# infer

Infer conditional variances of conditional variance models

## Description

returns the table or timetable `Tbl2`

= infer(`Mdl`

,`Tbl1`

)`Tbl2`

containing the inferred
conditional variances and innovations from evaluating the fully specified,
univariate conditional variance model `Mdl`

at the response
variable data in the table or timetable `Tbl1`

. When
`Mdl`

is a model fitted to the response data and returned by
`estimate`

, the inferred innovations are
residuals.* (since R2023a)*

`infer`

selects the response variable named in
`Mdl.SeriesName`

or the sole variable in
`Tbl1`

. To select a different response variable in
`Tbl1`

at which to evaluate the model, use the
`ResponseVariable`

name-value argument.

```
___
= infer(___,
```

specifies options using one or more name-value arguments in
addition to any of the input argument combinations in previous syntaxes.
`Name,Value`

)`infer`

returns the output argument combination for the
corresponding input arguments. For example, `infer(Mdl,Y,V0=v0)`

initializes the
conditional variance model of `Mdl`

using the presample
conditional variance data in `v0`

.

## Examples

### Infer GARCH Model Conditional Variances From Numeric Response Data

Infer conditional variances from a GARCH(1,1) model with known coefficients. Specify response data as a numeric vector.

Specify a GARCH(1,1) model with known parameters. Simulate 101 conditional variances and responses (innovations) from the model. Set aside the first observation from each series to use as presample data.

Mdl = garch(Constant=0.01,GARCH=0.8,ARCH=0.15); rng("default") % For reproducibility [vS,yS] = simulate(Mdl,101); y0 = yS(1); v0 = vS(1); y = yS(2:end); v = vS(2:end); figure tiledlayout(2,1) nexttile plot(v) title("Conditional Variances") nexttile plot(y) title("Innovations")

Infer the conditional variances of `y`

without using presample data. Compare them to the known (simulated) conditional variances.

vI = infer(Mdl,y); figure plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vI,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, No Presample") hold off

Notice the transient response (discrepancy) in the early time periods due to the absence of presample data.

Infer conditional variances using the set-aside presample innovation, `y0`

. Compare them to the known (simulated) conditional variances.

vE = infer(Mdl,y,E0=y0); figure plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vE,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, Innovations Presample") hold off

There is a slightly reduced transient response in the early time periods.

Infer conditional variances using the presample of conditional variance data, `v0`

. Compare them to the known (simulated) conditional variances.

vO = infer(Mdl,y,V0=v0); figure plot(v) plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vO,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, Conditional Variance Presample") hold off

There is a much smaller transient response in the early time periods.

Infer conditional variances using both the presample innovation and conditional variance. Compare them to the known (simulated) conditional variances.

vEO = infer(Mdl,y,E0=y0,V0=v0); figure plot(v) plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vEO,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, Presamples") hold off

When you use sufficient presample innovations and conditional variances, the inferred conditional variances are exact (there is no transient response).

### Infer EGARCH Model Conditional Variances

Infer conditional variances from an EGARCH(1,1) model with known coefficients. When you use, and then do not use presample data, compare the results from `infer`

.

Specify an EGARCH(1,1) model with known parameters. Simulate 101 conditional variances and responses (innovations) from the model. Set aside the first observation from each series to use as presample data.

Mdl = egarch(Constant=0.001,GARCH=0.8, ... ARCH=0.15,Leverage=-0.1); rng("default") % For reproducibility [vS,yS] = simulate(Mdl,101); y0 = yS(1); v0 = vS(1); y = yS(2:end); v = vS(2:end); figure tiledlayout(2,1) nexttile plot(v) title("Conditional Variances") nexttile plot(y) title("Innovations")

Infer the conditional variances of `y`

without using any presample data. Compare them to the known (simulated) conditional variances.

vI = infer(Mdl,y); figure plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vI,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, No Presample") hold off

Notice the transient response (discrepancy) in the early time periods due to the absence of presample data.

Infer conditional variances using the set-aside presample innovation, `y0`

. Compare them to the known (simulated) conditional variances.

vE = infer(Mdl,y,E0=y0); figure plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vE,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, Presample E") hold off

There is a slightly reduced transient response in the early time periods.

Infer conditional variances using the set-aside presample variance, `v0`

. Compare them to the known (simulated) conditional variances.

vO = infer(Mdl,y,V0=v0); figure plot(v) plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vO,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, Presample V") hold off

The transient response is almost eliminated.

