estimate
Syntax
Description
estimates parameters of the threshold-switching dynamic regression model
EstMdl
= estimate(Mdl
,tt0
,Y
)Mdl
. estimate
fits the model to the response
data Y
, and initializes the estimation procedure by treating the
parameter values of the fully specified threshold transitions tt0
as
initial values. estimate
implements a version of the conditional
least-squares algorithm described in [4].
uses additional
options specified by one or more name-value arguments. For example, EstMdl
= estimate(Mdl
,tt0
,Y
,Name,Value
)estimate(Mdl,tt0,Y,IterationPlot=true)
displays a plot of the loglikelihood versus the iteration step, after the algorithm
terminates.
Examples
Fit SETAR Model to Simulated Data
Assess estimation accuracy using simulated data from a known data-generating process (DGP). This example uses arbitrary parameter values.
Create Model for DGP
Create a discrete threshold transition at mid-level 1
.
ttDGP = threshold(1)
ttDGP = threshold with properties: Type: 'discrete' Levels: 1 Rates: [] StateNames: ["1" "2"] NumStates: 2
ttDGP
is a threshold
object representing the state-switching mechanism of the DGP.
Create the following fully specified self-exciting TAR (SETAR) model for the DGP.
State 1: .
State 2: .
.
Specify the submodels by using arima
.
mdl1DGP = arima(Constant=0); mdl2DGP = arima(Constant=2); mdlDGP = [mdl1DGP mdl2DGP];
Because the innovations distribution is invariant across states, the tsVAR
software ignores the value of the submodel innovations variance (Variance
property).
Create a threshold-switching model for the DGP. Specify the model-wide innovations variance.
MdlDGP = tsVAR(ttDGP,mdlDGP,Covariance=1);
MdlDGP
is a tsVAR
object representing the DGP.
Simulate Response Paths from DGP
Generate a random response path of length 100 from the DGP. By default, simulate
assumes a SETAR model with delay . In other words, the threshold variable is .
rng(1) % For reproducibiliy
y = simulate(MdlDGP,100);
y
is a 100-by-1 vector of representing the simulated response path.
Create Model for Estimation
Create a partially specified threshold-switching model that has the same structure as the data-generating process, but specify the transition mid-level, submodel coefficients, and model-wide constant as unknown for estimation.
tt = threshold(NaN); mdl1 = arima('Constant',NaN); mdl2 = arima('Constant',NaN); Mdl = tsVAR(tt,[mdl1,mdl2],'Covariance',NaN);
Mdl
is a partially specified tsVAR
object representing a template for estimation. NaN
-valued elements of the Switch
and Submodels
properties indicate estimable parameters.
Mdl
is agnostic of the threshold variable; tsVAR
object functions enable you to specify threshold variable characteristics or data.
Create Threshold Transitions Containing Initial Values
The estimation procedure requires initial values for all estimable threshold transition parameters.
Fully specify a threshold transition that has the same structure as tt
, but set the mid-level to 0
.
tt0 = threshold(0);
tt0
is a fully specified threshold
object.
Estimate Model
Fit the model to the simulated path. By default, the model is self-exciting and the delay of the threshold variable is .
EstMdl = estimate(Mdl,tt0,y)
EstMdl = tsVAR with properties: Switch: [1x1 threshold] Submodels: [2x1 varm] NumStates: 2 NumSeries: 1 StateNames: ["1" "2"] SeriesNames: "1" Covariance: 1.0225
EstMdl
is a fully specified tsVAR
object representing the estimated SETAR model.
Display an estimation summary of the submodels.
summarize(EstMdl)
Description 1-Dimensional tsVAR Model with 2 Submodels Switch Transition Type: discrete Estimated Levels: 1.128 Fit Effective Sample Size: 99 Number of Estimated Parameters: 2 Number of Constrained Parameters: 0 LogLikelihood: -141.574 AIC: 287.149 BIC: 292.339 Submodels Estimate StandardError TStatistic PValue ________ _____________ __________ __________ State 1 Constant(1) -0.12774 0.13241 -0.96474 0.33467 State 2 Constant(1) 2.1774 0.16829 12.939 2.7264e-38
The estimates are close to their true values.
Plot the estimated switching mechanism with the threshold data, which is the response data.
figure
ttplot(EstMdl.Switch,'Data',y)
Tune Delay Parameter for SETAR Estimation
estimate
does not fit the delay parameter to the data; you must specify its value when you call estimate
. This example shows how to tune .
