1

I am familiar with ARCH-type models to estimate the conditional volatility of some variable of interest in a univariate setting.

I know that there also exists the concept of multivariate ARCH-type models. However, as far as I understand, such models are used to derive conditional covariance matrices (correct me if I am wrong, my knowledge is very sparse in this area).

However, I am interested in modelling the conditional volatility of a variable in a panel setting (thousands of univariate time series). My approach thus far is as follows:

  1. I specify and estimate a conditional mean model for my panel: $y_{i,t} = \alpha_0 + \alpha_1x_{i,t-1} + e_{i,t}$.

  2. Assuming that this model is true, I then use the squared residuals, $\epsilon_{i,t}^2$, as a proxy for the variance and specify a conditional variance model: $\epsilon_{i,t}^2 = \beta_0 + \beta_1 \epsilon_{i,t-1}^2 + v_{i,t}$, with $v_{i,t} = \epsilon_{i,t}^2-\sigma_{i,t}^2$ ($\sigma_{i,t}^2$ being the true conditional variance). This corresponds to an AR(1) representation of an ARCH(1) model as far as I understand.

Thereby, model 1 allows me to forecast the conditional mean of $y$ and model 2 allows me to forecast the conditional variance of $y$. As suggested multiple times here, I realize that doing 1. and 2. in a single step is advantageous. However, the software packages I know that can do so only apply to a univariate time series setting.

My question is: Is my approach, although very naive, reasonable? Are there any "standard" procedures of applying ARCH-type models in a panel setting?

Richard Hardy
  • 54,375
  • 10
  • 95
  • 219
shenflow
  • 750
  • 8
  • 20
  • Some material on the topic is available online. Have you looked at the existing approaches? A lot depends on what kind of structure you are willing to assume for the data. E.g. your conditional mean model seems to ignore the panel nature of the data, as $\alpha_0$ and $\alpha_1$ can be estimated by pooling the data. Regarding the conditional variance, you could assume each time series has unique volatility dynamics and thus estimate $i$ different univariate GARCH models. You could assume all share the same dynamics, then you would somehow pool the data to obtain a single univar. GARCH model. – Richard Hardy Mar 16 '21 at 14:57
  • There could be other structures in between, e.g. an individual-specific ($i$-specific) intercept of the conditional variance equation but common (across $i$) slope coefficients. Regarding software use, some versions of panel GARCH could be special cases of the multivariate GARCH. If so, you could estimate them using existing multivariate GARCH functions by supplying these restrictions as options. (Are you sure your software package does not offer multivariate GARCH models?) – Richard Hardy Mar 16 '21 at 15:00
  • I have looked at existing approaches but it did not help me. You are right, my model does not include fixed effects - that is fine for now. Sadly, I can not estimate univariate GARCH models, since the time series only consist of around 10-30 observations each. I thought about pooling, but in my specific case, it does not make sense unfortunately. I agree that there might be fixed effects in the conditional variance equation aswell. I know about 'rmgarch' in R. But as far as I understood, multivariate GARCH does not serve my case - could you please explain how it does in your opinion? – shenflow Mar 16 '21 at 15:28
  • The series are indeed too short for anything from `rmgarch`. But it seems even univariate GARCH models are likely to have some trouble with such short time series. Fitting an individual model for each $i$ without borrowing information across $i$s may yield rather unstable estimates. On the other hand, even if you found a way to borrow information (a proper panel GARCH model), the limited length of the time series will be a fundamental obstance. Intuitively, I doubt you can exchange the cross sectional information for time series information to any significant degree here, though I am not sure. – Richard Hardy Mar 16 '21 at 15:41
  • Actually, the latter issue depends on how similar the individuals are. If they are virtually the same, then expanding the cross-sectional dimension will not help much. But if they are sufficiently different, it is roughly the same as expanding the time series dimension. Then it can help estimate the model's parameters more efficiently. If you are in a latter situation and manage to figure out a way to restrict the fitting of the model to yield the same coefficients across individuals, you would get what is needed. Bayesian estimation would allow for a soft version of such a restriction. – Richard Hardy Mar 16 '21 at 16:33
  • @shenflow Did you arrive at a solution to this problem? – GlaceCelery Sep 22 '21 at 11:23
  • @GlaceCelery no unfortunately not. – shenflow Sep 22 '21 at 13:51

0 Answers0