Infer conditional variances using both the presample innovation and conditional variance. Compare them to the known (simulated) conditional variances.

vEO = infer(Mdl,y,E0=y0,V0=v0); figure plot(v) plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vEO,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, Presamples") hold off

When you use sufficient presample innovations and conditional variances, the inferred conditional variances are exact (there is no transient response).

### Infer GJR Model Conditional Variances

Infer conditional variances from a GJR(1,1) model with known coefficients. When you use, and then do not use presample data, compare the results from `infer`

.

Specify a GJR(1,1) model with known parameters. Simulate 101 conditional variances and responses (innovations) from the model. Set aside the first observation from each series to use as presample data.

Mdl = gjr(Constant=0.01,GARCH=0.8,ARCH=0.14, ... Leverage=0.1); rng("default") % For reproducibility [vS,yS] = simulate(Mdl,101); y0 = yS(1); v0 = vS(1); y = yS(2:end); v = vS(2:end); figure tiledlayout(2,1) nexttile plot(v) title("Conditional Variances") nexttile plot(y) title("Innovations")

Infer the conditional variances of `y`

without using any presample data. Compare them to the known (simulated) conditional variances.

vI = infer(Mdl,y); figure plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vI,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, No Presample") hold off

Notice the transient response (discrepancy) in the early time periods due to the absence of presample data.

Infer conditional variances using the set-aside presample innovation, `y0`

. Compare them to the known (simulated) conditional variances.

vE = infer(Mdl,y,E0=y0); figure plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vE,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, Presample E") hold off

There is a slightly reduced transient response in the early time periods.

Infer conditional variances using the set-aside presample conditional variance, `vO`

. Compare them to the known (simulated) conditional variances.

vO = infer(Mdl,y,V0=v0); figure plot(v) plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vO,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, Presample V") hold off

There is a much smaller transient response in the early time periods.

Infer conditional variances using both the presample innovation and conditional variance. Compare them to the known (simulated) conditional variances.

vEO = infer(Mdl,y,E0=y0,V0=v0); figure plot(v) plot(1:100,v,"r",LineWidth=2) hold on plot(1:100,vEO,"k:",LineWidth=1.5) legend("Simulated","Inferred",Location="northeast") title("Inferred Conditional Variances, Presamples") hold off

When you use sufficient presample innovations and conditional variances, the inferred conditional variances are exact (there is no transient response).

### Conduct Likelihood Ratio Test for EGARCH Fit Comparison Specifying Data in Timetables

*Since R2023a*

Infer the loglikelihood objective function values for an EGARCH(1,1) and EGARCH(2,1) model fit to the average weekly closing NASDAQ returns. To identify which model is the more parsimonious, adequate fit, conduct a likelihood ratio test. Specify data in timetables.

Load the U.S. equity indices data `Data_EquityIdx.mat`

.

`load Data_EquityIdx`

The timetable `DataTimeTable`

contains the daily NASDAQ closing prices, among other indices.

Compute the weekly average closing prices of all timetable variables.

`DTTW = convert2weekly(DataTimeTable,Aggregation="mean");`

Compute the weekly returns and their sample mean.

DTTRet = price2ret(DTTW); DTTRet.Interval = []; T = height(DTTRet)

T = 626

When you plan to supply a timetable, you must ensure it has all the following characteristics:

The selected response variable is numeric and does not contain any missing values.

The timestamps in the

`Time`

variable are regular, and they are ascending or descending.

Remove all missing values from the timetable, relative to the NASDAQ returns series.

```
DTTRet = rmmissing(DTTRet,DataVariables="NASDAQ");
numobs = height(DTTRet)
```

numobs = 626

Because all sample times have observed NASDAQ returns, `rmmissing`

does not remove any observations.

Determine whether the sampling timestamps have a regular frequency and are sorted.

`areTimestampsRegular = isregular(DTTRet,"weeks")`

`areTimestampsRegular = `*logical*
1

areTimestampsSorted = issorted(DTTRet.Time)

`areTimestampsSorted = `*logical*
1

`areTimestampsRegular = 1`

indicates that the timestamps of `DTTRet`

represent a regular weekly sample. `areTimestampsSorted = 1`

indicates that the timestamps are sorted.

Reserve the first two observations to use as a presample.

