filter

Filter disturbances through vector error-correction (VEC) model

Syntax

Y = filter(Mdl,Z)

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

[Y,E] = filter(___)

Description

Y = filter(Mdl,Z) returns the multivariate response series Y, which results from filtering the underlying multivariate disturbance series Z. The Z series are associated with the model innovations process through the fully specified VEC(p – 1) model Mdl.

Y = filter(Mdl,Z,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, 'X',X,'Scale',false specifies X as exogenous predictor data for the regression component and refraining from scaling the disturbances by the lower triangular Cholesky factor of the model innovations covariance matrix.

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

Examples

Filter Disturbances Through VEC Model

Consider a VEC model for the following seven macroeconomic series. Then, fit the model to the data and filter disturbances through the fitted model.

• Gross domestic product (GDP)

• GDP implicit price deflator

• Paid compensation of employees

• Nonfarm business sector hours of all persons

• Effective federal funds rate

• Personal consumption expenditures

• Gross private domestic investment

Suppose that a cointegrating rank of 4 and one short-run term are appropriate, that is, consider a VEC(1) model.

Load the Data_USEconVECModel data set.

load Data_USEconVECModel

For more information on the data set and variables, enter Description at the command line.

Determine whether the data needs to be preprocessed by plotting the series on separate plots.

figure;
subplot(2,2,1)
plot(FRED.Time,FRED.GDP);
title('Gross Domestic Product');
ylabel('Index');
xlabel('Date');
subplot(2,2,2)
plot(FRED.Time,FRED.GDPDEF);
title('GDP Deflator');
ylabel('Index');
xlabel('Date');
subplot(2,2,3)
plot(FRED.Time,FRED.COE);
title('Paid Compensation of Employees');
ylabel('Billions of $');
xlabel('Date');
subplot(2,2,4)
plot(FRED.Time,FRED.HOANBS);
title('Nonfarm Business Sector Hours');
ylabel('Index');
xlabel('Date');

figure;
subplot(2,2,1)
plot(FRED.Time,FRED.FEDFUNDS);
title('Federal Funds Rate');
ylabel('Percent');
xlabel('Date');
subplot(2,2,2)
plot(FRED.Time,FRED.PCEC);
title('Consumption Expenditures');
ylabel('Billions of $');
xlabel('Date');
subplot(2,2,3)
plot(FRED.Time,FRED.GPDI);
title('Gross Private Domestic Investment');
ylabel('Billions of $');
xlabel('Date');

Stabilize all series, except the federal funds rate, by applying the log transform. Scale the resulting series by 100 so that all series are on the same scale.

FRED.GDP = 100*log(FRED.GDP);      
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);       
FRED.HOANBS = 100*log(FRED.HOANBS); 
FRED.PCEC = 100*log(FRED.PCEC);     
FRED.GPDI = 100*log(FRED.GPDI);

Create a VEC(1) model using the shorthand syntax. Specify the variable names.

Mdl = vecm(7,4,1);
Mdl.SeriesNames = FRED.Properties.VariableNames

Mdl = 
  vecm with properties:

             Description: "7-Dimensional Rank = 4 VEC(1) Model with Linear Time Trend"
             SeriesNames: "GDP"  "GDPDEF"  "COE"  ... and 4 more
               NumSeries: 7
                    Rank: 4
                       P: 2
                Constant: [7×1 vector of NaNs]
              Adjustment: [7×4 matrix of NaNs]
           Cointegration: [7×4 matrix of NaNs]
                  Impact: [7×7 matrix of NaNs]
   CointegrationConstant: [4×1 vector of NaNs]
      CointegrationTrend: [4×1 vector of NaNs]
                ShortRun: {7×7 matrix of NaNs} at lag [1]
                   Trend: [7×1 vector of NaNs]
                    Beta: [7×0 matrix]
              Covariance: [7×7 matrix of NaNs]

Mdl is a vecm model object. All properties containing NaN values correspond to parameters to be estimated given data.

Estimate the model using the entire data set and the default options. By default, estimate uses the first p = 2 observations as presample data.

