This paper presents a set of new run-time tests for speculative parallelization of loops that defy parallelization based on static analysis alone. It presents a novel method for speculative array privatization that is not only more efficient than previous methods when the speculation is correct, but also does not require rolling back the computation in case the variable is found not to be privatizable. We present another method for speculative parallelization which can overcome all loop-carried anti and output dependences, with even lower overhead than previous techniques which could not break such dependences. Again, in order to ameliorate the problem of paying a heavy penalty for speculatively parallelizing loops that turn out to be serial, we present a technique that enables early detection of loop-carried dependences. Our experimental results from a preliminary implementation of these tests on an IBM G30 SMP machine show a significant reduction in the penalty paid for mis-speculation, from roughly 50% to between 2% and 18% of the serial execution time. For parallel loops, we obtain about the same, and often, even better performance relative to the previous methods, making our techniques extremely attractive.