The Post-Double-Selection Method

Estimating Equation

The growth estimating equation is specified as follows:

d ln y_i,t =X

g∈G

δ_g^ex· [d ln F M A_ig,t] +X

g∈G

δ_g^im· [d ln CM A_ig,t] + ηw_i,t+ φs_i,t+ D_t+ ε_i,t,

where d ln F M A_ig,t=P

k∈Gλ^ex_ik,td ln F M A_ik,t, d ln CM A_ig,t=P

k∈Gλ^ex_ik,td ln CM A_ik,t are the log-differenced market access terms aggregated up to the cluster level, and D_t are the time fixed effects.

The vector w_i,t collects the industry-level initial equilibrium variables such as initial import and export shares (λ^im_ik,t and λ^ex_ik,t), weighted initial firm and consumer market access (λ^ex_ik,t· ln F M A_ik,t and λ^im_ik,t· ln CM A_ik,t), the squares ((λ^im_ik,t)² , (λ^ex_ik,t)², (λ^ex_ik,t· ln F M A_ik,t)² and (λ^im_ik,t· ln CM A_ik,t)²) and the interactions ((λ^ex_ik,t)²· ln F M A_ik,t

and (λ^im_ik,t)²· ln CM Aik,t). The vector si,t collects the interactions between the initial equilibrium variables and the industry-level foreign shocks, such as (λ^ex_ik,t)²·d ln F M Aik,t

and (λ^im_ik,t)² · d ln CM A_ik,t.

Since our estimating equation has a large number of controls relative to the sample

size, the OLS estimation is infeasible, and dimension reduction is necessary. We estimate the above growth equation by implementing the “post-double-selection”

method.

Post-Double-Selection Method

The post-double-selection procedure works in two steps. In the double-selection step, LASSO is applied to select controls variables that are useful for predicting the depen-dent and independepen-dent variables respectively. In the post-selection step, coefficients are estimated via an OLS regression of dependent variables on the independent variables and the selected controls.

First, let’s rewrite the estimation equation as follows:

d ln y_i,t = d_i,tδ + x_i,tβ_y+ µ_i,t,

where d_i,t denotes the vector of treatment variables d ln F M A_ig,t and d ln CM A_ig,t, and x_i,t is the vector of control variables.

Applying LASSO directly to our estimation equation above might lead to the omitted-variable bias if the LASSO procedure drops a control variable that is highly correlated with the treatment but the coefficient associated with the control is nonzero.

To learn about the relationship between the treatment variables and the controls, let’s introduce a reduced-form equation

d_i,t = x_i,tβ_d+ v_i,t for each element d_i,t of the vector d_i,t.

Substituting the reduced-form d_i,t into the growth estimation equation we get d ln y_i,t = x_i,t(β_dδ + β_y) + (v_i,tδ + µ_i,t)

d_i,t = x_i,tβ_d+ v_i,t. ∀d_i,t

Both equations are used for variable selection. The first equation is used to select a set of variables that are useful for predicting the dependent variable d ln y_i,t and the second equation is used to select a set of controls that are useful for predicting each of the treatment variables di,t. The reduced form system could be further rewritten as

z_i,t = x_i,tβ + ε_i,t

where z_i,t is the vector of dependent variable d ln y_i,t and all treatment variables d_i,t. A feasible double-selection procedure via LASSO is then defined as follows

minβ E(z_i,t− x_i,tβ)²+ λ

n||Lβ||₁

where L = diag(l₁, l₂, . . . , l_p) is a diagonal matrix of penalty loadings and λ is the penalty level. The LASSO estimator is used for variable selection by simply selecting the controls with nonzero estimated coefficients.

The double-selection procedure first selects a set of controls that are useful for predicting the independent variable d ln yi,t and treatment variables di,t. Then in the post-LASSO step, we estimate δ_g^ex and δ_g^im by ordinary least squares regression of d ln y_i,t on d_i,t and the union of the variables selected for predicting d ln y_i,t and d_i,t.

