The use of a validate/confirmation button is usually tied to whether or not the clicking of that button is required for the user's choices to take effect.
I would ask questions such as:
Is the data being filtered by these options visible and changing live on this page?
If so, the button would make zero sense.
1b. Is the click of the button required in order for the user's selection to take effect? If so, would clicking outside the box, which also closes it also have the same effect of saving the data?
Why is a drop down being used to begin with? Drop downs are typically not intuitive for multi-selection options.
2a. Where you list "option 1, option 2" because two options are selected, what happens when 3 options are selected? Does the drop down become wider? If not, then the user cannot view all selected options and may think only 2 are selected. Where you say "option 1, option 2" when the drop down is closed, this is typically either the currently selected option, or a category of the available options if no option is selected.
2b. Is the drop down being used simply because there is not enough space to view all the options all the time, or this might feel cluttered? If so, perhaps there is a similar way to hide/show the options, without a drop down. I would reccomend this, assuming you are not restricted by the use of a button to actually save the data and have the user's option take effect.
Deciding a category that each group of options falls into. Either: display this category name when all the options to select are not visible, OR reliably display all currently selected options. (i.e. showing only 1 and 2 when 1, 2, and 3 are selected is misleading).
When deciding if you should use a 'validate' or 'save' button, consider if this extra click is necessary for task completion.
Keep number of necessary clicks low.