# summarize

Display ARIMA model estimation results

## Description

example

summarize(Mdl) displays a summary of the ARIMA model Mdl.

• If Mdl is an estimated model returned by estimate, then summarize prints estimation results to the MATLAB® Command Window. The display includes an estimation summary and a table of parameter estimates with corresponding standard errors, t statistics, and p-values. The estimation summary includes fit statistics, such as the Akaike Information Criterion (AIC), and the estimated innovations variance.

• If Mdl is an unestimated model returned by arima, then summarize prints the standard object display (the same display that arima prints during model creation).

example

results = summarize(Mdl) returns one of the following variables and does not print to the Command Window.

• If Mdl is an estimated model, then results is a structure containing estimation results.

• If Mdl is an unestimated model, then results is an arima model object that is equal to Mdl.

## Input Arguments

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

## Output Arguments

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Model summary, returned as a structure array or an arima model object.

• If Mdl is an estimated model, then results is a structure array containing the fields in this table.

FieldDescription
DescriptionModel summary description (string)
SampleSizeEffective sample size (numeric scalar)
NumEstimatedParametersNumber of estimated parameters (numeric scalar)
LogLikelihoodOptimized loglikelihood value (numeric scalar)
AICAkaike Information Criterion (numeric scalar)
BICBayesian Information Criterion (numeric scalar)
TableMaximum likelihood estimates of the model parameters with corresponding standard errors, t statistics (estimate divided by standard error), and p-values (assuming normality); a table with rows corresponding to model parameters
VarianceTable

Maximum likelihood estimate of the model variance with corresponding standard errors, t statistics (estimate divided by standard error), and p-values (assuming normality).

If Mdl.Variance is constant, then VarianceTable is a table containing one row.

If Mdl.Variance is an estimated conditional variance model (for example, a garch model), then VarianceTable is a table whose rows correspond to estimated variance model parameters.

• If Mdl is an unestimated model, then results is an arima model object that is equal to Mdl.

## Examples

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Print the results from estimating an ARMA model using simulated data.

Simulate data from an ARMA(1,1) model using known parameter values.

MdlSim = arima('Constant',0.01,'AR',0.8,'MA',0.14,...
'Variance',0.1);
rng 'default';
Y = simulate(MdlSim,100);

Fit an ARMA(1,1) model to the simulated data, turning off the print display.

Mdl = arima(1,0,1);
EstMdl = estimate(Mdl,Y,'Display','off');

Print the estimation results.

summarize(EstMdl)

ARIMA(1,0,1) Model (Gaussian Distribution)

Effective Sample Size: 100
Number of Estimated Parameters: 4
LogLikelihood: -41.296
AIC: 90.592
BIC: 101.013

Value      StandardError    TStatistic      PValue
________    _____________    __________    __________

Constant    0.044537      0.046038        0.96741         0.33334
AR{1}        0.82289      0.071163         11.563      6.3104e-31
MA{1}        0.12032       0.10182         1.1817         0.23731
Variance     0.13373      0.017879         7.4794       7.466e-14

Load the NASDAQ data included with Econometrics™ toolbox. Convert the daily close composite index series to a return series. For numerical stability, convert the returns to percentage returns. Specify an AR(1) and GARCH(1,1) composite model. This is a model of the form

${r}_{t}=c+{\varphi }_{1}{r}_{t-1}+{\epsilon }_{t},$

where ${\epsilon }_{t}={\sigma }_{t}{z}_{t}$,

${\sigma }_{t}^{2}=\kappa +{\gamma }_{1}{\sigma }_{t-1}^{2}+{\alpha }_{1}{\epsilon }_{t-1}^{2},$

and ${z}_{t}$ is an independent and identically distributed standardized Gaussian process.

nasdaq = DataTable.NASDAQ;
r = 100*price2ret(nasdaq);
T = length(r);

Mdl = arima('ARLags',1,'Variance',garch(1,1));

Fit the model Mdl to the return series r by using estimate. Use the presample observations that estimate chooses by default.

EstMdl = estimate(Mdl,r,'Display','params');

ARIMA(1,0,0) Model (Gaussian Distribution):

Value      StandardError    TStatistic      PValue
________    _____________    __________    __________

Constant    0.072632      0.018047         4.0245      5.7085e-05
AR{1}        0.13816      0.019893          6.945      3.7847e-12

GARCH(1,1) Conditional Variance Model (Gaussian Distribution):

Value      StandardError    TStatistic      PValue
________    _____________    __________    __________

Constant    0.022377      0.0033201        6.7399      1.5852e-11
GARCH{1}     0.87312      0.0091019        95.928               0
ARCH{1}      0.11865       0.008717        13.611      3.4339e-42

Create a variable named results that contains the estimation results by using summarize.

results = summarize(EstMdl)
results = struct with fields:
Description: "ARIMA(1,0,0) Model (Gaussian Distribution)"
SampleSize: 3027
NumEstimatedParameters: 5
LogLikelihood: -4.7414e+03
AIC: 9.4929e+03
BIC: 9.5230e+03
Table: [2x4 table]
VarianceTable: [3x4 table]

Extract the parameter estimate summary tables from the estimation results structure array by using dot notation. The Table field contains the conditional mean model parameter estimates and inferences. The VarianceTable field contains the conditional variance model parameter estimates and inferences.

meanEstTbl = results.Table
meanEstTbl=2×4 table
Value      StandardError    TStatistic      PValue
________    _____________    __________    __________

Constant    0.072632      0.018047         4.0245      5.7085e-05
AR{1}        0.13816      0.019893          6.945      3.7847e-12

varianceEstTbl = results.VarianceTable
varianceEstTbl=3×4 table
Value      StandardError    TStatistic      PValue
________    _____________    __________    __________

Constant    0.022377      0.0033201        6.7399      1.5852e-11
GARCH{1}     0.87312      0.0091019        95.928               0
ARCH{1}      0.11865       0.008717        13.611      3.4339e-42

### Functions

Introduced in R2018a