Type:	Package
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>.
Version:	0.2.4
BugReports:	https://github.com/Natsiopoulos/ARDL/issues
License:	GPL-3
URL:	https://github.com/Natsiopoulos/ARDL
Encoding:	UTF-8
LazyData:	true
Depends:	R (≥ 3.5.0)
Suggests:	strucchange, tseries, qpcR, sandwich, testthat (≥ 3.0.0)
Imports:	aod, dplyr, dynlm, gridExtra, ggplot2, lmtest, msm, stringr, zoo
RoxygenNote:	7.2.3
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2023-08-20 09:29:37 UTC; axter
Author:	Kleanthis Natsiopoulos [aut, cre], Nickolaos Tzeremes [aut]
Maintainer:	Kleanthis Natsiopoulos <klnatsio@gmail.com>
Repository:	CRAN
Date/Publication:	2023-08-21 08:02:41 UTC

ARDL: 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.

Author(s)

Maintainer: Kleanthis Natsiopoulos klnatsio@gmail.com (ORCID)

Authors:

• Nickolaos Tzeremes bus9nt@econ.uth.gr (ORCID)

See Also

Useful links:

• https://github.com/Natsiopoulos/ARDL

• Report bugs at https://github.com/Natsiopoulos/ARDL/issues

The UK earnings equation data from Natsiopoulos and Tzeremes (2022)

Description

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).

Usage

NT2022

Format

A time-series object with 196 rows and 9 variables. Time period from 1971:Q1 until 2019:Q4.

time

time variable

w

real wage

Prod

labor productivity

UR

unemployment rate

Wedge

wedge effect

Union

union power

D7475

income policies 1974:Q1-1975:Q4

D7579

income policies 1975:Q1-1979:Q4

UnionR

union membership

Details

An object of class "zooreg" "zoo".

Source

http://qed.econ.queensu.ca/jae/datasets/natsiopoulos001/

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, 37, 5, 1079–1090. doi:10.1002/jae.2919

The UK earnings equation data from Pesaran et al. (2001)

Description

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).

Usage

PSS2001

Format

A time-series object with 112 rows and 7 variables. Time period from 1970:Q1 until 1997:Q4.

w

real wage

Prod

labor productivity

UR

unemployment rate

Wedge

wedge effect

Union

union power

D7475

income policies 1974:Q1-1975:Q4

D7579

income policies 1975:Q1-1979:Q4

Details

An object of class "zooreg" "zoo".

Source

http://qed.econ.queensu.ca/jae/datasets/pesaran001/ http://qed.econ.queensu.ca/jae/datasets/natsiopoulos001/

References

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

ARDL model regression

Description

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'.

Usage

ardl(...)

## S3 method for class 'uecm'
ardl(object, ...)

## Default S3 method:
ardl(formula, data, order, start = NULL, end = NULL, ...)

Arguments

Details

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.

Value

ardl returns an object of class c("dynlm", "lm", "ardl"). In addition, attributes 'order', 'data', 'parsed_formula' and 'full_formula' are provided.

Mathematical Formula

The general form of an ARDL(p,q_{1},\dots,q_{k}) is:

y_{t} = c_{0} + c_{1}t + \sum_{i=1}^{p}b_{y,i}y_{t-i} + \sum_{j=1}^{k}\sum_{l=0}^{q_{j}}b_{j,l}x_{j,t-l} + \epsilon_{t}

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

uecm, recm

Examples

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() function

Automatic ARDL model selection

Description

It searches for the best ARDL order specification, according to the selected criterion, taking into account the constraints provided.

Usage

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,
  ...
)

Arguments

Value

auto_ardl returns a list which contains:

Searching algorithm

The algorithm performs the optimization process starting from multiple starting points concerning the autoregressive order p. 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:

starting_order = c(3, 0, 1, 2):

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

ardl

Examples

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 Wald-test for no cointegration

Description

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.

