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The orbital period of the hot Jupiter WASP-4b appears to be decreasing at a rate of $-8.64 \pm 1.26$ msec/yr, based on transit-timing measurements spanning 12 years. Proposed explanations for the period change include tidal orbital decay, apsidal precession, and acceleration of the system along the line of sight. To investigate further, we performed new radial velocity measurements and speckle imaging of WASP-4. The radial-velocity data show that the system is accelerating towards the Sun at a rate of $-0.0422\pm 0.0028$ m/s/day. The associated Doppler effect should cause the apparent period to shrink at a rate of $-5.94 \pm 0.39$ msec/yr, comparable to the observed rate. Thus, the observed change in the transit period is mostly or entirely produced by the line-of-sight acceleration of the system. This acceleration is probably caused by a wide-orbiting companion of mass 10-300$\,M_{\rm Jup}$ and orbital distance 10-100$\,$AU, based on the magnitude of the radial-velocity trend and the non-detection of any companion in the speckle images. We expect that the orbital periods of 1 out of 3 hot Jupiters will change at rates similar to WASP-4b, based on the hot-Jupiter companion statistics of Knutson et al. (2014). Continued radial velocity monitoring of hot Jupiters is therefore essential to distinguish the effects of tidal orbital decay or apsidal precession from line-of-sight acceleration.