Create the following fully specified SETAR model for the DGP.
State 1: , where .
State 2: , where .
The system is in state 1 when , and it is in state 2 otherwise.
ttDGP = threshold(0); mdl1DGP = arima(Constant=0,Variance=1); mdl2DGP = arima(Constant=2,Variance=2); mdlDGP = [mdl1DGP; mdl2DGP]; MdlDGP = tsVAR(ttDGP,mdlDGP);
Generate a random response path of length 200 from the DGP. Specify that the threshold variable delay is 4
.
rng(1) % For reproducibiliy
y = simulate(MdlDGP,200,Delay=4);
plot(y)
Create a partially specified threshold-switching model that has the same structure as the data-generating process, but specify the transition mid-level, submodel coefficients, and state-specific variances as unknown for estimation.
tt = threshold(NaN); mdl = arima(0,0,0); Mdl = tsVAR(tt,[mdl; mdl]);
Fully specify a threshold transition that has the same structure as tt, but set the mid-level to 0.5
.
tt0 = threshold(0.5);
Tune by choosing a set of plausible values for it, and by fitting the SETAR model to the simulated data for each value in the set. Choose the model that maximizes the loglikelihood.
d = 1:8; nd = numel(d); logL = zeros(nd,1); % Preallocate for loglikelihoods EstMdl = cell(nd,1); % Preallocate for estimated models for j = 1:nd [EstMdl{j},logL(j)] = estimate(Mdl,tt0,y,Delay=d(j)); end
Extract the model that maximizes the loglikelihood.
[~,optDelay] = max(logL)
optDelay = 4
OptEstMdl = EstMdl{optDelay};
The optimal delay is 4
, which matches the delay of the DGP.
Display an estimation summary of the optimal model, and display its estimated switching mechanism.
summarize(OptEstMdl)
Description 1-Dimensional tsVAR Model with 2 Submodels Switch Transition Type: discrete Estimated Levels: -0.063 Fit Effective Sample Size: 196 Number of Estimated Parameters: 2 Number of Constrained Parameters: 0 LogLikelihood: -336.184 AIC: 676.367 BIC: 682.923 Submodels Estimate StandardError TStatistic PValue ________ _____________ __________ __________ State 1 Constant(1) -0.33535 0.24271 -1.3817 0.16707 State 2 Constant(1) 2.0073 0.10647 18.853 2.7794e-79
The estimates are close to the parameters of the DGP.
Plot Loglikelihood Evolution During Multivariate SETAR Estimation
Create the following fully specified SETAR model for the DGP.
State 1: .
State 2: .
State 3:.
.
The system is in state 1 when , the system is in state 2 when 3, and the system is in state 3 otherwise.
t = [-3 3]; ttDGP = threshold(t); constant1 = [-1; -4]; constant2 = [1; 4]; constant3 = [1; 4]; AR1 = [-0.5 0.1; 0.2 -0.75]; AR3 = [0.5 0.1; 0.2 0.75]; Sigma = [2 -1; -1 1]; mdl1DGP = varm(Constant=constant1,AR={AR1}); mdl2DGP = varm(Constant=constant2); mdl3DGP = varm(Constant=constant3,AR={AR3}); mdlDGP = [mdl1DGP; mdl2DGP; mdl3DGP]; MdlDGP = tsVAR(ttDGP,mdlDGP,Covariance=Sigma);
Generate a random response path of length 500 from the DGP. Specify that second response variable with a delay of 4 as the threshold variable.
rng(10) % For reproducibiliy
y = simulate(MdlDGP,500,Index=2,Delay=4);
Create a partially specified threshold-switching model that has the same structure as the DGP, but specify the transition mid-level, submodel coefficients, and model-wide covariance as unknown for estimation.
tt = threshold([NaN; NaN]); mdlar = varm(2,1); mdlc = varm(2,0); Mdl = tsVAR(tt,[mdlar; mdlc; mdlar],Covariance=nan(2));
Fully specify a threshold transition that has the same structure as tt
, but set the mid-levels to -1 and 1.
t0 = [-1 1]; tt0 = threshold(t0);
Fit the threshold-switching model to the simulated series. Specify the threshold variable . Plot the loglikelihood after each iteration of the threshold search algorithm.