Usage

bounds_f_test(
  object,
  case,
  alpha = NULL,
  pvalue = TRUE,
  exact = FALSE,
  R = 40000,
  test = c("F", "Chisq"),
  vcov_matrix = NULL
)

Arguments

Value

A list with class "htest" containing the following components:

Hypothesis testing

\Delta y_{t} = c_{0} + c_{1}t + \pi_{y}y_{t-1} + \sum_{j=1}^{k}\pi_{j}x_{j,t-1} + \sum_{i=1}^{p-1}\psi_{y,i}\Delta y_{t-i} + \sum_{j=1}^{k}\sum_{l=1}^{q_{j}-1} \psi_{j,l}\Delta x_{j,t-l} + \sum_{j=1}^{k}\omega_{j}\Delta x_{j,t} + \epsilon_{t}

Cases 1, 3, 5:

\mathbf{H_{0}:} \pi_{y} = \pi_{1} = \dots = \pi_{k} = 0

\mathbf{H_{1}:} \pi_{y} \neq \pi_{1} \neq \dots \neq \pi_{k} \neq 0

Case 2:

\mathbf{H_{0}:} \pi_{y} = \pi_{1} = \dots = \pi_{k} = c_{0} = 0

\mathbf{H_{1}:} \pi_{y} \neq \pi_{1} \neq \dots \neq \pi_{k} \neq c_{0} \neq 0

Case 4:

\mathbf{H_{0}:} \pi_{y} = \pi_{1} = \dots = \pi_{k} = c_{1} = 0

\mathbf{H_{1}:} \pi_{y} \neq \pi_{1} \neq \dots \neq \pi_{k} \neq c_{1} \neq 0

alpha, bounds and 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).

Cases

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.

Case 1:

• No intercept and no trend.

• case inputs: 1 or "n" where "n" stands for none.

Case 2:

• Restricted intercept and no trend.

• case inputs: 2 or "rc" where "rc" stands for restricted constant.

Case 3:

• Unrestricted intercept and no trend.

• case inputs: 3 or "uc" where "uc" stands for unrestricted constant.

Case 4:

• Unrestricted intercept and restricted trend.

• case inputs: 4 or "ucrt" where "ucrt" stands for unrestricted constant and restricted trend.

Case 5:

• 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.

References

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

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

bounds_t_test ardl uecm

Examples

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 for no cointegration

Description

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).

Usage

bounds_t_test(
  object,
  case,
  alpha = NULL,
  pvalue = TRUE,
  exact = FALSE,
  R = 40000,
  vcov_matrix = NULL
)

Arguments

Value

A list with class "htest" containing the following components:

Hypothesis testing

\Delta y_{t} = c_{0} + c_{1}t + \pi_{y}y_{t-1} + \sum_{j=1}^{k}\pi_{j}x_{j,t-1} + \sum_{i=1}^{p-1}\psi_{y,i}\Delta y_{t-i} + \sum_{j=1}^{k}\sum_{l=1}^{q_{j}-1} \psi_{j,l}\Delta x_{j,t-l} + \sum_{j=1}^{k}\omega_{j}\Delta x_{j,t} + \epsilon_{t}

\mathbf{H_{0}:} \pi_{y} = 0

\mathbf{H_{1}:} \pi_{y} \neq 0

alpha, bounds and 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).

Cases

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.

Case 1:

• No intercept and no trend.

• case inputs: 1 or "n" where "n" stands for none.

Case 2:

• Restricted intercept and no trend.

• case inputs: 2 or "rc" where "rc" stands for restricted constant.

Case 3:

• Unrestricted intercept and no trend.

• case inputs: 3 or "uc" where "uc" stands for unrestricted constant.

Case 4:

• Unrestricted intercept and restricted trend.

• case inputs: 4 or "ucrt" where "ucrt" stands for unrestricted constant and restricted trend.

Case 5:

• 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.

References

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

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

bounds_f_test ardl uecm

Examples

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)

ARDL formula specification builder

Description

It creates the ARDL specification according to the given "formula" and their corresponding "orders".

Usage

build_ardl_formula(parsed_formula, order)

Arguments

Value

build_ardl_formula returns a list containing the full formula and the independent and dependent parts of the formula separated. The full formula is ready to be used as input in the dynlm function.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

build_uecm_formula, build_recm_formula

RECM formula specification builder

Description

It creates the RECM (Restricted Error Correction Model) specification according to the given "formula" and the corresponding "order" of the underlying ARDL.

Usage

build_recm_formula(parsed_formula, order, case)

Arguments

Value

build_recm_formula returns a list containing the full formula and the independent and dependent parts of the formula separated. The full formula is ready to be used as input in the dynlm function providing also the 'ect' (error correction term) to the data.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

build_ardl_formula, build_uecm_formula

UECM formula specification builder

Description

It creates the UECM (Unrestricted Error Correction Model) specification according to the given "formula" and the corresponding "order" of the underlying ARDL.

