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EstMdl = estimate(Mdl,y)
[EstMdl,EstParamCov,logL,info]
= estimate(Mdl,y)
[EstMdl,EstParamCov,logL,info] = estimate(Mdl,y,Name,Value)
uses
maximum likelihood to estimate the parameters of the ARIMA(p,D,q)
model EstMdl
= estimate(Mdl
,y
)Mdl
given the observed univariate time
series y
. EstMdl
is an arima
model that stores the
results.
[EstMdl,
additionally returns EstParamCov
,logL
,info
]
= estimate(Mdl,y)EstParamCov
,
the variancecovariance matrix associated with estimated parameters, logL
,
the optimized loglikelihood objective function, and info
,
a data structure of summary information.
[EstMdl,EstParamCov,logL,info] = estimate(Mdl,y,
estimates
the model with additional options specified by one or more Name,Value
)Name,Value
pair
arguments.
Mdl
— ARIMA or ARIMAX modelarima
modelARIMA or ARIMAX model, specified as an arima
model returned by arima
or estimate
.
estimate
treats nonNaN
elements
in Mdl
as equality constraints and does not estimate
the corresponding parameters.
y
— Single path of response dataSingle path of response data to which the model is fit, specified
as a numeric column vector. The last observation of y
is
the latest.
Data Types: double
Specify optional commaseparated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.
'AR0'
— Initial estimates of nonseasonal autoregressive coefficientsInitial estimates of the nonseasonal autoregressive coefficients
for the ARIMA model, specified as the commaseparated pair consisting
of 'AR0'
and a numeric vector.
The number of coefficients in AR0
must equal
the number of lags associated with nonzero coefficients in the nonseasonal
autoregressive polynomial, ARLags
.
By default, estimate
derives initial estimates
using standard time series techniques.
Data Types: double
'Beta0'
— Initial estimates of regression coefficientsInitial estimates of regression coefficients for the regression
component, specified as the commaseparated pair consisting of 'Beta0'
and
a numeric vector.
The number of coefficients in Beta0
must
equal the number of columns of X
.
By default, estimate
derives initial estimates
using standard time series techniques.
Data Types: double
'Constant0'
— Initial ARIMA model constant estimateInitial ARIMA model constant estimate, specified as the commaseparated
pair consisting of 'Constant0'
and a scalar.
By default, estimate
derives initial estimates
using standard time series techniques.
Data Types: double
'Display'
— Command Window display option'params'
(default)  'diagnostics'
 'full'
 'iter'
 'off'
 string vector  cell vector of character vectorsCommand Window display option, specified as the commaseparated
pair consisting of 'Display'
and a value or any
combination of values in this table.
Value  estimate Displays 

'diagnostics'  Optimization diagnostics 
'full'  Maximum likelihood parameter estimates, standard errors, t statistics, iterative optimization information, and optimization diagnostics 
'iter'  Iterative optimization information 
'off'  No display in the Command Window 
'params'  Maximum likelihood parameter estimates, standard errors, and t statistics 
For example:
To run a simulation where you are fitting many models,
and therefore want to suppress all output, use 'Display','off'
.
To display all estimation results and the optimization
diagnostics, use 'Display',{'params','diagnostics'}
.
Data Types: char
 cell
 string
'DoF0'
— Initial tdistribution degreesoffreedom parameter estimate10
(default)  positive scalarInitial tdistribution degreesoffreedom
parameter estimate, specified as the commaseparated pair consisting
of 'DoF0'
and a positive scalar. DoF0
must
exceed 2.
Data Types: double
'E0'
— Presample innovationsPresample innovations that have mean 0 and provide initial values
for the ARIMA(p,Dq)
model, specified as the commaseparated pair consisting of 'E0'
and
a numeric column vector.
E0
must contain at least Mdl.Q
rows.
If you use a conditional variance model, such as a garch
model,
then the software might require more than Mdl.Q
presample
innovations.
If E0
contains extra rows, then estimate
uses the latest Mdl.Q
presample innovations. The
last row contains the latest presample innovation.
By default, estimate
sets the necessary presample
innovations to 0
.
Data Types: double
'MA0'
— Initial estimates of nonseasonal moving average coefficientsInitial estimates of nonseasonal moving average coefficients
for the ARIMA(p,Dq)
model, specified as the commaseparated pair consisting of 'MA0'
and
a numeric vector.
The number of coefficients in MA0
must equal
the number of lags associated with nonzero coefficients in the nonseasonal
moving average polynomial, MALags
.
By default, estimate
derives initial estimates
using standard time series techniques.
