filter

Filter disturbances through vector autoregression (VAR) model

Syntax

Y = filter(Mdl,Z)

[Y,E] = filter(Mdl,Z)

Tbl2 = filter(Mdl,Tbl1,Presample=Presample)

[___] = filter(___,Name=Value)

Description

Y = filter(Mdl,Z) returns the numeric array Y containing the multivariate response series, which results from filtering the underlying input numeric array Z containing the multivariate disturbance series. The series in Z are associated with the model innovations process through the fully specified VAR(p) model Mdl.

[Y,E] = filter(Mdl,Z) returns the numeric array containing the multivariate model innovations series E.

Tbl2 = filter(Mdl,Tbl1,Presample=Presample) returns the table or timetable Tbl2 containing the multivariate response series, which results from filtering the underlying multivariate disturbance series in the input table or timetable Tbl1. filter initializes the response series using the required table or timetable of presample data in Presample. Variables in Tbl1 are associated with the model innovations process through Mdl. (since R2022b)

filter selects the variables in Mdl.SeriesNames or all variables in Tbl1. To select different disturbance variables in Tbl1 to filter through the model, use the DisturbanceVariables name-value argument. filter selects the same variables for Presample by default, but you can select different variables by using the PresampleResponseVariables name-value argument.

[___] = filter(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. filter returns the output argument combination for the corresponding input arguments. For example, filter(Mdl,Z,Y0=PS,X=Exo) filters the numeric array of disturbances Z through the VAR(p) model Mdl, and specifies the numeric array of presample response data PS and the numeric matrix of exogenous predictor data Exo for the model regression component.

Examples

Filter Numeric Matrix of Disturbances Through VAR(4) Model

Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate data. Then, simulate responses by filtering a random series of Gaussian distributed disturbances through the estimated model. Supply the disturbances as a numeric matrix.

Load the Data_USEconModel data set.

load Data_USEconModel

Plot the two series on separate plots.

figure;
plot(DataTimeTable.Time,DataTimeTable.CPIAUCSL);
title("Consumer Price Index")
ylabel("Index")
xlabel("Date")

figure;
plot(DataTimeTable.Time,DataTimeTable.UNRATE);
title("Unemployment Rate")
ylabel("Percent")
xlabel("Date")

Stabilize the CPI by converting it to a series of growth rates, include a missing value at the beginning of the series to maintain time consistency among the series. Create a new data set containing the transformed variables, and remove all rows containing at least one missing observation.

RCPI = [NaN; price2ret(DataTimeTable.CPIAUCSL)];
UNRATE = DataTimeTable.UNRATE;
DTT = timetable(RCPI,UNRATE,RowTimes=DataTimeTable.Time);
DTT = rmmissing(DTT);

Create a default VAR(4) model by using the shorthand syntax.

Mdl = varm(2,4);

Estimate the model using the entire data set.

EstMdl = estimate(Mdl,DTT.Variables);

EstMdl is a fully specified, estimated varm model object.

Generate a numobs-by-2 series of random Gaussian distributed values, where numobs is the number of observations in the data.

numobs = size(DTT,1);
rng(1) % For reproducibility
Z = mvnrnd(zeros(Mdl.NumSeries,1),eye(Mdl.NumSeries),numobs);

To simulate responses, filter the disturbances through the estimated model.

Y = filter(EstMdl,Z);

Y is a 245-by-2 matrix of simulated responses. The first and second columns contain the simulated CPI growth rate and unemployment rate, respectively.

Plot the simulated and true responses.

figure
plot(DTT.Time,Y(:,1))
hold on
plot(DTT.Time,DTT.RCPI)
ylabel("Growth Rate")
xlabel("Date")
title("CPI Growth Rate")
legend("Simulation","True")

figure
plot(DTT.Time,Y(:,2))
hold on
plot(DTT.Time,DTT.UNRATE)
ylabel("Percent")
xlabel("Date")
title("Unemployment Rate")
legend("Simulation","True")
hold off

Filter Multiple Disturbance Paths

Estimate a VAR(4) model of the consumer price index (CPI), the unemployment rate, and the gross domestic product (GDP). Include a linear regression component containing the current quarter and the last four quarters of government consumption expenditures and investment (GCE). Pass multiple multivariate Gaussian disturbance paths through the estimated model.

