Introduction to kardl

Huseyin Karamelikli and Huseyin Utku Demir

2025-10-06

Introduction

The kardl package is an R tool for estimating symmetric and asymmetric Autoregressive Distributed Lag (ARDL) and Nonlinear ARDL (NARDL) models, designed for econometricians and researchers analyzing cointegration and dynamic relationships in time series data. It offers flexible model specifications, allowing users to include deterministic variables, asymmetric effects for short- and long-run dynamics, and trend components. The package supports customizable lag structures, model selection criteria (AIC, BIC, AICc, HQ), and parallel processing for computational efficiency. Key features include:

This vignette demonstrates how to use the kardl() function to estimate an asymmetric ARDL model, perform diagnostic tests, and visualize results, using economic data from Turkey.

Installation

kardl in R can easily be installed from its CRAN repository:

install.packages("kardl")
library(kardl)

Alternatively, you can use the devtools package to load directly from GitHub:

# Install required packages
install.packages(c("stats", "msm", "lmtest", "nlWaldTest", "car", "strucchange", "utils"))
# Install kardl from GitHub
install.packages("devtools")
devtools::install_github("karamelikli/kardl")

Load the package:

library(kardl)

Estimating an Asymmetric ARDL Model

This example estimates an asymmetric ARDL model to analyze the dynamics of exchange rate pass-through to domestic prices in Turkey, using a sample dataset (imf_example_data) with variables for Consumer Price Index (CPI), Exchange Rate (ER), Producer Price Index (PPI), and a COVID-19 dummy variable.

Step 1: Data Preparation

Assume imf_example_data contains monthly data for CPI, ER, PPI, and a COVID dummy variable. We prepare the data by ensuring proper formatting and adding the dummy variable. We retrieve data from the IMF’s International Financial Statistics (IFS) dataset and prepare it for analysis.

Note: The imf_example_data is a placeholder for demonstration purposes. You should replace it with your actual dataset. The data can be loaded by readxl or other data import functions.

Step 2: Define the Model Formula

We define the model formula using R’s formula syntax, incorporating asymmetric effects and deterministic variables. We use asym() for variables with both short- and long-run asymmetry, deterministic() for fixed effects, and trend for a linear time trend.

# Define the model formula
MyFormula <- CPI ~ ER + PPI + asym(ER + PPI) + deterministic(covid) + trend

Step 3: Model Estimation

We estimate the ARDL model using different mode settings to demonstrate flexibility in lag selection. The kardl() function supports various modes: "grid", "grid_custom", "quick", or a user-defined lag vector.

Using mode = "grid"

The "grid" mode evaluates all lag combinations up to maxlag and provides console feedback.