Data Types: double
'Options'
— Optimization optionsoptimoptions
optimization controller  optimset
optimization controllerOptimization options, specified as the commaseparated pair
consisting of 'Options'
and an optimoptions
or optimset
optimization
controller. For details on altering the default values of the optimizer,
see optimoptions
, optimset
, or fmincon
in Optimization Toolbox™.
For example, to change the constraint tolerance to 1e6
,
set Options = optimoptions(@fmincon,'ConstraintTolerance',1e6,'Algorithm','sqp')
.
Then, pass Options
into estimate
using 'Options',Options
.
By default, estimate
uses the same default
options as fmincon
, except Algorithm
is 'sqp'
and ConstraintTolerance
is 1e7
.
'SAR0'
— Initial estimates of seasonal autoregressive coefficientsInitial estimates of seasonal autoregressive coefficients for
the ARIMA(p,Dq)
model, specified as the commaseparated pair consisting of 'SAR0'
and
a numeric vector.
The number of coefficients in SAR0
must equal
the number of lags associated with nonzero coefficients in the seasonal
autoregressive polynomial, SARLags
.
By default, estimate
derives initial estimates
using standard time series techniques.
Data Types: double
'SMA0'
— Initial estimates of seasonal moving average coefficientsInitial estimates of seasonal moving average coefficients for
the ARIMA(p,Dq)
model, specified as the commaseparated pair consisting of 'SMA0'
and
a vector.
The number of coefficients in SMA0
must equal
the number of lags with nonzero coefficients in the seasonal moving
average polynomial, SMALags
.
By default, estimate
derives initial estimates
using standard time series techniques.
Data Types: double
'V0'
— Presample conditional variancesPresample conditional variances that provide initial values
for any conditional variance model, specified as the commaseparated
pair consisting of 'V0'
and a numeric column vector
with positive entries.
The software requires V0
to have at least
the number of observations required to initialize the variance model.
If the number of rows in V0
exceeds the number
necessary, then estimate
only uses the latest observations.
The last row contains the latest observation.
If the variance of the model is constant, then V0
is
unnecessary.
By default, estimate
sets the necessary presample
conditional variances to the average of the squared inferred residuals.
Data Types: double
'Variance0'
— Initial estimates of variances of innovationsInitial estimates of variances of innovations for the ARIMA(p,Dq)
model, specified as the commaseparated pair consisting of 'Variance0'
and
a positive scalar or a cell vector of positive scalars. If Variance0
is
a cell vector, then the conditional variance model must recognize
the parameter names as valid coefficients.
By default, estimate
derives initial estimates
using standard time series techniques.
Data Types: double
 cell
'X'
— Exogenous predictorsExogenous predictors in the regression model, specified as the
commaseparated pair consisting of 'X'
and a matrix.
The columns of X
are separate, synchronized
time series, with the last row containing the latest observations.
If you do not specify Y0
, then the number
of rows of X
must be at least numel(y,2)
+ Mdl.P
. Otherwise, the number of rows of X
should
be at least the length of y
.
If the number of rows of X
exceeds the number
necessary, then estimate
uses the latest observations
and synchronizes X
with the response series y
.
By default, estimate
does not estimate the
regression coefficients regardless of their presence in Mdl
.
Data Types: double
'Y0'
— Presample response dataPresample response data that provides initial values for the
ARIMA(p,Dq)
model, specified as the commaseparated pair consisting of 'Y0'
and
a numeric column vector.
Y0
is a column vector with at least Mdl.P
rows.
If the number of rows in Y0
exceeds Mdl.P
, estimate
only
uses the latest Mdl.P
observations. The last row
contains the latest observation.
By default, estimate
backward forecasts for
the necessary amount of presample observations.
Data Types: double
Notes

EstMdl
— Model containing parameter estimatesarima
modelModel containing parameter estimates, returned as an arima
model. estimate
uses
maximum likelihood to calculate all parameter estimates not constrained
by Mdl
(that is, all parameters in Mdl
that
you set to NaN
).
EstParamCov
— Variancecovariance matrix of maximum likelihood estimatesVariancecovariance matrix of maximum likelihood estimates of model parameters known to the optimizer, returned as a matrix.
The rows and columns contain the covariances of the parameter estimates. The standard errors of the parameter estimates are the square root of the entries along the main diagonal.