Load the Data_USEconModel data set. Compute the real GDP.

load Data_USEconModel
DataTimeTable.RGDP = DataTimeTable.GDP./DataTimeTable.GDPDEF*100;

Plot all variables on separate plots.

figure
tiledlayout(2,2)
nexttile
plot(DataTimeTable.Time,DataTimeTable.CPIAUCSL);
ylabel('Index')
title('Consumer Price Index')
nexttile
plot(DataTimeTable.Time,DataTimeTable.UNRATE);
ylabel('Percent')
title('Unemployment Rate')
nexttile
plot(DataTimeTable.Time,DataTimeTable.RGDP);
ylabel('Output')
title('Real Gross Domestic Product')
nexttile
plot(DataTimeTable.Time,DataTimeTable.GCE);
ylabel('Billions of $')
title('Government Expenditures')

Stabilize the CPI, GDP, and GCE by converting each to a series of growth rates. Synchronize the unemployment rate series with the others by removing its first observation.

inputVariables = {'CPIAUCSL' 'RGDP' 'GCE'};
Data = varfun(@price2ret,DataTimeTable,'InputVariables',inputVariables);
Data.Properties.VariableNames = inputVariables;
Data.UNRATE = DataTimeTable.UNRATE(2:end);

Expand the GCE rate series to a matrix that includes its current value and up through four lagged values. Remove the GCE variable from Data.

rgcelag4 = lagmatrix(Data.GCE,0:4);
Data.GCE = [];

Create a default VAR(4) model by using the shorthand syntax.

Mdl = varm(3,4);
Mdl.SeriesNames = {'rcpi' 'unrate' 'rgdpg'};

Estimate the model using the entire sample. Specify the GCE matrix as data for the regression component.

EstMdl = estimate(Mdl,Data.Variables,'X',rgcelag4);

Generate 1000 paths of numobs observations from a 3-D Gaussian distribution. numobs is the number of observations in the data without any missing values.

numpaths = 1000;
numseries = Mdl.NumSeries;
idx = all(~ismissing([Data array2table(rgcelag4)]),2);
numobs = sum(idx);
rng(1);
Z = mvnrnd(zeros(Mdl.NumSeries,1),eye(Mdl.NumSeries),numobs*numpaths);
Z = reshape(Z,[numobs,3,numpaths]);

Filter the disturbances through the estimated model. Supply the predictor data. Return the innovations (scaled disturbances).

[Y,E] = filter(EstMdl,Z,'X',rgcelag4);

Y and E are 244-by-3-by-1000 matrices of filtered responses and scaled disturbances, respectively. The columns correspond to the CPI growth rate, unemployment rate, and GDP growth rate, respectively. filter applies the same predictor data to all paths.

For each time point, compute the mean vector of the filtered responses among all paths.

MeanFilt = mean(Y,3);

MeanFilt is a 244-by-3 matrix containing the average of the responses at each time point.

Plot the filtered responses, their averages, and the data.

Data = Data(idx,:);

figure
tiledlayout(2,2)
for j = 1:Mdl.NumSeries
    nexttile
    plot(Data.Time,squeeze(Y(:,j,:)),'Color',[0.8,0.8,0.8])
    title(Mdl.SeriesNames{j});
    hold on
    h1 = plot(Data.Time,Data{:,j});
    h2 = plot(Data.Time,MeanFilt(:,j));
    hold off
end

hl = legend([h1 h2],'Data','Mean');
hl.Location = 'none';
hl.Position = [0.6 0.25 hl.Position(3:4)];

Filter Timetable of Disturbances Through VAR(4) Model

Since R2022b

Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate data. Then, simulate responses by filtering multiple random paths of Gaussian distributed disturbances through the estimated model. Supply the disturbances in a timetable. This example is based on Filter Numeric Matrix of Disturbances Through VAR(4) Model.

