# threshold

Create threshold transitions

## Description

`threshold`

creates threshold transitions from the specified levels and
transition type, either discrete or smooth. Use a `threshold`

object to specify
the switching mechanism of a threshold-switching dynamic regression model (`tsVAR`

).

To study a threshold transitions model, pass a fully specified `threshold`

object to an object function. You can specify
transition levels and rates as unknown parameters (`NaN`

values), which you
can estimate when you fit a `tsVAR`

model to data by using `estimate`

.

Alternatively, to create a random switching mechanism, governed by a discrete-time Markov
chain, for a Markov-switching dynamic regression model, see `dtmc`

and `msVAR`

.

## Creation

### Description

creates the threshold transitions object `tt`

= threshold(`levels`

)`tt`

for discrete state transitions specified by
the transition mid-levels `levels`

.

sets properties using
name-value argument syntax. For example, `tt`

= threshold(`levels`

,`Name,Value`

)```
threshold([0
1],Type="exponential",Rates=[0.5 1.5])
```

specifies smooth, exponential
transitions at mid-levels `0`

and `1`

with rates
`0.5`

and `1.5`

, respectively.

### Input Arguments

`levels`

— Transition mid-levels *t*_{1},
*t*_{2},…
*t*_{n}

increasing numeric vector

Transition mid-levels *t*_{1},
*t*_{2},…
*t*_{n}, specified as an
increasing numeric vector.

Levels *t*_{j} separate
threshold variable data into *n* + 1 states represented by the
intervals
(−∞,*t*_{1}),[*t*_{1},*t*_{2}),…
[*t*_{n},∞).

`NaN`

entries indicate estimable transition levels. The `estimate`

function of
`tsVAR`

treats
the known elements of `levels`

as equality constraints during
optimization.

`threshold`

stores `levels`

in the
Levels property

**Data Types: **`double`

## Properties

You can set most properties when you create a model by using name-value argument syntax.
You can modify only `StateNames`

by using dot notation. For example,
create a two-state, logistic transition at 0, and then label the first and second states
`Depression`

and `Recession`

,
respectively.

tt = threshold(0,Type="logistic"); tt.StateNames = ["Depression" "Recession"];

`Type`

— Type of transitions

`"discrete"`

(default) | `"normal"`

| `"logistic"`

| `"exponential"`

| `"custom"`

This property is read-only.

Type of transitions, specified as a character vector or string scalar.

The transition function
*F*(*z _{t}*,

*t*,

_{j}*r*) associates a transition type with each threshold level

_{j}*t*, where

_{j}*z*is a threshold variable and

_{t}*r*is a level-specific transition rate. Each function

_{j}*F*is bounded between 0 and 1. This table contains the supported types of transitions:

Value | Description |
---|---|

`"discrete"` (default) | Discrete transitions: $$F({z}_{t},{t}_{j})=\{\begin{array}{ll}0\hfill & ,{z}_{t}<{t}_{j}\hfill \\ 1\hfill & ,{z}_{t}>={t}_{j}\hfill \end{array}.$$ Discrete transitions do not have transition rates. |

`"normal"` | Cumulative normal transitions:
t,_{j}r)
=
_{j}`normcdf(` . |

`"logistic"` | Logistic transitions: $$F({z}_{t},{t}_{j},{r}_{j})=\frac{1}{1+{e}^{-{r}_{j}({z}_{t}-{t}_{j})}}.$$ |

`"exponential"` | Exponential transitions: $$F({z}_{t},{t}_{j},{r}_{j})=1-{e}^{-{r}_{j}{({z}_{t}-{t}_{j})}^{2}}.$$ |

`"custom"` | Custom transition function specified by the function handle of the TransitionFunction property. |

**Example: **
`"normal"`

**Data Types: **`char`

| `string`

`Levels`

— Transition mid-levels *t*_{1}, *t*_{2},…
*t*_{n}

increasing numeric vector

This property is read-only.

Transition mid-levels *t*_{1},
*t*_{2},…
*t*_{n}, specified as an
increasing numeric vector. The `levels`

input argument sets
`Levels`

.

**Data Types: **`double`

`Rates`

— Transition rates *r*_{1}, *r*_{2},…
*r*_{n}

`ones(n,1)`

(default) | positive numeric vector

This property is read-only.

Transition rates *r*_{1},
*r*_{2},…
*r*_{n}, specified as a
positive numeric vector of length *n*, the number of levels. Each rate
corresponds to a level in `Levels`

.

`NaN`

values indicate estimable rates. The `estimate`

function of
`tsVAR`

treats the
known elements of `Rates`

as equality constraints during
optimization.

`threshold`

ignores rates for discrete transitions.

**Example: **`[0.5 1.5]`

**Data Types: **`double`

`NumStates`

— Number of states

positive scalar

This property is read-only.

Number of states, specified as a positive scalar and derived from
`Levels`

.

**Data Types: **`double`

`StateNames`

— Unique state labels

`string(1:(n + 1))`

(default) | string vector | cell vector of character vectors | numeric vector

Unique state labels, specified as a string vector, cell vector of character vectors,
or numeric vector of length `numStates`

.

`StateNames(1)`

names the state corresponding to
(−∞,*t*_{1}), `StateName(2)`

names the state corresponding to
[*t*_{1},*t*_{2}),…
and `StateNames(n + 1)`

names the state corresponding to
[*t*_{n},∞).

**Example: **```
["Depression" "Recession" "Stagnant"
"Boom"]
```

