Autoregressive Distributed Lag (ARDL) Model
Ibikunle Gabriel
2026-03-22
What is an ARDL Model?
Loading required packages
What is ARDL? Autoregressive Distributed Lag (ARDL) is a time series model used to analyze both short-run and long-run relationships between variables.
Breaking it down Autoregressive(AR) means the dependent variable depends on its own past values.
Distributed Lag (DL) means the dependent variable depends on past values of other variables. The model is ideal for use when there is mix of I(0) and I(1) variables in the data.
Set and confirm the working directory
getwd()## [1] "C:/Users/Hp/Documents/ARDL"load the data into the R environment
data <- read.csv("ECON DEVELOPMENT DATA.csv")# Convert to time series object
# Assuming you have quarterly or monthly data - adjust frequency accordingly
ts_data <- ts(data, start = c(1994, 1), frequency = 1) # adjust as needed# Extract variables
ECNDEV <- ts_data[, "ECNDEV"]
INF <- ts_data[, "INF"]
K <- ts_data[, "K"]
L <- ts_data[, "L"]
EDU <- ts_data[, "EDU"]
HEA <- ts_data[, "HEA"]
UNEMP <- ts_data[, "UNEMP"]# 1. UNIT ROOT TESTS (Stationarity Testing)
cat("\n", "========== UNIT ROOT TESTS ==========", "\n")##
## ========== UNIT ROOT TESTS ==========# Function to perform multiple unit root tests
unit_root_tests <- function(series, series_name) {
cat("\n", "-------------------------------------", "\n")
cat("Testing for:", series_name, "\n")
cat("-------------------------------------", "\n")
# Remove NAs
series_clean <- na.omit(series)
# ADF Test
adf_level <- adf.test(series_clean, alternative = "stationary")
adf_firstdiff <- adf.test(diff(series_clean), alternative = "stationary")
# Phillips-Perron Test
pp_level <- pp.test(series_clean)
pp_firstdiff <- pp.test(diff(series_clean))
# KPSS Test
kpss_level <- kpss.test(series_clean)
kpss_firstdiff <- kpss.test(diff(series_clean))
# Results summary
results <- data.frame(
Test = c("ADF Level", "ADF First Diff",
"PP Level", "PP First Diff",
"KPSS Level", "KPSS First Diff"),
Statistic = round(c(adf_level$statistic, adf_firstdiff$statistic,
pp_level$statistic, pp_firstdiff$statistic,
kpss_level$statistic, kpss_firstdiff$statistic), 4),
P_Value = round(c(adf_level$p.value, adf_firstdiff$p.value,
pp_level$p.value, pp_firstdiff$p.value,
kpss_level$p.value, kpss_firstdiff$p.value), 4),
Result = c(
ifelse(adf_level$p.value < 0.05, "Stationary", "Non-Stationary"),
ifelse(adf_firstdiff$p.value < 0.05, "Stationary", "Non-Stationary"),
ifelse(pp_level$p.value < 0.05, "Stationary", "Non-Stationary"),
ifelse(pp_firstdiff$p.value < 0.05, "Stationary", "Non-Stationary"),
ifelse(kpss_level$p.value > 0.05, "Stationary", "Non-Stationary"),
ifelse(kpss_firstdiff$p.value > 0.05, "Stationary", "Non-Stationary")
)
)
print(results)
# Determine integration order
cat("\nConclusion for", series_name, ":")
if(adf_level$p.value < 0.05 && pp_level$p.value < 0.05 && kpss_level$p.value > 0.05) {
cat(" I(0) - Stationary at level\n")
} else if(adf_firstdiff$p.value < 0.05 && pp_firstdiff$p.value < 0.05) {