Load and Preprocess Data

Load the Data_USEconModel data set.

load Data_USEconModel

Preprocess the data. Regularize the time base.

RCPI = [NaN; price2ret(DataTimeTable.CPIAUCSL)];
UNRATE = DataTimeTable.UNRATE;
DTT = timetable(RCPI,UNRATE,RowTimes=DataTimeTable.Time);
DTT = rmmissing(DTT);

dt = DTT.Time;
dt = dateshift(dt,"start","quarter");
DTT.Time = dt;

Fit Model to Data

Estimate the model using the entire data set.

Mdl = varm(2,4);
Mdl.SeriesNames = DTT.Properties.VariableNames;
EstMdl = estimate(Mdl,DTT);

Simulate Paths of Disturbances

Generate a numobs-by-numseries-by-numpaths array of independent random Gaussian distributed values, where numobs is the number of observations in the data, numseries the number of response series 2, and numpaths is 100.

rng(1) % For reproducibility
numobs = height(DTT);
numseries = EstMdl.NumSeries;
numpaths = 100;

Z = mvnrnd(zeros(numseries,1),eye(numseries),numobs*numpaths);
Z = reshape(Z,numobs,numseries,numpaths);

DTT.ZRCPI = squeeze(Z(:,1,:));
DTT.ZUNRATE = squeeze(Z(:,2,:));
head(DTT)

    Time        RCPI       UNRATE       ZRCPI          ZUNRATE   
    _____    __________    ______    ____________    ____________

    Q1-48     0.0038371       4      1×100 double    1×100 double
    Q2-48      0.027284     3.6      1×100 double    1×100 double
    Q3-48     0.0086581     3.8      1×100 double    1×100 double
    Q4-48     -0.012807       4      1×100 double    1×100 double
    Q1-49    -0.0058382       5      1×100 double    1×100 double
    Q2-49    0.00041815     6.2      1×100 double    1×100 double
    Q3-49    -0.0071324     6.6      1×100 double    1×100 double
    Q4-49    -0.0059122     6.6      1×100 double    1×100 double

Filter Disturbances Through Model

When you filter disturbances by using a timetable, filter requires a presample. Split the timetable into presample and in-sample data sets. The presample data is the initial EstMdl.P observations, and the in-sample data set contains the remaining observations.

Presample = DTT(1:EstMdl.P,:);
InSample = DTT((EstMdl.P + 1):end,:);

Simulate response paths by filtering the in-sample disturbances through the estimated model. Specify the variable names of the disturbance series, the presample data, and the response variable names in the presample.

Tbl2 = filter(EstMdl,InSample,DisturbanceVariables=["ZRCPI" "ZUNRATE"], ...
    Presample=Presample,PresampleResponseVariables=EstMdl.SeriesNames);
size(Tbl2)

ans = 1×2

   241     8

head(Tbl2)

    Time        RCPI       UNRATE       ZRCPI          ZUNRATE       RCPI_Responses    UNRATE_Responses    RCPI_Innovations    UNRATE_Innovations
    _____    __________    ______    ____________    ____________    ______________    ________________    ________________    __________________

    Q1-49    -0.0058382       5      1×100 double    1×100 double     1×100 double       1×100 double        1×100 double         1×100 double   
    Q2-49    0.00041815     6.2      1×100 double    1×100 double     1×100 double       1×100 double        1×100 double         1×100 double   
    Q3-49    -0.0071324     6.6      1×100 double    1×100 double     1×100 double       1×100 double        1×100 double         1×100 double   
    Q4-49    -0.0059122     6.6      1×100 double    1×100 double     1×100 double       1×100 double        1×100 double         1×100 double   
    Q1-50     0.0012698     6.3      1×100 double    1×100 double     1×100 double       1×100 double        1×100 double         1×100 double   
    Q2-50      0.010101     5.4      1×100 double    1×100 double     1×100 double       1×100 double        1×100 double         1×100 double   
    Q3-50       0.01908     4.4      1×100 double    1×100 double     1×100 double       1×100 double        1×100 double         1×100 double   
    Q4-50      0.025954     4.3      1×100 double    1×100 double     1×100 double       1×100 double        1×100 double         1×100 double   

Tbl2 is a 241-by-2 matrix of in-sample data, paths of simulated disturbances, paths of filtered responses (variables names appended with _Responses), and paths of innovations (variables with name appended with _Innovations).

