Common Target Parameters

DML estimators can estimate a wide range of parameters of interest in economics. These include averages of treatment effects and regression coefficients. Below you will find a selection of commonly encountered target parameters. For each target parameter, we define the variables of interest, the nuisance quantities and the Neyman-orthogonal scores which underlie DML estimations.

Treatment Effects

Average Treatment Effect

Observation:
$W_i \equiv (Y_i, D_i, X_i)$, where:

• $Y_i$: outcome
• $D_i$: binary treatment variable
• $X_i$: controls

Nuisance Parameter:
$\eta \equiv (m, g^{(0)}, g^{(1)})$ with true values:

• $m_0(X_i) \equiv E[D_i \vert X_i]$
• $g^{(d)}_0(X_i) \equiv E[Y_i \vert D_i=0, X_i], \, d \in {0,1}$

Neyman Orthogonal Score:
\(\begin{align} \psi(W_i; \theta, \eta) &= \frac{D_i(Y_i - g^{(1)}(X_i))}{m(X_i)} - \frac{(1 - D_i)(Y_i - g^{(0)}(X_i))}{1 - m(X_i)} \\ &\quad + g^{(1)}(X_i) - g^{(0)}(X_i) - \theta \end{align}\)

Software:

• R: ddml, DoubleML
• Stata: ddml
• Python: DoubleML

Average Treatment Effect on the Treated

Observation:
$W_i \equiv (Y_i, D_i, X_i)$, where:

• $Y_i$: outcome
• $D_i$: binary treatment variable
• $X_i$: controls

Nuisance Parameter:
$\eta \equiv (m, g^{(0)}, p)$ with true values:

• $m_0(X_i) \equiv E[D_i \vert X_i]$
• $g^{(0)}_0(X_i) \equiv E[Y_i \vert D_i=0, X_i]$
• $p_0 \equiv E[D_i]$

Neyman Orthogonal Score:
\(\begin{align} \psi(W_i; \theta, \eta) = \frac{D_i(Y_i - g^{(0)}(X_i))}{p} - \frac{m(X_i)(1-D_i)(Y_i-g^{(0)}(X_i))}{p(1-m(X_i))} -\frac{D_i}{p}\theta \end{align}\)

Software:

• R: ddml, DoubleML
• Stata: ddml
• Python: DoubleML

Local Average Treatment Effect or Average Causal Response

Observation:
$W_i \equiv (Y_i, D_i, X_i, Z_i)$, where:

• $Y_i$: outcome
• $D_i$: treatment variable
• $X_i$: controls
• $Z_i$: binary instrument

Nuisance Parameter:
$\eta \equiv (r, \ell^{(0)}, \ell^{(1)}, p^{(0)}, p^{(1)})$ with true values:

• $r_0(X_i) \equiv E[Z_i \vert X_i]$
• $\ell^{(z)}_0(X_i) \equiv E[Y_i \vert Z_i=z, X_i], \, z \in {0,1}$
• $p^{(z)}_0(X_i) \equiv E[D_i \vert Z_i=z, X_i], \, z \in {0,1}$

Neyman Orthogonal Score:
\(\begin{align} \psi(W_i; \theta, \eta) &= \frac{Z_i(Y_i-\ell^{(1)}(X_i))}{r(X_i)} - \frac{(1-Z_i)(Y_i - \ell^{(0)}(X_i))}{1 - r(X_i)} + \ell^{(1)}(X_i) - \ell^{(0)}(X_i) \\ &\quad - \left(\frac{Z_i(D_i-p^{(1)}(X_i))}{r(X_i)} - \frac{(1-Z_i)(D_i - p^{(0)}(X_i))}{1 - r(X_i)} + p^{(1)}(X_i) - p^{(0)}(X_i)\right)\theta \end{align}\)

Software:

• R: ddml, DoubleML
• Stata: ddml
• Python: DoubleML

Group-Time Average Treatment Effects on the Treated

Observation:
$W_i \equiv (Y_{it}, D_{it}, G_i, X_i)_{t=0}^T$, where:

• $Y_{it}$: outcome at time $t$
• $D_{it}$: binary treatment indicator at time $t$
• $G_i$: time of first treatment
• $X_i$: controls
• Define $\Delta Y_{it}\equiv Y_{it}-Y_{it-1}$

Nuisance Parameter:
$\eta \equiv (h^{(0)}, h^{(1)}, \ell^{(0)}, p)$ with true values:

• $h_0^{(0)}(X_i) \equiv P(G_i>t \vert X_i)$
• $h_0^{(1)}(X_i) \equiv P(D_{it}=1, G_i=g \vert X_i)$
• $\ell_0^{(0)}(X_i) \equiv E[\Delta Y_{it} \vert G_i>t, X_i]$
• $p_0 \equiv P(D_{it}=1, G_i=g)$

Neyman Orthogonal Score:
\(\begin{align} \psi(W_i; \theta, \eta) &= \frac{\mathbb{1}\{D_{it}=1, G_i=g\}(\Delta Y_{it} - \ell^{(0)}(X_i))}{p} \\ &\quad - \frac{h^{(1)}(X_{i})\mathbb{1}\{G_i>t\}(\Delta Y_{it} - \ell^{(0)}(X_i))}{ph^{(0)}(X_i)} \\ &\quad - \frac{\mathbb{1}\{D_{it}=1, G_i=g\}}{p}\theta \end{align}\)

Software:

• R: ddml+did

Notes:
Discuss estimation if control is never-treated.

Regression Coefficients

Partially Linear Regression Coefficient

Observation:
$W_i \equiv (Y_i, D_i, X_i)$, where:

• $Y_i$: outcome
• $D_i$: treatment variable
• $X_i$: controls

Nuisance Parameter:
$\eta \equiv (m, \ell)$ with true values:

• $m_0(X_i) \equiv E[D_i \vert X_i]$
• $\ell_0(X_i) \equiv E[Y_i \vert X_i]$

Neyman Orthogonal Score:
\(\begin{align} \psi(W_i; \theta, \eta) &= \left(D_i - m_0(X_i)\right)\left(Y_i - \ell_0(X_i)\right) \\ &\quad - \left(D_i - m_0(X_i)\right)\left(D_i - m_0(X_i)\right)^\top \theta \end{align}\)

Software:

• R: ddml, DoubleML
• Stata: ddml
• Python: DoubleML

Partially Linear IV Regression Coefficient

Observation:
$W_i \equiv (Y_i, D_i, X_i, Z_i)$, where:

• $Y_i$: outcome
• $D_i$: treatment variable
• $X_i$: controls
• $Z_i$: instrument

Nuisance Parameter:
$\eta \equiv (m, \ell, r)$ with true values:

• $m_0(X_i) \equiv E[D_i \vert X_i]$
• $\ell_0(X_i) \equiv E[Y_i \vert X_i]$
• $r_0(X_i) \equiv E[Z_i \vert X_i]$

Neyman Orthogonal Score:
\(\begin{align} \psi(W_i; \theta, \eta) &= (Z_i - r(X_i))(Y_i - \ell(X_i)) \\ &\quad -(Z_i - r(X_i))(D_i - m(X_i))^\top\theta \end{align}\)

Software:

• R: ddml, DoubleML
• Stata: ddml
• Python: DoubleML

Flexible Partially Linear IV Regression Coefficient

Observation:
$W_i \equiv (Y_i, D_i, X_i, Z_i)$, where:

• $Y_i$: outcome
• $D_i$: treatment variable
• $X_i$: controls
• $Z_i$: instrument

Nuisance Parameter:
$\eta \equiv (m, \ell, p)$ with true values:

• $m_0(X_i) \equiv E[D_i \vert X_i]$
• $\ell_0(X_i) \equiv E[Y_i \vert X_i]$
• $p_0(Z_i, X_i) \equiv E[D_i \vert Z_i, X_i]$

Neyman Orthogonal Score:
\(\begin{align} \psi(W_i; \theta, \eta) &= (p(Z_i,X_i) - m(X_i))(Y_i - \ell(X_i)) \\ &\quad - (p(Z_i,X_i) - m(X_i))(D_i - m(X_i))^\top\theta \end{align}\)

Software:

• R: ddml, DoubleML
• Stata: ddml
• Python: DoubleML

Fixed Effects Partially Linear Regression Coefficient

Observation:
$W_i \equiv (Y_{it}, D_{it}, X_{it})_{t=0}^T$, where:

• $Y_{it}$: outcome at time $t$
• $D_{it}$: treatment variable at time $t$
• $X_{it}$: controls at time $t$
• Define $\Delta Y_{it} \equiv Y_{it} - Y_{it-1}$, $\Delta D_{it} \equiv D_{it} - D_{it-1}$

Nuisance Parameter:
$\eta \equiv (m_t, \ell_t)_{t=1}^T$ with true values:

• $m_{t0}(X_{it}, X_{it-1}) \equiv E[\Delta D_{it} \vert X_{it}, X_{it-1}]$
• $\ell_{t0}(X_{it}, X_{it-1}) \equiv E[\Delta Y_{it} \vert X_{it}, X_{it-1}]$

Neyman Orthogonal Score:
\(\begin{align} \psi(W_i; \theta, \eta) &= \sum_{t=1}^T \left(\Delta D_{it} - m_t(X_{it}, X_{it-1})\right)\left(\Delta Y_{it} - \ell_t(X_{it}, X_{it-1})\right) \\ &\quad - \sum_{t=1}^T \left(\Delta D_{it} - m_t(X_{it}, X_{it-1})\right)\left(\Delta D_{it} - m_t(X_{it}, X_{it-1})\right)^\top \theta \end{align}\)

Software:

• R: ddml, DoubleML
• Stata: ddml
• Python: DoubleML

Fixed Effects Partially Linear IV Regression Coefficient

Observation:
$W_i \equiv (Y_{it}, D_{it}, Z_{it}, X_{it})_{t=0}^T$, where:

• $Y_{it}$: outcome at time $t$
• $D_{it}$: treatment variable at time $t$
• $Z_{it}$: instrument at time $t$
• $X_{it}$: controls at time $t$
• Define $\Delta Y_{it} \equiv Y_{it} - Y_{it-1}$, $\Delta D_{it} \equiv D_{it} - D_{it-1}$, $\Delta Z_{it} \equiv Z_{it} - Z_{it-1}$

Nuisance Parameter:
$\eta \equiv (m_t, \ell_t, r_t)_{t=1}^T$ with true values:

• $m_{t0}(X_{it}, X_{it-1}) \equiv E[\Delta D_{it} \vert X_{it}, X_{it-1}]$
• $\ell_{t0}(X_{it}, X_{it-1}) \equiv E[\Delta Y_{it} \vert X_{it}, X_{it-1}]$
• $r_{t0}(X_{it}, X_{it-1}) \equiv E[\Delta Z_{it} \vert X_{it}, X_{it-1}]$

Neyman Orthogonal Score:
\(\begin{align} \psi(W_i; \theta, \eta) &= \sum_{t=1}^T \left(\Delta Z_{it} - r_t(X_{it}, X_{it-1})\right)\left(\Delta Y_{it} - \ell_t(X_{it}, X_{it-1})\right) \\ &\quad - \sum_{t=1}^T \left(\Delta Z_{it} - r_t(X_{it}, X_{it-1})\right)(\Delta D_{it} - m_t(X_{it}, X_{it-1}))^\top \theta \end{align}\)

Software:

• R: ddml, DoubleML
• Stata: ddml
• Python: DoubleML