filter

Filter disturbances using ARIMA or ARIMAX model

Syntax

Y = filter(Mdl,Z)

Y = filter(Mdl,Z,Name,Value)

[Y,E] = filter(___)

[Y,E,V] = filter(___)

Description

Y = filter(Mdl,Z) returns the response series Y resulting from filtering the underlying disturbance series Z. Z is associated with the model innovations process through the ARIMA model Mdl.

Y = filter(Mdl,Z,Name,Value) uses additional options specified by one or more name-value arguments. For example, filter(Mdl,Z,'X',X,'Z0',Z0) specifies exogenous predictor data X and presample disturbances Z0 to initialize the model.

[Y,E] = filter(___) returns the model innovations series E using any of the input arguments in the previous syntaxes.

[Y,E,V] = filter(___) returns the conditional variances V for a composite conditional mean and variance model (for example, an ARIMA and GARCH composite model).

Examples

Compute 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 (IRF). Generate a disturbance series with a one-time, unit impulse, and then filter it.

z = [1; zeros(19,1)];
y = filter(Mdl,z);

y is a 20-by-1 response path resulting from filtering the disturbance path z through the model. y represents the IRF. The filter function requires one presample observation to initialize the model. By default, filter uses the unconditional mean of the process, which is 0.

y = y/y(1);

Normalize the IRF such that the first element is 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.

Alternatively, you can use the impulse function to plot the IRF 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';

Simulate and Filter Multiple Paths

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

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

Generate 20 random paths from the model.

rng(1); % For reproducibility
[ySim,eSim,vSim] = simulate(Mdl,100,'NumPaths',20);

ySim, eSim, and vSim are 100-by-20 matrices of 20 simulated response, innovation, and conditional variance paths of length 100, respectively. Because Mdl does not have a conditional variance model, vSim is a matrix completely composed of the value of Mdl.Variance.

Obtain disturbance paths by standardizing the simulated innovations.

zSim = eSim./sqrt(vSim);

Filter the disturbance paths through the model.

[yFil,eFil] = filter(Mdl,zSim);

yFil and eFil are 100-by-20 matrices. The columns are independent paths generated from filtering corresponding disturbance paths in zSim through the model Mdl.

Confirm that the outputs of simulate and filter are identical.

sameE = norm(eSim - eFil) <eps

sameE = logical
   1

sameY = norm(ySim - yFil) < eps

sameY = logical
   1

The logical values 1 confirm the outputs are effectively identical.

Filter Disturbances Through Composite Conditional Mean and Variance Model

Create the composite AR(1)/GARCH(1,1) model

$$\begin{array}{l}{y}_t=1+0.5y_{t-1}+{\epsilon }_t\\ {\epsilon }_t={\sigma }_t{z}_t\\ {\sigma }_t^2=0.2+0.1{\sigma }_{t-1}^2+0.05{\epsilon }_{t-1}^2\\ z_t\sim N(0,1).\end{array}$$

Create the composite model.

CVMdl = garch('Constant',0.2,'GARCH',0.1,'ARCH',0.05);
Mdl = arima('Constant',1,'AR',0.5,'Variance',CVMdl)

Mdl = 
  arima with properties:

     Description: "ARIMA(1,0,0) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 1
               D: 0
               Q: 0
        Constant: 1
              AR: {0.5} at lag [1]
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: [GARCH(1,1) Model]

Mdl is an arima object. The property Mdl.Variance contains a garch object that represents the conditional variance model.

Generate a random series of 100 standard Gaussian of disturbances.

rng(1) % For reproducibility
z = randn(100,1);

Filter the disturbances through the model. Return and plot the simulated conditional variances.

[y,e,v] = filter(Mdl,z);
plot(z)

Input Arguments

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.

Z — Underlying disturbance series

numeric column vector | numeric matrix

Underlying disturbance series z_t, specified as a length numObs numeric column vector or a numObs-by-numPaths numeric matrix. numObs is the length of the time series (sample size). numPaths is the number of separate, independent disturbance paths.

z_t drives the innovation process ε_t. For a variance process σ_t², the innovation process is

$${\epsilon }_t={\sigma }_t{z}_t.$$

Each row corresponds to a period. The last row contains the latest set of disturbances.

Each column corresponds to a separate, independent path of disturbances. filter assumes that disturbances across any row occur simultaneously.

Z is the continuation of the presample disturbances Z0.

Data Types: double

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: filter(Mdl,Z,'X',X,'Z0',Z0) specifies exogenous predictor data X and presample disturbances Z0 to initialize the model.

Y0 — Presample response data

numeric column vector | numeric matrix

Presample response data providing initial values for the model, specified as a numeric column vector or a numeric matrix.

Each row of Y0 corresponds to a period in the presample. The following conditions apply:

• The last row contains the latest presample responses.

• To initialize the AR components, Y0 must have at least Mdl.P rows.

• If Y0 has more rows than is required to initialize the model, filter uses only the latest required rows.

Columns of Y0 are separate, independent presample paths. The following conditions apply:

• If Y0 is a column vector, filter applies it to each path.

• If Y0 is a matrix, filter applies Y0(:,j) to initialize path j. Y0 must have at least numPaths columns; filter uses only the first numPaths columns of Y0.

