corr

Model-implied temporal correlations of state-space model

Description

The corr function returns model-implied temporal correlations and covariances of the state or measurement variables in a stationary, time-invariant state-space model. To determine whether the model captures characteristics present in the data, You can compare model-implied associations of present and lagged variables to sample analogues. Other state-space model tools to characterize the dynamics of a specified system include the following:

• The impulse response function (IRF), computed by irf and plotted by irfplot, traces the effects of a shock to a state disturbance on the measurement variables in the system.

• The forecast error variance decomposition (FEVD), computed by fevd, provides information about the relative importance of each state disturbance in affecting the forecast error variance of all measurement variables in the system.

Fully Specified State-Space Model

example

Cyy = corr(Mdl) returns Corr(yt,yt – 1), the model-implied temporal correlation of each measurement variable of the fully specified, standard, stationary state-space model Mdl.

example

Cyy = corr(Mdl,Name,Value) uses additional options specified by one or more name-value arguments. For example, 'Covariance',true,'NumLags',10 specifies returning temporal covariances Cov(yt,yth), h = 0 through 10.

example

[Cyy,Cxx,Cyx] = corr(___) also returns Corr(xt,xth), the correlations between the state variables and their self-lags Cxx, and Corr(yt,xth), the correlations between the state variables and their self lags Cxx and the cross-correlations between the measurement variables and lags of the state variables Cyx using any of the input argument combinations in the previous syntaxes. h is the value of the NumLags name-value argument. corr returns covariances when the value of the Covariance name-value argument is true.

Partially Specified State-Space Model

example

[Cyy,Cxx,Cyx] = corr(___,'Params',estParams) uses the partially specified, standard state-space model Mdl and substitutes the parameter estimates estParams for all unknown parameters in the model.

Examples

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Explicitly create the state-space model

$\begin{array}{l}{x}_{1,t}=0.9{x}_{1,t-1}+0.2{u}_{1,t}\\ {x}_{2,t}=0.1{x}_{1,t-1}+0.3{x}_{2,t-1}+{u}_{2,t}\\ {y}_{1,t}={x}_{1,t}\\ {y}_{2,t}={x}_{1,t}+{x}_{2,t}.\end{array}$

A = [0.9 0; 0.1 0.3];
B = [0.2 0; 0 1];
C = [1 0; 1 1];
Mdl = ssm(A,B,C,'StateType',[0 0])
Mdl =
State-space model type: ssm

State vector length: 2
Observation vector length: 2
State disturbance vector length: 2
Observation innovation vector length: 0
Sample size supported by model: Unlimited

State variables: x1, x2,...
State disturbances: u1, u2,...
Observation series: y1, y2,...
Observation innovations: e1, e2,...

State equations:
x1(t) = (0.90)x1(t-1) + (0.20)u1(t)
x2(t) = (0.10)x1(t-1) + (0.30)x2(t-1) + u2(t)

Observation equations:
y1(t) = x1(t)
y2(t) = x1(t) + x2(t)

Initial state distribution:

Initial state means
x1  x2
0   0

Initial state covariance matrix
x1    x2
x1  0.21  0.03
x2  0.03  1.10

State types
x1          x2
Stationary  Stationary

Mdl is an ssm model object. Because all parameters have known values, the object is fully specified.

Compute the temporal correlations of the measurement variables through lag 1.

Cyy = corr(Mdl)
Cyy =
Cyy(:,:,1) =

1.0000    0.4411
0.9000    0.4072

Cyy(:,:,2) =

0.4411    1.0000
0.3970    0.4212

Rows correspond to lags, columns correspond to the latest observation of the measurement variable in the correlation, and pages correspond to the lagged measurement variable. For example,$\mathrm{Corr}\left({\mathit{y}}_{1,\mathit{t}}{\mathit{y}}_{2,\mathit{t}-1}\right)$ is 0.3970.

Display a heatmap of the temporal correlations between the latest observation of each measurement variable and all lags of measurement variable 1.

Corryy1 = Cyy(:,:,1);
hm = heatmap(Corryy1);
ylabel('h');
hm.YDisplayLabels = ["0" "1"];
xlabel('i')
hm.XDisplayLabels = ["1" "2"];
title('Corr(y_{i,t},y_{1,t - h})') Display a heatmap of the temporal correlations between the current observation of measurement variable 1 and all lags of all measurement variables.

