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. Convert serial dates to datetimes.
load Data_GDP dt = datetime(dates,'ConvertFrom','datenum');
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(dt(3:230),FS(:,2),'r') 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(dt(3:230),SS(:,2),'r') 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 pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
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
)
. If the number of rows exceeds j
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.
Version History
Introduced in R2019b
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