Take for example the Consumption-based model with a power utility function estimated by Cochrane in his paper "A Cross-Sectional Test of an Investment-Based Asset Pricing Model" (1996). The following refers to table 7 and figure 6.
When we use GMM to estimate the unconditional model, where we only use the base assets as moment conditions and conduct a J-test to test, if the pricing errors are close enough to zero, the model cannot be rejected, although of its poor empirical performance. We can see that for example by plotting the exspected returns against the predicted returns.
In contrast, if we use the base assets and additionally managed portfolios, a conditinal estimation, the J-test clearly rejects the model, which is also supported by the poor preditcions.
Now, it is very unclear to me why this is the case. How can it be that the J-test cannot reject the unconditional estimation, despite the poor performance but the conditional estimation?