  The Arellano-Bond estimator is a fundamental method for dynamic panel data
models, widely used in practice. However, the estimator is severely biased when
the data's time series dimension $T$ is long due to the large degree of
overidentification. We show that weak dependence along the panel's time series
dimension naturally implies approximate sparsity of the most informative moment
conditions, motivating the following approach to remove the bias: First, apply
LASSO to the cross-section data at each time period to construct most
informative (and cross-fitted) instruments, using lagged values of suitable
covariates. This step relies on approximate sparsity to select the most
informative instruments. Second, apply a linear instrumental variable estimator
after first differencing the dynamic structural equation using the constructed
instruments. Under weak time series dependence, we show the new estimator is
consistent and asymptotically normal under much weaker conditions on $T$'s
growth than the Arellano-Bond estimator. Our theory covers models with high
dimensional covariates, including multiple lags of the dependent variable,
common in modern applications. We illustrate our approach by applying it to
weekly county-level panel data from the United States to study opening K-12
schools and other mitigation policies' short and long-term effects on
COVID-19's spread.
