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

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

  1. 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

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

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

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

  1. 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