# Set model options
kardl_set(criterion = "BIC", differentAsymLag = TRUE, data=imf_example_data)
# Estimate model with grid mode
kardl_model <- kardl(data=imf_example_data,model= MyFormula, maxlag = 4, mode = "grid")
# View results
kardl_model
## ==============================================================
## KARDL Results
## The lags of the model is:
##                       AIC               BIC              AICc                HQ
## lag           2,1,0,3,0         1,1,0,3,0         1,0,0,0,0         2,1,0,3,0
## value -5.55341824428271 -5.39678297453557 -4.65456961489876 -5.49030935302091
## 
## The estimated model's formula is:
## L0.d.CPI~L1.CPI+L1.ER_POS+L1.ER_NEG+L1.PPI_POS+L1.PPI_NEG+L1.d.CPI+L0.d.ER_POS+L1.d.ER_POS+L0.d.ER_NEG+L0.d.PPI_POS+L1.d.PPI_POS+L2.d.PPI_POS+L3.d.PPI_POS+L0.d.PPI_NEG+covid+trend
## 
## The coeffients estimation's results are:
##   (Intercept)        L1.CPI     L1.ER_POS     L1.ER_NEG    L1.PPI_POS 
## -0.0386633545 -0.0121524327  0.0110491088  0.0252653399  0.0517030999 
##    L1.PPI_NEG      L1.d.CPI   L0.d.ER_POS   L1.d.ER_POS   L0.d.ER_NEG 
##  0.0451043051  0.3340367351  0.1111220281  0.0937503220 -0.0026590963 
##  L0.d.PPI_POS  L1.d.PPI_POS  L2.d.PPI_POS  L3.d.PPI_POS  L0.d.PPI_NEG 
##  0.0474101941  0.0021468066 -0.0519928228 -0.0517409420  0.0057550123 
##         covid         trend 
##  0.0033274526 -0.0002952073 
## ===============================================================
# Display model summary
summary(kardl_model)
## ==============================================================
## KARDL Summary
## 
## 
## Call:
## lm(formula = as.formula(fmodel), data = data)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.050478 -0.008129 -0.000904  0.006918  0.102836 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.0386634  0.0227251  -1.701 0.089572 .  
## L1.CPI       -0.0121524  0.0047287  -2.570 0.010494 *  
## L1.ER_POS     0.0110491  0.0051559   2.143 0.032652 *  
## L1.ER_NEG     0.0252653  0.0079538   3.177 0.001594 ** 
## L1.PPI_POS    0.0517031  0.0096244   5.372 1.25e-07 ***
## L1.PPI_NEG    0.0451043  0.0107631   4.191 3.35e-05 ***
## L1.d.CPI      0.3340367  0.0399191   8.368 7.54e-16 ***
## L0.d.ER_POS   0.1111220  0.0180412   6.159 1.63e-09 ***
## L1.d.ER_POS   0.0937503  0.0181990   5.151 3.88e-07 ***
## L0.d.ER_NEG  -0.0026591  0.0474028  -0.056 0.955291    
## L0.d.PPI_POS  0.0474102  0.0160401   2.956 0.003284 ** 
## L1.d.PPI_POS  0.0021468  0.0144158   0.149 0.881684    
## L2.d.PPI_POS -0.0519928  0.0143424  -3.625 0.000322 ***
## L3.d.PPI_POS -0.0517409  0.0137160  -3.772 0.000183 ***
## L0.d.PPI_NEG  0.0057550  0.0135155   0.426 0.670452    
## covid         0.0033275  0.0050899   0.654 0.513621    
## trend        -0.0002952  0.0002472  -1.194 0.233112    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01483 on 448 degrees of freedom
##   (5 observations deleted due to missingness)
## Multiple R-squared:  0.6516, Adjusted R-squared:  0.6392 
## F-statistic: 52.38 on 16 and 448 DF,  p-value: < 2.2e-16
## 
## ===============================================================

Using User-Defined Lags

Specify custom lags to bypass automatic lag selection:

kardl_model2 <- kardl(data=imf_example_data, MyFormula, mode = c(2, 1, 1, 3, 0))
# View results
kardl_model2$properLag
##     CPI  ER_POS  ER_NEG PPI_POS PPI_NEG 
##       2       1       1       3       0

Using All Variables

Use the . operator to include all variables except the dependent variable:

kardl_set(data=imf_example_data)
kardl(model =  CPI ~ . + deterministic(covid), mode = "grid")
## ==============================================================
## KARDL Results
## The lags of the model is:
##                       AIC               BIC              AICc                HQ
## lag               1,2,3             1,0,0             1,0,0             1,2,0
## value -3.30318556115875 -3.22112627740211 -2.97831477517044 -3.26494116658086
## 
## The estimated model's formula is:
## L0.d.ER~L1.ER+L1.CPI+L1.PPI+L1.d.ER+L0.d.CPI+L0.d.PPI+covid
## 
## The coeffients estimation's results are:
##  (Intercept)        L1.ER       L1.CPI       L1.PPI      L1.d.ER     L0.d.CPI 
## -0.069790957 -0.025177677  0.022530297 -0.003051024  0.078941014  0.742452528 
##     L0.d.PPI        covid 
## -0.019897958  0.020270715 
## ===============================================================

Visualizing Lag Criteria

The LagCriteria component contains lag combinations and their criterion values. We visualize these to compare model selection criteria (AIC, BIC, HQ).

library(dplyr)
library(tidyr)
library(ggplot2)
# Convert LagCriteria to a data frame
LagCriteria <- as.data.frame(kardl_model[["LagCriteria"]])
colnames(LagCriteria) <- c("lag", "AIC", "BIC", "AICc", "HQ")
LagCriteria <- LagCriteria %>% mutate(across(c(AIC, BIC, HQ), as.numeric))

# Pivot to long format
LagCriteria_long <- LagCriteria %>%
  select(-AICc) %>%
  pivot_longer(cols = c(AIC, BIC, HQ), names_to = "Criteria", values_to = "Value")

# Find minimum values
min_values <- LagCriteria_long %>%
  group_by(Criteria) %>%
  slice_min(order_by = Value) %>%
  ungroup()

# Plot
ggplot(LagCriteria_long, aes(x = lag, y = Value, color = Criteria, group = Criteria)) +
  geom_line() +
  geom_point(data = min_values, aes(x = lag, y = Value), color = "red", size = 3, shape = 8) +
  geom_text(data = min_values, aes(x = lag, y = Value, label = lag), vjust = 1.5, color = "black", size = 3.5) +
  labs(title = "Lag Criteria Comparison", x = "Lag Configuration", y = "Criteria Value") +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

Step 4: Long-Run Coefficients

We calculate long-run coefficients using kardl_longrun(), which standardizes coefficients by dividing them by the negative of the dependent variable’s long-run parameter.

