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Sagittarius A* (Sgr A*) is the variable radio, near-infrared (NIR), and X-ray source associated with accretion onto the Galactic center black hole. We have analyzed a comprehensive submillimeter (including new observations simultaneous with NIR monitoring), NIR, and 2-8 keV dataset. Submillimeter variations tend to lag those in the NIR by $\sim$30 minutes. An approximate Bayesian computation (ABC) fit to the X-ray first-order structure function shows significantly less power at short timescales in the X-rays than in the NIR. Less X-ray variability at short timescales combined with the observed NIR-X-ray correlations means the variability can be described as the result of two strictly correlated stochastic processes, the X-ray process being the low-pass-filtered version of the NIR process. The NIR--X-ray linkage suggests a simple radiative model: a compact, self-absorbed synchrotron sphere with high-frequency cutoff close to NIR frequencies plus a synchrotron self-Compton scattering component at higher frequencies. This model, with parameters fit to the submillimeter, NIR, and X-ray structure functions, reproduces the observed flux densities at all wavelengths, the statistical properties of all light curves, and the time lags between bands. The fit also gives reasonable values for physical parameters such as magnetic flux density $B\approx13$ G, source size $L \approx2.2R_{S}$, and high-energy electron density $n_{e}\approx4\times10^{7}$ cm$^{-3}$. An animation illustrates typical light curves, and we make public the parameter chain of our Bayesian analysis, the model implementation, and the visualization code.