EstMdl = estimate(Mdl,tt0,y,IterationPlot=true,Index=2,Delay=4);
The plot displays the evolution of the loglikelihood as the estimation procedure searches for optimal levels. The procedure terminates when one of the stopping criteria is satisfied.
Display an estimation summary of the model.
summarize(EstMdl)
Description 2-Dimensional tsVAR Model with 3 Submodels Switch Transition Type: discrete Estimated Levels: -2.877 2.991 Fit Effective Sample Size: 496 Number of Estimated Parameters: 14 Number of Constrained Parameters: 0 LogLikelihood: -1478.416 AIC: 2984.831 BIC: 3043.723 Submodels Estimate StandardError TStatistic PValue ________ _____________ __________ ___________ State 1 Constant(1) -1.0369 0.13859 -7.4821 7.312e-14 State 1 Constant(2) -4.0179 0.09959 -40.345 0 State 1 AR{1}(1,1) -0.43289 0.063204 -6.8491 7.4332e-12 State 1 AR{1}(2,1) 0.20344 0.045419 4.4791 7.4957e-06 State 1 AR{1}(1,2) 0.076123 0.024477 3.1099 0.0018714 State 1 AR{1}(2,2) -0.75607 0.01759 -42.983 0 State 2 Constant(1) 1.0545 0.14882 7.0858 1.382e-12 State 2 Constant(2) 4.1131 0.10694 38.46 1.9763e-323 State 3 Constant(1) 1.0621 0.095701 11.098 1.2802e-28 State 3 Constant(2) 3.8707 0.068772 56.284 0 State 3 AR{1}(1,1) 0.47396 0.058016 8.1694 3.0997e-16 State 3 AR{1}(2,1) 0.23013 0.041691 5.5199 3.3927e-08 State 3 AR{1}(1,2) 0.10561 0.018233 5.7924 6.9371e-09 State 3 AR{1}(2,2) 0.7568 0.013102 57.761 0
Specify Presample Data for STAR Model Estimation
Consider a smooth threshold-switching (STAR) model for the real US GDP growth rate, where each submodel is AR(4) and the threshold variable is the unemployment growth rate.
Create a partially specified threshold transition for the unemployment growth rate. Specify the normal cdf transition function, and an unknown, estimable mid-level and rate. Label the states "Contraction"
and "Expansion"
.
tt = threshold(NaN,Type="normal",Rates=NaN, ... StateNames=["Contraction" "Expansion"]);
tt
is a partially specified threshold object, and it is agnostic of the variable and data it represents.
Create a threshold-switching model for the real US GDP growth rate. Label the series "rRGDP"
.
mdl = arima(4,0,0);
submdls = [mdl; mdl];
Mdl = tsVAR(tt,submdls,SeriesNames="rRGDP");
Load the quarterly US macroeconomic data set Data_USEconModel.mat
. Compute the real GDP percent growth and unemployment growth.
load Data_USEconModel DataTimeTable = rmmissing(DataTimeTable,DataVariables=["GDP" "GDPDEF" "UNRATE"]); RGDP = DataTimeTable.GDP./DataTimeTable.GDPDEF; rRGDP = price2ret(RGDP)*100; % Response data gUNRATE = diff(DataTimeTable.UNRATE); % Exogenous thresold data dates = DataTimeTable.Time(2:end);
Suppose the estimation period includes growth rates from the first quarter of 1950. Identify the estimation period in the date base.
startEst = datetime(1950,1,1); idxEst = dates >= startEst;
To initialize the estimation procedure, fully specify a threshold transition that has the same structure as tt
, but set the mid-level to -0.5
with a rate of 100
.
tt0 = threshold(-0.5,Type=tt.Type,Rates=100,StateNames=tt.StateNames);
Fit the STAR model to the estimation period of the US GDP growth rate series. Specify the following parameters:
Set
Y0
to the responses before the estimation period to initialize the AR submodel components.Set
Type
to "exogenous" to characterize the threshold variable.Set Z to the threshold variable data
gUNRATE
in the estimation period.Set
MaxRate
to150
to expand the search for the optimal transition function rate.Plot the evolution of the loglikelihood.