Corry1y = squeeze(Cyy(:,1,:));
hm = heatmap(Corry1y);
ylabel('h');
hm.YDisplayLabels = ["0" "1"];
xlabel('j')
hm.XDisplayLabels = ["1" "2"];
title('Corr(y_{1,t},y_{j,t - h})') Display a heatmap of the temporal correlations between the current observation of all measurement variables and the first lag of all measurement variables.

Corryylag1 = squeeze(Cyy(2,:,:));
hm = heatmap(Corryylag1);
ylabel('i');
hm.YDisplayLabels = ["1" "2"];
xlabel('j')
hm.XDisplayLabels = ["1" "2"];
title('Corr(y_{i,t},y_{j,t - 1})') Explicitly create the state-space model

$\begin{array}{l}{x}_{1,t}=0.9{x}_{1,t-1}+0.2{u}_{1,t}\\ {x}_{2,t}=0.1{x}_{1,t-1}-0.3{x}_{2,t-1}+{u}_{2,t}\\ {y}_{1,t}={x}_{1,t}+{\epsilon }_{1,t}\\ {y}_{2,t}={x}_{1,t}+{x}_{2,t}+{\epsilon }_{2,t}.\end{array}$

A = [0.9 0; 0.1 -0.3];
B = [0.2 0; 0 1];
C = [1 0; 1 1];
D = eye(2);
Mdl = ssm(A,B,C,D,'StateType',[0 0]);

Mdl is an ssm model object.

Compute the temporal correlations of the measurement variables from lag 0 through 20.

numlags = 20;
Cyy = corr(Mdl,'NumLags',numlags);

Cyy is a 21-by-2-by-2 array representing the 20-period temporal correlations of the measurement variables.

Display Cyy(:,2,2), which is the model-implied autocorrelation of ${\mathit{y}}_{2,\mathit{t}}$.

acfy2 = Cyy(:,2,2)
acfy2 = 21×1

1.0000
-0.0466
0.1267
0.0634
0.0723
0.0605
0.0558
0.0498
0.0449
0.0404
⋮

Generate a random path of measurements of length 200 from the model.

rng(1); % For reproducibility
Y = simulate(Mdl,200);

Compute the sample autocorrelation function (ACF) of each variable for 2q0 lags.

sacfy1 = autocorr(Y(:,1),'NumLags',numlags);
sacfy2 = autocorr(Y(:,2),'NumLags',numlags);

Visually compare the model-implied and sample ACF of each measurement variable.

acfy1 = Cyy(:,1,1);
plot([acfy1 sacfy1])
xticklabels(0:numlags)
ylabel("Autocorrelation")
xlabel("Lags")
legend(["ACF(y_{1,t})" "Sample ACF(y_{1,t})"])
title("ACF(y_{1,t})")
axis tight plot([acfy2 sacfy2])
xticklabels(0:numlags)
ylabel("Autocorrelation")
xlabel("Lags")
legend(["ACF(y_{2,t})" "Sample ACF(y_{2,t})"])
title("ACF(y_{2,t})")
axis tight Explicitly create the state-space model

$\begin{array}{l}{x}_{1,t}=0.9{x}_{1,t-1}+0.2{u}_{1,t}\\ {x}_{2,t}=0.1{x}_{1,t-1}-0.3{x}_{2,t-1}+{u}_{2,t}\\ {y}_{1,t}={x}_{1,t}+{\epsilon }_{1,t}\\ {y}_{2,t}={x}_{1,t}+{x}_{2,t}+{\epsilon }_{2,t}.\end{array}$

A = [0.9 0; 0.1 -0.3];
B = [0.2 0; 0 1];
C = [1 0; 1 1];
D = eye(2);
Mdl = ssm(A,B,C,D,'StateType',[0 0]);

Mdl is an ssm model object.

Compute the temporal correlations of the measurement and state variables, as well as their cross-correlations.

[Cyy,Cxx,Cyx] = corr(Mdl);

Each output variable is a 2-by-2-by-2 array containing temporal correlations from lag 0 to 1. Cyy contains the correlations between the measurement variables, Cxx contains the correlations among the state variables, and Cyx contains cross-correlations between the current observation of the measurement variables and lagged state variables.

Plot a heatmap of the correlations between ${\mathit{x}}_{1,\mathit{t}}$ and the lags of all state variables.

