In the HJM model, in case we have 3 factors, do these factors have an economic meaning at all ?
One of the motivations for multifactor models such as Two-Factor-HW, HJM and LMM (Lobor-Market-Model) is derived from the properties of the yield curve. One can run a Principal-Component-Analysis on yield-curve data in order to analyse the number of independant factors contributing to yield curve movements.
It has been shown that there are generally three types of movements a yield curve can undergo (and thus also three factors driving it's movements):
One factor models such as Hull-White can only capture parallel shifts of the yield curve. Two factor models can either reproduce the tilt or the hump. With a three factor model all three common yield curve patterns can be modelled.
Fixed Income Markets and Their Derivatives (See p. 136 ) mentions/explains that the first two componetns of a PCA explain up to 95-98% of the variations of the yield curve. The remaining variation is mostly picked up by the fird factor. (I will also refer you to Interest Rate Risk Modeling: The Fixed Income Valuation Course as an alternative source)