Common mistake 1 with ARDL & bounds test

Reporting the bounds test while the residuals are autocorrelated. Fix the lags first.

Common mistake 2 with ARDL & bounds test

Choosing the wrong case (1–5) for the deterministic terms. Case 3 is the usual choice.

Common mistake 3 with ARDL & bounds test

Including an I(2) variable, which invalidates the critical values entirely.

Common mistake 4 with ARDL & bounds test

Reporting an inconclusive bounds test as a rejection. The test has a genuine middle region between the I(0) and I(1) bounds, and landing in it means the answer depends on integration orders the test deliberately does not assume.

EconStatsCraft › ARDL & bounds test

ARDL & bounds test

The autoregressive distributed-lag model with the Pesaran–Shin–Smith bounds test. Unlike Johansen it works when your regressors are a mix of I(0) and I(1), which is why it dominates applied dissertations.

What it needs

Dependent variable
Independent variables

The intuition

Johansen needs everything to be I(1); in practice you are rarely sure. Pesaran, Shin and Smith found that a plain autoregressive distributed-lag model, rewritten in error-correction form, permits a test for a long-run relationship whose critical values are valid whatever mixture of I(0) and I(1) you happen to have - hence two bounds rather than one critical value.

When to use it

Small samples, and the very common situation where you are not certain about the integration order of every regressor. This is why ARDL dominates applied dissertations.

Where the estimator comes from

Rewrite the ARDL as \Delta y_t=\alpha+\rho y_{t-1}+\theta' x_{t-1}+\sum_i \varphi_i \Delta y_{t-i}+\sum_j \psi_j' \Delta x_{t-j}+\varepsilon_t. The null of no long-run relationship is \rho=0 and \theta=0 jointly, tested with an F statistic. Its distribution is non-standard and depends on the integration orders, so PSS tabulate a LOWER bound assuming everything is I(0) and an UPPER bound assuming everything is I(1). Above the upper bound you reject regardless; below the lower bound you cannot reject regardless; in between you genuinely do not know without more information.

The model

\Delta y_t=\alpha+\rho\,y_{t-1}+\boldsymbol{\theta}^{\prime}\mathbf{x}_{t-1}+\sum_{i=1}^{p}\varphi_i\,\Delta y_{t-i}+\sum_{j=0}^{q}\boldsymbol{\psi}_j^{\prime}\,\Delta\mathbf{x}_{t-j}+\varepsilon_t

What it assumes

No regressor is I(2). This is the one hard requirement - check it.
The residuals are serially uncorrelated. If Breusch–Godfrey rejects, the bounds test is invalid.
The chosen lag structure is adequate.
The long-run relationship is stable over the sample. The bounds test assumes one set of coefficients throughout, and CUSUM on the error-correction residuals is the usual check that it holds.
The sample is large enough for the bounds critical values, which are asymptotic. Narayan's small-sample values exist precisely because the standard ones over-reject in short series.

Statistical properties

Valid for any mix of I(0) and I(1) regressors, but NOT if anything is I(2).
Performs better than Johansen in small samples, which is why dissertations favour it.
The long-run coefficients are −θ/ρ, with standard errors from the delta method.
Requires serially uncorrelated residuals; the lag structure exists to deliver that.
The five "cases" differ in how the constant and trend are restricted, and each has its own critical values.

How to read the output

F above the I(1) upper bound → a long-run relationship exists.
F below the I(0) lower bound → none.
Between the two → inconclusive; you need the integration order of each regressor to decide.
The long-run coefficients are the permanent effect of a permanent change in x.
The error-correction term shows how fast the system returns to that long run.

A worked reading

F = 23.39 against a 5% I(1) upper bound of 4.32 - comfortably rejected, so a long-run relationship exists. The ECM coefficient of −0.42 says 42% of any disequilibrium is corrected within one quarter.

Symptom → diagnosis → what to do

If you see this	It means	Do this
A regressor is I(2)	Critical values invalid	Difference that series first
Breusch–Godfrey rejects	Bounds test invalid	Increase the lag order until residuals are clean
F falls between the bounds	Genuinely inconclusive	Establish each regressor's integration order, then re-read
ECM term positive or above 1 in magnitude	No stable long-run relationship	Re-specify
Very long lags chosen	Over-fitting a short sample	Prefer BIC over AIC

Common mistakes

Reporting the bounds test while the residuals are autocorrelated. Fix the lags first.
Choosing the wrong case (1–5) for the deterministic terms. Case 3 is the usual choice.
Including an I(2) variable, which invalidates the critical values entirely.
Reporting an inconclusive bounds test as a rejection. The test has a genuine middle region between the I(0) and I(1) bounds, and landing in it means the answer depends on integration orders the test deliberately does not assume.

Where to read this properly

Pesaran, Shin & Smith (2001) - the bounds test
Pesaran & Shin (1999)
Narayan (2005) - small-sample critical values

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ARDL & bounds test

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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