Cx1x = squeeze(Cxx(:,1,:));
hm = heatmap(Cx1x);
ylabel('h');
hm.YDisplayLabels = ["0" "1"];
xlabel('j')
hm.XDisplayLabels = ["1" "2"];
title('Corr(x_{1,t},x_{j,t - h})') Plot a heatmap of the cross-correlations between all measurement variables and the lags of ${\mathit{x}}_{2,\mathit{t}}$.

Cyx2 = Cyx(:,:,2);
hm = heatmap(Cyx2);
ylabel('h');
hm.YDisplayLabels = ["0" "1"];
xlabel('i')
hm.XDisplayLabels = ["1" "2"];
title('Corr(y_{i,t},x_{2,t - h})') Simulate data from a known model, fit a model to the data, and then compare sample and model-implied covariances.

Simulate Data

Explicitly create the state-space model

$\begin{array}{l}{x}_{1,t}=0.9{x}_{1,t-1}+0.2{u}_{1,t}\\ {x}_{2,t}=0.1{x}_{1,t-1}-0.3{x}_{2,t-1}+{u}_{2,t}\\ {y}_{1,t}={x}_{1,t}+{\epsilon }_{1,t}\\ {y}_{2,t}={x}_{1,t}+{x}_{2,t}+{\epsilon }_{2,t}.\end{array}$

ADGP = [0.9 0; 0.1 -0.3];
BDGP = [0.2 0; 0 1];
CDGP = [1 0; 1 1];
DDGP = eye(2);

Generate a random path of measurements of length 500 from the model.

rng(1); % For reproducibility
numobs = 500;
Y = simulate(DGP,numobs);

Fit Model to Data

Create a state-space model template to fit to the data by replacing each nonzero state parameter of the data-generating process with a NaN value.

A = [NaN 0; NaN NaN];
B = [NaN 0; 0 NaN];
Mdl = ssm(A,B,CDGP,DDGP,'StateType',[0  0]);

Fit the model template to the data. Specify a random set of positive starting values. Return the vector of estimated parameters.

[~,estParams] = estimate(Mdl,Y,abs(rand(5,1)));
Method: Maximum likelihood (fminunc)
Sample size: 500
Logarithmic  likelihood:     -1694.08
Akaike   info criterion:      3398.15
Bayesian info criterion:      3419.23
|     Coeff       Std Err   t Stat     Prob
---------------------------------------------------
c(1) |  0.91506       0.04229   21.63954   0
c(2) | -0.25898       0.25406   -1.01933  0.30805
c(3) | -0.15383       0.08243   -1.86621  0.06201
c(4) | -0.16808       0.04926   -3.41221  0.00064
c(5) |  1.19275       0.06842   17.43153   0
|
|   Final State   Std Dev    t Stat    Prob
x(1) | -0.12293       0.30568   -0.40217  0.68756
x(2) | -0.80608       0.79263   -1.01697  0.30917

Compute Covariances

Compute model-implied temporal covariances of the measurement variables by passing the state-space model template and estimated parameters to corr. Return the covariances instead of the correlations.

Covyy = corr(Mdl,'Params',estParams,'Covariance',true)
Covyy =
Covyy(:,:,1) =

1.1737    0.1376
0.1589    0.1195

Covyy(:,:,2) =

0.1376    2.5676
0.1259   -0.1297

Covyy is a 2-by-2-by-2 array of temporal covariances of the measurement variables.

Compute the sample covariances of the measurement variables and their first lags.

AugData = lagmatrix(Y,[0 1]);
SCovyy = cov(AugData(2:end,:));

Compare Covariances

Compare the model-implied temporal covariances and the sample covariances.

names = ["y_1t" "y_2t" "y_1t-1" "y_2t-1"];

Covy1y = squeeze(Covyy(:,1,:))';
Covy2y = squeeze(Covyy(:,2,:))';
CovyLag1 = [Covy1y(:,2) Covy2y(:,2) Covy1y(:,1) Covy2y(:,1)]';
ModelCovariances = array2table([Covy1y(:) Covy2y(:) CovyLag1],'RowNames',names,...
'VariableNames',names)
ModelCovariances=4×4 table
y_1t        y_2t      y_1t-1      y_2t-1
_______    ________    _______    ________

y_1t       1.1737     0.13759    0.15891      0.1259
y_2t      0.13759      2.5676    0.11949    -0.12972
y_1t-1    0.15891     0.11949     1.1737     0.13759
y_2t-1     0.1259    -0.12972    0.13759      2.5676

