# forecast

Forecast univariate ARIMA or ARIMAX model responses or conditional variances

## Syntax

## Description

`[`

returns the `Y`

,`YMSE`

]
= forecast(`Mdl`

,`numperiods`

,`Y0`

)`numperiods`

-by-1 numeric vector of consecutive forecasted
responses `Y`

and the corresponding numeric vector of forecast mean
square errors (MSE) `YMSE`

of the fully specified, univariate ARIMA model
`Mdl`

. The presample response data in the numeric vector
`Y0`

initializes the model to generate forecasts.

returns the table or timetable `Tbl2`

= forecast(`Mdl`

,`numperiods`

,`Tbl1`

)`Tbl2`

containing a variable for each of
the paths of response, forecast MSE, and conditional variance series resulting from
forecasting the ARIMA model `Mdl`

over a `numperiods`

forecast horizon. `Tbl1`

is a table or timetable containing a variable
for required presample response data to initialize the model for forecasting.
`Tbl1`

can optionally contain variables of presample data for
innovations, conditional variances, and predictors.* (since R2023b)*

`forecast`

selects the response variable named in
`Mdl.SeriesName`

or the sole variable in `Tbl1`

. To
select a different response variable in `Tbl1`

to initialize the model,
use the `PresampleResponseVariable`

name-value argument.

`[___] = forecast(___,`

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

)`forecast`

returns the output argument combination for the
corresponding input arguments. For example, `forecast(Mdl,10,Y0,X0=Exo0,XF=Exo)`

specifies
the presample and forecast sample exogenous predictor data to `Exo0`

and
`Exo`

, respectively, to forecast a model with a regression component
(an ARIMAX model).

## Examples

### Forecast Conditional Mean Response Vector

Forecast the conditional mean response of simulated data over a 30-period horizon. Supply a vector of presample response data and return a vector of forecasts.

Simulate 130 observations from a multiplicative seasonal moving average (MA) model with known parameter values.

Mdl = arima(MA={0.5 -0.3},SMA=0.4,SMALags=12,Constant=0.04, ... Variance=0.2); rng(200,"twister") Y = simulate(Mdl,130);

Fit a seasonal MA model to the first 100 observations, and reserve the remaining 30 observations to evaluate forecast performance.

MdlTemplate = arima(MALags=1:2,SMALags=12); EstMdl = estimate(MdlTemplate,Y(1:100));

ARIMA(0,0,2) Model with Seasonal MA(12) (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ __________ Constant 0.20403 0.069064 2.9542 0.0031344 MA{1} 0.50212 0.097298 5.1606 2.4619e-07 MA{2} -0.20174 0.10447 -1.9312 0.053464 SMA{12} 0.27028 0.10907 2.478 0.013211 Variance 0.18681 0.032732 5.7073 1.148e-08

`EstMdl`

is a new `arima`

model that contains estimated parameters (that is, a fully specified model).

Forecast the fitted model into a 30-period horizon. Specify the estimation period data as a presample.

[YF,YMSE] = forecast(EstMdl,30,Y(1:100)); YF(15)

ans = 0.2040

YMSE(15)

ans = 0.2592

`YF`

is a 30-by-1 vector of forecasted responses, and `YMSE`

is a 30-by-1 vector of corresponding MSEs. The 15-period-ahead forecast is 0.2040 and its MSE is 0.2592.

Visually compare the forecasts to the holdout data.

figure h1 = plot(Y,Color=[.7,.7,.7]); hold on h2 = plot(101:130,YF,"b",LineWidth=2); h3 = plot(101:130,YF + 1.96*sqrt(YMSE),"r:",LineWidth=2); plot(101:130,YF - 1.96*sqrt(YMSE),"r:",LineWidth=2); legend([h1 h2 h3],"Observed","Forecast","95% confidence interval", ... Location="NorthWest") title("30-Period Forecasts and 95% Confidence Intervals") hold off

### Forecast NYSE Composite Index

*Since R2023b*

Forecast the weekly average NYSE closing prices over a 15-week horizon. Supply presample data in a timetable and return a timetable of forecasts.

**Load Data**

Load the US equity index data set `Data_EquityIdx`

.

```
load Data_EquityIdx
T = height(DataTimeTable)
```

T = 3028

The timetable `DataTimeTable`

includes the time series variable `NYSE`

, which contains daily NYSE composite closing prices from January 1990 through December 2001.

Plot the daily NYSE price series.

```
figure
plot(DataTimeTable.Time,DataTimeTable.NYSE)
title("NYSE Daily Closing Prices: 1990 - 2001")
```

**Prepare Timetable for Estimation**

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 NYSE price series.

```
DTT = rmmissing(DataTimeTable,DataVariables="NYSE");
T_DTT = height(DTT)
```

T_DTT = 3028

Because all sample times have observed NYSE prices, `rmmissing`

does not remove any observations.

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

`areTimestampsRegular = isregular(DTT,"days")`

`areTimestampsRegular = `*logical*
0

areTimestampsSorted = issorted(DTT.Time)

`areTimestampsSorted = `*logical*
1

`areTimestampsRegular = 0`

indicates that the timestamps of `DTT`

are irregular. `areTimestampsSorted = 1`

indicates that the timestamps are sorted. Business day rules make daily macroeconomic measurements irregular.

Remedy the time irregularity by computing the weekly average closing price series of all timetable variables.

DTTW = convert2weekly(DTT,Aggregation="mean"); areTimestampsRegular = isregular(DTTW,"weeks")

`areTimestampsRegular = `*logical*
1

T_DTTW = height(DTTW)

T_DTTW = 627

`DTTW`

is regular.

```
figure
plot(DTTW.Time,DTTW.NYSE)
title("NYSE Daily Closing Prices: 1990 - 2001")
```

**Create Model Template for Estimation**

Suppose that an ARIMA(1,1,1) model is appropriate to model NYSE composite series during the sample period.

