ARDL and ARDL Cointegration

A Practical Guide with R

Zahid Asghar

2026-09-12

Outline

Review: Why Cointegration Matters
ARDL Models: The Basics
ARDL Bounds Testing Approach
Step-by-Step Implementation in R
Interpreting Results
Advantages over Johansen Method
Practical Example with Real Data

Review: Cointegration Basics

What is Cointegration?

Recall from Johansen Method:

Two or more non-stationary series (I(1)) can have a stationary linear combination
This implies a long-run equilibrium relationship
Short-run deviations are temporary

Example: \[Y_t = \beta_0 + \beta_1 X_t + u_t\]

If \(Y_t \sim I(1)\) and \(X_t \sim I(1)\), but \(u_t \sim I(0)\), then \(Y\) and \(X\) are cointegrated.

Note

Cointegration prevents spurious regression problems!

Johansen Method: Key Requirements

You already know Johansen cointegration requires:

All variables must be I(1) - same order of integration
System of equations - VAR framework
Pre-testing for unit roots is essential
Trace test or Maximum eigenvalue test
Assumes symmetric adjustment to equilibrium

Important

Question: What if some variables are I(0) and others are I(1)?
Answer: This is where ARDL comes in!

ARDL Models

What is ARDL?

AutoRegressive Distributed Lag Model

General form with one dependent and one independent variable:

\[Y_t = \alpha_0 + \sum_{i=1}^{p}\beta_i Y_{t-i} + \sum_{j=0}^{q}\gamma_j X_{t-j} + \varepsilon_t\]

AutoRegressive (AR): lags of dependent variable \(Y_{t-1}, Y_{t-2}, ...\)
Distributed Lag (DL): current and lagged values of \(X_t, X_{t-1}, X_{t-2}, ...\)
p: lag order for Y (AR part)
q: lag order for X (DL part)
Denoted as ARDL(p, q)

Why Use ARDL?

Flexibility in modeling dynamic relationships:

Captures short-run dynamics (through differenced terms)
Estimates long-run relationships (through level terms)
Allows different lag lengths for different variables
Can include exogenous variables easily
Works with small sample sizes (minimum ~30 observations)

Tip

ARDL is particularly useful for macroeconomic data where relationships have both immediate and lagged effects.

ARDL with Multiple Variables

With multiple independent variables:

\[Y_t = \alpha_0 + \sum_{i=1}^{p}\beta_i Y_{t-i} + \sum_{j=0}^{q_1}\gamma_{1j} X_{1,t-j} + \sum_{j=0}^{q_2}\gamma_{2j} X_{2,t-j} + \sum_{j=0}^{q_3}\gamma_{3j} X_{3,t-j} + \varepsilon_t\]

Denoted as ARDL(p, q₁, q₂, q₃)
Each variable can have different optimal lag length
Example: ARDL(2, 1, 3, 0) means:
- 2 lags of Y
- 1 lag of X₁
- 3 lags of X₂
- 0 lags (contemporaneous only) of X₃

ARDL Bounds Testing Approach

The Innovation: Pesaran, Shin & Smith (2001)

The PSS (2001) Bounds Testing Approach

Revolutionary paper: “Bounds testing approaches to the analysis of level relationships”
Key innovation: Test for cointegration without pre-testing for unit roots
Works with variables that are:
- I(0) - stationary
- I(1) - first difference stationary
- Mix of I(0) and I(1)

Warning

Important restriction: Variables cannot be I(2) or higher!
You still need to check this.

How Does It Work?

Two-step transformation:

Step 1: Original ARDL model in levels

Step 2: Transform to Error Correction Model (ECM) representation

\[\Delta Y_t = \beta_0 + \sum_{i=1}^{p-1}\beta_i \Delta Y_{t-i} + \sum_{j=0}^{q-1}\gamma_j \Delta X_{t-j} + \underbrace{\theta_0 Y_{t-1} + \theta_1 X_{t-1}}_{\text{Long-run relationship}} + \varepsilon_t\]

Note

The lagged level terms (\(Y_{t-1}, X_{t-1}\)) capture the long-run relationship!