K-fold Cross Validation

The penalty level λ controls the degree of penalization. Practical choices for λ to prevent overfitting are provided in Belloni et al. (2012, 2014a,b). We follow the online appendix of Belloni et al. (2014a) and choose λ by K-fold cross validation.

The K-fold cross-validation works as follows:

i. Randomly split the data (y_i,t, x_i,t, d_i,t) into K subsets of equal size, S₁, S₂, . . . , S_K ii. Set the potential tuning parameter set to be [λ^RT − 100 : grid : λ^RT + 100],

where λ^RT = 2.2√

nΦ(1 − γ/2p) is the rule of thumb tuning parameter suggested in Belloni et al. (2012, 2014b), γ = 0.1/log(p), n is the number of observations, p the number of variables, and grid = 10.

iii. Given λ, for k = 1, 2, . . . , K:

(a) (Training on (y_i,t, x_i,t, d_i,t), i /∈ S_k) Leave the kth subset out, and imple-ment the post-double-selection method with tuning parameter λ on the K − 1 subsets. Denote the estimated coefficients as ˆδ^−k(λ) and ˆβ^−k_y (λ).

(b) (Validating on (yi,t, xi,t, di,t), i ∈ Sk) Given ˆδ^−k(λ) and ˆβ_y^−k(λ) compute the error in predicting the kth subset,

e_k(λ) =X

i∈S_k

(d ln y_i,t− d_i,tδˆ^−k(λ) − x_i,tβˆ_y^−k(λ))².

iv. This gives the cross-validation error

CV (λ) = 1 K

K

X

1

e_k(λ).

v. For each value of the tuning parameter λ ∈ [λ^RT − 100, λ^RT + 100], repeat steps 3-4 and choose the tuning parameter that minimizes the CV (λ).

Table C.3 Control Variables Selected in the Double-Selection Procedure via LASSO:

Baseline Estimation

Controls Included Controls Selected

Baseline Developed Countries Developing Countries

λ^ex_ik,t λ^ex_ik104,t λ^ex_ik64,t

λ^ex_ik176,t λ^ex_ik125,t λ^ex_ik180,t λ^ex_ik155,t λ^ex_ik165,t λ^ex_ik178,t λ^ex_ik210,t λ^ex_ik241,t λ^ex_ik244,t

λ^im_ik,t

λ^ex_iq,t λ^ex_iq2,t

λ^im_iq,t

λ^ex_ik,t· ln F M A_ik,t λ^ex_ik114,t· ln F M A_ik114,t λ^ex_ik92,t· ln F M A_ik92,t λ^ex_ik143,t· ln F M Aik143,t λ^ex_ik180,t· ln F M Aik180,t

λ^ex_ik179,t· ln F M A_ik179,t λ^ex_ik180,t· ln F M A_ik180,t

λ^im_ik,t· ln CM A_ik,t λ^im_ik80,t· ln CM A_ik80,t λ^im_ik71,t· ln CM A_ik71,t λ^im_ik166,t· ln CM Aik166,t λ^im_ik166,t· ln CM Aik166,t

λ^im_ik172,t· ln CM Aik172,t λ^im_ik180,t· ln CM Aik180,t

λ^im_ik176,t· ln CM Aik176,t λ^im_ik205,t· ln CM Aik205,t

λ^im_ik180,t· ln CM A_ik180,t λ^im_ik230,t· ln CM A_ik230,t λ^im_ik186,t· ln CM A_ik186,t

λ^im_ik214,t· ln CM A_ik214,t λ^im_ik221,t· ln CM A_ik221,t λ^im_ik222,t· ln CM A_ik222,t

P

k∈qλ^ex_ik,t· ln F M A_ik,t P

k∈qλ^im_ik,t· ln CM A_ik,t

Number of Controls Selected 16 16 0

Estimates Figures Figure 3.1 Figure 3.2 Figure 3.2

Notes: Industries in our sample are relabeled by number from 1 to 281 for coding purpose, i.e. k = 1, 2, . . . , 281. The numbers in the subscripts refers to the corresponding industries.