,
Nickolaos Tzeremes [aut] | Title: | ARDL, ECM and Bounds-Test for Cointegration |
|---|---|
| Description: | 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) <doi:10.1002/jae.616> 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>. |
| Authors: | Kleanthis Natsiopoulos [aut, cre]
,
Nickolaos Tzeremes [aut] |
| Maintainer: | Kleanthis Natsiopoulos <[email protected]> |
| License: | GPL-3 |
| Version: | 0.2.4 |
| Built: | 2024-10-29 04:02:58 UTC |
| Source: | https://github.com/natsiopoulos/ardl |
A simple way to construct complex ARDL specifications providing just the
model order additional to the model formula. It uses
dynlm under the hood. ardl is a generic function
and the default method constructs an 'ardl' model while the other method
takes a model of class 'uecm' and converts in into an
'ardl'.
ardl(...) ## S3 method for class 'uecm' ardl(object, ...) ## Default S3 method: ardl(formula, data, order, start = NULL, end = NULL, ...)ardl(...) ## S3 method for class 'uecm' ardl(object, ...) ## Default S3 method: ardl(formula, data, order, start = NULL, end = NULL, ...)
... |
Additional arguments to be passed to the low level regression fitting functions. |
object |
An object of |
formula |
A "formula" describing the linear model. Details for model specification are given under 'Details'. |
data |
A time series object (e.g., "ts", "zoo" or "zooreg") or a data
frame containing the variables in the model. In the case of a data frame,
it is coerced into a |
order |
A specification of the order of the ARDL model. A numeric vector of the same length as the total number of variables (excluding the fixed ones, see 'Details'). It should only contain positive integers or 0. An integer could be provided if all variables are of the same order. |
start |
Start of the time period which should be used for fitting the model. |
end |
End of the time period which should be used for fitting the model. |
The formula should contain only variables that exist in the data
provided through data plus some additional functions supported by
dynlm (i.e., trend()).
You can also specify fixed variables that are not supposed to be lagged (e.g.
dummies etc.) simply by placing them after |. For example, y ~
x1 + x2 | z1 + z2 where z1 and z2 are the fixed variables and
should not be considered in order. Note that the | notion
should not be confused with the same notion in dynlm where it
introduces instrumental variables.
ardl returns an object of class
c("dynlm", "lm", "ardl"). In addition, attributes 'order', 'data',
'parsed_formula' and 'full_formula' are provided.
The general form of an is:
Kleanthis Natsiopoulos, [email protected]
data(denmark) ## Estimate an ARDL(3,1,3,2) model ------------------------------------- ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) summary(ardl_3132) ## Add dummies or other variables that should stay fixed --------------- d_74Q1_75Q3 <- ifelse(time(denmark) >= 1974 & time(denmark) <= 1975.5, 1, 0) # the date can also be setted as below d_74Q1_75Q3_ <- ifelse(time(denmark) >= "1974 Q1" & time(denmark) <= "1975 Q3", 1, 0) identical(d_74Q1_75Q3, d_74Q1_75Q3_) den <- cbind(denmark, d_74Q1_75Q3) ardl_3132_d <- ardl(LRM ~ LRY + IBO + IDE | d_74Q1_75Q3, data = den, order = c(3,1,3,2)) summary(ardl_3132_d) compare <- data.frame(AIC = c(AIC(ardl_3132), AIC(ardl_3132_d)), BIC = c(BIC(ardl_3132), BIC(ardl_3132_d))) rownames(compare) <- c("no dummy", "with dummy") compare ## Estimate an ARDL(3,1,3,2) model with a linear trend ----------------- ardl_3132_tr <- ardl(LRM ~ LRY + IBO + IDE + trend(LRM), data = denmark, order = c(3,1,3,2)) # Alternative time trend specifications: # time(LRM) 1974 + (0, 1, ..., 55)/4 time(data) # trend(LRM) (1, 2, ..., 55)/4 (1:n)/freq # trend(LRM, scale = FALSE) (1, 2, ..., 55) 1:n ## Subsample ARDL regression (start after 1975 Q4) --------------------- ardl_3132_sub <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2), start = "1975 Q4") # the date can also be setted as below ardl_3132_sub2 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2), start = c(1975,4)) identical(ardl_3132_sub, ardl_3132_sub2) summary(ardl_3132_sub) ## Ease of use --------------------------------------------------------- # The model specification of the ardl_3132 model can be created as easy as order=c(3,1,3,2) # or else, it could be done using the dynlm package as: library(dynlm) m <- 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(m$coefficients, ardl_3132$coefficients) # The full formula can be extracted from the ARDL model, and this is equal to ardl_3132$full_formula m2 <- dynlm(ardl_3132$full_formula, data = ardl_3132$data) identical(m$coefficients, m2$coefficients) ## Post-estimation testing --------------------------------------------- # See examples in the help file of the uecm() functiondata(denmark) ## Estimate an ARDL(3,1,3,2) model ------------------------------------- ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) summary(ardl_3132) ## Add dummies or other variables that should stay fixed --------------- d_74Q1_75Q3 <- ifelse(time(denmark) >= 1974 & time(denmark) <= 1975.5, 1, 0) # the date can also be setted as below d_74Q1_75Q3_ <- ifelse(time(denmark) >= "1974 Q1" & time(denmark) <= "1975 Q3", 1, 0) identical(d_74Q1_75Q3, d_74Q1_75Q3_) den <- cbind(denmark, d_74Q1_75Q3) ardl_3132_d <- ardl(LRM ~ LRY + IBO + IDE | d_74Q1_75Q3, data = den, order = c(3,1,3,2)) summary(ardl_3132_d) compare <- data.frame(AIC = c(AIC(ardl_3132), AIC(ardl_3132_d)), BIC = c(BIC(ardl_3132), BIC(ardl_3132_d))) rownames(compare) <- c("no dummy", "with dummy") compare ## Estimate an ARDL(3,1,3,2) model with a linear trend ----------------- ardl_3132_tr <- ardl(LRM ~ LRY + IBO + IDE + trend(LRM), data = denmark, order = c(3,1,3,2)) # Alternative time trend specifications: # time(LRM) 1974 + (0, 1, ..., 55)/4 time(data) # trend(LRM) (1, 2, ..., 55)/4 (1:n)/freq # trend(LRM, scale = FALSE) (1, 2, ..., 55) 1:n ## Subsample ARDL regression (start after 1975 Q4) --------------------- ardl_3132_sub <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2), start = "1975 Q4") # the date can also be setted as below ardl_3132_sub2 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2), start = c(1975,4)) identical(ardl_3132_sub, ardl_3132_sub2) summary(ardl_3132_sub) ## Ease of use --------------------------------------------------------- # The model specification of the ardl_3132 model can be created as easy as order=c(3,1,3,2) # or else, it could be done using the dynlm package as: library(dynlm) m <- 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(m$coefficients, ardl_3132$coefficients) # The full formula can be extracted from the ARDL model, and this is equal to ardl_3132$full_formula m2 <- dynlm(ardl_3132$full_formula, data = ardl_3132$data) identical(m$coefficients, m2$coefficients) ## Post-estimation testing --------------------------------------------- # See examples in the help file of the uecm() function
It searches for the best ARDL order specification, according to the selected criterion, taking into account the constraints provided.
auto_ardl( formula, data, max_order, fixed_order = -1, starting_order = NULL, selection = "AIC", selection_minmax = c("min", "max"), grid = FALSE, search_type = c("horizontal", "vertical"), start = NULL, end = NULL, ... )auto_ardl( formula, data, max_order, fixed_order = -1, starting_order = NULL, selection = "AIC", selection_minmax = c("min", "max"), grid = FALSE, search_type = c("horizontal", "vertical"), start = NULL, end = NULL, ... )
formula |
A "formula" describing the linear model. Details for model
specification are given under 'Details' in the help file of the
|
data |
A time series object (e.g., "ts", "zoo" or "zooreg") or a data
frame containing the variables in the model. In the case of a data frame,
it is coerced into a |
max_order |
It sets the maximum order for each variable where the search
is taking place. A numeric vector of the same length as the total number of
variables (excluding the fixed ones, see 'Details' in the help file of the
|
fixed_order |
It allows setting a fixed order for some variables. The
algorithm will not search for any other order than this. A numeric vector
of the same length as the total number of variables (excluding the fixed
ones). It should contain positive integers or 0 to set as a constraint. A
-1 should be provided for any variable that should not be constrained.
|
starting_order |
Specifies the order for each variable from which each
search will start. It is a numeric vector of the same length as the total
number of variables (excluding the fixed ones). It should contain positive
integers or 0 or only one integer could be provided if the starting order
for all variables is the same. Default is set to NULL. If unspecified
( |
selection |
A character string specifying the selection criterion
according to which the candidate models will be ranked. Default is
|
selection_minmax |
A character string that indicates whether the
criterion in |
grid |
If |
search_type |
A character string describing the search type. If
"horizontal" (default), the searching algorithm increases or decreases by 1
the order of each variable in each iteration. When the order of the last
variable has been accessed, it begins again from the first variable until
it converges. If "vertical", the searching algorithm increases or decreases
by 1 the order of a variable until it converges. Then it continues the same
for the next variable. The two options result to very similar top orders.
