JOSS-paper/paper.md

In Natsiopoulos/ARDL: ARDL, ECM and Bounds-Test for Cointegration

title: 'ARDL: An R package for the analysis of level relationships' tags: - ARDL - bounds test - cointegration - econometrics - R date: "10 August 2022" output: pdf_document: default word_document: default html_document: df_print: paged authors: - name: Kleanthis Natsiopoulos orcid: 0000-0003-1180-2984 affiliation: 1 - name: Nickolaos G. Tzeremes orcid: 0000-0002-6938-3404 affiliation: 1 bibliography: paper.bib affiliations: - name: Department of Economics, University of Thessaly, 28th October, 78, 38333, Volos, Greece index: 1

Summary

Autoregressive Distributed Lag (ARDL) and Error Correction Models (ECM) [@Pesaran1999] are widely used in various economic, environmental, political science applications etc. These are very flexible because of the autoregressive (AR) and the distributed lag (DL, essentially AR terms of the independent variables) terms in the ARDL model. Also, they are used in the context of cointegration analysis as a platform to test and analyze the levels (long-run) relationship between variables. One of the most popular such tests is the bounds test proposed by @Pesaran2001 which allows testing for cointegration while at the same time estimates the level relationship. ARDL [@Natsiopoulos2022; @ardl-pkg] is an R package that aims to help users in the modeling process of ARDL and ECM and it also provides the tools towards the bounds test for cointegration.

Statement of need

ARDL is implemented in such a way that researchers can use it as a full featured tool for this specific type of analysis and students of all levels can be aware of how each piece of code works through the analytical manual and the examples which cover every functionality of the package. One of the greatest advantages of the package is that there is no need for the user to know the complex mathematical equations and write them by hand. Instead, the correct specification of the model is created automatically, just by providing the order of the ARDL. Some recent examples of publications that used the package are @baruah2022 and @Qiu2021.

It is very important to note that the ARDL package is the first software which managed to fully replicate the original paper by @Pesaran2001, verifying the validity and accuracy of the package [@Natsiopoulos2022].

State of the field

ARDL distinguishes itself from other related R packages like dLagM [@dlagm-pkg] and dynamac [@dynamac-pkg] in the sense that it is specifically designed to address a particular problem throughout every phase of modeling, testing and interpretation. It includes a rich set of dedicated tools, accompanied by an analytical manual that describes the mathematical process of every function.

On the other hand, dLagM and dynamac provide additional plotting functionalities and post-estimation diagnostics (serial correlation tests etc). The design of the ARDL packages differs from the pre-mentioned packages as it is not an all-in-one package. It is designed so that every exported object is of commonly used R classes so that it can be easily combined with other packages, each of which are also dedicated to a specific part of the post-estimation part of the analysis.

In addition, the ARDL packages natively supports time-series data, thus sub-sample and balanced sample estimations are possible. It also allows to specify any level of significance for the bounds F-test and t-tests, and includes p-values and exact sample critical values for any possible combination. Also, the estimation of long-run, short-run and interim multipliers accompanied by standard errors and p-values as well as the cointegrating equation are some of the available features.

In the following example, we show how the estimation of an ARDL model can be performed using the ARDL package, as opposed to when we use a package specialized in dynamic time series modeling (i.e. dynlm [@dynlm-pkg] which is also the package that the ARDL package uses under the hood).

For this example we use the Danish data on money income prices and interest rates, from @Johansen1990. We model the logarithm of real money, M2 (LRM) using the independent variables LRY, IBO and IDE (see ?denmark), and we estimate the ARDL(3,1,3,2).

Using the ARDL package we would write the following code:

library(ARDL)

ardl(LRM ~ LRY + IBO + IDE, data = denmark, order = c(3,1,3,2))

While using the dynlm package:

library(dynlm)

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)

Note that if someone wanted to estimate the same regression using the native lm function this would be even harder as the lags and differences are not supported inside the lm environment.