The Bounds Test

Hypothesis Testing:

\[H_0: \theta_0 = \theta_1 = 0 \quad \text{(No cointegration)}\] \[H_1: \theta_0 \neq 0 \text{ or } \theta_1 \neq 0 \quad \text{(Cointegration exists)}\]

Test using F-statistic (or Wald test)

But: F-statistic has non-standard distribution
Solution: PSS provide critical value bounds
- Lower bound I(0): assumes all variables are I(0)
- Upper bound I(1): assumes all variables are I(1)

Decision Rules

Compare your F-statistic with critical value bounds:

F < Lower bound → No cointegration (reject relationship)
Lower < F < Upper → Inconclusive (need unit root tests)
F > Upper bound → Cointegration exists! (proceed to estimate)

Critical Values

PSS (2001) provide tables based on:

Number of variables (k)
Sample size
Significance level (1%, 5%, 10%)
Model specification (with/without trend)

Example critical values at 5% level (k=3, n=large):

Bound Type	Critical Value
I(0)	3.23
I(1)	4.35

Tip

Modern R packages calculate these automatically!

Step-by-Step ARDL Process

The Complete Workflow

Step 1: Check Order of Integration

Ensure no I(2) variables using ADF/PP tests
Variables can be I(0), I(1), or mixed

Step 2: Estimate ARDL Model

Select optimal lag lengths using AIC/BIC
Estimate the unrestricted ARDL model

Step 3: Bounds Test

Test for cointegration using F-statistic
Compare with critical bounds

Workflow Continued

Step 4: If Cointegration Exists

Estimate the long-run relationship:

\[Y_t = \beta_0 + \beta_1 X_{1,t} + \beta_2 X_{2,t} + \beta_3 X_{3,t} + u_t\]

Step 5: Error Correction Model (ECM)

\[\Delta Y_t = \text{short-run dynamics} + \delta \cdot ECT_{t-1} + \varepsilon_t\]

Where \(ECT_{t-1} = Y_{t-1} - \hat{\beta}_0 - \hat{\beta}_1 X_{1,t-1} - \hat{\beta}_2 X_{2,t-1} - \hat{\beta}_3 X_{3,t-1}\)

Important

\(\delta\) should be negative and significant for valid cointegration!

Implementing ARDL in R

Required Packages

# Install if needed
install.packages("ARDL")
install.packages("dynamac")
install.packages("dplyr")
install.packages("tseries")
install.packages("urca")
install.packages("ggplot2")

# Load packages
library(ARDL)      # Main ARDL package
library(dynamac)   # Alternative ARDL package
library(dplyr)     # Data manipulation
library(tseries)   # Time series tests
library(urca)      # Unit root tests
library(ggplot2)   # Visualization

Load and Prepare Data

Let’s use a practical example: Relationship between Consumption, Income, and Interest Rate

# Example with simulated data (replace with your real data)
set.seed(123)
n <- 100

# Create time series
time <- 1:n
income <- 100 + 0.5*time + cumsum(rnorm(n, 0, 2))
interest <- 5 + 0.01*time + rnorm(n, 0, 0.5)
consumption <- 20 + 0.7*income - 2*interest + rnorm(n, 0, 3)

# Create data frame
data <- data.frame(
  time = time,
  consumption = consumption,
  income = income,
  interest = interest
)

head(data)

Step 1: Unit Root Tests

Check that no variables are I(2):

# ADF test function
adf_test <- function(x) {
  test <- ur.df(x, type = "trend", lags = 4)
  return(test@teststat[1])
}

# Test levels
cat("ADF Test - Levels:\n")
cat("Consumption:", adf_test(data$consumption), "\n")
cat("Income:", adf_test(data$income), "\n")
cat("Interest:", adf_test(data$interest), "\n")

# Test first differences
cat("\nADF Test - First Differences:\n")
cat("D.Consumption:", adf_test(diff(data$consumption)), "\n")
cat("D.Income:", adf_test(diff(data$income)), "\n")
cat("D.Interest:", adf_test(diff(data$interest)), "\n")

Note

Critical value at 5%: approximately -3.45 (with trend)

Step 2: Select Optimal Lags

# Auto-select lags using AIC
ardl_model <- auto_ardl(
  consumption ~ income + interest,
  data = data,
  max_order = 5,    # Maximum lag to consider
  selection = "AIC" # Can also use "BIC"
)

