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KARDL

Introduction

Thekardl package is an R tool for estimating symmetricand asymmetric Autoregressive Distributed Lag (ARDL) and Nonlinear ARDL(NARDL) models, designed for econometricians and researchers analyzingcointegration and dynamic relationships in time series data. It offersflexible model specifications, allowing users to include deterministicvariables, asymmetric effects for short- and long-run dynamics, andtrend components. The package supports customizable lag structures,model selection criteria (AIC, BIC, AICc, HQ), and parallel processingfor computational efficiency. Key features include:

• Flexible Formula Specification: Useasym(),asymL(), andasymS() tomodel asymmetric effects in short- and long-run dynamics, anddeterministic() for dummy variables.
• Lag Optimization: Choose between automatic lagselection ("quick","grid","grid_custom") or user-defined lags.
• Dynamic Analysis: Compute long-run coefficients,perform cointegration tests (PSS F, PSS t, Narayan, Banerjee, ECM), andtest for heteroskedasticity (ARCH).

This vignette demonstrates how to use thekardl()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 CRANrepository:

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

Alternatively, you can use thedevtools package to loaddirectly from GitHub:

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

Load the package:

library(kardl)

Estimating an AsymmetricARDL Model

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

Step 1: Data Preparation

Assumeimf_example_data contains monthly data for CPI,ER, PPI, and a COVID dummy variable. We prepare the data by ensuringproper formatting and adding the dummy variable. We retrieve data fromthe IMF’s International Financial Statistics (IFS) dataset and prepareit for analysis.

Note: Theimf_example_data is a placeholder fordemonstration purposes. You should replace it with your actual dataset.The data can be loaded byreadxl or other data importfunctions.

Step 2: Define the ModelFormula

We define the model formula using R’s formula syntax, incorporatingasymmetric effects and deterministic variables. We useasym() for variables with both short- and long-runasymmetry,deterministic() for fixed effects, andtrend for a linear time trend.

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

Step 3: Model Estimation

We estimate the ARDL model using differentmode settingsto demonstrate flexibility in lag selection. Thekardl()function supports various modes:"grid","grid_custom","quick", or a user-defined lagvector.

Usingmode = "grid"

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

# Set model optionskardl_set(criterion ="BIC",differentAsymLag =TRUE,data=imf_example_data)# Estimate model with grid modekardl_model<-kardl(data=imf_example_data,model= MyFormula,maxlag =4,mode ="grid")# View resultskardl_model# Display model summarysummary(kardl_model)

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 resultskardl_model2$properLag

Using All Variables

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

kardl_set(data=imf_example_data)kardl(model =  CPI~ .+deterministic(covid),mode ="grid")

Visualizing Lag Criteria

TheLagCriteria component contains lag combinations andtheir criterion values. We visualize these to compare model selectioncriteria (AIC, BIC, HQ).

library(dplyr)library(tidyr)library(ggplot2)# Convert LagCriteria to a data frameLagCriteria<-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 formatLagCriteria_long<- LagCriteria%>%select(-AICc)%>%pivot_longer(cols =c(AIC, BIC, HQ),names_to ="Criteria",values_to ="Value")# Find minimum valuesmin_values<- LagCriteria_long%>%group_by(Criteria)%>%slice_min(order_by = Value)%>%ungroup()# Plotggplot(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 usingkardl_longrun(), which standardizes coefficients bydividing them by the negative of the dependent variable’s long-runparameter.

# Long-run coefficientsmylong<-kardl_longrun(kardl_model)mylong

Step 5: Asymmetry Test

Theasymmetrytest() function performs Wald tests toassess 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# View long-run hypothesesast$Lhypotheses# Detailed resultssummary(ast)

Step 6: Cointegration Tests

We perform cointegration tests to assess long-term relationshipsusingpssf(),psst(),narayan(),banerjee(), andrecmt().

PSS F Bound Test

Thepssf() function tests for cointegration using thePesaran, 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,"."))cat(paste0("\nWe found '", A$Cont,"' at ", A$siglvl,"."))A$criticalValuessummary(A)

PSS t Bound Test

Thepsst() function tests the significance of the laggeddependent variable’s coefficient.

