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estimate

Fit vector error-correction (VEC) model to data

Syntax

EstMdl = estimate(Mdl,Y)
EstMdl = estimate(Mdl,Y,Name,Value)
[EstMdl,EstSE] = estimate(___)
[EstMdl,EstSE,logL,E] = estimate(___)

Description

example

EstMdl = estimate(Mdl,Y) returns a fully specified VEC(p – 1) model. This model stores the estimated parameter values resulting from fitting the VEC(p – 1) model Mdl to the observed multivariate response series Y using maximum likelihood.

example

EstMdl = estimate(Mdl,Y,Name,Value) uses additional options specified by one or more Name,Value pair arguments. For example, 'Model','H1*','X',X specifies the H1* Johansen form of the deterministic terms and X as exogenous predictor data for the regression component.

example

[EstMdl,EstSE] = estimate(___) returns the estimated, asymptotic standard errors of the estimated parameters using any of the input argument in the previous syntaxes.

example

[EstMdl,EstSE,logL,E] = estimate(___) returns the optimized loglikelihood objective function value (logL) and the multivariate residuals (E).

Examples

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Consider a VEC model for the following seven macroeconomic series, and then fit the model to the data.

  • 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.

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.

Display a short summary from the estimation.

results = summarize(EstMdl)
results = 

  struct with fields:

               Description: "7-Dimensional Rank = 4 VEC(1) Model"
                     Model: "H1"
                SampleSize: 238
    NumEstimatedParameters: 112
             LogLikelihood: -1.4939e+03
                       AIC: 3.2118e+03
                       BIC: 3.6007e+03
                     Table: [133x4 table]
                Covariance: [7x7 double]
               Correlation: [7x7 double]

The Table field of results is a table of parameter estimates and corresponding statistics.

Consider the model and data in Estimate VEC Model, and suppose that the estimation sample starts at Q1 of 1980.

Load the Data_USEconVECModel data set and preprocess the data.

load Data_USEconVECModel
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);

Identify the index corresponding to the start of the estimation sample.

estIdx = FRED.Time(2:end) > '1979-12-31';

Create a default VEC(1) model using the shorthand syntax. Assume that the appropriate cointegration rank is 4. Specify the variable names.

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

Estimate the model using the estimation sample. Specify all observations before the estimation sample as presample data. Also, specify estimation of the H Johansen form of the VEC model, which includes all deterministic parameters.

Y0 = FRED{~estIdx,:};
EstMdl = estimate(Mdl,FRED{estIdx,:},'Y0',Y0,'Model',"H")
EstMdl = 

  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: [17.5698 3.74759 -20.1998 ... and 4 more]'
              Adjustment: [7×4 matrix]
           Cointegration: [7×4 matrix]
                  Impact: [7×7 matrix]
   CointegrationConstant: [85.4825 -57.3569 -81.7344 ... and 1 more]'
      CointegrationTrend: [-0.0264185 -0.00275396 -0.0249583 ... and 1 more]'
                ShortRun: {7×7 matrix} at lag [1]
                   Trend: [0.000514564 -0.000291183 0.00179965 ... and 4 more]'
                    Beta: [7×0 matrix]
              Covariance: [7×7 matrix]

Because the VEC model order p is 2, estimate uses only the last two observations (rows) in Y0 as a presample.

Consider the model and data in Estimate VEC Model.

Load the Data_USEconVECModel data set and preprocess the data.

load Data_USEconVECModel
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);

The Data_Recessions data set contains the beginning and ending serial dates of recessions. Load this data set. Convert the matrix of date serial numbers to a datetime array.

load Data_Recessions
dtrec = datetime(Recessions,'ConvertFrom','datenum');

Create a dummy variable that identifies periods in which the U.S. was in a recession or worse. Specifically, the variable should be 1 if FRED.Time occurs during a recession, and 0 otherwise.

isin = @(x)(any(dtrec(:,1) <= x & x <= dtrec(:,2)));
isrecession = double(arrayfun(isin,FRED.Time));

Create a VEC(1) model using the shorthand syntax. Assume that the appropriate cointegration rank is 4. You do not have to specify the presence of a regression component when creating the model. Specify the variable names.

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

Estimate the model using the entire sample. Specify the predictor identifying whether the observation was measured during a recession. Return the standard errors.

[EstMdl,EstSE] = estimate(Mdl,FRED.Variables,'X',isrecession);

Display the regression coefficient for each equation and the corresponding standard errors.

EstMdl.Beta
EstSE.Beta
ans =

   -1.1975
   -0.0187
   -0.7530
   -0.7094
   -0.5932
   -0.6835
   -4.4839


ans =

    0.1547
    0.0581
    0.1507
    0.1278
    0.2471
    0.1311
    0.7150

EstMdl.Beta and EstSE.Beta are 7-by-1 vectors. Rows correspond to response variables in EstMdl.SeriesNames and columns correspond to predictors.

