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This paper studies vector autoregressive models with parsimoniously time-varying parameters. The parameters are assumed to follow parsimonious random walks, where parsimony stems from the assumption that increments to the parameters have a non-zero probability of being exactly equal to zero. We estimate the sparse and high-dimensional vector of changes to the parameters with the Lasso and the adaptive Lasso. The parsimonious random walk allows the parameters to be modelled non parametrically, so that our model can accommodate constant parameters, an unknown number of structural breaks, or parameters varying randomly. We characterize the finite sample properties of the Lasso by deriving upper bounds on the estimation and prediction errors that are valid with high probability, and provide asymptotic conditions under which these bounds tend to zero with probability tending to one. We also provide conditions under which the adaptive Lasso is able to achieve perfectmodel selection. We investigate by simulations the properties of the Lasso and the adaptive Lasso in settings where the parameters are stable, experience structural breaks, or follow a parsimonious random walk. We use our model to investigate the monetary policy response to inflation and business cycle fluctuations in the US by estimating a parsimoniously time varying parameter Taylor rule. We document substantial changes in the policy response of the Fed in the 1970s and 1980s, and since 2007, but also document the stability of this response in the rest of the sample.

Vector Autoregressions with Parsimoniously Time Varying Parameters and an Application to Monetary Policy