Plot the paths of simulated responses with the true responses.

figure
p1 = plot(Tbl2.Time,Tbl2.RCPI_Responses,Color=[0.5 0.5 0.5]);
hold on
p2 = plot(Tbl2.Time,Tbl2.RCPI,LineWidth=2);
ylabel("Growth Rate")
xlabel("Date")
title("CPI Growth Rate")
legend([p1(1) p2],["Simulated" "Observed"])
hold off

figure
p1 = plot(Tbl2.Time,Tbl2.UNRATE_Responses,Color=[0.5 0.5 0.5]);
hold on
p2 = plot(Tbl2.Time,Tbl2.UNRATE,LineWidth=2);
ylabel("Percent")
xlabel("Date")
title("Unemployment Rate")
legend([p1(1) p2],["Simulated" "Observed"])
hold off

Input Arguments

Mdl — VAR model

varm model object

VAR model, specified as a varm model object created by varm or estimate. Mdl must be fully specified.

Z — Underlying multivariate disturbance series

numeric matrix | numeric array

Underlying multivariate disturbance series z_t associated with the model innovations process ε_t, specified as a numobs-by-numseries numeric matrix or a numobs-by-numseries-by-numpaths numeric array.

numobs is the sample size. numseries is the number of disturbance series (Mdl.NumSeries). numpaths is the number of disturbance paths.

Rows correspond to sampling times, and the last row contains the latest set of disturbances.

Columns correspond to individual disturbance series for response variables.

Pages correspond to separate, independent paths. For a numeric matrix, Z is a single numseries-dimensional path of disturbance series. For a 3-D array, each page of Z represents a separate numseries-dimensional path. Among all pages, disturbances in corresponding rows occur at the same time.

The Scale name-value argument specifies whether to scale the disturbances before filter filters them through Mdl. For more details, see Scale.

Data Types: double

Tbl1 — Time series data

table | timetable

Since R2022b

Time series data containing observed disturbance variables z_t, associated with the model innovations process ε_t, or predictor variables x_t, specified as a table or timetable with numvars variables and numobs rows. You can optionally select numseries disturbance variables or numpreds predictor variables by using the DisturbanceVariables or PredictorVariables name-value arguments, respectively.

Each selected disturbance variable is a numobs-by-numpaths numeric matrix, and each predictor variable is a numeric vector. Each row is an observation, and measurements in each row occur simultaneously.

Each path (column) of a disturbance variable is independent of all others, but path j of all presample and in-sample variables correspond, for j = 1,…,numpaths. Each selected predictor variable contains one path, which filter applies to all paths.

If Tbl1 is a timetable, it must represent a sample with a regular datetime time step (see isregular), and the datetime vector Tbl1.Time must be ascending or descending.

If Tbl1 is a table, the last row contains the latest observation.

The Scale name-value argument specifies whether to scale the disturbances before filter filters them through Mdl. For more details, see Scale.

Presample — Presample data

table | timetable

Since R2022b

Presample data that provides initial values for the model Mdl, specified as a table or timetable, the same type as Tbl1, with numprevars variables and numpreobs rows. Presample is required when you supply a table or timetable of data Tbl1.

Each row is a presample observation, and measurements in each row, among all paths, occur simultaneously. numpreobs must be at least Mdl.P. If you supply more rows than necessary, filter uses the latest Mdl.P observations only.

Each variable is a numpreobs-by-numprepaths numeric matrix. Variables correspond to the response series associated with the respective disturbance in Tbl1. To control presample variable selection, see the optional PresampleResponseVariables name-value argument.

For each variable, columns are separate, independent paths.

• If variables are vectors, filter applies them to each path in Tbl1 to produce the filtered responses in Tbl2. Therefore, all paths of filtered responses derive from common initial conditions.

