forecast

Forecast univariate ARIMA or ARIMAX model responses or conditional variances

Syntax

[Y,YMSE]
= forecast(Mdl,numperiods,Y0)

[Y,YMSE,V]
= forecast(Mdl,numperiods,Y0)

Tbl2 = forecast(Mdl,numperiods,Tbl1)

[___] = forecast(___,Name=Value)

Description

[Y,YMSE] = forecast(Mdl,numperiods,Y0) returns the 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.

example

[Y,YMSE,V] = forecast(Mdl,numperiods,Y0) also forecasts a numperiods-by-1 numeric vector of conditional variances V from a composite conditional mean and variance model (for example, an ARIMA and GARCH composite model).

example

Tbl2 = forecast(Mdl,numperiods,Tbl1) returns the table or timetable 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.

example

[___] = forecast(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. 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).

example

Examples

collapse all

Forecast Conditional Mean Response Vector

Open Live Script

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

Figure contains an axes object. The axes object with title 30-Period Forecasts and 95% Confidence Intervals contains 4 objects of type line. These objects represent Observed, Forecast, 95% confidence interval.

Forecast NYSE Composite Index

Since R2023b

Open Live Script

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")

Figure contains an axes object. The axes object with title NYSE Daily Closing Prices: 1990 - 2001 contains an object of type line.

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")

Figure contains an axes object. The axes object with title NYSE Daily Closing Prices: 1990 - 2001 contains an object of type line.

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

Figure contains an axes object. The axes object with title NYSE Weekly Average Closing Price contains 4 objects of type line. These objects represent Observations, Forecasts, 95% forecast intervals.

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

Open Live Script

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

$y_{t} = 1 + 0.3 y_{t - 1} + 2 x_{t} + ε_{t},$

where $ε_{t}$ is a standard Gaussian random variable, and $x_{t}$ 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 $y_{0}$ 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(3) = 3.8232 is the 3-period-ahead forecast of the ARX(1) model.

Forecast Composite Conditional Mean and Variance Model

Since R2023b

Open Live Script

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.

$\begin{array}{l} y_{t} = 0.073 + 0.138 y_{t - 1} + ε_{t} \\ σ_{t}^{2} = 0.022 + 0.873 σ_{t - 1}^{2} + 0.119 ε_{t - 1}, \end{array}$

where $ε_{t}$ 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 contains an axes object. The axes object with title NASDAQ Weekly Average Percent Return Series contains 2 objects of type line. These objects represent Observed, Forecasted.

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

Figure contains an axes object. The axes object with title Conditional Variance Series contains 2 objects of type line. These objects represent Observed, Forecasted.

Forecast Multiple Paths

Open Live Script

Forecast multiple response and conditional variance paths from a known composite conditional mean and variance model: a SAR $(1, 0, 0) {(1, 1, 0)}_{4}$ 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:

$σ_{t}^{2} = 0.1 + 0.2 ε_{t}^{2} .$

Create an arima model object that represents this quarterly SAR $(1, 0, 0) {(1, 1, 0)}_{4}$ model:

$(1 - 0.5 L) (1 - 0.2 L^{4}) (1 - L^{4}) y_{t} = 1 + ε_{t},$

where $ε_{t}$ 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 $t$ -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

Figure contains an axes object. The axes object with title Response paths, xlabel Time (quarters) contains 11 objects of type patch, line. This object represents Forecast period.

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

Figure contains an axes object. The axes object with title Conditional Variance Paths, xlabel Time (quarters) contains 11 objects of type patch, line. This object represents Forecast period.

Input Arguments

collapse all

`Mdl` — Fully specified ARIMA model
`arima` model object

Fully specified ARIMA model, specified as an arima model object created by arima or estimate.

The properties of Mdl cannot contain NaN values.

`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

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_t, conditional variances σ_t², or predictors x_t, to initialize the model, specified as a table or timetable with 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 responses
At 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

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_t²
positive numeric column vector | positive numeric matrix

Presample conditional variances σ_t² 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

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(j) = true selects variable j from Tbl1.Properties.VariableNames

The selected variable must be a numeric vector and cannot contain missing values (NaNs).

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

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(j) = true selects variable j from Tbl1.Properties.VariableNames

The selected variable must be a numeric matrix and cannot contain missing values (NaNs).

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_t² to select from `Tbl1`
string scalar | character vector | integer | logical vector

Presample conditional variance variable σ_t² 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(j) = true selects variable j from Tbl1.Properties.VariableNames

The selected variable must be a numeric vector and cannot contain missing values (NaNs).

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

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(j) = true selects variable j from Tbl1.Properties.VariableNames

The selected variables must be numeric vectors and cannot contain missing values (NaNs).

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(t,:) contains the t-period-ahead predictor forecasts. 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 (NaNs). forecast applies the specified predictor variables to all forecasted paths.

If InSample is a timetable, the following conditions apply:

InSample must represent a sample with a regular datetime time step (see isregular).
The datetime vector InSample.Time must be ascending or descending.
Tbl1 must immediately precede InSample, with respect to the sampling frequency.

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

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(j) = true selects variable j from InSample.Properties.VariableNames

The selected variables must be numeric vectors and cannot contain missing values (NaNs).

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

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`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(t,:) contains the t-period-ahead forecasts, or the conditional mean forecast of all paths for time point t in the forecast period.

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(t,:) contains the forecast error variances of all paths for time point t in the forecast period.

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 j contains the conditional variance forecasts of period j. Otherwise, V is a matrix composed of the constant Mdl.Variance.