**Data Types: **`string`

`TransitionFunction`

— Custom transition function

`[]`

(default) | function handle

This property is read-only.

Custom transition function
*F*(*z _{t}*,

*t*,

_{j}*r*), specified as a function handle. The handle must specify a function with the following syntax:

_{j}function f =transitionfcn(z,tj,rj)

You can replace the name

.`transitionfcn`

`z`

is a numeric vector of threshold variable data.`tj`

is a numeric scalar threshold level.`rj`

is a numeric scalar rate.`f`

is a numeric vector of in the interval [0,1].

When `Type`

is not `"custom"`

,
`threshold`

ignores `TransitionFunction`

.

**Example: **`@transitionfcn`

**Data Types: **`function_handle`

## Object Functions

## Examples

### Create Discrete Threshold Transition

Create a threshold transition at mid-level ${\mathit{t}}_{1}=0$.

t1 = 0; tt = threshold(t1)

tt = threshold with properties: Type: 'discrete' Levels: 0 Rates: [] StateNames: ["1" "2"] NumStates: 2

`tt`

is a `threshold`

object representing a discrete threshold transition at mid-level `0`

. tt is fully specified because its properties do not contain `NaN`

values. Therefore, you can pass `tt`

to any `threshold`

object function.

Given a univariate threshold variable, `tt`

divides the range of the variable into two distinct states, which `tt`

labels `"1"`

and `"2"`

. `tt`

also specifies the switching mechanism of a threshold-switching autoregressive (TAR) model, represented by a `tsVAR`

object. Given values of the observed univariate transition variable:

The TAR model is in state

`"1"`

when the transition variable is in the interval $\left(-\infty ,0\right)$.The TAR model is in state

`"2"`

when the transition variable is in the interval $$[0,\infty )$$.

### Plot Smooth Threshold Transitions

This example shows how to create two logistic threshold transitions with different transition rates, and then display a gradient plot of the transitions.

Load the yearly Canadian inflation and interest rates data set. Extract the inflation rate based on consumer price index (`INF_C`

) from the table, and plot the series.

load Data_Canada INF_C = DataTable.INF_C; plot(dates,INF_C); axis tight

Assume the following characteristics of the inflation rate series:

Rates below 2% are low.

Rates at least 2% and below 8% are medium.

Rates at least 8% are high.

A logistic transition function describes the transition between states well.

Transition between low and medium rates are faster than transitions between medium and high.

Create threshold transitions to describe the Canadian inflation rates.

t = [2 8]; % Thresholds r = [3.5 1.5]; % Transition rates statenames = ["Low" "Med" "High"]; tt = threshold(t,Type="logistic",Rates=r,StateNames=statenames)

tt = threshold with properties: Type: 'logistic' Levels: [2 8] Rates: [3.5000 1.5000] StateNames: ["Low" "Med" "High"] NumStates: 3

Plot the threshold transitions; show the gradient of the transition function between the states, and overlay the data.

figure ttplot(tt,Data=INF_C)

### Prepare Switching Mechanism of Threshold-Switching Model for Estimation

A threshold-switching dynamic regression (`tsVAR`

) model has two main components:

Threshold transitions, which represent the switching mechanism between states. Mid-levels and transition rates are estimable.

A collection of autoregressive models describing the dynamic system among states. Submodel coefficients and covariances are estimable.

