  Measuring the causal impact of an advertising campaign on sales is an
essential task for advertising companies. Challenges arise when companies run
advertising campaigns in multiple stores which are spatially correlated, and
when the sales data have a low signal-to-noise ratio which makes the
advertising effects hard to detect. This paper proposes a solution to address
both of these challenges. A novel Bayesian method is proposed to detect weaker
impacts and a multivariate structural time series model is used to capture the
spatial correlation between stores through placing a $\mathcal{G}$-Wishart
prior on the precision matrix. The new method is to compare two posterior
distributions of a latent variable---one obtained by using the observed data
from the test stores and the other one obtained by using the data from their
counterfactual potential outcomes. The counterfactual potential outcomes are
estimated from the data of synthetic controls, each of which is a linear
combination of sales figures at many control stores over the causal period.
Control stores are selected using a revised Expectation-Maximization variable
selection (EMVS) method. A two-stage algorithm is proposed to estimate the
parameters of the model. To prevent the prediction intervals from being
explosive, a stationarity constraint is imposed on the local linear trend of
the model through a recently proposed prior. The benefit of using this prior is
discussed in this paper. A detailed simulation study shows the effectiveness of
using our proposed method to detect weaker causal impact. The new method is
applied to measure the causal effect of an advertising campaign for a consumer
product sold at stores of a large national retail chain.