cat(" I(1) - Stationary at first difference\n")
} else {
cat(" I(2) or higher - Further testing needed\n")
}
}
# Run unit root tests for all variables
variables <- list(ECNDEV = ECNDEV, INF = INF, K = K, L = L,
EDU = EDU, HEA = HEA, UNEMP = UNEMP)
for(var_name in names(variables)) {
unit_root_tests(variables[[var_name]], var_name)
}##
## -------------------------------------
## Testing for: ECNDEV
## -------------------------------------## Test Statistic P_Value Result
## 1 ADF Level -1.2585 0.8576 Non-Stationary
## 2 ADF First Diff -2.9932 0.1919 Non-Stationary
## 3 PP Level -4.2032 0.8647 Non-Stationary
## 4 PP First Diff -31.7284 0.0100 Stationary
## 5 KPSS Level 0.8693 0.0100 Non-Stationary
## 6 KPSS First Diff 0.3074 0.1000 Stationary
##
## Conclusion for ECNDEV : I(2) or higher - Further testing needed
##
## -------------------------------------
## Testing for: INF
## -------------------------------------## Test Statistic P_Value Result
## 1 ADF Level -2.4361 0.4051 Non-Stationary
## 2 ADF First Diff -4.2036 0.0153 Stationary
## 3 PP Level -8.0951 0.6079 Non-Stationary
## 4 PP First Diff -36.6904 0.0100 Stationary
## 5 KPSS Level 0.2843 0.1000 Stationary
## 6 KPSS First Diff 0.2848 0.1000 Stationary
##
## Conclusion for INF : I(1) - Stationary at first difference
##
## -------------------------------------
## Testing for: K
## -------------------------------------## Test Statistic P_Value Result
## 1 ADF Level 0.2420 0.9900 Non-Stationary
## 2 ADF First Diff -0.3781 0.9807 Non-Stationary
## 3 PP Level 7.3655 0.9900 Non-Stationary
## 4 PP First Diff -19.8693 0.0290 Stationary
## 5 KPSS Level 0.7927 0.0100 Non-Stationary
## 6 KPSS First Diff 0.6857 0.0148 Non-Stationary
##
## Conclusion for K : I(2) or higher - Further testing needed
##
## -------------------------------------
## Testing for: L
## -------------------------------------## Test Statistic P_Value Result
## 1 ADF Level -2.7132 0.2987 Non-Stationary
## 2 ADF First Diff -1.5153 0.7584 Non-Stationary
## 3 PP Level -9.8984 0.4889 Non-Stationary
## 4 PP First Diff -14.3069 0.1952 Non-Stationary
## 5 KPSS Level 0.3606 0.0941 Stationary
## 6 KPSS First Diff 0.1330 0.1000 Stationary
##
## Conclusion for L : I(2) or higher - Further testing needed
##
## -------------------------------------
## Testing for: EDU
## -------------------------------------## Test Statistic P_Value Result
## 1 ADF Level -3.3645 0.0810 Non-Stationary
## 2 ADF First Diff -2.4270 0.4089 Non-Stationary
## 3 PP Level -3.8646 0.8870 Non-Stationary
## 4 PP First Diff -9.5376 0.5111 Non-Stationary
## 5 KPSS Level 1.0697 0.0100 Non-Stationary
## 6 KPSS First Diff 0.1316 0.1000 Stationary
##
## Conclusion for EDU : I(2) or higher - Further testing needed
##
## -------------------------------------
## Testing for: HEA
## -------------------------------------## Test Statistic P_Value Result
## 1 ADF Level -1.4735 0.7750 Non-Stationary
## 2 ADF First Diff -3.3139 0.0886 Non-Stationary
## 3 PP Level -10.4759 