields a more parsimonious model. HQ provides another compromise. None of them, by itself, establishes that the selected regression has adequately uncorrelated residuals.

Worked Example: Comparing AIC and BIC on a Common Sample

Suppose a hypothetical series has $n=100$ usable observations. Each candidate includes an intercept and the lagged level, with $p$ lagged differences. Thus $k_p=p+2$. All candidates use the same observations and deterministic terms. Their residual mean squares are $RSS_p/n=0.90$, $0.82$, and $0.77$ for $p=0,1,2$, respectively. Compute the criteria and choose the minimum.

For $p=0$, $k_0=2$. Since $100\ln(0.90)=-10.53605$,

AIC(0)=-10.53605+2(2)=-6.53605\approx-6.54, \quad BIC(0)=-10.53605+2\ln(100)=-1.32571\approx-1.33.

For $p=1$, $k_1=3$, and $100\ln(0.82)=-19.84509$:

AIC(1)=-19.84509+2(3)=-13.84509\approx-13.85, \quad BIC(1)=-19.84509+3\ln(100)=-6.02958\approx-6.03.

For $p=2$, $k_2=4$, and $100\ln(0.77)=-26.13648$:

AIC(2)=-26.13648+2(4)=-18.13648\approx-18.14, \quad BIC(2)=-26.13648+4\ln(100)=-7.71580\approx-7.72.

Both criteria attain their minimum at $p=2$ among these three candidates. This is a provisional selection: the $p=2$ residuals still need to be checked for serial correlation. The calculation also depends on using the same $n$. If each model is fitted on a different sample, differences in $RSS_p/n$ and in the likelihood can reflect the changed observations as well as the changed lag order.

Residual Diagnostics After Selection

After choosing a candidate $p$, inspect the residual autocorrelation function and, where appropriate, a portmanteau diagnostic. If residuals retain a pattern, the lag specification may not have absorbed the dependence. A common portmanteau summary is the Ljung–Box statistic computed from residual autocorrelations $\hat{\rho}_k$:

Q^*=n(n+2)\sum_{k=1}^{m}\frac{\hat{\rho}_k^2}{n-k}.

Here $m$ is the number of residual autocorrelation lags included. Under a suitable white-noise null and regularity conditions, this statistic has an approximate chi-square reference distribution. For fitted regressions, the degrees-of-freedom convention and calibration depend on the model and implementation. Do not mechanically apply a reference distribution without checking that it is appropriate for the fitted ADF regression. Residual plots and autocorrelation summaries are useful diagnostics even when a formal calibration needs care.

Worked Example: Interpreting Residual Autocorrelations

Consider a hypothetical ADF regression fitted with $n=80$ observations and $p=1$. The first four residual autocorrelations are $\hat{\rho}_1=0.31$, $\hat{\rho}_2=0.22$, $\hat{\rho}_3=0.08$, and $\hat{\rho}_4=0.04$. Calculate the Ljung–Box summary through $m=4$.

Substituting the values gives

\begin{aligned} Q^* &=80(82)\left( \frac{0.31^2}{79}+\frac{0.22^2}{78} +\frac{0.08^2}{77}+\frac{0.04^2}{76} \right)\\ &=6560(0.00194114)\approx12.73. \end{aligned}

The first two residual autocorrelations are notably larger than the later two, and the combined statistic is driven mostly by those first two terms. As a rough visual reference, $1.96/\sqrt{80}\approx0.219$; the first correlation exceeds that value and the second is just above it. This bound is only a heuristic for inspecting residual autocorrelations, not a formal test of this fitted regression. Together, the pattern and the statistic suggest checking whether a longer lag specification is warranted. They do not establish that the unit-root null is true or false, and a formal portmanteau p-value should use a calibration appropriate to the regression.

A useful diagnostic sequence is to select a candidate, inspect its residuals, and reconsider the order if meaningful autocorrelation remains. Then refit and check again. Increasing $p$ mechanically until every residual autocorrelation looks small is not a complete strategy: it can produce an unnecessarily long regression, particularly in a short series.

