Sample data for a complex structure, with periodic stable \(\beta_{X,t}\) and random walk \(\beta_{C,t}\)

Sample data for a complex structure, with periodic stable \(\beta_{X,t}\) and random walk \(\beta_{C,t}\)

Usage

data_complex

Format

A data frame with 1000 rows and 4 variables, the data generation process follows the following equation $$Y_t = 40+0.5Y_{t-1}-\beta_{X,t}X_t-0.5X_{t-1}-\beta_{C}C_t+v_t, v_t \sim \text{i.i.d }\mathcal{N}(0,1)\\ \text{where } \beta_{X,t}= -I(t\in[0,400])-2.5I(t\in(400,700]-I(t\in(700,1000])), \\\beta_{C,t} =\beta_{C,t-1}+1.5+w_{\beta,t}, w_{\beta,t}\sim \mathcal{N}(0,0.05)$$

Date

A simulated date

y

outcome, with data generating process as above

y_1

one step laggged out come

c

continous covarite

x

continous treatment

x_1

one step lagged continous treatment