Create an ARIMA(1,1,1) model template for estimation. Set the response series name to `NYSE`

.

```
Mdl = arima(1,1,1);
Mdl.SeriesName = "NYSE";
```

`Mdl`

is a partially specified `arima`

model object.

**Partition Data**

`estimate`

and `forecast`

require `Mdl.P`

presample observations to initialize the model for estimaiton and forecasting.

Partition the data into three sets:

A presample set for estimation

An in-sample set, to which you fit the model and initialize the model for forecasting

A holdout sample of length 15 to measure the model's predictive performance

numpreobs = Mdl.P; % Required presample length numperiods = 15; % Forecast horizon DTTW0 = DTTW(1:numpreobs,:); % Estimation presample DTTW1 = DTTW((numpreobs+1):(end-numperiods),:); % In-sample for estimation and presample for forecasting DTTW2 = DTTW((end-numperiods+1):end,:); % Holdout sample

**Fit Model to Data**

Fit an ARIMA(1,1,1) model to the in-sample weekly average NYSE closing prices. Specify the presample timetable and the presample response variable name.

`EstMdl = estimate(Mdl,DTTW1,Presample=DTTW0,PresampleResponseVariable="NYSE");`

ARIMA(1,1,1) Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ ___________ Constant 0.31873 0.23754 1.3418 0.17965 AR{1} 0.41132 0.2371 1.7348 0.082779 MA{1} -0.31232 0.24486 -1.2755 0.20212 Variance 55.472 1.8496 29.992 1.2638e-197

`EstMdl`

is a fully specified, estimated `arima`

model object.

**Forecast Conditional Mean**

Forecast the weekly average NASDQ closing prices 15 weeks beyond the estimation sample using the fitted model. Use the estimatoin sample data as a presample to initialize the forecast. Specify the response variable name in the presample data.

Tbl2 = forecast(EstMdl,numperiods,DTTW1)

`Tbl2=`*15×3 timetable*
Time NYSE_Response NYSE_MSE NYSE_Variance
___________ _____________ ________ _____________
28-Sep-2001 521.34 55.472 55.472
05-Oct-2001 519.89 122.47 55.472
12-Oct-2001 519.62 194.53 55.472
19-Oct-2001 519.82 268.72 55.472
26-Oct-2001 520.23 343.8 55.472
02-Nov-2001 520.71 419.24 55.472
09-Nov-2001 521.23 494.83 55.472
16-Nov-2001 521.76 570.49 55.472
23-Nov-2001 522.3 646.17 55.472
30-Nov-2001 522.84 721.86 55.472
07-Dec-2001 523.38 797.56 55.472
14-Dec-2001 523.92 873.26 55.472
21-Dec-2001 524.46 948.96 55.472
28-Dec-2001 525 1024.7 55.472
04-Jan-2002 525.55 1100.4 55.472

`Tbl2`

is a 15-by-3 timetable containing the forecasted weekly average closing price forecasts `NYSE_Response`

, corresponding forecast MSEs `NYSE_MSE`

, and the model's constant variance `NYSE_Variance`

(`EstMdl.Variance = 55.8147`

).

Plot the forecasts and approximate 95% forecast intervals.

Tbl2.NYSE_Lower = Tbl2.NYSE_Response - 1.96*sqrt(Tbl2.NYSE_MSE); Tbl2.NYSE_Upper = Tbl2.NYSE_Response + 1.96*sqrt(Tbl2.NYSE_MSE); figure h1 = plot([DTTW1.Time((end-75):end); DTTW2.Time], ... [DTTW1.NYSE((end-75):end); DTTW2.NYSE],Color=[.7,.7,.7]); hold on h2 = plot(Tbl2.Time,Tbl2.NYSE_Response,"k",LineWidth=2); h3 = plot(Tbl2.Time,Tbl2{:,["NYSE_Lower" "NYSE_Upper"]},"r:",LineWidth=2); legend([h1 h2 h3(1)],"Observations","Forecasts","95% forecast intervals", ... Location="NorthWest") title("NYSE Weekly Average Closing Price") hold off

The process is nonstationary, so the width of each forecast interval grows with time. The model tends to unestimate the weekly average closing prices.

### Forecast ARX Model

Forecast the following known autoregressive model with one lag and an exogenous predictor (ARX(1)) model into a 10-period forecast horizon:

$${y}_{t}=1+0.3{y}_{t-1}+2{x}_{t}+{\epsilon}_{t},$$

where ${\epsilon}_{\mathit{t}}$ is a standard Gaussian random variable, and ${\mathit{x}}_{\mathit{t}}$ is an exogenous Gaussian random variable with a mean of 1 and a standard deviation of 0.5.

Create an `arima`

model object that represents the ARX(1) model.

Mdl = arima(Constant=1,AR=0.3,Beta=2,Variance=1);

To forecast responses from the ARX(1) model, the `forecast`

function requires:

One presample response ${\mathit{y}}_{0}$ to initialize the autoregressive term

Future exogenous data to include the effects of the exogenous variable on the forecasted responses

Set the presample response to the unconditional mean of the stationary process:

$$E({y}_{t})=\frac{1+2(1)}{1-0.3}.$$

For the future exogenous data, draw 10 values from the distribution of the exogenous variable.

```
rng(1,"twister");
y0 = (1 + 2)/(1 - 0.3);
xf = 1 + 0.5*randn(10,1);
```

Forecast the ARX(1) model into a 10-period forecast horizon. Specify the presample response and future exogenous data.

fh = 10; yf = forecast(Mdl,fh,y0,XF=xf)

`yf = `*10×1*
3.6367
5.2722
3.8232
3.0373
3.0657
3.3470
3.4454
4.2120
4.0667
4.8065

`yf(3)`

= `3.8232`

is the 3-period-ahead forecast of the ARX(1) model.

