filter

Filter disturbances through regression model with ARIMA errors

Syntax

Y = filter(Mdl,Z)

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

Tbl2 = filter(Mdl,Tbl1)

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

Description

Y = filter(Mdl,Z) returns a numeric array of one or more response series Y resulting from filtering the numeric array of one or more underlying disturbance series Z through the fully specified, univariate regression model with ARIMA errors Mdl. Z is associated with the error model innovations process that drives the specified regression model with ARIMA errors.

[Y,E,U] = filter(Mdl,Z) also returns numeric arrays of one or more series of error model innovations E and unconditional disturbances U, resulting from filtering the disturbance paths Z through the model Mdl.

Tbl2 = filter(Mdl,Tbl1) returns the table or timetable Tbl2 containing the results from filtering the paths of disturbances in the input table or timetable Tbl1 through Mdl. The disturbance variable in Tbl1 is associated with the model innovations process that drives Mdl. (since R2023b)

filter selects the variable Mdl.SeriesName, or the sole variable in Tbl1, as the disturbance variable to filter through the model. To select a different variable in Tbl1 to filter through the model, use the DisturbanceVariable 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,X=Pred,Z0=PSZ) specifies the predictor data Pred for the model regression component and the observed errors in the presample period PSZ to initialize the model.

Examples

Filter Disturbance Vector to Compute Impulse Response Function

Compute the impulse response function (IRF) of an innovation shock to the regression model with ARMA(2,1) errors. Supply the innovation shock as a vector.

The IRF assesses the dynamic behavior of a system to a one-time shock. Typically, the magnitude of the shock is 1. Alternatively, it might be more meaningful to examine an IRF of an innovation shock with a magnitude of one standard deviation.

In regression models with ARIMA errors,

• The IRF is invariant to the behavior of the predictors and the intercept.

• The IRF of the model is defined as the impulse response of the unconditional disturbances as governed by the ARIMA error component.

Create the following regression model with ARMA(2,1) errors:

$$\begin{array}{c}{y}_t=u_t\\ u_t=0.5u_{t-1}-0.8u_{t-2}+{\epsilon }_t-0.5{\epsilon }_{t-1},\end{array}$$

where $${\epsilon }_t$$ is Gaussian with variance 0.1.

Mdl = regARIMA(Intercept=0,AR={0.5 -0.8},MA=-0.5, ...
    Variance=0.1);

When you construct an impulse response function for a regression model with ARIMA errors, you must set Intercept to 0.

Simulate the first 30 responses of the impulse response function by generating an error series with a one-time impulse with magnitude equal to one standard deviation, and then filter it. Also, use impulse to compute the IRF.

z = [sqrt(Mdl.Variance); zeros(29,1)];  % Shock of 1 std
yFltr = filter(Mdl,z);
yImpls = impulse(Mdl,30);

When you construct an IRF of a regression model with ARIMA errors containing a regression component, do not specify the predictor matrix, X, in filter.

Plot the IRFs.

figure
tiledlayout(2,1)
nexttile
stem((0:numel(yFltr)-1)',yFltr,"filled")
title("Impulse Response to Shock of One Standard Deviation")
nexttile
stem((0:numel(yImpls)-1)',yImpls,"filled")
title("Impulse Response to Unit Shock")

The IRF given a shock of one standard deviation is a scaled version of the IRF returned by impulse.

Compute Step Response

Compute the step response function of a regression model with ARMA(2,1) errors.

The step response assesses the dynamic behavior of a system to a persistent shock. Typically, the magnitude of the shock is 1. Alternatively, it might be more meaningful to examine a step response of a persistent innovation shock with a magnitude of one standard deviation. This example plots the step response of a persistent innovations shock in a model without an intercept and predictor matrix for regression. However, note that filter is flexible in that it accepts a persistent innovations or predictor shock that you construct using any magnitude, then filters it through the model.

Specify the following regression model with ARMA(2,1) errors:

$$\begin{array}{c}{y}_t=u_t\\ u_t=0.5u_{t-1}-0.8u_{t-2}+{\epsilon }_t-0.5{\epsilon }_{t-1},\end{array}$$

where $${\epsilon }_t$$ is Gaussian with variance 0.1.