The default ("horizontal"), sometimes is a little more accurate, but the
"vertical" is almost 2 times faster. Not applicable if |
start |
Start of the time period which should be used for fitting the model. |
end |
End of the time period which should be used for fitting the model. |
... |
Additional arguments to be passed to the low level regression fitting functions. |
auto_ardl returns a list which contains:
best_model |
An object of class |
best_order |
A numeric vector with the order of the best model selected |
top_orders |
A data.frame with the orders of the top 20 models |
The algorithm performs the optimization process
starting from multiple starting points concerning the autoregressive order
. The searching algorithm will perform a complete search, each time
starting from a different starting order. These orders are presented in the
tables below, for grid = FALSE and different values of
starting_order.
starting_order = NULL:
| ARDL(p) | -> | p | q1 | q2 | ... | qk |
| ARDL(1) | -> | 1 | 1 | 1 | ... | 1 |
| ARDL(2) | -> | 2 | 2 | 2 | ... | 2 |
| : | -> | : | : | : | : | : |
| ARDL(P) | -> | P | P | P | ... | P |
starting_order = c(3, 0, 1, 2):
| p | q1 | q2 | q3 |
| 3 | 0 | 1 | 2 |
| 4 | 0 | 1 | 2 |
| : | : | : | : |
| P | 0 | 1 | 2 |
Kleanthis Natsiopoulos, [email protected]
data(denmark) ## Find the best ARDL order -------------------------------------------- # Up to 5 for the autoregressive order (p) and 4 for the rest (q1, q2, q3) # Using the defaults search_type = "horizontal", grid = FALSE and selection = "AIC" # ("Not run" indications only for testing purposes) ## Not run: model1 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4)) model1$top_orders ## Same, with search_type = "vertical" ------------------------------- model1_h <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), search_type = "vertical") model1_h$top_orders ## Find the global optimum ARDL order ---------------------------------- # It may take more than 10 seconds model_grid <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), grid = TRUE) ## Different selection criteria ---------------------------------------- # Using BIC as selection criterion instead of AIC model1_b <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), selection = "BIC") model1_b$top_orders # Using other criteria like adjusted R squared (the bigger the better) adjr2 <- function(x) { summary(x)$adj.r.squared } model1_adjr2 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), selection = "adjr2", selection_minmax = "max") model1_adjr2$top_orders # Using functions from other packages as selection criteria if (requireNamespace("qpcR", quietly = TRUE)) { library(qpcR) model1_aicc <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), selection = "AICc") model1_aicc$top_orders adjr2 <- function(x){ Rsq.ad(x) } model1_adjr2 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), selection = "adjr2", selection_minmax = "max") model1_adjr2$top_orders ## DIfferent starting order -------------------------------------------- # The searching algorithm will start from the following starting orders: # p q1 q2 q3 # 1 1 3 2 # 2 1 3 2 # 3 1 3 2 # 4 1 3 2 # 5 1 3 2 model1_so <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), starting_order = c(1,1,3,2)) # Starting from p=3 (don't search for p=1 and p=2) # Starting orders: # p q1 q2 q3 # 3 1 3 2 # 4 1 3 2 # 5 1 3 2 model1_so_3 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), starting_order = c(3,1,3,2)) # If starting_order = NULL, the starting orders for each iteration will be: # p q1 q2 q3 # 1 1 1 1 # 2 2 2 2 # 3 3 3 3 # 4 4 4 4 # 5 5 5 5 } ## Add constraints ----------------------------------------------------- # Restrict only the order of IBO to be 2 model1_ibo2 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), fixed_order = c(-1,-1,2,-1)) model1_ibo2$top_orders # Restrict the order of LRM to be 3 and the order of IBO to be 2 model1_lrm3_ibo2 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), fixed_order = c(3,-1,2,-1)) model1_lrm3_ibo2$top_orders ## Set the starting date for the regression (data starts at "1974 Q1") - # Set regression starting date to "1976 Q1" model1_76q1 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), start = "1976 Q1") start(model1_76q1$best_model) ## End(Not run)data(denmark) ## Find the best ARDL order -------------------------------------------- # Up to 5 for the autoregressive order (p) and 4 for the rest (q1, q2, q3) # Using the defaults search_type = "horizontal", grid = FALSE and selection = "AIC" # ("Not run" indications only for testing purposes) ## Not run: model1 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4)) model1$top_orders ## Same, with search_type = "vertical" ------------------------------- model1_h <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), search_type = "vertical") model1_h$top_orders ## Find the global optimum ARDL order ---------------------------------- # It may take more than 10 seconds model_grid <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), grid = TRUE) ## Different selection criteria ---------------------------------------- # Using BIC as selection criterion instead of AIC model1_b <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), selection = "BIC") model1_b$top_orders # Using other criteria like adjusted R squared (the bigger the better) adjr2 <- function(x) { summary(x)$adj.r.squared } model1_adjr2 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), selection = "adjr2", selection_minmax = "max") model1_adjr2$top_orders # Using functions from other packages as selection criteria if (requireNamespace("qpcR", quietly = TRUE)) { library(qpcR) model1_aicc <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), selection = "AICc") model1_aicc$top_orders adjr2 <- function(x){ Rsq.ad(x) } model1_adjr2 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), selection = "adjr2", selection_minmax = "max") model1_adjr2$top_orders ## DIfferent starting order -------------------------------------------- # The searching algorithm will start from the following starting orders: # p q1 q2 q3 # 1 1 3 2 # 2 1 3 2 # 3 1 3 2 # 4 1 3 2 # 5 1 3 2 model1_so <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), starting_order = c(1,1,3,2)) # Starting from p=3 (don't search for p=1 and p=2) # Starting orders: # p q1 q2 q3 # 3 1 3 2 # 4 1 3 2 # 5 1 3 2 model1_so_3 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), starting_order = c(3,1,3,2)) # If starting_order = NULL, the starting orders for each iteration will be: # p q1 q2 q3 # 1 1 1 1 # 2 2 2 2 # 3 3 3 3 # 4 4 4 4 # 5 5 5 5 } ## Add constraints ----------------------------------------------------- # Restrict only the order of IBO to be 2 model1_ibo2 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), fixed_order = c(-1,-1,2,-1)) model1_ibo2$top_orders # Restrict the order of LRM to be 3 and the order of IBO to be 2 model1_lrm3_ibo2 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), fixed_order = c(3,-1,2,-1)) model1_lrm3_ibo2$top_orders ## Set the starting date for the regression (data starts at "1974 Q1") - # Set regression starting date to "1976 Q1" model1_76q1 <- auto_ardl(LRM ~ LRY + IBO + IDE, data = denmark, max_order = c(5,4,4,4), start = "1976 Q1") start(model1_76q1$best_model) ## End(Not run)
bounds_f_test performs the Wald bounds-test for no cointegration
Pesaran et al. (2001). It is a Wald test on the parameters of a UECM
(Unrestricted Error Correction Model) expressed either as a Chisq-statistic
or as an F-statistic.
