Can I use PCA after lasso variable selection? I have a data regarding life satisfaction, of more than 2000 observations and 265 variables (most are categorical variables). I want to build a model, estimating the effects of society problems on the life satisfaction in the USA. 
So firstly, I use lasso2 to choose the variables which are most suitable for the regression, as well as reduce the dimension. It then results in 98 selected variables, however this is still a very large number of variables for a regression. Therefore, should I use PCA or Factor Analysis to combine these 98 variables into some factors, and then use these factors to regress? 
Is it possible anyway? I am afraid that, PCA/ FA and LASSO do the same thing, which is dimension reduction, and we can not use them at the same time. If so, which methods should I use after LASSO, in order to create factors, and use the factors to build a model? 
Thank you very much! 
 A: PCA and LASSO are two very different approaches and usually employed to very different ends. 
Lasso regression is not a dimension reduction technique but can be used for variable selection. Typically, when people use lasso (or other regularization techniques) the goal of the exercise is to attain a model which is good for prediction, although it also will tell you which variables are related to your response, how, and to what degree. The regularization that ridge employs helps account for multi-collinearity, and restricts the coefficients so they don't fit noise in your data, and (ideally) only fits the true signal produced by those effects. 
Principle components analysis (PCA) on the other hand is a dimension reduction technique, meaning that it takes n dimensions in the data, identifies the orthogonal axes of variation in the n-dimensional cloud, starting with the axis of greatest variation, and continues in the same manner until it has n axes. In this case there is no response variable and these models are not typically used for prediction, so much as understanding the relationships between all the variables simultaneously. There are multivariate techniques that can be used for regression though, such as linear discriminant analysis (LDA). 
What is the goal of your study? Is it purely prediction? Are you trying to understand the underlying relationships between variables? Some combination of both? If it is the latter then I would suggest you build separate models as predictive ones are not often easily interpreted when you have so many factors. This is because techniques that are good for prediction usually involve variable selection which is heavily frowned upon for purposes other than prediction as it promotes 'fishing expeditions' as you hunt for 'significant' relationships. In contrast, techniques which do not have a response variable (i.e. PCA) are used for hypothesis generation and do not rely on 'significance'.
If you clarify your goals more I am happy to offer advice on how to proceed. 