# Long-run coefficients
mylong <- kardl_longrun(kardl_model)
mylong
## ___________________________________________________
## Adjusted KARDL Results (long-run results)
##                 Estimate Std. Error   t value     Pr(>|t|)    
## (Intercept) -3.1815321  0.6750740 -4.712864 3.265356e-06 ***
## L1.ER_POS    0.9092096  0.2083414  4.364038 1.586695e-05 ***
## L1.ER_NEG    2.0790356  0.5669276  3.667197 2.746682e-04 ***
## L1.PPI_POS   4.2545473  1.7282246  2.461802 1.419976e-02   *
## L1.PPI_NEG   3.7115453  1.3351815  2.779806 5.668108e-03  **
## ___________________________________________________
##   Signif. codes: 0.001 = ***, 0.01 = **, 0.05 = *, 0.1 = .
## ___________________________________________________

Step 5: Asymmetry Test

The asymmetrytest() function performs Wald tests to assess short- and long-run asymmetry in the model.

ast <- imf_example_data %>% kardl(CPI ~ ER + PPI + asym(ER + PPI) + deterministic(covid) + trend, mode = c(1, 2, 3, 0, 1)) %>% asymmetrytest()
ast
## Asymmetries in the long run
## 
##             F         P
## ER  2.1649064 0.1418984
## PPI 0.7152717 0.3981528
## ________________________________
## Asymmetries in the short run
## 
##             F         P
## ER  1.6798112 0.1956198
## PPI 0.7582018 0.3843603
# View long-run hypotheses
ast$Lhypotheses
## $H0
## [1] "- Coef(L1.ER_POS)/Coef(L1.CPI) = - Coef(L1.ER_NEG)/Coef(L1.CPI)"  
## [2] "- Coef(L1.PPI_POS)/Coef(L1.CPI) = - Coef(L1.PPI_NEG)/Coef(L1.CPI)"
## 
## $H1
## [1] "- Coef(L1.ER_POS)/Coef(L1.CPI) ≠ - Coef(L1.ER_NEG)/Coef(L1.CPI)"  
## [2] "- Coef(L1.PPI_POS)/Coef(L1.CPI) ≠ - Coef(L1.PPI_NEG)/Coef(L1.CPI)"
# Detailed results
summary(ast)
## Asymmetries in the long run
## H0: - Coef(L1.ER_POS)/Coef(L1.CPI) = - Coef(L1.ER_NEG)/Coef(L1.CPI)
## H1: - Coef(L1.ER_POS)/Coef(L1.CPI) ≠ - Coef(L1.ER_NEG)/Coef(L1.CPI)
## 
## H0: - Coef(L1.PPI_POS)/Coef(L1.CPI) = - Coef(L1.PPI_NEG)/Coef(L1.CPI)
## H1: - Coef(L1.PPI_POS)/Coef(L1.CPI) ≠ - Coef(L1.PPI_NEG)/Coef(L1.CPI)
## 
##             F         P df1 df2
## ER  2.1649064 0.1418984   1 446
## PPI 0.7152717 0.3981528   1 446
## _____________________________
## 
## Asymmetries in the long run
## H0: Coef(L0.d.ER_POS) + Coef(L1.d.ER_POS) + Coef(L2.d.ER_POS) = Coef(L0.d.ER_NEG) + Coef(L1.d.ER_NEG) + Coef(L2.d.ER_NEG) + Coef(L3.d.ER_NEG)
## H1: Coef(L0.d.ER_POS) + Coef(L1.d.ER_POS) + Coef(L2.d.ER_POS) ≠ Coef(L0.d.ER_NEG) + Coef(L1.d.ER_NEG) + Coef(L2.d.ER_NEG) + Coef(L3.d.ER_NEG)
## 
## H0: Coef(L0.d.PPI_POS) = Coef(L0.d.PPI_NEG) + Coef(L1.d.PPI_NEG)
## H1: Coef(L0.d.PPI_POS) ≠ Coef(L0.d.PPI_NEG) + Coef(L1.d.PPI_NEG)
## 
##             F         P DF    Sum.of.Sq ResDF1 ResDF2      RSS1      RSS2
## ER  1.6798112 0.1956198  1 0.0003860722    447    446 0.1028906 0.1025045
## PPI 0.7582018 0.3843603  1 0.0001742581    447    446 0.1026788 0.1025045

Step 6: Cointegration Tests

We perform cointegration tests to assess long-term relationships using pssf(), psst(), narayan(), banerjee(), and recmt().

