Difference between heteroscedasticity and ARCH effects? What is the difference between heteroscedasticity and ARCH effects?
For example in R you can do a Breusch-Pagan Test to test for heteroscedasticity, and a Lagrange Multiplier (LM) test for autoregressive conditional heteroscedasticity (ARCH) effects. Do these tests not test for the exact same thing?
 A: There are different forms of heteroskedasticity and ARCH is one of them.
In a cross-sectional setting, conditional heteroskedasticity usually means that $\text{Var}(\varepsilon_i|X_i)=\sigma_i^2=\sigma^2(X_i)$ where $\varepsilon_i$ is the error term of observation $i$ and $X_i$ is the vector of regressor values for observation $i$. Thus, the conditional variance is a function of the regressors.
Example: $\text{Var}(\varepsilon_i|X_i)=c|X_{1i}|$ for some positive constant $c$.
ARCH is a specific kind of conditional heteroskedasticity that applies only to time series data (or data that has a time series dimension). As is clear from the title, it is autoregressive. Here, $\text{Var}(\varepsilon_t|I_{t-1})=\sigma_t^2=\sigma^2(I_{t-1})$ where $\varepsilon_t$ is the error term of the time period $t$ and $I_{t-1}$ is the information available at time $t-1$, e.g. the set of historical values of the dependent variable: $I_{t-1}=\{y_{t-1},y_{t-2},\dots,y_1,y_0\}$.
Example: the popular GARCH(1,1) model has $\text{Var}(\varepsilon_t|I_{t-1})=\sigma_t^2=\omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2$.
Note that in the case of ARCH, the conditional variance does not vary with the regressors. (You could have a combination of ARCH patterns and variance dependence on regressors, but this would not be a pure ARCH any longer.)
The tests you are referring to do not test the same thing. Breusch-Pagan tests for the first kind of heteroskedasticity, while ARCH-LM tests for the second kind (ARCH).