Mdl = regARIMA(Intercept=0,AR={0.5 -0.8},MA=-0.5, ...
    Variance=0.1);

Compute the first 30 responses to a sequence of unit errors by generating an error series of one standard deviation, and then filtering it.

z = sqrt(Mdl.Variance)*ones(30,1); % Persistent shock of one std
y = filter(Mdl,z);            
y = y/y(1);  % Normalize relative to y(1)

Plot the step response function.

figure
stem((0:numel(y)-1)',y,"filled")
title("Step Response for Persistent Shock of One STD")

The step response settles around 0.4.

Filter Timetable of Disturbances Through Estimated Model

Fit a regression model with ARMA(1,1) errors by regressing the US consumer price index (CPI) quarterly changes onto the US gross domestic product (GDP) growth rate. Supply a timetable of data and specify the series for the fit. Then, filter paths of disturbances in a timetable through the fitted model.

Load and Transform Data

Load the US macroeconomic data set. Compute the series of GDP quarterly growth rates and CPI quarterly changes.

load Data_USEconModel
DTT = price2ret(DataTimeTable,DataVariables="GDP");
DTT.GDPRate = 100*DTT.GDP;
DTT.CPIDel = diff(DataTimeTable.CPIAUCSL);
T = height(DTT) 

T = 
248

figure
tiledlayout(2,1)
nexttile
plot(DTT.Time,DTT.GDPRate)
title("GDP Rate")
ylabel("Percent Growth")
nexttile
plot(DTT.Time,DTT.CPIDel)
title("Index")

The series appear stationary, albeit heteroscedastic.

Prepare Timetable for Estimation

When you plan to supply a timetable, you must ensure it has all the following characteristics:

• The selected response variable is numeric and does not contain any missing values.

• The timestamps in the Time variable are regular, and they are ascending or descending.

Remove all missing values from the timetable.

DTT = rmmissing(DTT);
T_DTT = height(DTT)

T_DTT = 
248

Because each sample time has an observation for all variables, rmmissing does not remove any observations.

Determine whether the sampling timestamps have a regular frequency and are sorted.

areTimestampsRegular = isregular(DTT,"quarters")

areTimestampsRegular = logical
   0

areTimestampsSorted = issorted(DTT.Time)

areTimestampsSorted = logical
   1

areTimestampsRegular = 0 indicates that the timestamps of DTT are irregular. areTimestampsSorted = 1 indicates that the timestamps are sorted. Macroeconomic series in this example are timestamped at the end of the month. This quality induces an irregularly measured series.

Remedy the time irregularity by shifting all dates to the first day of the quarter.

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

areTimestampsRegular = logical
   1

DTT is regular.

Create Model Template for Estimation

Suppose that a regression model of CPI quarterly changes onto the GDP rate, with ARMA(1,1) errors, is appropriate.

Create a model template for a regression model with ARMA(1,1) errors template.

Mdl = regARIMA(1,0,1)

Mdl = 
  regARIMA with properties:

     Description: "ARMA(1,1) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 1
               Q: 1
              AR: {NaN} at lag [1]
             SAR: {}
              MA: {NaN} at lag [1]
             SMA: {}
        Variance: NaN

Mdl is a partially specified regARIMA object.

Fit Model to Data

Fit a regression model with ARMA(1,1) errors to the data. Specify the entire series GDP rate and CPI quarterly changes series, and specify the response and predictor variable names.

EstMdl = estimate(Mdl,DTT,ResponseVariable="GDPRate", ...
    PredictorVariables="CPIDel");

 
    Regression with ARMA(1,1) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic      PValue  
                 ________    _____________    __________    __________

    Intercept      0.0162      0.0016077        10.077      6.9995e-24
    AR{1}         0.60515       0.089912        6.7305      1.6906e-11
    MA{1}        -0.16221        0.11051       -1.4678         0.14216
    Beta(1)      0.002221     0.00077691        2.8587       0.0042532
    Variance     0.000113     7.2753e-06        15.533      2.0838e-54

EstMdl is a fully specified, estimated regARIMA object.

Filter Random Gaussian Disturbance Paths

Generate 2 random, independent series of length T_DTT from the standard Gaussian distribution. Store the matrix of series as one variable in DTT.

rng(1,"twister") % For reproducibility
DTT.Z = randn(T_DTT,2);

DTT contains a new variable called Z containing a T_DTT-by-2 matrix of two disturbance paths.

Filter the paths of disturbances through the estimated model. Specify the table variable name containing the disturbance paths.

