uncond_moments_int: Calculate the unconditional mean, variance, the first p...
In gmvarkit: Estimate Gaussian and Student's t Mixture Vector Autoregressive Models

uncond_moments_int

R Documentation

Calculate the unconditional mean, variance, the first p autocovariances, and the first p autocorrelations of a GMVAR, StMVAR, or G-StMVAR process

Description

uncond_moments_int calculates the unconditional mean, variance, the first p autocovariances, and the first p autocorrelations of the specified GMVAR, StMVAR, or G-StMVAR process.

Usage

uncond_moments_int(
  p,
  M,
  d,
  params,
  model = c("GMVAR", "StMVAR", "G-StMVAR"),
  parametrization = c("intercept", "mean"),
  constraints = NULL,
  same_means = NULL,
  weight_constraints = NULL,
  structural_pars = NULL
)

Arguments

`p`	a positive integer specifying the autoregressive order of the model.
`M`	For GMVAR and StMVAR models: a positive integer specifying the number of mixture components. For G-StMVAR models: a size `(2\times 1)` integer vector specifying the number of GMVAR type components `M1` in the first element and StMVAR type components `M2` in the second element. The total number of mixture components is `M=M1+M2`.
`d`	the number of time series in the system.
`params`	a real valued vector specifying the parameter values. For unconstrained models: Should be size `((M(pd^2+d+d(d+1)/2+2)-M1-1)x1)` and have the form `\theta` `=` (`\upsilon``_{1}`, ...,`\upsilon``_{M}`, `\alpha_{1},...,\alpha_{M-1},``\nu``)`, where `\upsilon``_{m}` `= (\phi_{m,0},``\phi``_{m},\sigma_{m})` `\phi``_{m}` `= (vec(A_{m,1}),...,vec(A_{m,p})` and `\sigma_{m} = vech(\Omega_{m})`, m=1,...,M, `\nu``=(\nu_{M1+1},...,\nu_{M})` `M1` is the number of GMVAR type regimes. For constrained models: Should be size `((M(d+d(d+1)/2+2)+q-M1-1)x1)` and have the form `\theta` `= (\phi_{1,0},...,\phi_{M,0},``\psi`, `\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},``\nu`), where `\psi` `(qx1)` satisfies (`\phi``_{1},...,` `\phi``_{M}) =` `C \psi` where `C` is a `(Mpd^2xq)` constraint matrix. For same_means models: Should have the form `\theta` `= (``\mu`,`\psi`, `\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},``\nu``)`, where `\mu``= (\mu_{1},...,\mu_{g})` where `\mu_{i}` is the mean parameter for group `i` and `g` is the number of groups. If AR constraints are employed, `\psi` is as for constrained models, and if AR constraints are not employed, `\psi` `=` (`\phi``_{1},...,``\phi``_{M})`. For models with weight_constraints: Drop `\alpha_1,...,\alpha_{M-1}` from the parameter vector. For structural models: Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector should have the form `\theta` `= (\phi_{1,0},...,\phi_{M,0},``\phi``_{1},...,``\phi``_{M}, vec(W),``\lambda``_{2},...,``\lambda``_{M},\alpha_{1},...,\alpha_{M-1},``\nu``)`, where `\lambda``_{m}=(\lambda_{m1},...,\lambda_{md})` contains the eigenvalues of the `m`th mixture component. If AR parameters are constrained: Replace `\phi``_{1},...,` `\phi``_{M}` with `\psi` `(qx1)` that satisfies (`\phi``_{1},...,` `\phi``_{M}) =` `C \psi`, as above. If same_means: Replace `(\phi_{1,0},...,\phi_{M,0})` with `(\mu_{1},...,\mu_{g})`, as above. If `W` is constrained: Remove the zeros from `vec(W)` and make sure the other entries satisfy the sign constraints. If `\lambda_{mi}` are constrained via `C_lambda`: Replace `\lambda``_{2},...,` `\lambda``_{M}` with `\gamma` `(rx1)` that satisfies (`\lambda``_{2}` `,...,` `\lambda``_{M}) =` `C_{\lambda} \gamma` where `C_{\lambda}` is a `(d(M-1) x r)` constraint matrix. If `\lambda_{mi}` are constrained via `fixed_lambdas`: Drop `\lambda``_{2},...,` `\lambda``_{M}` from the parameter vector. Above, `\phi_{m,0}` is the intercept parameter, `A_{m,i}` denotes the `i`th coefficient matrix of the `m`th mixture component, `\Omega_{m}` denotes the error term covariance matrix of the `m`:th mixture component, and `\alpha_{m}` is the mixing weight parameter. The `W` and `\lambda_{mi}` are structural parameters replacing the error term covariance matrices (see Virolainen, 2022). If `M=1`, `\alpha_{m}` and `\lambda_{mi}` are dropped. If `parametrization=="mean"`, just replace each `\phi_{m,0}` with regimewise mean `\mu_{m}`. `vec()` is vectorization operator that stacks columns of a given matrix into a vector. `vech()` stacks columns of a given matrix from the principal diagonal downwards (including elements on the diagonal) into a vector. In the GMVAR model, `M1=M` and `\nu` is dropped from the parameter vector. In the StMVAR model, `M1=0`. In the G-StMVAR model, the first `M1` regimes are GMVAR type and the rest `M2` regimes are StMVAR type. In StMVAR and G-StMVAR models, the degrees of freedom parameters in `\nu` should be strictly larger than two. The notation is similar to the cited literature.