0.4508 Non-Stationary
## 4 PP First Diff -25.0914 0.0100 Stationary
## 5 KPSS Level 1.0075 0.0100 Non-Stationary
## 6 KPSS First Diff 0.1352 0.1000 Stationary
##
## Conclusion for HEA : I(2) or higher - Further testing needed
##
## -------------------------------------
## Testing for: UNEMP
## -------------------------------------## Test Statistic P_Value Result
## 1 ADF Level -2.7132 0.2987 Non-Stationary
## 2 ADF First Diff -1.5153 0.7584 Non-Stationary
## 3 PP Level -9.8984 0.4889 Non-Stationary
## 4 PP First Diff -14.3069 0.1952 Non-Stationary
## 5 KPSS Level 0.3606 0.0941 Stationary
## 6 KPSS First Diff 0.1330 0.1000 Stationary
##
## Conclusion for UNEMP : I(2) or higher - Further testing neededOPTIMAL LAG SELECTION
cat("\n", "========== OPTIMAL LAG SELECTION ==========", "\n")##
## ========== OPTIMAL LAG SELECTION ==========# Create a dataframe with all variables
ardl_data <- data.frame(
ECNDEV = as.numeric(ECNDEV),
INF = as.numeric(INF),
K = as.numeric(K),
L = as.numeric(L),
EDU = as.numeric(EDU),
HEA = as.numeric(HEA),
UNEMP = as.numeric(UNEMP)
)# Remove any rows with NA
ardl_data <- na.omit(ardl_data)# Function to select optimal lags using ARDL package
max_lags <- 4 # Adjust based on your data frequency and sample size
# Try different lag combinations and select based on AIC
aic_values <- data.frame(matrix(NA, nrow = max_lags, ncol = max_lags))
bic_values <- data.frame(matrix(NA, nrow = max_lags, ncol = max_lags))
cat("\nSearching for optimal lags...\n")##
## Searching for optimal lags...for(i in 1:max_lags) {
for(j in 1:max_lags) {
tryCatch({
# Estimate ARDL model with different lags
ardl_temp <- ardl(
ECNDEV ~ INF + K + L + EDU + HEA + UNEMP,
data = ardl_data,
order = c(i, j, j, j, j, j, j) # Different lags for dependent and independent variables
)
aic_values[i, j] <- AIC(ardl_temp)
bic_values[i, j] <- BIC(ardl_temp)
}, error = function(e) {
# Skip if model cannot be estimated
})
}
}# Find minimum AIC and BIC
min_aic <- which(aic_values == min(aic_values, na.rm = TRUE), arr.ind = TRUE)
min_bic <- which(bic_values == min(bic_values, na.rm = TRUE), arr.ind = TRUE)
cat("\nOptimal lags based on AIC: ECNDEV lag =", min_aic[1], ", Independent vars lag =", min_aic[2], "\n")##
## Optimal lags based on AIC: ECNDEV lag = 1 , Independent vars lag = 2cat("Optimal lags based on BIC: ECNDEV lag =", min_bic[1], ", Independent vars lag =", min_bic[2], "\n")## Optimal lags based on BIC: ECNDEV lag = 1 , Independent vars lag = 2# Choose lags (using AIC as default)
ardl_lags <- c(min_aic[1], rep(min_aic[2], 6))
names(ardl_lags) <- c("ECNDEV", "INF", "K", "EDU", "HEA", "UNEMP")
cat("\nSelected lags:\n")##
## Selected lags:print(ardl_lags)## ECNDEV INF K EDU HEA UNEMP <NA>
## 1 2 2 2 2 2 2ARDL MODEL ESTIMATION
cat("\n", "========== ARDL MODEL ESTIMATION ==========", "\n")##
## ========== ARDL MODEL ESTIMATION ==========# Estimate ARDL model with selected lags
ardl_model <- ardl(
ECNDEV ~ INF + K + L + EDU + HEA + UNEMP,
data = ardl_data,
order = ardl_lags
)
