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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 specified econometric models. For univariate,
discrete time series analysis, the available model objects are:

`arima`

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

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

— for exponential GARCH models`gjr`

— for Glosten-Jagannathan-Runkle models`regARIMA`

— for regression models with ARIMA errors

For multivariate, discrete time series analysis, the available model objects are:

`dssm`

— for state-space models with diffuse initial states optionally containing exogenous predictor variables`ssm`

— for state-space models optionally containing exogenous predictor variables`varm`

— for vector autogression models optionally containing exogenous predictor variables

Econometrics Toolbox also has functionality for 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.`customblm`

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

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

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

. Assign values to model properties 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(2,0,0) Model: -------------------- Distribution: Name = 'Gaussian' P: 2 D: 0 Q: 0 Constant: 0 AR: {0.8 -0.2} at Lags [1 2] SAR: {} MA: {} SMA: {} 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

where

Assume 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(1,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: NaN GARCH: {NaN} at Lags [1] ARCH: {NaN} at Lags [1]

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(1,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: NaN GARCH: {NaN} at Lags [1] ARCH: {NaN} at Lags [1]

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 = 1x2 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 = 1x2 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(2,0,0) Model: -------------------- Distribution: Name = 'Gaussian' P: 2 D: 0 Q: 0 Constant: 0 AR: {0.8 -0.2} at Lags [1 2] SAR: {} MA: {} SMA: {} 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(2,0,0) Model: -------------------- 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: {} Variance: 0.2

The variable `Mdl`

is updated accordingly.

Methods are functions that accept models 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`

All models have these three methods in common, but the model objects in the toolbox can have other methods.

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

model
vs. a `garch`

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

Find method reference pages for a specific model by entering,
for example, `doc arima.estimate`

.

`arima`

| `conjugateblm`

| `customblm`

| `diffuseblm`

| `dssm`

| `egarch`

| `garch`

| `gjr`

| `regARIMA`

| `semiconjugateblm`

| `ssm`

| `struct`

| `varm`

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