Tbl2 = filter(EstMdl,DTT,DisturbanceVariable="Z");
tail(Tbl2)

    Time     Interval        GDP         GDPRate      CPIDel              Z                     Y_Response              Y_ErrorInnovation           Y_RegressionInnovation  
    _____    ________    ___________    __________    ______    ______________________    ______________________    __________________________    __________________________

    Q2-07       91        0.00018278      0.018278     1.675     -0.36436      -0.7055    0.016068     0.0071243     -0.0038733     -0.0074997     -0.0001316     -0.0090757
    Q3-07       91        0.00016916      0.016916     1.359    -0.093312      -0.3311    0.015757     0.0084046    -0.00099194     -0.0035197    -0.00044331     -0.0077954
    Q4-07       94        6.1286e-05     0.0061286     3.355      0.48981      -1.5208    0.021299    -0.0041131      0.0052068      -0.016167      0.0050995      -0.020313
    Q1-08       91        9.3272e-05     0.0093272      1.93       1.4014      0.16528    0.033339     0.0082868       0.014898       0.001757       0.017139     -0.0079132
    Q2-08       91        0.00011103      0.011103     3.367     -0.27422     -0.48787     0.02124       0.00594     -0.0029151     -0.0051862      0.0050402       -0.01026
    Q3-08       92        8.9585e-05     0.0089585     1.641      0.67582      0.58697    0.026907      0.017072      0.0071842      0.0062397       0.010707     0.00087209
    Q4-08       92       -0.00016145     -0.016145    -7.098      0.19058     -0.90337     0.02354     0.0061124      0.0020259     -0.0096032        0.00734      -0.010088
    Q1-09       90       -8.6878e-05    -0.0086878     1.137      0.67036      0.37101    0.027439      0.015597      0.0071262       0.003944       0.011239    -0.00060284

size(Tbl2)

ans = 1×2

   248     8

Tbl2 is a 248-by-8 timetable containing all variables in DTT, and the two filtered response paths Y_Response, error model innovation paths Y_ErrorInnovation, and unconditional disturbance paths Y_RegressionInnovation.

Filter Error Paths Through Regression Model with SARIMA Errors

Simulate 100 independent paths of responses by filtering 100 independent paths of errors ${\mathit{z}}_{\mathit{t}}$, where innovations ${\epsilon }_{\mathit{t}}=\sigma {\mathit{z}}_{\mathit{t}}$, through the following regression model with SARIMA${\left(2,1,1\right)}_{12}$ errors.

$$\begin{array}{c}{y}_t=X\left[\begin{array}{c}1.5\\ -2\end{array}\right]+u_t\\ \left(1-0.2L-0.1L^2\right)\left(1-L\right)\left(1-0.01L^{12}\right)\left(1-L^{12}\right)u_t=\left(1+0.5L\right)\left(1+0.02L^{12}\right){\epsilon }_t,\end{array}$$

where $${\epsilon }_t$$ follows a $\mathit{t}$-distribution with 15 degrees of freedom.

Distribution = struct("Name","t","DoF",15);
Mdl = regARIMA(AR={0.2 0.1},SAR=0.01,SARLags=12, ...
    MA=0.5,SMA=0.02,SMALags=12,D=1,Seasonality=12, ...
    Beta=[1.5; -2],Intercept=0,Variance=0.1, ...
    Distribution=Distribution)

Mdl = 
  regARIMA with properties:

     Description: "Regression with ARIMA(2,1,1) Error Model Seasonally Integrated with Seasonal AR(12) and MA(12) (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = 15
       Intercept: 0
            Beta: [1.5 -2]
               P: 27
               D: 1
               Q: 13
              AR: {0.2 0.1} at lags [1 2]
             SAR: {0.01} at lag [12]
              MA: {0.5} at lag [1]
             SMA: {0.02} at lag [12]
     Seasonality: 12
        Variance: 0.1

Simulate a length 25 path of data from the standard bivariate normal distribution for the predictor variables in the regression component.

rng(1,"twister")  % For reproducibility
numObs = 25;
Pred = randn(numObs,2);

Simulate 100 independent paths of errors of length 25 from the standard normal distribution.

numPaths = 100;
Z = randn(numObs,numPaths);

Simulate 100 independent response paths from model by filtering the paths of errors through the model. Supply the predictor data for the regression component.

