diff --git a/matlab/kalman/likelihood/kalman_filter_d.m b/matlab/kalman/likelihood/kalman_filter_d.m
index 7a7cdef13..f93566f61 100644
--- a/matlab/kalman/likelihood/kalman_filter_d.m
+++ b/matlab/kalman/likelihood/kalman_filter_d.m
@@ -5,7 +5,7 @@ function [dLIK,dlik,a,Pstar] = kalman_filter_d(Y, start, last, a, Pinf, Pstar, k
 %    Y           [double]      pp*smpl matrix of (detrended) data, where pp is the number of observed variables.
 %    start       [integer]     scalar, first observation.
 %    last        [integer]     scalar, last observation.
-%    a           [double]      mm*1 vector, levels of the state variables.
+%    a           [double]      mm*1 vector, levels of the predicted initial state variables (E_{0}(alpha_1)).
 %    Pinf        [double]      mm*mm matrix used to initialize the covariance matrix of the state vector.
 %    Pstar       [double]      mm*mm matrix used to initialize the covariance matrix of the state vector.
 %    kalman_tol  [double]      scalar, tolerance parameter (rcond) of F_star.
@@ -25,7 +25,8 @@ function [dLIK,dlik,a,Pstar] = kalman_filter_d(Y, start, last, a, Pinf, Pstar, k
 % OUTPUTS
 %    LIK         [double]      scalar, minus loglikelihood
 %    lik         [double]      smpl*1 vector, log density of each vector of observations.
-%    a           [double]      mm*1 vector, current estimate of the state vector.
+%    a           [double]      mm*1 vector, current estimate of the state vector tomorrow 
+%                                                               (E_{T}(alpha_{T+1})).
 %    Pstar       [double]      mm*mm matrix, covariance matrix of the state vector.
 %
 % REFERENCES
diff --git a/matlab/kalman/likelihood/kalman_filter_fast.m b/matlab/kalman/likelihood/kalman_filter_fast.m
index db3dec3b7..331019d95 100644
--- a/matlab/kalman/likelihood/kalman_filter_fast.m
+++ b/matlab/kalman/likelihood/kalman_filter_fast.m
@@ -20,7 +20,7 @@ function [LIK, LIKK, a, P] = kalman_filter_fast(Y,start,last,a,P,kalman_tol,ricc
 %! @item last
 %! Integer scalar, last period (@var{last}-@var{first} has to be inferior to T).
 %! @item a
-%! Vector (@var{mm}*1) of doubles, initial mean of the state vector.
+%! Vector (@var{mm}*1) of doubles, levels of the predicted initial state variables (E_{0}(alpha_1)).
 %! @item P
 %! Matrix (@var{mm}*@var{mm}) of doubles, initial covariance matrix of the state vector.
 %! @item kalman_tol
@@ -59,7 +59,7 @@ function [LIK, LIKK, a, P] = kalman_filter_fast(Y,start,last,a,P,kalman_tol,ricc
 %! @item likk
 %! Column vector of doubles, values of the density of each observation.
 %! @item a
-%! Vector (@var{mm}*1) of doubles, mean of the state vector at the end of the (sub)sample.
+%! Vector (@var{mm}*1) of doubles, mean of the state vector at the end of the (sub)sample (E_{T}(alpha_{T+1})).
 %! @item P
 %! Matrix (@var{mm}*@var{mm}) of doubles, covariance of the state vector at the end of the (sub)sample.
 %! @end table
diff --git a/matlab/kalman/likelihood/kalman_filter_ss.m b/matlab/kalman/likelihood/kalman_filter_ss.m
index d8dc80468..545258300 100644
--- a/matlab/kalman/likelihood/kalman_filter_ss.m
+++ b/matlab/kalman/likelihood/kalman_filter_ss.m
@@ -17,7 +17,7 @@ function [LIK, likk, a] = kalman_filter_ss(Y,start,last,a,T,K,iF,log_dF,Z,pp,Zfl
 %! @item last
 %! Integer scalar, last period (@var{last}-@var{first} has to be inferior to T).
 %! @item a
-%! Vector (mm*1) of doubles, initial mean of the state vector.
+%! Vector (mm*1) of doubles, levels of the predicted initial state variables (E_{0}(alpha_1)).