bounds_f_test( object, case, alpha = NULL, pvalue = TRUE, exact = FALSE, R = 40000, test = c("F", "Chisq"), vcov_matrix = NULL )bounds_f_test( object, case, alpha = NULL, pvalue = TRUE, exact = FALSE, R = 40000, test = c("F", "Chisq"), vcov_matrix = NULL )
object |
An object of |
case |
An integer from 1-5 or a character string specifying whether the 'intercept' and/or the 'trend' have to participate in the short-run or the long-run relationship (cointegrating equation) (see section 'Cases' below). |
alpha |
A numeric value between 0 and 1 indicating the significance
level of the critical value bounds. If |
pvalue |
A logical indicating whether you want the p-value to be
provided. The default is |
exact |
A logical indicating whether you want asymptotic (T = 1000) or
exact sample size critical value bounds and p-value. The default is
|
R |
An integer indicating how many iterations will be used if
|
test |
A character vector indicating whether you want the Wald test to be expressed as 'F' or as 'Chisq' statistic. Default is "F". |
vcov_matrix |
The estimated covariance matrix of the random variable
that the test uses to estimate the test statistic. The default is
|
A list with class "htest" containing the following components:
method |
a character string indicating what type of test was performed. |
alternative |
a character string describing the alternative hypothesis. |
statistic |
the value of the test statistic. |
null.value |
the value of the population parameters |
data.name |
a character string giving the name(s) of the data. |
parameters |
numeric vector containing the critical value bounds. |
p.value |
the p-value of the test. |
PSS2001parameters |
numeric vector containing the critical value bounds as presented by Pesaran et al. (2001). See section 'alpha, bounds and p-value' below for details. |
tab |
data.frame containing the statistic, the critical value bounds, the alpha level of significance and the p-value. |
In this section it is explained how the critical value bounds and p-values are obtained.
If exact = FALSE, then the asymptotic (T = 1000) critical
value bounds and p-value are provided.
Only the asymptotic critical value bounds and p-values, and only for k <= 10 are precalculated, everything else has to be computed.
Precalculated critical value bounds and p-values were simulated
using set.seed(2020) and R = 70000.
Precalculated critical value bounds exist only for alpha
being one of the 0.005, 0.01, 0.025, 0.05, 0.075, 0.1, 0.15 or 0.2,
everything else has to be computed.
If alpha is one of the 0.1, 0.05, 0.025 or 0.01 (and
exact = FALSE and k <= 10), PSS2001parameters shows
the critical value bounds presented in Pesaran et al. (2001)
(less precise).
According to Pesaran et al. (2001), we distinguish the long-run relationship (cointegrating equation) (and thus the bounds-test and the Restricted ECMs) between 5 different cases. These differ in terms of whether the 'intercept' and/or the 'trend' are restricted to participate in the long-run relationship or they are unrestricted and so they participate in the short-run relationship.
No intercept and no trend.
case inputs: 1 or "n" where "n" stands for none.
Restricted intercept and no trend.
case inputs: 2 or "rc" where "rc" stands for restricted
constant.
Unrestricted intercept and no trend.
case inputs: 3 or "uc" where "uc" stands for unrestricted
constant.
Unrestricted intercept and restricted trend.
case inputs: 4 or "ucrt" where "ucrt" stands for
unrestricted constant and restricted trend.
Unrestricted intercept and unrestricted trend.
case inputs: 5 or "ucut" where "ucut" stands for
unrestricted constant and unrestricted trend.
Note that you can't restrict (or leave unrestricted) a parameter that doesn't
exist in the input model. For example, you can't compute recm(object,
case=3) if the object is an ARDL (or UECM) model with no intercept. The same
way, you can't compute bounds_f_test(object, case=5) if the object is
an ARDL (or UECM) model with no linear trend.
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
Kleanthis Natsiopoulos, [email protected]
data(denmark) ## How to use cases under different models (regarding deterministic terms) ## Construct an ARDL(3,1,3,2) model with different deterministic terms - # Without constant ardl_3132_n <- ardl(LRM ~ LRY + IBO + IDE -1, data = denmark, order = c(3,1,3,2)) # With constant ardl_3132_c <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) # With constant and trend ardl_3132_ct <- ardl(LRM ~ LRY + IBO + IDE + trend(LRM), data = denmark, order = c(3,1,3,2)) ## F-bounds test for no level relationship (no cointegration) ---------- # For the model without a constant bounds_f_test(ardl_3132_n, case = 1) # or bounds_f_test(ardl_3132_n, case = "n") # For the model with a constant # Including the constant term in the long-run relationship (restricted constant) bounds_f_test(ardl_3132_c, case = 2) # or bounds_f_test(ardl_3132_c, case = "rc") # Including the constant term in the short-run relationship (unrestricted constant) bounds_f_test(ardl_3132_c, case = "uc") # or bounds_f_test(ardl_3132_c, case = 3) # For the model with constant and trend # Including the constant term in the short-run and the trend in the long-run relationship # (unrestricted constant and restricted trend) bounds_f_test(ardl_3132_ct, case = "ucrt") # or bounds_f_test(ardl_3132_ct, case = 4) # For the model with constant and trend # Including the constant term and the trend in the short-run relationship # (unrestricted constant and unrestricted trend) bounds_f_test(ardl_3132_ct, case = "ucut") # or bounds_f_test(ardl_3132_ct, case = 5) ## Note that you can't restrict a deterministic term that doesn't exist # For example, the following tests will produce an error: ## Not run: bounds_f_test(ardl_3132_c, case = 1) bounds_f_test(ardl_3132_ct, case = 3) bounds_f_test(ardl_3132_c, case = 4) ## End(Not run) ## Asymptotic p-value and critical value bounds (assuming T = 1000) ---- # Include critical value bounds for a certain level of significance # F-statistic is larger than the I(1) bound (for a=0.05) as expected (p-value < 0.05) bft <- bounds_f_test(ardl_3132_c, case = 2, alpha = 0.05) bft bft$tab # Traditional but less precise critical value bounds, as presented in Pesaran et al. (2001) bft$PSS2001parameters # F-statistic is slightly larger than the I(1) bound (for a=0.005) # as p-value is slightly smaller than 0.005 bounds_f_test(ardl_3132_c, case = 2, alpha = 0.005) ## Exact sample size p-value and critical value bounds ----------------- # Setting a seed is suggested to allow the replication of results # 'R' can be increased for more accurate resutls # F-statistic is smaller than the I(1) bound (for a=0.01) as expected (p-value > 0.01) # Note that the exact sample p-value (0.01285) is very different than the asymptotic (0.004418) # It can take more than 30 seconds ## Not run: set.seed(2020) bounds_f_test(ardl_3132_c, case = 2, alpha = 0.01, exact = TRUE) ## End(Not run) ## "F" and "Chisq" statistics ------------------------------------------ # The p-value is the same, the test-statistic and critical value bounds are different but analogous bounds_f_test(ardl_3132_c, case = 2, alpha = 0.01) bounds_f_test(ardl_3132_c, case = 2, alpha = 0.01, test = "Chisq")data(denmark) ## How to use cases under different models (regarding deterministic terms) ## Construct an ARDL(3,1,3,2) model with different deterministic terms - # Without constant ardl_3132_n <- ardl(LRM ~ LRY + IBO + IDE -1, data = denmark, order = c(3,1,3,2)) # With constant ardl_3132_c <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) # With constant and trend ardl_3132_ct <- ardl(LRM ~ LRY + IBO + IDE + trend(LRM), data = denmark, order = c(3,1,3,2)) ## F-bounds test for no level relationship (no cointegration) ---------- # For the model without a constant bounds_f_test(ardl_3132_n, case = 1) # or bounds_f_test(ardl_3132_n, case = "n") # For the model with a constant # Including the constant term in the long-run relationship (restricted constant) bounds_f_test(ardl_3132_c, case = 2) # or bounds_f_test(ardl_3132_c, case = "rc") # Including the constant term in the short-run relationship (unrestricted constant) bounds_f_test(ardl_3132_c, case = "uc") # or bounds_f_test(ardl_3132_c, case = 3) # For the model with constant and trend # Including the constant term in the short-run and the trend in the long-run relationship # (unrestricted constant and restricted trend) bounds_f_test(ardl_3132_ct, case = "ucrt") # or bounds_f_test(ardl_3132_ct, case = 4) # For the model with constant and trend # Including the constant term and the trend in the short-run relationship # (unrestricted constant and unrestricted trend) bounds_f_test(ardl_3132_ct, case = "ucut") # or bounds_f_test(ardl_3132_ct, case = 5) ## Note that you can't restrict a deterministic term that doesn't exist # For example, the following tests will produce an error: ## Not run: bounds_f_test(ardl_3132_c, case = 1) bounds_f_test(ardl_3132_ct, case = 3) bounds_f_test(ardl_3132_c, case = 4) ## End(Not run) ## Asymptotic p-value and critical value bounds (assuming T = 1000) ---- # Include critical value bounds for a certain level of significance # F-statistic is larger than the I(1) bound (for a=0.05) as expected (p-value < 0.05) bft <- bounds_f_test(ardl_3132_c, case = 2, alpha = 0.05) bft bft$tab # Traditional but less precise critical value bounds, as presented in Pesaran et al. (2001) bft$PSS2001parameters # F-statistic is slightly larger than the I(1) bound (for a=0.005) # as p-value is slightly smaller than 0.005 bounds_f_test(ardl_3132_c, case = 2, alpha = 0.005) ## Exact sample size p-value and critical value bounds ----------------- # Setting a seed is suggested to allow the replication of results # 'R' can be increased for more accurate resutls # F-statistic is smaller than the I(1) bound (for a=0.01) as expected (p-value > 0.01) # Note that the exact sample p-value (0.01285) is very different than the asymptotic (0.004418) # It can take more than 30 seconds ## Not run: set.seed(2020) bounds_f_test(ardl_3132_c, case = 2, alpha = 0.01, exact = TRUE) ## End(Not run) ## "F" and "Chisq" statistics ------------------------------------------ # The p-value is the same, the test-statistic and critical value bounds are different but analogous bounds_f_test(ardl_3132_c, case = 2, alpha = 0.01) bounds_f_test(ardl_3132_c, case = 2, alpha = 0.01, test = "Chisq")
bounds_t_test performs the t-bounds test for no cointegration
Pesaran et al. (2001). It is a t-test on the parameters of a UECM
(Unrestricted Error Correction Model).
