After you have a potential model for your data, you must specify the model to
MATLAB^{®} to proceed with your analysis. Econometrics
Toolbox™ has model objects for storing discrete-time econometric models.

For univariate series, the available model objects are:

`arima`

— for integrated, autoregressive, moving average (ARIMA) models optionally containing exogenous predictor variables`garch`

— for generalized autoregressive conditional heteroscedasticity models (GARCH)`egarch`

— for exponential GARCH models`gjr`

— for Glosten-Jagannathan-Runkle models`regARIMA`

— for regression models with ARIMA errors

For multivariate series, the available model objects are:

Econometrics Toolbox supports univariate Bayesian linear regression analysis. Bayesian linear regression model objects specify the joint prior distribution of the regression coefficients and disturbance variance. The available prior model objects are:

`conjugateblm`

— for the normal-inverse-gamma conjugate prior model. The regression coefficients and disturbance variance are dependent random variables.`semiconjugateblm`

— for the normal-inverse-gamma semiconjugate prior model. The regression coefficients and disturbance variance are independent random variables.`diffuseblm`

— the joint prior distribution is proportional to the inverse of the disturbance variance.`empiricalblm`

— the joint prior distribution is specified by a random sample from the joint posterior distribution.`customblm`

— the joint prior distribution is specified in a custom function that you declare.

To perform Bayesian variable selection, available prior model objects are:

`mixconjugateblm`

— for performing stochastic search variable selection (SSVS). The regression coefficients and disturbance variance are dependent random variables (the prior and posterior distributions are conjugate).`mixsemiconjugateblm`

— for performing SSVS. The regression coefficients and disturbance variance are independent random variables (the prior and posterior distributions are semiconjugate).`lassoblm`

— for performing Bayesian lasso regression.

Econometrics Toolbox supports modelling and analyzing discrete or continuous state Markov models. Available model objects are:

`dtmc`

— for discrete-time Markov chain models characterized by transition matrices.`ssm`

— for continuous, multivariate state-space models optionally containing exogenous predictor variables`dssm`

— for continuous, multivariate state-space models with diffuse initial states optionally containing exogenous predictor variables

To create a model object, specify the form of your model to one of the model
functions (e.g., `arima`

or `garch`

). The function
creates the model object of the corresponding type in the MATLAB workspace, as shown in the figure.

You can work with model objects as you would with any other variable in MATLAB. For example, you can assign the object variable a name, view it in the MATLAB Workspace, and display its value in the Command Window by typing its name.

This image shows a workspace containing an `arima`

model named
`Mdl`

.

A model object holds all the information necessary to estimate, simulate, and forecast econometric models. This information includes the:

Parametric form of the model

Number of model parameters (e.g., the degree of the model)

Innovation distribution (Gaussian or Student’s

*t*)Amount of presample data needed to initialize the model

Such pieces of information are *properties* of
the model, which are stored as fields *within* the model object.
In this way, a model object resembles a MATLAB data structure (`struct`

array).

The five model types—`arima`

, `garch`

,
`egarch`

, `gjr`

, and
`regARIMA`

—have properties according to the econometric
models they support. Each property has a predefined name, which you cannot
change.

For example, `arima`

supports conditional mean models
(multiplicative and additive AR, MA, ARMA, ARIMA, and ARIMAX processes). Every
`arima`

model object has these properties, shown with their
corresponding names.

Property Name | Property Description |
---|---|

`Constant` | Model constant |

`AR` | Nonseasonal AR coefficients |

`MA` | Nonseasonal MA coefficients |

`SAR` | Seasonal AR coefficients (in a multiplicative model) |

`SMA` | Seasonal MA coefficients (in a multiplicative model) |

`D` | Degree of nonseasonal differencing |

`Seasonality` | Degree of seasonal differencing |

`Variance` | Variance of the innovation distribution |

`Distribution` | Parametric family of the innovation distribution |

`P` | Amount of presample data needed to initialize the AR component of the model |

`Q` | Amount of presample data needed to initialize the MA component of the model |

When a model object exists in the workspace, double-click its name in the Workspace window to open the Variable Editor. The Variable Editor shows all model properties and their names.

Notice that in addition to a name, each property has a value.

Specify a model by assigning values to model properties. You do not need, nor are you able, to specify a value for every property. The constructor function assigns default values to any properties you do not, or cannot, specify.

It is good practice to be aware of the default property values for any model you create.

In addition to having a predefined name, each model property has a predefined data type. When assigning or modifying a property’s value, the assignment must be consistent with the property data type.

For example, the `arima`

properties have these data
types.

Property Name | Property Data Type |
---|---|

`Constant` | Scalar |

`AR` | Cell array |

`MA` | Cell array |

`SAR` | Cell array |

`SMA` | Cell array |

`D` | Nonnegative integer |

`Seasonality` | Nonnegative integer |

`Variance` | Positive scalar |

`Distribution` | `struct` array |

`P` | Nonnegative integer (you cannot specify) |

`Q` | Nonnegative integer (you cannot specify) |

To illustrate assigning property values, consider specifying the AR(2) model

$${y}_{t}=0.8{y}_{t-1}-0.2{y}_{t-2}+{\epsilon}_{t},$$

where the innovations are independent and identically distributed normal random variables with mean 0 and variance 0.2. Because the equation is a conditional mean model, use `arima`

to create an object that represents the model. Assign values to model properties by using name-value pair arguments.

This model has two AR coefficients, 0.8 and -0.2. Assign these values to the property `AR`

as a cell array, `{0.8,-0.2}`

. Assign the value `0.2`

to `Variance`

, and `0`

to `Constant`

. You do not need to assign a value to `Distribution`

because the default innovation distribution is `'Gaussian'`

. There are no MA terms, seasonal terms, or degrees of integration, so do not assign values to these properties. You cannot specify values for the properties `P`

and `Q`

.

