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, there are five available model objects:
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 you can create
a state-space model using an ssm
model
obvect.
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.
Tip 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 = 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(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 accept arima
, garch
, egarch
, gjr
,
and regARIMA
models:
estimate
infer
forecast
simulate
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
| egarch
| garch
| gjr
| regARIMA
| ssm
| struct