bounds_t_test( object, case, alpha = NULL, pvalue = TRUE, exact = FALSE, R = 40000, vcov_matrix = NULL )bounds_t_test( object, case, alpha = NULL, pvalue = TRUE, exact = FALSE, R = 40000, vcov_matrix = NULL )
object |
An object of |
case |
An integer (1, 3 or 5) or a character string specifying whether the 'intercept' and/or the 'trend' have to participate in the short-run relationship (see section 'Cases' below). Note that the t-bounds test can't be applied for cases 2 and 4. |
alpha |
A numeric value between 0 and 1 indicating the significance
level of the critical value bounds. If |
pvalue |
A logical indicating whether you want the p-value to be
provided. The default is |
exact |
A logical indicating whether you want asymptotic (T = 1000) or
exact sample size critical value bounds and p-value. The default is
|
R |
An integer indicating how many iterations will be used if
|
vcov_matrix |
The estimated covariance matrix of the random variable
that the test uses to estimate the test statistic. The default is
|
A list with class "htest" containing the following components:
method |
a character string indicating what type of test was performed. |
alternative |
a character string describing the alternative hypothesis. |
statistic |
the value of the test statistic. |
null.value |
the value of the population parameters |
data.name |
a character string giving the name(s) of the data. |
parameters |
numeric vector containing the critical value bounds. |
p.value |
the p-value of the test. |
PSS2001parameters |
numeric vector containing the critical value bounds as presented by Pesaran et al. (2001). See section 'alpha, bounds and p-value' below for details. |
tab |
data.frame containing the statistic, the critical value bounds, the alpha level of significance and the p-value. |
In this section it is explained how the critical value bounds and p-values are obtained.
If exact = FALSE, then the asymptotic (T = 1000) critical
value bounds and p-value are provided.
Only the asymptotic critical value bounds and p-values, and only for k <= 10 are precalculated, everything else has to be computed.
Precalculated critical value bounds and p-values were simulated
using set.seed(2020) and R = 70000.
Precalculated critical value bounds exist only for alpha
being one of the 0.005, 0.01, 0.025, 0.05, 0.075, 0.1, 0.15 or 0.2,
everything else has to be computed.
If alpha is one of the 0.1, 0.05, 0.025 or 0.01 (and
exact = FALSE and k <= 10), PSS2001parameters shows
the critical value bounds presented in Pesaran et al. (2001)
(less precise).
According to Pesaran et al. (2001), we distinguish the long-run relationship (cointegrating equation) (and thus the bounds-test and the Restricted ECMs) between 5 different cases. These differ in terms of whether the 'intercept' and/or the 'trend' are restricted to participate in the long-run relationship or they are unrestricted and so they participate in the short-run relationship.
No intercept and no trend.
case inputs: 1 or "n" where "n" stands for none.
Restricted intercept and no trend.
case inputs: 2 or "rc" where "rc" stands for restricted
constant.
Unrestricted intercept and no trend.
case inputs: 3 or "uc" where "uc" stands for unrestricted
constant.
Unrestricted intercept and restricted trend.
case inputs: 4 or "ucrt" where "ucrt" stands for
unrestricted constant and restricted trend.
Unrestricted intercept and unrestricted trend.
case inputs: 5 or "ucut" where "ucut" stands for
unrestricted constant and unrestricted trend.
Note that you can't restrict (or leave unrestricted) a parameter that doesn't
exist in the input model. For example, you can't compute recm(object,
case=3) if the object is an ARDL (or UECM) model with no intercept. The same
way, you can't compute bounds_f_test(object, case=5) if the object is
an ARDL (or UECM) model with no linear trend.
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
Kleanthis Natsiopoulos, [email protected]
data(denmark) ## How to use cases under different models (regarding deterministic terms) ## Construct an ARDL(3,1,3,2) model with different deterministic terms - # Without constant ardl_3132_n <- ardl(LRM ~ LRY + IBO + IDE -1, data = denmark, order = c(3,1,3,2)) # With constant ardl_3132_c <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) # With constant and trend ardl_3132_ct <- ardl(LRM ~ LRY + IBO + IDE + trend(LRM), data = denmark, order = c(3,1,3,2)) ## t-bounds test for no level relationship (no cointegration) ---------- # For the model without a constant bounds_t_test(ardl_3132_n, case = 1) # or bounds_t_test(ardl_3132_n, case = "n") # For the model with a constant # Including the constant term in the short-run relationship (unrestricted constant) bounds_t_test(ardl_3132_c, case = "uc") # or bounds_t_test(ardl_3132_c, case = 3) # For the model with constant and trend # Including the constant term and the trend in the short-run relationship # (unrestricted constant and unrestricted trend) bounds_t_test(ardl_3132_ct, case = "ucut") # or bounds_t_test(ardl_3132_ct, case = 5) ## Note that you can't use bounds t-test for cases 2 and 4, or use a wrong model # For example, the following tests will produce an error: ## Not run: bounds_t_test(ardl_3132_n, case = 2) bounds_t_test(ardl_3132_c, case = 4) bounds_t_test(ardl_3132_ct, case = 3) ## End(Not run) ## Asymptotic p-value and critical value bounds (assuming T = 1000) ---- # Include critical value bounds for a certain level of significance # t-statistic is larger than the I(1) bound (for a=0.05) as expected (p-value < 0.05) btt <- bounds_t_test(ardl_3132_c, case = 3, alpha = 0.05) btt btt$tab # Traditional but less precise critical value bounds, as presented in Pesaran et al. (2001) btt$PSS2001parameters # t-statistic doesn't exceed the I(1) bound (for a=0.005) as p-value is greater than 0.005 bounds_t_test(ardl_3132_c, case = 3, alpha = 0.005) ## Exact sample size p-value and critical value bounds ----------------- # Setting a seed is suggested to allow the replication of results # 'R' can be increased for more accurate resutls # t-statistic is smaller than the I(1) bound (for a=0.01) as expected (p-value > 0.01) # Note that the exact sample p-value (0.009874) is very different than the asymptotic (0.005538) # It can take more than 90 seconds ## Not run: set.seed(2020) bounds_t_test(ardl_3132_c, case = 3, alpha = 0.01, exact = TRUE) ## End(Not run)data(denmark) ## How to use cases under different models (regarding deterministic terms) ## Construct an ARDL(3,1,3,2) model with different deterministic terms - # Without constant ardl_3132_n <- ardl(LRM ~ LRY + IBO + IDE -1, data = denmark, order = c(3,1,3,2)) # With constant ardl_3132_c <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) # With constant and trend ardl_3132_ct <- ardl(LRM ~ LRY + IBO + IDE + trend(LRM), data = denmark, order = c(3,1,3,2)) ## t-bounds test for no level relationship (no cointegration) ---------- # For the model without a constant bounds_t_test(ardl_3132_n, case = 1) # or bounds_t_test(ardl_3132_n, case = "n") # For the model with a constant # Including the constant term in the short-run relationship (unrestricted constant) bounds_t_test(ardl_3132_c, case = "uc") # or bounds_t_test(ardl_3132_c, case = 3) # For the model with constant and trend # Including the constant term and the trend in the short-run relationship # (unrestricted constant and unrestricted trend) bounds_t_test(ardl_3132_ct, case = "ucut") # or bounds_t_test(ardl_3132_ct, case = 5) ## Note that you can't use bounds t-test for cases 2 and 4, or use a wrong model # For example, the following tests will produce an error: ## Not run: bounds_t_test(ardl_3132_n, case = 2) bounds_t_test(ardl_3132_c, case = 4) bounds_t_test(ardl_3132_ct, case = 3) ## End(Not run) ## Asymptotic p-value and critical value bounds (assuming T = 1000) ---- # Include critical value bounds for a certain level of significance # t-statistic is larger than the I(1) bound (for a=0.05) as expected (p-value < 0.05) btt <- bounds_t_test(ardl_3132_c, case = 3, alpha = 0.05) btt btt$tab # Traditional but less precise critical value bounds, as presented in Pesaran et al. (2001) btt$PSS2001parameters # t-statistic doesn't exceed the I(1) bound (for a=0.005) as p-value is greater than 0.005 bounds_t_test(ardl_3132_c, case = 3, alpha = 0.005) ## Exact sample size p-value and critical value bounds ----------------- # Setting a seed is suggested to allow the replication of results # 'R' can be increased for more accurate resutls # t-statistic is smaller than the I(1) bound (for a=0.01) as expected (p-value > 0.01) # Note that the exact sample p-value (0.009874) is very different than the asymptotic (0.005538) # It can take more than 90 seconds ## Not run: set.seed(2020) bounds_t_test(ardl_3132_c, case = 3, alpha = 0.01, exact = TRUE) ## End(Not run)
Creates the cointegrating equation (long-run level relationship) providing an 'ardl', 'uecm' or 'recm' model.