Before you create a threshold-switching model, you must specify its threshold transitions by using `theshold`

. If you plan to fit a threshold-switching model to data, you can fully specify its threshold transitions if you know all mid-levels and transition rates. If you need to estimate some or all mid-levels and rates, you can enter `NaN`

values as placeholders for unknown parameters. `estimate`

treats all specified parameters as equality constraints during estimation. Regardless, to fit threshold transition parameters to data, you must specify a partially specified `threshold`

object as the switching mechanism of a threshold-switching model.

**Prepare All Estimable Parameters for Estimation**

Consider a smooth transition autoregressive (STAR) model that switches between three states (two thresholds) with an exponential transition function.

Create the switching mechanism. Specify that all estimable parameters are unknown.

t1 = [NaN NaN]; % Two unknown mid-levels r1 = [NaN NaN]; % Two unknown transition rates tt1 = threshold(t1,Type="exponential",Rates=r1)

tt1 = threshold with properties: Type: 'exponential' Levels: [NaN NaN] Rates: [NaN NaN] StateNames: ["1" "2" "3"] NumStates: 3

`tt`

is a partially specified `threshold`

object to pass to `tsVAR`

as the switching mechanism. The `estimate`

function of `tsVAR`

fits the two mid-levels and transition rates to the data with any unknown submodel parameters in the threshold-switching model.

**Specify Equality Constraints**

Consider a STAR model, which has a switching mechanism with the following qualities:

The thresholds are at -1 and 1.

The transition function is exponential.

The transition rate between the first and second state is 0.5, but the rate between the second and third states is unknown.

Create the switching mechanism.

```
t2 = [-1 1];
r2 = [0.5 NaN];
tt2 = threshold(t2,Type="exponential",Rates=r2)
```

tt2 = threshold with properties: Type: 'exponential' Levels: [-1 1] Rates: [0.5000 NaN] StateNames: ["1" "2" "3"] NumStates: 3

`tt`

is a partially specified `threshold`

object to pass to `tsVAR`

as the switching mechanism. The `estimate`

function of `tsVAR`

does the following:

Treat the mid-levels

`tt.Levels`

and the first transition rate`tt.Rates(1)`

as equality constraintsFit the second transition rate

`tt.Rates(2)`

to the data with any unknown submodel parameters in the threshold-switching model

### Specify Custom Transition Functions

Create smooth threshold transitions with the following qualities:

Mid-levels are at -1, 1, and 2.

The transition function is the Student's $\mathit{t}$ cdf, which allows for a more gradual mixing than the normal cdf.

The transition rates, which are the degrees of freedom of the distribution, are 3, 10, and 100.

```
t = [-1 1 2];
r = [3 10 100];
ttransfcn = @(z,ti,ri)tcdf(z,ri);
tt = threshold(t,Type="custom",TransitionFunction=ttransfcn,Rates=r)
```

tt = threshold with properties: Type: 'custom' Levels: [-1 1 2] Rates: [3 10 100] StateNames: ["1" "2" "3" "4"] NumStates: 4

Plot graphs of each transition function.

```
figure
ttplot(tt,Type="graph")
legend(string(tt.Levels))
```

## More About

### State Transition

In threshold-switching dynamic regression models (`tsVAR`

), a *state
transition* occurs when a threshold variable
*z _{t}* crosses a transition mid-level.

Discrete transitions result in an abrupt change in the submodel computing the response.
Smooth transitions create weighted combinations of submodel responses that change
continuously with the value of *z _{t}*, and state
changes indicate a shift in the dominant submodel.

The smooth transition weights are determined by a transition function
*F*(*z _{t}*,

*t*,

_{j}*r*), where

_{j}*t*is threshold

_{j}*j*and

*r*is transition rate

_{j}*j*(see Type). Discrete, normal, and logistic transition functions separate states at small and large values of

*z*. Exponential transitions separate states at small and large values of |

_{t}*z*|. Exponential transitions model economic variables with "inner" and "outer" states, such as deviations from purchasing power parity.

_{t}## Tips

To widen a smooth transition band to show a more gradual mixing of states, decrease the transition rate by specifying the

`Rates`

name-value argument when you create threshold transitions.

## References

[1] Enders, Walter. *Applied Econometric Time Series*. New York: John Wiley & Sons, Inc., 2009.

[2] 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.

[3] van Dijk, Dick.
*Smooth Transition Models: Extensions and Outlier Robust Inference*.
Rotterdam, Netherlands: Tinbergen Institute Research Series, 1999.

## Version History

**Introduced in R2021b**

## See Also

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### Functions

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