ARDL
Overview
ARDL
creates complex autoregressive distributed lag (ARDL) models and
constructs the underlying unrestricted and restricted error correction
model (ECM) automatically, just by providing the order. It also performs
the bounds-test for cointegration as described in Pesaran et
al. (2001) and
provides the multipliers and the cointegrating equation. The validity
and the accuracy of this package have been verified by successfully
replicating the results of Pesaran et al. (2001) in Natsiopoulos and
Tzeremes (2022) doi:10.1002/jae.2919.
Why ARDL
?
- Estimate complex ARDL models just providing the ARDL order
- Estimate the conditional ECM just providing the underlying ARDL model or the order
- Estimate the long-run multipliers
- Apply the bound test for no cointegration (Pesaran et al., 2001)
- Both the F-test and the t-test are available
- The p-value is also available along with the critical value bounds for specific level of statistical significance
- Exact p-values and critical value bounds are available, along with the asymptotic ones
Installation
# You can install the released version of ARDL from CRAN:
install.packages("ARDL")
# Or the latest development version from GitHub:
install.packages("devtools")
devtools::install_github("Natsiopoulos/ARDL")
Usage
This is a basic example which shows how to use the main functions of the
ARDL
package.
Assume that we want to model the LRM
(logarithm of real money, M2) as
a function of LRY
, IBO
and IDE
(see ?denmark
). The problem is
that applying an OLS regression on non-stationary data would result into
a spurious regression. The estimated parameters would be consistent only
if the series were cointegrated.
library(ARDL)
data(denmark)
First, we find the best ARDL specification. We search up to order 5.
models <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = 5)
#> Warning: The `x` argument of `as_tibble.matrix()` must have unique column names if `.name_repair` is omitted as of tibble 2.0.0.
#> Using compatibility `.name_repair`.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.
# The top 20 models according to the AIC
models$top_orders
#> LRM LRY IBO IDE AIC
#> 1 3 1 3 2 -251.0259
#> 2 3 1 3 3 -250.1144
#> 3 2 2 0 0 -249.6266
#> 4 3 2 3 2 -249.1087
#> 5 3 2 3 3 -248.1858
#> 6 2 2 0 1 -247.7786
#> 7 2 1 0 0 -247.5643
#> 8 2 2 1 1 -246.6885
#> 9 3 3 3 3 -246.3061
#> 10 2 2 1 2 -246.2709
#> 11 2 1 1 1 -245.8736
#> 12 2 2 2 2 -245.7722
#> 13 1 1 0 0 -245.6620
#> 14 2 1 2 2 -245.1712
#> 15 3 1 2 2 -245.0996
#> 16 1 0 0 0 -244.4317
#> 17 1 1 0 1 -243.7702
#> 18 5 5 5 5 -243.3120
#> 19 4 1 3 2 -243.0728
#> 20 4 1 3 3 -242.4378
# The best model was found to be the ARDL(3,1,3,2)
ardl_3132 <- models$best_model
ardl_3132$order
#> [1] 3 1 3 2
summary(ardl_3132)
#>
#> Time series regression with "zooreg" data:
#> Start = 1974 Q4, End = 1987 Q3
#>
#> Call:
#> dynlm::dynlm(formula = full_formula, data = data, start = start,
#> end = end)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.029939 -0.008856 -0.002562 0.008190 0.072577
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.6202 0.5678 4.615 4.19e-05 ***
#> L(LRM, 1) 0.3192 0.1367 2.336 0.024735 *
#> L(LRM, 2) 0.5326 0.1324 4.024 0.000255 ***
#> L(LRM, 3) -0.2687 0.1021 -2.631 0.012143 *
#> LRY 0.6728 0.1312 5.129 8.32e-06 ***
#> L(LRY, 1) -0.2574 0.1472 -1.749 0.088146 .