• Otherwise, for each variable Vark and each path j, filter applies Presample.Vark(:,j) to produce Tbl2.Vark(:,j). Variables must have at least numpaths columns, and filter uses only the first numpaths columns.

If Presample is a timetable, all the following conditions must be true:

• Presample must represent a sample with a regular datetime time step (see isregular).

• The inputs Tbl1 and Presample must be consistent in time such that Presample immediately precedes Tbl1 with respect to the sampling frequency and order.

• The datetime vector of sample timestamps Presample.Time must be ascending or descending.

If Presample is a table, the last row contains the latest presample observation.

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,Y0=PS,X=Exo) filters the numeric array of disturbances Z through the VAR(p) model Mdl, and specifies the numeric array of presample response data PS and the numeric matrix of exogenous predictor data Exo for the model regression component.

DisturbanceVariables — Variables to select from Tbl1 to treat as disturbance variables z_t

string vector | cell vector of character vectors | vector of integers | logical vector

Since R2022b

Variables to select from Tbl1 to treat as disturbance variables z_t, specified as one of the following data types:

• String vector or cell vector of character vectors containing numseries variable names in Tbl1.Properties.VariableNames

• A length numseries vector of unique indices (integers) of variables to select from Tbl1.Properties.VariableNames

• A length numvars logical vector, where DisturbanceVariables(j) = true selects variable j from Tbl1.Properties.VariableNames, and sum(DisturbanceVariables) is numseries

The selected variables must be numeric vectors (single path) or matrices (columns represent multiple independent paths) of the same width, and cannot contain missing values (NaN).

If the number of variables in Tbl1 matches Mdl.NumSeries, the default specifies all variables in Tbl1. If the number of variables in Tbl1 exceeds Mdl.NumSeries, the default matches variables in Tbl1 to names in Mdl.SeriesNames.

Example: DisturbanceVariables=["GDP" "CPI"]

Example: DisturbanceVariables=[true false true false] or DisturbanceVariable=[1 3] selects the first and third table variables as the disturbance variables.

Data Types: double | logical | char | cell | string

Y0 — Presample responses

numeric matrix | numeric array

Presample responses that provide initial values for the model Mdl, specified as a numpreobs-by-numseries numeric matrix or a numpreobs-by-numseries-by-numprepaths numeric array. Use Y0 only when you supply a numeric array of disturbance data Z.

numpreobs is the number of presample observations. numprepaths is the number of presample response paths.

Each row is a presample observation, and measurements in each row, among all pages, occur simultaneously. The last row contains the latest presample observation. Y0 must have at least Mdl.P rows. If you supply more rows than necessary, filter uses the latest Mdl.P observations only.

Each column corresponds to the response series associated with the respective disturbance in Z.

Pages correspond to separate, independent paths.

• If Y0 is a matrix, filter applies it to each path (page) to produce the filtered responses Y. Therefore, all paths in Y derive from common initial conditions.

• Otherwise, filter applies Y0(:,:,j) to produce Y(:,:,j). Y0 must have at least numpaths pages, and filter uses only the first numpaths pages.

By default, filter sets any necessary presample observations.

• For stationary VAR processes without regression components, filter uses the unconditional mean $$\mu ={\Phi }^{-1}(L)c.$$

• For nonstationary processes or models containing a regression component, filter sets presample observations to an array composed of zeros.

Data Types: double

PresampleResponseVariables — Variables to select from Presample to use for presample response data

string vector | cell vector of character vectors | vector of integers | logical vector

Since R2022b

Variables to select from Presample to use for presample data, specified as one of the following data types:

• String vector or cell vector of character vectors containing numseries variable names in Presample.Properties.VariableNames

• A length numseries vector of unique indices (integers) of variables to select from Presample.Properties.VariableNames

• A length numvars logical vector, where PresampleResponseVariables(j) = true selects variable j from Presample.Properties.VariableNames, and sum(PresampleResponseVariables) is numseries

The selected variables must be numeric vectors (single path) or matrices (columns represent multiple independent paths) of the same width, and cannot contain missing values (NaN).

