9.1 Impulse response function

IRF for ARDL $(1,0)$ is given as

$\begin{equation} IRF=\left\{\begin{array}{cl} 0 & for~t=0~(period~zero)\\\beta_1 \beta_3^{t-1} & for~t=1,2,...,\infty~(all~future~periods)\end{array}\right. \tag{9.8} \end{equation}$

The impulse response function IRF of the partial adjustment model ARDL $(1,0)$ can be represented in the table and on the figure

TABLE 9.1: Responses of $y_t$ with impulse in $x_t$ according to ARDL $(1,0)$
Period	Response
$~~~~~~~0~~~~~~~$	$0$
$1$	$\beta_1$
$2$	$\beta_1 \beta_3$
$3$	$\beta_1 \beta_3^2$
$4$	$\beta_1 \beta_3^3$
$5$	$\beta_1 \beta_3^4$
$\cdots$	$~~~~~~~~~~\cdots~~~~~~~~~~$

The sum of the responses in all future periods represents the long-term effect of the variable $x_t$ on the variable $y_t$ , i.e. the sum of the geometric series converges to

$\begin{equation} \sum_{t=1}^\infty \beta_1 \beta_3^{t-1}=\frac{\beta_1}{1-\beta_3} \tag{9.9} \end{equation}$

Let’s consider following two examples

$~~~~y_t=2+0.5x_t+0.6y_{t-1}+u_t~~~~~~~~~~~~~~~~~~~~~~~y_t=2+0.5x_t-0.6y_{t-1}+u_t$

The impacts of the variable $x_t$ on the variable $y_t$ decline geometrically under the condition that $|\beta_3|<1$ , which means that the rate at which these impacts decay depends on the value of the parameter $\beta_3$
The sign of the parameter $\beta_3$ influences the shape of the impulse response function
In the long-run, the variables are expected to be in equilibrium or steady state
The long-run equilibrium equation of the partial adjustment model ARDL $(1,0)$ is

$\begin{equation} y^E=\frac{\beta_0}{1-\beta_3}+\frac{\beta_1}{1-\beta_3}x^E \tag{9.10} \end{equation}$

Exercise 39. Based on the partial adjustment model ARDL $(1,0)$

$y_t=2+0.5x_t+0.6y_{t-1}+u_t$

determine:

Short-run effect of $x_t$ on $y_t$
Long-run effect of $x_t$ on $y_t$
Speed of decreasing responses
Equilibrium equation

Solution

Short-run effect is

$0.5$ (the response in the first period

$\beta_1=0.5$ ). Long-run effect is

$1.25$ computed by formula

$\frac{\beta_1}{1-\beta_3}=\frac{0.5}{1-0.6}=1.25$ . Responses in each period are decreasing at speed rate of

$40\%$ according to

$(\beta_3-1)100\%=(0.6-1)100\%=-40\%$ . Equilibrium equation is

$y^E=5+1.25x^E$ .

IRF for a ARDL $(1,1)$ is given as

$\begin{equation} IRF=\left\{\begin{array}{cl} 0 & for~t=0~(period~zero)\\ \beta_1 & for~t=1~(first~period)\\ (\beta_2+ \beta_1\beta_3)\beta_3^{t-1} & for~t=2,3,...,\infty~(all~future~periods)\end{array}\right. \tag{9.11} \end{equation}$

The impulse response function IRF of the ARDL $(1,1)$ can also be represented in the table and on the figure

TABLE 9.2: Responses of $y_t$ with impulse in $x_t$ according to ARDL $(1,1)$
Period	Response
$~~~~~~~0~~~~~~~$	$0$
$1$	$\beta_1$
$2$	$\beta_2+\beta_1 \beta_3$
$3$	$(\beta_2+\beta_1 \beta_3)\beta_3$
$4$	$(\beta_2+\beta_1 \beta_3)\beta_3^2$
$5$	$(\beta_2+\beta_1 \beta_3)\beta_3^3$
$\cdots$	$~~~~~~~~~~\cdots~~~~~~~~~~$

Likewise, the sum of responses in all future periods represents the long-term effect of the variable $x_t$ on the variable $y_t$