SampleCovariances = array2table(SCovyy,'RowNames',names,'VariableNames',names)
SampleCovariances=4×4 table
y_1t        y_2t       y_1t-1      y_2t-1
________    ________    ________    ________

y_1t        1.2459     0.22058     0.11689    0.070475
y_2t       0.22058      2.5332    0.074687    -0.17466
y_1t-1     0.11689    0.074687      1.2437     0.22158
y_2t-1    0.070475    -0.17466     0.22158      2.5419

The model-implied and sample covariances appear to be similar in magnitude. Note that the covariances are invariant to the reference time; for example, Cov(${\mathit{y}}_{1,\mathit{t}},{\mathit{y}}_{2,\mathit{t}}$) = Cov(${\mathit{y}}_{1,\mathit{t}-1},{\mathit{y}}_{2,\mathit{t}-1}$).

Input Arguments

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Standard, stationary state-space model, specified as an ssm model object returned by ssm or its estimate function.

• Temporal moments are well defined for stationary states. Therefore, corr issues an error if one or more of the following conditions apply:

• At least one state is nonstationary (Mdl.StateType contains at least one value of 2).

• At least one coefficient is time varying.

• Either the measurement or state variable is dimension varying.

• If Mdl is partially specified (that is, it contains unknown parameters), specify estimates of the unknown parameters by using the 'Params' name-value argument. Otherwise, corr issues an error.

• The initial covariance matrix Mdl.Cov0 is implied by the transition equation. Therefore, corr ignores Mdl.Cov0 and values corresponding to Cov0 in the value of Params.

Name-Value 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: 'Covariance',true,'NumLags',10 specifies returning Cov(yt,yt – 10), covariances of self- and cross-lags of the measurement variables from lags 0 through 10.

Maximum number of state or measurement variable lags to include in the computation, specified as a nonnegative integer. corr returns associations from lags 0 through NumLags.

Example: 'NumLags',10

Data Types: double

Estimates of the unknown parameters in the partially specified state-space model Mdl, specified as a numeric vector.

If Mdl is partially specified (contains unknown parameters specified by NaNs), you must specify Params. The estimate function returns parameter estimates of Mdl in the appropriate form. However, you can supply custom estimates by arranging the elements of Params as follows:

• If Mdl is an explicitly created model (Mdl.ParamMap is empty []), arrange the elements of Params to correspond to hits of a column-wise search of NaNs in the state-space model coefficient matrices, initial state mean vector, and covariance matrix.

• If Mdl is time invariant, the order is A, B, C, D, Mean0, and Cov0.

• If Mdl is time varying, the order is A{1} through A{end}, B{1} through B{end}, C{1} through C{end}, D{1} through D{end}, Mean0, and Cov0.

• If Mdl is an implicitly created model (Mdl.ParamMap is a function handle), the first input argument of the parameter-to-matrix mapping function determines the order of the elements of Params.

If Mdl is fully specified, corr ignores Params.

Example: Consider the state-space model Mdl with A = B = [NaN 0; 0 NaN] , C = [1; 1], D = 0, and initial state means of 0 with covariance eye(2). Mdl is partially specified and explicitly created. Because the model parameters contain a total of four NaNs, Params must be a 4-by-1 vector, where Params(1) is the estimate of A(1,1), Params(2) is the estimate of A(2,2), Params(3) is the estimate of B(1,1), and Params(4) is the estimate of B(2,2).

Data Types: double

Flag for returning the temporal covariances instead of the correlations, specified as a value in this table.

ValueDescription
falseOutput arguments represent temporal correlations
trueOutput arguments represent temporal covariances

Example: 'Covariance',true

Data Types: logical

Output Arguments

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Temporal associations of the measurement variables (correlations or covariances), returned as a (NumLags + 1)-by-n-by-n numeric array.

Cyy(h + 1,i,j) is the temporal association between yi,t and yj,th, for h = 0,1,2,...,NumLags, i = 1,2,...,n (number of measurement variables), and j = 1,2,...,n.

Temporal associations of the state variables, returned as a (NumLags + 1)-by-m-by-m numeric array.

Cxx(h + 1,i,j) is the temporal association between xi,t and xj,th, for h = 0,1,2,...,NumLags, i = 1,2,...,m (number of state variables), and j = 1,2,...,m.

