Cosmetic changes.

notes.tex

@@ -37,11 +37,11 @@
 
 \begin{document}
 
-
-The evaluation of the likelihood of a linear DSGE model rely on the Kalman
-filter. If the steady state of the DSGE model is provided in closed form, the
-Kalman filter is the part of the estimation where most of the time is spent. In
-this note we evaluate the gain we can obtain by considering an alternative
+Computing the likelihood of a linear DSGE model's heavily depends on the
+utilisation of the Kalman filter. When the DSGE model's steady state is already
+available in a closed form, the estimation process allocates a significant
+portion of its time to the Kalman filter. In this document, we aim to quantify
+the potential benefits that may arise from considering an alternative
 initialisation of the Kalman filter.\newline
 
 The reduced form DSGE model is given by:
@@ -59,11 +59,12 @@ variables, the vector of observed endogenous variables is:
 \[
 y^{\star}_t = Z y_t
 \]
-where $Z$ is a $p\times n$ selection matrix\footnote{Here we implicitly exclude,
-  without loss of generality, measurement errors.}. The likelihood, the density
-of the sample, can be expressed as a product of the condition (Gaussian)
+where $Z$ is a $p\times n$ selection matrix\footnote{Here, without loss of
+generality, we implicitly exclude measurement errors.}. The likelihood, the
+density of the sample, can be expressed as a product of the (Gaussian) conditional
 densities of the prediction errors $v_t$ which are determine recursively with
 the Kalman filter:
+
 \begin{equation}
   \label{eq:kalman:1}
   v_t = y_t^{\star} - \bar y^{\star}- Z \hat y_t
@@ -99,10 +100,12 @@ for the conditional covariance matrix of the endogenous variables:
   \label{eq:ricatti}
   P_{t+1} = A P_t A' - A P_t Z' \left(Z P_t Z \right)^{-1}Z A P_t A' + B \Sigma_{\varepsilon}B'
 \end{equation}
-This is the Ricatti equation, where most of the time is spent when evaluating
-the Kalman filter and where round off errors can accumulate dramatically. When
-iterating over these equations the Ricatti equation will eventually (provided
-the sample is large enough) converge to a fixed point $\bar P$ satisfying:
+This is the Riccati equation, which consumes a significant portion of the
+computation time during the evaluation of the Kalman filter. Additionally, it is
+a critical point where round-off errors can accumulate significantly. As we
+iterate through these equations, the Riccati equation will ultimately, given a
+sufficiently large sample size, converge to a fixed point denoted as $\bar P$,
+which satisfies the following:
 \begin{equation}
   \label{eq:ricatti:fp}
 A \bar P A' - \bar P - A \bar P Z' \left(Z \bar P Z \right)^{-1}Z A \bar P A' + B \Sigma_{\varepsilon}B' = 0
@@ -114,12 +117,12 @@ and the Kalman filter equations fall down into the following system:
     \hat y_{t+1} &= A \hat y_t + \bar K v_t
   \end{cases}
 \]
-referred as the steady state Kalman filter, with $\bar F = Z \bar P Z'$ and
-$\bar K = A \bar P A' Z' \bar F^{-1}$, on which iterations take obviously much
-less time than with system (\ref{eq:kalman:1})--(\ref{eq:kalman:5}). The number
-of iterations required to (approximately) converge to the steady state does not
-depend on the data but only on the persistence in the model (through matrix
-$A$).\newline
+referred to as the steady state Kalman filter, with $\bar F = Z \bar P Z'$ and a
+constant Kalman gain matrix $\bar K = A \bar P A' Z' \bar F^{-1}$. Iterations are
+then significantly faster compared to the system described by
+(\ref{eq:kalman:1})--(\ref{eq:kalman:5}). The number of iterations required to
+(approximately) converge to the steady state does not depend on the data but only
+on the persistence in the model (through matrix $A$).\newline
 
 Due to its recursive nature, the Kalman filter requires an initial condition
 $\hat y_0$ and $P_0$. Assuming that the model is stationary, a usual choice for
@@ -132,23 +135,24 @@ which is a Sylvester equation that can be easily solved.\newline
 
