Movatterモバイル変換

Estimating an Asymmetric ARDL 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 Model Formula

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.

Using`mode = "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

## ==============================================================## 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 summarysummary(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 resultskardl_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 thedependent 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

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

## ___________________________________________________## 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

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

## 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 hypothesesast$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 resultssummary(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 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,"."))

## 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

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,"."))

## 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

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,"."))

## 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

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,"."))

## 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

Therecmt() function tests for cointegration using anError 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 resultssummary(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

Thearchtest() function checks for autoregressiveconditional 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 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

##         CPI  asyP_ER_PP  asyN_ER_NN asyP_PPI_PP asyN_PPI_NN ##           2           1           0           3           0

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.

Movatterモバイル変換

Introduction to kardl

Huseyin Karamelikli and Huseyin Utku Demir

2025-10-06

Introduction

Installation

Estimating an Asymmetric ARDL Model

Step 1: Data Preparation

Step 2: Define the Model Formula

Step 3: Model Estimation

Using`mode = "grid"`

Using User-Defined Lags

Using All Variables

Visualizing Lag Criteria

Step 4: Long-Run Coefficients

Step 5: Asymmetry Test

Step 6: Cointegration Tests

PSS F Bound Test

PSS t Bound Test

Narayan Test

Banerjee Test

Restricted ECM Test

Step 7: ARCH Test

Step 8: Customizing Asymmetric Variables

Key Functions and Parameters

Conclusion

Movatterモバイル変換

Introduction to kardl

Huseyin Karamelikli and Huseyin Utku Demir

2025-10-06

Introduction

Installation

Estimating an Asymmetric ARDL Model

Step 1: Data Preparation

Step 2: Define the Model Formula

Step 3: Model Estimation

Usingmode = "grid"

Using User-Defined Lags

Using All Variables

Visualizing Lag Criteria

Step 4: Long-Run Coefficients

Step 5: Asymmetry Test

Step 6: Cointegration Tests

PSS F Bound Test

PSS t Bound Test

Narayan Test

Banerjee Test

Restricted ECM Test

Step 7: ARCH Test

Step 8: Customizing Asymmetric Variables

Key Functions and Parameters

Conclusion

Using`mode = "grid"`