infer

Infer ARIMA or ARIMAX model residuals or conditional variances

Syntax

[E,V] = infer(Mdl,Y)
[E,V,logL] = infer(Mdl,Y)
[E,V,logL] = infer(Mdl,Y,Name,Value)


Description

[E,V] = infer(Mdl,Y) infers residuals and conditional variances of a univariate ARIMA model fit to data Y.

[E,V,logL] = infer(Mdl,Y) additionally returns the loglikelihood objective function values.

[E,V,logL] = infer(Mdl,Y,Name,Value) infers the ARIMA or ARIMAX model residuals and conditional variances, and returns the loglikelihood objective function values, with additional options specified by one or more Name,Value pair arguments.

Input Arguments

Mdl — Fully specified ARIMA or ARIMAX model

arima model

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

The properties of Mdl cannot contain NaNs.

Y — Response data

numeric column vector | numeric matrix

Response data, specified as a numeric column vector or numeric matrix. If Y is a matrix, then it has numObs observations and numPaths separate, independent paths.

infer infers the residuals and variances of Y. Y represents the time series characterized by Mdl, and it is the continuation of the presample series Y0.

• If Y is a column vector, then it represents one path of the underlying series.

• If Y is a matrix, then it represents numObs observations of numPaths paths of an underlying time series.

infer assumes that observations across any row occur simultaneously. The last observation of any series is the latest.

Data Types: double

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'E0' — Presample innovations

0 (default) | numeric column vector | numeric matrix

Presample innovations that have mean 0 and provide initial values for the model, specified as the comma-separated pair consisting of 'E0' and a numeric column vector or numeric matrix.

E0 must contain at least numPaths columns and enough rows to initialize the ARIMA model and any conditional variance model. That is, E0 must contain at least Mdl.Q innovations, but can be greater if you use a conditional variance model. If the number of rows in E0 exceeds the number necessary, then infer only uses the latest observations. The last row contains the latest observation.

If the number of columns exceeds numPaths, then infer only uses the first numPaths columns. If E0 is a column vector, then infer applies it to each inferred path.

Data Types: double

'V0' — Presample conditional variances

numeric column vector | numeric matrix

Presample conditional variances providing initial values for any conditional variance model, specified as the comma-separated pair consisting of 'V0' and a numeric column vector or matrix with positive entries.

V0 must contain at least numPaths columns and enough rows to initialize the variance model. If the number of rows in V0 exceeds the number necessary, then infer only uses the latest observations. The last row contains the latest observation.

If the number of columns exceeds numPaths, then infer only uses the first numPaths columns. If V0 is a column vector, then infer applies it to each inferred path.

By default, infer sets the necessary observations to the unconditional variance of the conditional variance process.

Data Types: double

'X' — Exogenous predictor data

numeric matrix

Exogenous predictor data for the regression component, specified as the comma-separated 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 numObs + Mdl.P. Otherwise, the number of rows of X should be at least numObs. In either case, if the number of rows of X exceeds the number necessary, then infer uses only the latest observations.

By default, the conditional mean model does not have a regression coefficient.

Data Types: double

'Y0' — Presample response data

numeric column vector | numeric matrix

Presample response data that provides initial values for the model, specified as the comma-separated pair consisting of 'Y0' and a numeric column vector or numeric matrix. Y0 must contain at least Mdl.P rows and numPaths columns. If the number of rows in Y0 exceeds Mdl.P, then infer only uses the latest Mdl.P observations. The last row contains the latest observation. If the number of columns exceeds numPaths, then infer only uses the first numPaths columns. If Y0 is a column vector, then infer applies it to each inferred path.

By default, infer backcasts to obtain the necessary observations.

Data Types: double

Notes

• NaNs indicate missing values and infer removes them. The software merges the presample data and main data sets separately, then uses list-wise deletion to remove any NaNs. That is, infer sets PreSample = [Y0 E0 V0] and Data = [Y X], then it removes any row in PreSample or Data that contains at least one NaN.

• The removal of NaNs in the main data reduces the effective sample size. Such removal can also create irregular time series.

• infer assumes that you synchronize the response and predictor series such that the latest observation of each occurs simultaneously. The software also assumes that you synchronize the presample series similarly.

• The software applies all exogenous series in X to each response series in Y.

Output Arguments

E — Inferred residuals

numeric matrix

Inferred residuals, returned as a numeric matrix. E has numObs rows and numPaths columns.

V — Inferred conditional variances

numeric matrix

Inferred conditional variances, returned as a numeric matrix. V has numObs rows and numPaths columns.

logL — Loglikelihood objective function values

numeric vector

Loglikelihood objective function values associated with the model Mdl, returned as a numeric vector. logL has numPaths elements associated with the corresponding path in Y.

Data Types: double

Examples

Infer Residuals

Infer residuals from an AR model.

Specify an AR(2) model using known parameters.

Mdl = arima('AR',{0.5,-0.8},'Constant',0.002,...
	'Variance',0.8);

Simulate response data with 102 observations.

rng 'default';
Y = simulate(Mdl,102);

Use the first two responses as presample data, and infer residuals for the remaining 100 observations.

E = infer(Mdl,Y(3:end),'Y0',Y(1:2));
figure;
plot(E);
title 'Inferred Residuals';

Infer Conditional Variances

Infer the conditional variances from an AR(1) and GARCH(1,1) composite model.

Specify an AR(1) model using known parameters. Set the variance equal to a garch model.

Mdl = arima('AR',{0.8,-0.3},'Constant',0);
MdlVar = garch('Constant',0.0002,'GARCH',0.6,...
	'ARCH',0.2);
Mdl.Variance = MdlVar;

Simulate response data with 102 observations.

rng 'default';
Y = simulate(Mdl,102);

Infer conditional variances for the last 100 observations without using presample data.

[Ew,Vw] = infer(Mdl,Y(3:end));

Infer conditional variances for the last 100 observations using the first two observations as presample data.

[E,V] = infer(Mdl,Y(3:end),'Y0',Y(1:2));

Plot the two sets of conditional variances for comparison. Examine the first few observations to see the slight difference between the series at the beginning.

figure;
subplot(2,1,1);
plot(Vw,'r','LineWidth',2);
hold on;
plot(V);
legend('Without Presample','With Presample');
title 'Inferred Conditional Variances';
hold off

subplot(2,1,2);
plot(Vw(1:5),'r','LineWidth',2);
hold on;
plot(V(1:5));
legend('Without Presample','With Presample');
title 'Beginning of Series';
hold off

Infer Residuals from ARMAX Model

Infer residuals from an ARMAX model.

Specify an ARMA(1,2) model using known parameters for the response (MdlY) and an AR(1) model for the predictor data (MdlX).

MdlY = arima('AR',0.2,'MA',{-0.1,0.6},'Constant',...
    1,'Variance',2,'Beta',3);
MdlX = arima('AR',0.3,'Constant',0,'Variance',1);

Simulate response and predictor data with 102 observations.

rng 'default'; % For reproducibility
X = simulate(MdlX,102);
Y = simulate(MdlY,102,'X',X);

Use the first two responses as presample data, and infer residuals for the remaining 100 observations.

E = infer(MdlY,Y(3:end),'Y0',Y(1:2),'X',X);
figure;
plot(E);
title 'Inferred Residuals'; 

References

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