# Expected Value of an ARMA-GARCH Model

An ARMA(p,q) model is given by

$$\qquad \qquad Y_t = c + \sum\limits_{i=1}^{p}\varphi_iY_{t-i}+\sum\limits_{i=1}^{q}\theta_i\varepsilon_{t-i} + \varepsilon$$

with $$\varepsilon_t \sim N(0,\sigma^2)$$.

Let's say our model is simply an ARMA(1,1) model. The expected value for tomorrow's forecast then is

$$\qquad \qquad E[Y_{t+1}] = E[c+\varphi_1Y_t+\theta_1\varepsilon_t + \varepsilon_{t+1}]$$

$$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1\varepsilon_t + E[\varepsilon_{t+1}]$$

$$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1\varepsilon_t$$

with $$Y_t$$ being our time series. Correct so far?

Now, a GARCH(p,q) model is given by

$$\qquad \qquad X_t = \sigma_t\varepsilon_t$$

$$\qquad \qquad \sigma_t^2=\omega + \sum\limits_{i=1}^{q}\alpha_iX_{t-i}^2+\sum\limits_{i=1}^{p}\beta_i\sigma_{t-i}^2$$

with $$\varepsilon_t \sim N(0,\sigma^2)$$ again and with $$X_t$$ being our time series here right?

Now my questions arise when looking at an ARMA(n,m)-GHARCH(p,q) model:

$$\qquad \qquad Y_t = c + \sum\limits_{i=1}^{n}\varphi_iY_{t-i}+\sum\limits_{j=1}^{m}\theta_jX_{t-j}+X_t$$.

I understand that $$X_t$$ ist the underlying time series of the garch model, but isn't that simply the time series we are looking at and trying to just fit to a GARCH-Model?

Next, when looking at tomorrow's expected value from the ARMA(1,1)-GARCH(1,1) model, it is

$$\qquad \qquad E[Y_{t+1}] = ... = c+\varphi_1Y_t+\theta_1X_t + E[X_{t+1}]$$

$$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1X_t + E[\sigma_{t+1}\varepsilon_{t+1}]$$

$$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1X_t + E[\sigma_{t+1}]E[\varepsilon_{t+1}]$$

$$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1X_t + E[\sigma_{t+1}]*0$$

$$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1X_t$$

Aren't $$X_t$$ and $$Y_t$$ basically the same time series were we tried to apply these models to? And where is $$\sigma^2$$ now, I thought our goal was to use tomorrow's forecasted variance in our combined model.

• Jan 18, 2021 at 12:19
• The link has helped me a lot, thanks! So basically for my data above the X_t is the "time series" of the variance and the Y_t the time series of the returns of financial data? Jan 18, 2021 at 14:09
• Looking at your equations above, $X_t$ is the random part of $Y_t$, and $X_t$ is modelled by GARCH as you have specified above. Jan 18, 2021 at 18:41
• Thanks a lot, I think I understood the difference between ARMA and GARCH now, I was a little irritated because of the switched parameters $Y_t$ and $\varepsilon_t$. I understand both on their own now I guess, but I still don't get what the combination of them does, because that $\varepsilon$ from the ARMA-GARCH Model is for tomorrows forecast then $E[\varepsilon_{t+1}]=0$ because that follows from $\varepsilon = \sigma_tZ_t$ and $E[Z_t]=0$. I don't get how that $\varepsilon$ helps me with tomorrows forecast. Jan 19, 2021 at 0:46
• Pay attention to what information we are allowed to use at which point. $\mathbb{E}_t(\varepsilon_{t+1})=0$ but $\mathbb{E}_t(Z_t)=Z_t$ (a concrete realization of $Z_t$). Here I used subscript $_t$ for the expectation operator to denote it is conditional on information available at time $t$ (and similarly with $t+1$). Jan 19, 2021 at 8:49

Consider a dependent variable $$Y_t$$ which is an original time series or a transformation (such as log-returns) thereof.
When the cond. mean of $$Y_t$$ follows ARMA, the cond. variance of $$Y_t$$ can be constant, follow GARCH or whatever.
When the cond. variance of $$Y_t$$ follows GARCH, the cond. mean of $$Y_t$$ can be constant, follow ARMA or whatever.
Consider a basic example of ARMA(1,1)-GARCH(1,1): \begin{aligned} y_t &= c+\varphi_1 y_{t-1}+\varepsilon_t+\theta_1\varepsilon_{t-1}, \\ \varepsilon_t &= \sigma_t z_t, \\ \sigma^2_t &= \omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2, \\ z_t &\sim i.i.d(0,1). \end{aligned} Given the estimated model, a one-step-ahead estimate/forecast of the conditional mean and variance of $$Y_t$$ are the following: \begin{aligned} \mathbb{E_t}(y_{t+1}) &= c+\hat\varphi_1 y_{t}+\hat\theta_1\hat\varepsilon_{t}, \\ \text{Var}_t(y_{t+1}) &= c+\hat\alpha_1\hat\varepsilon_{t}^2+\hat\beta_1\hat\sigma_{t}^2. \end{aligned}