# View selected model
summary(ardl_model)

# Get the selected lag order
ardl_model$order
# Output example: (2, 1, 3) means ARDL(2,1,3)

Tip

AIC (Akaike) - better for prediction
BIC (Schwarz) - more parsimonious, better for small samples

Step 3: Bounds Test

# Perform bounds test for cointegration
bounds_test <- bounds_f_test(
  ardl_model,
  case = 3  # Case 3: unrestricted intercept, no trend
)

# View results
bounds_test

# The output shows:
# - F-statistic
# - Lower bound I(0)
# - Upper bound I(1)
# - Decision

Interpretation:

if (bounds_test$tab[1,1] > bounds_test$tab[1,3]) {
  cat("F-statistic > Upper bound → Cointegration exists!")
} else if (bounds_test$tab[1,1] < bounds_test$tab[1,2]) {
  cat("F-statistic < Lower bound → No cointegration")
} else {
  cat("Inconclusive → Need to check unit roots")
}

Step 4: Long-Run Relationship

If cointegration is confirmed:

# Extract long-run coefficients
lr_model <- multipliers(ardl_model, type = "lr")

# View long-run relationship
summary(lr_model)

# Long-run elasticities
lr_model$coefficients

Output interpretation:

Coefficient for income: long-run marginal propensity to consume
Coefficient for interest: long-run effect of interest rate on consumption
Standard errors and t-statistics: test significance of long-run effects

Step 5: Error Correction Model

# Estimate ECM representation
ecm_model <- ecm(ardl_model)

# View ECM results
summary(ecm_model)

# Extract error correction term coefficient
ect_coef <- ecm_model$coefficients["ect"]

cat("Error Correction Term coefficient:", ect_coef, "\n")

Key checks:

ECT coefficient should be negative (adjustment toward equilibrium)
ECT coefficient should be significant (p < 0.05)
Magnitude indicates speed of adjustment
- Example: -0.3 means 30% of disequilibrium corrected each period

Diagnostic Tests

Always check model validity:

# 1. Serial Correlation Test
bg_test <- bgtest(ardl_model, order = 4)
print(bg_test)

# 2. Heteroskedasticity Test
bp_test <- bptest(ardl_model)
print(bp_test)

# 3. Normality Test
jb_test <- jarque.bera.test(residuals(ardl_model))
print(jb_test)

# 4. Stability Test (CUSUM)
plot(stability(ardl_model, type = "cusum"))

Warning

If diagnostics fail, consider:
- Adding more lags
- Including dummy variables
- Transforming variables (log)

Interpreting Results

Understanding Coefficients

Short-Run Coefficients (from ARDL/ECM):

\(\Delta income\): immediate (impact) effect on consumption
\(\Delta interest\): immediate effect of interest rate change

Long-Run Coefficients:

Show equilibrium relationships
More policy-relevant for structural analysis

Error Correction Term

Example: \(ECT_{t-1} = -0.35\) (p-value < 0.01)

Interpretation:

Negative sign: Confirms cointegration (adjustment toward equilibrium)
Magnitude 0.35: 35% of disequilibrium is corrected each period
Time to equilibrium: Roughly \(1/0.35 \approx 2.86\) periods
Significance: The long-run relationship is statistically valid

Important

If ECT > 0 or insignificant, the cointegration relationship is questionable!

Visualizing Results

# Plot actual vs fitted values
plot(ardl_model)

# Plot residuals
residuals_df <- data.frame(
  time = 1:length(residuals(ardl_model)),
  residuals = residuals(ardl_model)
)

ggplot(residuals_df, aes(x = time, y = residuals)) +
  geom_line() +
  geom_hline(yintercept = 0, color = "red", linetype = "dashed") +
  labs(title = "ARDL Model Residuals",
       x = "Time", y = "Residuals") +
  theme_minimal()

# Plot dynamic forecasts
forecast_plot <- forecast(ardl_model, h = 10)
plot(forecast_plot)

ARDL vs Johansen Method

Comparison Table

Feature	ARDL Bounds Test	Johansen Method
Order of integration	I(0), I(1), or mixed	All must be I(1)
Pre-testing required	No (only check not I(2))	Yes (must confirm I(1))
Estimation	Single equation	System (VAR)
Complexity	Simpler	More complex
Small samples	Works well (n≥30)	Needs larger samples
Different lags	Yes, per variable	No, same for all
Endogeneity	Can handle with lags	Better for simultaneous equations

When to Use ARDL

ARDL is preferred when:

Mixed integration orders (some I(0), some I(1))
Small sample size (< 100 observations)
Single equation focus (one dependent variable of interest)
Different dynamics across variables
Simpler interpretation is desired
Preliminary analysis before full system estimation

When to Use Johansen

Johansen is preferred when:

Multiple cointegrating relationships suspected
All variables are I(1) (confirmed)
System-wide analysis needed
Simultaneous determination (multiple endogenous variables)
Large sample available
Bidirectional causality important

Note

Many researchers use both methods as robustness checks!

