Update scores.rst

Author: SvenKlaassen

doc/guide/scores.rst

@@ -288,12 +288,12 @@ Difference-in-Differences for Panel Data
 For the difference-in-differences model implemented in ``DoubleMLDID`` one can choose between
 ``score='observational'`` and ``score='experimental'``.
 
-``score='observational'`` implements the score function:
+``score='observational'`` implements the score function (dropping the unit index :math:`i`):
 
 .. math::
 
     \psi(W,\theta, \eta) 
-    :&= -\frac{D}{\mathbb{E}_n[D]}\theta + \left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right) \left(Y(1)-Y(0) - g(0,X)\right)
+    :&= -\frac{D}{\mathbb{E}_n[D]}\theta + \left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right) \left(Y_1 - Y_0 - g(0,X)\right)
 
     &= \psi_a(W; \eta) \theta + \psi_b(W; \eta)
 
@@ -303,21 +303,21 @@ where the components of the linear score are
 
     \psi_a(W; \eta) &=  - \frac{D}{\mathbb{E}_n[D]},
 
-    \psi_b(W; \eta) &= \left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right) \left(Y(1)-Y(0) - g(0,X)\right)
+    \psi_b(W; \eta) &= \left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right) \left(Y_1 - Y_0 - g(0,X)\right)
 
 and the nuisance elements :math:`\eta=(g, m)` are defined as
 
 .. math::
 
-    g_{0}(0, X) &= \mathbb{E}[Y(1)-Y(0)|D=0, X]
+    g_{0}(0, X) &= \mathbb{E}[Y_1 - Y_0|D=0, X]
 
     m_0(X) &= P(D=1|X).
 
 If ``in_sample_normalization='False'``, the score is set to
 
 .. math::
 
-    \psi(W,\theta,\eta) &= - \frac{D}{p}\theta + \frac{D - m(X)}{p(1-m(X))}\left(Y(1)-Y(0) -g(0,X)\right)
+    \psi(W,\theta,\eta) &= - \frac{D}{p}\theta + \frac{D - m(X)}{p(1-m(X))}\left(Y_1 - Y_0 -g(0,X)\right)
 
     &= \psi_a(W; \eta) \theta + \psi_b(W; \eta)
 
@@ -330,7 +330,7 @@ implements the score function:
 .. math::
 
     \psi(W,\theta, \eta) 
-    :=\; &-\theta + \left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right)\left(Y(1)-Y(0) -g(0,X)\right)
+    :=\; &-\theta + \left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right)\left(Y_1 - Y_0 -g(0,X)\right)
 
     &+ \left(1 - \frac{D}{\mathbb{E}_n[D]}\right) \left(g(1,X) - g(0,X)\right)
 
@@ -342,17 +342,17 @@ where the components of the linear score are
 
     \psi_a(W; \eta) \;=  &- 1,
 
-    \psi_b(W; \eta) \;= &\left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right)\left(Y(1)-Y(0) -g(0,X)\right)
+    \psi_b(W; \eta) \;= &\left(\frac{D}{\mathbb{E}_n[D]} - \frac{\frac{m(X) (1-D)}{1-m(X)}}{\mathbb{E}_n\left[\frac{m(X) (1-D)}{1-m(X)}\right]}\right)\left(Y_1 - Y_0 -g(0,X)\right)
 
     &+  \left(1 - \frac{D}{\mathbb{E}_n[D]}\right) \left(g(1,X) - g(0,X)\right)
 
 and the nuisance elements :math:`\eta=(g, m)` are defined as
 
 .. math::
 
-    g_{0}(0, X) &= \mathbb{E}[Y(1)-Y(0)|D=0, X]
+    g_{0}(0, X) &= \mathbb{E}[Y_1 - Y_0|D=0, X]
 
-    g_{0}(1, X) &= \mathbb{E}[Y(1)-Y(0)|D=1, X]
+    g_{0}(1, X) &= \mathbb{E}[Y_1 - Y_0|D=1, X]
 
     m_0(X) &= P(D=1|X).
 
@@ -361,7 +361,7 @@ Analogously, if ``in_sample_normalization='False'``,  the score is set to
 .. math::
 
     \psi(W,\theta, \eta) 
-    :=\; &-\theta +  \frac{D - m(X)}{p(1-m(X))}\left(Y(1)-Y(0) -g(0,X)\right)
+    :=\; &-\theta +  \frac{D - m(X)}{p(1-m(X))}\left(Y_1 - Y_0 -g(0,X)\right)
 
     &+ \left(1 - \frac{D}{p}\right) \left(g(1,X) - g(0,X)\right)
 
@@ -376,7 +376,7 @@ Difference-in-Differences for repeated cross-sections
 For the difference-in-differences model implemented in ``DoubleMLDIDCS`` one can choose between
 ``score='observational'`` and ``score='experimental'``.
 
-``score='observational'`` implements the score function:
+``score='observational'`` implements the score function (dropping the unit index :math:`i`):
 
 .. math::