# Markov Chain Monte Carlo (MCMC) Sampling for the Stochastic Volatility (SV) Model `svlm` is a wrapper around `svsample` with a formula interface. The name derives from SV and `lm` because a linear model with SV residuals is fitted. The function simulates from the joint posterior distribution of the regression coefficients and the SV parameters `mu`, `phi`, `sigma` (and potentially `nu` and `rho`), along with the latent log-volatilities `h_0,...,h_n` and returns the MCMC draws. ```r svlm( formula, data, draws = 10000, burnin = 1000, heavytails = FALSE, asymmetry = FALSE, priorspec = NULL, thin = 1, keeptime = "all", quiet = FALSE, startpara = NULL, startlatent = NULL, parallel = c("no", "multicore", "snow"), n_cpus = 1L, cl = NULL, n_chains = 1L, print_progress = "automatic", expert = NULL, ... ) ``` ## Arguments - `formula`: an object of class `"formula"`, as in `lm`. - `data`: an optional data frame, list or environment (or object coercible by `as.data.frame` to a data frame) containing the variables in the model. If not found in `data`, the variables are taken from `environment(formula)`, typically the environment from which `svlm` is called. - `draws`: single number greater or equal to 1, indicating the number of draws after burn-in (see below). Will be automatically coerced to integer. The default value is 10000. - `burnin`: single number greater or equal to 0, indicating the number of draws discarded as burn-in. Will be automatically coerced to integer. The default value is 1000. - `heavytails`: if `TRUE`, then the residuals of the linear model will follow a t-distribution conditional on the latent volatility process. This model is usually called SV-t. If `priorspec` is given, then `heavytails` is ignored. - `asymmetry`: if `TRUE`, then the residuals of the linear model will follow an SV process with leverage. If `priorspec` is given, then `heavytails` is ignored. - `priorspec`: using the smart constructor `specify_priors`, one can set the details of the prior distribution. - `thin`: single number greater or equal to 1, coercible to integer. Every `thinpara`th parameter and latent draw is kept and returned. The default value is 1, corresponding to no thinning of the parameter draws i.e. every draw is stored. - `keeptime`: Either 'all' (the default) or 'last'. Indicates which latent volatility draws should be stored. - `quiet`: logical value indicating whether the progress bar and other informative output during sampling should be omitted. The default value is `FALSE`, implying verbose output. - `startpara`: **optional** named list, containing the starting values for the parameter draws. If supplied, `startpara` may contain elements named `mu`, `phi`, `sigma`, `nu`, `rho`, `beta`, and `latent0`. The default value is equal to the prior mean. In case of parallel execution with `cl` provided, `startpara` can be a list of named lists that initialize the parallel chains. - `startlatent`: **optional** vector of length `length(y)`, containing the starting values for the latent log-volatility draws. The default value is `rep(-10, length(y))`. In case of parallel execution with `cl` provided, `startlatent` can be a list of named lists that initialize the parallel chains. - `parallel`: **optional** one of `"no"` (default), `"multicore"`, or `"snow"`, indicating what type of parallellism is to be applied. Option `"multicore"` is not available on Windows. - `n_cpus`: **optional** positive integer, the number of CPUs to be used in case of parallel computations. Defaults to `1L`. Ignored if parameter `cl` is supplied and `parallel != "snow"`. - `cl`: **optional** so-called SNOW cluster object as implemented in package `parallel`. Ignored unless `parallel == "snow"`. - `n_chains`: **optional** positive integer specifying the number of independent MCMC chains - `print_progress`: **optional** one of `"automatic"`, `"progressbar"`, or `"iteration"`, controls the output. Ignored if `quiet` is `TRUE`. - `expert`: **optional** named list of expert parameters. For most applications, the default values probably work best. Interested users are referred to the