Thank you Ivo Carneiro De Sousa for your valuable suggestions. 

Dear Navita Nathani 

Greetings,

Please see the attached file about your topic may be useful for you.

Best Regards

Navita -

Many times when people say "tests" in statistics, I think they are thinking "p-values," but p-values alone are misleading.  They are sample size dependent.  A small sample size gives you a larger p-value, and a larger sample size gives you a smaller p-value, all else held equal.  Consider this:

Press release for the American Statistical Association:

http://www.amstat.org/newsroom/pressreleases/P-ValueStatement.pdf

You may expect a degree of heteroscedasticity and/or a degree of autocorrelation, more than just considering a 'test' that says yes or no and seems more definitive than it is in reality.

Perhaps you should be using some data to select some candidate models, based on theory, and using other data to validate them/it.  For a given model, how well would it have worked compared to data you held out of the selection process to use for validation? 

Also, scatterplot graphics can help you to see what is happening.

By the way, time series data are not cross-sectional data.  You mentioned the former in the explanation to your question, and the latter in the question.  

I was thinking about regression use in either here.

You could be considering "longitudinal"/"panel" data.  You could research those terms.  There I think both are considered, but separately, cross-sectional and time series have different meanings, as discussed at the link below:

 https://www.researchgate.net/publication/275365705_When_Prediction_is_Not_Time_Series_Forecasting

Cheers - Jim

PS - Note that for regression, neither the y data nor any x data need be normally distributed.  They can, for example, be greatly skewed.  However, it might be desirable that the estimated residuals (or really better, the estimated random factors of the estimated residuals in weighted least squares regression) be close to normally distributed, though the central limit theorem and a large enough sample may help anyway.

Research When Prediction is Not Time Series Forecasting

It depends on the hypotheses you want to test.

In general you need a multi-level approach here.

can we use multicollinearity in cross sectional data?