B737-700
Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Air Transport Pilot (ATP)
Flight Crew Last 90 Days 100; Flight Crew Type 12500
Aircraft Equipment Problem Critical
Air Traffic Control Provided Assistance; Flight Crew Overcame Equipment Problem
Aircraft
Climbing out of ZZZ in moderate rain and passing through about 18;000 feet; with the FO flying; we got an airspeed disagree and altitude disagree on our displays; a Master Caution for Engine; autopilot (B) kicked off; and VNAV disappeared all within a few seconds. We compared instruments and determined that the Standby and Captain's displays matched with airspeed at 280 and close to the same altitude still climbing. We transferred aircraft control and ran the airspeed unreliable checklist/altitude unreliable checklist. After reporting our difficulties with ATC; we leveled at FL250; to finish checklist and run checklist for EEC alternate mode. At that time the FO instruments showed an airspeed of 130 KIAS and 25;500 feet. It took a while to get through checklist; but we were able to get autopilot-A back.  We talked about the bad weather at ZZZ and decided to continue to ZZZ1 due to VMC conditions there. On descent passing through 15;500 feet; FO instrumentation appeared to recover. Approach and landing were uneventful.  Priority handling was requested to ensure ATC was aware of our indication problems.
B737-700 Captain reported a loss of reliable airspeed and altitude on the FO's side during climbout.  The Captain's instruments were functioning properly; so the crew decided to continue to destination due to it having better weather conditions than their departure airport.