NOTES. This Script has 2 programs for simulations of TWFE scenarios Programs twfe_setup: Program used to define parameters for data simulation. twfe_data : Simulates data based on Defined parameters This Notes Provides some information of How TWFE_SETUP works I. Baseline: Outcome Without treatment ======================================= 1. Data size ------------- Two globals are needed: ids To define # of Individuals time To define # of periods we observe data Final Dataset is a balance Dataset with N (ids) individuals and T (time periods) 2. Exogenous variables, and individual effects ----------------------------------------------- The program creates 3 exogenous variables. x1 Time fixed variable x1_i ~ N(0,1) x2 Time varying variable set as a Stationary process x2_it=0.8*x2_i,t-1 + RUNIFORM(-.5,.5) x3 Time varying variable Random across time. X3_ut=rnormal And 1 individual unobserved and time invariant factor u Time fixed unobserved factor u_i ~ N(0,1) All X's are correlated with u_i. So one cannot use XTREG, RE 3. Trends ---------- Two types of trends can be simulated. Common trend and individual trend $xtrend Used to define Common trend slope. This can be any possitive or negative number $itrend Used to define individual trend slope. When used, the individual trend slope will be defined using a random draw from a uniform distribution between -$itrend to $itrend 4. Idiosyncratic error ----------------------- Idiosyncratic error v_it is assumed to distribution as N(0,1). But the size of the error can be modified using $noise. The final error has a distribution N(0,$noise) 5. Outcome without treatment ----------------------------- Given the information above, The data (without treatment) is created as follows: y0_it = 1+ x1_i + x2_it + x3_it + b_c * t + b_i * t + u_i + v_it * $noise \___________/ \_____________/ | \_____________/ Common Individual ind. Idiosyncratic trend trend effect error u_i ~ N(0,1) x1_i = 0.4 * u_i + (1-0.4^2)^.5 * ux1_i ux1_i ~ N(0,1) x2_i1 = 0.4 * u_i + (1-0.4^2)^.5 * ux2_i if t=1 x2_it = 0.8 * x2_i(t-1) + ux2_it if t>1 ux2_it ~ runiform(-.5,.5) x3_it = 0.4 * u_i + (1-0.4^2)^.5 * ux3_it ux3_it ~ rnormal(0,1) v_it ~ N(0,1) b_c = $xtrend b_i ~ runiform(-$itrend, $itrend) II. Treatment Timing ==================== 1. staggered design ------------------- Use $time_dd ==0 for Staggered design. Then consider the following: Treatment period is choosen at random using a uniform distribution. For example if we consider a $time length of 10 periods, observation "i" has 10% probability of being treated at period 1, 2, 3,..., 10. To allow for some observations to be "always treated" or "never treated", one can use global "out_time". This should be an integer >=0. For example if $out_time = 2, then observation "i" could be treated in any of the following periods: Observed periods ########################################################################## t' -1 0 1 2 3 4 5 6 7 8 9 10 11 12 <--- Observed time line t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 <--- For reference of TRUE timing ###### ######## Before After data is data is observed observed If an observation is treated at t'<1 then it appears always treated if an observation is treated at t'>10 then it appears never treated If $out_time = ($time/2), This can create the more usual DD structure, when $time is odd. 2. Standard DD design --------------------- if $time_dd !=0, simulated data will be like Standard DD. 50% units are treated at $time_dd III. Definition of Treatment Status and Treatment effect ======================================================= This section describes the different options for modifying the treatment effects 1. Treatment size ----------------- Treatment size (how much outcome increases) is constructed assuming two components TTE = TEFF0 + TEFF1 \____/ \_____/ \______/ Total T Eff0 Eff1 effect global ------ tsize_0 Defines size of TEFF0. This effect can be heterogenous by individual, but it is fixed across time. tsize_1 Defines Max size of TEFF1. This effect can be heterogenous by individual, but it can change across time treated, and when is an individual treated. Total Treatment (TTE) is the sum of the two types of effects 2. Treatment Heterogeneity across individual (i) ------------------------------------------------ The following parameters can be used to modify the heterogeneity of the treatment effects for individuals: TTE_i = het0_i * TEFF0 + het1_i * TEFF1 \_____/ \______/ \_____/ \______/ \_____/ Total T Ind. Eff0 Ind. Eff1 effect heter0 heter1 for i The factors "het0_i" and "het1_i" are used to induce heterogeneity on the treatment effects. This heterogeneity is obtained using a random draw from a uniform distributions as follows: het0_i = 1 + het0_i where het0_i~runiform(-$trhet0, $trhet0) het1_i = 1 + het1_i where het1_i~runiform(-$trhet1, $trhet1) Where: trhet0 defines the heterogeneity for TEFF0 trhet1 defines the heterogeneity for TEFF1 3. Treatment Heterogeneity across time (t) ------------------------------------------ The following parameters can be used to modify the heterogeneity of the treatment effects across time: TTE_it = het0_i * TEFF0 + het1_i * TEFF1 * het1_it * het2_it \______/ \______/ \_____/ \______/ \_____/ \______/ \______/ Total T Ind. Eff0 Ind. Eff1 time time effect heter0 heter1 heter1 heter2 Two parameters are used to determine heterogeneity. This affect het1_t and het2_t. HET1_IT: The first type of time heterogeneity determines if the treatment effect increases or decreases with the number of periods a unit is treated. This is determined using global tch_type. This can have 3 values: tch_type: 0 The effect does not change with #periods treated -> het1_t = 1 1 The effect increases with #periods treated. The effect grows at a decreasing rate. max effect at P->infinity -> het1_it = 1 - 1 / (#p_it+.1) For example at #p_it = 1, het1_i1 = 9.1 at #p_it = 2, het1_i2 = 52 at #p_it = 3, het1_i3 = 67 2 The effect decreases with #periods treated. The effect shrinks at a decreasig rate. When P -> infinity, TE is zero. (thank you Austin Nichols) -> het1_it = 1 / #p_it For example at #p_it = 1, het1_i1 = 100 at #p_it = 2, het1_i2 = 50 at #p_it = 3, het1_i3 = 33.33 HET2_IT: The second type of time heterogeneity determines if the treatment effect is larger or smaller if observation "i" was treated earlier or later compared to other observations. The timing depends on "t" not "t'" This is determined using global tch_early. This can have 3 values: 0 Effect is the same regardless of when was unit treated. -> het2_it = 1 1 Effect is larger for Units treated earlier (includes Out of time window treatment) -> het2_it = 0.5 + ( 1 / t0 ) where t0 is the time when unit "i" was treated 2 Effect is larger for Units treated later (includes Out of time window treatment) -> het2_it = 1 + ( 0.5 - 1 / t0 ) where t0 is the time when unit "i" was treated IV. Final Data Creation and Data Structure ========================================== Final Outcome is defined as follows: treat y_it = y0_it + TTE_it * treat_it where treat_it=1 if observation i is treated at time t *******************************************************************************/