Android PopupWindow elevation does not show shadow

As answered by an Android developer.

If the inflated view doesn't have a background set, or the popup window itself doesn't have a background set (or has a transparent background) then you won't get a shadow.

which was my case and seems to be yours, since you are not using setBackgroundDrawable.

This worked for me

popupWindow.setBackgroundDrawable(new ColorDrawable(Color.WHITE));

I've opened a new issue suggesting that they update the documentation (https://code.google.com/p/android/issues/detail?id=174919)

For others who visit this answer and missed what the OP already had, you should set the elevation to create a shadow:

if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.LOLLIPOP) {
    popupWindow.setElevation(20);
}

Depending on what your content view is, you might also need to set the background drawable, although this is not always necessary. If needed you can do as @Maragues suggested:

popupWindow.setBackgroundDrawable(new ColorDrawable(Color.WHITE));

To support pre-Lollipop devices you could use a 9-patch or image that includes the shadow within it.

Code

This is the code for the image above.

LayoutInflater inflater = (LayoutInflater) getSystemService(LAYOUT_INFLATER_SERVICE);
View popupView = inflater.inflate(R.layout.popup_window, null);
int width = LinearLayout.LayoutParams.WRAP_CONTENT;
int height = LinearLayout.LayoutParams.WRAP_CONTENT;
boolean focusable = true;
final PopupWindow popupWindow = new PopupWindow(popupView, width, height, focusable);
popupView.setOnTouchListener(new View.OnTouchListener() {
    @Override
    public boolean onTouch(View v, MotionEvent event) {
        popupWindow.dismiss();
        return true;
    }
});

if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.LOLLIPOP) {
    popupWindow.setElevation(20);
}

popupWindow.showAtLocation(anyView, Gravity.CENTER, 0, 0);

Note:

The elevation is in pixels when set in code, but usually in dp when set in xml. You should convert a dp value to pixels when setting it in code.