The rows and columns associated with any parameters held fixed
as equality constraints contain 0
s.
estimate
uses the outer product of gradients
(OPG) method to perform covariance matrix estimation.
estimate
orders the parameters in EstParamCov
as
follows:
Constant
Nonzero AR
coefficients at positive
lags
Nonzero SAR
coefficients at positive
lags
Nonzero MA
coefficients at positive
lags
Nonzero SMA
coefficients at positive
lags
Regression coefficients (when you specify X
in estimate
)
Variance parameters (scalar for constantvariance models, vector of additional parameters otherwise)
Degrees of freedom (t innovation distribution only)
Data Types: double
logL
— Optimized loglikelihood objective function valueOptimized loglikelihood objective function value, returned as a scalar.
Data Types: double
info
— Summary informationSummary information, returned as a structure.
Field  Description 

exitflag  Optimization exit flag (see fmincon in Optimization Toolbox) 
options  Optimization options controller (see optimoptions and fmincon in Optimization Toolbox) 
X  Vector of final parameter estimates 
X0  Vector of initial parameter estimates 
For example, you can display the vector of final estimates by
typing info.X
in the Command Window.
Data Types: struct
Fit an ARMA(2,1) model to simulated data.
Simulate 500 data points from the ARMA(2,1) model
where follows a Gaussian distribution with mean 0 and variance 0.1.
Mdl = arima('AR',{0.5,0.3},'MA',0.2,... 'Constant',0,'Variance',0.1); rng(5); % For reproducibility y = simulate(Mdl,500);
The simulated data is stored in the column vector Y
.
Specify an ARMA(2,1) model with no constant and unknown coefficients and variance.
ToEstMdl = arima(2,0,1); ToEstMdl.Constant = 0
ToEstMdl = ARIMA(2,0,1) Model:  Distribution: Name = 'Gaussian' P: 2 D: 0 Q: 1 Constant: 0 AR: {NaN NaN} at Lags [1 2] SAR: {} MA: {NaN} at Lags [1] SMA: {} Variance: NaN
Fit the ARMA(2,1) model to y
.
EstMdl = estimate(ToEstMdl,y);
ARIMA(2,0,1) Model:  Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic     Constant 0 Fixed Fixed AR{1} 0.494039 0.103213 4.78662 AR{2} 0.25348 0.0699297 3.62478 MA{1} 0.279583 0.107211 2.60778 Variance 0.100092 0.0066403 15.0734
The result is a new arima
model called EstMdl
. The estimates in EstMdl
resemble the parameter values that generated the simulated data.
Fit an integrated ARIMA(1,1,1) model to the daily close of the NASDAQ Composite Index.
Load the NASDAQ data included with the toolbox. Extract the first 1500 observations of the Composite Index (January 1990 to December 1995).
load Data_EquityIdx
nasdaq = DataTable.NASDAQ(1:1500);
Specify an ARIMA(1,1,1) model for fitting.
Mdl = arima(1,1,1);
The model is nonseasonal, so you can use shorthand syntax.
Fit the model to the first half of the data.
EstMdl = estimate(Mdl,nasdaq(1:750));
ARIMA(1,1,1) Model:  Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic     Constant 0.223401 0.184176 1.21297 AR{1} 0.114342 0.119438 0.957334 MA{1} 0.127636 0.119251 1.07031 Variance 18.9832 0.689994 27.5122
The result is a new arima
model (EstMdl
). The estimated parameters, their standard errors, and
statistics display in the Command Window.
Use the estimated parameters as initial values for fitting the second half of the data.
con0 = EstMdl.Constant; ar0 = EstMdl.AR{1}; ma0 = EstMdl.MA{1}; var0 = EstMdl.Variance; [EstMdl2,EstParamCov2,logL2,info2] = estimate(Mdl,.... nasdaq(751:end),'Constant0',con0,'AR0',ar0,... 'MA0',ma0,'Variance0',var0);
ARIMA(1,1,1) Model:  Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic     Constant 0.611438 0.326753 1.87125 AR{1} 0.15071 0.117818 1.27918 MA{1} 0.385688 0.109055 3.53665 Variance 36.4933 1.22699 29.7422
The parameter estimates are stored in the info
data structure. Display the final parameter estimates.
info2.X
ans = 0.6114 0.1507 0.3857 36.4933
Fit an ARIMAX model to a simulated time series without specifying initial values for the response or the parameters.
Define the ARIMAX(2,1,1) model
to eventually simulate a time series of length 500, where follows a Gaussian distribution with mean 0 and variance 0.1.