EstMdl = estimate(Mdl,FRED.Variables)

EstMdl = 
  vecm with properties:

             Description: "7-Dimensional Rank = 4 VEC(1) Model"
             SeriesNames: "GDP"  "GDPDEF"  "COE"  ... and 4 more
               NumSeries: 7
                    Rank: 4
                       P: 2
                Constant: [14.1329 8.77841 -7.20359 ... and 4 more]'
              Adjustment: [7×4 matrix]
           Cointegration: [7×4 matrix]
                  Impact: [7×7 matrix]
   CointegrationConstant: [-28.6082 109.555 -77.0912 ... and 1 more]'
      CointegrationTrend: [4×1 vector of zeros]
                ShortRun: {7×7 matrix} at lag [1]
                   Trend: [7×1 vector of zeros]
                    Beta: [7×0 matrix]
              Covariance: [7×7 matrix]

EstMdl is an estimated vecm model object. It is fully specified because all parameters have known values. By default, estimate imposes the constraints of the H1 Johansen VEC model form by removing the cointegrating trend and linear trend terms from the model. Parameter exclusion from estimation is equivalent to imposing equality constraints to zero.

Generate a numobs-by-7 series of random Gaussian distributed values, where numobs is the number of observations in the data minus p.

numobs = size(FRED,1) - Mdl.P;
rng(1) % For reproducibility
Z = randn(numobs,Mdl.NumSeries);

To simulate responses, filter the disturbances through the estimated model. Specify the first p = 2 observations as presample data.

Y = filter(EstMdl,Z,'Y0',FRED{1:2,:});

Y is a 238-by-7 matrix of simulated responses. Columns correspond to the variable names in EstMdl.SeriesNames.

Plot the simulated and true responses.

figure;
subplot(2,2,1)
plot(FRED.Time(3:end),[FRED.GDP(3:end) Y(:,1)]);
title('Gross Domestic Product');
ylabel('Index (scaled)');
xlabel('Date');
legend('Simulation','True','Location','Best')
subplot(2,2,2)
plot(FRED.Time(3:end),[FRED.GDPDEF(3:end) Y(:,2)]);
title('GDP Deflator');
ylabel('Index (scaled)');
xlabel('Date');
legend('Simulation','True','Location','Best')
subplot(2,2,3)
plot(FRED.Time(3:end),[FRED.COE(3:end) Y(:,3)]);
title('Paid Compensation of Employees');
ylabel('Billions of $ (scaled)');
xlabel('Date');
legend('Simulation','True','Location','Best')
subplot(2,2,4)
plot(FRED.Time(3:end),[FRED.HOANBS(3:end) Y(:,4)]);
title('Nonfarm Business Sector Hours');
ylabel('Index (scaled)');
xlabel('Date');
legend('Simulation','True','Location','Best')

figure;
subplot(2,2,1)
plot(FRED.Time(3:end),[FRED.FEDFUNDS(3:end) Y(:,5)]);
title('Federal Funds Rate');
ylabel('Percent');
xlabel('Date');
subplot(2,2,2)
plot(FRED.Time(3:end),[FRED.PCEC(3:end) Y(:,6)]);
title('Consumption Expenditures');
ylabel('Billions of $ (scaled)');
xlabel('Date');
subplot(2,2,3)
plot(FRED.Time(3:end),[FRED.GPDI(3:end) Y(:,7)]);
title('Gross Private Domestic Investment');
ylabel('Billions of $ (scaled)');
xlabel('Date');

Filter Multiple Disturbance Paths

Consider this VEC(1) model for three hypothetical response series.

$$\begin{array}{rcl}\Delta y_t& =& c+A{B}^{\prime }{y}_{t-1}+{\Phi }_1\Delta y_{t-1}+{\epsilon }_t\\ & =& \\ & =& \left[\begin{array}{c}-1\\ -3\\ -30\end{array}\right]+\left[\begin{array}{cc}-0.3& 0.3\\ -0.2& 0.1\\ -1& 0\end{array}\right]\left[\begin{array}{ccc}0.1& -0.2& 0.2\\ -0.7& 0.5& 0.2\end{array}\right]y_{t-1}+\left[\begin{array}{ccc}0& 0.1& 0.2\\ 0.2& -0.2& 0\\ 0.7& -0.2& 0.3\end{array}\right]\Delta y_{t-1}+{\epsilon }_t.\end{array}$$

The innovations are multivariate Gaussian with a mean of 0 and the covariance matrix

$$\Sigma =\left[\begin{array}{ccc}1.3& 0.4& 1.6\\ 0.4& 0.6& 0.7\\ 1.6& 0.7& 5\end{array}\right].$$

Create variables for the parameter values.

Adjustment = [-0.3 0.3; -0.2 0.1; -1 0];
Cointegration = [0.1 -0.7; -0.2 0.5; 0.2 0.2];
ShortRun = {[0. 0.1 0.2; 0.2 -0.2 0; 0.7 -0.2 0.3]};
Constant = [-1; -3; -30];
Trend = [0; 0; 0];
Covariance = [1.3 0.4 1.6; 0.4 0.6 0.7; 1.6 0.7 5];

Create a vecm model object representing the VEC(1) model using the appropriate name-value pair arguments.