### Forecast Composite Conditional Mean and Variance Model

*Since R2023b*

Consider the following AR(1) conditional mean model with a GARCH(1,1) conditional variance model for the weekly average NASDAQ rate series (as a percent) from January 2, 1990 through December 31, 2001.

$$\begin{array}{l}{y}_{t}=0.073+0.138{y}_{t-1}+{\epsilon}_{t}\\ {\sigma}_{t}^{2}=0.022+0.873{\sigma}_{t-1}^{2}+0.119{\epsilon}_{t-1},\end{array}$$

where ${\epsilon}_{\mathit{t}}$ is a series of independent random Gaussian variables with a mean of 0.

Create the model. Name the response series `NASDAQ`

.

```
CondVarMdl = garch(Constant=0.022,GARCH=0.873,ARCH=0.119);
Mdl = arima(Constant=0.073,AR=0.138,Variance=CondVarMdl);
Mdl.SeriesName = "NASDAQ";
```

Load the equity index data set. Remedy the time irregularity by computing the weekly average closing price series of all timetable variables.

load Data_EquityIdx DTTW = convert2weekly(DataTimeTable,Aggregation="mean");

Convert the weekly average NASDAQ closing price series to a percent return series.

RetTT = price2ret(DTTW); RetTT.NASDAQ = RetTT.NASDAQ*100;

Infer residuals and conditional variances from the model.

RetTT2 = infer(Mdl,RetTT); T = numel(RetTT);

Forecast the model over a 25-day horizon. Supply the entire data set as a presample (`forecast`

uses only the latest required observations to initialize the conditional mean and variance models). Supply variable names for the presample innovations and conditional variances. By default, `forecast`

uses the variable name `Mdl.SeriesName`

as the presample response variable.

fh = 25; ForecastTT = forecast(Mdl,fh,RetTT2,PresampleInnovationVariable="NASDAQ_Residual", ... PresampleVarianceVariable="NASDAQ_Variance");

Plot the forecasted responses and conditional variances with the observed series from June 2000.

pdates = RetTT2.Time > datetime(2000,6,1); figure plot(RetTT2.Time(pdates),RetTT2.NASDAQ(pdates)) hold on plot([RetTT2.Time(end); ForecastTT.Time], ... [RetTT2.NASDAQ(end); ForecastTT.NASDAQ_Response]) title("NASDAQ Weekly Average Percent Return Series") legend("Observed","Forecasted") axis tight grid on hold off

figure plot(RetTT2.Time(pdates),RetTT2.NASDAQ_Variance(pdates)) hold on plot([RetTT2.Time(end); ForecastTT.Time], ... [RetTT2.NASDAQ_Variance(end); ForecastTT.NASDAQ_Variance]) title("Conditional Variance Series") legend("Observed","Forecasted") axis tight grid on hold off

### Forecast Multiple Paths

Forecast multiple response and conditional variance paths from a known composite conditional mean and variance model: a SAR$\left(1,0,0\right){\left(1,1,0\right)}_{4}$ conditional mean model with an ARCH(1) conditional variance model. Specify multiple presample response paths.

Create a `garch`

model object that represents this ARCH(1) model:

$${\sigma}_{t}^{2}=0.1+0.2{\epsilon}_{t}^{2}.$$

Create an `arima`

model object that represents this quarterly SAR$\left(1,0,0\right){\left(1,1,0\right)}_{4}$ model:

$$(1-0.5L)(1-0.2{L}^{4})(1-{L}^{4}){y}_{t}=1+{\epsilon}_{t},$$

where ${\epsilon}_{\mathit{t}}$ is a standard Gaussian random variable.

CVMdl = garch(ARCH=0.2,Constant=0.1)

CVMdl = garch with properties: Description: "GARCH(0,1) Conditional Variance Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 0 Q: 1 Constant: 0.1 GARCH: {} ARCH: {0.2} at lag [1] Offset: 0

```
Mdl = arima(Constant=1,AR=0.5,Variance=CVMdl,Seasonality=4, ...
SARLags=4,SAR=0.2)
```

Mdl = arima with properties: Description: "ARIMA(1,0,0) Model Seasonally Integrated with Seasonal AR(4) (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 9 D: 0 Q: 0 Constant: 1 AR: {0.5} at lag [1] SAR: {0.2} at lag [4] MA: {} SMA: {} Seasonality: 4 Beta: [1×0] Variance: [GARCH(0,1) Model]

Because `Mdl`

contains 9 autoregressive terms and 1 ARCH term, `forecast`

requires `Mdl.P = 9`

responses and `CVMdl.Q`

= 1 conditional variance to generate each $\mathit{t}$-period-ahead forecast.

Generate 10 random paths of length 9 from the model.

```
rng(1,"twister")
numpreobs = Mdl.P;
numpaths = 10;
[Y0,~,V0] = simulate(Mdl,numpreobs,NumPaths=numpaths);
```

Forecast 10 paths of responses and conditional variances from the model into a 12-quarter forecast horizon. Specify the presample response paths `Y0`

and conditional variance paths V0.

fh = 12; [YF,~,VF] = forecast(Mdl,fh,Y0,V0=V0);

`YF`

and `VF`

are 12-by-10 matrices of independent forecasted response and conditional variance paths, respectively. `YF(j,k)`

is the `j`

-period-ahead forecast of path `k`

. Path `YF(:,k)`

represents the continuation of the presample path `Y0(:,k)`

. `forecast`

structures `VF`

similarly.

Plot the presample and forecasted responses.

