filter

Filter disturbances using ARIMA or ARIMAX model

Syntax

[Y,E,V] = filter(Mdl,Z)
[Y,E,V] = filter(Mdl,Z,Name,Value)


Description

[Y,E,V] = filter(Mdl,Z) filters disturbances, Z, to produce responses, innovations, and conditional variances of a univariate ARIMA(p,D,q) model.

[Y,E,V] = filter(Mdl,Z,Name,Value) filters disturbances using additional options specified by one or more Name,Value pair arguments.

Input Arguments

Mdl

ARIMA model, as created by arima or estimate. The input model cannot have any NaN values.

Z

numObs-by-NumPaths matrix of disturbances, zt, that drives the innovation process, εt. For a variance process σt2, the innovation process is given by $${\epsilon }_t={\sigma }_t{z}_t.$$ As a column vector, Z represents a path of the underlying disturbance series. As a matrix, Z represents numObs observations of NumPaths paths of the underlying disturbance series. filter assumes that observations across any row occur simultaneously. The last row contains the most recent observation. Note NaNs indicate missing values. filter removes them from Z using list-wise deletion. That is, filter removes any row of Z containing at least one NaN. This deletion reduces the effective sample size and can cause irregular time series.

Name-Value 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.

V0	Positive presample conditional variances that provide initial values for the model. If V0 is a column vector, then filter applies it to each output path. If V0 is a matrix, then it requires at least NumPaths columns. If the number of columns exceeds NumPaths, then filter uses the first NumPaths columns. V0 requires enough rows to initialize the moving average component of the ARIMA model and any conditional variance model. The required number of rows is at least Mdl.Q. If you include a conditional variance model, then filter might require more than Mdl.Q rows. If the number of rows in V0 exceeds the number necessary, then filter uses the most recent observations. The last row contains the most recent observation. Default: filter sets necessary presample observations to the unconditional variance of the conditional variance process.
X	Matrix of predictor data corresponding to a regression component in the conditional mean model. The columns of X are separate, synchronized time series, with the last row containing the most recent observations. The number of rows of X must be at least the number of rows of Z. When the number of rows of X exceeds the number necessary, then filter uses the most recent observations. Default: filter does not include a regression component in the conditional mean model regardless of the presence of regression coefficients in Mdl.
Y0	Presample response data, providing initial values for the model. If Y0 is a column vector, then filter applies it to each output path. If Y0 is a matrix, then it requires at least NumPaths columns. If the number of columns in Y0 exceeds NumPaths, then filter uses the first NumPaths columns. Y0 requires at least Mdl.P rows to initialize the model. If the number of rows in Y0 exceeds Mdl.P, then filter uses the most recent Mdl.P observations. The last row contains the most recent observation. Default: filter sets the necessary presample observations to the unconditional mean for stationary processes, and to 0 for nonstationary processes or processes with a regression component.
Z0	Presample disturbances, providing initial values for the input disturbance series, Z. If Z0 is a column vector, then filter applies it to each output path. If Z0 is a matrix, then it requires at least NumPaths columns. If the number of columns exceeds NumPaths, then filter uses the first NumPaths columns. Z0 requires a sufficient number of rows to initialize the moving average component of the ARIMA model and any conditional variance model. The required number of rows is at least Mdl.Q, but might be more if a conditional variance model is included. If the number of rows in Z0 exceeds the number necessary, then filter uses the most recent observations. The last row contains the most recent observation. Default: filter sets the necessary presample observations to 0.

Notes

• NaNs in the data indicate missing values and filter removes them. The software merges the presample data and main data sets separately, then uses list-wise deletion to remove any NaNs. That is, filter sets PreSample = [Y0 Z0 V0] and Data = [Z X], then it removes any row in PreSample or Data that contains at least one NaN.

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

• filter assumes that you synchronize presample data such that the most recent observation of each presample series occurs simultaneously.