Usage

build_uecm_formula(parsed_formula, order)

Arguments

Value

build_uecm_formula returns a list containing the full formula and the independent and dependent parts of the formula separated. The full formula is ready to be used as input in the dynlm function.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

build_ardl_formula, build_recm_formula

Cointegrating equation (long-run level relationship)

Description

Creates the cointegrating equation (long-run level relationship) providing an 'ardl', 'uecm' or 'recm' model.

Usage

coint_eq(object, case)

## S3 method for class 'recm'
coint_eq(object, ...)

## Default S3 method:
coint_eq(object, case)

Arguments

Value

coint_eq returns an numeric vector containing the cointegrating equation.

Cases

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.

Case 1:

• No intercept and no trend.

• case inputs: 1 or "n" where "n" stands for none.

Case 2:

• Restricted intercept and no trend.

• case inputs: 2 or "rc" where "rc" stands for restricted constant.

Case 3:

• Unrestricted intercept and no trend.

• case inputs: 3 or "uc" where "uc" stands for unrestricted constant.

Case 4:

• Unrestricted intercept and restricted trend.

• case inputs: 4 or "ucrt" where "ucrt" stands for unrestricted constant and restricted trend.

Case 5:

• 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.

References

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

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

plot_lr ardl uecm recm bounds_f_test bounds_t_test

Examples

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)

Delta method

Description

An internal generic function, customized for approximating the standard errors of the estimated multipliers.

Usage

delta_method(object, vcov_matrix = NULL)

## S3 method for class 'ardl'
delta_method(object, vcov_matrix = NULL)

## S3 method for class 'uecm'
delta_method(object, vcov_matrix = NULL)

Arguments

Details

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.

Value

delta_method returns a numeric vector of the same length as the number of the independent variables (excluding the fixed ones) in the model.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

multipliers, ardl, uecm

The Danish data on money income prices and interest rates

Description

This data set contains the series used by S. Johansen and K. Juselius for estimating a money demand function of Denmark.

Usage

denmark

Format

A time-series object with 55 rows and 5 variables. Time period from 1974:Q1 until 1987:Q3.

LRM

logarithm of real money, M2

LRY

logarithm of real income

LPY

logarithm of price deflator

IBO

bond rate

IDE

bank deposit rate

Details

An object of class "zooreg" "zoo".

Source

https://onlinelibrary.wiley.com/doi/10.1111/j.1468-0084.1990.mp52002003.x

References

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.

Critical value bounds stochastic simulation for Wald bounds-test for no cointegration

Description

f_bounds_sim simulates the critical value bounds for the Wald bounds-test for no cointegration Pesaran et al. (2001) expressed both as F-statistics and as Chisq-statistics.

Usage

f_bounds_sim(case, k, alpha, T, R = 40000)

Arguments

Value

f_bounds_sim returns a list containing two data frames. One with the critical value bounds for the F-statistic and one with the critical value bounds for the Chisq-statistic.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

t_bounds_sim bounds_f_test

F-test of regression's overall significance

Description

f_test_custom performs an overall significance F-test on a regression. It is used along with .lm.fit to get the F-statistic as this is about 10 times faster than extracting it from a regression using lm.

Usage

f_test_custom(dep_var, indep_vars, model_res, const = TRUE)

Arguments

Value

f_test_custom returns a list containing the F-statistic and the numerator's degrees of freedom.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

vcov_custom f_bounds_sim t_bounds_simindependent

Multipliers estimation

Description

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.

Usage

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)

Arguments

Details

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.

Value

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.

Mathematical Formula

Short-Run Multipliers:

As derived from an ARDL:

\frac{\partial y_{t}}{\partial x_{j,t}} = b_{j,0} \;\;\;\;\; j \in \{1,\dots,k\}

As derived from an Unrestricted ECM:

\frac{\partial y_{t}}{\partial x_{j,t}} = \omega_{j} \;\;\;\;\; j \in \{1,\dots,k\}

Constant and Linear Trend:

c_{0}

c_{1}

Delay & Interim Multipliers:

As derived from an ARDL:

Delay_{x_{j},s} = \frac{\partial y_{t+s}}{\partial x_{j,t}} = b_{j,s} + \sum_{i=1}^{min\{p,s\}} b_{y,i} \frac{\partial y_{t+(s-i)}}{\partial x_{j,t}} \;\;\;\;\; b_{j,s} = 0 \;\; \forall \;\; s > q