Mdl = arima('AR',{0.5,0.3},'MA',0.2,'D',1,... 'Constant',0,'Variance',0.1,'Beta',[1.5 2.6 0.3]); T = 500;
Simulate three stationary AR(1) series and presample values:
where follows a Gaussian distribution with mean 0 and variance 0.01 for i = {1,2,3}.
numObs = Mdl.P + T; MdlX1 = arima('AR',0.1,'Constant',0,'Variance',0.01); MdlX2 = arima('AR',0.2,'Constant',0,'Variance',0.01); MdlX3 = arima('AR',0.3,'Constant',0,'Variance',0.01); X1 = simulate(MdlX1,numObs); X2 = simulate(MdlX2,numObs); X3 = simulate(MdlX3,numObs); Xmat = [X1 X2 X3];
The simulated exogenous predictors are stored in the numObs
by3 matrix Xmat
.
Simulate 500 data points from the ARIMA(2,1,1) model.
y = simulate(Mdl,T,'X',Xmat);
The simulated response is stored in the column vector y
.
Create an ARIMA(2,1,1) model with known 0
valued constant and unknown coefficients and variance.
ToEstMdl = arima(2,1,1); ToEstMdl.Constant = 0
ToEstMdl = ARIMA(2,1,1) Model:  Distribution: Name = 'Gaussian' P: 3 D: 1 Q: 1 Constant: 0 AR: {NaN NaN} at Lags [1 2] SAR: {} MA: {NaN} at Lags [1] SMA: {} Variance: NaN
ToEstMdl
is an ARIMA(2,1,1) model. estimate
changes this designation to ARIMAX(2,1,1) when you pass the exogenous predictors into the X
argument. estimate
estimates all parameters with the value NaN
in ToEstMdl
.
Fit the ARIMAX(2,1,1) model to y
including regression matrix Xmat
.
EstMdl = estimate(ToEstMdl,y,'X',Xmat);
ARIMAX(2,1,1) Model:  Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic     Constant 0 Fixed Fixed AR{1} 0.416338 0.0460672 9.03763 AR{2} 0.274052 0.0406445 6.74265 MA{1} 0.334598 0.0572075 5.84885 Beta1 1.4194 0.142422 9.96619 Beta2 2.54199 0.133102 19.0981 Beta3 0.287669 0.14035 2.04965 Variance 0.0967773 0.00579104 16.7115
ToEstMdl
is a new arima
model designated as ARIMAX(2,1,1) since exogenous predictors enter the model. The estimates in ToEstMdl
resemble the parameter values that generated the simulated data.
Fit an ARIMAX model to a time series specifying initial values for the response and the parameters.
The Credit Defaults data set contains four variables:
Default rate on investmentgrade corporate bonds (IGD)
Percentage of investmentgrade bond issuers first rated 3 years ago (AGE)
Oneyearahead forecast of the change in corporate profits, adjusted for inflation (CPF)
Spread between corporate bond yields and those of comparable government bonds (SPR)
Assume that an ARIMAX(1,0,0) model is appropriate to fit IGD using AGE, CPF, and SPR as exogenous predictors. Load the Credit Defaults data set. Assign the response IGD to y
. Assign the predictors AGE, CPF, and SPR to the matrix X
.
load Data_CreditDefaults
X = Data(:,[1 3:4]);
T = size(X,1);
y = Data(:,5);
The response and exogenous predictor series should be stationary before you continue. If your response is not stationary, then specify the degree of integration in the arima
statement. If your exogenous predictors are not stationary, then you must difference them using diff
. The series in this example are stationary to not distract from its main purpose.
Separate the initial values from the main response and exogenous predictors. Choose initial values for the regression coefficients Beta0
.
y0 = y(1); yEst = y(2:T); XEst = X(2:end,:); Beta0 = [0.5 0.5 0.5];
y0
initializes the response series and yest
is the main response series for estimation. XEst
is the main exogenous predictor matrix for estimation.
Specify the model Mdl
to fit to the data.
Mdl = arima(1,0,0);
Fit the model to the data and specify the initial values.
EstMdl = estimate(Mdl,yEst,'X',XEst,... 'Y0',y0,'Beta0',Beta0);
ARIMAX(1,0,0) Model:  Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic     Constant 0.204768 0.266078 0.769582 AR{1} 0.0173111 0.565618 0.0306057 Beta1 0.0239329 0.0218416 1.09574 Beta2 0.0124602 0.00749915 1.66155 Beta3 0.0680874 0.074504 0.913876 Variance 0.00539462 0.00224392 2.4041
Suppose EstParamCov
is an estimated parameter
covariance matrix returned by estimate
. The software
sets the variances and covariances of parameters fixed during estimation
to 0
. Enter this command to count the number of
free parameters (numParams
) in a fitted model.
numParams = sum(any(EstParamCov))
This command counts the number of columns (or equivalently, rows) with any nonzero values.
[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
[2] Enders, W. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, 1995.
[3] Greene, W. H. Econometric Analysis. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1997.
[4] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
arima
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