# Create a class 'gsmvar' object defining a reduced form or structural GMVAR, StMVAR, or G-StMVAR model `GSMVAR` creates a class `'gsmvar'` object that defines a reduced form or structural GMVAR, StMVAR, or G-StMVAR model ```r GSMVAR( data, p, M, d, params, conditional = TRUE, model = c("GMVAR", "StMVAR", "G-StMVAR"), parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, calc_cond_moments, calc_std_errors = FALSE, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 ) ## S3 method for class 'gsmvar' logLik(object, ...) ## S3 method for class 'gsmvar' residuals(object, ...) ## S3 method for class 'gsmvar' summary(object, ..., digits = 2) ## S3 method for class 'gsmvar' plot(x, ..., type = c("both", "series", "density")) ## S3 method for class 'gsmvar' print(x, ..., digits = 2, summary_print = FALSE) ``` ## Arguments - `data`: a matrix or class `'ts'` object with `d>1` columns. Each column is taken to represent a single times series. `NA` values are not supported. Ignore if defining a model without data is desired. - `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 (2x1) 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`: number of times series in the system, i.e. `ncol(data)`. This can be used to define GSMVAR models without data and can be ignored if `data` is provided. - `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 c("$_{M},\n$", "$ 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. - `conditional`: a logical argument specifying whether the conditional or exact log-likelihood function should be used. - `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 $(Mpd^2 x q)$ 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`]' $(Mpd^2 x pd^2)$ where `I = diag(p*d^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; thatis, 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 $(d(M-1) x r)$ constraint matrix that satisfies (lambda $_{2}$$,...,$ lambda $_{M}) =$ C_{lambda} gamma where gamma is the new $(r x 1)$ parameter subject to which the model is estimated (similarly to AR parameter constraints). The entries of `C_lambda` must be either positive or zero . Ignore (or set to `NULL`) if the eigenvalues $\lambda_{mi}$ should not be constrained. * `fixed_lambdas` - a length $d(M-1)$ numeric vector (lambda $_{2}$$,...,$ 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). - `calc_cond_moments`: should conditional means and covariance matrices should be calculated? Default is `TRUE` if the model contains data and `FALSE` otherwise. - `calc_std_errors`: should approximate standard errors be calculated? - `stat_tol`: numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime has eigenvalues larger that `1 - stat_tol` the model is classified as non-stationary. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. - `posdef_tol`: numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. - `df_tol`: the parameter vector is considered to be outside the parameter space if all degrees of freedom parameters are not larger than `2 + df_tol`. - `object`: object of class `'gsmvar'` generated by `fitGSMVAR` or `GSMVAR`. - `...`: currently not used. - `digits`: number of digits to be printed. - `x`: object of class `'gsmvar'` generated by `fitGSMVAR` or `GSMVAR`. - `type`: which type figure should be produced? Or both? - `summary_print`: if set to `TRUE` then the print will include log-likelihood and information criteria values. ## Returns Returns an object of class `'gsmvar'` defining the specified reduced form or structural GMVAR, StMVAR, or G-StMVAR model. Can be used to work with other functions provided in `gmvarkit`. Note that the first autocovariance/correlation matrix in `$uncond_moments` is for the lag zero, the second one for the lag one, etc. ## Details If data is provided, then also multivariate quantile residuals (**Kalliovirta and Saikkonen 2010**) are computed and included in the returned object. If the function fails to calculate approximative standard errors and the parameter values are near the border of the parameter space, it might help to use smaller numerical tolerance for the stationarity and positive definiteness conditions. The first plot displays the time series together with estimated mixing weights. The second plot displays a (Gaussian) kernel density estimates of the individual series together with the marginal stationary density implied by the model. The colored regimewise stationary densities are multiplied with the mixing weight parameter estimates. ## Functions * `logLik(gsmvar)`: Log-likelihood method * `residuals(gsmvar)`: residuals method to extract multivariate quantile residuals * `summary(gsmvar)`: summary method * `plot(gsmvar)`: plot method for class 'gsmvar' * `print(gsmvar)`: print method ## About S3 methods If data is not provided, only the `print` and `simulate` methods are available. If data is provided, then in addition to the ones listed above, `predict` method is also available. See `?simulate.gsmvar` and `?predict.gsmvar` for details about the usage. ## Examples ```r # GMVAR(1, 2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) mod12 # GMVAR(1, 2), d=2 model without data mod12_2 <- GSMVAR(p=1, M=2, d=2, params=params12) mod12_2 # StMVAR(1, 2), d=2 model: mod12t <- GSMVAR(gdpdef, p=1, M=2, params=c(params12, 10, 20), model="StMVAR") mod12t # G-StMVAR(1, 1, 1), d=2 model: mod12gs <- GSMVAR(gdpdef, p=1, M=c(1, 1), params=c(params12, 20), model="G-StMVAR") mod12gs # GMVAR(2, 2), d=2 model with mean-parametrization: params22 <- c(0.869, 0.549, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.576, 1.168, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22, parametrization="mean") mod22 # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s ``` ## References * Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. **Journal of Econometrics**, 192 , 485-498. * Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. **Unpublished Revision of HECER Discussion Paper No. 247**. * Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. **Journal of Business & Economic Statistics**. * Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648. ## See Also `fitGSMVAR`, `add_data`, `swap_parametrization`, `GIRF`, `gsmvar_to_sgsmvar`, `stmvar_to_gstmvar`, `reorder_W_columns`, `swap_W_signs`, `update_numtols`

GSMVAR function