Y = [Y0; YF]; figure plot(Y) hold on h = gca; px = [numpreobs+0.5 h.XLim([2 2]) numpreobs+0.5]; py = h.YLim([1 1 2 2]); hp = patch(px,py,[0.9 0.9 0.9]); uistack(hp,"bottom"); axis tight legend("Forecast period") xlabel("Time (quarters)") title("Response paths") hold off

V = [V0; VF]; figure plot(V) hold on h = gca; px = [numpreobs+0.5 h.XLim([2 2]) numpreobs+0.5]; py = h.YLim([1 1 2 2]); hp = patch(px,py,[0.9 0.9 0.9]); uistack(hp,"bottom"); legend("Forecast period") axis tight xlabel("Time (quarters)") title("Conditional Variance Paths") hold off

## Input Arguments

`numperiods`

— Forecast horizon

positive integer

Forecast horizon, or the number of time points in the forecast period, specified as a positive integer.

**Data Types: **`double`

`Y0`

— Presample response data *y*_{t}

numeric column vector | numeric matrix

_{t}

Presample response data *y _{t}* used to
initialize the model for forecasting, specified as a

`numpreobs`

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

-by-`numpaths`

numeric matrix. When you supply `Y0`

, supply all optional data as
numeric arrays, and `forecast`

returns results in numeric
arrays.`numpreobs`

is the number of presample observations.
`numpaths`

is the number of independent presample paths, from which
`forecast`

initializes the resulting `numpaths`

forecasts (see Algorithms).

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.P`

to initialize
the model. If `numpreobs`

> `Mdl.P`

,
`forecast`

uses only the latest `Mdl.P`

rows.
For more details, see Time Base Partitions for Forecasting.

Columns of `Y0`

correspond to separate, independent presample
paths.

If

`Y0`

is a column vector, it represents a single path of the response series.`forecast`

applies it to each forecasted path. In this case, all forecast paths`Y`

derive from the same initial responses.If

`Y0`

is a matrix, each column represents a presample path of the response series.`numpaths`

is the maximum among the second dimensions of the specified presample observation matrices`Y0`

,`E0`

, and`V0`

.

**Data Types: **`double`

`Tbl1`

— Presample data

table | timetable

*Since R2023b*

Presample data containing required presample responses
*y _{t}*, and, optionally, innovations

*ε*, conditional variances

_{t}*σ*

_{t}^{2}, or predictors

*x*, to initialize the model, specified as a table or timetable with

_{t}`numprevars`

variables and
`numpreobs`

rows. You can select a response, innovation, conditional
variance, or multiple predictor variables from `Tbl1`

by using the
`PresampleResponseVariable`

,
`PresampleInnovationVariable`

,
`PresampleVarianceVariable`

, or
`PresamplePredictorVariables`

name-value argument,
respectively.`numpreobs`

is the number of presample observations.
`numpaths`

is the number of independent presample paths, from which
`forecast`

initializes the resulting `numpaths`

forecasts (see Algorithms).

For all selected variables except predictor variables, each variable contains a
single path (`numpreobs`

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

-by-`numpaths`

matrix) of presample
response, innovations, or conditional variance data.

Each selected predictor variable contains a single path of observations.
`forecast`

applies all selected predictor variables to each
forecasted path. When you do not specify presample innovation data for forecasting an
ARIMAX model, `forecast`

uses the presample predictor data to
infer presample innovations.

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

must be one of the following values:

At least

`Mdl.P`

when`Presample`

provides only presample responsesAt least

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

otherwise

When `Mdl.Variance`

is a conditional variance model,
`forecast`

can require more than the minimum required number of
presample values. If `numpreobs`

exceeds the minimum number,
`forecast`

uses the latest required number of observations
only.

If `Tbl1`

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

`Tbl1`

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

).The datetime vector of sample timestamps

`Tbl1.Time`

must be ascending or descending.

If `Tbl1`

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

Although `forecast`

requires presample response data,
`forecast`

sets default presample innovation and conditional
variance data as follows:

To infer necessary presample innovations from presample responses,

`numpreobs`

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

(see`infer`

). Additionally, for ARIMAX models,`forecast`

requires enough presample predictor data. If`numpreobs`

is less than`Mdl.P + Mdl.Q`

or you do not specify presample predictor data for ARIMAX forecasting,`forecast`

sets all necessary presample innovations to zero.To infer necessary presample variances from presample innovations,

`forecast`

requires a sufficient number of presample innovations to initialize the specified conditional variance model (see`infer`

). If you do not specify enough presample innovations to initialize the conditional variance model,`forecast`

sets the necessary presample variances to the unconditional variance of the specified variance process.

### 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: **`forecast(Mdl,10,Y0,X0=Exo0,XF=Exo)`

specifies the presample
and forecast sample exogenous predictor data to `Exo0`

and
`Exo`

, respectively, to forecast a model with a regression
component.

`E0`

— Presample innovations *ε*_{t}

numeric column vector | numeric matrix

_{t}

Presample innovations *ε _{t}* used to
initialize either the moving average (MA) component of the ARIMA model or the
conditional variance model, specified as a

`numpreobs`

-by-1 column
vector or `numpreobs`

-by-`numpaths`

numeric matrix.
Use `E0`

only when you supply the numeric array of presample response
data `Y0`

. `forecast`

assumes that the
presample innovations have a mean of zero.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`

to initialize
the model. If `Mdl.Variance`

is a conditional variance model (for
example, a `garch`

model object), `E0`

might require more than `Mdl.Q`

rows. If `numpreobs`

is greater than required, `forecast`

uses only the latest
required rows.

Columns of `E0`

correspond to separate, independent presample
paths.

If

`E0`

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

applies it to each forecasted path. In this case, all forecast paths`Y`

derive from the same initial innovations.If

`E0`

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

is the maximum among the second dimensions of the specified presample observation matrices`Y0`

,`E0`

, and`V0`

.

By default:

If you provide enough presample responses and, for ARIMAX models, presample predictor data (

`X0`

),`forecast`

infers necessary presample innovations from the presample data. In this case,`numpreobs`

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

(see`infer`

)Otherwise,

`forecast`

sets all necessary presample innovations to zero.

