*2.2. Hybrid Error Correction Model*

The Error Correction Model might show an error correction of the first difference exclusively, which is as follows:

$$
\Delta \mathbf{Y}\_t = \mathbf{Y}\_t - \mathbf{Y}\_{t-1} \tag{4}
$$

Error Correction Model can use for quantitative computation, and it is necessary to point out that it is the base of the Auto Regressive Distributive Lag Model (ARDL). We have used the Error Correction Model to condition that, if the ARDL sum coefficient is equal to 1, by decreasing the constant terms. Consequently, the coefficient of error correction term long-run association can attain if and only if the transformation at term grows at the constant rate, N. Hence, the coefficient mathematical model Error Correction Model can be presented as:

$$Y\_t = \beta \mathbf{0} + \beta\_1 Y\_{t-1} + \beta\_2 Z\_t + \beta\_3 Z\_{t-1} + \beta\_t \tag{5}$$

The proposed term *Yt*−<sup>1</sup> is deducted from the ARDL both sides:

$$Y\_t - Y\_{t-1} = \beta\_0 + \beta\_1 Y\_{t-1} + \beta\_2 Z\_t + \beta\_3 Z\_{t-1} - Y\_{t-1} + \beta\_t \tag{6}$$

$$
\Delta \mathbf{Y}\_t = \beta\_0 + \beta\_1 \mathbf{Y}\_{t-1} + \beta\_2 \mathbf{Z}\_t + \beta\_3 \mathbf{Z}\_{t-1} - \mathbf{Y}\_{t-1} + \mathfrak{G}\_t \tag{7}
$$

Through addition and deducting β2*Zt*−<sup>1</sup> in the right-hand side of the mathematical model. The new equation is as follows:

$$
\Delta Y\_t = \beta\_0 + \beta\_1 Y\_{t-1} + \beta\_2 Z\_t - \beta\_2 Z\_{t-1} + \beta\_3 Z\_{t-1} - \chi\_{t-1} + \beta\_2 Z\_{t-1} + \beta\_t \tag{8}
$$

or

$$
\Delta Y\_t = \beta\_0 + (\beta\_1 - 1)Y\_{t-1} + \beta\_2 Z\_t + (\beta\_2 + \beta\_3)Z\_{t-1} + \beta\_t \tag{9}
$$

To fulfill the condition of the Error Correction Model, the coefficient (*Zt*−1) must be analogous to the deducted coefficient *Yt*−1. So, the newly construed mathematical model is as follows:

$$
\beta\_1 - 1 = -(\beta\_2 - \beta\_3) \tag{10}
$$

$$
\beta\_1 + \beta\_2 + \beta\_3 = 1 \tag{11}
$$

Consequently, the term of error correction constant considers as:

$$
\Delta Y\_t = \beta\_0 + \beta\_2 Z\_t - \tau (Y\_{t-1} - Z\_{t-1}) + \mu\_t \tag{12}
$$

$$
\tau \, = - (\beta\_1 - 1) = \, \left(\beta\_2 + \beta\_3\right) \tag{13}
$$

If the variation in constant term increases at a continuous rate N, the association of the long-run phenomena is:

$$N = \beta \mathfrak{o} + \beta \mathfrak{z}N - \mathfrak{r}(\mathfrak{y}^\* - \mathbb{Z}^\*) \tag{14}$$

$$
\pi(y^\* - Z^\*) = \beta\_0 + (\beta\_2 - 1)N \tag{15}
$$

$$y^\* = \beta \alpha + \frac{(\beta\_2 - 1)N}{\pi} + Z^\* \tag{16}$$

then the original order having and without having the value of log is considered as:

$$y\_t^\* = KZ\_t^\* \tag{17}$$

If we take the log from both sides, then it will be as:

$$
\log y\_t^\* = \log K + \log \mathbf{Z}\_t^\* \tag{18}
$$

Through using the anti-log of the new model, the long run will consider:

$$y^\* = \exp\left[\frac{\beta\_0 + (\beta\_2 - 1)B}{\tau}\right] \tag{19}$$

where K represents the association between the variable Y and Z, the Error Correction Model is being used to measure the long-run relationship, and the Error Correction Model characterizes the previous imbalance in an existing factor. It can be:

$$
\Delta N\_t = \sum\_{i=1}^{N} \tau\_1 \Delta N\_{t-1} + \tau\_2 \Delta N\_{t-1} \beta\_2 N + \text{YZ}\_t + \text{ } \mu\_t \tag{20}
$$
