# filter

Filtered inference of operative latent states in Markov-switching dynamic regression data

## Description

returns filtered state probabilities `FS`

= filter(`Mdl`

,`Y`

)`FS`

from conducting optimal conditional inference of the probabilities of the operative latent states in the regime-switching data `Y`

. The Markov-switching dynamic regression model `Mdl`

models the data. `filter`

uses a recursive application of Bayes' rule, as in Hamilton [3].

uses additional
options specified by one or more name-value arguments. For example, `FS`

= filter(`Mdl`

,`Y`

,`Name,Value`

)`'Y0',Y0`

initializes the dynamic component of each submodel by using the presample response data `Y0`

.

## Examples

### Compute Filtered State Probabilities

Compute filtered state probabilities from a two-state Markov-switching dynamic regression model for a 1-D response process. This example uses arbitrary parameter values for the data-generating process (DGP).

**Create Fully Specified Model for DGP**

Create a two-state discrete-time Markov chain model for the switching mechanism.

P = [0.9 0.1; 0.2 0.8]; mc = dtmc(P);

`mc`

is a fully specified `dtmc`

object.

For each state, create an AR(0) (constant only) model for the response process. Store the models in a vector.

mdl1 = arima('Constant',2,'Variance',3); mdl2 = arima('Constant',-2,'Variance',1); mdl = [mdl1; mdl2];

`mdl1`

and `mdl2`

are fully specified `arima`

objects.

Create a Markov-switching dynamic regression model from the switching mechanism `mc`

and the vector of submodels `mdl`

.

Mdl = msVAR(mc,mdl);

`Mdl`

is a fully specified `msVAR`

object.

**Simulate Data from DGP**

`filter`

requires responses to compute filtered state probabilities. Generate one random response and state path, both of length 30, from the DGP.

```
rng(1000); % For reproducibility
[y,~,sp] = simulate(Mdl,30);
```

**Compute State Probabilities**

Compute filtered and smoothed state probabilities from the Markov-switching model given the simulated response data.

fs = filter(Mdl,y); ss = smooth(Mdl,y);

`fs`

and `ss`

are 30-by-2 matrices of filtered and smoothed state probabilities, respectively, for each period in the simulation horizon. Although the filtered state probabilities at time * t* (

`fs(`

`t`

`,:)`

) are based on the response data through time *(*

`t`

`y(1:`

`t`

`)`

), the smoothed state probabilities at time *(*

`t`

`ss(t,:)`

) are based on all observations.Plot the simulated state path and the filtered and smoothed state probabilities on the same graph.

figure plot(sp,'m') hold on plot(fs(:,2),'r') plot(ss(:,2),'g') yticks([0 1 2]) xlabel("Time") title("Observed States with Estimated State Probabilities") legend({'Simulated states','Filtered probability: state 2',... 'Smoothed probability: state 2'}) hold off

### Compute Filtered Probabilities of Recession

Consider a two-state Markov-switching dynamic regression model of the postwar US real GDP growth rate. The model has the parameter estimates presented in [1].

**Create Markov-Switching Dynamic Regression Model**

Create a fully specified discrete-time Markov chain model that describes the regime switching mechanism. Label the regimes.

P = [0.92 0.08; 0.26 0.74]; mc = dtmc(P,'StateNames',["Expansion" "Recession"]);

Create separate, fully specified AR(0) models for the two regimes.

sigma = 3.34; % Homoscedastic models across states mdl1 = arima('Constant',4.62,'Variance',sigma^2); mdl2 = arima('Constant',-0.48,'Variance',sigma^2); mdl = [mdl1 mdl2];

Create the Markov-switching dynamic regression model from the switching mechanism `mc`

and the state-specific submodels `mdl`

.

Mdl = msVAR(mc,mdl);

`Mdl`

is a fully specified `msVAR`

object.

**Load and Preprocess Data**

Load the US GDP data set.