Interim_{x_{j},s} = \sum_{i=0}^{s} Delay_{x_{j},s}

Constant and Linear Trend:

Delay_{intercept,s} = c_{0} + \sum_{i=1}^{min\{p,s\}} b_{y,i} Delay_{intercept,s-i} \;\;\;\;\; c_{0} = 0 \;\; \forall \;\; s \neq 0

Interim_{intercept,s} = \sum_{i=0}^{s} Delay_{intercept,s}

Delay_{trend,s} = c_{1} + \sum_{i=1}^{min\{p,s\}} b_{y,i} Delay_{trend,s-i} \;\;\;\;\; c_{1} = 0 \;\; \forall \;\; s \neq 0

Interim_{trend,s} = \sum_{i=0}^{s} Delay_{trend,s}

Long-Run Multipliers:

As derived from an ARDL:

\frac{\partial y_{t+\infty}}{\partial x_{j,t}} = \theta_{j} = \frac{\sum_{l=0}^{q_{j}}b_{j,l}}{1-\sum_{i=1}^{p}b_{y,i}} \;\;\;\;\; j \in \{1,\dots,k\}

Constant and Linear Trend:

\mu = \frac{c_{0}}{1-\sum_{i=1}^{p}b_{y,i}}

\delta = \frac{c_{1}}{1-\sum_{i=1}^{p}b_{y,i}}

As derived from an Unrestricted ECM:

\frac{\partial y_{t+\infty}}{\partial x_{j,t}} = \theta_{j} = \frac{\pi_{j}}{-\pi_{y}} \;\;\;\;\; j \in \{1,\dots,k\}

Constant and Linear Trend:

\mu = \frac{c_{0}}{-\pi_{y}}

\delta = \frac{c_{1}}{-\pi_{y}}

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

ardl, uecm, plot_delay

Examples

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)

Case parser

Description

It parses the 'case' and checks the integrity of the 'case' input and the compatibility with the formula.

Usage

parse_case(parsed_formula, case)

Arguments

Details

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.

Value

An integer from 1-5 representing the case.

References

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

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

parse_formula

Formula parser

Description

It parses the formula and separates the dependent, independent and fixed variables and also the constant and linear trends (if present).

Usage

parse_formula(formula, colnames_data)

Arguments

Details

The notation we follow (e.g., using y, x, z, w etc.) is according to Pesaran et al. (2001).

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.

Value

A list containing other lists with the names of the dependent, independent and fixed variables, the constant and linear trends and the number of variables in each category.

References

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

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

parse_order

Order parser

Description

It parses the order and checks the integrity of the order input.

Usage

parse_order(orders, order_name, var_names, kz, restriction = FALSE)

Arguments

Value

A numeric vector of the same length as the total number of variables (excluding the fixed ones).

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

parse_formula

Create plots for the delay multipliers

Description

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.

Usage

plot_delay(
  multipliers,
  facets_ncol = 2,
  interval = FALSE,
  interval_color = "blue",
  show.legend = FALSE,
  xlab = "Period",
  ylab = "Delay",
  ...
)

Arguments

Value

plot_delay returns a number of ggplot objects.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

multipliers

Examples

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)

Create plot for the long-run (cointegrating) equation

Description

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.

Usage

plot_lr(
  object,
  coint_eq,
  facets = FALSE,
  show_fitted = FALSE,
  show.legend = FALSE,
  xlab = "Time",
  ...
)

Arguments

Value

plot_lr returns a ggplot object.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

coint_eq

Examples

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)

Restricted ECM regression

Description

Creates the Restricted Error Correction Model (RECM). This is the conditional RECM, which is the RECM of the underlying ARDL.

Usage

recm(object, case)

Arguments

Details

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.

Value

recm returns an object of class c("dynlm", "lm", "recm"). In addition, attributes 'order', 'data', 'parsed_formula' and 'full_formula' are provided.