In summary, specify the model as follows:

Mdl = arima('AR',{0.8,-0.2},'Variance',0.2,'Constant',0)

Mdl = arima with properties: Description: "ARIMA(2,0,0) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 2 D: 0 Q: 0 Constant: 0 AR: {0.8 -0.2} at lags [1 2] SAR: {} MA: {} SMA: {} Seasonality: 0 Beta: [1×0] Variance: 0.2

The output displays the value of the created model, `Mdl`

. Notice that the property `Seasonality`

is not in the output. `Seasonality`

only displays for models with seasonal integration. The property is still present, however, as seen in the Variable Editor.

`Mdl`

has values for every `arima`

property, even though the specification included only three. `arima`

assigns default values for the unspecified properties. The values of `SAR`

, `MA`

, and `SMA`

are empty cell arrays because the model has no seasonal or MA terms. The values of `D`

and `Seasonality`

are `0`

because there is no nonseasonal or seasonal differencing. `arima`

sets:

`P`

equal to`2`

, the number of presample observations needed to initialize an AR(2) model.`Q`

equal to`0`

because there is no MA component to the model (i.e., no presample innovations are needed).

As another illustration, consider specifying the GARCH(1,1) model

$${y}_{t}={\epsilon}_{t},$$

where

$$\begin{array}{rcl}{\epsilon}_{t}& =& {\sigma}_{t}{z}_{t}\\ {\sigma}_{t}^{2}& =& \kappa +{\gamma}_{1}{\sigma}_{t-1}^{2}+{\alpha}_{1}{\epsilon}_{t-1}^{2}\end{array}$$

Assume $${z}_{t}$$ follows a standard normal distribution.

This model has one GARCH coefficient (corresponding to the lagged variance term) and one ARCH coefficient (corresponding to the lagged squared innovation term), both with unknown values. To specify this model, enter:

Mdl = garch('GARCH',NaN,'ARCH',NaN)

Mdl = garch with properties: Description: "GARCH(1,1) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: NaN GARCH: {NaN} at lag [1] ARCH: {NaN} at lag [1] Offset: 0

The default value for the constant term is also `NaN`

. Parameters with `NaN`

values need to be estimated or otherwise specified before you can forecast or simulate the model. There is also a shorthand syntax to create a default GARCH(1,1) model:

Mdl = garch(1,1)

Mdl = garch with properties: Description: "GARCH(1,1) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: NaN GARCH: {NaN} at lag [1] ARCH: {NaN} at lag [1] Offset: 0

The shorthand syntax returns a GARCH model with one GARCH coefficient and one ARCH coefficient, with default `NaN`

values.

The property values in an existing model are retrievable. Working with models resembles working with `struct`

arrays because you can access model properties using dot notation. That is, type the model name, then the property name, separated by `'.'`

(a period).

For example, consider the `arima`

model with this AR(2) specification:

Mdl = arima('AR',{0.8,-0.2},'Variance',0.2,'Constant',0);

To display the value of the property `AR`

for the created model, enter:

arCoefficients = Mdl.AR

`arCoefficients=`*1×2 cell array*
{[0.8000]} {[-0.2000]}

`AR`

is a cell array, so you must use cell-array syntax. The coefficient cell arrays are lag-indexed, so entering

secondARCoefficient = Mdl.AR{2}

secondARCoefficient = -0.2000

returns the coefficient at lag 2. You can also assign any property value to a new variable:

ar = Mdl.AR

`ar=`*1×2 cell array*
{[0.8000]} {[-0.2000]}

You can also modify model properties using dot notation. For example, consider this AR(2) specification:

Mdl = arima('AR',{0.8,-0.2},'Variance',0.2,'Constant',0)

Mdl = arima with properties: Description: "ARIMA(2,0,0) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 2 D: 0 Q: 0 Constant: 0 AR: {0.8 -0.2} at lags [1 2] SAR: {} MA: {} SMA: {} Seasonality: 0 Beta: [1×0] Variance: 0.2

The created model has the default Gaussian innovation distribution. Change the innovation distribution to a Student's *t* distribution with eight degrees of freedom. The data type for `Distribution`

is a `struct`

array.

Mdl.Distribution = struct('Name','t','DoF',8)

Mdl = arima with properties: Description: "ARIMA(2,0,0) Model (t Distribution)" Distribution: Name = "t", DoF = 8 P: 2 D: 0 Q: 0 Constant: 0 AR: {0.8 -0.2} at lags [1 2] SAR: {} MA: {} SMA: {} Seasonality: 0 Beta: [1×0] Variance: 0.2

The variable `Mdl`

is updated accordingly.

Object functions are functions that accept model objects as inputs. In Econometrics Toolbox, these functions, which represent steps in an econometrics analysis workflow, accept any of the model objects included in the toolbox:

`estimate`

`forecast`

`simulate`

Model that you can fit to data have these three methods in common, but the model objects in the toolbox can have other object functions.

Object functions can distinguish between model objects (e.g., an
`arima`

model vs. a `garch`

model). That is,
some object functions accept different optional inputs and return different outputs
depending on the type of model that is input.

Find object function reference pages for a specific model by entering, for
example, `doc arima/estimate`

.

`arima`

|`conjugateblm`

|`customblm`

|`diffuseblm`

|`dssm`

|`dtmc`

|`egarch`

|`garch`

|`gjr`

|`lassoblm`

|`mixconjugateblm`

|`mixsemiconjugateblm`

|`regARIMA`

|`semiconjugateblm`

|`ssm`

|`varm`

|`vecm`