DTTRet0 = DTTRet(1:2,:); DTTRet = DTTRet(3:end,:);

Fit an EGARCH(1,1) model to the returns. Supply in-sample and presample data in timetables, and specify `NASDAQ`

as the variable containing the presample innovations. Infer the loglikelihood objective function value.

MdlEGARCH11 = egarch(1,1); MdlEGARCH11.SeriesName = "NASDAQ"; EstMdlEGARCH11 = estimate(MdlEGARCH11,DTTRet, ... Presample=DTTRet0,PresampleInnovationVariable="NASDAQ");

EGARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ __________ Constant -0.48899 0.15218 -3.2133 0.0013123 GARCH{1} 0.95567 0.013348 71.598 0 ARCH{1} 0.2766 0.052276 5.2912 1.2154e-07 Leverage{1} -0.10593 0.025607 -4.1366 3.5244e-05

[TblEGARCH11,logLEGARCH11] = infer(EstMdlEGARCH11,DTTRet, ... Presample=DTTRet0,PresampleInnovationVariable="NASDAQ"); tail(TblEGARCH11)

Time NYSE NASDAQ NASDAQ_Variance NASDAQ_Residual ___________ ___________ ___________ _______________ _______________ 16-Nov-2001 0.0021092 0.0048052 3.4439e-05 0.0048052 23-Nov-2001 0.001451 0.00085891 3.0717e-05 0.00085891 30-Nov-2001 -0.00039051 0.0020552 2.4587e-05 0.0020552 07-Dec-2001 0.00087108 0.005263 2.0775e-05 0.005263 14-Dec-2001 -0.002694 -0.0012244 2.0067e-05 -0.0012244 21-Dec-2001 0.0019929 -0.00094985 1.7698e-05 -0.00094985 28-Dec-2001 0.0019952 -4.93e-05 1.5413e-05 -4.93e-05 04-Jan-2002 -0.00011742 -0.0012263 1.2447e-05 -0.0012263

`TblEGARCH11`

is a timetable of NASDAQ residuals `NASDAQ_Residual`

and conditional variances `NASDAQ_Variance`

, and all variables in the specified in-sample data `DTTRet`

. `logLEGARCH11`

is the loglikelihood of the estimated model `EstMdlEGARCH11`

evaluated at the specified presample and in-sample data.

Fit an EGARCH(2,1) model to the returns. Supply in-sample and presample data in timetables, and specify `NASDAQ`

as the variable containing the presample innovations. Infer the loglikelihood objective function value.

MdlEGARCH21 = egarch(2,1); MdlEGARCH21.SeriesName = "NASDAQ"; EstMdlEGARCH21 = estimate(MdlEGARCH21,DTTRet, ... Presample=DTTRet0,PresampleInnovationVariable="NASDAQ");

EGARCH(2,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant -0.48641 0.15765 -3.0854 0.002033 GARCH{1} 0.96882 0.27062 3.58 0.00034361 GARCH{2} -0.012906 0.26607 -0.048505 0.96131 ARCH{1} 0.27422 0.074054 3.7029 0.00021312 Leverage{1} -0.10486 0.035239 -2.9756 0.002924

[TblEGARCH21,logLEGARCH21] = infer(EstMdlEGARCH21,DTTRet, ... Presample=DTTRet0,PresampleInnovationVariable="NASDAQ");

Conduct a likelihood ratio test, with the more parsimonious EGARCH(1,1) model as the null model, and the EGARCH(2,1) model as the alternative. The degree of freedom for the test is 1, because the EGARCH(2,1) model has one more parameter than the EGARCH(1,1) model (an additional GARCH term).

[h,p] = lratiotest(logLEGARCH21,logLEGARCH11,1)

`h = `*logical*
0

p = 0.9565

The null hypothesis is not rejected (`h = 0`

). At the 0.05 significance level, the EGARCH(1,1) model is not rejected in favor of the EGARCH(2,1) model.

### Conduct Likelihood Ratio Test for GARCH and GJR Fit Comparison

A GARCH(*P*, *Q*) model is nested within a GJR(*P*, *Q*) model. Therefore, you can perform a likelihood ratio test to compare GARCH(*P*, *Q*) and GJR(*P*, *Q*) model fits.

Infer the loglikelihood objective function values for a GARCH(1,1) and GJR(1,1) model fit to NASDAQ Composite Index returns. Conduct a likelihood ratio test to identify which model is the more parsimonious, adequate fit.