`load Data_GDP`

`Data`

contains quarterly measurements of the US real GDP in the period 1947:Q1–2005:Q2. The period of interest in [1] is 1947:Q2–2004:Q2. For more details on the data set, enter `Description`

at the command line.

Transform the data to an annualized rate series by:

Converting the data to a quarterly rate within the estimation period

Annualizing the quarterly rates

qrate = diff(Data(2:230))./Data(2:229); % Quarterly rate arate = 100*((1 + qrate).^4 - 1); % Annualized rate

The transformation drops the first observation.

**Compute Filtered State Probabilities**

Compute filtered state probabilities for the data and model.

FS = filter(Mdl,arate); FS(end,:)

`ans = `*1×2*
0.9396 0.0604

`FS`

is a 228-by-2 matrix of filtered state probabilities. Rows correspond to periods in the data `arate`

, and columns correspond to the regimes.

Plot the filtered probabilities of recession, as in [1], Figure 6.

figure; plot(dates(3:230),FS(:,2),'r') datetick('x') title('Current Filter Probabilities and NBER Recessions') recessionplot

**Compute Smoothed State Probabilities**

Compute smoothed state probabilities, and then plot the smoothed probabilities of recession as in [1], Figure 6.

SS = smooth(Mdl,arate); figure plot(dates(3:230),SS(:,2),'r') datetick('x') recessionplot title('Full-Sample Smoothed Probabilities and NBER Recessions')

### Compute Filtered State Probabilities from Model with VARX Submodels

Compute filtered state probabilities from a three-state Markov-switching dynamic regression model for a 2-D VARX response process. This example uses arbitrary parameter values for the DGP.

**Create Fully Specified Model for DGP**

Create a three-state discrete-time Markov chain model for the switching mechanism.

P = [5 1 1; 1 5 1; 1 1 5]; mc = dtmc(P);

`mc`

is a fully specified `dtmc`

object. `dtmc`

normalizes the rows of `P`

so that they sum to `1`

.

For each state, create a fully specified VARX(0) model (constant and regression coefficient matrix only) for the response process. Specify different constant vectors across models. Specify the same regression coefficient for the two regressors, and specify the same covariance matrix. Store the VARX models in a vector.

% Constants C1 = [1;-1]; C2 = [3;-3]; C3 = [5;-5]; % Regression coefficient Beta = [0.2 0.1;0 -0.3]; % Covariance matrix Sigma = [1.8 -0.4; -0.4 1.8]; % VARX submodels mdl1 = varm('Constant',C1,'Beta',Beta,... 'Covariance',Sigma); mdl2 = varm('Constant',C2,'Beta',Beta,... 'Covariance',Sigma); mdl3 = varm('Constant',C3,'Beta',Beta,... 'Covariance',Sigma); mdl = [mdl1; mdl2; mdl3];

`mdl`

contains three fully specified `varm`

model objects.

For the DGP, create a fully specified Markov-switching dynamic regression model from the switching mechanism `mc`

and the submodels `mdl`

.

Mdl = msVAR(mc,mdl);

`Mdl`

is a fully specified `msVAR`

model.

**Simulate Data from DGP**

Simulate data for the two exogenous series by generating 30 observations from the standard 2-D Gaussian distribution.

```
rng(1) % For reproducibility
X = randn(30,2);
```

Generate one random response and state path, both of length 30, from the DGP. Specify the simulated exogenous data for the submodel regression components.

`[Y,~,SP] = simulate(Mdl,30,'X',X);`

`Y`

is a 30-by-2 matrix of one simulated response path. `SP`

is a 30-by-1 vector of one simulated state path.

**Compute State Probabilities**

Compute filtered and smoothed state probabilities from the DGP given the simulated response data.

FS = filter(Mdl,Y,'X',X); SS = smooth(Mdl,Y,'X',X);

`FS`

and `SS`

are 30-by-2 matrices of filtered and smoothed state probabilities, respectively, for each period in the simulation horizon.