To check whether the effects of recessions are significant, obtain summary statistics from summarize, and then display the results for Beta.

results = summarize(EstMdl);
isbeta = contains(results.Table.Properties.RowNames,'Beta');
betaresults = results.Table(isbeta,:)
whichsig = EstMdl.SeriesNames(betaresults.PValue < 0.05)
betaresults =

  7x4 table

                   Value      StandardError    TStatistic      PValue  
                 _________    _____________    __________    __________

    Beta(1,1)      -1.1975    0.15469           -7.7411      9.8569e-15
    Beta(2,1)    -0.018738    0.05806          -0.32273          0.7469
    Beta(3,1)     -0.75305    0.15071           -4.9966      5.8341e-07
    Beta(4,1)     -0.70936    0.12776           -5.5521      2.8221e-08
    Beta(5,1)      -0.5932    0.24712           -2.4004        0.016377
    Beta(6,1)     -0.68353    0.13107           -5.2151       1.837e-07
    Beta(7,1)      -4.4839      0.715           -6.2712      3.5822e-10


whichsig = 

  1x6 string array

    "GDP"    "COE"    "HOANBS"    "FEDFUNDS"    "PCEC"    "GPDI"

All series except GDPDEF appear to have a significant recessions effect.

Input Arguments

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VEC model containing unknown parameter values, specified as a vecm model object returned by vecm.

estimate treats non-NaN elements in Mdl as equality constraints and does not estimate the corresponding parameters.

Observed multivariate response series to which estimate fits the model, specified as a numobs-by-numseries numeric matrix. numobs is the sample size and numseries is Mdl.NumSeries, which is the number of response series. Rows correspond to observations, and columns correspond to individual response variables. Y represents the continuation of the presample response series in Y0. The last row contains the latest observation.

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 single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Model','H1*','Y0',Y0,'X',X includes only the deterministic terms in the H1 form of the VEC model, uses the matrix Y0 as presample responses required for estimation, and includes a linear regression component composed of the predictor data in X.

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Presample responses to initiate the model estimation, specified as the comma-separated pair consisting of 'Y0' and a numpreobs-by-numseries numeric matrix.

The columns of Y0 must correspond to the columns of Y. Y0 must have at least Mdl.P rows. If you supply more rows than necessary, estimate uses the latest Mdl.P observations only. The last row contains the latest observation.

By default, estimate uses Y(1:Mdl.P,:) as presample observations, and then fits the model to Y((Mdl.P + 1):end,:). estimate removes the first Mdl.P observations from Y and uses them as presample data. This action reduces the effective sample size.

Data Types: double

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. Rows correspond to observations, and columns correspond to individual predictor variables. All predictor variables are present in the regression component of each response equation.

estimate 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 numobsMdl.P observations to account for the presample removal.

In either case, if you supply more rows than necessary, estimate uses the latest observations only. The last row contains the latest observation.

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

Data Types: double

Johansen form of the VEC(p – 1) model deterministic terms [2], specified as the comma-separated pair consisting of 'Model' and a value in this table (for variable definitions, see Vector Error-Correction Model).

ValueError-Correction TermDescription
'H2'

AB´yt − 1

No intercepts or trends are present in the cointegrating relations, and no deterministic trends are present in the levels of the data.

If all response series have a mean of zero, then specify this model. Otherwise, do not specify this model.

'H1*'

A(B´yt−1+c0)

Intercepts are present in the cointegrating relations, and no deterministic trends are present in the levels of the data.

'H1'

A(B´yt−1+c0)+c1

Intercepts are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data.

'H*'A(B´yt−1+c0+d0t)+c1

Intercepts and linear trends are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data.

'H'A(B´yt−1+c0+d0t)+c1+d1t

Intercepts and linear trends are present in the cointegrating relations, and deterministic quadratic trends are present in the levels of the data.

If quadratic trends are not present in the data, this model can produce good in-sample fits, but poor out-of-sample forecasts.

During estimation, if the overall model constant, overall linear trend, cointegrating constant, or cointegrating linear trend parameters are not in the model, then estimate constrains them to zero. If you specify a different equality constraint, that is, if the properties corresponding to those deterministic terms being constrained to zero have a value other than a vector of NaN values or zeros, then estimate issues an error. To enforce supported equality constraints, choose the Johansen model containing the deterministic term that you want to constrain.

Example: 'Model','H1*'

Data Types: string | char

Estimation information display type, specified as the comma-separated pair consisting of 'Display' and a value in this table.