Temporal cross-associations between measurement and state variables, returned as a (NumLags + 1)-by-n-by-m numeric array.

Cyx(h + 1,i,j) is the temporal association between yi,t and xj,th, for h = 0,1,2,...,NumLags, i = 1,2,...,n, and j = 1,2,...,m.

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Model-Implied Temporal Associations

Model-implied temporal correlations and covariances measure self- and cross-lag associations between measurement and state variables in a state-space model, as prescribed by the model. To facilitate model specification, you can compare model-implied temporal correlations and covariances to sample analogues.

Consider the time-invariant state-space model at time t

$\begin{array}{l}{x}_{t}=A{x}_{t-1}+B{u}_{t}\\ {y}_{t}=C{x}_{t}+D{\epsilon }_{t}.\end{array}$

Consider a demeaned state-space model represented by state and measurement variables ${\stackrel{˜}{x}}_{0,t}$ and ${\stackrel{˜}{y}}_{0,t}$:

1. Append the state-space model with an appropriately sized constant state vector representing an intercept.

$\begin{array}{l}\left[\begin{array}{c}{x}_{0,t}\\ 1\end{array}\right]=\left[\begin{array}{cc}{A}_{0}& {A}_{1}\\ 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{0,t-1}\\ 1\end{array}\right]+\left[\begin{array}{c}B\\ 0\end{array}\right]{u}_{t}\\ {y}_{t}=\left[\begin{array}{cc}{C}_{0}& {C}_{1}\end{array}\right]\left[\begin{array}{c}{x}_{0,t}\\ 1\end{array}\right]+D{\epsilon }_{t}.\end{array}$

2. Demean the variables.

$\begin{array}{l}{\stackrel{˜}{x}}_{0,t}={x}_{0,t}-E\left({x}_{0,t}\right)={x}_{0,t}-{\left(I-{A}_{0}\right)}^{-1}{A}_{1}\\ {\stackrel{˜}{y}}_{0,t}={y}_{0,t}-E\left({y}_{0,t}\right)={y}_{0,t}-{C}_{1}-{C}_{0}{\left(I-{A}_{0}\right)}^{-1}{A}_{1}.\end{array}$

3. Demean the state-space model and drop constant terms that do not affect the covariance.

$\begin{array}{l}{\stackrel{˜}{x}}_{0,t}={A}_{0}{\stackrel{˜}{x}}_{0,t-1}+B{u}_{t}\\ {\stackrel{˜}{y}}_{0,t}={C}_{0}{\stackrel{˜}{x}}_{0,t}+D{\epsilon }_{t}.\end{array}$

Because the difference between the full state-space model and the demeaned model is the inclusion of constant states, $\text{Cov}\left({x}_{t},{x}_{t-h}\right)=\text{Cov}\left({\stackrel{˜}{x}}_{0,t},{\stackrel{˜}{x}}_{0,t-h}\right)={\Gamma }_{0,h},$ which implies

$Cxx\left(1,:,:\right)={\Gamma }_{0,0}={A}_{0}{\Gamma }_{0,0}{A}_{0}\prime +BB\prime .$

Let ${\stackrel{˜}{\Gamma }}_{0,0}$ be the solution to the equation. Using the demeaned state equation,

$Cxx\left(h+1,:,:\right)={\Gamma }_{0,h}={A}_{o}{\Gamma }_{h-1};h=1,2,....$

The preceding results imply the following:

• $Cyy\left(1,:,:\right)=\text{Cov}\left({y}_{t},{y}_{t}\right)={C}_{0}{\Gamma }_{0,0}{C}_{0}^{\prime }+D{D}^{\prime }.$

• $Cyy\left(h+1,:,:\right)=\text{Cov}\left({y}_{t},{y}_{t-h}\right)={C}_{0}{\Gamma }_{0,h}{C}_{0}^{\prime };h=1,2,....$

• $Cyx\left(h+1:,:\right)=\text{Cov}\left({y}_{t},{x}_{t-h}\right)={C}_{0}{\Gamma }_{0,h};h=0,1,2....$

Tips

• To obtain an association matrix of lead variables from an association matrix of lagged variables, use the identity

$C\left({a}_{t},{b}_{t+h}\right)=C{\left({a}_{t},{b}_{t-h}\right)}^{\prime },$

where:

• C is an association function, either Corr or Cov.

• at and bt are yt or xt.