Y = filter(Mdl,Z,X=Pred);

figure
plot(Y)
title("Simulated Response Paths")

Plot the 2.5th, 50th (median), and 97.5th percentiles of the simulated response paths.

lower = prctile(Y,2.5,2);
middle = median(Y,2);
upper = prctile(Y,97.5,2);

figure
plot(1:25,lower,"r:",1:25,middle,"k", ...
    1:25,upper,"r:")
title("Monte Carlo Summary of Responses")
legend("95% Interval","Median",Location="best")

Compare Responses from filter and simulate

Simulate responses using filter and simulate. Then compare the simulated responses.

Both filter and simulate filter a series of errors to produce output responses y, innovations e, and unconditional disturbances u. The difference is that simulate generates errors from Mdl.Distribution, whereas filter accepts a random array of errors that you generate from any distribution.

Specify the following regression model with ARMA(2,1) errors:

$$\begin{array}{l}\begin{array}{c}{y}_t=X_t\left[\begin{array}{c}0.1\\ -0.2\end{array}\right]+u_t\\ u_t=0.5u_{t-1}-0.8u_{t-2}+{\epsilon }_t-0.5{\epsilon }_{t-1},\end{array}\end{array}$$

where $${\epsilon }_t$$ is Gaussian with variance 0.1.

Mdl = regARIMA(Intercept=0,AR={0.5 -0.8},MA=-0.5, ...
    Beta=[0.1 -0.2],Variance=0.1);

Mdl is a fully specified regARIMA object.

Simulate a one path of bivariate standard normal data for the predictor variables. Then, simulate a path of responses and innovations from the regression model with ARMA(2,1) errors. Supply the simulated predictor data to simulate for the regression component.

rng(1,"twister")        % For reproducibility
Pred = randn(100,2);    % Simulate predictor data
[ySim,eSim] = simulate(Mdl,100,X=Pred);

ySim and eSIM are 100-by-1 vectors of simulated responses and innovations, respectively, from the model Mdl.

Produce model errors by standardizing the simulated innovations. Filter the simulated errors through the model. Supply the predictor data to filter.

z1 = eSim./sqrt(Mdl.Variance);
yFlt1 = filter(Mdl,z1,X=Pred);

yFlt1 is a 100-by-1 vector of responses resulting from filtering the simulated errors z1 through the model Mdl.

Confirm that the simulated responses from simulate and filter are identical by plotting the two series.

figure
h1 = plot(ySim);
hold on
h2 = plot(yFlt1,".");
title("Filtered and Simulated Responses")
legend([h1 h2],["Simulate" "Filter"],Location="best")
hold off

Alternatively, simulate responses by randomly generating your own errors and passing them into filter.

rng(1,"twister")
Pred = randn(100,2);
z2 = randn(100,1);
yFlt2 = filter(Mdl,z2,X=Pred);
figure
h1 = plot(ySim);
hold on
h2 = plot(yFlt2,".");
title("Filtered and Simulated Responses")
legend([h1 h2],["Simulate" "Filter"],Location="best")
hold off

This plot is the same as the previous plot, confirming that both simulation methods are equivalent.

filter multiplies the error, Z, by sqrt(Mdl.Variance) before filtering Z through the model. Therefore, if you want to specify a different distribution, set Mdl.Variance to 1, and then generate your own errors using, for example, random("unif",a,b) for the Uniform(a, b) distribution.

Input Arguments

Mdl — Fully specified regression model with ARIMA errors

regARIMA model object

Fully specified regression model with ARIMA errors, specified as a regARIMA model object created by regARIMA or estimate.

The properties of Mdl cannot contain NaN values.

Z — Error model disturbance series z_t that drives the innovations process ε_t

numeric column vector | numeric matrix

Error model disturbance series z_t that drives the error model innovations process ε_t, specified as a numobs-by-1 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. The innovations process ε_t = σz_t, where σ = sqrt(Mdl.Variance), the standard deviation of the innovations.

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

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

Z is the continuation of the presample disturbances Z0.

Data Types: double

Tbl1 — Time series data

table | timetable

Since R2023b

Time series data containing the error model disturbance series z_t that drives the error model innovations process ε_t, and, optionally, predictor variables x_t, specified as a table or timetable with numvars variables and numobs rows. You can optionally select the disturbance variable or numpreds predictor variables by using the DisturbanceVariable or PredictorVariables name-value arguments, respectively. The innovations process ε_t = σz_t, where σ = sqrt(Mdl.Variance), the standard deviation of the innovations.