Load the NASDAQ data included with the toolbox, and convert the index to returns. Set aside the first two observations to use as presample data.

```
load Data_EquityIdx
nasdaq = DataTable.NASDAQ;
r = price2ret(nasdaq);
r0 = r(1:2);
rn = r(3:end);
```

Fit a GARCH(1,1) model to the returns, and infer the loglikelihood objective function value.

Mdl1 = garch(1,1); EstMdl1 = estimate(Mdl1,rn,E0=r0);

GARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 2.005e-06 5.4298e-07 3.6926 0.00022197 GARCH{1} 0.88333 0.0084536 104.49 0 ARCH{1} 0.10924 0.0076666 14.249 4.5739e-46

[~,logL1] = infer(EstMdl1,rn,E0=r0);

Fit a GJR(1,1) model to the returns, and infer the loglikelihood objective function value.

Mdl2 = gjr(1,1); EstMdl2 = estimate(Mdl2,rn,E0=r0);

GJR(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant 2.4754e-06 5.6986e-07 4.3439 1.3997e-05 GARCH{1} 0.88101 0.0095108 92.632 0 ARCH{1} 0.064017 0.0091852 6.9696 3.1794e-12 Leverage{1} 0.089301 0.0099215 9.0007 2.2429e-19

[~,logL2] = infer(EstMdl2,rn,E0=r0);

Conduct a likelihood ratio test, with the more parsimonious GARCH(1,1) model as the null model, and the GJR(1,1) model as the alternative. The degree of freedom for the test is 1, because the GJR(1,1) model has one more parameter than the GARCH(1,1) model (a leverage term).

[h,p] = lratiotest(logL2,logL1,1)

`h = `*logical*
1

p = 4.5815e-10

The null hypothesis is rejected (`h = 1`

). At the 0.05 significance level, the GARCH(1,1) model is rejected in favor of the GJR(1,1) model.

## Input Arguments

`Y`

— Response data

numeric column vector | numeric matrix

Response data, specified as a `numobs`

-by-1 numeric
column vector or `numobs`

-by-`numpaths`

matrix.

As a column vector, `Y`

represents a single path of the
underlying series.

As a matrix, the rows of `Y`

correspond to periods and
the columns correspond to separate paths. The observations across any row
occur simultaneously.

`infer`

infers the conditional variances of
`Y`

. `Y`

usually represents an
innovation series with mean 0 and variances characterized by
`Mdl`

. It is the continuation of the presample
innovation series `E0`

. `Y`

can also
represent a time series of innovations with mean 0 plus an offset. If
`Mdl`

has a nonzero offset, then the software stores
its value in the `Offset`

property
(`Mdl.Offset`

).

The last observation of any series is the latest observation.

`Tbl1`

— Time series data

table | timetable

*Since R2023a*

Time series data containing response variable
*y _{t}*, at which

`infer`

evaluates the conditional variance
model `Mdl`

, specified as a table or timetable with
`numvars`

variables and `numobs`

rows.
You can optionally select a response variable by using the
`ResponseVariable`

name-value argument.The selected variable is a single path (`numobs`

-by-1
vector) or multiple paths
(`numobs`

-by-`numpaths`

matrix) of
`numobs`

observations of response data. Each row is an
observation, and measurements in each row occur simultaneously.

The selected response variable in `Tbl1`

is a
`numobs`

-by-`numpaths`

numeric matrix.
Each row is an observation, and measurements in each row occur
simultaneously.

Each path (column) of the selected variable is independent of the other paths.

If `Tbl1`

is a timetable, it must represent a sample
with a regular datetime time step (see `isregular`

), and the datetime
vector `Tbl1.Time`

must be strictly ascending or
descending.

If `Tbl1`

is a table, the last row contains the latest
observation.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`'E0',[1 1;0.5 0.5],'V0',[1 0.5;1 0.5]`

specifies two
equivalent presample paths of innovations and two, different presample paths of
conditional variances.

`ResponseVariable`

— Variable to select from `Tbl1`

to treat as response variable *y*_{t}

string scalar | character vector | integer | logical vector

_{t}

*Since R2023a*

Variable to select from `Tbl1`

to treat as the response variable
*y _{t}*, specified as one of the following
data types:

String scalar or character vector containing a variable name in

`Tbl1.Properties.VariableNames`

Variable index (integer) to select from

`Tbl1.Properties.VariableNames`

A length

`numvars`

logical vector, where`ResponseVariable(`

selects variable) = true`j`

from`j`

`Tbl1.Properties.VariableNames`

, and`sum(ResponseVariable)`

is`1`

The selected variable must be a numeric vector and cannot contain missing values (`NaN`

).