#> IBO -1.0785 0.3217 -3.353 0.001790 **
#> L(IBO, 1) -0.1062 0.5858 -0.181 0.857081
#> L(IBO, 2) 0.2877 0.5691 0.505 0.616067
#> L(IBO, 3) -0.9947 0.3925 -2.534 0.015401 *
#> IDE 0.1255 0.5545 0.226 0.822161
#> L(IDE, 1) -0.3280 0.7213 -0.455 0.651847
#> L(IDE, 2) 1.4079 0.5520 2.550 0.014803 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.0191 on 39 degrees of freedom
#> Multiple R-squared: 0.988, Adjusted R-squared: 0.9843
#> F-statistic: 266.8 on 12 and 39 DF, p-value: < 2.2e-16
Then we can estimate the UECM (Unrestricted Error Correction Model) of the underlying ARDL(3,1,3,2).
uecm_3132 <- uecm(ardl_3132)
summary(uecm_3132)
#>
#> Time series regression with "zooreg" data:
#> Start = 1974 Q4, End = 1987 Q3
#>
#> Call:
#> dynlm::dynlm(formula = full_formula, data = data, start = start,
#> end = end)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.029939 -0.008856 -0.002562 0.008190 0.072577
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.62019 0.56777 4.615 4.19e-05 ***
#> L(LRM, 1) -0.41685 0.09166 -4.548 5.15e-05 ***
#> L(LRY, 1) 0.41538 0.11761 3.532 0.00108 **
#> L(IBO, 1) -1.89172 0.39111 -4.837 2.09e-05 ***
#> L(IDE, 1) 1.20534 0.44690 2.697 0.01028 *
#> d(L(LRM, 1)) -0.26394 0.10192 -2.590 0.01343 *
#> d(L(LRM, 2)) 0.26867 0.10213 2.631 0.01214 *
#> d(LRY) 0.67280 0.13116 5.129 8.32e-06 ***
#> d(IBO) -1.07852 0.32170 -3.353 0.00179 **
#> d(L(IBO, 1)) 0.70701 0.46874 1.508 0.13953
#> d(L(IBO, 2)) 0.99468 0.39251 2.534 0.01540 *
#> d(IDE) 0.12546 0.55445 0.226 0.82216
#> d(L(IDE, 1)) -1.40786 0.55204 -2.550 0.01480 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.0191 on 39 degrees of freedom
#> Multiple R-squared: 0.7458, Adjusted R-squared: 0.6676
#> F-statistic: 9.537 on 12 and 39 DF, p-value: 3.001e-08
And also the RECM (Restricted Error Correction Model) of the underlying ARDL(3,1,3,2), allowing the constant to join the short-run relationship (case 2), instead of the long-run (case 3).
recm_3132 <- recm(uecm_3132, case = 2)
summary(recm_3132)
#>
#> Time series regression with "zooreg" data:
#> Start = 1974 Q4, End = 1987 Q3
#>
#> Call:
#> dynlm::dynlm(formula = full_formula, data = data, start = start,
#> end = end)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.029939 -0.008856 -0.002562 0.008190 0.072577
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> d(L(LRM, 1)) -0.26394 0.09008 -2.930 0.005405 **
#> d(L(LRM, 2)) 0.26867 0.09127 2.944 0.005214 **
#> d(LRY) 0.67280 0.11591 5.805 7.03e-07 ***
#> d(IBO) -1.07852 0.30025 -3.592 0.000837 ***
#> d(L(IBO, 1)) 0.70701 0.44359 1.594 0.118300
#> d(L(IBO, 2)) 0.99468 0.36491 2.726 0.009242 **
#> d(IDE) 0.12546 0.48290 0.260 0.796248
#> d(L(IDE, 1)) -1.40786 0.48867 -2.881 0.006160 **
#> ect -0.41685 0.07849 -5.311 3.63e-06 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.01819 on 43 degrees of freedom
#> (0 observations deleted due to missingness)
#> Multiple R-squared: 0.7613, Adjusted R-squared: 0.7113
#> F-statistic: 15.24 on 9 and 43 DF, p-value: 9.545e-11
Let’s test if there is a long-run levels relationship (cointegration) using the bounds test from Pesaran et al. (2001).
# The bounds F-test (under the case 2) rejects the NULL hypothesis (let's say, assuming alpha = 0.01) with p-value = 0.004418.
bounds_f_test(ardl_3132, case = 2)
#>
#> Bounds F-test (Wald) for no cointegration
#>
#> data: d(LRM) ~ L(LRM, 1) + L(LRY, 1) + L(IBO, 1) + L(IDE, 1) + d(L(LRM, 1)) + d(L(LRM, 2)) + d(LRY) + d(IBO) + d(L(IBO, 1)) + d(L(IBO, 2)) + d(IDE) + d(L(IDE, 1))
#> F = 5.1168, p-value = 0.004418
#> alternative hypothesis: Possible cointegration
#> null values:
#> k T
#> 3 1000
# The bounds t-test (under the case 3) rejects the NULL hypothesis (let's say, assuming alpha = 0.01) with p-value = 0.005538.
