9 Dynamic models and cointegration

• Different types of dynamic model can be used and each of them is a special case of autoregressive distributed lag model ARDL(p,q)
• ARDL(p,q) dynamic model combines two parts:

1. autoregressive part AR(p)
2. distributed lag part DL(q)

• Autoregressive part includes a lagged values of dependent variable $y_{t-1}$, $y_{t-2}$, $\dots$, $y_{t-p}$, while a distributed lag part includes a lagged values of independent variable $x_{t-1}$, $x_{t-2}$, $\dots$, $x_{t-q}$.

• Due to simplicity ARDL$(1,1)$ will be considered to explain commonly used other dynamic models in econometrics

$$\begin{equation} y_t=\beta_0+\beta_1x_t+\beta_2x_{t-1}+\beta_3y_{t-1}+u_t \tag{9.1} \end{equation}$$

1. If $\beta_2=0$ a model ARDL$(1,1)$ reduces to ARDL$(1,0)$ and it is known as a partial adjustment model PA

$$\begin{equation} y_t=\beta_0+\beta_1x_t+\beta_3y_{t-1}+u_t \tag{9.2} \end{equation}$$

2. If $\beta_3=0$ a model ARDL$(1,1)$ reduces to ARDL$(0,1)$ and it is known as a finite distributed lag model FDL

$$\begin{equation} y_t=\beta_0+\beta_1x_t+\beta_2x_{t-1}+u_t \tag{9.3} \end{equation}$$

3. If $\beta_2=\beta_3=0$ a model ARDL$(1,1)$ reduces to ARDL$(0,0)$ and it is known as a static model

$$\begin{equation} y_t=\beta_0+\beta_1x_t+u_t \tag{9.4} \end{equation}$$

4. If $\beta_1=-\beta_2$ and $\beta_3=1$ then ARDL$(1,1)$ is transformed into a model in the first differences

$$\begin{equation} \Delta y_t=\beta_0+\beta_1\Delta x_t+u_t \tag{9.5} \end{equation}$$

• In the static model $y_t=\beta_0+\beta_1 x_t+u_t$ the most common issue is first-order autocorrelation $u_t=\rho_1 u_{t-1}+v_t$ under the condition $|\rho_1|<1$

• By including the error terms first-order autocorrelation in the static model, the following is obtained:

$$\begin{equation} y_t=\beta_0+\beta_1 x_t+\rho_1 u_{t-1}+v_t \tag{9.6} \end{equation}$$

• Substituting $u_{t-1}=y_{t-1}-\beta_0-\beta_1 x_{t-1}$ a static model becomes dynamic model

$$\begin{equation} y_t=(1-\rho_1)\beta_0+\beta_1 x_t-\rho_1\beta_1 x_{t-1}+\rho_1 y_{t-1}+v_t \tag{9.7} \end{equation}$$

• Dynamic model (9.7) is ARDL$(1,1)$ which absorbs the error terms autocorrelation

• On the other hand, it would be incorrect to include the dependent variable with a one lag $y_{t-1}$ as RHS variable when the autocorrelation problem does not exist, because in that case the OLS estimator is biased

• Models ARDL$(1,1)$, ARDL$(1,0)$ and ARDL$(0,1)$ are of special interest to users because they provide information about the short-run and the long-run relationship between two variables $x_t$ and $y_t$, assuming that variables $x_t$ is strictly exogenous ($x_t$ couses $y_t$)

• Moreover, a unit change in the variable $x_t$ (impulse in the first period) results in changing the variable $y_t$ in all future periods (responses)

• This means that effects of variable $x_t$ on variable $y_t$ are distributed over time

• Function that give us such information is called impulse response function IRF