# multipliers: Multipliers estimation In ARDL: ARDL, ECM and Bounds-Test for Cointegration

## Description

multipliers is a generic function used to estimate short-run (impact), interim and long-run (total) multipliers, along with their corresponding standard errors, t-statistics and p-values.

## Usage

 1 2 3 4 5 6 7 multipliers(object, type = "lr", vcov_matrix = NULL) ## S3 method for class 'ardl' multipliers(object, type = "lr", vcov_matrix = NULL) ## S3 method for class 'uecm' multipliers(object, type = "lr", vcov_matrix = NULL) 

## Arguments

 object An object of class 'ardl' or 'uecm'. 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 100 for 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 vcov(object) (when vcov_matrix = NULL), but other estimations of the covariance matrix of the regression's estimated coefficients can also be used (e.g., using vcovHC or vcovHAC).

## 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.

Note that type = 0 is equivalent to type = "sr". Also, type = s will produce the same estimates as type = "lr" for those particular variable for which s >= from their ARDL order.

The delta method is used for approximating the standard errors (and thus the t-statistics and p-values) of the estimated multipliers.

## Value

multipliers returns 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.

## Mathematical Formula

Constant and Linear Trend:

As derived from an ARDL:

μ = \frac{c_{0}}{1-∑_{i=1}^{p}b_{y,i}}

δ = \frac{c_{1}}{1-∑_{i=1}^{p}b_{y,i}}

As derived from an Unrestricted ECM:

μ = \frac{c_{0}}{-π_{y}}

δ = \frac{c_{1}}{-π_{y}}

Short-Run Multipliers:

As derived from an ARDL:

\frac{\partial y_{t}}{\partial x_{j,t}} = \frac{b_{j,0}}{1-∑_{i=1}^{p}b_{y,i}} \;\;\;\;\; \forall j=1,…,k

As derived from an Unrestricted ECM:

\frac{\partial y_{t}}{\partial x_{j,t}} = \frac{ω_{j}}{-π_{y}} \;\;\;\;\; \forall j=1,…,k

Interim Multipliers:

As derived from an ARDL:

\frac{\partial y_{t+s}}{\partial x_{j,t}} = \frac{∑_{l=1}^{s}b_{j,l}}{1-∑_{i=1}^{p}b_{y,i}} \;\;\;\;\; \forall j=1,…,k \;\;\;\;\; s \in \{0,…,q_{j}\}

As derived from an Unrestricted ECM:

\frac{\partial y_{t+s}}{\partial x_{j,t}} = \frac{π_{j} + ψ_{j,s}}{-π_{y}} \;\;\;\;\; \forall j=1,…,k \;\;\;\;\; s \in \{1,…,q_{j}-1\}

Long-Run Multipliers:

As derived from an ARDL:

\frac{\partial y_{t+∞}}{\partial x_{j,t}} = θ_{j} = \frac{∑_{l=0}^{q_{j}}b_{j,l}}{1-∑_{i=1}^{p}b_{y,i}} \;\;\;\;\; \forall j=1,…,k

As derived from an Unrestricted ECM:

\frac{\partial y_{t+∞}}{\partial x_{j,t}} = θ_{j} = \frac{π_{j}}{-π_{y}} \;\;\;\;\; \forall j=1,…,k

## Author(s)

Kleanthis Natsiopoulos, klnatsio@gmail.com

ardl, uecm
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 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 interim multipliers of an ARDL(3,1,3,2) model ---------- # Note that the estimated interim multipliers match the long-run multipliers # for those variables that their ARDL order equals or exceeds the interim step mult_lr <- multipliers(uecm_3132, type = "lr") mult_1 <- multipliers(uecm_3132, type = 1) mult_2 <- multipliers(uecm_3132, type = 2) uecm_3132\$order mult_lr mult_1 mult_2