A<- kardl_model%>%psst(case =3,signif_level ="0.05")cat(paste0("The t statistic = ", A$statistic," where k = ", A$k,"."))cat(paste0("\nWe found '", A$Cont,"' at ", A$siglvl,"."))A$criticalValuessummary(A)

Narayan Test

Thenarayan() function is tailored for small samplesizes.

A<- kardl_model%>%narayan(case =3,signif_level ="0.05")cat(paste0("The F statistic = ", A$statistic," where k = ", A$k,"."))cat(paste0("\nWe found '", A$Cont,"' at ", A$siglvl,"."))A$criticalValuessummary(A)

Banerjee Test

Thebanerjee() 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$statistic,"."))cat(paste0("\nWe found '", A$Cont,"' at ", A$siglvl,"."))A$criticalValuessummary(A)

Restricted ECM Test

Therecmt() function tests for cointegration using anError Correction Model.

recmt_model<- imf_example_data%>%recmt(MyFormula,mode ="grid_custom",case =3)recmt_model# View resultssummary(recmt_model)

Step 7: ARCH Test

Thearchtest() function checks for autoregressiveconditional heteroskedasticity in the model residuals.

arch_result<-archtest(kardl_model$finalModel$model$residuals,q =2)summary(arch_result)

Step 8: CustomizingAsymmetric Variables

We demonstrate how to customize prefixes and suffixes for asymmetricvariables usingkardl_set().

# Set custom prefixes and suffixeskardl_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

Key Functions and Parameters

• kardl(data, model, maxlag, mode, ...):

  • data: A time series dataset (e.g., a data frame withCPI, ER, PPI).
  • model: A formula specifying the long-run equation,e.g.,y ~ x + z + asym(z) + asymL(x2 + x3) + asymS(x3 + x4) + deterministic(dummy1 + dummy2) + trend.Supports:
    • asym(): Asymmetric effects for both short- and long-rundynamics.
    • asymL(): Long-run asymmetric variables.
    • asymS(): Short-run asymmetric variables.
    • deterministic(): Fixed dummy variables.
    • trend: Linear time trend.
  • maxlag: Maximum number of lags (default: 4). Usesmaller values (e.g., 2) for small datasets, larger values (e.g., 8) forlong-term dependencies.
  • mode: Estimation mode:
    • "quick": Verbose output for interactive use.
    • "grid": Verbose output with lag optimization.
    • "grid_custom": Silent, efficient execution.
    • User-defined vector (e.g.,c(1, 2, 4, 5) orc(CPI = 2, ER_POS = 3, ER_NEG = 1, PPI = 3)).
  • Returns a list with components:inputs,finalModel,start_time,end_time,properLag,TimeSpan,OptLag,LagCriteria,type (“kardlmodel”).

• kardl_set(...): Configures optionslikecriterion (AIC, BIC, AICc, HQ),differentAsymLag,AsymPrefix,AsymSuffix,ShortCoef, andLongCoef. Usekardl_get() to retrieve settingsandkardl_reset() to restore defaults.

• kardl_longrun(model): Calculatesstandardized long-run coefficients, returningtype(“kardl_longrun”),coef,delta_se,results, andstarsDesc.

• asymmetrytest(model): Performs Waldtests for short- and long-run asymmetry, returningLhypotheses,Lwald,Shypotheses,Swald, andtype (“asymmetrytest”).

• pssf(model, case, signif_level):Performs the Pesaran, Shin, and Smith F Bound test for cointegration,supporting cases 1–5 and significance levels (“auto”, 0.01, 0.025, 0.05,0.1, 0.10).

• psst(model, case, signif_level):Performs the PSS t Bound test, focusing on the lagged dependentvariable’s coefficient.

• narayan(model, case, signif_level):Conducts the Narayan test for cointegration, optimized for small samples(cases 2–5).

• banerjee(model, signif_level):Performs the Banerjee cointegration test for small datasets (≤100observations).

• recmt(data, model, maxlag, mode, case, signif_level, ...):Conducts the Restricted ECM test for cointegration, with similarparameters tokardl() and case/significance leveloptions.

• archtest(resid, q): Tests for ARCHeffects in model residuals, returningtype,statistic,parameter,p.value,andFval.

