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Are there any formal tests for heteroscedasticity for non-normal data? I want to run the test on time series logged returns, so would it be okay to assume a linear relationship? To me it makes intuitive sense that the greater the return (rise or fall in price) the greater the variance.

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With financial data, the question you raised was first addressed in the 1980s by Robert Engle who later got a Nobel Prize for his contributions to econometrics, as well as Tim Bollerslev. They formalized exactly what you think makes sense: the greater the change in the asset price, the more volatility will endure. Models of these type are called autoregressive conditional heteroskedastity models, or ARCH for short, and they say that variance at time $t$ is a function of the magnitudes of shocks in prior periods: $$ \sigma_t^2 = \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \alpha_2 \epsilon_{t-2}^2 + \ldots + \nu_t $$ with understanding that coefficients in this model are positive. There's been an unimaginable number of extensions this model has received, including all soft of transforms, putting lags of $\sigma_t^2$ into the model, treating positive and negative shocks differently, specifying thresholds, etc. These are reduced form models, in the sense that they may not provide a clear message about the mechanism, but they fit financial data very well, and in particular generate the desirable heavy tail distributions even with $\epsilon_t$ are conditionally $N(0,\sigma_t^2)$.

The heteroskedasticity test in this context would of course be the test that $\alpha_1 = \alpha_2 = \ldots = 0$.

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One such test is the Brown-Forsythe test, which is derived from ANOVA.

The basic idea is that you pick a way to partition your data and then compare the mean absolute deviations from the median of each group. One common way is to put observations whose time is before the half-way point in one group, and those whose time is at/after the half-way point in another.

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For time series data, you can also examine the ACF and PACF of squared residuals of your model that assumes a constant variance. Formally, you could run a Ljung-Box or McLeod-Li test.

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