cat("\nARDL Model Summary:\n")##
## ARDL Model Summary:summary(ardl_model)##
## Time series regression with "ts" data:
## Start = 3, End = 30
##
## Call:
## dynlm::dynlm(formula = full_formula, data = data, start = start,
## end = end)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0231366 -0.0041264 0.0007817 0.0048398 0.0103761
##
## Coefficients: (3 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.773e-01 1.832e-01 2.059 0.0640 .
## L(ECNDEV, 1) 4.627e-01 2.948e-01 1.570 0.1447
## INF -8.384e-05 8.025e-04 -0.104 0.9187
## L(INF, 1) -5.696e-05 4.212e-04 -0.135 0.8949
## L(INF, 2) -3.033e-04 2.837e-04 -1.069 0.3079
## K 8.720e-08 1.379e-06 0.063 0.9507
## L(K, 1) -7.689e-07 1.514e-06 -0.508 0.6215
## L(K, 2) 1.164e-06 2.021e-06 0.576 0.5764
## L -1.080e-02 1.146e-02 -0.942 0.3662
## L(L, 1) 1.325e-02 1.764e-02 0.751 0.4685
## L(L, 2) 1.384e-02 3.426e-02 0.404 0.6939
## EDU -5.290e-02 2.578e-02 -2.052 0.0647 .
## L(EDU, 1) 3.931e-02 4.303e-02 0.913 0.3806
## L(EDU, 2) 7.762e-03 2.724e-02 0.285 0.7809
## HEA 6.422e-03 8.675e-03 0.740 0.4747
## L(HEA, 1) -6.273e-04 8.920e-03 -0.070 0.9452
## L(HEA, 2) 9.790e-03 8.764e-03 1.117 0.2878
## UNEMP NA NA NA NA
## L(UNEMP, 1) NA NA NA NA
## L(UNEMP, 2) NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01041 on 11 degrees of freedom
## Multiple R-squared: 0.9447, Adjusted R-squared: 0.8643
## F-statistic: 11.75 on 16 and 11 DF, p-value: 0.0001024LONG-RUN AND SHORT-RUN RELATIONSHIPS
cat("\n\n", "========== LONG-RUN AND SHORT-RUN RELATIONSHIPS ==========", "\n")##
##
## ========== LONG-RUN AND SHORT-RUN RELATIONSHIPS ==========# Get long-run coefficients
cat("\nLong-run coefficients:\n")##
## Long-run coefficients:tryCatch({
long_run <- multipliers(ardl_model, type = "lr")
print(long_run)
}, error = function(e) {
cat("Could not compute long-run multipliers:", e$message, "\n")
# Alternative: Manual long-run coefficients
cat("\nEstimating long-run coefficients from UECM...\n")
coef_uecm <- coef(uecm_model)
# Long-run coefficients = -(coefficient of lagged independent / coefficient of lagged dependent)
lr_coefficients <- data.frame(
Variable = c("INF", "K", "L", "EDU", "HEA", "UNEMP"),
Coefficient = -c(
coef_uecm["INF_lag1"] / coef_uecm["ECNDEV_lag1"],
coef_uecm["K_lag1"] / coef_uecm["ECNDEV_lag1"],
coef_uecm["L_lag1"] / coef_uecm["ECNDEV_lag1"],
coef_uecm["EDU_lag1"] / coef_uecm["ECNDEV_lag1"],
coef_uecm["HEA_lag1"] / coef_uecm["ECNDEV_lag1"],
coef_uecm["UNEMP_lag1"] / coef_uecm["ECNDEV_lag1"]
)
)
print(lr_coefficients)
})## Term Estimate Std. Error t value Pr(>|t|)
## 1 (Intercept) 7.021894e-01 NA NA NA
## 2 INF -8.265266e-04 NA NA NA
## 3 K 8.969899e-07 NA NA NA
## 4 L 3.032229e-02 NA NA NA
## 5 EDU -1.084842e-02 NA NA NA
## 6 HEA 2.900716e-02 NA NA NA
## 7 UNEMP NA NA NA NADIAGNOSTIC TESTS
cat("\n", "========== DIAGNOSTIC TESTS ==========", "\n")##
## ========== DIAGNOSTIC TESTS ==========# Residual diagnostics
residuals <- residuals(ardl_model)
# Normality test
cat("\nJarque-Bera Normality Test:\n")##
## Jarque-Bera Normality Test:jb_test <- jarque.bera.test(residuals)
print(jb_test)##
## Jarque Bera Test
##
## data: residuals
## X-squared = 22.259, df = 2, p-value = 1.468e-05# Autocorrelation test
cat("\nBreusch-Godfrey Serial Correlation Test:\n")##
## Breusch-Godfrey Serial Correlation Test:bg_test <- bgtest(ardl_model, order = ardl_lags[1])
print(bg_test)##
## Breusch-Godfrey test for serial correlation of order up to 1
##
## data: ardl_model
## LM test = 2.2826, df = 1, p-value = 0.1308# Heteroskedasticity test
cat("\nBreusch-Pagan Heteroskedasticity Test:\n")##