Practical Example

Real Data Example

Let’s analyze the relationship between GDP, Investment, and Government Spending using Pakistani data:

# Load data (example - use your own data)
library(readr)
pak_data <- read_csv("pakistan_macro.csv")

# Variables:
# - gdp: Real GDP (in billions)
# - investment: Gross Fixed Capital Formation
# - govt_spending: Government Expenditure

# Take logs for elasticity interpretation
pak_data <- pak_data %>%
  mutate(
    log_gdp = log(gdp),
    log_inv = log(investment),
    log_govt = log(govt_spending)
  )

Unit Root Tests

library(urca)

# ADF tests on levels
adf_gdp <- ur.df(pak_data$log_gdp, type = "trend", lags = 4)
adf_inv <- ur.df(pak_data$log_inv, type = "trend", lags = 4)
adf_govt <- ur.df(pak_data$log_govt, type = "trend", lags = 4)

# Summary
summary(adf_gdp)
summary(adf_inv)
summary(adf_govt)

# Check first differences if needed
adf_d_gdp <- ur.df(diff(pak_data$log_gdp), type = "trend", lags = 4)
summary(adf_d_gdp)

Expected result: Variables are I(1), good for ARDL!

Estimate ARDL Model

# Auto-select optimal ARDL model
ardl_pak <- auto_ardl(
  log_gdp ~ log_inv + log_govt,
  data = pak_data,
  max_order = 4,
  selection = "BIC"  # Use BIC for parsimony
)

# View selected order
ardl_pak$order
# Example output: (2, 1, 1) = ARDL(2,1,1)

# Full model summary
summary(ardl_pak)

Bounds Test Results

# Perform bounds test
bounds_result <- bounds_f_test(ardl_pak, case = 3)

# View results
print(bounds_result)

# Example output:
#           F-statistic  Lower.bound  Upper.bound
# 10%           5.234        2.72         3.77
# 5%            5.234        3.23         4.35
# 1%            5.234        4.29         5.61

Interpretation: F = 5.234 > Upper bound at 5% (4.35)
→ Strong evidence of cointegration!

Long-Run Results

# Extract long-run coefficients
lr_pak <- multipliers(ardl_pak, type = "lr")

# View results
summary(lr_pak)

# Example output:
#              Estimate  Std.Error  t-value  Pr(>|t|)
# log_inv        0.456      0.089    5.123    0.000 ***
# log_govt       0.234      0.067    3.493    0.001 **

Economic Interpretation:

1% increase in investment → 0.456% increase in GDP (long-run)
1% increase in govt spending → 0.234% increase in GDP (long-run)

ECM Results

# Estimate ECM
ecm_pak <- ecm(ardl_pak)

# View results
summary(ecm_pak)

# Example output:
#              Estimate  Std.Error  t-value  Pr(>|t|)
# ect           -0.428      0.098   -4.367    0.000 ***
# d(log_inv)     0.312      0.076    4.105    0.000 ***
# d(log_govt)    0.189      0.054    3.500    0.001 **

Interpretation:

ECT = -0.428: About 43% of disequilibrium corrected per period
Speed of adjustment: \(1/0.428 \approx 2.3\) periods to equilibrium
Short-run effects also significant