Mdl = vecm('Adjustment',Adjustment,'Cointegration',Cointegration,...
    'Constant',Constant,'ShortRun',ShortRun,'Trend',Trend,...
    'Covariance',Covariance)

Mdl = 
  vecm with properties:

             Description: "3-Dimensional Rank = 2 VEC(1) Model"
             SeriesNames: "Y1"  "Y2"  "Y3" 
               NumSeries: 3
                    Rank: 2
                       P: 2
                Constant: [-1 -3 -30]'
              Adjustment: [3×2 matrix]
           Cointegration: [3×2 matrix]
                  Impact: [3×3 matrix]
   CointegrationConstant: [2×1 vector of NaNs]
      CointegrationTrend: [2×1 vector of NaNs]
                ShortRun: {3×3 matrix} at lag [1]
                   Trend: [3×1 vector of zeros]
                    Beta: [3×0 matrix]
              Covariance: [3×3 matrix]

Mdl is, effectively, a fully specified vecm model object. That is, the cointegration constant and linear trend are unknown. However, they are not needed for simulating observations or forecasting, given that the overall constant and trend parameters are known.

Generate 1000 paths of 100 observations from a 3-D Gaussian distribution. numobs is the number of observations in the data without any missing values.

numobs = 100;
numpaths = 1000;
rng(1);
Z = randn(numobs,Mdl.NumSeries,numpaths);

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

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

Y and E are 100-by-3-by-1000 matrices of filtered responses and scaled disturbances, respectively.

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

MeanFilt = mean(Y,3);

MeanFilt is a 100-by-3 matrix containing the average of the filtered responses at each time point.

Plot the filtered responses and their averages.

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

Input Arguments

Mdl — VEC model

vecm model object

VEC model, specified as a vecm model object created by vecm or estimate. Mdl must be fully specified.

Z — Underlying multivariate disturbance series

numeric matrix | numeric array

Underlying multivariate disturbance series associated with the model innovations process, 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 pair argument specifies whether to scale the disturbances before filter filters them through Mdl. For more details, see Scale.

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.

Example: 'Scale',false,'X',X does not scale Z by the lower triangular Cholesky factor of the model covariance matrix before filtering, and uses the matrix X as predictor data in the regression component.

'Y0' — Presample responses

numeric matrix | numeric array

Presample responses that provide initial values for the model Mdl, specified as the comma-separated pair consisting of 'Y0' and a numpreobs-by-numseries numeric matrix or a numpreobs-by-numseries-by-numprepaths numeric array.

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

Rows correspond to presample observations, and 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.

Columns must correspond to the response series in Y.

Pages correspond to separate, independent paths.

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

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

Among all pages, observations in a particular row occur at the same time.

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

'X' — Predictor data

numeric matrix

Predictor data for the regression component in the model, specified as the comma-separated pair consisting of 'X' and a numeric matrix containing numpreds columns.

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

Rows correspond to observations, and 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.

Columns correspond to individual predictor variables. 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

'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 the comma-separated pair consisting of 'Scale' and true or false.

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

• If Scale is true, then E(:,:,j) = L*Z(:,:,j) and L = chol(Mdl.Covariance,'lower').

• If Scale is false, then E(:,:,j) = Z(:,:,j).

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 type of data reduction reduces the effective sample size.

Output Arguments

Y — Filtered multivariate response series

numeric matrix | numeric array

Filtered multivariate response series, 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.

E — Multivariate model innovations series

numeric matrix | numeric array

Multivariate model innovations series, 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.

Algorithms

• filter computes Y and 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.

     $$\Delta y_t={\widehat{\Phi }}^{-1}(L)\left(\widehat{c}+\widehat{d}t+\widehat{A}\widehat{B}\prime y_{t-1}+\widehat{\beta }{x}_t+e_t\right).$$

     For variable definitions, see Vector Error-Correction Model.

• 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 do not specify Y0, then t₀ = 0.

  • Otherwise, filter sets t₀ to size(Y0,1) – Mdl.P. Therefore, the times in the trend component are t = t₀ + 1, t₀ + 2,..., t₀ + numobs, where numobs is the effective sample size (size(Y,1) after filter removes missing values). 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 Y (excluding missing values) determines t₀.

References

[1] Hamilton, J. 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.