# Create a class 'stvar' object defining a reduced form or structural smooth transition VAR model `STVAR` creates a class `'stvar'` object that defines a reduced form or structural smooth transition VAR model ```r STVAR( data, p, M, d, params, weight_function = c("relative_dens", "logistic", "mlogit", "exponential", "threshold", "exogenous"), weightfun_pars = NULL, cond_dist = c("Gaussian", "Student", "ind_Student", "ind_skewed_t"), parametrization = c("intercept", "mean"), identification = c("reduced_form", "recursive", "heteroskedasticity", "non-Gaussianity"), AR_constraints = NULL, mean_constraints = NULL, weight_constraints = NULL, B_constraints = NULL, penalized = FALSE, penalty_params = c(0.05, 1), allow_unstab = FALSE, calc_std_errors = FALSE ) ## S3 method for class 'stvar' logLik(object, ...) ## S3 method for class 'stvar' residuals(object, ...) ## S3 method for class 'stvar' summary(object, ..., digits = 2, standard_error_print = FALSE) ## S3 method for class 'stvar' plot(x, ..., plot_type = c("trans_weights", "cond_mean")) ## S3 method for class 'stvar' print(x, ..., digits = 2, summary_print = FALSE, standard_error_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 - `M`: a positive integer specifying the number of regimes - `d`: number of times series in the system, i.e. `ncol(data)`. This can be used to define STVAR models without data and can be ignored if `data` is provided. - `params`: a real valued vector specifying the parameter values. Should have the form $\theta = (\phi_{1},...,\phi_{M},\varphi_1,...,\varphi_M,\sigma,\alpha,\nu)$, where (see exceptions below): * $\phi_{m} = $ the $(d \times 1)$ intercept (or mean) vector of the $m$th regime. * $\varphi_m = (vec(A_{m,1}),...,vec(A_{m,p}))$ $(pd^2 \times 1)$. * - **if `cond_dist="Gaussian"` or `"Student"`:**: $\sigma = (vech(\Omega_1),...,vech(\Omega_M))$ $(Md(d + 1)/2 \times 1)$. - **if `cond_dist="ind_Student"` or `"ind_skewed_t"`:**: $\sigma = (vec(B_1),...,vec(B_M)$ $(Md^2 \times 1)$. * $\alpha = $ the $(a\times 1)$ vector containing the transition weight parameters (see below). * - **if `cond_dist = "Gaussian")`:**: Omit $\nu$ from the parameter vector. - **if `cond_dist="Student"`:**: $\nu \> 2$ is the single degrees of freedom parameter. - **if `cond_dist="ind_Student"`:**: $\nu = (\nu_1,...,\nu_d)$ $(d \times 1)$, $\nu_i \> 2$. - **if `cond_dist="ind_skewed_t"`:**: $\nu = (\nu_1,...,\nu_d,\lambda_1,...,\lambda_d)$ $(2d \times 1)$, $\nu_i \> 2$ and $\lambda_i \in (0, 1)$. For models with... - **`weight_function="relative_dens"`:**: $\alpha = (\alpha_1,...,\alpha_{M-1})$ $(M - 1 \times 1)$, where $\alpha_m$ $(1\times 1), m=1,...,M-1$ are the transition weight parameters. - **`weight_function="logistic"`:**: $\alpha = (c,\gamma)$ $(2 \times 1)$, where $c\in\mathbb{R}$ is the location parameter and $\gamma >0$ is the scale parameter. - **`weight_function="mlogit"`:**: $\alpha = (\gamma_1,...,\gamma_M)$ $((M-1)k\times 1)$, where $\gamma_m$ $(k\times 1)$, $m=1,...,M-1$ contains the multinomial logit-regression coefficients of the $m$th regime. Specifically, for switching variables with indices in $I\subset\lbrace 1,...,d\rbrace$, and with $\tilde{p}\in\lbrace 1,...,p\rbrace$ lags included, $\gamma_m$ contains the coefficients for the vector $z_{t-1} = (1,\tilde{z}_{\min\lbrace I\rbrace},...,\tilde{z}_{\max\lbrace I\rbrace})$, where $\tilde{z}_{i} =(y_{it-1},...,y_{it-\tilde{p}})$, $i\in I$. So $k=1+|I|\tilde{p}$ where $|I|$ denotes the number of elements in $I$. - **`weight_function="exponential"`:**: $\alpha = (c,\gamma)$ $(2 \times 1)$, where $c\in\mathbb{R}$ is the location