If `Tbl1`

has one variable, the default specifies that variable. Otherwise, the default matches the variable to name in `Mdl.SeriesName`

.

**Example: **`ResponseVariable="StockRate2"`

**Example: **`ResponseVariable=[false false true false]`

or `ResponseVariable=3`

selects the third table variable as the response variable.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`E0`

— Presample innovation paths *ε*_{t}

numeric column vector | numeric matrix

_{t}

Presample innovation paths
*ε _{t}*, specified as a

`numpreobs`

-by-1 numeric column vector or a
`numpreobs`

-by-`numprepaths`

matrix. The presample innovations provide initial values for the
innovations process of the conditional variance model
`Mdl`

, and derive from a distribution with mean
0. Use `E0`

only when you supply the numeric array of
response data `Y`

.`numpreobs`

is the number of presample observations.
`numprepaths`

is the number of presample response
paths.

Each row is a presample observation, and measurements in each row
occur simultaneously. The last row contains the latest presample
observation. `numpreobs`

must be at least
`Mdl.Q`

. If `numpreobs`

>
`Mdl.Q`

, `infer`

uses the
latest required number of observations only. The last element or row
contains the latest observation.

If

`E0`

is a column vector, it represents a single path of the underlying innovation series.`infer`

applies it to each output path.If

`E0`

is a matrix, each column represents a presample path of the underlying innovation series.`numprepaths`

must be at least`numpaths`

. If`numprepaths`

>`numpaths`

,`infer`

uses the first`size(Y,2)`

columns only.

The defaults are:

For GARCH(

*P*,*Q*) and GJR(*P*,*Q*) models,`infer`

sets any necessary presample innovations to the square root of the average squared value of the offset-adjusted response series`Y`

.For EGARCH(

*P*,*Q*) models,`infer`

sets any necessary presample innovations to zero.

**Data Types: **`double`

`V0`

— Positive presample conditional variance paths *σ*_{t}^{2}

positive column vector | positive matrix

_{t}

Positive presample conditional variance paths
*σ _{t}*

^{2}, specified as a

`numpreobs`

-by-1 positive column vector
or `numpreobs`

-by-`numprepaths`

positive matrix. `V0`

provides initial values for the
conditional variances in the model. Use `V0`

only when
you supply the numeric array of disturbances
`Z`

.Each row is a presample observation, and measurements in each row occur simultaneously. The last row contains the latest presample observation.

For GARCH(

*P*,*Q*) and GJR(*P*,*Q*) models,`numpreobs`

must be at least`Mdl.P`

.For EGARCH(

*P*,*Q*) models,`numpreobs`

must be at least`max([Mdl.P Mdl.Q])`

.