Mathematical Formula

The formula of a Restricted ECM conditional to an ARDL(p,q_{1},\dots,q_{k}) is:

\Delta y_{t} = c_{0} + c_{1}t + \sum_{i=1}^{p-1}\psi_{y,i}\Delta y_{t-i} + \sum_{j=1}^{k}\sum_{l=1}^{q_{j}-1} \psi_{j,l}\Delta x_{j,t-l} + \sum_{j=1}^{k}\omega_{j}\Delta x_{j,t} + \pi_{y}ECT_{t} + \epsilon_{t}

\psi_{j,l} = 0 \;\; \forall \;\; q_{j} = 1, \psi_{j,l} = \omega_{j} = 0 \;\; \forall \;\; q_{j} = 0

Under Case 1:

• c_{0}=c_{1}=0

• ECT = y_{t-1} - (\sum_{j=1}^{k} \theta_{j} x_{j,t-1})

Under Case 2:

• c_{0}=c_{1}=0

• ECT = y_{t-1} - (\mu + \sum_{j=1}^{k}\theta_{j} x_{j,t-1})

Under Case 3:

• c_{1}=0

• ECT = y_{t-1} - (\sum_{j=1}^{k} \theta_{j} x_{j,t-1})

Under Case 4:

• c_{1}=0

• ECT = y_{t-1} - (\delta(t-1)+ \sum_{j=1}^{k} \theta_{j} x_{j,t-1})

Under Case 5:

• ECT = y_{t-1} - (\sum_{j=1}^{k} \theta_{j} x_{j,t-1})

In all cases, x_{j,t-1} in ECT is replaced by x_{j,t} \;\;\;\;\; \forall \;\; q_{j} = 0

Cases

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.

Case 1:

• No intercept and no trend.

• case inputs: 1 or "n" where "n" stands for none.

Case 2:

• Restricted intercept and no trend.

• case inputs: 2 or "rc" where "rc" stands for restricted constant.

Case 3:

• Unrestricted intercept and no trend.

• case inputs: 3 or "uc" where "uc" stands for unrestricted constant.

Case 4:

• Unrestricted intercept and restricted trend.

• case inputs: 4 or "ucrt" where "ucrt" stands for unrestricted constant and restricted trend.

Case 5:

• 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.

References

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

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

ardl uecm

Examples

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() function

Critical value bounds stochastic simulation for t-bounds test for no cointegration

Description

t_bounds_sim simulates the critical value bounds for the t-bounds test for no cointegration Pesaran et al. (2001).

Usage

t_bounds_sim(case, k, alpha, T, R = 40000)

Arguments

Value

t_bounds_sim returns a data frame with the critical value bounds for the t-statistic.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

f_bounds_sim bounds_t_test

Convert dynlm model (ardl, uecm, recm) to lm model

Description

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.

Usage

to_lm(object, fix_names = FALSE, data_class = NULL, ...)

Arguments

Value

to_lm returns an object of class "lm".

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

ardl, uecm, recm

Examples

## 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)

Unrestricted ECM regression

Description

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.

Usage

uecm(...)

## S3 method for class 'ardl'
uecm(object, ...)

## Default S3 method:
uecm(formula, data, order, start = NULL, end = NULL, ...)

Arguments

Details

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.

Value

uecm returns an object of class c("dynlm", "lm", "uecm"). In addition, attributes 'order', 'data', 'parsed_formula' and 'full_formula' are provided.

Mathematical Formula

The formula of an Unrestricted ECM conditional to an ARDL(p,q_{1},\dots,q_{k}) is:

\Delta y_{t} = c_{0} + c_{1}t + \pi_{y}y_{t-1} + \sum_{j=1}^{k}\pi_{j}x_{j,t-1} + \sum_{i=1}^{p-1}\psi_{y,i}\Delta y_{t-i} + \sum_{j=1}^{k}\sum_{l=1}^{q_{j}-1} \psi_{j,l}\Delta x_{j,t-l} + \sum_{j=1}^{k}\omega_{j}\Delta x_{j,t} + \epsilon_{t}

\psi_{j,l} = 0 \;\; \forall \;\; q_{j} \leq 1, \;\;\;\;\; \psi_{y,i} = 0 \;\; if \;\; p = 1

In addition, x_{j,t-1} and \Delta x_{j,t} cancel out becoming x_{j,t} \;\; \forall \;\; q_{j} = 0

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

ardl recm

Examples

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)

Variance-Covariance matrix of a regression

Description

vcov_custom creates the Variance-Covariance matrix of a regression. It is used instead of the vcov because the latter doesn't work with .lm.fit.

Usage

vcov_custom(indep_vars, model_res)

Arguments

Value

vcov_custom returns a Variance-Covariance matrix.

Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

See Also

f_test_custom f_bounds_sim t_bounds_sim