Plot the simulated state path and the filtered and smoothed state probabilities on subplots in the same figure.

figure subplot(3,1,1) plot(SP,'m') yticks([1 2 3]) legend({'Simulated states'}) subplot(3,1,2) plot(FS,'--') legend({'Filtered s1','Filtered s2','Filtered s3'}) subplot(3,1,3) plot(SS,'-') legend({'Smoothed s1','Smoothed s2','Smoothed s3'})

### Specify Presample Data

Consider the data in Compute Filtered Probabilities of Recession, but assume that the period of interest is 1960:Q1–2004:Q2. Also, consider adding an autoregressive term to each submodel.

**Create Partially Specified Model for Estimation**

Create a partially specified Markov-switching dynamic regression model for estimation. Specify AR(1) submodels.

P = NaN(2); mc = dtmc(P,'StateNames',["Expansion" "Recession"]); mdl = arima(1,0,0); Mdl = msVAR(mc,[mdl; mdl]);

Because the submodels are AR(1), each requires one presample observation to initialize its dynamic component for estimation.

**Create Fully Specified Model Containing Initial Values**

Create the model containing initial parameter values for the estimation procedure.

mc0 = dtmc(0.5*ones(2),'StateNames',["Expansion" "Recession"]); submdl01 = arima('Constant',1,'Variance',1,'AR',0.001); submdl02 = arima('Constant',-1,'Variance',1,'AR',0.001); Mdl0 = msVAR(mc0,[submdl01; submdl02]);

**Load and Preprocess Data**

Load the data. Transform the entire set to an annualized rate series.

```
load Data_GDP
qrate = diff(Data)./Data(1:(end - 1));
arate = 100*((1 + qrate).^4 - 1);
```

Identify the presample and estimation sample periods using the dates associated with the annualized rate series. Because the transformation applies the first difference, you must drop the first observation date from the original sample.

dates = datetime(dates(2:end),'ConvertFrom','datenum',... 'Format','yyyy:QQQ','Locale','en_US'); estPrd = datetime(["1960:Q2" "2004:Q2"],'InputFormat','yyyy:QQQ',... 'Format','yyyy:QQQ','Locale','en_US'); idxEst = isbetween(dates,estPrd(1),estPrd(2)); idxPre = dates < (estPrd(1));

**Estimate Model**

Fit the model to the estimation sample data. Specify the presample observation.

```
arate0 = arate(idxPre);
arateEst = arate(idxEst);
EstMdl = estimate(Mdl,Mdl0,arateEst,'Y0',arate0);
```

`EstMdl`

is a fully specified `msVAR`

object.

**Compute State Probabilities**

Compute filtered and smoothed state probabilities from the estimated model and data in the estimation period. Specify the presample observation. Plot the estimated probabilities of recession on subplots in the same figure.

FS = filter(EstMdl,arateEst,'Y0',arate0); SS = smooth(EstMdl,arateEst,'Y0',arate0); figure; subplot(2,1,1) plot(dates(idxEst),FS(:,2),'r') title("Current Filter Probabilities and NBER Recessions") recessionplot subplot(2,1,2) plot(dates(idxEst),SS(:,2),'r') title("Full-Sample Smoothed Probabilities and NBER Recessions") recessionplot

### Return Loglikelihood for Data

Consider the model and data in Compute Filtered Probabilities of Recession.

Create the fully specified Markov-switching model.

P = [0.92 0.08; 0.26 0.74]; mc = dtmc(P,'StateNames',["Expansion" "Recession"]); sigma = 3.34; mdl1 = arima('Constant',4.62,'Variance',sigma^2); mdl2 = arima('Constant',-0.48,'Variance',sigma^2); mdl = [mdl1; mdl2]; Mdl = msVAR(mc,mdl);

Load and preprocess the data.

```
load Data_GDP
qrate = diff(Data(2:230))./Data(2:229);
arate = 100*((1 + qrate).^4 - 1);
```

Compute filtered state probabilities and the loglikelihood for the data and model.