coint_eq(object, case) ## S3 method for class 'recm' coint_eq(object, ...) ## Default S3 method: coint_eq(object, case)coint_eq(object, case) ## S3 method for class 'recm' coint_eq(object, ...) ## Default S3 method: coint_eq(object, case)
object |
An object of |
case |
An integer from 1-5 or a character string specifying whether the
'intercept' and/or the 'trend' have to participate in the long-run level
relationship (cointegrating equation) (see section 'Cases' below). If the
input object is of class 'recm', |
... |
Currently unused argument. |
coint_eq returns an numeric vector containing the
cointegrating equation.
According to Pesaran et al. (2001), we distinguish the long-run relationship (cointegrating equation) (and thus the bounds-test and the Restricted ECMs) between 5 different cases. These differ in terms of whether the 'intercept' and/or the 'trend' are restricted to participate in the long-run relationship or they are unrestricted and so they participate in the short-run relationship.
No intercept and no trend.
case inputs: 1 or "n" where "n" stands for none.
Restricted intercept and no trend.
case inputs: 2 or "rc" where "rc" stands for restricted
constant.
Unrestricted intercept and no trend.
case inputs: 3 or "uc" where "uc" stands for unrestricted
constant.
Unrestricted intercept and restricted trend.
case inputs: 4 or "ucrt" where "ucrt" stands for
unrestricted constant and restricted trend.
Unrestricted intercept and unrestricted trend.
case inputs: 5 or "ucut" where "ucut" stands for
unrestricted constant and unrestricted trend.
Note that you can't restrict (or leave unrestricted) a parameter that doesn't
exist in the input model. For example, you can't compute recm(object,
case=3) if the object is an ARDL (or UECM) model with no intercept. The same
way, you can't compute bounds_f_test(object, case=5) if the object is
an ARDL (or UECM) model with no linear trend.
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
Kleanthis Natsiopoulos, [email protected]
plot_lr ardl uecm recm
bounds_f_test bounds_t_test
data(denmark) library(zoo) # for cbind.zoo() ## Estimate the Cointegrating Equation of an ARDL(3,1,3,2) model ------- # From an ARDL model (under case 2, restricted constant) ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) ce2_ardl <- coint_eq(ardl_3132, case = 2) # From an UECM (under case 2, restricted constant) uecm_3132 <- uecm(ardl_3132) ce2_uecm <- coint_eq(uecm_3132, case = 2) # From a RECM (under case 2, restricted constant) # Notice that if a RECM has already been estimated under a certain case, # the 'coint_eq()' can't be under different case, so no 'case' argument needed. recm_3132 <- recm(uecm_3132, case = 2) # The RECM is already under case 2, so the 'case' argument is no needed ce2_recm <- coint_eq(recm_3132) identical(ce2_ardl, ce2_uecm, ce2_recm) ## Check for a degenerate level relationship --------------------------- # The bounds F-test under both cases reject the Null Hypothesis of no level relationship. bounds_f_test(ardl_3132, case = 2) bounds_f_test(ardl_3132, case = 3) # The bounds t-test also rejects the NUll Hypothesis of no level relationship. bounds_t_test(ardl_3132, case = 3) # But when the constant enters the long-run equation (case 3) # this becomes a degenerate relationship. ce3_ardl <- coint_eq(ardl_3132, case = 3) plot_lr(ardl_3132, coint_eq = ce2_ardl, show.legend = TRUE) plot_lr(ardl_3132, coint_eq = ce3_ardl, show.legend = TRUE) plot_lr(ardl_3132, coint_eq = ce3_ardl, facets = TRUE, show.legend = TRUE)data(denmark) library(zoo) # for cbind.zoo() ## Estimate the Cointegrating Equation of an ARDL(3,1,3,2) model ------- # From an ARDL model (under case 2, restricted constant) ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) ce2_ardl <- coint_eq(ardl_3132, case = 2) # From an UECM (under case 2, restricted constant) uecm_3132 <- uecm(ardl_3132) ce2_uecm <- coint_eq(uecm_3132, case = 2) # From a RECM (under case 2, restricted constant) # Notice that if a RECM has already been estimated under a certain case, # the 'coint_eq()' can't be under different case, so no 'case' argument needed. recm_3132 <- recm(uecm_3132, case = 2) # The RECM is already under case 2, so the 'case' argument is no needed ce2_recm <- coint_eq(recm_3132) identical(ce2_ardl, ce2_uecm, ce2_recm) ## Check for a degenerate level relationship --------------------------- # The bounds F-test under both cases reject the Null Hypothesis of no level relationship. bounds_f_test(ardl_3132, case = 2) bounds_f_test(ardl_3132, case = 3) # The bounds t-test also rejects the NUll Hypothesis of no level relationship. bounds_t_test(ardl_3132, case = 3) # But when the constant enters the long-run equation (case 3) # this becomes a degenerate relationship. ce3_ardl <- coint_eq(ardl_3132, case = 3) plot_lr(ardl_3132, coint_eq = ce2_ardl, show.legend = TRUE) plot_lr(ardl_3132, coint_eq = ce3_ardl, show.legend = TRUE) plot_lr(ardl_3132, coint_eq = ce3_ardl, facets = TRUE, show.legend = TRUE)
This data set contains the series used by S. Johansen and K. Juselius for estimating a money demand function of Denmark.
denmarkdenmark
A time-series object with 55 rows and 5 variables. Time period from 1974:Q1 until 1987:Q3.
logarithm of real money, M2
logarithm of real income
logarithm of price deflator
bond rate
bank deposit rate
An object of class "zooreg" "zoo".
https://onlinelibrary.wiley.com/doi/10.1111/j.1468-0084.1990.mp52002003.x
Johansen, S. and Juselius, K. (1990), Maximum Likelihood Estimation and Inference on Cointegration – with Applications to the Demand for Money, Oxford Bulletin of Economics and Statistics, 52, 2, 169–210.
multipliers is a generic function used to estimate short-run (impact),
delay, interim and long-run (total) multipliers, accompanied by their
corresponding standard errors, t-statistics and p-values.
