From a statistical point of view, DSGE models are unobserved components models: only a few variables are observed. Filtering or smoothing provide estimate of the unobserved variables given the observations.

The model is put in state space form

\[\begin{align*} y^o_t &= M s_t + N\epsilon_t\\ s_t &= Ts_{t-1} + R\eta_t \end{align*}\]

where $y^o_t$ represents observed variable at period t. The coefficient matrices of the transition equation, T and R are provided by the solution of the linear(-isze) rational expectation model. $\epsilon_t$ are possible measurement errors and $\eta_t$ the structural shocks. Most often matrix M is a selection matrix.

Filtering provides estimates conditional only on past observations:

\[\mathbb{E}(y^{no}_t|Y^o_{t-1})\]

where $y^{no}_t$ are unobserved variables at period t and $Y^o_{t-1}$ represent the set observations until period t-1 included.

Smoothing provides estimates of unobserved variables conditional on the entire sample of observations:

\[\mathbb{E}(y^{no}_t|Y^o_T)\]

where $Y^o_T$ represents the all observations in the sample.

Dynare command

varobs

Observed variables are declared with the varobs command

Command: varobs VARIABLE_NAME...;

This command lists the name of observed endogenous variables for the estimation procedure. These variables must be available in the data file (see estimation_cmd <estim-comm>).

Alternatively, this command is also used in conjunction with the partial_information option of stoch_simul, for declaring the set of observed variables when solving the model under partial information.

Only one instance of varobs is allowed in a model file. If one needs to declare observed variables in a loop, the macro processor can be used as shown in the second example below.

Example

     varobs C y rr;

observation_trends

It is possible to declare a deterministic linear trend that is removed for the computations and added back in the results

Block: observation_trends ;

This block specifies linear trends for observed variables as functions of model parameters. In case the loglinear option is used, this corresponds to a linear trend in the logged observables, i.e. an exponential trend in the level of the observables.

Each line inside of the block should be of the form:

    VARIABLE_NAME(EXPRESSION);

In most cases, variables shouldn't be centered when observation_trends is used.

Example

     observation_trends;
     Y (eta);
     P (mu/eta);
     end;

calib_smoother

This command triggers the computation of the filter and smoother for calibrated models

Command: calib_smoother [VARIABLE_NAME]...;

Command: calib_smoother (OPTIONS...)[VARIABLE_NAME]...;

This command computes the smoothed variables (and possible the filtered variables) on a calibrated model.

A datafile must be provided, and the observable variables declared with varobs. The smoother is based on a first-order approximation of the model.

By default, the command computes the smoothed variables and shocks and stores the results in oo_.SmoothedVariables and oo_.SmoothedShocks. It also fills oo_.UpdatedVariables.

Options

• datafile = FILENAME: file containing the observation in CSV format.

• filtered_vars: triggers the computation of filtered variables.

• first_obs = INTEGER: first observation

• diffuse_filter = INTEGER: use a diffuse filter for nonstationary models.

Julia functions

calibsmoother!(; context::Context=context,
                 datafile::String = "",
                     data::AxisArrayTable = AxisArrayTable(Matrix{Float64}(undef, 0, 0), Undated[], Symbol[]),
                 first_obs::PeriodSinceEpoch = Undated(typemin(Int)),
                 last_obs::PeriodSinceEpoch = Undated(typemin(Int)),
                 nobs::Int = 0
               )