[FS,logL] = filter(Mdl,arate); logL

logL = -640.3016

## Input Arguments

`Y`

— Observed response data

numeric matrix

Observed response data, specified as a `numObs`

-by-`numSeries`

numeric matrix.

`numObs`

is the sample size. `numSeries`

is the number of response variables (`Mdl.NumSeries`

).

Rows correspond to observations, and the last row contains the latest observation. Columns correspond to individual response variables.

`Y`

represents the continuation of the presample response series in `Y0`

.

**Data Types: **`double`

### Name-Value Arguments

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

**Example:**

`'Y0',Y0,'X',X`

initializes the dynamic component of each submodel in `Mdl`

by using the presample response data `Y0`

, and includes a linear regression component in each submodel composed of the predictor data in `X`

and the specified regression coefficients.`Y0`

— Presample response data

numeric matrix

Presample response data, specified as the comma-separated pair consisting of `'Y0'`

and a `numPreSampleObs`

-by-`numSeries`

numeric matrix.

The number of presample observations `numPreSampleObs`

must be sufficient to initialize the AR terms of all submodels. If `numPreSampleObs`

exceeds the AR order of any state, `filter`

uses the latest observations. By default, `Y0`

is the initial segment of `Y`

, which reduces the effective sample size.

**Data Types: **`double`

`S0`

— Initial state probabilities

nonnegative numeric vector

Initial state probabilities, specified as the comma-separated pair consisting of `'S0'`

and a nonnegative numeric vector of length `numStates`

.

`filter`

normalizes `S0`

to produce a distribution.

By default, `S0`

is a steady-state distribution computed by `asymptotics`

.

**Example: **`'S0',[0.2 0.2 0.6]`

**Example: **`'S0',[0 1]`

specifies state 2 as the initial state.

**Data Types: **`double`

`X`

— Predictor data

numeric matrix | cell vector of numeric matrices

Predictor data used to evaluate regression components in all submodels of `Mdl`

, specified as the comma-separated pair consisting of `'X'`

and a numeric matrix or a cell vector of numeric matrices.

To use a subset of the same predictors in each state, specify `X`

as a matrix with `numPreds`

columns and at least `numObs`

rows. Columns correspond to distinct predictor variables. Submodels use initial columns of the associated matrix, in order, up to the number of submodel predictors. The number of columns in the `Beta`

property of `Mdl.SubModels(`

determines the number of exogenous variables in the regression component of submodel * j*)

`j`

. If the number of rows exceeds `numObs`

, then `filter`

uses the latest observations.To use different predictors in each state, specify a cell vector of such matrices with length `numStates`

.

By default, `filter`

ignores regression components in `Mdl`

.

**Data Types: **`double`

## Output Arguments

`FS `

— Filtered state probabilities

nonnegative numeric matrix

Filtered state probabilities, returned as a `numObs`

-by-`numStates`

nonnegative numeric matrix.

`logL`

— Estimated loglikelihood

numeric scalar

Estimated loglikelihood of the response data `Y`

, returned as a numeric scalar.

## Algorithms

`filter`

proceeds iteratively from an initial estimate of the state distribution `S0`

to estimates in `FS`

by using forecasts from the current data history at each time step. `smooth`

refines current estimates of the state distribution that `filter`

produces by iterating backward from the full sample history `Y`

.

## References

[1] Chauvet, M., and J. D. Hamilton. "Dating Business Cycle Turning Points." In *Nonlinear Analysis of Business Cycles (Contributions to Economic Analysis, Volume 276)*. (C. Milas, P. Rothman, and D. van Dijk, eds.). Amsterdam: Emerald Group Publishing Limited, 2006.

[2]
Hamilton, J. D. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle." *Econometrica*. Vol. 57, 1989, pp. 357–384.

[3]
Hamilton, J. D. "Analysis of Time Series Subject to Changes in Regime." *Journal of Econometrics*. Vol. 45, 1990, pp. 39–70.

[4]
Hamilton, James D. *Time Series Analysis*. Princeton, NJ: Princeton University Press, 1994.

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