$\begin{equation} \sum_{t=1}^\infty \frac{\Delta y_{T+t}}{\Delta x_t}=\frac{\beta_1+\beta_2}{1-\beta_3} \tag{9.12} \end{equation}$

Let’s consider two additional examples

$y_t=7+0.4x_t+0.1x_{t-1}+0.5y_{t-1}+u_t~~~~~~~~~~~~~~~~~~~~~~~y_t=7+0.4x_t+0.1x_{t-1}-0.5y_{t-1}+u_t$

Equilibrium or long-run equation based on ARDL $(1,1)$ is

$\begin{equation} y^E=\frac{\beta_0}{1-\beta_3}+\frac{\beta_1+\beta_2}{1-\beta_3}x^E \tag{9.13} \end{equation}$

Equilibrium state may exist only if variables are related in the long-run
If variables are related in the long-run then they are related in the short-run also (not necessary conversely)
If variables are related both in the short-run and in the long-run it is evidence of cointegration

Exercise 40. Using model ARDL $(1,1)$

$y_t=5+0.3x_t+0.2x_{t-1}+0.7y_{t-1}+u_t$

determine:

Short-run effect of $x_t$ on $y_t$
Long-run effect of $x_t$ on $y_t$
Speed of decreasing responses
Equilibrium equation

Solution

Short-run effect is

$0.3$ (the response in the first period

$\beta_1=0.3$ ). Long-run effect is

$1.67$ computed by formula

$\frac{\beta_1+\beta_2}{1-\beta_3}=\frac{0.3+0.2}{1-0.7}=1.67$ . Responses in each period are decreasing at speed rate of

$30\%$ according to

$(\beta_3-1)100\%=(0.7-1)100\%=-30\%$ . Equilibrium equation is

$y^E=16.67+1.67x^E$ .

Exercise 41. Using the same time-series data from the object indicators estimate three models: ARDL

$(0,0)$ , ARDL

$(1,0)$ and ARDL

$(1,1)$ with names static, dynamic1 and dynamic2, respectively. For that purpose apply dynlm() commad from the package dynlm, assuming that variable

$x_t$ is production and variable

$y_t$ is growth. Present the results of estimated models in a single table by modelsummary() command. Which model fits the data better? For all three models check if the first-order autocorrleation problem exist by employing bgtest() command from the package lmtest. Based on partial adjustment model calculate the short-run and the long-run effect of the production on the growth rate.

Solution

In this example an evidence of autocorrelation problem in static model is found according to Breusch-Godfrey test of order up to

$1$ . Opposite to that, both dynamic models dynamic1 and dynamic2 do not exhibit autocorrelation problem as it is absorbed by including lagged growth on the RHS

$(y_{t-1})$ . Note that within dynlm() command a lagged variable is computed using a lag operator L(). Partial adjustment model dynamic1 fits the data better in terms of

$R^2$ and statistical significance of the estimated parameters. According to that model a short-run effect of production to the growth rate is

$0.166\%$ , and the long-run effect is

$0.574\%$ (both effects are expressed in

$\%$ as both variable

$x_t$ and

$y_t$ are given in percentages).

# Installing and loading dynlm package
install.packages("dynlm")
library(dynlm)

# Estimating three model by applying dynlm() command
static=dynlm(growth~production,data=indicators)
dynamic1=dynlm(growth~production+L(growth),data=indicators)
dynamic2=dynlm(growth~production+L(production)+L(growth),data=indicators)

# Presenting results in a single table
library(modelsummary) # loading the package modelsummary (if you are runing a new session)
modelsummary(list(static,dynamic1,dynamic2),stars=TRUE,fmt=4)

library(lmtest) # loading the package lmtest (if you are runing a new session)
bgtest(static)
bgtest(dynamic1)
bgtest(dynamic2)

# Short-run effect is the second coefficient from "dynamic1" model
coef(dynamic1)[2]

# Long-run effect is the ratio between the second coefficient and 1 minus the third coefficient from "dynamic1" model
coef(dynamic1)[2]/(1-coef(dynamic1)[3])

$~~~$