• All predictor series in X (i.e., columns of X) are applied to each disturbance series in Z to produce NumPaths response series Y.

Output Arguments

Y	numObs-by-NumPaths matrix of simulated responses. Y is the continuation of the presample series Y0.
E	numObs-by-NumPaths matrix of simulated innovations with conditional variances, V. Each column is a scaled series of innovations (or disturbances) such that E = sqrt(V)*Z.
V	numObs-by-NumPaths matrix of conditional variances of the innovations in E such that E = sqrt(V)*Z. V is the continuation of the presample series V0.

Examples

Simulate and Filter

Specify a mean zero ARIMA(2,0,1) model.

Mdl = arima('Constant',0,'AR',{0.5,-0.8},'MA',-0.5,...
    'Variance',0.1);

Simulate the model using Monte Carlo simulation. Then, standardize the simulated innovations and filter them.

rng(1); % For reproducibility
[y,e,v] = simulate(Mdl,100);
Z = e./sqrt(v);
[Y,E,V] = filter(Mdl,Z);

Confirm that the outputs of simulate and filter are identical.

isequal(y,Y)

ans = logical
   1

The logical value 1 confirms the two outputs are identical.

Simulate an Impulse Response Function

Specify a mean zero ARIMA(2,0,1) model.

Mdl = arima('Constant',0,'AR',{0.5,-0.8},'MA',-0.5,...
    'Variance',0.1);

Simulate the first 20 responses of the impulse response function. Generate a disturbance series with a one-time, unit impulse, and then filter it. Set all presample observations equal to zero. Normalize the impulse response function to ensure that the first element is 1.

Z = [1;zeros(19,1)];
Y = filter(Mdl,Z,'Y0',zeros(Mdl.P,1));
Y = Y/Y(1);

Plot the impulse response function.

figure;
stem((0:numel(Y)-1)',Y,'filled');
title 'Impulse Response';

The impulse response assesses the dynamic behavior of a system to a one-time, unit impulse. You can also use the impulse method to plot the impulse response function for an ARIMA process.

Simulate Step Response

Assess the dynamic behavior of a system to a persistent change in a variable by plotting a step response.

Specify a mean zero ARIMA(2,0,1) process.

Mdl = arima('Constant',0,'AR',{0.5,-0.8},'MA',-0.5,...
    'Variance',0.1);

Simulate the first 20 responses to a sequence of unit disturbances. Generate a disturbance series of ones, and then filter it. Set all presample observations equal to zero.

Z = ones(20,1);
Y = filter(Mdl,Z,'Y0',zeros(Mdl.P,1));
Y = Y/Y(1);

The last step normalizes the step response function to ensure that the first element is 1.

Plot the step response function.

figure;
stem((0:numel(Y)-1)',Y,'filled');
title 'Step Response';

Simulate Responses from ARIMAX Model

Create models for the response and predictor series. Set an ARIMAX(2,1,3) model to the response MdlY, and an AR(1) model to the MdlX.

MdlY = arima('AR',{0.1 0.2},'D',1,'MA',{-0.1 0.1 0.05},...
'Constant',1,'Variance',0.5, 'Beta',2);
MdlX = arima('AR',0.5,'Constant',0,'Variance',0.1);

Simulate a length 100 predictor series x and a series of iid normal disturbances z having mean zero and variance 1.

rng(1);
z = randn(100,1);
x = simulate(MdlX,100);

Filter the disturbances z using MdlY to produce the response series y, and plot y.

y = filter(MdlY,z,'X',x);
figure;
plot(y);
title 'Filter to simulate ARIMA(2,1,3)';
xlabel 'Time';
ylabel 'Response';

Alternatives

• filter generalizes simulate. That is, both filter a series of disturbances to produce output responses, innovations, and conditional variances. However, simulate autogenerates a series of mean zero, unit variance, independent and identically distributed (iid) disturbances according to the distribution in Mdl. In contrast, filter lets you directly specify your own disturbances.

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.