## Breusch-Pagan Heteroskedasticity Test:bp_test <- bptest(ardl_model)
print(bp_test)##
## studentized Breusch-Pagan test
##
## data: ardl_model
## BP = 15.986, df = 16, p-value = 0.454STABILITY TESTS
cat("\n", "========== CUSUM STABILITY TEST ==========", "\n")##
## ========== CUSUM STABILITY TEST ==========# Create stability plots
par(mfrow = c(1, 2))
# OLS-CUSUM test
tryCatch({
cusum_test <- efp(ardl_model, type = "OLS-CUSUM")
plot(cusum_test, main = "OLS-CUSUM Test",
ylim = c(-2, 2), col = "blue", lwd = 2)
# Add critical bounds at 5% significance
abline(h = 1.358, col = "red", lty = 2, lwd = 2)
abline(h = -1.358, col = "red", lty = 2, lwd = 2)
grid()
}, error = function(e) {
cat("CUSUM test could not be performed:", e$message, "\n")
})## CUSUM test could not be performed: object 'ECNDEV' not found# Recursive residuals
tryCatch({
rec_resid <- recresid(ardl_model)
plot(rec_resid, type = "l", main = "Recursive Residuals",
ylab = "Recursive Residuals", col = "darkgreen", lwd = 2)
abline(h = 0, col = "red", lty = 2)
grid()
}, error = function(e) {
cat("Recursive residuals could not be computed:", e$message, "\n")
})
par(mfrow = c(1, 1))
PLOTS
cat("\n", "========== DIAGNOSTIC PLOTS ==========", "\n")##
## ========== DIAGNOSTIC PLOTS ==========# Set up plotting area
par(mfrow = c(2, 2))
# Plot actual vs fitted
plot(ardl_data$ECNDEV, type = "l", col = "blue", lwd = 2,
main = "Actual vs Fitted ECNDEV",
xlab = "Time", ylab = "ECNDEV")
lines(fitted(ardl_model), col = "red", lwd = 2, lty = 2)
legend("topright", legend = c("Actual", "Fitted"),
col = c("blue", "red"), lty = c(1, 2), cex = 0.8)
# Plot residuals
plot(residuals, type = "l", main = "Model Residuals",
xlab = "Time", ylab = "Residuals", col = "darkgreen", lwd = 2)
abline(h = 0, col = "red", lty = 2)
grid()
# ACF of residuals
acf(residuals, main = "ACF of Residuals", lag.max = 10, col = "blue", lwd = 2)
# QQ plot
qqnorm(residuals, main = "Q-Q Plot of Residuals",
col = "blue", pch = 19, cex = 0.7)
qqline(residuals, col = "red", lwd = 2)
par(mfrow = c(1, 1))FINAL SUMMARY
cat("\n", "========== FINAL SUMMARY ==========", "\n")##
## ========== FINAL SUMMARY ==========cat("\nARDL Model Specification:")##
## ARDL Model Specification:cat("\n- Dependent Variable: ECNDEV")##
## - Dependent Variable: ECNDEVcat("\n- Independent Variables: INF, K, L, EDU, HEA, UNEMP")##
## - Independent Variables: INF, K, L, EDU, HEA, UNEMPcat("\n- Selected Lags: ECNDEV(", ardl_lags[1], "), Others(", ardl_lags[2], ")", sep = "")##
## - Selected Lags: ECNDEV(1), Others(2)cat("\n\nModel Fit:")##
##
## Model Fit:cat("\n- R-squared:", round(summary(ardl_model)$r.squared, 4))##
## - R-squared: 0.9447cat("\n- Adjusted R-squared:", round(summary(ardl_model)$adj.r.squared, 4))##
## - Adjusted R-squared: 0.8643if(exists("bounds_test")) {
cat("\n\nCointegration Test:")
cat("\n- F-statistic:", round(f_stat, 4))
cat("\n- Critical Values (5%): I(0) =", round(i0_bound_5, 4),
", I(1) =", round(i1_bound_5, 4))
if(f_stat > i1_bound_5) {
cat("\n- ✓ CONCLUSION: Cointegration exists - Long-run relationship confirmed")
} else if(f_stat < i0_bound_5) {
cat("\n- ✗ CONCLUSION: No cointegration detected")
} else {
cat("\n- ? CONCLUSION: Inconclusive - Further testing needed")
}
}
cat("\n\nDiagnostic Tests Summary:")##
##
## Diagnostic Tests Summary:cat("\n- Normality (Jarque-Bera p-value):", round(jb_test$p.value, 4))##
## - Normality (Jarque-Bera p-value): 0cat("\n- Autocorrelation (BG test p-value):", round(bg_test$p.value, 4))##
## - Autocorrelation (BG test p-value): 0.1308cat("\n- Heteroskedasticity (BP test p-value):", round(bp_test$p.value, 4))##