EstMdl = estimate(Mdl,tt0,rRGDP(idxEst),Y0=rRGDP(~idxEst), ... Z=gUNRATE(idxEst),Type="exogenous",MaxRate=150,IterationPlot=true);
Display an estimation summary and plot the estimated switching mechanism with the threshold data.
summarize(EstMdl)
Description 1-Dimensional tsVAR Model with 2 Submodels Switch Transition Type: normal Estimated Levels: 0.208 Estimated Rates: 75.566 Fit Effective Sample Size: 237 Number of Estimated Parameters: 10 Number of Constrained Parameters: 0 LogLikelihood: -285.022 AIC: 590.043 BIC: 624.724 Submodels Estimate StandardError TStatistic PValue _________ _____________ __________ __________ State 1 Constant(1) 1.0473 0.11549 9.068 1.2125e-19 State 1 AR{1}(1,1) 0.15792 0.068783 2.2959 0.021679 State 1 AR{2}(1,1) 0.059888 0.066409 0.9018 0.36716 State 1 AR{3}(1,1) -0.10455 0.070384 -1.4854 0.13744 State 1 AR{4}(1,1) -0.098037 0.063249 -1.55 0.12114 State 2 Constant(1) -0.12491 0.15383 -0.81197 0.41681 State 2 AR{1}(1,1) 0.15366 0.13993 1.0981 0.27215 State 2 AR{2}(1,1) -0.027925 0.16754 -0.16667 0.86763 State 2 AR{3}(1,1) -0.24366 0.12778 -1.9068 0.056543 State 2 AR{4}(1,1) -0.075389 0.14024 -0.53759 0.59086
figure ttplot(EstMdl.Switch,Data=gUNRATE(idxEst)) dEst = dates(idxEst); xticklabels(string(dEst(xticks)))
The large rate suggests little mixing occurs between the models of each regime. When the quarterly unempoyment growth is less than 0.206%, the dominant model for the US GDP growth rate is in EstMdl.Submodels(1)
. Otherwise, the dominant submodel is in EstMdl.Submodels(2)
.
Many of the coefficients are insignificant, which can suggest a search for a simpler model.
Perform Constrained Estimation
This example shows how to specify equality constraints on estimable submodel coefficients and threshold parameters.
Consider the model for the real US GDP growth rate in Specify Presample Data for STAR Model Estimation. Assume the submodels are AR(4), but they include only the model constant (time trend in differenced series) and fourth lag. Also, suppose the transition function rate 3 for the unemployment growth series.
Create the partially specified threshold transition for the unemployment growth rate. Specify the known transition rate.
rConstraint = 3; tt = threshold(NaN,Type="normal",Rates=rConstraint, ... StateNames=["Contraction" "Expansion"]);
Create a threshold-switching model for the real US GDP growth rate. Explicitly specify the AR lags to include in the submodels by using the ARLags
option.
mdl = arima(ARLags=4);
submdls = [mdl; mdl];
Mdl = tsVAR(tt,submdls,SeriesNames="rRGDP");
Load the quarterly US macroeconomic data set Data_USEconModel.mat
. Compute the real GDP percent growth and unemployment growth.
load Data_USEconModel DataTimeTable = rmmissing(DataTimeTable,DataVariables=["GDP" "GDPDEF" "UNRATE"]); RGDP = DataTimeTable.GDP./DataTimeTable.GDPDEF; rRGDP = price2ret(RGDP)*100; % Response data gUNRATE = diff(DataTimeTable.UNRATE); % Exogenous thresold data dates = DataTimeTable.Time(2:end);
Suppose the estimation period includes growth rates from the first quarter of 1950. Identify the estimation period in the date base.
startEst = datetime(1950,1,1); idxEst = dates >= startEst;
Fully specify a threshold transition that has the same structure as tt
, including the equality constraint on the transition function rate. Set the initial mid-level to 0.
tt0 = threshold(0,Type=tt.Type,Rates=rConstraint,StateNames=tt.StateNames);
Fit the STAR model to the estimation period of the US GDP growth rate series. Specify the following parameters:
Set
Y0
to the responses before the estimation period to initialize the AR submodel components.Set
Type
to"exogenous"
to characterize the threshold variable.Set
Z
to the threshold variable datagUNRATE
in the estimation period.Set
MaxRate
to 150 to expand the search for the optimal transition function rate.Plot the evolution of the loglikelihood.
