Infer ARIMA or ARIMAX model residuals or conditional variances

```
[E,V] =
infer(Mdl,Y)
```

[E,V,logL]
= infer(Mdl,Y)

[E,V,logL] = infer(Mdl,Y,Name,Value)

`[`

infers
residuals and conditional variances of a univariate ARIMA model fit
to data `E`

,`V`

] =
infer(`Mdl`

,`Y`

)`Y`

.

`[`

additionally
returns the loglikelihood objective function values.`E`

,`V`

,`logL`

]
= infer(`Mdl`

,`Y`

)

`[E,V,logL] = infer(Mdl,Y,`

infers
the ARIMA or ARIMAX model residuals and conditional variances, and
returns the loglikelihood objective function values, with additional
options specified by one or more `Name,Value`

)`Name,Value`

pair
arguments.

`Y`

— Response datanumeric column vector | numeric matrix

Response data, specified as a numeric column vector or numeric matrix. If `Y`

is a matrix, then it has `numObs`

observations and
`numPaths`

separate, independent paths.

`infer`

infers the residuals and variances of `Y`

. `Y`

represents
the time series characterized by `Mdl`

, and it is
the continuation of the presample series `Y0`

.

If

`Y`

is a column vector, then it represents one path of the underlying series.If

`Y`

is a matrix, then it represents`numObs`

observations of`numPaths`

paths of an underlying time series.

`infer`

assumes that observations across any
row occur simultaneously. The last observation of any series is the
latest.

**Data Types: **`double`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`'E0'`

— Presample innovations`0`

(default) | numeric column vector | numeric matrixPresample innovations that have mean 0 and provide initial values
for the model, specified as the comma-separated pair consisting of `'E0'`

and
a numeric column vector or numeric matrix.

`E0`

must contain at least `numPaths`

columns
and enough rows to initialize the ARIMA model and any conditional
variance model. That is, `E0`

must contain at least `Mdl.Q`

innovations,
but can be greater if you use a conditional variance model. If the
number of rows in `E0`

exceeds the number necessary,
then `infer`

only uses the latest observations. The
last row contains the latest observation.

If the number of columns exceeds `numPaths`

,
then `infer`

only uses the first `numPaths`

columns.
If `E0`

is a column vector, then `infer`

applies
it to each inferred path.

**Data Types: **`double`

`'V0'`

— Presample conditional variancesnumeric column vector | numeric matrix

Presample conditional variances providing initial values for
any conditional variance model, specified as the comma-separated pair
consisting of `'V0'`

and a numeric column vector
or matrix with positive entries.

`V0`

must contain at least `numPaths`

columns
and enough rows to initialize the variance model. If the number of
rows in `V0`

exceeds the number necessary, then `infer`

only
uses the latest observations. The last row contains the latest observation.

If the number of columns exceeds `numPaths`

,
then `infer`

only uses the first `numPaths`

columns.
If `V0`

is a column vector, then `infer`

applies
it to each inferred path.

By default, `infer`

sets the necessary observations
to the unconditional variance of the conditional variance process.

**Data Types: **`double`

`'X'`

— Exogenous predictor datanumeric matrix

Exogenous predictor data for the regression component, specified as
the comma-separated pair consisting of `'X'`

and a
matrix.

The columns of `X`

are separate, synchronized time
series, with the last row containing the latest observations.