`numpreobs`

must be at least
`max([Mdl.P Mdl.Q])`

. If
`numpreobs`

> ```
max([Mdl.P
Mdl.Q])
```

, `infer`

uses the latest
required number of observations only. The last element or row contains
the latest observation.

If

`V0`

is a column vector, it represents a single path of the conditional variance series.`infer`

applies it to each output path.If

`V0`

is a matrix, each column represents a presample path of the conditional variance series.`numprepaths`

must be at least`numpaths`

. If`numprepaths`

>`numpaths`

,`infer`

uses the first`size(Y,2)`

columns only.

By default, `infer`

sets any necessary
presample conditional variances to the unconditional variance of the
process.

**Data Types: **`double`

`Presample`

— Presample data

table | timetable

*Since R2023a*

Presample data containing paths of innovation
*ε _{t}* or conditional
variance

*σ*

_{t}^{2}series to initialize the model, specified as a table or timetable, the same type as

`Tbl1`

, with
`numprevars`

variables and
`numpreobs`

rows. Use
`Presample`

only when you supply a table or
timetable of data `Tbl1`

.Each selected variable is a single path
(`numpreobs`

-by-1 vector) or multiple paths
(`numpreobs`

-by-`numprepaths`

matrix) of `numpreobs`

observations representing the
presample of `numpreobs`

observations of the innovation
or conditional variance series for
`ResponseVariable`

, the selected response variable in
`Tbl1`

.

Each row is a presample observation, and measurements in each row
occur simultaneously. `numpreobs`

must be one of the
following values:

`Mdl.Q`

when`Presample`

provides only presample innovations.`Mdl.P`

when`Presample`

provides only presample conditional variances.`max([Mdl.P Mdl.Q])`

when`Presample`

provides both presample innovations and conditional variances

If `numpreobs`

exceeds the minimum
number, `infer`

uses the latest required number
of observations only.

If `Presample`

is a timetable, all the following
conditions must be true:

`Presample`

must represent a sample with a regular datetime time step (see`isregular`

).The inputs

`Tbl1`

and`Presample`

must be consistent in time such that`Presample`

immediately precedes`Tbl1`

with respect to the sampling frequency and order.The datetime vector of sample timestamps

`Presample.Time`

must be ascending or descending.

If `Presample`

is a table, the last row contains
the latest presample observation.

The defaults are:

For GARCH(

*P*,*Q*) and GJR(*P*,*Q*) models,`infer`

sets any necessary presample innovations to the square root of the average squared value of the offset-adjusted response series`Y`

.For EGARCH(

*P*,*Q*) models,`infer`

sets any necessary presample innovations to zero.`infer`

sets any necessary presample conditional variances to the unconditional variance of the process.

If you specify the `Presample`

, you must specify
the presample innovation or conditional variance variable names by using
the `PresampleInnovationVariable`

or
`PresampleVarianceVariable`

name-value
argument.

`PresampleInnovationVariable`

— Variable of `Presample`

containing presample innovation paths *ε*_{t}

string scalar | character vector | integer | logical vector

_{t}

*Since R2023a*

Variable of `Presample`

containing presample innovation paths *ε _{t}*, specified as one of the following data types:

String scalar or character vector containing a variable name in

`Presample.Properties.VariableNames`

Variable index (integer) to select from

`Presample.Properties.VariableNames`

A length

`numprevars`

logical vector, where`PresampleInnovationVariable(`

selects variable) = true`j`

from`j`

`Presample.Properties.VariableNames`

, and`sum(PresampleInnovationVariable)`

is`1`

The selected variable must be a numeric matrix and cannot contain missing values (`NaN`

).

If you specify presample innovation data by using the `Presample`

name-value argument, you must specify `PresampleInnovationVariable`

.

**Example: **`PresampleInnovationVariable="StockRateInnov0"`

**Example: **`PresampleInnovationVariable=[false false true false]`

or `PresampleInnovationVariable=3`

selects the third table variable as the presample innovation variable.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`PresampleVarianceVariable`

— Variable of `Presample`

containing data for the presample conditional variances *σ*_{t}^{2}

string scalar | character vector | integer | logical vector

_{t}

*Since R2023a*

Variable of `Presample`

containing data for the presample conditional
variances
*σ _{t}*

^{2}, specified as one of the following data types:

String scalar or character vector containing a variable name in

`Presample.Properties.VariableNames`

Variable index (positive integer) to select from

`Presample.Properties.VariableNames`

A logical vector, where

`PresampleVarianceVariable(`

selects variable) = true`j`

from`j`

`Presample.Properties.VariableNames`

The selected variable must be a numeric vector and cannot contain missing values
(`NaN`

s).

If you specify presample conditional variance data by using the `Presample`

name-value argument, you must specify `PresampleVarianceVariable`

.

**Example: **`PresampleVarianceVariable="StockRateVar0"`

**Example: **`PresampleVarianceVariable=[false false true false]`

or `PresampleVarianceVariable=3`

selects the third table variable as the presample conditional variance variable.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

**Notes:**

`NaN`

values in`Y`

,`E0`

, and`V0`

indicate missing values.`infer`

removes missing values from specified data by list-wise deletion.For the presample,

`infer`

horizontally concatenates`E0`

and`V0`

, and then it removes any row of the concatenated matrix containing at least one`NaN`