PresampleResponseNames does not need to contain the same names as in Tbl1; filter uses the data in selected variable PresampleResponseVariables(j) as a presample for the response variable corresponding to DisturbanceVariables(j).

The default specifies the same response variables as those selected from Tbl1 (see DisturbanceVariables).

Example: PresampleResponseVariables=["GDP" "CPI"]

Example: PresampleResponseVariables=[true false true false] or PresampleResponseVariable=[1 3] selects the first and third table variables for presample data.

Data Types: double | logical | char | cell | string

X — Predictor data x_t

numeric matrix

Predictor data x_t for the regression component in the model, specified as a numeric matrix containing numpreds columns. Use X only when you supply a numeric array of disturbance data Z.

numpreds is the number of predictor variables (size(Mdl.Beta,2)).

Each row corresponds to an observation, and measurements in each row occur simultaneously. The last row contains the latest observation. X must have at least as many observations as Z. If you supply more rows than necessary, filter uses only the latest observations. filter does not use the regression component in the presample period.

Each column is an individual predictor variable. All predictor variables are present in the regression component of each response equation.

filter applies X to each path (page) in Z; 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

PredictorVariables — Variables to select from Tbl1 to treat as exogenous predictor variables x_t

string vector | cell vector of character vectors | vector of integers | logical vector

Since R2022b

Variables to select from Tbl1 to treat as exogenous predictor variables x_t, specified as one of the following data types:

• String vector or cell vector of character vectors containing numpreds variable names in Tbl1.Properties.VariableNames

• A length numpreds vector of unique indices (integers) of variables to select from Tbl1.Properties.VariableNames

• A length numvars logical vector, where PredictorVariables(j) = true selects variable j from Tbl1.Properties.VariableNames, and sum(PredictorVariables) is numpreds

The selected variables must be numeric vectors and cannot contain missing values (NaN).

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

Example: PredictorVariables=["M1SL" "TB3MS" "UNRATE"]

Example: PredictorVariables=[true false true false] or PredictorVariable=[1 3] selects the first and third table variables to supply the predictor data.

Data Types: double | logical | char | cell | string

Scale — Flag indicating whether to scale disturbances

true (default) | false

Flag indicating whether to scale disturbances by the lower triangular Cholesky factor of the model covariance matrix, specified as a value in this table. In the table:

• Z is the input array of disturbance data Z or the specified disturbance variables in the input Tbl1.

• E is the output array of innovations E or the innovation variables in the output Tbl2.

Value	Description
true	E(:,:,j) = L*Z(:,:,j), where L = chol(Mdl.Covariance,"lower")
false	No scale, E(:,:,j) = Z(:,:,j)

For each page j = 1,...,numpaths, filter filters the numobs-by-numseries matrix of innovations E(:,:,j) through the VAR(p) model Mdl using the specified scale.

Example: Scale=false

Data Types: logical

Note

• NaN values in Z, Y0, and X indicate missing values. filter removes missing values from the data by list-wise deletion.

  1. If Z is a 3-D array, then filter horizontally concatenates the pages of Z to form a numobs-by-numpaths*numseries matrix.

  2. If a regression component is present, then filter horizontally concatenates X to Z to form a numobs-by-(numpaths*numseries + numpreds) matrix. filter assumes that the last rows of each series occur at the same time.

  3. filter removes any row that contains at least one NaN from the concatenated data.

  4. filter applies steps 1 and 3 to the presample paths in Y0.

  This process ensures that the filtered responses and innovations of each path are the same size and are based on the same observation times. In the case of missing observations, the results obtained from multiple paths of Z can differ from the results obtained from each path individually.

  This data reduction reduces the effective sample size.

• filter issues an error when any table or timetable input contains missing values.

Output Arguments

Y — Filtered multivariate response series y_t

numeric matrix | numeric array

Filtered multivariate response series y_t, returned as a numobs-by-numseries numeric matrix or a numobs-by-numseries-by-numpaths numeric array. Y represents the continuation of the presample responses in Y0.

filter returns Y only when you supply the input Z.

