Create Threshold-Switching Dynamic Regression Models

This example shows how to create fully and partially specified threshold-switching dynamic regression models by using tsVAR.

The main components of a threshold-switching model are:

• A threshold transition object (threshold), which specifies the states and the switching mechanism among them.

• A vector of submodels, which describe the response in each state. Each submodel can be a univariate ARX (arima) or a multivariate VARX (varm) model. All submodels must contain the same response series.

Create Fully Specified Univariate Model

A model is fully specified when all its parameters are set to known values. For example, a model must be fully specified to simulate or forecast responses from it. The tsVAR function enables you to specify all parameter values, and the estimate function always returns a fully specified model.

Create and fully specify a simple univariate, three-state, mean-variance-switching, smooth threshold autoregressive (STAR) model with the following characteristics:

• The threshold transitions are at mid-levels 2 and 8, the transition function is logistic, the transition rate from state 1 to 2 is 3.5, and the rate from state 2 to 3 is 1.5.

• The state 1 submodel is ${\mathit{y}}_{1,\mathit{t}}=-5+{\epsilon }_{1,\mathit{t}}$, where ${\epsilon }_{1,\mathit{t}}\sim {\rm N}\left(0,0.1\right)$.

• The state 2 submodel is ${\mathit{y}}_{2,\mathit{t}}={\epsilon }_{2,\mathit{t}}$, where ${\epsilon }_{2,\mathit{t}}\sim {\rm N}\left(0,0.3\right)$.

• The state 3 submodel is ${\mathit{y}}_{3,\mathit{t}}=5+{\epsilon }_{3,\mathit{t}}$, where ${\epsilon }_{3,\mathit{t}}\sim {\rm N}\left(0,0.5\right)$.

ttLPS = threshold([2 8],Type="logistic",Rates=[3.5 1.5]);

mdl1 = arima(Constant=-5,Variance=0.1);
mdl2 = arima(Constant=0,Variance=0.3);
mdl3 = arima(Constant=5,Variance=0.5);

Mdl1 = tsVAR(ttLPS,[mdl1 mdl2 mdl3],SeriesNames="y")

Mdl1 = 
  tsVAR with properties:

         Switch: [1x1 threshold]
      Submodels: [3x1 varm]
      NumStates: 3
      NumSeries: 1
     StateNames: ["1"    "2"    "3"]
    SeriesNames: "y"
     Covariance: []

Mdl1.Submodels(1)

ans = 
  varm with properties:

     Description: "1-Dimensional VAR(0) Model"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 0
        Constant: -5
              AR: {}
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: 0.1

Mdl1 is a fully specified tsVAR model object (no properties contain NaN entries). tsVAR converts the arima submodels to 1-D varm submodels of the same form. The default covariance (Mdl1.Covariance) is empty, which indicates that the software generates the innovations from the submodel covariance specification (Mdl1.Submodels.Covariance) of the current state. When you specify a nonempty value for the Covariance name-value argument, a model-wide covariance generates innovations independent of the current state.

Create Fully Specified Multivariate Model

You can create a multivariate threshold-switching model in much the same way as a univariate model.

Create a bivariate, three-state STAR model with the following characteristics:

• The threshold transitions are at mid-levels 2 and 8, the transition function is logistic, the transition rate from state 1 to 2 is 3.5, and the rate from state 2 to 3 is 1.5.

• The state 1 submodel is ${\mathit{y}}_{1,\mathit{t}}=\left[\begin{array}{c}1\\ -1\end{array}\right]+\left[\begin{array}{c}1\\ -1\end{array}\right]{\mathit{x}}_{1,\mathit{t}}+{\epsilon }_{1,\mathit{t}}$, where ${\mathit{x}}_{1,\mathit{t}}$ is a 1-D exogenous predictor series and ${\epsilon }_{1,\mathit{t}}\sim {\rm N}\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}1& -0.1\\ -0.1& 1\end{array}\right]\right)$.

• The state 2 submodel is ${\mathit{y}}_{2,\mathit{t}}=\left[\begin{array}{c}2\\ -2\end{array}\right]+\left[\begin{array}{cc}0.5& 0.1\\ 0.5& 0.5\end{array}\right]{\mathit{y}}_{2,\mathit{t}-1}+\left[\begin{array}{cc}2& 2\\ -2& -2\end{array}\right]{\mathit{x}}_{2,\mathit{t}}+{\epsilon }_{2,\mathit{t}}$, where ${\mathit{x}}_{2,\mathit{t}}$ is a 2-D exogenous predictor series and ${\epsilon }_{2,\mathit{t}}\sim {\rm N}\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}2& -0.2\\ -0.2& 2\end{array}\right]\right)$.

• The state 3 submodel is ${\mathit{y}}_{3,\mathit{t}}=\left[\begin{array}{c}3\\ -3\end{array}\right]+\left[\begin{array}{cc}0.25& 0\\ 0& 0\end{array}\right]{\mathit{y}}_{3,\mathit{t}-1}+\left[\begin{array}{cc}0& 0\\ 0.25& 0\end{array}\right]{\mathit{y}}_{3,\mathit{t}-2}+\left[\begin{array}{ccc}3& 3& 3\\ -3& -3& -3\end{array}\right]{\mathit{x}}_{3,\mathit{t}}+{\epsilon }_{3,\mathit{t}}$, where ${\mathit{x}}_{3,\mathit{t}}$ is a 3-D exogenous predictor series and ${\epsilon }_{3,\mathit{t}}\sim {\rm N}\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}3& -0.3\\ -0.3& 3\end{array}\right]\right)$.

