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I have a panel data of stock returns and I want to estimate the covariance matrix of these returns throughout time. I also want to use exponential smoothing. The scheme is as follows.

$$\hat{\Sigma}_t = \lambda\hat{\Sigma}_{t-1} + (1-\lambda)(r_t - \hat{\mu}_t)(r_t - \hat{\mu}_t)^{\top} \tag{1}$$

Here $\hat{\Sigma}_t$ and $\hat{\mu}_t$ are my estimates of the covariance matrix of returns and the mean return at time $t$.

In Chapter 10 of Tsay "Analysis of Financial Time Series" the author describes the existing literature on this and gives the following example. He has two index returns, one from Hong Kong and one from Japan. He considers the univariate case first, that is he estimates the variances of the two indices separately. To this end he fits a GARCH(1,1) model to both these index returns.

The estimated conditional variance models are as follows.

$$\sigma_{1,t}^2 = 0.038 + 0.855\sigma_{1,t-1}^2 + 0.143(r_t - 0.109)^2$$ $$\sigma_{2,t}^2 = 0.044 + 0.861\sigma_{2,t-1}^2 + 0.127(r_t - 0.109)^2$$

I am guessing that $0.855$ and $0.861$ also happen to be the optimal smoothing parameters. Can someone confirm this?

In both cases the coefficients (excluding the constant) sum up to $0.998$, which is awfully close to $1$. This is very convenient since this is what we need for exponential smoothing, i.e. the weights of the previous estimate and the current data have to sum up to $1$. I find it very hard to believe that this is some coincidence.

Can someone please explain the link between fitting a GARCH model and the optimal exponential smoothing parameter? Furthermore, can someone maybe explain what the author may have done to find those estimates that fit right into the exponential smoothing framework? What would the author have done if the fitted GARCH model had parameters $0.5$ and $0.2$?

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Can someone please explain the link between fitting a GARCH model and the optimal exponential smoothing parameter?

There are quite a few versions of exponential smoothing methods (for a taxonomy, see Table 7.8 in Section 7.6 of Hyndman & Athanasopoulos "Forecasting: Principles and Practice"). I assume you are talking about the simple exponential smoothing (coded $(N,N)$ in the table) where

$$ \hat x_t = \lambda x_{t-1} + (1-\lambda)\hat x_{t-1} $$

using the notation of the weigted average form of the model as introduced in Section 7.1 of the textbook.

Meanwhile, the GARCH(1,1) model in its simplest form (w.r.t. the conditional mean equation) is

$$ \begin{aligned} r_t &= \sigma_t\varepsilon_t; \\ \sigma_t^2 &= \omega + \alpha r_{t-1}^2 + \beta\sigma_{t-1}^2; \\ \varepsilon_t &\sim i.i.d(0,1). \\ \end{aligned} $$

The conditional variance equation of the GARCH(1,1) model is indeed similar to the simple exponential smoothing, especially once we put hats on the conditional variances to denote predicted values ($\hat\sigma_t^2$) and estimated values ($\hat\sigma_{t-1}^2$):

$$ \hat\sigma_t^2 = \omega + \alpha r_{t-1}^2 + \beta\hat\sigma_{t-1}^2. $$

However, this equation exhibits a few differences from the simple exponential smoothing:

  1. It includes a constant $\omega$.
  2. It includes the lagged squared return $r_{t-1}^2$ in place of the lagged conditional variance $\sigma_{t-1}^2$. Including $\sigma_{t-1}^2$ would make the equation infeasible in practice since the conditional variance is not observed, while the return $r_{t-1}$ is.
  3. Its coefficients need not sum to zero.

An IGARCH(1,1) model would come the closest to exponential smoothing. There we have $\omega=0$ and $\beta=1-\alpha$, hence

$$ \hat\sigma_t^2 = \alpha r_{t-1}^2 + (1-\alpha)\hat\sigma_{t-1}^2; $$

but still there is $r_{t-1}^2$ in place of $\sigma_{t-1}^2$. $r_{t-1}^2$ is a proxy for $\sigma_{t-1}^2$, albeit a noisy one, so this looks pretty much like simple exponential smoothing. But strictly speaking, I don't think you can directly claim this:

I am guessing that $0.855$ and $0.861$ also happen to be the optimal smoothing parameters.

However, if $\sigma_{t-1}^2$ were observable and you could estimate a simple exponential smoothing model for it, you would probably get parameter estimates similar to the ones from IGARCH(1,1); namely, $\hat\lambda$ would be close to $\hat\alpha$.

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  • $\begingroup$ Thank you very much for your detailed answer. What would be the harm, if there is any, in fitting a GARCH(1,1) model to my data set and use the conditional variance evolution to predict the variance? For example, suppose that my fit gives $\sigma_t^2 = 0.001 + 0.1\sigma_{t-1}^2 + 0.2r_t^2$. Then I initialize $\sigma_1^2 = y_1^2$ and use this recursive relation to estimate the most recent variance. Do people do this sort of stuff and how do they justify it? Coming from probability theory I find it a bit difficult to decide where to draw the line in data analysis regarding mathematical precision $\endgroup$ – Calculon Aug 13 '16 at 20:34
  • $\begingroup$ @Calculon, now you have the contemporaneous $r_t^2$ on the right hand side. The conditional expectation of $r_t^2$ is $\sigma_t^2$. I don't know if that makes sense. In exponential smoothing you have only lagged values on the right hand side. $\endgroup$ – Richard Hardy Aug 13 '16 at 20:43
  • $\begingroup$ I am so sorry. That was a typo. I meant $r_{t-1}^2$. $\endgroup$ – Calculon Aug 13 '16 at 20:45
  • $\begingroup$ OK, then you just have a GARCH(1,1) model, right? The only difference then seems to be how one initializes. Typically, one would use the unconditional variance as the starting point. $\endgroup$ – Richard Hardy Aug 13 '16 at 20:46
  • $\begingroup$ How would one initialize if exponential smoothing were being used? $\endgroup$ – Calculon Aug 13 '16 at 20:47

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