# We also provide the critical value bounds for alpha = 0.01.
tbounds <- bounds_t_test(uecm_3132, case = 3, alpha = 0.01)
tbounds
#>
#> Bounds t-test for no cointegration
#>
#> data: d(LRM) ~ L(LRM, 1) + L(LRY, 1) + L(IBO, 1) + L(IDE, 1) + d(L(LRM, 1)) + d(L(LRM, 2)) + d(LRY) + d(IBO) + d(L(IBO, 1)) + d(L(IBO, 2)) + d(IDE) + d(L(IDE, 1))
#> t = -4.5479, Lower-bound I(0) = -3.43, Upper-bound I(1) = -4.37,
#> p-value = 0.005538
#> alternative hypothesis: Possible cointegration
#> null values:
#> k T
#> 3 1000
# Here is a more clear view of the main results.
tbounds$tab
#> statistic lower.bound upper.bound alpha p.value
#> t -4.547939 -3.43 -4.37 0.01 0.005538316
Here we have the short-run and the long-run multipliers (with standard errors, t-statistics and p-values).
multipliers(ardl_3132, type = "sr")
#> term estimate std.error t.statistic p.value
#> 1 (Intercept) 6.2856579 0.7719160 8.1429302 6.107445e-10
#> 2 LRY 1.6139988 0.1239310 13.0233660 8.809795e-16
#> 3 IBO -2.5872899 0.5202961 -4.9727264 1.364936e-05
#> 4 IDE 0.3009803 0.9950853 0.3024668 7.639036e-01
multipliers(ardl_3132)
#> term estimate std.error t.statistic p.value
#> 1 (Intercept) 6.2856579 0.7719160 8.142930 6.107445e-10
#> 2 LRY 0.9964676 0.1239310 8.040503 8.358472e-10
#> 3 IBO -4.5381160 0.5202961 -8.722180 1.058619e-10
#> 4 IDE 2.8915201 0.9950853 2.905801 6.009239e-03
Now let’s graphically check the estimated long-run relationship
(cointegrating equation) against the dependent variable LRM
.
ce <- coint_eq(ardl_3132, case = 2)
library(zoo) # for cbind.zoo
library(xts) # for xts
den <- cbind.zoo(LRM = denmark[,"LRM"], ce)
den <- xts(den)
# make the plot
den <- xts(den)
plot(den, legend.loc = "right")
Ease of use
Let’s see what it takes to build the above ARDL(3,1,3,2) model.
Using the ARDL
package (literally one line of code):
ardl_model <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2))
Without the ARDL
package: (Using the dynlm
package,
because striving with the lm
function would require extra data
transformation to behave like time-series)
library(dynlm)
dynlm_ardl_model <- dynlm(LRM ~ L(LRM, 1) + L(LRM, 2) + L(LRM, 3) + LRY + L(LRY, 1) +
IBO + L(IBO, 1) + L(IBO, 2) + L(IBO, 3) +
IDE + L(IDE, 1) + L(IDE, 2), data = denmark)
identical(ardl_model$coefficients, dynlm_ardl_model$coefficients)
#> [1] TRUE
An ARDL model has a relatively simple structure, although the difference in typing effort is noticeable.
Not to mention the complex transformation for an ECM. The extra typing is the least of your problems trying to do this. First you would need to figure out the exact structure of the model!
Using the ARDL
package (literally one line of code):
uecm_model <- uecm(ardl_model)
Without the ARDL
package:
dynlm_uecm_model <- dynlm(d(LRM) ~ L(LRM, 1) + L(LRY, 1) + L(IBO, 1) +
L(IDE, 1) + d(L(LRM, 1)) + d(L(LRM, 2)) +
d(LRY) + d(IBO) + d(L(IBO, 1)) + d(L(IBO, 2)) +
d(IDE) + d(L(IDE, 1)), data = denmark)
identical(uecm_model$coefficients, dynlm_uecm_model$coefficients)
#> [1] TRUE
References
Kleanthis Natsiopoulos and Nickolaos G. Tzeremes (2022). ARDL bounds test for cointegration: Replicating the Pesaran et al. (2001) results for the UK earnings equation using R. Journal of Applied Econometrics. https://doi.org/10.1002/jae.2919
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