I have try to conduct a second order factor consisting of five first order factors. However, some of the first order factors are related to each other negatively, while others are related to each other positively. I checked some of the books but I could not find the answer. When I conduct the analyses I get higher goodness of fit statistics which are above the cut points. So, does it make sense to make analyses with those factors?
Sure. Nothing prevents the second order factor from loading on the first order factors in such a way that some of the covariances are negative. However, a second order model makes very stringent assumptions. Your first order model allows for 10 correlations between the factors. The second order model claims that these can all be computed from 5 loadings (or 4 loadings and a variance). That's a tall order. And any discrepancies in the first order model (say, manifest 1 from Factor 1 has a residual correlation with manifest 3 of Factor 4), will kick in to further reduce the fit of your model. A measure has to be very well designed and reflective of genuine underlying factors to fit a second order model.