Each row is an observation, and measurements in each row occur simultaneously. The selected disturbance variable is a single path (numobs-by-1 vector) or multiple paths (numobs-by-numpaths matrix) of numobs observations of disturbance data.

Each path (column) of the selected disturbance variable is independent of the other paths, but path j of all presample and in-sample variables correspond, for j = 1,…,numpaths. Each selected predictor variable is a numobs-by-1 numeric vector representing one path. The filter function includes all predictor variables in the model when it filters each disturbance path. Variables in Tbl1 represent the continuation of corresponding variables in Presample.

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 strictly ascending or descending.

If Tbl1 is a table, the last row contains the latest 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,X=Pred,Z0=PSZ) specifies the predictor data Pred for the model regression component and the observed errors in the presample period PSZ to initialize the model.

DisturbanceVariable — Disturbance variable z_t to select from Tbl1

string scalar | character vector | integer | logical vector

Since R2023b

Disturbance variable z_t to select from Tbl1 containing the disturbance data to filter through Mdl, specified as one of the following data types:

• String scalar or character vector containing a variable name in Tbl1.Properties.VariableNames

• Variable index (positive integer) to select from Tbl1.Properties.VariableNames

• A logical vector, where DisturbanceVariable(j) = true selects variable j from Tbl1.Properties.VariableNames

The selected variable must be a numeric vector and cannot contain missing values (NaNs).

If Tbl1 has one variable, the default specifies that variable. Otherwise, the default matches the variable to names in Mdl.SeriesName.

Example: DisturbanceVariable="StockRateDist"

Example: DisturbanceVariable=[false false true false] or DisturbanceVariable=3 selects the third table variable as the disturbance variable.

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

X — Predictor data

numeric matrix

Predictor data for the model regression component, specified as a numobs-by-numpreds numeric matrix. numpreds is the number of predictor variables (numel(Mdl.Beta)). Use X only when you supply the numeric array of disturbance data Z.

X must have at least numobs rows. The last row contains the latest predictor data. If X has more than numobs rows, filter uses only the latest numobs rows. Each row of X corresponds to each period in Z (period for which filter filters errors; the period after the presample).

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

PredictorVariables — Predictor variables x_t to select from Tbl1

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

Predictor variables x_t to select from Tbl1 containing the predictor data for the model regression component, 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 vector of unique indices (positive integers) of variables to select from Tbl1.Properties.VariableNames

• A logical vector, where PredictorVariables(j) = true selects variable j from Tbl1.Properties.VariableNames

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

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

Z0 — Presample disturbance data z_t

numeric column vector | numeric matrix

Presample disturbance data z_t to initialize the error model, specified as a numpreobs-by-1 numeric column vector or a numpreobs-by-numprepaths numeric matrix. Use Z0 only when you supply the numeric array of disturbance data Z.

Each row is a presample observation (sampling time), and measurements in each row occur simultaneously. The last row contains the latest presample observation. numpreobs must be at least Mdl.Q to initialize the error model moving average (MA) component. If numpreobs is larger than required, filter uses the latest required observations only.

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

• If Z0 is a column vector, it represents a single disturbance path. filter applies it to each output path.

• If Z0 is a matrix, each column represents a presample disturbance path. filter applies Z0(:,j) to initialize path j. numprepaths must be at least numpaths. If numprepaths > numpaths, filter uses the first size(Z,2) columns only.

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

Data Types: double

U0 — Presample regression innovation data (unconditional disturbances) u_t

numeric column vector | numeric matrix

Presample regression innovation data (unconditional disturbances) u_t to initialize the error model, specified as a numpreobs-by-1 numeric column vector or a numpreobs-by-numprepaths numeric matrix. Use U0 only when you supply the numeric array of disturbance data Z.

Each row is a presample observation (sampling time), and measurements in each row occur simultaneously. The last row contains the latest presample observation. numpreobs must be at least Mdl.P to initialize the error model autoregressive (AR) component. If numpreobs is larger than required, filter uses the latest required observations only.

Columns of U0 are separate, independent presample paths. The following conditions apply:

• If U0 is a column vector, it represents a single path. filter applies it to each path.