EstMdl = estimate(Mdl,tt0,rRGDP(idxEst),Y0=rRGDP(~idxEst), ... Z=gUNRATE(idxEst),Type="exogenous",IterationPlot=true);
Display an estimation summary and plot the estimated switching mechanism with the threshold data.
summarize(EstMdl)
Description 1-Dimensional tsVAR Model with 2 Submodels Switch Transition Type: normal Estimated Levels: 0.228 Estimated Rates: 3.000 Fit Effective Sample Size: 237 Number of Estimated Parameters: 4 Number of Constrained Parameters: 6 LogLikelihood: -267.892 AIC: 543.784 BIC: 557.656 Submodels Estimate StandardError TStatistic PValue _________ _____________ __________ __________ State 1 Constant(1) 1.5928 0.087805 18.14 1.543e-73 State 1 AR{1}(1,1) 0 0 NaN NaN State 1 AR{2}(1,1) 0 0 NaN NaN State 1 AR{3}(1,1) 0 0 NaN NaN State 1 AR{4}(1,1) -0.11789 0.067607 -1.7438 0.081194 State 2 Constant(1) -0.73667 0.15808 -4.6601 3.1609e-06 State 2 AR{1}(1,1) 0 0 NaN NaN State 2 AR{2}(1,1) 0 0 NaN NaN State 2 AR{3}(1,1) 0 0 NaN NaN State 2 AR{4}(1,1) -0.040337 0.1299 -0.31053 0.75616
figure ttplot(EstMdl.Switch,Data=gUNRATE(idxEst)) dEst = dates(idxEst); xticklabels(string(dEst(xticks)))
Estimate Model Containing Regression Component
Consider the following data-generating process (DGP) for a 2-D time-varying TAR (TVTAR) model containing an exogenous regression component.
State 1: , where and is the 2-by-2 identity matrix.
State 2: , where .
State 3: , where .
The exogenous threshold variable represents a linear time trend over the sample period. In this example, is the sampling period normalized by the sample size.
The system is in state 1 when , the system is in state 2 when , and the system is in state 3 otherwise.
Create a TVTAR model that represents the DGP.
tDGP = [0.3 0.6]; ttDGP = threshold(tDGP); % Constant vectors C1 = [1;-1]; C2 = [2;-2]; C3 = [3;-3]; % Common Autoregression coefficient AR = {[0.6 0.1; 0.4 0.2]}; % Regression coefficient vectors Beta1 = [0.2;-0.4]; Beta2 = [0.6;-1.0]; Beta3 = [0.9;-1.3]; % Innovations covariance matrices Sigma1 = 0.5*eye(2); Sigma2 = eye(2); Sigma3 = 1.5*eye(2); % VAR Submodels mdl1 = varm(Constant=C1,AR=AR,Beta=Beta1,Covariance=Sigma1); mdl2 = varm(Constant=C2,AR=AR,Beta=Beta2,Covariance=Sigma2); mdl3 = varm(Constant=C3,AR=AR,Beta=Beta3,Covariance=Sigma3); mdlDGP = [mdl1; mdl2; mdl3]; DGP = tsVAR(ttDGP,mdlDGP);
For the exogenous predictors , simulate a 2-D 1000-period path from the Gaussian distribution with mean 0 and standard deviation 10.
rng(1); % For reproducibility
numPeriods = 1000;
X = 10*randn(numPeriods,2);
For the threshold variable, create a 1000-element vector of equally spaced elements from 0 to 1.
z = linspace(0,1,numPeriods)';
Simulate a 1000-period path from the DGP.
Y = simulate(DGP,numPeriods,X=X,Type="exogenous",Z=z);
Create a TVTAR model template for estimation. Specify all estimable parameters as unknown by using NaN
s.
t = [NaN NaN]; tt = threshold(t); mdl = varm(2,1); % Unknown contsant and AR coefficient by default mdl.Beta = NaN(2,1); % Configure exogenous regression component for estimation Mdl = tsVAR(tt,[mdl; mdl; mdl]);
Specify initial values of 0.25 and 0.7 for the regime thresholds.
t0 = [.25 .7]; tt0 = threshold(t0);
The largest lag among all models is 1. Therefore, the estimation procedure requires one presample observation to initialize the model.
Identify indices for the required presample, and identify indices for the estimation sample.
p = mdl.P; idxPre = 1:p; idxEst = (p + 1):numPeriods;
Fit the TVTAR model to the data. Specify presample responses Y0
, characterize the threshold variable Type
and provide its data Z
, and specify the exogenous data X
.
