ARDL Model (Autoregressive Distributed Lag)

ARDL is a time-series regression model that combines lags of the dependent variable and lags of the independent variables in a single equation. ARDL is appropriate when variables have a mixed order of integration — some stationary at level I(0), some at first difference I(1) — and it allows testing for cointegration through the bounds test of Pesaran, Shin and Smith (2001).

In EcoLab, ARDL belongs to the Time Series group and generates reproducible code for Stata, R and Python. See also Estimation & Modeling and EcoLab Overview.

---

When should you use ARDL?

Use ARDL when all of the following hold:

• The data is time series (a single country/unit over time), typically a small-to-moderate sample.
• Variables are integrated of order no higher than I(1) (no variable is I(2)).
• The order of integration is mixed I(0)/I(1) — a case where Engle–Granger or Johansen is hard to apply.
• You need to estimate both the long-run relationship and the short-run dynamics via an error correction model (ECM).

Do not use ARDL when any variable is I(2), or when the data is multi-unit panel data (consider FEM/REM or panel ARDL/PMG).

---

Model specification

General form of ARDL($p$, $q$) with one independent variable:

$$Y_t = c + \sum_{i=1}^{p} \varphi_i \, Y_{t-i} + \sum_{j=0}^{q} \beta_j \, X_{t-j} + \varepsilon_t$$

where $p$ is the lag order of the dependent variable $Y$ and $q$ is the lag order of the independent variable $X$. The error correction form (ARDL-ECM) separates the long-run relationship from short-run adjustment:

$$\Delta Y_t = c + \alpha \, (Y_{t-1} - \theta \, X_{t-1}) + \sum_i \gamma_i \, \Delta Y_{t-i} + \sum_j \delta_j \, \Delta X_{t-j} + \varepsilon_t$$

$\theta$ is the long-run coefficient; $\alpha$ (the error correction term) must be negative and statistically significant to confirm the existence of a cointegrating relationship.

---

Assumptions and testing steps

1. Order of integration: unit-root tests (ADF, Phillips–Perron, KPSS) to ensure no variable is I(2).
2. Optimal lag selection: based on information criteria (AIC/BIC).
3. Bounds test: the F-statistic compared against the upper/lower critical bounds to conclude on cointegration.
4. Diagnostics: autocorrelation (Breusch–Godfrey), heteroskedasticity (Breusch–Pagan/White), normality of residuals (Jarque–Bera).
5. Stability: CUSUM and CUSUMSQ tests.

---

Running ARDL in EcoLab

1. In the Data Collection module, connect EcoData or public sources (World Bank, IMF, FRED) to obtain the time series of the dependent and independent variables.
2. In the Modeling module, click Add model → choose the Time Series group → ARDL.
3. Select the dependent variable $Y$, the independent variables $X$, and let the system pick lags by AIC/BIC (or set them manually).
4. Click Run model. Review the Estimation tab (short-run/long-run coefficients), the Diagnostics tab (bounds test, CUSUM), and the Replication Code tab for the Stata/R/Python script.

---

Input / output example

Input (illustrative): annual series 1990–2023 with growth (GDP growth %), fdi (FDI/GDP %), open (trade openness %), inf (inflation %).

Output (format, illustrative figures — not real results):

Component	Coefficient	Std. error	p-value
Long run: fdi	0.42	0.15	0.012
Long run: open	0.08	0.04	0.061
ECM ($\alpha$)	−0.55	0.13	0.001
Bounds F-stat	6.10	(above upper I(1) bound)	cointegrated

Interpretation: a negative and significant ECM coefficient confirms cointegration; the speed of adjustment toward long-run equilibrium is about 55% per year.

---

Replication code

Stata

* --- ARDL Model ---
tsset year

* Estimate ARDL with automatic lag selection
* lags(. 2 2 2): auto for Y, max 2 for each X
ardl growth fdi open inf, lags(. 2 2 2)

* Bounds test for cointegration (Pesaran et al. 2001)
estat btest

* Error Correction Model (ECM) form
ardl growth fdi open inf, lags(. 2 2 2) ec

* Post-estimation diagnostics
estat bgodfrey     // autocorrelation
estat hettest      // heteroskedasticity

R

# --- ARDL Model ---
library(ARDL)

# Estimate ARDL with specified lag orders
model <- ardl(growth ~ fdi + open + inf, data = df,
              order = c(2, 2, 2, 2))
summary(model)

# Bounds test for cointegration
bounds <- bounds_f_test(model, case = 3)
print(bounds)

# Error Correction Model (long-run coefficients)
ecm <- recm(model, case = 3)
summary(ecm)

Python

from statsmodels.tsa.ardl import ARDL
import statsmodels.api as sm

# Estimate ARDL(2, 2, 2, 2)
y = df['growth']
X = df[['fdi', 'open', 'inf']]

model = ARDL(endog=y, lags=2, exog=X, order=2)
result = model.fit()
print(result.summary())

# Bounds test (manual F-test on lagged levels)
# Compare F-statistic against PSS critical values
print("AIC:", result.aic)
print("BIC:", result.bic)

---

Limitations and notes

• ARDL cannot handle I(2) variables; check beforehand with unit-root tests.
• Sensitive to lag selection; compare AIC and BIC.
• Too small a sample makes the bounds test unreliable.
• ARDL is a single-equation model; if you suspect multi-directional simultaneity, consider VAR/VECM.

---

Video tutorial

Video Tutorial: Running ARDL in EcoLab

See also

• Worked example: FDI and economic growth in Vietnam (ARDL)
• FEM and REM for panel data
• Estimation & Econometric Modeling