To convert this ARDL(3,1,3,2) model into an ECM, using the ARDL package we would write:

uecm_model <- uecm(ardl_model)

While using the dynlm package:

dynlm(d(LRM) ~ L(LRM, 1) + L(LRY, 1) + L(IBO, 1) +
                           L(IDE, 1) + d(L(LRM, 1)) + d(L(LRM, 2)) +
                           d(LRY) + d(IBO) + d(L(IBO, 1)) + d(L(IBO, 2)) +
                           d(IDE) + d(L(IDE, 1)), data = denmark)

The API of dynamac has a similar approach with dynlm and one has to define every independent variable one by one. The API of dLagM requires less typing and the way for one to define an ARDL model is by setting the maximum lags for the dependent and the independent variables (the same maximum order for all the independent ones as in a typical VAR model) and then remove one by one the lags that are not required from each independent variable.

As this is a quick presentation of just one of the functionalities of the package, we are not going to list all the key advantages of the ARDL package or compare them with the limitations in other open source or proprietary software. We refere to these two aforementioned packages as these seem to be the R packages that argue that they support the same functionalities as the ARDL package. We believe that using the ARDL package is simpler, faster and less prone to errors.

Breaking down the ARDL package

Model estimation

The package does not explicitly connects the modeling with the bounds test, as the ARDL and ECM model may well be used independently in other research approaches. This way each function is dedicated to perform a very specific part of the whole process.

An $ARDL(p, q_1, ..., q_k)$ model can be fully specified by its order. The ardl function implements the following formula:

\begin{equation}\label{eq:ardl} 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} \end{equation}

An Unrestricted ECM (UECM) has a 1:1 relationship with the $ARDL(p, q_1, ..., q_k)$ model. The uecm function estimates it using the formula:

\begin{equation}\label{eq:uecm} \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} \end{equation}

The Restricted ECM (RECM) can also be fully described by the order of the underlying ARDL and the case for potential restriction in the deterministic parameters (constant and linear trend), using the function recm.

Model relationships

\autoref{fig:relationships} shows the relationships between the regression model objects. The bounds tests and all the other functions of the package also have this kind of interconnectivity as they inherit all the necessary information.

{ width=70% }

Where $ARDL(p, q_1, ..., q_k)$ represent the ARDL order and Case is the restriction of the deterministic parameters. When an arrow points from one model or information (order or case) to another model, it can be interpreted as the first one can fully describe the second one. When it crosses the case it means that this information is also needed.

Bounds test

The functions bounds_f_test and bounds_t_test return a typical htest object and they perform a Wald or t test respectively on an UECM. The input of the functions can also be an ARDL model as they are interconnected as described above. The hypothesis tests for the bounds F-test and t-test are based on (\ref{eq:uecm}).

For the bounds F-test the null hypothesis is $\mathbf{H_{0}:} \pi_{y} = \pi_{1} = \dots = \pi_{k} = c_{0} = c_{1} = 0$, where the restriction of the deterministic trends $c_{0}$ and $c_{1}$ depends on the case chosen.

The null hypothesis of the bounds t-test is $\mathbf{H_{0}:} \pi_{y} = 0$.

Making inference after cointegration

After the modeling part and if a cointegrating relationship can be established, the long-run (but also the short-run or interim) multipliers can be computed using the multipliers function. Note that it is irrelevant whether they are estimated based on an ARDL (\ref{eq:ardl}) or an UECM (\ref{eq:uecm}), in terms of the results.

Lastly, the cointegrating relationship vector can be constructed using the function coint_eq.

Conclusion

Using the intuitive API of the ARDL package, even the most complex ARDL, UECM or RECM model can be fully specified using the $ARDL(p, q_{1}, ..., q_{k})$ order and the case for the restriction of the constant and linear deterministic parameters. The interconnection between the objects allows for further post-estimation testing. Finally, the detailed estimation formulas are provided in order to enhance the understanding of every step of the analysis giving a reference point for the equations in publications.

Acknowledgments

We want to acknowledge Nikolaos Chatsios for designing the package's logo and the diagram in \autoref{fig:relationships}.

References

Natsiopoulos/ARDL documentation built on Sept. 2, 2023, 2:33 a.m.