EstMdl = estimate(Mdl,tt0,Y(idxEst,:),Y0=Y(idxPre,:), ... Type="exogenous",Z=z(idxEst),X=X(idxEst,:));
Display an estimation summary separately for each state, and display the estimated threshold transitions.
summarize(EstMdl,1)
Description 2-Dimensional VAR Submodel, State 1 Submodel Estimate StandardError TStatistic PValue ________ _____________ __________ ___________ State 1 Constant(1) 0.6284 0.1463 4.2952 1.7456e-05 State 1 Constant(2) -0.86147 0.1664 -5.1771 2.2538e-07 State 1 AR{1}(1,1) 0.64569 0.046378 13.922 4.6483e-44 State 1 AR{1}(2,1) 0.41777 0.052749 7.92 2.3752e-15 State 1 AR{1}(1,2) 0.11447 0.029747 3.8481 0.00011901 State 1 AR{1}(2,2) 0.18968 0.033833 5.6064 2.0657e-08 State 1 Beta(1,1) 0.12661 0.01005 12.598 2.1562e-36 State 1 Beta(2,1) -0.28306 0.01143 -24.765 2.1435e-135
summarize(EstMdl,2)
Description 2-Dimensional VAR Submodel, State 2 Submodel Estimate StandardError TStatistic PValue ________ _____________ __________ __________ State 2 Constant(1) 2.0016 0.20741 9.6503 4.9029e-22 State 2 Constant(2) -1.919 0.2359 -8.1345 4.1377e-16 State 2 AR{1}(1,1) 0.59692 0.033401 17.871 1.9848e-71 State 2 AR{1}(2,1) 0.37274 0.03799 9.8117 1.0027e-22 State 2 AR{1}(1,2) 0.1173 0.023148 5.0674 4.0337e-07 State 2 AR{1}(2,2) 0.18002 0.026327 6.8376 8.0532e-12 State 2 Beta(1,1) 0.6689 0.014264 46.896 0 State 2 Beta(2,1) -1.122 0.016223 -69.16 0
summarize(EstMdl,3)
Description 2-Dimensional VAR Submodel, State 3 Submodel Estimate StandardError TStatistic PValue ________ _____________ __________ ___________ State 3 Constant(1) 3.1135 0.10796 28.841 6.5868e-183 State 3 Constant(2) -3.135 0.12279 -25.533 8.5619e-144 State 3 AR{1}(1,1) 0.61547 0.011379 54.089 0 State 3 AR{1}(2,1) 0.40065 0.012942 30.958 1.9801e-210 State 3 AR{1}(1,2) 0.10685 0.0089014 12.004 3.4042e-33 State 3 AR{1}(2,2) 0.19124 0.010124 18.889 1.3979e-79 State 3 Beta(1,1) 0.92011 0.0080742 113.96 0 State 3 Beta(2,1) -1.3378 0.0091834 -145.68 0
EstMdl.Switch
ans = threshold with properties: Type: 'discrete' Levels: [0.3096 0.5887] Rates: [] StateNames: ["1" "2" "3"] NumStates: 3
Input Arguments
Mdl
— Partially specified threshold-switching dynamic regression model
tsVAR
model object
Partially specified threshold-switching dynamic regression model used to indicate
constrained and estimable model parameters, specified as a tsVAR
model
object returned by tsVAR
. Properties of
Mdl
describe the model structure and specify the parameters.
estimate
treats specified parameters as equality constraints
during estimation.
estimate
fits unspecified (NaN
-valued)
parameters to the data Y
.
Because estimate
computes innovations covariances after
estimation, you cannot constrain them. If Mdl.Covariance
is nonempty
(see tsVAR
),
estimate
provides an estimate in
EstMdl.Covariance
. Otherwise, for each
in
i
1:Mdl.NumStates
, estimate
provides the
innovations covariance estimate of state
in i
EstMdl.Submodels(
.i
).Covariance
tt0
— Fully specified threshold transitions
threshold
object
Fully specified threshold transitions used to initialize the estimation procedure,
specified as a threshold
object returned by threshold
.
Properties of a fully specified model object do not contain NaN
values.
tt0
is a copy of Mdl.Switch
with
NaN
values replaced by initial values for levels
(Mdl.Switch.Levels
) and rates
(Mdl.Switch.Rates
). Levels (tt0.Levels
) and rates
(tt0.Rates
) must be within the limits set by the name-value
arguments 'MaxLevel'
and 'MaxRate'
.