Data Types: double

`Tbl2` — Paths of MMSE forecasts of responses y_t, corresponding forecast MSEs, and MMSE forecasts of conditional variances σ_t² of future model innovations ε_t
table | timetable

Since R2023b

Paths of MMSE forecasts of responses y_t, corresponding forecast MSEs, and MMSE forecasts of conditional variances σ_t² of future model innovations ε_t, returned as a table or timetable, the same data type as 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 responseName_Response, where responseName is 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.responseName_Response represents the continuation of the corresponding presample response path in Tbl1 (Tbl2.responseName_Response(1,:) occurs in the next time point, with respect to the periodicity Tbl1, after the last presample response). Tbl2.responseName_Response(j,k) contains the j-period-ahead forecasted response of path 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.responseName_Response. forecast names the forecast MSEs responseName_MSE, where responseName is 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 responseName_Variance, where responseName is 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.responseName_Variance represents a continuation of the presample conditional variance process, either supplied by Tbl1 or set by default (Tbl2.responseName_Variance(1,:) occurs in the next time point, with respect to the periodicity Tbl1, after the last presample conditional variance). Tbl2.responseName_Variance(j,k) contains the j-period-ahead forecasted conditional variance of path 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

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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 = ${[\begin{matrix} y_{T - K - 1} & y_{T - K} \end{matrix}]}^{'}$ 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 = $[\begin{matrix} x_{1, (T - K + 1) : T} & x_{2, (T - K + 1) : T} & x_{3, (T - K + 1) : T} \end{matrix}]$ 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:
1. 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.
2. forecast applies listwise deletion to the combined presample matrix by removing all rows containing at least one NaN.
3. 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

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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)

You could optionally supply presample responses using the '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.

forecast

Syntax

Description

Examples

Forecast Conditional Mean Response Vector

Forecast NYSE Composite Index

Forecast ARX Model

Forecast Composite Conditional Mean and Variance Model

Forecast Multiple Paths

Input Arguments

Mdl — Fully specified ARIMA model arima model object

numperiods — Forecast horizon positive integer

Y0 — Presample response data yt numeric column vector | numeric matrix

Tbl1 — Presample data table | timetable

Name-Value Arguments

E0 — Presample innovations εt numeric column vector | numeric matrix

V0 — Presample conditional variances σt2 positive numeric column vector | positive numeric matrix

PresampleResponseVariable — Response variable yt to select from Tbl1 string scalar | character vector | integer | logical vector

PresampleInnovationVariable — Presample innovation variable of εt to select from Tbl1 string scalar | character vector | integer | logical vector

PresampleVarianceVariable — Presample conditional variance variable σt2 to select from Tbl1 string scalar | character vector | integer | logical vector

X0 — Presample predictor data numeric matrix

PresamplePredictorVariables — Presample exogenous predictor variables xt to select from Tbl1 string vector | cell vector of character vectors | vector of integers | logical vector

XF — Forecasted (or future) predictor data numeric matrix

InSample — Forecasted (future) predictor data table | timetable

PredictorVariables — Exogenous predictor variables xt to select from InSample string vector | cell vector of character vectors | vector of integers | logical vector

Output Arguments

Y — Minimum mean square error (MMSE) conditional mean forecasts numeric column vector | numeric matrix

YMSE — MSE of forecasted responses numeric column vector | numeric matrix

V — MMSE forecasts of conditional variances of future model innovations numeric column vector | numeric matrix

Tbl2 — Paths of MMSE forecasts of responses yt, corresponding forecast MSEs, and MMSE forecasts of conditional variances σt2 of future model innovations εt table | timetable

More About

Time Base Partitions for Forecasting

Algorithms

References

Version History

R2023b: forecast accepts input data in tables and timetables, and returns results in tables and timetables

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

See Also

Objects

Functions

Topics

`Mdl` — Fully specified ARIMA model
`arima` model object

`numperiods` — Forecast horizon
positive integer

`Y0` — Presample response data y_t
numeric column vector | numeric matrix

`Tbl1` — Presample data
table | timetable

`E0` — Presample innovations ε_t
numeric column vector | numeric matrix

`V0` — Presample conditional variances σ_t²
positive numeric column vector | positive numeric matrix

`PresampleResponseVariable` — Response variable y_t to select from `Tbl1`
string scalar | character vector | integer | logical vector

`PresampleInnovationVariable` — Presample innovation variable of ε_t to select from `Tbl1`
string scalar | character vector | integer | logical vector

`PresampleVarianceVariable` — Presample conditional variance variable σ_t² to select from `Tbl1`
string scalar | character vector | integer | logical vector

`X0` — Presample predictor data
numeric matrix

`PresamplePredictorVariables` — Presample exogenous predictor variables x_t to select from `Tbl1`
string vector | cell vector of character vectors | vector of integers | logical vector

`XF` — Forecasted (or future) predictor data
numeric matrix

`InSample` — Forecasted (future) predictor data
table | timetable

`PredictorVariables` — Exogenous predictor variables x_t to select from `InSample`
string vector | cell vector of character vectors | vector of integers | logical vector

`Y` — Minimum mean square error (MMSE) conditional mean forecasts
numeric column vector | numeric matrix

`YMSE` — MSE of forecasted responses
numeric column vector | numeric matrix

`V` — MMSE forecasts of conditional variances of future model innovations
numeric column vector | numeric matrix

`Tbl2` — Paths of MMSE forecasts of responses y_t, corresponding forecast MSEs, and MMSE forecasts of conditional variances σ_t² of future model innovations ε_t
table | timetable

R2023b: `forecast` accepts input data in tables and timetables, and returns results in tables and timetables