• States 1 through 3 have names "Low", "Med", and "High".

ttLPS = threshold([2 8],Type="logistic",Rates=[3.5 1.5], ...
    StateNames=["Low" "Med" "High"]);

% Constants (numSeries x 1 vectors)
C1 = [1;-1];
C2 = [2;-2];
C3 = [3;-3];

% Autoregression coefficients (numSeries x numSeries matrices)
AR1 = {};                             % 0 lags
AR2 = {[0.5 0.1; 0.5 0.5]};           % 1 lag
AR3 = {[0.25 0; 0 0], [0 0; 0.25 0]}; % 2 lags

% Regression coefficients (numSeries x numRegressors matrices)
Beta1 = [1;-1];           % 1 regressor
Beta2 = [2 2;-2 -2];      % 2 regressors
Beta3 = [3 3 3;-3 -3 -3]; % 3 regressors

% Innovations covariances (numSeries x numSeries matrices)
Sigma1 = [1 -0.1; -0.1 1];
Sigma2 = [2 -0.2; -0.2 2];
Sigma3 = [3 -0.3; -0.3 3];

% Submodels
mdl1 = varm(Constant=C1,AR=AR1,Beta=Beta1,Covariance=Sigma1);
mdl2 = varm(Constant=C2,AR=AR2,Beta=Beta2,Covariance=Sigma2);
mdl3 = varm(Constant=C3,AR=AR3,Beta=Beta3,Covariance=Sigma3);

Mdl2 = tsVAR(ttLPS,[mdl1 mdl2 mdl3],SeriesNames=["Y1" "Y2"])

Mdl2 = 
  tsVAR with properties:

         Switch: [1x1 threshold]
      Submodels: [3x1 varm]
      NumStates: 3
      NumSeries: 2
     StateNames: ["Low"    "Med"    "High"]
    SeriesNames: ["Y1"    "Y2"]
     Covariance: []

Create Partially Specified Model

A model is partially specified when it contains at least one unknown parameter. A partially specified model represents a model template for estimation. The model template specifies the following:

• Structural parameters that are not estimable (for example, the number of states, transition function type, and the number of submodel-specific lags)

• The unknown, estimable parameters (NaN-valued entries corresponding to threshold mid-levels, transition rates, submodel coefficients, and innovations variance)

• Known parameters held fixed at their values during estimation.

Before you can use a partially specified model, you must fit it to data by using estimate. The tsVAR function enables you to specify mandatory structural parameter values, equality constraints on parameters, and which parameters are unknown and estimable.

Create a partially specified model with the same structure as Mdl1. In other words, set all estimable parameters of Mdl1 to NaN. Use the short-hand syntax of arima to create the submodels.

ttLPS1 = threshold([NaN NaN],Type="logistic",Rates=[NaN NaN]);

mdl = arima(0,0,0); % Constant-only ARIMA model

Mdl3 = tsVAR(ttLPS1,[mdl mdl mdl],SeriesNames="y");
Mdl3.Switch

ans = 
  threshold with properties:

          Type: 'logistic'
        Levels: [NaN NaN]
         Rates: [NaN NaN]
    StateNames: ["1"    "2"    "3"]
     NumStates: 3

Mdl3.Submodels(1)

ans = 
  varm with properties:

     Description: "1-Dimensional VAR(0) Model"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 0
        Constant: NaN
              AR: {}
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: NaN

Mdl3 is a partially specified tsVAR object. NaN entries in Mdl3.Switch and Mdl3.Submodels correspond to estimable parameters.

estimate treats any specified parameters as equality constraints during estimation. Therefore, you can experiment with particular values or set values that have economic significance while estimate fits the others. Also, you can estimate a model-wide covariance by using the Covariance name-value argument.

Suppose the threshold mid-levels of Mdl3 are 2 and 8, and all submodels have a common, unknown covariance. Create a model that has the constraints.

ttLPS2 = threshold([2 8],Type="logistic",Rates=[NaN NaN]);
Mdl4 = tsVAR(ttLPS2,[mdl mdl mdl],SeriesNames="y",Covariance=NaN)

Mdl4 = 
  tsVAR with properties:

         Switch: [1x1 threshold]
      Submodels: [3x1 varm]
      NumStates: 3
      NumSeries: 1
     StateNames: ["1"    "2"    "3"]
    SeriesNames: "y"
     Covariance: NaN

Mdl4.Switch

ans = 
  threshold with properties:

          Type: 'logistic'
        Levels: [2 8]
         Rates: [NaN NaN]
    StateNames: ["1"    "2"    "3"]
     NumStates: 3

Mdl4.Submodels(1)

ans = 
  varm with properties:

     Description: "1-Dimensional VAR(0) Model"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 0
        Constant: NaN
              AR: {}
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: NaN

When you pass Mdl4 to estimate with data, the function fits the transition rates, model constants, and model-wide covariance. When Mdl4.Covariance is nonempty, the software ignores all submodel covariance specifications.

Threshold Variable Specification

To this point, although a specified tsVAR object contains information about the threshold transitions of its threshold variable, it is agnostic of the threshold variable itself. However, when you operate on a tsVAR object using estimate, simulate, or forecast, you can specify the threshold variable type, either "endogenous" (the default) or "exogenous", by using the Type name-value argument.

The threshold variable of a self-exciting threshold autoregressive (SETAR) model is one of the response variables, specified by the Index name-value argument. The Delay name-value argument specifies the lag of the variable at which the software determines whether to switch states. The following pseudocode illustrates simulating a 10 observation path of a univariate SETAR model Mdl with a delay of 4.

y = simulate(Mdl,10,Delay=4);

A model with an exogenous threshold variable additionally requires threshold variable data, which you can specify by using the Z name-value argument. The following pseudocode illustrates simulating a 10 observation path of a univariate TAR model Mdl, which uses the exogenous data in z to determine whether to switch states.

y = simulate(Mdl,10,Type="exogenous",Z=z);