Tip
For broad coverage of the parameter space, run the algorithm from multiple
instances of tt0
. For more details, see Algorithms.
Y
— Observed response data
numeric matrix
Observed response data to which estimate
fits the model, 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
ax
— Axes on which to plot loglikelihood for each iteration
Axes
object
Axes on which to plot the loglikelihood for each iteration when the
'IterationPlot'
name-value argument is true
,
specified as an Axes
object.
By default, estimate
plots to the current axes
(gca
).
If 'IterationPlot'
is false
,
estimate
ignores ax
.
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: Type="exogenous",Z=z,IterationPlot=true
specifies data
z
for the exogenous threshold variable and plots the loglikelihood for
each iteration.
Type
— Type of threshold variable data
"endogenous"
(default) | "exogenous"
Type of threshold variable data, specified as a value in this table.
Value | Description |
---|---|
"endogenous" | The model is self-exciting with threshold variable data generated by response j, where
|
"exogenous" | The threshold variable is exogenous to the system. The name-value argument 'Z' specifies the threshold variable data and is required. |
Example: Type="exogenous",Z=z
specifies the data z
for the exogenous threshold variable.
Example: Type="endogenous",Index=2,Delay=4
specifies the endogenous threshold variable as y2,t−4, whose data is Y(:,2)
.
Data Types: char
| string
| cell
Y0
— Presample response data
numeric matrix
Presample response data, specified as a
numPreSampleObs
-by-numSeries
numeric
matrix.
The number of presample observations numPreSampleObs
must be
sufficient to initialize the AR terms of all submodels. For models of type
"endogenous"
, the number of presample observations must also be
sufficient to initialize the delayed response. If numPreSampleObs
exceeds the number necessary to initialize the model, estimate
uses only the latest observations. The last row contains the latest
observations.
By default, Y0
is the initial segment of
Y
, which reduces the effective sample size.
Data Types: double
Z
— Threshold variable data zt
[]
(default) | numeric vector
Threshold variable data for models of type "exogenous"
,
specified as a numeric vector of length at least numObs
. If the
length exceeds numobs
, estimate
uses only
the latest observations. The last row contains the latest observation.
Data Types: double
Delay
— Threshold variable delay d in yj,t−d
1
(default) | positive integer
Threshold variable delay d in
yj,t−d,
for models of type "endogenous"
, specified as a positive
integer.
Example: Delay=4
specifies that the threshold variable is
y2,t−d,
where j is the value of Index
.
Data Types: double
Index
— Threshold variable index j in yj,t−d
1
(default) | scalar in 1:Mdl.NumSeries
Threshold variable index j in
yj,t−d
for models of type "endogenous"
, specified as a scalar in
1:Mdl.NumSeries
.
estimate
ignores Index
for univariate
AR models.
Example: Index=2
specifies that the threshold variable is
y2,t−d,
where d is the value of Delay
.
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 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 estimate
uses the latest
observations.
To use different predictors in each state, specify a cell vector of such matrices
with length numStates
.
By default, estimate
ignores regression components in
Mdl
.
Data Types: double
MaxLevel
— Maximum deviation of estimated threshold levels
0
(default) | positive numeric scalar
Maximum deviation of estimated threshold levels above or below the threshold variable data, z or yj,t−d, specified as a positive numeric scalar representing the percent of the range.
The default value 0
means that initial and estimated levels
must be within the range of the data. A value greater than 0
expands the range of the search.
Example: MaxLevel=10
expands the search for levels 10% above and
below the range of the threshold variable data.
Data Types: double
MaxRate
— Maximum estimated threshold transition rates
positive numeric scalar | positive numeric vector
Maximum estimated threshold transition rates, specified as a positive numeric
scalar or vector with length equal to the number of rates in
Mdl.Switch.Rates
.
For a scalar, all estimated rates are bounded by MaxRate
. For a
vector, the estimate of
Mdl.Switch.Rate(
is bounded by
j
)MaxRate(
.j
)
estimate
estimates discrete transition levels as logistic
transitions at MaxRate
.
The default value is 10
for each rate.
Example: MaxLevel=3
bounds all estimated rates in
Mdl.Switch.Rates
by 3
.