PSS F Bound Test

The pssf() function tests for cointegration using the Pesaran, Shin, and Smith F Bound test.

A <- kardl_model %>% pssf(case = 3, signif_level = "0.05")
cat(paste0("The F statistic = ", A$statistic, " where k = ", A$k, "."))
## The F statistic = 11.2705611215356 where k = 2.
cat(paste0("\nWe found '", A$Cont, "' at ", A$siglvl, "."))
## 
## We found 'Cointegration' at 0.05.
A$criticalValues
##      k  0.10L  0.10U  0.05L  0.05U 0.025L 0.025U  0.01L  0.01U 
##   2.00   4.19   5.06   4.87   5.85   5.49   6.59   6.34   7.52
summary(A)
## Method: PesaranF. Case: 5
## H0: Coef(L1.CPI) = Coef(L1.ER_POS) = Coef(L1.ER_NEG) = Coef(L1.PPI_POS) = Coef(L1.PPI_NEG) = 0
## H1: Coef(L1.CPI) ≠ Coef(L1.ER_POS) ≠ Coef(L1.ER_NEG) ≠ Coef(L1.PPI_POS) ≠ Coef(L1.PPI_NEG)≠ 0
## The F statistics = 11.2705611215356 where k = 2. 
## The proboblity is = 0.0000000003
## The results: There is Cointegration
## Significance level: 0.05
## ___________________________________________________
## Critical values are as follows:
##      k  0.10L  0.10U  0.05L  0.05U 0.025L 0.025U  0.01L  0.01U 
##   2.00   4.19   5.06   4.87   5.85   5.49   6.59   6.34   7.52

PSS t Bound Test

The psst() function tests the significance of the lagged dependent variable’s coefficient.

A <- kardl_model %>% psst(case = 3, signif_level = "0.05")
cat(paste0("The t statistic = ", A$statistic, " where k = ", A$k, "."))
## The t statistic = -2.56993172541683 where k = 4.
cat(paste0("\nWe found '", A$Cont, "' at ", A$siglvl, "."))
## 
## We found 'No Cointegration' at 0.05.
A$criticalValues
##      k  0.10L  0.10U  0.05L  0.05U 0.025L 0.025U  0.01L  0.01U 
##   3.00  -3.13  -3.84  -3.41  -4.16  -3.65  -4.42  -3.96  -4.73
summary(A)
## Method: Pesarant. Case: 5
## H0: Coef(L1.CPI) = 0
## H1: Coef(L1.CPI)≠ 0
## The t statistics = -2.56993172541683 where k = 4.
## The results: There is No Cointegration
## Significance level: 0.05
## ___________________________________________________
## Critical values are as follows:
##      k  0.10L  0.10U  0.05L  0.05U 0.025L 0.025U  0.01L  0.01U 
##   3.00  -3.13  -3.84  -3.41  -4.16  -3.65  -4.42  -3.96  -4.73

Narayan Test

The narayan() function is tailored for small sample sizes.

A <- kardl_model %>% narayan(case = 3, signif_level = "0.05")
cat(paste0("The F statistic = ", A$statistic, " where k = ", A$k, "."))
## The F statistic = 11.2705611215356 where k = 2.
cat(paste0("\nWe found '", A$Cont, "' at ", A$siglvl, "."))
## 
## We found 'Cointegration' at 0.05.
A$criticalValues
##      k  0.10L  0.10U  0.05L  0.05U 0.025L 0.025U  0.01L  0.01U 
##  2.000  4.307  5.223  5.067  6.103     NA     NA  6.730  8.053
summary(A)
## Method: Narayan. Case: 5
## H0: Coef(L1.CPI) = Coef(L1.ER_POS) = Coef(L1.ER_NEG) = Coef(L1.PPI_POS) = Coef(L1.PPI_NEG) = 0
## H1: Coef(L1.CPI) ≠ Coef(L1.ER_POS) ≠ Coef(L1.ER_NEG) ≠ Coef(L1.PPI_POS) ≠ Coef(L1.PPI_NEG)≠ 0
## The F statistics = 11.2705611215356 where k = 2.
## The proboblity is = 0.0000000003
## The results: There is Cointegration
## Significance level: 0.05
## ___________________________________________________
## Critical values are as follows:
##      k  0.10L  0.10U  0.05L  0.05U 0.025L 0.025U  0.01L  0.01U 
##  2.000  4.307  5.223  5.067  6.103     NA     NA  6.730  8.053

Banerjee Test

The banerjee() function is designed for small datasets (≤100 observations).