Underfitting, Overfitting, and Practical Selection

With too small a $p$, residual serial correlation can remain. The disturbance assumptions supporting standard ADF inference may then be inadequate, so the test's size and power can be unreliable. With too large a $p$, extra coefficients consume degrees of freedom and can increase uncertainty in the estimate of $\gamma$. In finite samples, this often reduces the ability to reject a false unit-root null. Overfitting is not automatically harmless just because the extra lag coefficients are estimated: it can make the test less informative and affect finite-sample performance.

A common practical approach is to set a defensible maximum lag, compare candidate orders using an information criterion, and diagnose the residuals of the selected model. Another approach begins at the maximum and removes the highest lag when its coefficient is not sufficiently informative, applying a stated sequential rule. Neither approach is universally best. The maximum, criterion, residual check, and deterministic specification should be reported so the choice can be assessed.

Worked Example: The Cost of a Large Maximum Lag

Suppose a hypothetical sample contains $T=60$ observations, and the regression includes an intercept, a lagged level, and $p$ lagged differences. The first usable regression date is $t=p+2$, because the model needs $\Delta y_t$, $y_{t-1}$, and $\Delta y_{t-p}$. If $p=2$, the regression uses $t=4,\ldots,60$, or $57$ observations. If $p=8$, it uses $t=10,\ldots,60$, or $51$ observations.

For $p=2$, there are $k=2+2=4$ regression coefficients, leaving $57-4=53$ residual degrees of freedom. For $p=8$, $k=8+2=10$, leaving $51-10=41$ residual degrees of freedom. If the aim is to compare information criteria fairly, one can instead fit every candidate on the common sample $t=10,\ldots,60$, so each has $n=51$. On that sample, the $p=2$ regression has $51-4=47$ residual degrees of freedom, while $p=8$ still has $51-10=41$.

The common-sample comparison removes one source of unfairness, but it does not remove the cost of six additional lag coefficients: the larger model still uses six degrees of freedom. A high maximum lag is therefore not a free safeguard against underfitting. Choose it in light of the series length and plausible short-run dynamics, then assess the selected regression's residuals.

Common pitfall: “The AIC is lowest, so the ADF regression is correctly specified and its unit-root conclusion is settled.” The correction is that an information criterion chooses a relative fit-complexity trade-off among the candidates supplied; it does not test whether residual autocorrelation remains. Check the residuals, use the appropriate ADF critical values, and report the lag-selection rule. A selected lag order is not evidence by itself for or against a unit root.

A Practical Workflow

Fix the deterministic terms.
Choose the intercept and trend specification appropriate to the series and question; keep it unchanged while comparing lag orders.

Set candidate orders and sample.
Choose a defensible maximum lag. Use a common estimation sample when comparing information criteria, and record the effective sample size.

Compare criteria.
Calculate or inspect AIC, BIC, or HQ using a consistent parameter-count convention. Treat the chosen order as a candidate, not a final diagnostic.

Check residual dependence.
Inspect residual autocorrelations and use a portmanteau check only with a suitable reference convention. If dependence remains, reconsider the lag order or specification.

Conduct and report the unit-root test.
Use the ADF statistic and critical values matched to the deterministic terms, and report the selected $p$, selection method, sample, and residual diagnostics.

The central balance is between controlling short-run serial dependence and preserving enough information to estimate the unit-root regression precisely. Information criteria help organize that choice; residual diagnostics check whether the resulting specification is plausible. Neither replaces the other.

Check Your Understanding

Try these questions before reviewing the explanations above.

In the ADF regression, what does $p$ count, and which coefficient is tested for a unit root?
Why should candidate lag orders use the same deterministic terms and, where feasible, the same estimation sample when comparing information criteria?
For $n=100$, $RSS/n=0.77$, and $k=4$, calculate AIC and BIC using the formulas in this tutorial.
What does a visible pattern in residual autocorrelations suggest, and why does it not by itself determine the unit-root conclusion?
Give one finite-sample cost of choosing too many lagged differences and one inferential risk of choosing too few.