.For in-sample data

`Y`

,`infer`

removes any row containing at least one`NaN`

.

This type of data reduction reduces the effective sample size and can create an irregular time series.

For numeric data inputs,

`infer`

assumes that you synchronize the presample data such that the latest observations occur simultaneously.`infer`

issues an error when any table or timetable input contains missing values.

## Output Arguments

`V`

— Conditional variances

numeric column vector | numeric matrix

Conditional variances inferred from the response data
`Y`

, returned as a numeric column vector or matrix.
`infer`

returns `V`

only
when you supply the input `Y`

.

The dimensions of `V`

and `Y`

are
equivalent. If `Y`

is a matrix, then the columns of
`V`

are the inferred conditional variance paths
corresponding to the columns of `Y`

.

Rows of `V`

are periods corresponding to the periodicity
of `Y`

.

`logL`

— Loglikelihood objective function values

numeric scalar | numeric vector

`Tbl2`

— Inferred conditional variance *σ*_{t}^{2} and
innovation *ε*_{t} paths

table | timetable

_{t}

_{t}

*Since R2023b*

Inferred conditional variance
*σ _{t}*

^{2}and innovation

*ε*paths, returned as a table or timetable, the same data type as

_{t}`Tbl1`

.
`infer`

returns `Tbl2`

only
when you supply the input `Tbl1`

. When
`Mdl`

is an estimated model returned by
`estimate`

, the returned, inferred innovations are
residuals.`Tbl2`

contains the following variables:

The inferred conditional variance paths, which are in a

`numobs`

-by-`numpaths`

numeric matrix, with rows representing observations and columns representing independent paths. Each path represents the continuation of the corresponding path of presample conditional variances in`Presample`

.`infer`

names the filtered conditional variance variable in`Tbl2`

, where_Variance`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`StockReturns`

,`Tbl2`

contains a variable for the corresponding inferred conditional variance paths with the name`StockReturns_Variance`

.The inferred innovation paths, which are in a

`numobs`

-by-`numpaths`

numeric matrix, with rows representing observations and columns representing independent paths. Each path corresponds to the input response path in`Tbl1`

and represents the continuation of the corresponding presample innovations path in`Presample`

.`infer`

names the inferred innovations variable in`Tbl2`

, where_Residual`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`StockReturns`

,`Tbl2`

contains a variable for the corresponding inferred innovations paths with the name`StockReturns_Residual`

.All variables

`Tbl1`

.

If `Tbl1`

is a timetable, row times of
`Tbl1`

and `Tbl2`

are
equal.

## Algorithms

If you do not specify presample data (`E0`

and
`V0`

, or `Presample`

),
`infer`

derives the necessary presample observations from
the unconditional, or long-run, variance of the offset-adjusted response process.

For all conditional variance model types, required presample conditional variances are the sample average of the squared disturbances of the offset-adjusted specified response data (

`Y`

or`Tbl1`

).For GARCH(

*P*,*Q*) and GJR(*P*,*Q*) models, the required presample innovations are the square root of the average squared value of the offset-adjusted response data.For EGARCH(

*P*,*Q*) models, the required presample innovaitons are`0`

.

These specifications minimize initial transient effects.

## References

[1] Bollerslev, T. “Generalized Autoregressive Conditional
Heteroskedasticity.” *Journal of Econometrics.* Vol. 31,
1986, pp. 307–327.

[2] Bollerslev, T. “A Conditionally Heteroskedastic Time Series Model for
Speculative Prices and Rates of Return.” *The Review of Economics and
Statistics*. Vol. 69, 1987, pp. 542–547.

[3] 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.

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

[5] Engle, R. F. “Autoregressive Conditional Heteroskedasticity with
Estimates of the Variance of United Kingdom Inflation.”
*Econometrica*. Vol. 50, 1982, pp. 987–1007.

[6] Glosten, L. R., R. Jagannathan, and D. E. Runkle. “On
the Relation between the Expected Value and the Volatility of the Nominal Excess Return
on Stocks.” *The Journal of Finance*. Vol. 48, No. 5, 1993,
pp. 1779–1801.

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

## Version History

**Introduced in R2012a**

### R2023a: `infer`

accepts input data in tables and timetables, and returns results in tables and timetables

In addition to accepting input data (in-sample and presample) in numeric arrays,
`infer`

accepts input data in tables or regular
timetables. When you supply data in a table or timetable, the following conditions
apply:

`infer`

chooses the default in-sample response series on which to operate, but you can use the specified optional name-value argument to select a different series.If you specify optional presample innovation or conditional variance data to initialize the model, you must also specify the presample innovation or conditional variance series name.

`infer`

returns results in a table or timetable.

Name-value arguments to support tabular workflows include:

`ResponseVariable`

specifies the variable name of the response paths in the input data, from which`infer`

infers conditional variances and innovations.`Presample`

specifies the input table or timetable of presample innovation and conditional variance data.`PresampleInnovationVariable`

specifies the variable name of the innovation paths to select from`Presample`

.`PresampleVarianceVariable`

specifies the variable name of the conditional variance paths to select from`Presample`

.

## See Also

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