Dear Tuan,

How many cases are there in total and how many do not fit?

You could exclude some cases. I suggest you analyse why they are different and to be excluded as well.

Hope, that helps.

Best,

Ines

Sometimes, "log" transforming the outcome variable can do some magic. I'll suggest you give it a try.

so if there is a pattern to your error, perhaps you can identity a missing variable that would account for the patches of over / under-prediction... so to give a very simplistic example, let's say you regressed per capita income vs city size and discovered that all the large cities that are ports were under predictions (the per capita income in reality is higher)... maybe there is a need for a "coastal city economy" type of effect and this would be useful

transformation of your response or predictors may change this.

You may also try a robust linear regression method and compare it to your results. It may be more reliable. 

If all these suggestions do not help, you need a model that accounts for heteroscedasticity, they usual deal with an additional matrix of weights. Just google "Regression Models with Heteroscedastic Errors"

I like Mortons answer the most,

If your variance varies between groups, you should find out why this happens. If you can't figure out what the reason is, you stil might report your results (because the relationship between your predictor and your dependent variable can still be true) but warn your readers that the inferences can not be drawn from the results. (meaning: hypothesis testing is not possible).

Tuan,

It is rare to see a perfectly homoskedastic model because that would mean that the error is true random noise, which in turn means that a) the model is indeed linear, and b) it includes all relevant variables that explain the DV. These are very restrictive conditions, so some heteroskedasticity may need to be accepted. Question is: how much heteroskedasticity should you allow? If the QQ plot is nonlinear, first you should rule out collineariy by checking VIF values. This lets you verify that the IVs are nearly orthogonal. Then check the adj R-sq value. A strongly non-linear QQ plot and low R-sq might indicate an omitted variable bias. If yes, you need to refer to theoretic basis of your model to examine what additional variables could be included in the regression. Alternatively, if the QQ plot is a strong S-curve, you can do log or polynomial (typically square root) transformation of IVs (if your theory supports) and see if QQ plot straightens up. For valid transformation, the R-sq should improve. Of course you will have to explain what the transformed variables mean in theoretic terms.

Try transforming your data using appropriate method (log, inverse, etc.), then check the assumptions again. If this fails, use a non-linear regression approach. 

Tq for various feedback and suggestions. Over my reading I'd found levene test to test the homoscedasticity..

Constant variance of error(or homoscedasticity) seems to be unrealistic, so the model has serious problems. There are models that do not require dispersion parameters because they are implicit in the relation between media, maximum variable value and minimum variable value of the positive univariate sample taken.  So far, nobody has given me any convincing proof of the need of variance in its conventional definition and handling, neither about its use in continuous distributions. In the same way that media may change from sample to sample, variance may change for those univariate samples. So problem may be bigger for multivariate models that assume linear relationships of several variables. This problems ask better explanations before teaching or applying those premises and methods. Thanks, Emilio