  Using grand canonical Quantum Monte Carlo (QMC) simulations combined with
Maximum Entropy analytic continuation, as well as analytical methods, we
examine the one- and two-particle dynamical properties of the Hubbard model on
two coupled chains at half-filling. The one-particle spectral weight function,
$A({\bf k},\omega)$, undergoes a qualitative change with interchain hopping
$t_\perp$ associated with a transition from a four-band insulator to a two-band
insulator. A simple analytical model based on the propagation of exact rung
singlet states gives a good description of the features at large $t_\perp$. For
smaller $t_\perp$, $A({\bf k}, \omega)$ is similar to that of the
one-dimensional model, with a coherent band of width the effective
antiferromagnetic exchange $J$ reasonably well-described by renormalized
spin-wave theory. The coherent band rides on a broad background of width
several times the parallel hopping integral $t$, an incoherent structure
similar to that found in calculations on both the one- and two-dimensional
models. We also present QMC results for the two-particle spin and charge
excitation spectra, and relate their behavior to the rung singlet picture for
large $t_\perp$ and to the results of spin-wave theory for small $t_\perp$.