Diagnostic Checks

# Load required package
library(lmtest)

# 1. Breusch-Godfrey Test (Serial Correlation)
bgtest(ardl_pak, order = 4)
# H0: No serial correlation

# 2. Breusch-Pagan Test (Heteroskedasticity)
bptest(ardl_pak)
# H0: Homoskedastic errors

# 3. Jarque-Bera Test (Normality)
jarque.bera.test(residuals(ardl_pak))
# H0: Errors are normally distributed

# 4. CUSUM Test (Stability)
cusum_test <- stability(ardl_pak, type = "cusum")
plot(cusum_test)

Visualizing the Relationship

# Plot actual vs fitted
library(ggplot2)

fitted_vals <- fitted(ardl_pak)
actual_vals <- pak_data$log_gdp[1:length(fitted_vals)]

plot_df <- data.frame(
  time = 1:length(actual_vals),
  actual = actual_vals,
  fitted = fitted_vals
)

ggplot(plot_df, aes(x = time)) +
  geom_line(aes(y = actual, color = "Actual"), size = 1) +
  geom_line(aes(y = fitted, color = "Fitted"), size = 1, linetype = "dashed") +
  labs(title = "ARDL Model: Actual vs Fitted Values",
       subtitle = "Log GDP in Pakistan",
       x = "Time Period", y = "Log GDP",
       color = "Series") +
  theme_minimal() +
  theme(legend.position = "bottom")

Advanced Topics

ARDL with Structural Breaks

If you suspect structural breaks (e.g., policy changes, crises):

# Add dummy variables
pak_data$crisis_dummy <- ifelse(pak_data$year >= 2008 & 
                                pak_data$year <= 2010, 1, 0)

# Estimate ARDL with dummy
ardl_break <- auto_ardl(
  log_gdp ~ log_inv + log_govt + crisis_dummy,
  data = pak_data,
  max_order = 4
)

# Check if dummy is significant
summary(ardl_break)

ARDL with Non-linear Effects

Include squared terms for non-linear relationships:

# Create squared term
pak_data$log_inv_sq <- pak_data$log_inv^2

# Estimate model
ardl_nonlin <- auto_ardl(
  log_gdp ~ log_inv + log_inv_sq + log_govt,
  data = pak_data,
  max_order = 4
)

summary(ardl_nonlin)

Test for non-linearity: If \(\beta_{inv^2}\) is significant, relationship is non-linear

NARDL: Non-linear ARDL

Account for asymmetric effects (positive vs negative changes):

library(nardl)

# Separate positive and negative changes
nardl_model <- nardl(
  log_gdp ~ log_inv + log_govt,
  data = pak_data,
  p = 2, q = 1  # ARDL(2,1) structure
)

# Test for asymmetry
asymmetry_test <- nardl_model$pss

# View results
summary(nardl_model)

Note

NARDL is useful when effects differ in expansions vs recessions

Common Mistakes & Tips

Common Mistakes

Not checking for I(2) variables
- Always test! I(2) invalidates bounds test
Ignoring diagnostic tests
- Serial correlation biases standard errors
- Check residuals carefully
Over-fitting with too many lags
- Use information criteria (AIC/BIC)
- Keep model parsimonious
Misinterpreting ECT sign
- Must be negative for valid cointegration
- Positive ECT = explosive process

Common Mistakes (continued)

Not testing coefficient restrictions
- Test long-run coefficients significance
- Use Wald tests when needed
Ignoring structural breaks
- Check data for known shocks/policy changes
- Use CUSUM/CUSUMSQ tests
Wrong case selection in bounds test
- Case 1: No intercept, no trend
- Case 2: Restricted intercept, no trend
- Case 3: Unrestricted intercept, no trend ← Most common
- Case 4: Unrestricted intercept, restricted trend
- Case 5: Unrestricted intercept, unrestricted trend

Best Practices

Always plot your data first
- Check for trends, breaks, outliers
Use log transformations when appropriate
- For multiplicative relationships
- To interpret coefficients as elasticities
Report multiple model specifications
- Different lag lengths
- With/without additional controls
Conduct robustness checks
- Compare ARDL with Johansen
- Try different lag selection criteria
Interpret economically, not just statistically
- Do magnitudes make sense?
- Are signs consistent with theory?

Reporting ARDL Results

In your paper/thesis, report:

Unit root test results (confirm not I(2))
Selected ARDL specification and lag selection criteria
Bounds test results with F-statistic and decision
Long-run coefficients with standard errors
Short-run coefficients and ECT from ECM
Diagnostic test results (serial correlation, heteroskedasticity, normality, stability)
Economic interpretation of all key coefficients

Tip

Create a comprehensive table with all results for easy reference!