A <- kardl_model %>% banerjee(signif_level = "0.05")
cat(paste0("The ECM parameter = ", A$coef, ", k = ", A$k, ", t statistic = ", A$statistic, "."))
## The ECM parameter = -0.00867941225158909, k = 4, t statistic = -2.07931701669587.
cat(paste0("\nWe found '", A$Cont, "' at ", A$siglvl, "."))
## 
## We found 'No Cointegration' at 0.05.
A$criticalValues
##  0.01  0.05  0.10  0.25 
## -5.07 -4.38 -4.02 -3.46
summary(A)
## Method: Banerjee. Case: NULL
## H0: Coef(L1.CPI) = 0
## H1: Coef(L1.CPI)≠ 0
## The coeffient of ECM is -0.00867941225158909 , t=-2.07931701669587 k=4
## The results: There is No Cointegration
## Significance level: 0.05
## ___________________________________________________
## Critical values are as follows:
##  0.01  0.05  0.10  0.25 
## -5.07 -4.38 -4.02 -3.46

Restricted ECM Test

The recmt() function tests for cointegration using an Error Correction Model.

recmt_model <- imf_example_data %>% recmt(MyFormula, mode = "grid_custom", case = 3)
recmt_model
## 
## RESM test. Case: 5
## The t statistics = 3.73402572442152 where k = 4.
## 
## Warnings:
## Warning: Trend is used in the model. The case was set to 5.
# View results
summary(recmt_model)
## Method: recmt. Case: 5
## H0: Coef(EcmRes) = 0
## H1: Coef(EcmRes)≠ 0
## Attention: This function was performed in two steps.
## In the first step, the ECM was calculated and in the second step, the PSS test was performed.
## 
## The coeffient of ECM is 0.00261085240978681 , t=3.73402572442152 k=4
## The results: There is No Cointegration
## Significance level: 
## ___________________________________________________
## Critical values are as follows:
##      k  0.10L  0.10U  0.05L  0.05U 0.025L 0.025U  0.01L  0.01U 
##   3.00  -3.13  -3.84  -3.41  -4.16  -3.65  -4.42  -3.96  -4.73

Step 7: ARCH Test

The archtest() function checks for autoregressive conditional heteroskedasticity in the model residuals.

arch_result <- archtest(kardl_model$finalModel$model$residuals, q = 2)
summary(arch_result)
## ARCH test
## 
## F = 8.372629, p-value = 0.0002681842, df1 = 2, df2 = 460
## 
## 
## Call:
## lm(formula = as.formula(paste0("resid~", paste("resid", 1:q, 
##     sep = "", collapse = "+"))), data = ndata)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0006394 -0.0001738 -0.0001317 -0.0000137  0.0103827 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.747e-04  3.091e-05   5.652 2.79e-08 ***
## resid1       1.902e-01  4.660e-02   4.081 5.28e-05 ***
## resid2      -2.159e-02  4.649e-02  -0.464    0.643    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0006081 on 460 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.03512,    Adjusted R-squared:  0.03093 
## F-statistic: 8.373 on 2 and 460 DF,  p-value: 0.0002682

Step 8: Customizing Asymmetric Variables

We demonstrate how to customize prefixes and suffixes for asymmetric variables using kardl_set().

# Set custom prefixes and suffixes
kardl_reset()
kardl_set(AsymPrefix = c("asyP_", "asyN_"), AsymSuffix = c("_PP", "_NN"))
kardl_custom <- kardl(data=imf_example_data, MyFormula, mode = "grid_custom")
kardl_custom$properLag
##         CPI  asyP_ER_PP  asyN_ER_NN asyP_PPI_PP asyN_PPI_NN 
##           2           1           0           3           0

Key Functions and Parameters

For detailed documentation, use ?kardl, ?kardl_set, ?kardl_longrun, ?asymmetrytest, ?pssf, ?psst, ?narayan, ?banerjee, ?recmt, or ?archtest.

Conclusion

The kardl package is a versatile tool for econometric analysis, offering robust support for symmetric and asymmetric ARDL/NARDL modeling, cointegration tests, stability diagnostics, and heteroskedasticity checks. Its flexible formula specification, lag optimization, and support for parallel processing make it ideal for studying complex economic relationships. For more information, visit https://github.com/karamelikli/kardl or contact the authors at hakperest@gmail.com.