multipliers(object, type = "lr", vcov_matrix = NULL, se = FALSE) ## S3 method for class 'ardl' multipliers(object, type = "lr", vcov_matrix = NULL, se = FALSE) ## S3 method for class 'uecm' multipliers(object, type = "lr", vcov_matrix = NULL, se = FALSE)multipliers(object, type = "lr", vcov_matrix = NULL, se = FALSE) ## S3 method for class 'ardl' multipliers(object, type = "lr", vcov_matrix = NULL, se = FALSE) ## S3 method for class 'uecm' multipliers(object, type = "lr", vcov_matrix = NULL, se = FALSE)
object |
An object of |
type |
A character string describing the type of multipliers. Use "lr" for long-run (total) multipliers (default), "sr" or 0 for short-run (impact) multipliers or an integer between 1 and 200 for delay and interim multipliers. |
vcov_matrix |
The estimated covariance matrix of the random variable
that the transformation function uses to estimate the standard errors (and
so the t-statistics and p-values) of the multipliers. The default is
|
se |
A logical indicating whether you want standard errors for delay
multipliers to be provided. The default is FALSE. Note that this parameter
does not refer to the standard errors for the long-run and short-run
multipliers, for which are always calculated. IMPORTANT: Calculating standard
errors for long periods of delays may cause your computer to run out of
memory and terminate your R session, losing important unsaved work. As a rule
of thumb, try not to exceed |
The function invokes two different methods, one for
objects of class 'ardl' and one for objects of
class 'uecm'. This is because of the different (but equivalent)
transformation functions that are used for each class/model ('ardl' and
'uecm') to estimate the multipliers.
type = 0 is equivalent to type = "sr".
Note that the interim multipliers are the cumulative sum of the delays, and that the sum of the interim multipliers (for long enough periods) and thus a distant enough interim multiplier match the long-run multipliers.
The delay (interim) multiplier can be interpreted as the effect on the dependent variable in period t+s, resulting from an instant (sustained) shock to an independent variable in period t.
The delta method is used for approximating the standard errors (and thus the t-statistics and p-values) of the estimated long-run and delay multipliers.
multipliers returns (for long and short run multipliers) a
data.frame containing the independent variables (including possibly
existing intercept or trend and excluding the fixed variables) and their
corresponding standard errors, t-statistics and p-values. For delay and
interim multipliers it returns a list with a data.frame for each variable,
containing the delay and interim multipliers for each period.
Short-Run Multipliers:
Delay & Interim Multipliers:
Long-Run Multipliers:
Kleanthis Natsiopoulos, [email protected]
data(denmark) ## Estimate the long-run multipliers of an ARDL(3,1,3,2) model --------- # From an ARDL model ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) mult_ardl <- multipliers(ardl_3132) mult_ardl # From an UECM uecm_3132 <- uecm(ardl_3132) mult_uecm <- multipliers(uecm_3132) mult_uecm all.equal(mult_ardl, mult_uecm) ## Estimate the short-run multipliers of an ARDL(3,1,3,2) model -------- mult_sr <- multipliers(uecm_3132, type = "sr") mult_0 <- multipliers(uecm_3132, type = 0) all.equal(mult_sr, mult_0) ## Estimate the delay & interim multipliers of an ARDL(3,1,3,2) model -- mult_lr <- multipliers(uecm_3132, type = "lr") mult_inter80 <- multipliers(uecm_3132, type = 80) mult_lr sum(mult_inter80$`(Intercept)`$Delay) mult_inter80$`(Intercept)`$Interim[nrow(mult_inter80$`(Intercept)`)] sum(mult_inter80$LRY$Delay) mult_inter80$LRY$Interim[nrow(mult_inter80$LRY)] sum(mult_inter80$IBO$Delay) mult_inter80$IBO$Interim[nrow(mult_inter80$IBO)] sum(mult_inter80$IDE$Delay) mult_inter80$IDE$Interim[nrow(mult_inter80$IDE)] plot(mult_inter80$LRY$Delay, type='l') plot(mult_inter80$LRY$Interim, type='l') mult_inter12 <- multipliers(uecm_3132, type = 12, se = TRUE) plot_delay(mult_inter12, interval = 0.95)data(denmark) ## Estimate the long-run multipliers of an ARDL(3,1,3,2) model --------- # From an ARDL model ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) mult_ardl <- multipliers(ardl_3132) mult_ardl # From an UECM uecm_3132 <- uecm(ardl_3132) mult_uecm <- multipliers(uecm_3132) mult_uecm all.equal(mult_ardl, mult_uecm) ## Estimate the short-run multipliers of an ARDL(3,1,3,2) model -------- mult_sr <- multipliers(uecm_3132, type = "sr") mult_0 <- multipliers(uecm_3132, type = 0) all.equal(mult_sr, mult_0) ## Estimate the delay & interim multipliers of an ARDL(3,1,3,2) model -- mult_lr <- multipliers(uecm_3132, type = "lr") mult_inter80 <- multipliers(uecm_3132, type = 80) mult_lr sum(mult_inter80$`(Intercept)`$Delay) mult_inter80$`(Intercept)`$Interim[nrow(mult_inter80$`(Intercept)`)] sum(mult_inter80$LRY$Delay) mult_inter80$LRY$Interim[nrow(mult_inter80$LRY)] sum(mult_inter80$IBO$Delay) mult_inter80$IBO$Interim[nrow(mult_inter80$IBO)] sum(mult_inter80$IDE$Delay) mult_inter80$IDE$Interim[nrow(mult_inter80$IDE)] plot(mult_inter80$LRY$Delay, type='l') plot(mult_inter80$LRY$Interim, type='l') mult_inter12 <- multipliers(uecm_3132, type = 12, se = TRUE) plot_delay(mult_inter12, interval = 0.95)
This data set contains the series used by Natsiopoulos and Tzeremes (2022) for re-estimating the UK earnings equation. The clean format of the data retrieved from the Data Archive of Natsiopoulos and Tzeremes (2022).
NT2022NT2022
A time-series object with 196 rows and 9 variables. Time period from 1971:Q1 until 2019:Q4.
time variable
real wage
labor productivity
unemployment rate
wedge effect
union power
income policies 1974:Q1-1975:Q4
income policies 1975:Q1-1979:Q4
union membership
An object of class "zooreg" "zoo".
http://qed.econ.queensu.ca/jae/datasets/natsiopoulos001/
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, 37, 5, 1079–1090. doi:10.1002/jae.2919
Creates plots for the delay multipliers and their uncertainty intervals based
on their estimated standard errors. This is a basic
ggplot with a few customizable parameters.
plot_delay( multipliers, facets_ncol = 2, interval = FALSE, interval_color = "blue", show.legend = FALSE, xlab = "Period", ylab = "Delay", ... )plot_delay( multipliers, facets_ncol = 2, interval = FALSE, interval_color = "blue", show.legend = FALSE, xlab = "Period", ylab = "Delay", ... )
multipliers |
A list returned from |
facets_ncol |
If a positive integer, it indicates the number of the columns in the facet. If FALSE, each plot is created separately. The default is 2. |
interval |
If FALSE (default), no uncertainty intervals are drawn. If a positive integer, the intervals are this number times the standard error. If a number between 0 and 1 (e.g. 0.95), the equivalent confidence interval is drawn (e.g. 95% CI). In case of the confidence intervals, they are based on the Gaussian distribution. |
interval_color |
The color of the uncertainty intervals. Default is "blue". |
show.legend |
A logical indicating whether the interval legend is shown. Default is FALSE. |
xlab, ylab
|
Names displayed at the x and y axes respectively. Default is "Period" and "Delay" respectively. |
... |
Currently unused argument. |
plot_delay returns a number of ggplot
objects.