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

Options

The options for the KARDL package are set by thekardl_set() function in R. The default values are set inthekardl_set list. You can change the options by using thekardl_set() function with the desired parameters. Thefollowing options are available:

The details of the options are as follows:

1. data

data is a data frame or time series object containingthe variables to be used in the model estimation. The default value isNULL, which means that the user must provide a dataset whencalling thekardl() function.

Details

Thedata parameter is essential for thekardl() function to perform model estimation. It shouldcontain all the variables specified in the model formula, including thedependent variable and any independent variables, asymmetric components,and deterministic variables defined in the formula. The trend will begenerated automatically if specified in the formula. Input data can bein the form of a data frame, tibble, or time series object (e.g.,ts,xts,zoo).

When providing thedata, ensure that: - The dataset isclean and free of missing values for the variables used in the model. -The variables are appropriately formatted (e.g., numeric for continuousvariables). - The time series data is ordered correctly, especially ifthe analysis involves lagged variables.

2. model

model is a formula object specifying the model to beestimated. The default value isNULL, which means that theuser must provide a model formula when calling thekardl()function.

Details

Themodel parameter defines the structure of the ARDL orNARDL model to be estimated. It should include the dependent variable onthe left side of the formula and the independent variables, asymmetriccomponents, deterministic variables, and trend (if applicable) on theright side. The formula can include: -asym(): To specifyvariables with asymmetric effects in both short- and long -run dynamics.-asymL(): To specify variables with asymmetric effectsonly in the long-run -dynamics. -asymS(): To specifyvariables with asymmetric effects only in the short-run -dynamics. -deterministic(): To include fixed dummy variables (e.g.,seasonal d -ummies, event dummies). -trend: To include alinear time trend in the model. When constructing themodelformula, ensure that: - All variables used in the formula are present inthedata provided. - The formula is syntactically correctand follows R’s formula conventions. - The use of asymmetric anddeterministic functions is appropriate for the research question anddata characteristics.

3. maxlag

maxlag is an integer value specifying the maximum numberof lags to be considered for the model estimation. The default value is4.

Details

Themaxlag parameter sets the upper limit for the numberof lags that thekardl() function will evaluate whenoptimizing the lag structure of the model. This is particularlyimportant when using modes like"grid" or"grid_custom", where the function systematically testsdifferent lag combinations up to the specified maximum. When choosing avalue formaxlag, consider the following: -DataFrequency: For monthly data, amaxlag of 4 isoften sufficient to capture short-term dynamics. For quarterly data, alowermaxlag ( e.g., 2) may be appropriate, while for dailydata, a highermaxlag (e.g., 8 or more) might be necessary.-Sample Size: A largermaxlag increasesthe number of parameters to estimate, which can be problematic withsmall sample sizes. Ensure that the sample size is adequate to supportthe number of lags being considered. -ModelComplexity: Highermaxlag values lead to morecomplex models, which may overfit the data. Balance the need forcapturing dynamics with the risk of overfitting. -ComputationalResources: Evaluating a large number of lag combinations can becomputationally intensive. Consider the available resources and timeconstraints when settingmaxlag.

4. mode

mode is a character string or numeric vector specifyingthe mode of the model estimation. The default value is"quick". The available options are:
-“quick”: This mode provides a fast estimation of themodel without optimizing the lags. It is suitable for initialexplorations or when the user has a predefined lag structure. -“grid”: This mode performs a grid search over allpossible lag combinations up to the specifiedmaxlag. Itprovides verbose output, including the lag criteria for eachcombination, and is useful for thorough lag optimization. -“grid_custom”: Similar to"grid", but withsilent execution. It is more efficient for large datasets or when theuser wants to avoid console output during the lag optimization process.-User-defined vector: The user can specify a customlag structure by providing a numeric vector (e.g.,c(1, 2, 4, 5)) or a named vector (e.g.,c(CPI = 2, ER_POS = 3, ER_NEG = 1, PPI = 3)). This allowsfor complete control over the lag selection process.

