# Does $\delta$ parameter in GARCH-M stay unchanged when the process is scaled?

Assume we have a GARCH-M(1,1) process as follows:

$$y_t = \beta y_{t-1} + \delta h_t + \epsilon_t, \quad \epsilon_t \sim N(0, h_t)$$

$$h_t = a_0 + a_1 \epsilon^2_{t-1} + b_1 h_{t-1}.$$

If we scale $$y_t$$ to create $$x_t = 100\times y_t$$, I recently found out in this question Do parameters stay unchanged when GARCH is scaled? that $$a_0$$ would scale quadratically with the scaling value. How would the $$\delta$$ parameter change though in the GARCH-M process?

## 1 Answer

$$\delta$$ would scale inversely proportionally to the scaling of $$y_t$$.
Take the original model

\begin{aligned} y_t &= \beta y_{t-1} + \delta h_t + \epsilon_t, \\ h_t &= a_0 + a_1 \epsilon^2_{t-1} + b_1 h_{t-1}, \\ \epsilon_t &\sim i.i.N(0,h_t) \\ \end{aligned}

and scale $$y_t$$ with a positive constant $$c$$. This turns the model into

\begin{aligned} (cy_t) &= \beta (cy_{t-1}) + \frac{\delta}{c} (c^2 h_t) + (c\epsilon_t), \\ (c^2 h_t) &= (c^2 a_0) + a_1 (c\epsilon_{t-1})^2 + b_1 (c^2 h_{t-1}), \\ (c\epsilon_t) &\sim i.i.N(0,c^2 h_t) \\ \end{aligned}

which can be expressed as

\begin{aligned} \tilde y_t &= \beta \tilde y_{t-1} + \tilde\delta \tilde h_t + \tilde\epsilon_t, \\ \tilde h_t &= \tilde a_0 + a_1 \tilde\epsilon_{t-1}^2 + b_1 \tilde h_{t-1}, \\ \tilde\epsilon_t &\sim i.i.N(0,\tilde h_t) \\ \end{aligned}

where

\begin{aligned} \tilde y_t &= c y_t, \\ \tilde\delta &= \frac{\delta}{c}, \\ \tilde\epsilon_t &= c\epsilon_t, \\ \tilde a_0 &= c^2 a_0 \quad \text{and} \\ \tilde h_t &= c^2 h_t. \end{aligned}

The second line of the equation block above is what you are interested in.