Kleanthis Natsiopoulos, [email protected]
ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) delay_mult <- multipliers(ardl_3132, type = 12, se = TRUE) ## Simply plot the delay multipliers ----------------------------------- plot_delay(delay_mult) ## Rearrange them ------------------------------------------------------ plot_delay(delay_mult, facets_ncol = 1) ## Add 1 standard deviation uncertainty intervals ---------------------- plot_delay(delay_mult, interval = 1) ## Add 95% confidence intervals, change color and add legend ----------- plot_delay(delay_mult, interval = 0.95, interval_color = "darkgrey", show.legend = TRUE)ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) delay_mult <- multipliers(ardl_3132, type = 12, se = TRUE) ## Simply plot the delay multipliers ----------------------------------- plot_delay(delay_mult) ## Rearrange them ------------------------------------------------------ plot_delay(delay_mult, facets_ncol = 1) ## Add 1 standard deviation uncertainty intervals ---------------------- plot_delay(delay_mult, interval = 1) ## Add 95% confidence intervals, change color and add legend ----------- plot_delay(delay_mult, interval = 0.95, interval_color = "darkgrey", show.legend = TRUE)
Creates a plot for the long-run relationship in comparison with the dependent
variable, and the fitted values of the model. This is a basic
ggplot with a few customizable parameters.
plot_lr( object, coint_eq, facets = FALSE, show_fitted = FALSE, show.legend = FALSE, xlab = "Time", ... )plot_lr( object, coint_eq, facets = FALSE, show_fitted = FALSE, show.legend = FALSE, xlab = "Time", ... )
object |
An object of |
coint_eq |
The objected returned from |
facets |
A logical indicating whether the long-run relationship appears in a separate plot. Default is FALSE. |
show_fitted |
A logical indicating whether the fitted values are shown. Default is FALSE. |
show.legend |
A logical indicating whether the legend is shown. Default is FALSE. |
xlab |
Name displayed at the x axis. Default is "Time". |
... |
Currently unused argument. |
plot_lr returns a ggplot object.
Kleanthis Natsiopoulos, [email protected]
ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) ce2 <- coint_eq(ardl_3132, case = 2) plot_lr(ardl_3132, coint_eq = ce2) ## Compare fitted values and place long-run relationship separately ---- ce3 <- coint_eq(ardl_3132, case = 3) plot_lr(ardl_3132, coint_eq = ce3, facets = TRUE, show_fitted = TRUE, show.legend = TRUE)ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) ce2 <- coint_eq(ardl_3132, case = 2) plot_lr(ardl_3132, coint_eq = ce2) ## Compare fitted values and place long-run relationship separately ---- ce3 <- coint_eq(ardl_3132, case = 3) plot_lr(ardl_3132, coint_eq = ce3, facets = TRUE, show_fitted = TRUE, show.legend = TRUE)
This data set contains the series used by Pesaran et al. (2001) for estimating the UK earnings equation. The clean format of the data retrieved from the Data Archive of Natsiopoulos and Tzeremes (2022).
PSS2001PSS2001
A time-series object with 112 rows and 7 variables. Time period from 1970:Q1 until 1997:Q4.
real wage
labor productivity
unemployment rate
wedge effect
union power
income policies 1974:Q1-1975:Q4
income policies 1975:Q1-1979:Q4
An object of class "zooreg" "zoo".
http://qed.econ.queensu.ca/jae/datasets/pesaran001/ http://qed.econ.queensu.ca/jae/datasets/natsiopoulos001/
M. Hashem Pesaran, Richard J. Smith, and Yongcheol Shin, (2001), "Bounds Testing Approaches to the Analysis of Level Relationships", Journal of Applied Econometrics, 16, 3, 289–326.
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, 37, 5, 1079–1090. doi:10.1002/jae.2919
Creates the Restricted Error Correction Model (RECM). This is the conditional RECM, which is the RECM of the underlying ARDL.
recm(object, case)recm(object, case)
object |
An object of |
case |
An integer from 1-5 or a character string specifying whether the 'intercept' and/or the 'trend' have to participate in the short-run or the long-run relationship (cointegrating equation) (see section 'Cases' below). |
Note that the statistical significance of 'ect' in a RECM should not be
tested using the corresponding t-statistic (or the p-value) because it
doesn't follow a standard t-distribution. Instead, the
bounds_t_test should be used.
recm returns an object of class
c("dynlm", "lm", "recm"). In addition, attributes 'order', 'data',
'parsed_formula' and 'full_formula' are provided.
The formula of a Restricted ECM conditional to
an is:
In all cases, in is replaced by
According to Pesaran et al. (2001), we distinguish the long-run relationship (cointegrating equation) (and thus the bounds-test and the Restricted ECMs) between 5 different cases. These differ in terms of whether the 'intercept' and/or the 'trend' are restricted to participate in the long-run relationship or they are unrestricted and so they participate in the short-run relationship.
No intercept and no trend.
case inputs: 1 or "n" where "n" stands for none.
Restricted intercept and no trend.
case inputs: 2 or "rc" where "rc" stands for restricted
constant.
Unrestricted intercept and no trend.
case inputs: 3 or "uc" where "uc" stands for unrestricted
constant.
Unrestricted intercept and restricted trend.
case inputs: 4 or "ucrt" where "ucrt" stands for
unrestricted constant and restricted trend.
Unrestricted intercept and unrestricted trend.
case inputs: 5 or "ucut" where "ucut" stands for
unrestricted constant and unrestricted trend.
Note that you can't restrict (or leave unrestricted) a parameter that doesn't
exist in the input model. For example, you can't compute recm(object,
case=3) if the object is an ARDL (or UECM) model with no intercept. The same
way, you can't compute bounds_f_test(object, case=5) if the object is
an ARDL (or UECM) model with no linear trend.
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
Kleanthis Natsiopoulos, [email protected]
data(denmark) ## Estimate the RECM, conditional to it's underlying ARDL(3,1,3,2) ----- # Indirectly from an ARDL ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) recm_3132 <- recm(ardl_3132, case = 2) # Indirectly from an UECM uecm_3132 <- uecm(ardl_3132) recm_3132_ <- recm(uecm_3132, case = 2) identical(recm_3132, recm_3132_) summary(recm_3132) ## Error Correction Term (ect) & Speed of Adjustment ------------------- # The coefficient of the ect, # shows the Speed of Adjustment towards equilibrium. # Note that this can be also be obtained from an UECM, # through the coefficient of the term L(y, 1) (where y is the dependent variable). tail(recm_3132$coefficients, 1) uecm_3132$coefficients[2] ## Post-estimation testing --------------------------------------------- # See examples in the help file of the uecm() functiondata(denmark) ## Estimate the RECM, conditional to it's underlying ARDL(3,1,3,2) ----- # Indirectly from an ARDL ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) recm_3132 <- recm(ardl_3132, case = 2) # Indirectly from an UECM uecm_3132 <- uecm(ardl_3132) recm_3132_ <- recm(uecm_3132, case = 2) identical(recm_3132, recm_3132_) summary(recm_3132) ## Error Correction Term (ect) & Speed of Adjustment ------------------- # The coefficient of the ect, # shows the Speed of Adjustment towards equilibrium. # Note that this can be also be obtained from an UECM, # through the coefficient of the term L(y, 1) (where y is the dependent variable). tail(recm_3132$coefficients, 1) uecm_3132$coefficients[2] ## Post-estimation testing --------------------------------------------- # See examples in the help file of the uecm() function
Takes a dynlm model of class 'ardl',
'uecm' or 'recm' and converts it into an lm model. This
can help using the model as a regular lm model with
functions that are not compatible with dynlm models such
as the predict function to forecast.
to_lm(object, fix_names = FALSE, data_class = NULL, ...)to_lm(object, fix_names = FALSE, data_class = NULL, ...)
object |
An object of |
fix_names |
A logical, indicating whether the variable names should be
rewritten without special functions and character in the names such as "d()"
or "L()". When |
data_class |
If "ts", it converts the data class to
|
... |
Currently unused argument. |
to_lm returns an object of class
"lm".