**Data Types: **`double`

`V0`

— Presample conditional variances *σ*_{t}^{2}

positive numeric column vector | positive numeric matrix

_{t}

Presample conditional variances
*σ _{t}*

^{2}used to initialize the conditional variance model, specified as a

`numpreobs`

-by-1 positive column vector or
`numpreobs`

-by-`numpaths`

positive matrix. Use
`V0`

only when you supply the numeric array of presample response
data `Y0`

. If the model variance `Mdl.Variance`

is
constant, `forecast`

ignores `V0`

.Rows of `V0`

correspond to periods in the presample, and the last
row contains the latest presample conditional variance. `numpreobs`

must be enough to initialize the conditional variance model (see `forecast`

). If `numpreobs`

exceeds the minimum number,
`forecast`

uses only the latest observations.

Columns of `V0`

correspond to separate, independent paths.

If

`V0`

is a column vector,`forecast`

applies it to each forecasted path. In this case, the conditional variance model of all forecast paths`Y`

derives from the same initial conditional variances.If

`V0`

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

is the maximum among the second dimensions of the specified presample observation matrices`Y0`

,`E0`

, and`V0`

.

By default:

If you specify enough presample innovations

`E0`

to initialize the conditional variance model`Mdl.Variance`

,`forecast`

infers any necessary presample conditional variances by passing the conditional variance model and`E0`

to the`infer`

function.If you do not specify

`E0`

, but you specify enough presample responses and, for ARIMAX models, presample predictor data,`Y0`

to infer enough presample innovations,`forecast`

infers any necessary presample conditional variances from the inferred presample innovations.If you do not specify enough presample data,

`forecast`

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

**Data Types: **`double`

`PresampleResponseVariable`

— Response variable *y*_{t} to select from `Tbl1`

string scalar | character vector | integer | logical vector

_{t}

*Since R2023b*

Response variable *y _{t}* to select from

`Tbl1`

containing the presample response data, specified as one
of the following data types:String scalar or character vector containing a variable name in

`Tbl1.Properties.VariableNames`

Variable index (positive integer) to select from

`Tbl1.Properties.VariableNames`

A logical vector, where

`PresampleResponseVariable(`

selects variable) = true`j`

from`j`

`Tbl1.Properties.VariableNames`

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

s).

If `Tbl1`

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

.

**Example: **`PresampleResponseVariable="StockRate"`

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

or
`PresampleResponseVariable=3`

selects the third table variable as
the response variable.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`PresampleInnovationVariable`

— Presample innovation variable of *ε*_{t} to select from `Tbl1`

string scalar | character vector | integer | logical vector

_{t}

*Since R2023b*

Presample innovation variable of *ε _{t}* to
select from

`Tbl1`

containing presample innovation data, specified as
one of the following data types:String scalar or character vector containing a variable name in

`Tbl1.Properties.VariableNames`

Variable index (positive integer) to select from

`Tbl1.Properties.VariableNames`

A logical vector, where

`PresampleInnovationVariable(`

selects variable) = true`j`

from`j`

`Tbl1.Properties.VariableNames`

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

s).

If you specify presample innovation data in `Tbl1`

, you must
specify `PresampleInnovationVariable`

.

**Example: **`PresampleInnovationVariable="StockRateDist0"`

**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`

— Presample conditional variance variable *σ*_{t}^{2} to select
from `Tbl1`

string scalar | character vector | integer | logical vector

_{t}

Presample conditional variance variable
*σ _{t}*

^{2}to select from

`Tbl1`

containing presample conditional variance data, specified
as one of the following data types:String scalar or character vector containing a variable name in

`Tbl1.Properties.VariableNames`

Variable index (positive integer) to select from

`Tbl1.Properties.VariableNames`

A logical vector, where

`PresampleVarianceVariable(`

selects variable) = true`j`

from`j`

`Tbl1.Properties.VariableNames`

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

s).

If you specify presample conditional variance data in `Tbl1`

,
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`

`X0`

— Presample predictor data

numeric matrix

Presample predictor data used to infer the presample innovations
`E0`

, specified as a
`numpreobs`

-by-`numpreds`

numeric matrix. Use
`X0`

only when you supply the numeric array of presample response
data `Y0`

and your model contains a regression component.
`numpreds`

= `numel(Mdl.Beta)`

.

Rows of `X0`

correspond to periods in the presample, and the last
row contains the latest set of presample predictor observations. Columns of
`X0`

represent separate time series variables, and they correspond
to the columns of `XF`

and `Mdl.Beta`

.

If you do not specify `E0`

, `X0`

must have at
least `numpreobs`

– `Mdl.P`

rows so that
`forecast`

can infer presample innovations. If the number of
rows exceeds the minimum number required to infer presample innovations,
`forecast`

uses only the latest required presample predictor
observations. A best practice is to set `X0`

to the same predictor
data matrix used in the estimation, simulation, or inference of
`Mdl`

. This setting ensures that `forecast`

infers presample innovations `E0`

correctly.

If you specify `E0`

, `forecast`

ignores
`X0`

.

If you specify `X0`

but you do not specify forecasted predictor
data `XF`

, `forecast`

issues an
error.

By default, `forecast`

drops the regression component from the model when it infers presample innovations, regardless of the value of the regression coefficient `Mdl.Beta`

.

**Data Types: **`double`

`PresamplePredictorVariables`

— Presample exogenous predictor variables *x*_{t} to select from `Tbl1`

string vector | cell vector of character vectors | vector of integers | logical vector

_{t}

*Since R2023b*

Presample exogenous predictor variables
*x _{t}* to select from

`Tbl1`

containing presample exogenous predictor data, specified as one of the following data types:String vector or cell vector of character vectors containing

`numpreds`

variable names in`Tbl1.Properties.VariableNames`

A vector of unique indices (positive integers) of variables to select from

`Tbl1.Properties.VariableNames`

A logical vector, where

`PresamplePredictorVariables(`

selects variable) = true`j`

from`j`

`Tbl1.Properties.VariableNames`

The selected variables must be numeric vectors and cannot contain missing values
(`NaN`

s).