ValueDescription
'off'estimate does not display estimation information at the command line.
'table'estimate displays a table of estimation information. Rows correspond to parameters, and columns correspond to estimates, standard errors, t statistics, and p values.
'full'In addition to a table of summary statistics, estimate displays the estimated innovations covariance and correlation matrices, loglikelihood value, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and other estimation information.

Data Types: string | char

Maximum number of solver iterations allowed, specified as the comma-separated pair consisting of 'MaxIterations' and a positive numeric scalar.

estimate dispatches MaxIterations to mvregress.

Data Types: double

Note

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

  • For the presample, estimate removes any row containing at least one NaN.

  • For the estimation sample, estimate removes any row of the concatenated data matrix [Y X] containing at least one NaN.

This type of data reduction reduces the effective sample size.

Output Arguments

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Estimated VEC(p – 1) model, returned as a vecm model object. EstMdl is a fully specified vecm model.

Estimated, asymptotic standard errors of the estimated parameters, returned as a structure array containing the fields in this table.

FieldDescription
ConstantStandard errors of the overall model constants (c) corresponding to the estimates in EstMdl.Constant, an Mdl.NumSeries-by-1 numeric vector
AdjustmentStandard errors of the adjustment speeds (A) corresponding to the estimates in EstMdl.Adjustment, an Mdl.NumSeries-by-Mdl.Rank numeric vector
ImpactStandard errors of the impact coefficient (Π) corresponding to the estimates in EstMdl.Impact, an Mdl.NumSeries-by-Mdl.NumSeries numeric vector
ShortRunStandard errors of the short-run coefficients (Φ) corresponding to estimates in EstMdl.ShortRun, a cell vector with elements corresponding to EstMdl.ShortRun
BetaStandard errors of regression coefficients (β) corresponding to the estimates in EstMdl.Beta, an Mdl.NumSeries-by-numpreds numeric matrix
TrendStandard errors of the overall linear time trends (d) corresponding to the estimates in EstMdl.Trend, an Mdl.NumSeries-by-1 numeric vector

If estimate applies equality constraints during estimation by fixing any parameters to a value, then corresponding standard errors of those parameters are 0.

Optimized loglikelihood objective function value, returned as a numeric scalar.

Multivariate residuals from the fitted model, returned as a numeric matrix containing numseries columns.

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

  • Otherwise, E has numobsMdl.P rows to account for the presample removal.

More About

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Vector Error-Correction Model

A vector error-correction (VEC) model is a multivariate, stochastic time series model consisting of a system of m = numseries equations of m distinct, differenced response variables. Equations in the system can include an error-correction term, which is a linear function of the responses in levels used to stabilize the system. The cointegrating rank, r, is the number of cointegrating relations that exist in the system.

Each response equation can include an autoregressive polynomial composed of first differences of the response series (short-run polynomial of degree p – 1), a constant, a time trend, exogenous predictor variables, and a constant and time trend in the error-correction term.

A VEC(p – 1) model in difference-equation notation and in reduced form can be expressed in two ways:

  • This equation is the component form of a VEC model, where the cointegration adjustment speeds and cointegration matrix are explicit, whereas the impact matrix is implied.

    Δyt=A(Byt1+c0+d0t)+c1+d1t+Φ1Δyt1+...+Φp1Δyt(p1)+βxt+εt=c+dt+AByt1+Φ1Δyt1+...+Φp1Δyt(p1)+βxt+εt.

    The cointegrating relations are B'yt – 1 + c0 + d0t and the error-correction term is A(B'yt – 1 + c0 + d0t).

  • This equation is the impact form of a VEC model, where the impact matrix is explicit, whereas the cointegration adjustment speeds and cointegration matrix are implied.

    Δyt=Πyt1+A(c0+d0t)+c1+d1t+Φ1Δyt1+...+Φp1Δyt(p1)+βxt+εt=c+dt+Πyt1+Φ1Δyt1+...+Φp1Δyt(p1)+βxt+εt.

In the equations:

  • yt is an m-by-1 vector of values corresponding to m response variables at time t, where t = 1,...,T.

  • Δyt = ytyt – 1. The structural coefficient is the identity matrix.

  • r is the number of cointegrating relations and, in general, 0 < r < m.

  • A is an m-by-r matrix of adjustment speeds.

  • B is an m-by-r cointegration matrix.

  • Π is an m-by-m impact matrix with a rank of r.

  • c0 is an r-by-1 vector of constants (intercepts) in the cointegrating relations.

  • d0 is an r-by-1 vector of linear time trends in the cointegrating relations.

  • c1 is an m-by-1 vector of constants (deterministic linear trends in yt).

  • d1 is an m-by-1 vector of linear time trend values (deterministic quadratic trends in yt).

  • c = Ac0 + c1 and is the overall constant.

  • d = Ad0 + d1 and is the overall time trend coefficient.

  • Φj is an m-by-m matrix of short-run coefficients, where j = 1,...,p – 1 and Φp – 1 is not a matrix containing only zeros.