Details

Themode parameter determines how thekardl() function approaches the estimation of the ARDL orNARDL model. Each mode has its advantages and is suited to differentscenarios: - Use"quick" for rapid assessments when the lagstructure is already known or when computational speed is a priority. -Use"grid" for comprehensive lag optimization, especiallywhen the optimal lag structure is unknown. This mode is ideal forexploratory analysis and model selection. - Use"grid_custom" for efficient lag optimization withoutconsole output, particularly for large datasets or when running multiplemodels in batch mode. - Use a user-defined vector when the user hasspecific knowledge about the appropriate lags for each variable,allowing for tailored model specifications. When using"grid" or"grid_custom", ensure that themaxlag parameter is set appropriately to balance thethoroughness of the search with computational feasibility.

5. criterion

criterion is a character string specifying the criterionto be used for selecting the optimal lags. The default value is"AIC". The available options are:

• User defined function: For detailed information onthe model selection criteria used in the methods, see the documentationfor themodelCriterion function.
• AIC: Akaike Information Criterion (AIC). Thiscriterion balances model fit and complexity, favoring models with alower AIC value. It is widely used in time series analysis and modelselection.
• BIC: Schwarz Criterion (SC), also known as theBayesian Information Criterion (BIC). This criterion imposes a stricterpenalty for model complexity compared to AIC, often leading to simplermodels when data size is large.
• AICc: Corrected Akaike Information Criterion. Thisis a modification of AIC that accounts for small sample sizes. It ismore reliable than AIC when the number of observations is limited.
• HQ: Hannan-Quinn Criterion. This criterion issimilar to AIC and BIC but uses a logarithmic penalty term that growsmore slowly than BIC. It is often used in econometric applications.

Details

For detailed information on the model selection criteria used in themethods, see the documentation for themodelCriterionfunction.

The choice of the criterion can significantly impact the selected laglength and, consequently, the performance of the model. Each criterionhas its strengths and is suited to specific scenarios:

• Use"AIC" for general purposes, especially whenprioritizing a good fit over simplicity.
• Use"BIC" when you prefer a more parsimonious model,particularly with large datasets.
• Use"AICc" when working with small sample sizes toavoid overfitting.
• Use"HQ" for a balance between AIC and BIC, often ineconometrics or time series models.

Ensure that the selected criterion aligns with the goals of youranalysis and the characteristics of your data.

Examples

Default criterion (AIC)

kardl_set(criterion ="AIC")kardl(data, MyFormula)

Using BIC for lag selection

kardl_set(criterion ="BIC")kardl(data, MyFormula)

Using AICc for small samplesizes

kardl_set(criterion ="AICc")data%>%kardl(MyFormula)

Using HQ criterion

kardl_set(criterion ="HQ")kardl(data, MyFormula)

6. differentAsymLag

differentAsymLag is a logical value (TRUEorFALSE) indicating whether positive and negativeasymmetric variables should be assigned different lags during theestimation process. The default value isFALSE, meaningthat both positive and negative components will use the same lag.

Details

Asymmetric decomposition separates a variable into its positive andnegative changes. In some models, it may be desirable to assigndifferent lags to these components to capture distinct dynamicbehaviors. SettingdifferentAsymLag = TRUE allows thefunction to optimize lags for positive and negative componentsindependently. WhendifferentAsymLag = FALSE, bothcomponents will share the same lag.

This parameter is particularly useful when:

• Positive and negative changes in a variable are expected to havedifferent impacts over time.
• The user wants more flexibility in modeling asymmetricresponses.

Attention!

• WhendifferentAsymLag = TRUE, ensure that the model hassufficient data to estimate separate lags reliably.
• For models with limited observations or a high number of variables,differentAsymLag = FALSE may be more robust andcomputationally efficient.

Examples

kadrl_set(differentAsymLag =FALSE)kardl(data, MyFormula)

kardl_set(differentAsymLag =TRUE)kardl(data, MyFormula)

kardl_set(AsymPrefix =c())

kardl_set(AsymPrefix =c("POS_","NEG_"))

kardl_set(AsymPrefix =c("Change_","Fall_"),AsymSuffix =c("_High","_Low"))

kardl_set(batch ="1/1")kardl(data, MyFormula)

kardl_set(batch ="2/4")kardl(data, MyFormula)

kardl_set(batch ="3/6")kardl(data, MyFormula)