Kleanthis Natsiopoulos, [email protected]
## Convert ARDL into lm ------------------------------------------------ ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) ardl_3132_lm <- to_lm(ardl_3132) summary(ardl_3132)$coefficients summary(ardl_3132_lm)$coefficients ## Convert UECM into lm ------------------------------------------------ uecm_3132 <- uecm(ardl_3132) uecm_3132_lm <- to_lm(uecm_3132) summary(uecm_3132)$coefficients summary(uecm_3132_lm)$coefficients ## Convert RECM into lm ------------------------------------------------ recm_3132 <- recm(ardl_3132, case = 2) recm_3132_lm <- to_lm(recm_3132) summary(recm_3132)$coefficients summary(recm_3132_lm)$coefficients ## Use the lm model to forecast ---------------------------------------- # Forecast using the in-sample data insample_data <- ardl_3132$model head(insample_data) predicted_values <- predict(ardl_3132_lm, newdata = insample_data) # The predicted values are expected to be the same as the fitted values ardl_3132$fitted.values predicted_values # Convert to ts class for the plot predicted_values <- ts(predicted_values, start = c(1974,4), frequency=4) plot(denmark$LRM, lwd=4) #The input dependent variable lines(ardl_3132$fitted.values, lwd=4, col="blue") #The fitted values lines(predicted_values, lty=2, lwd=2, col="red") #The predicted values ## Convert to lm for post-estimation testing --------------------------- # Ramsey's RESET test for functional form library(lmtest) # for resettest() library(strucchange) # for efp(), and sctest() ## Not run: # This produces an error. # resettest() cannot use data of class 'zoo' such as the 'denmark' data # used to build the original model resettest(uecm_3132, type = c("regressor")) ## End(Not run) uecm_3132_lm <- to_lm(uecm_3132, data_class = "ts") resettest(uecm_3132_lm, power = 2) # CUSUM test for structural change detection ## Not run: # This produces an error. # efp() does not understand special functions such as "d()" and "L()" efp(uecm_3132$full_formula, data = uecm_3132$model) ## End(Not run) uecm_3132_lm_names <- to_lm(uecm_3132, fix_names = TRUE) fluctuation <- efp(uecm_3132_lm_names$full_formula, data = uecm_3132_lm_names$model) sctest(fluctuation) plot(fluctuation)## Convert ARDL into lm ------------------------------------------------ ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) ardl_3132_lm <- to_lm(ardl_3132) summary(ardl_3132)$coefficients summary(ardl_3132_lm)$coefficients ## Convert UECM into lm ------------------------------------------------ uecm_3132 <- uecm(ardl_3132) uecm_3132_lm <- to_lm(uecm_3132) summary(uecm_3132)$coefficients summary(uecm_3132_lm)$coefficients ## Convert RECM into lm ------------------------------------------------ recm_3132 <- recm(ardl_3132, case = 2) recm_3132_lm <- to_lm(recm_3132) summary(recm_3132)$coefficients summary(recm_3132_lm)$coefficients ## Use the lm model to forecast ---------------------------------------- # Forecast using the in-sample data insample_data <- ardl_3132$model head(insample_data) predicted_values <- predict(ardl_3132_lm, newdata = insample_data) # The predicted values are expected to be the same as the fitted values ardl_3132$fitted.values predicted_values # Convert to ts class for the plot predicted_values <- ts(predicted_values, start = c(1974,4), frequency=4) plot(denmark$LRM, lwd=4) #The input dependent variable lines(ardl_3132$fitted.values, lwd=4, col="blue") #The fitted values lines(predicted_values, lty=2, lwd=2, col="red") #The predicted values ## Convert to lm for post-estimation testing --------------------------- # Ramsey's RESET test for functional form library(lmtest) # for resettest() library(strucchange) # for efp(), and sctest() ## Not run: # This produces an error. # resettest() cannot use data of class 'zoo' such as the 'denmark' data # used to build the original model resettest(uecm_3132, type = c("regressor")) ## End(Not run) uecm_3132_lm <- to_lm(uecm_3132, data_class = "ts") resettest(uecm_3132_lm, power = 2) # CUSUM test for structural change detection ## Not run: # This produces an error. # efp() does not understand special functions such as "d()" and "L()" efp(uecm_3132$full_formula, data = uecm_3132$model) ## End(Not run) uecm_3132_lm_names <- to_lm(uecm_3132, fix_names = TRUE) fluctuation <- efp(uecm_3132_lm_names$full_formula, data = uecm_3132_lm_names$model) sctest(fluctuation) plot(fluctuation)
uecm is a generic function used to construct Unrestricted Error
Correction Models (UECM). The function invokes two different
methods. The default method works exactly like
ardl. The other method requires an object of
class 'ardl'. Both methods create the conditional UECM,
which is the UECM of the underlying ARDL.
uecm(...) ## S3 method for class 'ardl' uecm(object, ...) ## Default S3 method: uecm(formula, data, order, start = NULL, end = NULL, ...)uecm(...) ## S3 method for class 'ardl' uecm(object, ...) ## Default S3 method: uecm(formula, data, order, start = NULL, end = NULL, ...)
... |
Additional arguments to be passed to the low level regression fitting functions. |
object |
An object of |
formula |
A "formula" describing the linear model. Details for model specification are given under 'Details'. |
data |
A time series object (e.g., "ts", "zoo" or "zooreg") or a data
frame containing the variables in the model. In the case of a data frame,
it is coerced into a |
order |
A specification of the order of the underlying ARDL model (e.g.,
for the UECM of an ARDL(1,0,2) model it should be |
start |
Start of the time period which should be used for fitting the model. |
end |
End of the time period which should be used for fitting the model. |
The formula should contain only variables that exist in the data
provided through data plus some additional functions supported by
dynlm (i.e., trend()).
You can also specify fixed variables that are not supposed to be lagged (e.g.
dummies etc.) simply by placing them after |. For example, y ~
x1 + x2 | z1 + z2 where z1 and z2 are the fixed variables and
should not be considered in order. Note that the | notion
should not be confused with the same notion in dynlm where it
introduces instrumental variables.
uecm returns an object of class
c("dynlm", "lm", "uecm"). In addition, attributes 'order', 'data',
'parsed_formula' and 'full_formula' are provided.
The formula of an Unrestricted ECM conditional
to an is:
In addition, and cancel out
becoming
Kleanthis Natsiopoulos, [email protected]
data(denmark) ## Estimate the UECM, conditional to it's underlying ARDL(3,1,3,2) ----- # Indirectly ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) uecm_3132 <- uecm(ardl_3132) # Directly uecm_3132_ <- uecm(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) identical(uecm_3132, uecm_3132_) summary(uecm_3132) ## Post-estimation testing --------------------------------------------- library(lmtest) # for bgtest(), bptest(), and resettest() library(tseries) # for jarque.bera.test() library(strucchange) # for efp(), and sctest() # Breusch-Godfrey test for higher-order serial correlation bgtest(uecm_3132, order = 4) # Breusch-Pagan test against heteroskedasticity bptest(uecm_3132) # Ramsey's RESET test for functional form ## Not run: # This produces an error. # resettest() cannot use data of class 'zoo' such as the 'denmark' data # used to build the original model resettest(uecm_3132, type = c("regressor")) ## End(Not run) uecm_3132_lm <- to_lm(uecm_3132, data_class = "ts") resettest(uecm_3132_lm, power = 2) # Jarque-Bera test for normality jarque.bera.test(residuals(uecm_3132)) # CUSUM test for structural change detection ## Not run: # This produces an error. # efp() does not understand special functions such as "d()" and "L()" efp(uecm_3132$full_formula, data = uecm_3132$model) ## End(Not run) uecm_3132_lm_names <- to_lm(uecm_3132, fix_names = TRUE) fluctuation <- efp(uecm_3132_lm_names$full_formula, data = uecm_3132_lm_names$model) sctest(fluctuation) plot(fluctuation)data(denmark) ## Estimate the UECM, conditional to it's underlying ARDL(3,1,3,2) ----- # Indirectly ardl_3132 <- ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) uecm_3132 <- uecm(ardl_3132) # Directly uecm_3132_ <- uecm(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2)) identical(uecm_3132, uecm_3132_) summary(uecm_3132) ## Post-estimation testing --------------------------------------------- library(lmtest) # for bgtest(), bptest(), and resettest() library(tseries) # for jarque.bera.test() library(strucchange) # for efp(), and sctest() # Breusch-Godfrey test for higher-order serial correlation bgtest(uecm_3132, order = 4) # Breusch-Pagan test against heteroskedasticity bptest(uecm_3132) # Ramsey's RESET test for functional form ## Not run: # This produces an error. # resettest() cannot use data of class 'zoo' such as the 'denmark' data # used to build the original model resettest(uecm_3132, type = c("regressor")) ## End(Not run) uecm_3132_lm <- to_lm(uecm_3132, data_class = "ts") resettest(uecm_3132_lm, power = 2) # Jarque-Bera test for normality jarque.bera.test(residuals(uecm_3132)) # CUSUM test for structural change detection ## Not run: # This produces an error. # efp() does not understand special functions such as "d()" and "L()" efp(uecm_3132$full_formula, data = uecm_3132$model) ## End(Not run) uecm_3132_lm_names <- to_lm(uecm_3132, fix_names = TRUE) fluctuation <- efp(uecm_3132_lm_names$full_formula, data = uecm_3132_lm_names$model) sctest(fluctuation) plot(fluctuation)