E — Multivariate model innovation series ε_t

numeric matrix | numeric array

Multivariate model innovations series ε_t, returned as a numobs-by-numseries numeric matrix or a numobs-by-numseries-by-numpaths numeric array. For details on the value of E, see Scale.

filter returns E only when you supply the input Z.

Tbl2 — Multivariate filtered response and innovation series

table | timetable

Since R2022b

Multivariate filtered response y_t and innovation series ε_t, returned as a table or timetable, the same data type as Tbl1. filter returns Tbl2 only when you supply the input Tbl1.

Tbl2 contains the following variables:

• The filtered response variables y_t. Each filtered response variable is a numobs-by-numpaths numeric matrix, with rows representing observations and columns representing independent paths, each corresponding to the input observations and paths in Tbl1. filter names the filtered response for disturbance variable DisturbanceJ in Tbl1 DisturbanceJ_Responses. For example, if one of the selected disturbance variables in Tbl1 to filter is GDP, Tbl2 contains a variable for the corresponding filtered responses with the name GDP_Responses.

• The innovation variables ε_t. Each innovation variable is a numobs-by-numpaths numeric matrix, with rows representing observations and columns representing independent paths, each corresponding to the input observations and paths in Tbl1. filter names the innovation variable for disturbance variable DisturbanceJ in Tbl1 DisturbanceJ_Innovations. For example, if one of the selected disturbance variables in Tbl1 to filter is GDP, Tbl2 contains a variable for the corresponding innovations with the name GDP_Innovations.

• All variables Tbl1.

If Tbl1 is a timetable, Tbl1 and Tbl2 have the same row order, either ascending or descending.

Algorithms

• filter computes filtered responses Y and innovations E using this process for each page j in Z.

  1. If Scale is true, then E(:,:,j) = L*Z(:,:,j), where L = chol(Mdl.Covariance,'lower'). Otherwise, E(:,:,j) = Z(:,:,j). Set e_t = E(:,:,j).

  2. Y(:,:,j) is y_t in this system of equations.

     $$y_t={\widehat{\Phi }}^{-1}(L)\left(\widehat{c}+\widehat{\delta }t+e_t\right).$$

     For variable definitions, see More About.

• filter generalizes simulate. Both functions filter a disturbance series through a model to produce responses and innovations. However, whereas simulate generates a series of mean-zero, unit-variance, independent Gaussian disturbances Z to form innovations E = L*Z, filter enables you to supply disturbances from any distribution.

• filter uses this process to determine the time origin t₀ of models that include linear time trends:

  • If you specify Z and you do not specify a presample by using the Y0 name-value argument, t₀ = 0.

  • Otherwise, if you specify Tbl1 or you supply a presample, filter sets t₀ to size(Y0,1) – Mdl.P, where Y0 contains presample data, the value of Y0 or Presample. Therefore, the times in the trend component are t = t₀ + 1, t₀ + 2,..., t₀ + numobs, where numobs is the effective sample size (size(Z,1), after filter removes missing values, or height(Tbl1)). This convention is consistent with the default behavior of model estimation, in which estimate removes the first Mdl.P responses, reducing the effective sample size. Although filter explicitly uses the first Mdl.P presample responses in Y0 to initialize the model, the total number of observations in Y0 and the input data (Z, excluding missing values, or Tbl1) determines t₀.

References

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

[2] Johansen, S. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.

[3] Juselius, K. The Cointegrated VAR Model. Oxford: Oxford University Press, 2006.

[4] Lütkepohl, H. New Introduction to Multiple Time Series Analysis. Berlin: Springer, 2005.

Version History

Introduced in R2017a

R2022b: filter accepts input data in tables and timetables, and return results in tables and timetables

In addition to accepting input data in numeric arrays, filter accepts input data in tables and timetables. filter chooses default series on which to operate, but you can use the following name-value arguments to select variables.

• DisturbanceVariables specifies the disturbance series names in the input data to filter through the model.

• Presample specifies the input table or regular timetable of presample response data.

• PresampleResponseVariables specifies the response series names from Presample.

• PredictorVariables specifies the predictor series names in the input data for a model regression component.