By default, filter sets any necessary presample observations by using one of the following methods:

• For a model with a stationary AR process and without a regression component, filter sets all presample responses to the unconditional mean of the model.

• For a model that represents a nonstationary process or that contains a regression component, filter sets all necessary presample responses to zero.

Data Types: double

Z0 — Presample disturbance data

numeric column vector | numeric matrix

Presample disturbances providing initial values for the input disturbance series Z, specified as a numeric column vector or a numeric matrix.

Each row of Z0 corresponds to a period in the presample. The following conditions apply:

• The last row contains the latest presample disturbances.

• To initialize the MA components, Z0 must have at least Mdl.Q rows.

• If Mdl.Variance is a conditional variance model (for example, a garch model object), Z0 can require more rows than Mdl.Q to initialize the model.

• If Z0 has more rows than is required to initialize the model, filter uses only the latest required rows.

Columns of Z0 are separate, independent presample paths. The following conditions apply:

• If Z0 is a column vector, filter applies it to each path.

• If Z0 is a matrix, filter applies Z0(:,j) to initialize path j. Z0 must have at least numPaths columns; filter uses only the first numPaths columns of Z0.

By default, filter sets the necessary presample disturbances to 0.

Data Types: double

V0 — Presample conditional variance data

positive numeric column vector | positive numeric matrix

Presample conditional variance data used to initialize the conditional variance model, specified as a positive numeric column vector or a positive numeric matrix. If the conditional variance Mdl.Variance is constant, filter ignores V0.

Each row of V0 corresponds to a period in the presample. The following conditions apply:

• The last row contains the latest presample conditional variances.

• To initialize the conditional variance model, V0 must have at least Mdl.Q rows. For details, see the filter function of conditional variance models.

• If V0 has more rows than is required to initialize the conditional variance model, filter uses only the latest required rows.

Columns of V0 are separate, independent presample paths. The following conditions apply:

• If V0 is a column vector, filter applies it to each simulated path.

• If V0 is a matrix, filter applies V0(:,j) to initialize simulating path j. V0 must have at least numPaths columns; filter uses only the first numPaths columns of V0.

By default, filter sets all necessary presample 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 in the model, specified as a numObs-by-numPreds numeric matrix.

numPreds is the number of predictor variables (numel(Mdl.Beta)).

Each row of X corresponds to the corresponding period in Z (period for which filter filters disturbances; the period after the presample). The following conditions apply:

• The last row contains the latest predictor data.

• If the specified predictor data has more then numObs rows, filter uses only the latest numObs rows.

• filter does not use the regression component in the presample period.

Columns of X are separate predictor variables.

filter applies X to each filtered path; that is, X represents one path of observed predictors.

By default, filter excludes the regression component, regardless of its presence in Mdl.

Data Types: double

Note

• NaNs in input data indicate missing values. filter uses listwise deletion to delete all sampled times (rows) in the input data containing at least one missing value. Specifically, filter performs these steps:

  1. Synchronize, or merge, the presample data sets Y0, Z0, and V0 to create the set Presample, in other words, Presample = [Y0 Z0 V0].

  2. Synchronize the filter data Z and X to create the set Data, in other words, Data = [Z X].

  3. Remove all rows from Presample and Data containing at least one NaN.

  Listwise deletion applied to the filter data can reduce the sample size and create irregular time series.

  filter merges data sets that have a different number of rows according to the latest observation, and then pads the shorter series with enough default values to form proper matrices.

• filter assumes that you synchronize the data sets so that the latest observations occur simultaneously. The software also assumes that you synchronize the presample series similarly.

Output Arguments

Y — Simulated response paths y_t

numeric column vector | numeric matrix

Simulated response paths y_t, returned as a length numObs column vector or a numObs-by-numPaths numeric matrix.

For each t = 1, …, numObs, the simulated response at time t Y(t,:) corresponds to the filtered disturbance at time t Z(t,:) and response path j Y(:,j) corresponds to the filtered disturbance path j Z(:,j).

Y represents the continuation of the presample response paths in Y0.

E — Simulated paths of model innovations ε_t

numeric column vector | numeric matrix

Simulated paths of model innovations ε_t with conditional variances $${\sigma }_t^2$$ (V), returned as a length numObs column vector or a numObs-by-numPaths numeric matrix. The dimensions of Y and E correspond.

Columns of E are scaled disturbance paths (innovations) such that, for a particular path

$${\epsilon }_t={\sigma }_t{z}_t.$$

V — Conditional variance paths

numeric column vector | numeric matrix

Conditional variance paths $${\sigma }_t^2$$, returned as a length numObs column vector or numObs-by-numPaths numeric matrix. The dimensions of Y and V correspond.

If Z is a matrix, then the columns of V are the filtered conditional variance paths corresponding to the columns of Z.

Columns of V are conditional variance paths of corresponding paths of innovations ε_t (E) such that, for a particular path

$${\epsilon }_t={\sigma }_t{z}_t.$$

V represents the continuation of the presample conditional variance paths in V0.

Alternative Functionality

filter generalizes simulate; both functions 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 enables you to directly specify custom disturbances.

References

[1] 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.

[2] Enders, Walter. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, Inc., 1995.

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

Version History

Introduced in R2012b