Modeling Known Regime Shifts with Interactions and Indicators
Summary
The document describes an attempt to model electricity prices before, during, and after the introduction of a carbon price. It presents a regression approach for a known structural break: retain baseline coefficients and interact selected explanatory variables with a regime indicator so those coefficients can change in the designated period. It also considers a model with only a regime indicator, which captures a level shift rather than a change in the relationship between variables.
The example uses electricity price as the response and carbon price as a predictor, but reports that neither specification explains the observed increase fully. The question raises how to interpret the models and why residuals remain. No answer, fitted estimates, or diagnostic evidence is included, so the document does not resolve whether the models are correctly specified. Its setup suggests that a single predictor or a simple level shift may omit other drivers or dynamics, but these possibilities are not examined.
Key ideas
- A known break can be modeled by interacting a regime indicator with selected predictors.
- Interaction terms allow predictor coefficients to differ across regimes.
- A standalone regime indicator represents a level shift rather than changing predictor slopes.
- Residual patterns remain unexplained in the example, and the document offers no resolution or model diagnostics.
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Full text
# modelling known regime shifts
# modelling known regime shifts
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:
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
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
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.