 %! @item T
 %! Matrix (mm*mm) of doubles, transition matrix of the state equation.
 %! @item K
@@ -42,7 +42,7 @@ function [LIK, likk, a] = kalman_filter_ss(Y,start,last,a,T,K,iF,log_dF,Z,pp,Zfl
 %! @item likk
 %! Column vector of doubles, values of the density of each observation.
 %! @item a
-%! Vector (mm*1) of doubles, mean of the state vector at the end of the (sub)sample.
+%! Vector (mm*1) of doubles, current estimate of the state vector tomorrow (E_{T}(alpha_{T+1})).
 %! @end table
 %! @sp 2
 %! @strong{This function is called by:}
diff --git a/matlab/kalman/likelihood/missing_observations_kalman_filter.m b/matlab/kalman/likelihood/missing_observations_kalman_filter.m
index a63c2f3f5..2d9c19028 100644
--- a/matlab/kalman/likelihood/missing_observations_kalman_filter.m
+++ b/matlab/kalman/likelihood/missing_observations_kalman_filter.m
@@ -8,7 +8,7 @@ function  [LIK, lik, a, P] = missing_observations_kalman_filter(data_index,numbe
 %    Y                            [double]    pp*smpl matrix of data.
 %    start                        [integer]   scalar, index of the first observation.
 %    last                         [integer]   scalar, index of the last observation.
-%    a                            [double]    pp*1 vector, initial level of the state vector.
+%    a                            [double]    pp*1 vector, levels of the predicted initial state variables (E_{0}(alpha_1)).
 %    P                            [double]    pp*pp matrix, covariance matrix of the initial state vector.
 %    kalman_tol                   [double]    scalar, tolerance parameter (rcond).
 %    riccati_tol                  [double]    scalar, tolerance parameter (riccati iteration).
@@ -25,7 +25,7 @@ function  [LIK, lik, a, P] = missing_observations_kalman_filter(data_index,numbe
 % OUTPUTS
 %    LIK        [double]    scalar, MINUS loglikelihood
 %    lik        [double]    vector, density of observations in each period.
-%    a          [double]    mm*1 vector, estimated level of the states.
+%    a          [double]    mm*1 vector, current estimate of the state vector tomorrow (E_{T}(alpha_{T+1})).
 %    P          [double]    mm*mm matrix, covariance matrix of the states.
 %
 %
diff --git a/matlab/kalman/likelihood/missing_observations_kalman_filter_d.m b/matlab/kalman/likelihood/missing_observations_kalman_filter_d.m
index ec8c286a4..cafcc2938 100644
--- a/matlab/kalman/likelihood/missing_observations_kalman_filter_d.m
+++ b/matlab/kalman/likelihood/missing_observations_kalman_filter_d.m
@@ -12,7 +12,7 @@ function [dLIK,dlik,a,Pstar] = missing_observations_kalman_filter_d(data_index,n
 %    Y                            [double]      pp*smpl matrix of (detrended) data, where pp is the number of observed variables.
 %    start                        [integer]     scalar, first observation.
 %    last                         [integer]     scalar, last observation.
-%    a                            [double]      mm*1 vector, levels of the state variables.
+%    a                            [double]      mm*1 vector, levels of the predicted initial state variables (E_{0}(alpha_1)).
 %    Pinf                         [double]      mm*mm matrix used to initialize the covariance matrix of the state vector.
 %    Pstar                        [double]      mm*mm matrix used to initialize the covariance matrix of the state vector.
 %    kalman_tol                   [double]      scalar, tolerance parameter (rcond).
@@ -30,7 +30,7 @@ function [dLIK,dlik,a,Pstar] = missing_observations_kalman_filter_d(data_index,n
 % OUTPUTS
 %    dLIK        [double]    scalar, MINUS loglikelihood
 %    dlik        [double]    vector, density of observations in each period.
-%    a           [double]    mm*1 vector, estimated level of the states.
+%    a           [double]    mm*1 vector, current estimate of the state vector tomorrow (E_{T}(alpha_{T+1})).
 %    Pstar       [double]    mm*mm matrix, covariance matrix of the states.
 %
 % REFERENCES