Residual analysis for the lasso estimator I obtained the residuals of the lasso estimator for the full prostate data as following. I'm asked the question: "Was a residual analysis done, Are the residuals homoscedastic?"
Is there a way to check the homoscedasticity of the residuals of the lasso estimates?
require(readr)
require(glmnet) 
data <- read_delim("http://www-stat.stanford.edu/~hastie/ElemStatLearn/datasets/prostate.data",
                   "\t", escape_double = FALSE, trim_ws = TRUE)

# remove the columns of row number and train set index
data <- data[, c(-1,-11)]

# Design matrix and response vector
X <- as.matrix(data[,1:8])
y <- data$lpsa

# Cross validation
set.seed(1)
lasso.cv <- cv.glmnet(X, y)

# Prediction
yhat <- predict(lasso.cv, newx = X)

# residuals
res <- y - yhat

 A: I am not sure you would expect the residuals to be homoscedastic in a regularised regression such as lasso, since you have moved away from ordinary least squares and its particular properties.
You can judge whether there is heteroscedasticity by seeing if there is any relationship between the residuals and the predicted values.
An initial approach might be plot the residuals against the predicted values as below for example using  the code plot(res ~ yhat) and look for an obvious pattern; for example in OLS regression, a fan or a U-shape might suggest the structural form of your assumed model might be wrong.
Visually here it looks as if lower predicted values are more likely to be associated with negative residuals while higher predicted values more likely to be associated with positive residuals.  That is indeed the case, and is a result of the shrinkage that comes from regularisation, causing the predicted values to tend to be closer to the mean and so the residuals at the extremes to be less centred around the $x$-axis.