 In this note we consider instead the fixed point of the Ricatti equation
 $\bar P$ as an initial condition for the covariance matrix of the endogenous
-variables. The computational advantage is obvious: we directly jump to the
-steady state Kalman filter and never go through
+variables. This approach offers a clear computational advantage:  we directly jump to the
+steady state Kalman filter without the need to go through
 (\ref{eq:kalman:1})--(\ref{eq:kalman:5}). Clearly this will reduce the time
 required to compute the likelihood, but this will also change the evaluation of
-the likelihood. Using a medium scaled model we quantify the saved the computing
-time and evaluate the consequences for the estimation of the parameters. Dynare
+the likelihood. To assess the impact of this alternative initial condition, we
+conduct experiments using a medium-sized model, quantifying the time savings and
+evaluating their implications on parameter estimation.  Dynare
 provides a MEX file, linked to the subroutine \verb+sb02od+ from the
 \href{http://www.slicot.org/}{Slicot library} (version 5.0), to solve
 (\ref{eq:ricatti:fp}) for $\bar{P}$. The use of $\bar P$ as an initial condition
 for the Kalman filter is triggered by option \verb+lik_init=4+ in the
 \verb+estimation+ command.\newline
 
-We consider the estimation of the Quest-III model (113 endogenous variables, 19
+We consider\footnote{The test environment consists of an 11th Gen Intel(R) Core(TM) i9-11900K @ 3.50GHz with 32 GB memory , using MATLAB R2023b under Debian GNU/Linux.} the estimation of the Quest-III model (113 endogenous variables, 19
 innovations, 17 observed variables) as an example. All the codes to reproduce
-the results are available available in a
-\href{https://git.dynare.org/stepan-a/quest-iii-estimation}{Git repository}. The
-model is estimated under both settings (initialization with $\bar P$, RFP, or
+the results are available in a
+\href{https://git.ithaca.fr/stepan-a/quest-iii-estimation}{Git repository}. The
+model is estimated under both settings (initialization with $\bar P$, or
 with $P^{\star}$)\footnote{To reproduce the results run \verb+Q3.mod+, provided
   in the root of the Git repository, with macro options
   \verb+-DUSE_RICATTI_FIXED_POINT=true+ and
@@ -159,10 +163,10 @@ fixed point of the Ricatti equation ($\bar P$). Because the objective functions
 are different, the optimiser probably do not behave identically in both cases.
 If we just evaluate the likelihood on the same point (the prior mean), instead
 of running \verb+mode_compute=5+, we find that the evaluation with $P^{\star}$
-is just 21\% slower than the evaluation with $\bar P$, suggesting that
-initialising with the fixed point of the Ricatti equation makes the objective
-easier to optimise. This is expected since it is well known that
-iterations on (\Ref{eq:ricatti}) are not well behaved. The values of the logged
+is just 21\% slower than the evaluation with $\bar P$, indicating that initialising
+with the fixed point of the Riccati equation renders the objective more amenable to
+optimisation.  This is expected since it is widely recognised that
+iterations on (\Ref{eq:ricatti}) exhibit erratic behaviour. The values of the logged
 posterior kernel at the estimated posterior mode are close (7235.92 with
 $\bar P$ and 7237.68 with $P^{\star}$). But much larger differences can be
 observed elsewhere. For instance, at the prior mean the values are -275.58 with
@@ -178,7 +182,7 @@ plain curves for the initialisation with $P^{\star}$, red dashed curves for the
 initialisation with $\bar P$). The conclusion seems to be that even if the
 likelihood functions are different, inference on the parameters is only
 marginally affected. At least for a first attempt, we can safely consider the
-alternative initialisation of the Kalman filter. We should try also to compare
+alternative initialisation of the Kalman filter. We should also try to compare
 posterior densities, running samplers under both settings, this would bring a
 more definitive answer (at least in the case of Quest-III). Ideally we should
 also build a Monte-Carlo exercise, on a smaller model. All this is left for