• If U0 is a matrix, each column represents a presample path. filter applies U0(:,j) to initialize path j. numprepaths must be at least numpaths. If numprepaths > numpaths, filter uses the first size(Z,2) columns only.

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

Data Types: double

Presample — Presample data

table | timetable

Since R2023b

Presample data containing paths of disturbance z_t or regression innovation (unconditional disturbance) u_t series to initialize the model, specified as a table or timetable, the same type as Tbl1, with numprevars variables and numpreobs rows. Use Presample only when you supply a table or timetable of data Tbl1.

Each selected variable is a single path (numpreobs-by-1 vector) or multiple paths (numpreobs-by-numprepaths matrix) of numpreobs observations representing the presample of the error model disturbance or regression innovation series for DisturbanceVariable, the selected error model disturbance variable in Tbl1.

Each row is a presample observation, and measurements in each row occur simultaneously. numpreobs must be one of the following values:

• At least Mdl.P when Presample provides only presample regression innovations to initialize the error model AR component

• At least Mdl.Q when Presample provides only presample error model disturbances to initialize the error model MA component

• At least max([Mdl.P Mdl.Q]) otherwise

If you supply more rows than necessary, filter uses the latest required number of observations only.

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.

By default, filter sets necessary presample error model disturbances and regression innovations to zero.

If you specify the Presample, you must specify the presample error model disturbance or regression innovation variable name by using the PresampleDisturbanceVariable or PresampleRegressionDisturbanceVariable name-value argument.

PresampleDisturbanceVariable — Error model disturbance variable z_t to select from Presample

string scalar | character vector | integer | logical vector

Since R2023b

Error model disturbance variable z_t to select from Presample containing the presample error model disturbance data, specified as one of the following data types:

• String scalar or character vector containing the variable name to select from Presample.Properties.VariableNames

• Variable index (positive integer) to select from Presample.Properties.VariableNames

• A logical vector, where PresampleDisturbanceVariable(j) = true selects variable j from Presample.Properties.VariableNames

The selected variable must be a numeric vector and cannot contain missing values (NaNs).

If you specify presample error model disturbance data by using the Presample name-value argument, you must specify PresampleDisturbanceVariable.

Example: PresampleDisturbanceVariable="GDP_Z"

Example: PresampleDisturbanceVariable=[false false true false] or PresampleDisturbanceVariable=3 selects the third table variable for presample error model disturbance data.

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

PresampleRegressionDisturbanceVariable — Regression model innovation variable to select from Presample

string scalar | character vector | integer | logical vector

Since R2023b

Regression model innovation variable, associated with unconditional disturbances u_t, to select from Presample containing data for the presample regression model innovations, specified as one of the following data types:

• String scalar or character vector containing a variable name in Presample.Properties.VariableNames

• Variable index (positive integer) to select from Presample.Properties.VariableNames

• A logical vector, where PresampleRegressionDisturbanceVariable(j) = true selects variable j from Presample.Properties.VariableNames

The selected variable must be a numeric vector and cannot contain missing values (NaNs).

If you specify presample regression model innovation data by using the Presample name-value argument, you must specify PresampleRegressionDisturbanceVariable.

Example: PresampleRegressionDisturbanceVariable="StockRateU"

Example: PresampleRegressionDisturbanceVariable=[false false true false] or PresampleRegressionDisturbanceVariable=3 selects the third table variable as the presample regression model innovation data.

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

Note

• NaN values in Z, X, Z0 and U0 indicate missing values. filter removes missing values from specified data by listwise deletion.

  • For the presample, filter horizontally concatenates the possibly jagged arrays Z0 and U0 with respect to the last rows, and then it removes any row of the concatenated matrix containing at least one NaN.

  • For in-sample data, filter horizontally concatenates the possibly jagged arrays Z and X, and then it removes any row of the concatenated matrix containing at least one NaN.

  This type of data reduction reduces the effective sample size and can create an irregular time series.

• For numeric data inputs, filter assumes that you synchronize the presample data such that the latest observations occur simultaneously.

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

• All predictor variables (columns) in X are associated with each input error series to produce numpaths output series.

Output Arguments

Y — Simulated response paths y_t

numeric column vector | numeric matrix

Simulated response paths y_t, returned as a numobs-by-1 column vector or a numobs-by-numpaths numeric matrix. filter returns Y only when you supply the input Z.