If you do not specify `Y0`

, then the number of rows
of `X`

must be at least ```
numObs +
Mdl.P
```

. Otherwise, the number of rows of
`X`

should be at least `numObs`

.
In either case, if the number of rows of `X`

exceeds
the number necessary, then `infer`

uses
only the latest observations.

By default, the conditional mean model does not have a regression coefficient.

**Data Types: **`double`

`'Y0'`

— Presample response datanumeric column vector | numeric matrix

Presample response data that provides initial values for the model, specified as the
comma-separated pair consisting of `'Y0'`

and a numeric
column vector or numeric matrix. `Y0`

must contain at
least `Mdl.P`

rows and `numPaths`

columns. If the number of rows in `Y0`

exceeds
`Mdl.P`

, then `infer`

only uses the latest `Mdl.P`

observations. The last row contains the latest observation. If the
number of columns exceeds `numPaths`

, then `infer`

only uses the first
`numPaths`

columns. If `Y0`

is a
column vector, then `infer`

applies it
to each inferred path.

By default, `infer`

backcasts to obtain the necessary
observations.

**Data Types: **`double`

`NaN`

s indicate missing values and`infer`

removes them. The software merges the presample data and main data sets separately, then uses list-wise deletion to remove any`NaN`

s. That is,`infer`

sets`PreSample`

=`[Y0 E0 V0]`

and`Data`

=`[Y X]`

, then it removes any row in`PreSample`

or`Data`

that contains at least one`NaN`

.The removal of

`NaN`

s in the main data reduces the effective sample size. Such removal can also create irregular time series.`infer`

assumes that you synchronize the response and predictor series such that the latest observation of each occurs simultaneously. The software also assumes that you synchronize the presample series similarly.The software applies all exogenous series in

`X`

to each response series in`Y`

.

`E`

— Inferred residualsnumeric matrix

Inferred residuals, returned as a numeric matrix. `E`

has `numObs`

rows
and `numPaths`

columns.

`V`

— Inferred conditional variancesnumeric matrix

Inferred conditional variances, returned as a numeric matrix. `V`

has `numObs`

rows
and `numPaths`

columns.

`logL`

— Loglikelihood objective function valuesnumeric vector

Loglikelihood objective function values associated with the
model `Mdl`

, returned as a numeric vector. `logL`

has `numPaths`

elements
associated with the corresponding path in `Y`

.

**Data Types: **`double`

Infer residuals from an AR model.

Specify an AR(2) model using known parameters.

Mdl = arima('AR',{0.5,-0.8},'Constant',0.002,... 'Variance',0.8);

Simulate response data with 102 observations.

```
rng 'default';
Y = simulate(Mdl,102);
```

Use the first two responses as presample data, and infer residuals for the remaining 100 observations.

E = infer(Mdl,Y(3:end),'Y0',Y(1:2)); figure; plot(E); title 'Inferred Residuals';

Infer the conditional variances from an AR(1) and GARCH(1,1) composite model.

Specify an AR(1) model using known parameters. Set the variance equal to a `garch`

model.

Mdl = arima('AR',{0.8,-0.3},'Constant',0); MdlVar = garch('Constant',0.0002,'GARCH',0.6,... 'ARCH',0.2); Mdl.Variance = MdlVar;

Simulate response data with 102 observations.

```
rng 'default';
Y = simulate(Mdl,102);
```

Infer conditional variances for the last 100 observations without using presample data.

[Ew,Vw] = infer(Mdl,Y(3:end));

Infer conditional variances for the last 100 observations using the first two observations as presample data.

`[E,V] = infer(Mdl,Y(3:end),'Y0',Y(1:2));`

Plot the two sets of conditional variances for comparison. Examine the first few observations to see the slight difference between the series at the beginning.

figure; subplot(2,1,1); plot(Vw,'r','LineWidth',2); hold on; plot(V); legend('Without Presample','With Presample'); title 'Inferred Conditional Variances'; hold off subplot(2,1,2); plot(Vw(1:5),'r','LineWidth',2); hold on; plot(V(1:5)); legend('Without Presample','With Presample'); title 'Beginning of Series'; hold off

Infer residuals from an ARMAX model.

Specify an ARMA(1,2) model using known parameters for the response (`MdlY`

) and an AR(1) model for the predictor data (`MdlX`

).

MdlY = arima('AR',0.2,'MA',{-0.1,0.6},'Constant',... 1,'Variance',2,'Beta',3); MdlX = arima('AR',0.3,'Constant',0,'Variance',1);

Simulate response and predictor data with 102 observations.

rng 'default'; % For reproducibility X = simulate(MdlX,102); Y = simulate(MdlY,102,'X',X);

Use the first two responses as presample data, and infer residuals for the remaining 100 observations.

E = infer(MdlY,Y(3:end),'Y0',Y(1:2),'X',X); figure; plot(E); title 'Inferred Residuals';

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. *Time
Series Analysis: Forecasting and Control* 3rd ed. Englewood
Cliffs, NJ: Prentice Hall, 1994.

[2] Enders, W. *Applied Econometric Time Series*.
Hoboken, NJ: John Wiley & Sons, 1995.

[3] Hamilton, J. D. *Time Series Analysis*.
Princeton, NJ: Princeton University Press, 1994.

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