literature provided in the References section. If `expert` is provided, it may contain the following named elements: - **interweave**: Logical value. If `TRUE` (the default), then ancillarity-sufficiency interweaving strategy (ASIS) is applied to improve on the sampling efficiency for the parameters. Otherwise one parameterization is used. - **correct_model_misspecification**: Logical value. If `FALSE` (the default), then auxiliary mixture sampling is used to sample the latent states. If `TRUE`, extra computations are made to correct for model misspecification either ex-post by reweighting or on-line using a Metropolis-Hastings step. - ``...``: Any extra arguments will be forwarded to `updatesummary`, controlling the type of statistics calculated for the posterior draws. ## Returns The value returned is a list object of class `svdraws` holding - **para**: `mcmc.list` object containing the **parameter** draws from the posterior distribution. - **latent**: `mcmc.list` object containing the **latent instantaneous log-volatility** draws from the posterior distribution. - **latent0**: `mcmc.list` object containing the **latent initial log-volatility** draws from the posterior distribution. - **tau**: `mcmc.list` object containing the **latent variance inflation factors** for the sampler with conditional t-innovations **(optional)**. - **beta**: `mcmc.list` object containing the **regression coefficient** draws from the posterior distribution **(optional)**. - **y**: the left hand side of the observation equation, usually the argument `y`. In case of an AR(`k`) specification, the first `k` elements are removed. - **runtime**: `proc_time` object containing the run time of the sampler. - **priors**: a `priorspec` object containing the parameter values of the prior distributions for `mu`, `phi`, `sigma`, `nu`, `rho`, and `beta`s, and the variance of specification for `latent0`. - **thinning**: `list` containing the thinning parameters, i.e. the arguments `thinpara`, `thinlatent` and `keeptime`. - **summary**: `list` containing a collection of summary statistics of the posterior draws for `para`, `latent`, and `latent0`. - **meanmodel**: `character` containing information about how `designmatrix` was employed. - **svlm**: a flag for the use of `svlm` - **model_terms**: helper object that represents the formula - **formula**: argument `formula` - **xlevels**: helper object that is needed to interpret the formula To display the output, use `print`, `summary` and `plot`. The `print` method simply prints the posterior draws (which is very likely a lot of output); the `summary` method displays the summary statistics currently stored in the object; the `plot` method `plot.svdraws` gives a graphical overview of the posterior distribution by calling `volplot`, `traceplot` and `densplot` and displaying the results on a single page. ## Details For details concerning the algorithm please see the paper by Kastner and Frühwirth-Schnatter (2014) and Hosszejni and Kastner (2019). ## Examples ```r # Simulate data n <- 50L dat <- data.frame(x = runif(n, 3, 4), z = runif(n, -1, -0.5)) designmatrix <- matrix(c(dat$x, dat$x^2, log10(dat$x), dat$z), ncol = 4) betas <- matrix(c(-1, 1, 2, 0), ncol = 1) y <- designmatrix %*% betas + svsim(n)$y dat$y <- y # Formula interface res <- svlm(y ~ 0 + x + I(x^2) + log10(x) + z, data = dat) # Prediction predn <- 10L preddat <- data.frame(x = runif(predn, 3, 4), z = runif(predn, -1, -0.5)) pred <- predict(res, newdata = preddat, steps = predn) ``` ## References Kastner, G. and Frühwirth-Schnatter, S. (2014). Ancillarity-sufficiency interweaving strategy (ASIS) for boosting MCMC estimation of stochastic volatility models. **Computational Statistics & Data Analysis**, 76 , 408--423, tools:::Rd_expr_doi("10.1016/j.csda.2013.01.002") . Hosszejni, D. and Kastner, G. (2019). Approaches Toward the Bayesian Estimation of the Stochastic Volatility Model with Leverage. **Springer Proceedings in Mathematics & Statistics**, 296 , 75--83, tools:::Rd_expr_doi("10.1007/978-3-030-30611-3_8") . ## See Also `svsample`, `svsim`, `specify_priors`

svlm function