If you specify presample predictor data, you must also specify in-sample predictor
data by using the `InSample`

and
`PredictorVariables`

name-value arguments.

By default, `forecast`

excludes the regression component,
regardless of its presence in `Mdl`

.

**Example: **```
PresamplePredictorVariables=["M1SL" "TB3MS"
"UNRATE"]
```

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

or
`PredictorVariable=[1 3]`

selects the first and third table
variables to supply the predictor data.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

`XF`

— Forecasted (or future) predictor data

numeric matrix

Forecasted (or future) predictor data, specified as a numeric matrix with
`numpreds`

columns. `XF`

represents the evolution
of specified presample predictor data `X0`

forecasted into the
future (the forecast period). Use `XF`

only when you supply the
numeric array of presample response data `Y0`

.

Rows of `XF`

correspond to time points in the future;
`XF(`

contains the
* t*,:)

*-period-ahead predictor forecasts.*

`t`

`XF`

must have at least `numperiods`

rows. If the number of rows exceeds
`numperiods`

, `forecast`

uses only the first
(earliest) `numperiods`

forecasts. For more details, see Time Base Partitions for Forecasting.Columns of `XF`

are separate time series variables, and they
correspond to the columns of `X0`

and
`Mdl.Beta`

.

By default, the `forecast`

function generates forecasts from `Mdl`

without a regression component, regardless of the value of the regression coefficient `Mdl.Beta`

.

`InSample`

— Forecasted (future) predictor data

table | timetable

*Since R2023b*

Forecasted (future) predictor data for the exogenous regression component of the
model, specified as a table or timetable. `InSample`

contains
`numvars`

variables, including `numpreds`

predictor variables *x _{t}*.

`forecast`

returns the forecasted variables in the output
table or timetable `Tbl2`

, which is commensurate with
`InSample`

.

Each row corresponds to an observation in the forecast horizon, the first row is
the earliest observation, and measurements in each row, among all paths, occur
simultaneously. `InSample`

must have at least
`numperiods`

rows to cover the forecast horizon. If you supply
more rows than necessary, `forecast`

uses only the first
`numperiods`

rows.

Each selected predictor variable is a numeric vector without missing values
(`NaN`

s). `forecast`

applies the specified
predictor variables to all forecasted paths.

If `InSample`

is a timetable, the following conditions apply:

If `InSample`

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

By default, `forecast`

does not include the regression
component in the model, regardless of the value of `Mdl.Beta`

.

`PredictorVariables`

— Exogenous predictor variables *x*_{t} to select from `InSample`

string vector | cell vector of character vectors | vector of integers | logical vector

_{t}

*Since R2023b*

Exogenous predictor variables *x _{t}* to
select from

`InSample`

containing exogenous predictor data in the
forecast horizon, specified as one of the following data types:String vector or cell vector of character vectors containing

`numpreds`

variable names in`InSample.Properties.VariableNames`

A vector of unique indices (positive integers) of variables to select from

`InSample.Properties.VariableNames`

A logical vector, where

`PredictorVariables(`

selects variable) = true`j`

from`j`

`InSample.Properties.VariableNames`

The selected variables must be numeric vectors and cannot contain missing values
(`NaN`

s).

By default, `forecast`

excludes the regression component,
regardless of its presence in `Mdl`

.

**Example: **```
PredictorVariables=["M1SL" "TB3MS"
"UNRATE"]
```

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

or
`PredictorVariable=[1 3]`

selects the first and third table
variables to supply the predictor data.

**Data Types: **`double`

| `logical`

| `char`

| `cell`

| `string`

**Note**

For numeric array inputs, `forecast`

assumes that you
synchronize all specified presample data sets so that the latest observation of each
presample series occurs simultaneously. Similarly, `forecast`

assumes that the first observation in the forecasted predictor data
`XF`

occurs in the time point immediately after the last observation
in the presample predictor data `X0`

.

## Output Arguments

`Y`

— Minimum mean square error (MMSE) conditional mean forecasts

numeric column vector | numeric matrix

Minimum mean square error (MMSE) conditional mean forecasts
*y _{t}*, returned as a

`numperiods`

-by-1 column vector or a
`numperiods`

-by-`numpaths`

numeric matrix.
`Y`

represents a continuation of `Y0`

(`Y(1,:)`

occurs in the time point immediately after
`Y0(end,:)`

). `forecast`

returns
`Y`

only when you supply numeric presample data
`Y0`

.`Y(`

contains the
* t*,:)

*, or the conditional mean forecast of all paths for time point*

*-period-ahead forecasts*`t`

*in the forecast period.*

`t`

`forecast`

determines `numpaths`

from the
number of columns in the presample data sets `Y0`

,
`E0`

, and `V0`

. For details, see Algorithms. If each
presample data set has one column, `Y`

is a column vector.

**Data Types: **`double`

`YMSE`

— MSE of forecasted responses

numeric column vector | numeric matrix

MSE of the forecasted responses `Y`

(forecast error variances),
returned as a `numperiods`

-by-1 column vector or a
`numperiods`

-by-`numpaths`

numeric matrix.
`forecast`

returns `YMSE`

only when you
supply numeric presample data `Y0`

.

`YMSE(`

contains the forecast error variances of all paths for time point * t*,:)

*in the forecast period.*

`t`

`forecast`

determines `numpaths`

from the
number of columns in the presample data sets `Y0`

,
`E0`

, and `V0`

. For details, see Algorithms. If you do
not specify any presample data sets, or if each data set is a column vector,
`YMSE`

is a column vector.

The square roots of `YMSE`

are the standard errors of the forecasts `Y`

.

**Data Types: **`double`

`V`

— MMSE forecasts of conditional variances of future model innovations

numeric column vector | numeric matrix

MMSE forecasts of the conditional variances of future model innovations, returned as
a `numperiods`

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

-by-`numpaths`

numeric matrix.
`forecast`

returns `V`

only when you supply
numeric presample data `Y0`

.

