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I wish to model a price time series with a known regime shift: electricity price before during and after the introduction of a carbon price. The time series looks like this:

enter image description here

you can see the jump in price between 2012 and 2014. This paper states that

"when the sample regimes are known. If the first regime includes the first $i_0$ observations of $P$ total observations, and the structural change is confined to the first $k$ parameters, $\beta_j$ , $j = 1 \dots k$, of $K$ total parameters, we can define a dichotomous variable $D$ such that

\begin{equation} \quad D = \begin{cases} 0 \quad \text{ if } \ i = 1, \dots, i_0 \\ 1 \quad \text{ if } \ i = i_0 + 1, \dots, P\end{cases} \end{equation}

Our model is then

\begin{equation} \quad y_i = \sum_{j=1}^{K} \ x_{ij} \beta_j + \sum_{j=1}^k \ (x_{ij} D_i) \gamma_j + e_i \end{equation}

where the change in the structural parameter associated with the $x$’s in the second regime is represented by the $\gamma_j.$"

I have tried to run this model in the following form

\begin{equation} \quad D = \begin{cases} 0 \quad \text{ if } \ i = 1, \dots, i_0, i_1 + 1, \dots, P \\ 1 \quad \text{ if } \ i = i_0 + 1, \dots, i_1\end{cases} \end{equation}

\begin{equation} \quad y_i = x_i \beta + (x_i D_i) \gamma + e_i \end{equation}

where the $y$'s are electricity price in AUD/MWh, and the $x$'s are carbon price in AUD/tonne CO2. This model results in the following residuals

enter image description here

I have also attempted to run a simple model with the dichotomous variable $D$

\begin{equation} \quad y_i = D_i \gamma + e_i \end{equation}

which produces the following residuals

enter image description here

Neither model seems to completely account for the increase in electricity price due to the carbon price regime. I am wondering why this is? Am I using the models correctly?

I am also a bit unclear as to how each model accounts for the regime switch and how they differ. Any insight is appreciated.

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