# Out of sample ARCH forecast

I have estimated a conditional mean model for a time series:

$$x_t = x_{t-1} + \epsilon_t$$.

Say I have estimated it using periods 1 to 10. I can do an out of sample conditional mean forecast by multiplying the estimated coefficients with the respective variable values of period 11, i.e. I can strictly out of sample forecast period 12.

However, how do I do this with an ARCH model that is based on the conditional mean model mentioned above, i.e. a conditional variance model:

$$\epsilon^2_t = \alpha_0 + \alpha_1 \epsilon^2_{t-1}$$.

Say I have estimated it using periods 1 to 10 again. If I again use this information in period 11 to generate a forecast in period 12, it is not strictly out of sample. This is because $$\epsilon_{11}$$ is based on the non out of sample conditional mean forecast for period 11 (it is not out of sample, because the forecast is partially based on data from period 10 to forecast for period 11, and period 10 is not out of sample).

In other words: my first real "out of sample" ARCH-Model input is the squared error of the abovementioned out of sample conditional mean forecast. Hence, I can not provide an out of sample conditional variance for period 12. Is this correct?

EDIT: By (strictly) out of sample I mean that the data used for forecasting is different than the data used for estimation.

Let us start from the more familiar AR model and then move on to ARCH.

When forecasting with AR models, we use actual values of the variable whenever available and their forecasts when not available. We also use estimated values of errors whenever available and conditional expectations (i.e. zero) when not available
E.g. AR(2): $$\quad y_t=\varphi_1 y_{t-1}+\varphi_2 y_{t-2}+\varepsilon_t.\quad$$ Suppose our available sample is from $$1$$ to $$T$$. Then we have actual values $$y_1,\dots,y_T$$ and estimated errors $$\hat\varepsilon_1,\dots,\hat\varepsilon_T$$. Beyond that we use forecasts of $$y$$ that we construct iteratively and conditional expectations of $$\varepsilon$$ that are zero. E.g.

\begin{aligned} \hat y_{T+1\mid T} &= \hat\varphi_1 y_T+\hat\varphi_2 y_{T-1} + 0 \\ \hat y_{T+2\mid T} &= \hat\varphi_1\hat y_{T+1\mid T}+\hat\varphi_2 y_{T} + 0 \\ \hat y_{T+3\mid T} &= \hat\varphi_1\hat y_{T+2\mid T}+\hat\varphi_2\hat y_{T+1\mid T} + 0 \\ \dots \\ \hat y_{T+n\mid T} &= \hat\varphi_1\hat y_{T+n-1\mid T}+\hat\varphi_2\hat y_{T+n-2\mid T} + 0 \\ \dots \end{aligned}

When forecasting with ARCH models, we use estimated values of squared errors whenever available and estimated conditional expectations of squared errors (i.e. $$\hat\sigma_{\dots\mid T}^2$$) when not available.
E.g. ARCH(1): $$\quad\sigma_t^2=\omega+\alpha_1\varepsilon_{t-1}^2+\alpha_2\varepsilon_{t-2}^2.\quad$$ Suppose our available sample is from $$1$$ to $$T$$. Then we have estimated values $$\hat\varepsilon_1^2,\dots,\hat\varepsilon_T^2$$. Beyond that we use estimated conditional expectations $$\hat\sigma_{\dots\mid T}^2$$ that we obtain iteratively. E.g.

\begin{aligned} \hat \sigma_{T+1\mid T}^2 &= \hat\omega+\hat\alpha_1\hat\varepsilon_T^2+\hat\alpha_2\hat\varepsilon_{T-1}^2 \\ \hat \sigma_{T+2\mid T}^2 &= \hat\omega+\hat\alpha_1\hat \sigma_{T+1\mid T}^2+\hat\alpha_2\hat\varepsilon_T^2 \\ \hat \sigma_{T+3\mid T}^2 &= \hat\omega+\hat\alpha_1\hat \sigma_{T+2\mid T}^2+\hat\alpha_2\hat \sigma_{T+1\mid T}^2 \\ \\ \dots \\ \hat \sigma_{T+n\mid T}^2 &= \hat\omega+\hat\alpha_1\hat \sigma_{T+n-1\mid T}^2+\hat\alpha_2\hat \sigma_{T+n-2\mid T}^2 \\ \dots \end{aligned}

• Thanks for the reply. I get the mechanism you are explaining. However, I am specifically interested in out of sample (o-o-s) forecasts, i.e. forecasts that are based on data which has not been used to estimate the respective formulas (e.g. AR, or ARCH) coefficients. As I tried to exemplify in my question, such an o-o-s forecast is a bit more tricky in the case of ARCH (in my opinion), as the regressor value itself is the outcome of an estimation (mean model), which in turn affected the variance model estimation. Hence, to be truely out of sample, this regressor needs to be estimated o-o-s. Commented Jan 25, 2021 at 14:21
• @shenflow, I am providing exactly that. What I provide are not in-sample fitted values (they should not be called forecasts) but out of sample forecasts. Regarding this regressor needs to be estimated o-o-s, we use $\hat\sigma_{T+h\mid T}^2$ instead of $\hat\varepsilon_{T+h}^2$ for time indices $h>0$. Does that help? Commented Jan 25, 2021 at 14:37
• Okay, but consider the first row of your AR forecast example. You use $y_T$ and $y_{T-1}$ to forecast $y_{T+1}$. Yet, both of these values have been used to estimate $\hat{\varphi_1}$ and $\hat{\varphi_2}$. Thus, this is not an out of sample (strictly speaking) forecast, as the data used to estimate the coefficients and the data used to generate the forecast overlap. Commented Jan 25, 2021 at 14:46
• @shenflow, OK, now I see what your concern is. And it is an interesting one! The fact that $y_T$ was used both for estimating the model's coefficients and for constructing the forecast $\hat y_{T+1\mid T}$ reflects an intrinsic feature of autoregressive models. However, I do not think it is problematic. After all, no forecast could ever be viewed as "completely out of sample" as long as it is based on parameter estimates that were calculated from the sample data. (The data points are all hiding in the parameter estimates!) So this applies to absolutely all models that rely on data. Commented Jan 25, 2021 at 15:07
• @shenflow, in other words, the data used to estimate the coefficients and the data used to generate the forecast overlap is always the case, since the parameter estimates involved in constructing any time series forecast (or even cross sectional prediction) are functions of the sample data. Commented Jan 25, 2021 at 15:09