Fig. 2

Assignment of F(t)× "intervention" and F(t)× "post-time" during the lag period for different lag lengths L. \({\widehat\beta}_3\) and corresponding 95% CIs with different models with 1000 simulation repetitions. Panel a and b plot the \({\widehat{\beta }}_{3}\) and corresponding 95% CIs with different models. The outcome time series of Panel a and b are 2&ReLU and N2&ReLU type with lag length L = 4 and adding white noises \(N\left(0,{3}^{2}\right)\). Different colors represent different models: pink represents the CSR model, red represents the OSR-ReLU model, and blue represents the OSR-Sig model. The horizontal plane represents the true value of the long-term effect of the intervention on the outcome variable for that type (+ 2 or -2). The dots in the figure represent the point estimates \({\widehat{\beta }}_{3}\) of the long-term effects, and the vertical lines represent the 95% CIs for the corresponding estimates