When `Mdl.Variance`

is a conditional variance model, row

contains the conditional variance
forecasts of period `j`

. Otherwise,
`j`

`V`

is a matrix composed of the constant
`Mdl.Variance`

.

`forecast`

determines `numpaths`

from the
number of columns in the presample data sets `Y0`

,
`E0`

, and `V0`

. For details, see Algorithms. If you do
not specify any presample data sets, or if each data set is a column vector,
`YMSE`

is a column vector.

**Data Types: **`double`

`Tbl2`

— Paths of MMSE forecasts of responses *y*_{t}, corresponding forecast MSEs, and MMSE forecasts of conditional variances
*σ*_{t}^{2} of future
model innovations *ε*_{t}

table | timetable

_{t}

_{t}

_{t}

*Since R2023b*

Paths of MMSE forecasts of responses *y _{t}*,
corresponding forecast MSEs, and MMSE forecasts of conditional variances

*σ*

_{t}^{2}of future model innovations

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

_{t}`Tbl1`

.
`forecast`

returns `Tbl2`

only when you
supply the input `Tbl1`

.`Tbl2`

contains the following variables:

The forecasted response paths, which are in a

`numperiods`

-by-`numpaths`

numeric matrix, with rows representing periods in the forecast horizon and columns representing independent paths, each corresponding to the input presample response paths in`Tbl1`

.`forecast`

names the forecasted response variable

, where_Response`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`GDP`

,`Tbl2`

contains a variable for the corresponding forecasted response paths with the name`GDP_Response`

.Each path in

`Tbl2.`

represents the continuation of the corresponding presample response path in_Response`responseName`

`Tbl1`

(`Tbl2.`

occurs in the next time point, with respect to the periodicity_Response(1,:)`responseName`

`Tbl1`

, after the last presample response).`Tbl2.`

contains the_Response(`responseName`

,`j`

)`k`

-period-ahead forecasted response of path`j`

.`k`

The forecast MSE paths, which are in a

`numperiods`

-by-`numpaths`

numeric matrix, with rows representing periods in the forecast horizon and columns representing independent paths, each corresponding to the forecasted responses in`Tbl2.`

._Response`responseName`

`forecast`

names the forecast MSEs

, where_MSE`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`GDP`

,`Tbl2`

contains a variable for the corresponding forecast MSE with the name`GDP_MSE`

.The forecasted conditional variance paths, which are in a

`numperiods`

-by-`numpaths`

numeric matrix, with rows representing periods in the forecast horizon and columns representing independent paths.`forecast`

names the forecasted conditional variance variable

, where_Variance`responseName`

is`responseName`

`Mdl.SeriesName`

. For example, if`Mdl.SeriesName`

is`StockReturns`

,`Tbl2`

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

.Each path in

`Tbl2.`

represents a continuation of the presample conditional variance process, either supplied by_Variance`responseName`

`Tbl1`

or set by default (`Tbl2.`

occurs in the next time point, with respect to the periodicity_Variance(1,:)`responseName`

`Tbl1`

, after the last presample conditional variance).`Tbl2.`

contains the_Variance(`responseName`

,`j`

)`k`

-period-ahead forecasted conditional variance of path`j`

.`k`

When you supply

`InSample`

,`Tbl2`

contains all variables in`InSample`

.

If `Tbl1`

is a timetable, the following conditions hold:

The row order of

`Tbl2`

, either ascending or descending, matches the row order of`Tbl1`

.`Tbl2.Time(1)`

is the next time after`Tbl1.Time(end)`

relative the sampling frequency, and`Tbl2.Time(2:numobs)`

are the following times relative to the sampling frequency.

## More About

### Time Base Partitions for Forecasting

*Time base partitions for forecasting* are two
disjoint, contiguous intervals of the time base; each interval contains time series data for
forecasting a dynamic model. The *forecast period* (forecast horizon)
is a `numperiods`

length partition at the end of the time base during
which the `forecast`

function generates the forecasts
`Y`

from the dynamic model `Mdl`

. The
*presample period* is the entire partition occurring before the
forecast period. The `forecast`

function can require observed
responses, innovations, or conditional variances in the presample period
(`Y0`

, `E0`

, and `V0`

, or
`Tbl1`

) to initialize the dynamic model for forecasting. The model
structure determines the types and amounts of required presample observations.

A common practice is to fit a dynamic model to a portion of the data set, and then validate the predictability of the model by comparing its forecasts to observed responses. During forecasting, the presample period contains the data to which the model is fit, and the forecast period contains the holdout sample for validation. Suppose that *y _{t}* is an observed response series;

*x*

_{1,t},

*x*

_{2,t}, and

*x*

_{3,t}are observed exogenous series; and time

*t*= 1,…,

*T*. Consider forecasting responses from a dynamic model of

*y*

_{t}containing a regression component with

`numperiods`

= *K*periods. Suppose that the dynamic model is fit to the data in the interval [1,

*T*–

*K*] (for more details, see

`estimate`

). This figure shows the time base partitions for forecasting.For example, to generate the forecasts `Y`

from an ARX(2) model, `forecast`

requires:

Presample responses

`Y0`

= $${\left[\begin{array}{cc}{y}_{T-K-1}& {y}_{T-K}\end{array}\right]}^{\prime}$$ to initialize the model. The 1-period-ahead forecast requires both observations, whereas the 2-periods-ahead forecast requires*y*_{T – K}and the 1-period-ahead forecast`Y(1)`

. The`forecast`

function generates all other forecasts by substituting previous forecasts for lagged responses in the model.Future exogenous data

`XF`

= $$\left[\begin{array}{ccc}{x}_{1,\left(T-K+1\right):T}& {x}_{2,\left(T-K+1\right):T}& {x}_{3,\left(T-K+1\right):T}\end{array}\right]$$ for the model regression component. Without specified future exogenous data, the`forecast`

function ignores the model regression component, which can yield unrealistic forecasts.