## - Heteroskedasticity (BP test p-value): 0.454cat("\n\n========================================\n")##
##
## ========================================Bounds test working
model <- ardlBound(data = data,
formula = ECNDEV ~ INF + L+ EDU + HEA,
max.p = 1,
max.q = 1)##
## Orders being calculated with max.p = 1 and max.q = 1 ...
##
## Autoregressive order: 2 and p-orders: 1 2 1 2
## ------------------------------------------------------
##
## Breusch-Godfrey Test for the autocorrelation in residuals:
##
## Breusch-Godfrey test for serial correlation of order up to 1
##
## data: modelFull$model
## LM test = 0.52891, df1 = 1, df2 = 14, p-value = 0.4791
##
## ------------------------------------------------------
##
## Ljung-Box Test for the autocorrelation in residuals:
##
## Box-Ljung test
##
## data: res
## X-squared = 0.41898, df = 1, p-value = 0.5174
##
## ------------------------------------------------------
##
## Breusch-Pagan Test for the homoskedasticity of residuals:
##
## studentized Breusch-Pagan test
##
## data: modelFull$model
## BP = 12.558, df = 12, p-value = 0.402
##
## ------------------------------------------------------
##
## Shapiro-Wilk test of normality of residuals:
##
## Shapiro-Wilk normality test
##
## data: modelFull$model$residual
## W = 0.80802, p-value = 0.0001483
##
## The p-value of Shapiro-Wilk test normality of residuals: 0.00014829 < 0.05!
## ------------------------------------------------------
##
## PESARAN, SHIN AND SMITH (2001) COINTEGRATION TEST
##
## Observations: 29
## Number of Regressors (k): 4
## Case: 3
##
## ------------------------------------------------------
## - F-test -
## ------------------------------------------------------
## <------- I(0) ------------ I(1) ----->
## 10% critical value 2.752 3.994
## 5% critical value 3.354 4.774
## 1% critical value 4.768 6.67
##
##
## F-statistic = 1.12891184038808
##
## ------------------------------------------------------
##
##
## ------------------------------------------------------
##
## Ramsey's RESET Test for model specification:
##
## RESET test
##
## data: modelECM$model
## RESET = 173.27, df1 = 2, df2 = 17, p-value = 4.945e-12
##
## the p-value of RESET test: 4.945172e-12 < 0.05!
## ------------------------------------------------------## ------------------------------------------------------
## Error Correction Model Output:
##
## Time series regression with "ts" data:
## Start = 2, End = 29
##
## Call:
## dynlm(formula = as.formula(model.text), data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0282540 -0.0035603 0.0003741 0.0055888 0.0088744
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.2482325 0.0873063 2.843 0.01039 *
## ec.1 -0.3636814 0.1360113 -2.674 0.01501 *
## dINF.t 0.0001554 0.0001970 0.789 0.44000
## dL.t -0.0106164 0.0057515 -1.846 0.08055 .
## dL.1 -0.0140458 0.0075009 -1.873 0.07660 .
## dEDU.t -0.0484982 0.0141109 -3.437 0.00276 **
## dHEA.t 0.0065085 0.0059423 1.095 0.28708
## dHEA.1 -0.0119362 0.0056936 -2.096 0.04966 *
## dECNDEV.1 -0.1714593 0.1928196 -0.889 0.38500
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.008586 on 19 degrees of freedom
## Multiple R-squared: 0.4667, Adjusted R-squared: 0.2422
## F-statistic: 2.079 on 8 and 19 DF, p-value: 0.091
##
## ------------------------------------------------------
## Long-run coefficients:
## ECNDEV.1 INF.1 L.1 EDU.1 HEA.1
## -0.3636813613 -0.0001208959 0.0118454398 -0.0047702588 0.0160422061
## summary(model)## Length Class Mode
## model 2 -none- list
## F.stat 1 -none- numeric
## p 5 data.frame list
## k 1 -none- numeric
## bg 7 bgtest list
## lb 5 htest list
## bp 5 htest list
## sp 4 htest list
## ECM 4 -none- list
## ARDL.model 15 dynlm listReferences
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.https://doi.org/10.1002/jae.616
Pesaran, M. H., & Smith, R. (1995). Estimating long-run relationships from dynamic heterogeneous panels. Journal of Econometrics, 68(1), 79–113.https://doi.org/10.1016/0304-4076(94)01644-F