For each t = 1, …, numobs, the simulated responses at time t Y(t,:) correspond to the filtered errors at time t Z(t,:) and response path j Y(:,j) corresponds to the filtered disturbance path j Z(:,j) when Z is a matrix.

Y represents the continuation of presample inputs.

E — Simulated, mean-zero innovations paths ε_t

numeric column vector | numeric matrix

Simulated, mean-zero innovations paths ε_t of the error model, returned as a numobs-by-1 column vector or a numobs-by-numpaths numeric matrix. filter returns E only when you supply the input Z.

The dimensions of Y and E correspond.

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

U — Simulated unconditional disturbance paths u_t

numeric column vector | numeric matrix

Simulated unconditional disturbance paths u_t, returned as a numobs-by-1 column vector or a numobs-by-numpaths numeric matrix. filter returns U only when you supply the input Z.

The dimensions of Y and U correspond.

Tbl2 — Simulated response y_t, error model innovation ε_t, and unconditional disturbance u_t paths

table | timetable

Since R2023b

Simulated response y_t, error model innovation ε_t, and unconditional disturbance u_t paths, 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 paths, which are in a numobs-by-numpaths numeric matrix, with rows representing observations and columns representing independent paths, each corresponding to the input observations and paths of the error model disturbance variable in Tbl1. filter names the simulated response variable in Tbl2 responseName_Response, where responseName is Mdl.SeriesName. For example, if Mdl.SeriesName is StockReturns, Tbl2 contains a variable for the corresponding simulated response paths with the name StockReturns_Response.

• The simulated error model innovation paths, which are in a numobs-by-numpaths numeric matrix, with rows representing observations and columns representing independent paths, each corresponding to the input observations and paths of the error model disturbance variable in Tbl1. filter names the simulated error model innovation variable in Tbl2 responseName_ErrorInnovation, where responseName is Mdl.SeriesName. For example, if Mdl.SeriesName is StockReturns, Tbl2 contains a variable for the corresponding simulated error model innovation paths with the name StockReturns_ErrorInnovation.

• The simulated unconditional disturbance paths, which are in a numobs-by-numpaths numeric matrix, with rows representing observations and columns representing independent paths, each corresponding to the input observations and paths of the error model disturbance variable in Tbl1. filter names the simulated unconditional disturbance variable in Tbl2 responseName_RegressionInnovation, where responseName is Mdl.SeriesName. For example, if Mdl.SeriesName is StockReturns, Tbl2 contains a variable for the corresponding simulated unconditional disturbance paths with the name StockReturns_RegressionInnovation.

• All variables Tbl1.

If Tbl1 is a timetable, row times of Tbl1 and Tbl2 are equal.

Alternative Functionality

filter generalizes simulate. Both filter a series of errors to produce responses Y, innovations E, and unconditional disturbances U. However, simulate autogenerates a series of mean zero, unit variance, independent and identically distributed (iid) errors according to the distribution in Mdl. In contrast, filter requires that you specify your own errors, which can come from any distribution.

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] Davidson, R., and J. G. MacKinnon. Econometric Theory and Methods. Oxford, UK: Oxford University Press, 2004.

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

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

[5] Pankratz, A. Forecasting with Dynamic Regression Models. John Wiley & Sons, Inc., 1991.

[6] Tsay, R. S. Analysis of Financial Time Series. 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005.

Version History

Introduced in R2013b

R2023b: filter accepts input data in tables and timetables

In addition to accepting input data (in-sample and presample data) in numeric arrays, filter accepts input data in tables or regular timetables. When you supply data in a table or timetable, the following conditions apply:

• filter chooses the default in-sample error model disturbance series on which to operate, but you can use the specified optional name-value argument to select a different series.

• If you specify optional presample error model disturbance or regression model innovation data to initialize the model, you must also specify the appropriate presample variable names.

• filter returns results in a table or timetable.

Name-value arguments to support tabular workflows include:

• DisturbanceVariable specifies the name of the disturbance series in the input data to filter through the model.

• PredictorVariables specifies the names of the predictor series to select from the input data for the model regression component.

• Presample specifies the input table or timetable of presample error model disturbance or regression innovation data.

• PresampleDisturbanceVariable specifies the name of the error model disturbance series to select from Presample.

• PresampleRegressionDisturbanceVariable specifies the name of the regression model innovation series to select from Presample.