`model`	is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first `M1` components are GMVAR type and the rest `M2` components are StMVAR type.
`parametrization`	`"intercept"` or `"mean"` determining whether the model is parametrized with intercept parameters `\phi_{m,0}` or regime means `\mu_{m}`, m=1,...,M.
`constraints`	a size `(Mpd^2 \times q)` constraint matrix `C` specifying general linear constraints to the autoregressive parameters. We consider constraints of form (`\phi``_{1},...,``\phi``_{M}) =` `C \psi`, where `\phi``_{m}` `= (vec(A_{m,1}),...,vec(A_{m,p}) (pd^2 x 1), m=1,...,M`, contains the coefficient matrices and `\psi` `(q x 1)` contains the related parameters. For example, to restrict the AR-parameters to be the same for all regimes, set `C`= [`I:...:I`]' `(Mpd^2 \times pd^2)` where `I = diag(pd^2)`. Ignore (or set to `NULL`) if linear constraints should not* be employed.
`same_means`	Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if `M=3`, the argument `list(1, 2:3)` restricts the mean parameters of the second and third regime to be the same but the first regime has freely estimated (unconditional) mean. Ignore or set to `NULL` if mean parameters should not be restricted to be the same among any regimes. This constraint is available only for mean parametrized models; that is, when `parametrization="mean"`.
`weight_constraints`	a numeric vector of length `M-1` specifying fixed parameter values for the mixing weight parameters `\alpha_m, \ m=1,...,M-1`. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.
`structural_pars`	If `NULL` a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of the following elements: `W` - a `(dxd)` matrix with its entries imposing constraints on `W`: `NA` indicating that the element is unconstrained, a positive value indicating strict positive sign constraint, a negative value indicating strict negative sign constraint, and zero indicating that the element is constrained to zero. `C_lambda` - a `(d(M-1) \times r)` constraint matrix that satisfies (`\lambda``_{2},...,` `\lambda``_{M}) =` `C_{\lambda} \gamma` where `\gamma` is the new `(r \times 1)` parameter subject to which the model is estimated (similarly to AR parameter constraints). The entries of `C_lambda` must be either positive or zero. Ignore (or set to `NULL`) if the eigenvalues `\lambda_{mi}` should not be constrained. `fixed_lambdas` - a length `d(M-1)` numeric vector (`\lambda``_{2},...,` `\lambda``_{M})` with elements strictly larger than zero specifying the fixed parameter values for the parameters `\lambda_{mi}` should be constrained to. This constraint is alternative `C_lambda`. Ignore (or set to `NULL`) if the eigenvalues `\lambda_{mi}` should not be constrained. See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is `W` times a time-varying diagonal matrix with positive diagonal entries).

Details

The unconditional moments are based on the stationary distribution of the process.

Value

Returns a list with three components:

$uncond_mean: a length d vector containing the unconditional mean of the process.
$autocovs: an (d x d x p+1) array containing the lag 0,1,...,p autocovariances of the process. The subset [, , j] contains the lag j-1 autocovariance matrix (lag zero for the variance).
$autocors: the autocovariance matrices scaled to autocorrelation matrices.

References

Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.
Virolainen S. 2025. A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics. 43:1, 44-54.
Virolainen S. in press. A Gaussian and Student’s mixture vector autoregressive model with an application to monetary policy shocks. Econometrics and Statistics.