  • xt is a k-by-1 vector of values corresponding to k exogenous predictor variables.

  • β is an m-by-k matrix of regression coefficients.

  • εt is an m-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively an m-by-m covariance matrix Σ. For ts, εt and εs are independent.

Condensed and in lag operator notation, the system is

Φ(L)(1L)yt=A(Byt1+c0+d0t)+c1+d1t+βxt+εt=c+dt+AByt1+βxt+εt

where Φ(L)=IΦ1Φ2...Φp1, I is the m-by-m identity matrix, and Lyt = yt – 1.

If m = r, then the VEC model is a stable VAR(p) model in levels of the responses. If r = 0, then the error-correction term is a matrix of zeros, and the VEC(p – 1) model is a stable VAR(p – 1) model in first differences of the responses.

Johansen Form

The Johansen forms of a VEC Model differ with respect to the presence of deterministic terms. As detailed in [2], the estimation procedure differs among the forms. Consequently, allowable equality constraints on the deterministic terms during estimation differ among forms. For more details, see The Role of Deterministic Terms.

This table describes the five Johansen forms and supported equality constraints.

FormError-Correction TermDeterministic CoefficientsEquality Constraints
H2

AB´yt − 1

c = 0 (Constant).

d = 0 (Trend).

c0 = 0 (CointegrationConstant).

d0 = 0 (CointegrationTrend).

You can fully specify B.

All deterministic coefficients are zero.

H1*

A(B´yt−1+c0)

c = Ac0.

d = 0.

d0 = 0.

If you fully specify either B or c0, then you must fully specify the other.

MATLAB® derives the value of c from c0 and A.

All deterministic trends are zero.

H1

A(B´yt−1 + c0) + c1

c = Ac0 + c1.

d = 0.

d0 = 0.

You can fully specify B.

You can specify a mixture of NaN and numeric values for c.

MATLAB derives the value of c0 from c and A.

All deterministic trends are zero.

H*

A(B´yt−1 + c0 + d0t) + c1

c = Ac0 + c1.

d = Ad0.

If you fully specify either B or d0, then you must fully specify the other.

You can specify a mixture of NaN and numeric values for c.

MATLAB derives the value of c0 from c and A.

MATLAB derives the value of d from A and d0.

H

A(B´yt−1+c0+d0t)+c1+d1t

c = Ac0 + c1.

d = A.d0 + d1.

You can fully specify B.

You can specify a mixture of NaN and numeric values for c and d.

MATLAB derives the values of c0 and d0 from c, d, and A.

Algorithms

  • If 1 ≤ Mdl.RankMdl.NumSeries1, as with most VEC models, then estimate performs parameter estimation in two steps.

    1. estimate estimates the parameters of the cointegrating relations, including any restricted intercepts and time trends, by the Johansen method [2].

      • The form of the cointegrating relations corresponds to one of the five parametric forms considered by Johansen in [2] (see 'Model'). For more details, see jcitest and jcontest.

      • The adjustment speed parameter (A) and the cointegration matrix (B) in the VEC(p – 1) model cannot be uniquely identified. However, the product Π = A*Bʹ is identifiable. In this estimation step, B = V1:r, where V1:r is the matrix composed of all rows and the first r columns of the eigenvector matrix V. V is normalized so that Vʹ*S11*V = I. For more details, see [2].

    2. estimate constructs the error-correction terms from the estimated cointegrating relations. Then, estimate estimates the remaining terms in the VEC model by constructing a vector autoregression (VAR) model in first differences and including the error-correction terms as predictors. For models without cointegrating relations (Mdl.Rank = 0) or with a cointegrating matrix of full rank (Mdl.Rank = Mdl.Numseries), estimate performs this VAR estimation step only.

  • You can remove stationary series, which are associated with standard unit vectors in the space of cointegrating relations, from cointegration analysis. To pretest individual series for stationarity, use adftest, pptest, kpsstest, and lmctest. As an alternative, you can test for standard unit vectors in the context of the full model by using jcontest.

  • If 1Mdl.RankMdl.NumSeries1, the asymptotic error covariances of the parameters in the cointegrating relations (which include B, c0, and d0 corresponding to the Cointegration, CointegrationConstant, and CointegrationTrend properties, respectively) are generally non-Gaussian. Therefore, estimate does not estimate or return corresponding standard errors.

    In contrast, the error covariances of the composite impact matrix, which is defined as the product A*Bʹ, are asymptotically Gaussian. Therefore, estimate estimates and returns its standard errors. Similar caveats hold for the standard errors of the overall constant and linear trend (A*c0 and A*d0corresponding to the Constant and Trend properties, respectively) of the H1* and H* Johansen forms.

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.

Introduced in R2017b

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