Dynamic models containing either a moving average component or a conditional variance model can require presample innovations or conditional variances. Given enough presample responses, `forecast`

infers the required presample innovations and conditional variances. If such a model also contains a regression component, then `forecast`

must have enough presample responses and exogenous data to infer the required presample innovations and conditional variances. This figure shows the arrays of required observations for this case, with corresponding input and output arguments.

## Algorithms

The

`forecast`

function sets the number of sample paths (`numpaths`

) to the maximum number of columns among the specified presample data sets:For input numeric arrays of presample data,

`numpaths`

is the maximum width among`E0`

,`V0`

, and`Y0`

.For an input table or timetable of presample data,

`numpaths`

is the maximum width among the variables representing the presample responses`PresampleResponseVariable`

, innovations`PresampleInnovationVariable`

, and conditional variances`PresampleVarianceVariable`

.

All specified presample data sets must have either one column or

`numpaths`

> 1 columns. Otherwise,`forecast`

issues an error. For example, if you supply`Y0`

and`E0`

, and`Y0`

has five columns representing five paths, then`E0`

can have one column or five columns. If`E0`

has one column,`forecast`

applies`E0`

to each path.`NaN`

values in presample and future data sets indicate missing data. For input numeric arrays,`forecast`

removes missing data from the presample data sets following this procedure:`forecast`

horizontally concatenates the specified presample data sets`Y0`

,`E0`

,`V0`

, and`X0`

so that the latest observations occur simultaneously. The result can be a jagged array because the presample data sets can have a different number of rows. In this case,`forecast`

prepads variables with an appropriate number of zeros to form a matrix.`forecast`

applies listwise deletion to the combined presample matrix by removing all rows containing at least one`NaN`

.`forecast`

extracts the processed presample data sets from the result of step 2, and removes all prepadded zeros.

`forecast`

applies a similar procedure to the forecasted predictor data`XF`

. After`forecast`

applies listwise deletion to`XF`

, the result must have at least`numperiods`

rows. Otherwise,`forecast`

issues an error.List-wise deletion reduces the sample size and can create irregular time series.

`forecast`

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

`forecast`

computes the MSEs`YMSE`

of the conditional mean forecasts`Y`

, the function treats the specified predictor data sets as exogenous, nonstochastic, and statistically independent of the model innovations. Therefore,`YMSE`

reflects only the variance associated with the ARIMA component of the input model`Mdl`

.

## References

[1] Baillie, Richard T., and Tim Bollerslev. “Prediction in Dynamic Models with Time-Dependent Conditional Variances.” *Journal of Econometrics* 52, (April 1992): 91–113. https://doi.org/10.1016/0304-4076(92)90066-Z.

[2] Bollerslev, Tim. “Generalized Autoregressive Conditional Heteroskedasticity.” *Journal of Econometrics* 31 (April 1986): 307–27. https://doi.org/10.1016/0304-4076(86)90063-1.

[3] Bollerslev, Tim. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” *The Review of Economics and Statistics* 69 (August 1987): 542–47. https://doi.org/10.2307/1925546.

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

[5] Enders, Walter. *Applied Econometric Time Series*. Hoboken, NJ: John Wiley & Sons, Inc., 1995.

[6] Engle, Robert. F. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” *Econometrica* 50 (July 1982): 987–1007. https://doi.org/10.2307/1912773.

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

## Version History

**Introduced in R2012a**

### R2023b: `forecast`

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

In addition to accepting input presample and in-sample data in numeric arrays,
`forecast`

accepts input data in tables or regular timetables. Use
`Tbl1`

to supply presample data and `InSample`

to
provide in-sample (future) predictor data for the forecast horizon.

When you supply data in a table or timetable, the following conditions apply:

`forecast`

chooses the default presample response series on which to operate, but you can use the optional`PresampleResponseVariable`

name-value argument to select a different variable.`forecast`

returns results in a table or timetable.

Name-value arguments to support tabular workflows include:

`PresampleResponseVariable`

specifies the variable name of the presample response paths in the input presample data`Tbl1`

to initialize the response series for the forecast.`PresampleInnovationVariable`

specifies the variable name of the innovation paths in the input presample data`Tbl1`

to initialize the model for the forecast.`PresampleVarianceVariable`

specifies the variable name of the conditional variance paths in the input presample data`Tbl1`

to initialize the conditional variance series for the forecast.`PresamplePredictorVariables`

specifies the variable names of the predictor data in the input presample data`Tbl1`

for the model exogenous regression component.`PredictorVariables`

specifies the variable names of the predictor data in the input in-sample data`InSample`

for the model exogenous regression component in the forecast horizon.

### R2019a: Univariate time series models require specification of presample response data to forecast responses

The `forecast`

function now has a third input argument for you to
supply presample response data.

forecast(Mdl,numperiods,Y0) forecast(Mdl,numperiods,Y0,Name,Value)

Before R2019a, the syntaxes were:

forecast(Mdl,numperiods) forecast(Mdl,numperiods,Name,Value)

`'Y0'`

name-value argument.There are no plans to remove the previous syntaxes or the `'Y0'`

name-value argument at this time. However, you are encouraged to supply presample responses
because, to forecast responses from a dynamic model, `forecast`

must
initialize models containing lagged responses. Without specified presample responses,
`forecast`

initializes models by using reasonable default values,
but these values might not support all workflows.

For stationary models without a regression component, all presample responses are the unconditional mean of the process, by default.

For nonstationary models or models containing a regression component, all presample responses are

`0`

, by default.

**Update Code**

Update your code by specifying presample responses in the third input argument.

If you do not supply presample responses, then `forecast`

provides
default presample values that might not support all workflows.

## See Also

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