  We give an algebraic construction of standard modules (infinite dimensional
modules categorifying the PBW basis of the underlying quantized enveloping
algebra) for Khovanov-Lauda-Rouquier algebras in all finite types. This allows
us to prove in an elementary way that these algebras satisfy the homological
properties of an `affine quasi-hereditary algebra.' In simply-laced types these
properties were established originally by Kato via a geometric approach. We
also construct some Koszul-like projective resolutions of standard modules
corresponding to multiplicity-free positive roots.