parameter and $\gamma >0$ is the scale parameter. - **`weight_function="threshold"`:**: $\alpha = (r_1,...,r_{M-1})$ $(M-1 \times 1)$, where $r_1,...,r_{M-1}$ are the thresholds. - **`weight_function="exogenous"`:**: Omit $\alpha$ from the parameter vector. - **AR_constraints:**: Replace $\varphi_1,...,\varphi_M$ with $\psi$ as described in the argument `AR_constraints`. - **mean_constraints:**: Replace $\phi_{1},...,\phi_{M}$ with $(\mu_{1},...,\mu_{g})$ where $\mu_i, \ (d\times 1)$ is the mean parameter for group $i$ and $g$ is the number of groups. - **weight_constraints:**: If linear constraints are imposed, replace $\alpha$ with $\xi$ as described in the argument `weigh_constraints`. If weight functions parameters are imposed to be fixed values, simply drop $\alpha$ from the parameter vector. - **`identification="heteroskedasticity"`:**: $\sigma = (vec(W),\lambda_2,...,\lambda_M)$, where $W$ $(d\times d)$ and $\lambda_m$ $(d\times 1)$, $m=2,...,M$, satisfy $\Omega_1=WW'$ and $\Omega_m=W\Lambda_mW'$, $\Lambda_m=diag(\lambda_{m1},...,\lambda_{md})$, $\lambda_{mi}>0$, $m=2,...,M$, $i=1,...,d$. - **B_constraints:**: For models identified by heteroskedasticity, replace $vec(W)$ with $\tilde{vec}(W)$ that stacks the columns of the matrix $W$ in to vector so that the elements that are constrained to zero are not included. For models identified by non-Gaussianity, replace $vec(B_1),...,vec(B_M)$ with similarly with vectorized versions $B_m$ so that the elements that are constrained to zero are not included. Above, $\phi_{m}$ is the intercept parameter, $A_{m,i}$ denotes the $i$th coefficient matrix of the $m$th regime, $\Omega_{m}$ denotes the positive definite error term covariance matrix of the $m$th regime, and $B_m$ is the invertible $(d\times d)$ impact matrix of the $m$th regime. $\nu_m$ is the degrees of freedom parameter of the $m$th regime. If `parametrization=="mean"`, just replace each $\phi_{m}$ 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. $Bvec()$ is a vectorization operator that stacks the columns of a given impact matrix $B_m$ into a vector so that the elements that are constrained to zero by the argument `B_constraints` are excluded. - `weight_function`: What type of transition weights $\alpha_{m,t}$ should be used? - **`"relative_dens"`:**: c("$\\alpha_{m,t}=\n$", "$ \\frac{\\alpha_mf_{m,dp}(y_{t-1},...,y_{t-p+1})}{\\sum_{n=1}^M\\alpha_nf_{n,dp}(y_{t-1},...,y_{t-p+1})}$"), where $\alpha_m\in (0,1)$ are weight parameters that satisfy $\sum_{m=1}^M\alpha_m=1$ and $f_{m,dp}(\cdot)$ is the $dp$-dimensional stationary density of the $m$th regime corresponding to $p$ consecutive observations. Available for Gaussian conditional distribution only. - **`"logistic"`:**: $M=2$, $\alpha_{1,t}=1-\alpha_{2,t}$, and $\alpha_{2,t}=[1+\exp\lbrace -\gamma(y_{it-j}-c) \rbrace]^{-1}$, where $y_{it-j}$ is the lag $j$ observation of the $i$th variable, $c$ is a location parameter, and $\gamma > 0$ is a scale parameter. - **`"mlogit"`:**: c("$\\alpha_{m,t}=\\frac{\\exp\\lbrace \\gamma_m'z_{t-1} \\rbrace}\n$", "$ {\\sum_{n=1}^M\\exp\\lbrace \\gamma_n'z_{t-1} \\rbrace}$"), where $\gamma_m$ are coefficient vectors, $\gamma_M=0$, and $z_{t-1}$ $(k\times 1)$ is the vector containing a constant and the (lagged) switching variables. - **`"exponential"`:**: $M=2$, $\alpha_{1,t}=1-\alpha_{2,t}$, and $\alpha_{2,t}=1-\exp\lbrace -\gamma(y_{it-j}-c) \rbrace$, where $y_{it-j}$ is the lag $j$ observation of the $i$th variable, $c$ is a location parameter, and $\gamma > 0$ is a scale parameter. - **`"threshold"`:**: $\alpha_{m,t} = 1$ if $r_{m-1}

STVAR function