ARDL (version 0.2.3)

recm: 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)

Value

recm returns an object of class

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

Arguments

object

An object of class 'ardl' or 'uecm'.

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).

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.

Author

Kleanthis Natsiopoulos,

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.

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

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]