Example: MaxLevel=[5 10]
bounds
Mdl.Switch.Rates(1)
by 5
and bounds
Mdl.Switch.Rates(2)
by 10
.
Data Types: double
MaxItertations
— Limit on number of iterations of threshold search algorithm
100
(default) | positive integer
Limit on the number of iterations of the threshold search algorithm, specified as a positive integer.
Example: MaxIterations=1000
Data Types: double
Tolerance
— Convergence tolerance
1e-6
(default) | positive numeric scalar
Convergence tolerance, specified as a positive numeric scalar. The search
algorithm terminates after an iteration in which the optimality tolerance on
loglikelihood objective logL
changes by less than the value of
Tolerance
.
Example: Tolerance=1e-4
Data Types: double
IterationPlot
— Flag indicating whether to show plot of loglikelihood versus iteration step
false
(default) | true
Flag indicating whether to show a plot of the loglikelihood versus iteration step at algorithm termination, specified as a value in this table.
Value | Description |
---|---|
true | When the algorithm terminates, estimate shows a
plot of the loglikelihood at each iteration step. |
false | estimate does not show a plot. |
Example: IterationPlot=true
Data Types: logical
Output Arguments
EstMdl
— Estimated threshold-switching dynamic regression model
tsVAR
object
Estimated threshold-switching dynamic regression model, returned as a tsVAR
object.
EstMdl
is a copy of Mdl
that has
NaN
values replaced with parameter estimates.
EstMdl
is fully specified.
logL
— Estimated loglikelihood
numeric scalar
Estimated loglikelihood of the data in Y
from the conditional
least-squares problem, returned as a numeric scalar and evaluated at the optimal
parameter values in the threshold transitions EstMdl.Switch
. For more
details, see Algorithms.
h
— Handle to iteration plot
graphics object
Handle to the iteration plot, returned as a graphics object when IterationPlot
is true
.
h
contains a unique plot identifier, which you can use to query or modify properties of the plot.
Tips
Several factors can lead to poor estimates of model parameters. The factors include:
Threshold data does not cross levels or sufficiently sample submodels.
Mdl
contains more estimable parameters than the sample size supports.High-rate transitions are indistinguishable.
Self-exciting autoregressive models have multiple sources of endogenous dynamics.
To improve estimates, perform these actions:
Control the parameter search by experimenting with the
'MaxLevel'
and'MaxRate'
name-value arguments.Consider a parsimonious model with initial levels within the range of threshold variable data.
Constrain specific parameters to potentially improve performance.
To estimate the delay d in a self-exciting model, compare the loglikelihood
logL
with different specifications for the'Delay'
name-value argument. In practice, usually d is limited to a range of reasonable values.You can estimate a simple time-varying STAR (TVSTAR) model by specifying exogenous threshold data z = t, which is a linear time trend over the sample period [3].
Algorithms
estimate
searches over levels and rates forEstMdl.Switch
while solving a conditional least-squares problem for submodel parameters, maximizing the data likelihood, as described in [4].logL
is conditional on the optimal parameter values inEstMdl.Switch
.Models with smooth transitions (STAR models) represent the response as a weighted mixture of conditional means from all submodels ([4], Eqn. 3.6).
estimate
determines the weights by the value of the threshold variable, z or yj,t−d, relative to threshold levels.estimate
treats models with discrete transitions (TAR models) as logistic STAR models with transition rates set to'MaxRate'
in order to disambiguate search levels that fall between threshold variable data.estimate
handles two types of equality constraints.estimate
computes innovations variances and covariances after estimation. Therefore, you cannot constrain them.
References
[1] Chan, Kung-Sik, and Howell Tong. “On Estimating Thresholds in Autoregressive Models.” Journal of Time Series Analysis 7 (May 1986): 179–90. https://doi.org/10.1111/j.1467-9892.1986.tb00501.x.
[2] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[3] Teräsvirta, Tima. "Modelling Economic Relationships with Smooth Transition Regressions." In A. Ullahand and D.E.A. Giles (eds.), Handbook of Applied Economic Statistics, 507–552. New York: Marcel Dekker, 1998.
[4] van Dijk, Dick. Smooth Transition Models: Extensions and Outlier Robust Inference. Rotterdam, Netherlands: Tinbergen Institute Research Series, 1999.
Version History
Introduced in R2021b
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