Conclusion & Resources

Summary

Key Takeaways:

ARDL is flexible - works with I(0), I(1), or mixed
Bounds testing avoids unit root pre-testing debates
Simpler than Johansen for single equation models
Provides both short-run dynamics and long-run equilibrium
R implementation is straightforward with ARDL package
Always check diagnostics and stability
Economic interpretation matters most!

When to Use What?

Quick Decision Guide:

flowchart TD
    A[Start: Check Your Data] --> B{Variables <br/> mixed I0/I1?}
    B -->|Yes| C[Use ARDL]
    B -->|No, All I1| D{Small sample <br/> n < 80?}
    D -->|Yes| C
    D -->|No| E{Single equation <br/> focus?}
    E -->|Yes| F[ARDL preferred <br/> Simpler]
    E -->|No| G{Multiple <br/> cointegrating <br/> relations?}
    G -->|Yes| H[Use Johansen]
    G -->|No| F
    
    style C fill:#90EE90
    style H fill:#FFB6C1
    style F fill:#90EE90
    
    C --> I[ARDL Bounds Test]
    H --> J[Johansen Test]
    F --> K[Either works <br/> Use both for robustness]

flowchart TD
    A[Start: Check Your Data] --> B{Variables <br/> mixed I0/I1?}
    B -->|Yes| C[Use ARDL]
    B -->|No, All I1| D{Small sample <br/> n < 80?}
    D -->|Yes| C
    D -->|No| E{Single equation <br/> focus?}
    E -->|Yes| F[ARDL preferred <br/> Simpler]
    E -->|No| G{Multiple <br/> cointegrating <br/> relations?}
    G -->|Yes| H[Use Johansen]
    G -->|No| F
    
    style C fill:#90EE90
    style H fill:#FFB6C1
    style F fill:#90EE90
    
    C --> I[ARDL Bounds Test]
    H --> J[Johansen Test]
    F --> K[Either works <br/> Use both for robustness]

Key Decision Points:

Mixed Integration?
- I(0) and I(1) mixed → ARDL only
Sample Size?
- Small (n<80) → ARDL better
- Large (n>100) → Both work
Research Focus?
- Single equation → ARDL simpler
- System dynamics → Johansen
Multiple Cointegration?
- Multiple relations → Johansen
- Single relation → ARDL

Tip

Best Practice: Use both methods as robustness checks when possible!

R Packages Summary

Essential packages:

# ARDL analysis
install.packages("ARDL")      # Main package
install.packages("dynamac")   # Alternative

# Unit root tests
install.packages("urca")      # Comprehensive tests
install.packages("tseries")   # ADF, PP, KPSS

# Diagnostics
install.packages("lmtest")    # LM tests
install.packages("car")       # VIF, other tests

# Visualization
install.packages("ggplot2")   # Publication-quality plots

Additional Resources

Key Papers:

Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326.
Pesaran, M. H., & Shin, Y. (1999). An autoregressive distributed lag modelling approach to cointegration analysis.

Online Resources:

ARDL R package documentation: help(package = "ARDL")
Econometrics tutorials: https://www.econometrics-with-r.org/
Time series in R: https://otexts.com/fpp3/

Practice Exercise

Try this on your own:

Load any macroeconomic dataset with 3-4 variables
Test for unit roots (ensure not I(2))
Estimate an ARDL model using auto_ardl()
Conduct bounds test for cointegration
If cointegrated:
- Estimate long-run relationship
- Estimate ECM
- Interpret the speed of adjustment
Run diagnostic tests
Visualize actual vs fitted values

Tip

Start with Pakistani data from World Bank or State Bank of Pakistan!

Questions?

Thank you!

Contact:

Email: your.email@university.edu
Office Hours: [Your hours]

Remember:

Practice makes perfect. Start with simple examples and gradually increase complexity!

References

Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326.
Pesaran, M. H., & Shin, Y. (1999). An autoregressive distributed lag modelling approach to cointegration analysis. Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial Symposium, 371-413.
Nkoro, E., & Uko, A. K. (2016). Autoregressive Distributed Lag (ARDL) cointegration technique: application and interpretation. Journal of Statistical and Econometric Methods, 5(4), 63-91.
Jordan, S., & Philips, A. Q. (2018). Cointegration testing and dynamic simulations of autoregressive distributed lag models. The Stata Journal, 18(4), 902-923.