You need to use robust/sandwich standard errors to resolve heteroscedasticity problem. An alternative way is to use bootstrapping. If data are within-subject correlated, then nothing to worry about. Otherwise, you can constrain the error terms to follow an AR(1) process. Alternatively, you can introduce an autoregressive independent variable, but this process typically bias the parameter estimates. The latter method allows you to use a dynamic multilevel model (https://doi.org/10.1177%2F0963721416666518).

Hi Sanjib Sherpa,

Heteroskedasticity and autocorrelation are between independent and dependent variables, including multilevel model. To overcome this problem, it is better that you use E-view statistical package. In the E-views, you can determine Newey-West Heteroskedasticity consistent covariance estimates, if the data structure is time-series. If the data are cross-sectional, you can determine White Heteroskedasticity consistent covariance estimates. Also, you have to check whether the variables have a co-integration relationship between the independent and dependent variables to permit autocorrelation in the model.

Hope this will help you. Good luck in your efforts.

I do like Eviews. It includes tests for auto-correlation & heteroskedssicity. Heteroskedasticity is unequal variances among residuals, so is fundamenttaly an issue with the dependent variable. Autocorrllation is when rediduals are correlated with one another over time (for time series) or among observations (for cross-sectionms. Both indicate issues with the estimated models. Ask if the models are appriate gog noth theory and data. Unequal variances may indicate different populations. Correlation among observations should be mofelef

Should be modeled as influence of past on present (inn time series) or influence of sequential observations on one another (in cross-section)

Hi Sanjib,

at least for the heteroscedasticity part I would say:

At least for a model with categorical predictors this could be overcome. For example, if there is a control vs. treatment group experimental design, it is natural to assume that there is a larger variance within the treatment group if one goes with the idea of "random effect" (the treatment has stronger influence on some participants than on others). In the control group this is not the case, because there is no treatment effect, 'normal variance' so to say.

If this is a description which resembles your situation (between subject design), then you can deal with this, by estimating participant intercepts: by condition instead of taking the grand mean as reference (i.e., separate intercept variances for control vs treatment). In most programs I think this could be easily achieved by multiplying the factors in the random-intercept term with a dummy coding variable that reflects the factor level by hand ... (in R for example), in Stata ... I can not tell, maybe somebody else can (btw. I think this argument is valid in a lot of studies, but nobody does this, knows this, or will notice it if you don't take care of it..., but it is, of course statistically accurate).

However, things change in a within participant design, like "before-after" comparison, or with continuous predictor variables, or more complex designs. Maybe if you provide more info... :) - Since you mention auto-correlation it implies somehow that there are more than two observations per subject (maybe on a time scale)?

Hope this helps a bit.

Best

René

Random effects, that is multilevel models are specifically designed to model dependency (ie autocorrelation) and heteroscedasticity (at any level) and give more correct standard errors ; see

Article Modelling Complexity: Analysing Between-Individual and Betwe...

Article Do multilevel models ever give different results?.

Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...

Article Fixed and Random effects models: making an informed choice

Thank you all for the response. As stated by Kelvyn, robust cluster SE random effect model controls for both autocorrelation and heteroscedasticity. And MLM= Random effect model which means MLM- robust address these issues but i could not find any article that applied MLM discussing these two issues. I have not also found the article that discussed how to solve omitted variable bias in MLM.

Yes, multilevel modeling can account for both heteroscedasticity and autoccorrelation. [May not be the best approach, ( perhabs PSCE, HAC, ? ) nor the model...I really don't know the sort of data you are modelling.] See : https://www.google.com/url?sa=t&source=web&rct=j&url=https://stats.idre.ucla.edu/sas/faq/how-can-i-fit-a-multilevel-model-with-heteroskedasticity-in-my-residual-variances/&ved=2ahUKEwj8lZL7r77rAhUMY8AKHRjCCdsQFjAJegQIARAB&usg=AOvVaw1bQEAp1DEdEX1l5OTN5yT3 *Intuitively, clustered standard errors allow researchers to deal with two issues: the correlation of observation in the same group, and the correlation over time of the same units (e.g., students or classes or countries over time). The clustering is performed using the variable specified as the model’s fixed effects! New on STATA 16 that might help - see https://www.stata.com/manuals/xtxtreg.pdf

Statistical methodology for handling omitted variables is presented in a multilevel modeling framework was discussed by Kim J. S., Frees E. W.(2006). "Omitted variables in multilevel models". Article on Mendeley's https://www.mendeley.com/catalogue/be8ef5be-1946-3e25-a263-b963feccec80

Sanjib Sherpa To be clear I am not suggesting that robust standard errors are the way to go. I see autocorrelation and heteroscedasticity as substantially interesting and not a problem requiring a technical fix. Finding differences with robust SE is telling you there is a problem and is not a solution see

https://www.researchgate.net/post/When_should_I_use_multilevel_modellings_vs_cluster_robust_standard_errors

What is required is terms in the model that explicitly account for both heteroscedasticity and dependence.

On the former, see Article Modelling Complexity: Analysing Between-Individual and Betwe...

Book Developing multilevel models for analysing contextuality, he...

In terms of dependence , the multilevel random effects model analyses differences between entities which are equivalent to similarity (autocorrelation) within.

This paper has an explicit discussion via Variance partitioning Coefficients and the intra class correlation coefficient

Article Ethnic Residential Segregation: a Multi-Level, Multi-Group, ...

This volume considers autocorrelation over time and over space.

Book Developing multilevel models for analysing contextuality, he...

@Kelvyn Jones, Sanjib, et.al. -

(Hmm. My @ key does not seem to work.)

Kelvyn, you noted that "...variance (at any level) being a function of predictor variable...."

Heteroscedasticity is a function of prediction size, not any one predictor in multiple regression. (See https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity for one level.) Is that what you meant?

You also noted that "I see autocorrelation and heteroscedasticity as substantially interesting and not a problem requiring a technical fix." I think that that is a very important comment for everyone to remember.

Also, I think the use of scatterplots can be helpful.

Cheers - Jim