infer

Infer vector autoregression model (VAR) innovations

Syntax

E = infer(Mdl,Y)

E = infer(Mdl,Y,Name,Value)

[E,logL] = infer(___)

Description

E = infer(Mdl,Y) returns the inferred multivariate innovations series from evaluating the fully specified VAR(p) model Mdl at the response data Y.

E = infer(Mdl,Y,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, you can specify presample responses or exogenous predictor data.

[E,logL] = infer(___) returns the loglikelihood objective function value evaluated at E using any of the input arguments in the previous syntaxes.

Examples

Infer VAR(4) Model Innovations

Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate data. Then, infer the model innovations using the estimated model.

Load the Data_USEconModel data set.

load Data_USEconModel

Plot the two series on separate plots.

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

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

Stabilize the CPI by converting it to a series of growth rates. Synchronize the two series by removing the first observation from the unemployment rate series.

rcpi = price2ret(DataTable.CPIAUCSL);
unrate = DataTable.UNRATE(2:end);

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

Mdl = varm(2,4);

Estimate the model using the entire data set.

EstMdl = estimate(Mdl,[rcpi unrate]);

EstMdl is a fully specified, estimated varm model object.

Infer innovations from the estimated model.

E = infer(EstMdl,[rcpi unrate]);

E is a 241-by-2 matrix of inferred innovations. The first and second columns contain the residuals corresponding to the CPI growth rate and unemployment rate, respectively.

Alternatively, you can return residuals when you call estimate by supplying an output variable in the fourth position.

Plot the residuals on separate plots. Synchronize the residuals with the dates by removing any missing observations from the data and removing the first Mdl.P dates.

idx = all(~isnan([rcpi unrate]),2);
datesr = DataTable.Time(idx);

figure;
plot(datesr((Mdl.P + 1):end),E(:,1));
ylabel('Consumer price index');
xlabel('Date');
title('Residual plot');
hold on
plot([min(datesr) max(datesr)],[0 0],'r--');
hold off

figure;
plot(datesr((Mdl.P + 1):end),E(:,2));
ylabel('Unemployment rate');
xlabel('Date');
title('Residual plot');
hold on
plot([min(datesr) max(datesr)],[0 0],'r--');
hold off

The residuals corresponding to the CPI growth rate exhibit heteroscedasticity because the series appears to cycle through periods of higher and lower variance.

Infer Innovations from Model Containing Regression Component

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). Infer model innovations.

Load the Data_USEconModel data set. Compute the real GDP.

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

Plot all variables on separate plots.

figure;
subplot(2,2,1)
plot(DataTable.Time,DataTable.CPIAUCSL);
ylabel('Index');
title('Consumer Price Index');
subplot(2,2,2)
plot(DataTable.Time,DataTable.UNRATE);
ylabel('Percent');
title('Unemployment Rate');
subplot(2,2,3)
plot(DataTable.Time,DataTable.RGDP);
ylabel('Output');
title('Real Gross Domestic Product');
subplot(2,2,4)
plot(DataTable.Time,DataTable.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,DataTable,'InputVariables',inputVariables);
Data.Properties.VariableNames = inputVariables;
Data.UNRATE = DataTable.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 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);

Infer innovations from the estimated model. Supply the predictor data. Return the loglikelihood objective function value.

[E,logL] = infer(EstMdl,Data.Variables,'X',rgcelag4);
logL

logL = 1.7056e+03

E is a 240-by-3 matrix of inferred innovations. The columns contain the residuals corresponding to the CPI growth rate, unemployment rate, and GDP growth rate, respectively.

Plot the residuals on separate plots. Synchronize the residuals with the dates by removing any missing observations from the data and removing the first Mdl.P dates.

idx = all(~isnan([Data.Variables rgcelag4]),2);
datesr = DataTable.Time(idx);

figure;
for j = 1:Mdl.NumSeries
    subplot(2,2,j)
    plot(datesr((Mdl.P + 1):end),E(:,j));
    ylabel(Mdl.SeriesNames{j});
    xlabel('Date');
    title('Residual plot');
    hold on
    plot([min(datesr) max(datesr)],[0 0],'r--');
    hold off
end

The residuals corresponding to the CPI and GDP growth rates exhibit heteroscedasticity because the CPI series appears to cycle through periods of higher and lower variance. Also, the first half of the GDP series seems to have higher variance than the latter half.

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.

Y — Response data

numeric matrix | numeric array

Response data, 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 response series (Mdl.NumSeries). numpaths is the number of response paths.

Rows correspond to observations, and the last row contains the latest observation. Y represents the continuation of the presample response series in Y0.

Columns must correspond to the response variable names in Mdl.SeriesNames.

Pages correspond to separate, independent numseries-dimensional paths. Among all pages, responses in a particular row occur at the same time.

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: 'Y0',Y0,'X',X uses the matrix Y0 as presample responses and the matrix X as predictor data in the regression component.

'Y0' — Presample responses

numeric matrix | numeric array

Presample responses providing initial values for the model, 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, infer uses only the latest Mdl.P observations.

Columns must correspond to the columns of Y.

Pages correspond to separate independent paths.

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

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

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

By default, infer uses Y(1:Mdl.P,:) as presample observations. This action reduces the effective sample size.

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. infer does not use the regression component in the presample period. Therefore, X must have at least as many observations as are used after the presample period.

• If you specify Y0, then X must have at least numobs rows (see Y).

• Otherwise, X must have at least numobs – Mdl.P observations to account for the presample removal.

In either case, if you supply more rows than necessary, infer uses the latest observations only.

Columns correspond to individual predictor variables. All predictor variables are present in the regression component of each response equation.

infer applies X to each path (page) in Y; that is, X represents one path of observed predictors.

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

Data Types: double

Note

NaN values in Y, Y0, and X indicate missing values. infer removes missing values from the data by list-wise deletion.

1. If Y is a 3-D array, then infer horizontally concatenates the pages of Y to form a numobs-by-(numpaths*numseries + numpreds) matrix.

2. If a regression component is present, then infer horizontally concatenates X to Y to form a numobs-by-numpaths*numseries + 1 matrix. infer assumes that the last rows of each series occur at the same time.

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

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

This process ensures that the inferred output 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 Y can differ from the results obtained from each path individually.

This type of data reduction reduces the effective sample size.

Output Arguments

E — Inferred multivariate innovations series

numeric matrix | numeric array

Inferred multivariate innovations series, returned as either a numeric matrix, or as a numeric array that contains columns and pages corresponding to Y.

• If you specify Y0, then E has numobs rows (see Y).

• Otherwise, E has numobs – Mdl.P rows to account for the presample removal.

logL — Loglikelihood objective function value

numeric scalar | numeric vector

Loglikelihood objective function value associated with the VAR model Mdl, returned as a numeric scalar or a numpaths-element numeric vector. logL(j) corresponds to the response path in Y(:,:,j).

Algorithms

• infer infers innovations by evaluating the VAR model Mdl with respect to the innovations using the supplied data Y, Y0, and X. The inferred innovations are

  $${\widehat{\epsilon }}_t=\widehat{\Phi }(L)y_t-\widehat{c}-\widehat{\beta }{x}_t-\widehat{\delta }t.$$

• infer 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, infer 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 infer 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 infer 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, 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.