--- title: "Bayesian Error Correction Models with Priors on the Cointegration Space" author: "Franz X. Mohr" date: "`r Sys.Date()`" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Bayesian Error Correction Models with Priors on the Cointegration Space} %\VignetteEngine{knitr::rmarkdown} \usepackage[utf8]{inputenc} --- ```{r setup, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) ``` ## Introduction This vignette provides the code to set up and estimate a Bayesian vector error correction (BVEC) model with the `bvartools` package. The presented Gibbs sampler is based on the approach of Koop et al. (2010), who propose a prior on the cointegration space. The estimated model has the following form $$\Delta y_t = \Pi y_{t - 1} + \sum_{l = 1}^{p - 1} \Gamma_l \Delta y_{t - l} + C d_t + u_t,$$ where $\Pi = \alpha \beta^{\prime}$ with cointegration rank $r$, $u_t \sim N(0, \Sigma)$ and $d_t$ contains an intercept and seasonal dummies. For an introduction to vector error correction models see [https://www.r-econometrics.com/timeseries/vecintro/](https://www.r-econometrics.com/timeseries/vecintro/). ## Data To illustrate the workflow of the analysis, data set E6 from Lütkepohl (2006) is used, which contains data on German long-term interest rates and inflation from 1972Q2 to 1998Q4. ```{r data, fig.align='center', fig.height=5, fig.width=4.5} library(bvartools) data("e6") plot(e6) # Plot the series ``` The `gen_vec` function produces the data matrices `Y`, `W` and `X` for the BVEC estimator, where `Y` is the matrix of dependent variables, `W` is a matrix of potentially cointegrated regressors, and `X` is the matrix of non-cointegration regressors. ```{r} data <- gen_vec(e6, p = 4, r = 1, const = "unrestricted", season = "unrestricted", iterations = 5000, burnin = 1000) ``` Argument `p` represents the lag order of the VAR form of the model and `r` is the cointegration rank of `Pi`. Function `gen_vec` requires to specify the inclusion of intercepts, trends and seasonal dummies separately. This allows to decide on whether they enter the cointegration term or the non-cointegration part of the model. ## Priors Function `add_priors` adds the necessary prior specifications to object `data`. We For the current application we use non-informative priors. ```{r} data <- add_priors(data, coint = list(v_i = 0, p_tau_i = 1), coef = list(v_i = 0, v_i_det = 0), sigma = list(df = 0, scale = .0001)) ``` ## Estimation ### User-specific algorithm The following code produces posterior draws using the algortihm described in Koop et al. (2010). ```{r flat prior} # Reset random number generator for reproducibility set.seed(7654321) # Obtain data matrices y <- t(data$data$Y) w <- t(data$data$W) x <- t(data$data$X) r <- data$model$rank # Set rank tt <- ncol(y) # Number of observations k <- nrow(y) # Number of endogenous variables k_w <- nrow(w) # Number of regressors in error correction term k_x <- nrow(x) # Number of differenced regressors and unrestrictec deterministic terms k_gamma <- k * k_x # Total number of non-cointegration coefficients k_alpha <- k * r # Number of elements in alpha k_beta <- k_w * r # Number of elements in beta # Priors a_mu_prior <- data$priors$noncointegration$mu # Prior means a_v_i_prior <- data$priors$noncointegration$v_i # Inverse of the prior covariance matrix v_i <- data$priors$cointegration$v_i p_tau_i <- data$priors$cointegration$p_tau_i sigma_df_prior <- data$priors$sigma$df # Prior degrees of freedom sigma_scale_prior <- data$priors$sigma$scale # Prior covariance matrix sigma_df_post <- tt + sigma_df_prior # Posterior degrees of freedom # Initial values beta <- matrix(0, k_w, r) beta[1:r, 1:r] <- diag(1, r) sigma_i <- diag(1 / .0001, k) g_i <- sigma_i iterations <- data$model$iterations # Number of iterations of the Gibbs sampler burnin <- data$model$burnin # Number of burn-in draws draws <- iterations + burnin # Total number of draws # Data containers draws_alpha <- matrix(NA, k_alpha, iterations) draws_beta <- matrix(NA, k_beta, iterations) draws_pi <- matrix(NA, k * k_w, iterations) draws_gamma <- matrix(NA, k_gamma, iterations) draws_sigma <- matrix(NA, k^2, iterations) # Start Gibbs sampler for (draw in 1:draws) { # Draw conditional mean parameters temp <- post_coint_kls(y = y, beta = beta, w = w, x = x, sigma_i = sigma_i, v_i = v_i, p_tau_i = p_tau_i, g_i = g_i, gamma_mu_prior = a_mu_prior, gamma_v_i_prior = a_v_i_prior) alpha <- temp$alpha beta <- temp$beta Pi <- temp$Pi gamma <- temp$Gamma # Draw variance-covariance matrix u <- y - Pi %*% w - matrix(gamma, k) %*% x sigma_scale_post <- solve(tcrossprod(u) + v_i * alpha %*% tcrossprod(crossprod(beta, p_tau_i) %*% beta, alpha)) sigma_i <- matrix(rWishart(1, sigma_df_post, sigma_scale_post)[,, 1], k) sigma <- solve(sigma_i) # Update g_i g_i <- sigma_i # Store draws if (draw > burnin) { draws_alpha[, draw - burnin] <- alpha draws_beta[, draw - burnin] <- beta draws_pi[, draw - burnin] <- Pi draws_gamma[, draw - burnin] <- gamma draws_sigma[, draw - burnin] <- sigma } } ``` Obtain point estimates of cointegration variables: ```{r beta} beta <- apply(t(draws_beta) / t(draws_beta)[, 1], 2, mean) # Obtain means for every row beta <- matrix(beta, k_w) # Transform mean vector into a matrix beta <- round(beta, 3) # Round values dimnames(beta) <- list(dimnames(w)[[1]], NULL) # Rename matrix dimensions beta # Print ``` ### `bvec` objects The `bvec` function can be used to collect output of the Gibbs sampler in a standardised object, which can be used further for forecasting, impulse response analysis or forecast error variance decomposition. ```{r bvec-object} # Number of non-deterministic coefficients k_nondet <- (k_x - 4) * k # Generate bvec object bvec_est <- bvec(y = data$data$Y, w = data$data$W, x = data$data$X[, 1:6], x_d = data$data$X[, -(1:6)], Pi = draws_pi, r = 1, Gamma = draws_gamma[1:k_nondet,], C = draws_gamma[(k_nondet + 1):nrow(draws_gamma),], Sigma = draws_sigma) ``` Posterior draws an be inspected with `plot`: ```{r, message=FALSE, warning=FALSE, fig.align='center', fig.height=5, fig.width=10} plot(bvec_est) ``` Obtain summaries of posterior draws ```{r} summary(bvec_est) ``` ### Default algorithm As an alternative to a user-specific algorithm function `draw_posterior` can be used estimate such a model as well: ```{r, eval = FALSE} bvec_est <- draw_posterior(data) ``` ## Evaluation Posterior draws can be thinned with function `thin`: ```{r thin} bvec_est <- thin(bvec_est, thin = 5) ``` The function `bvec_to_bvar` can be used to transform the VEC model into a VAR in levels: ```{r vec2var} bvar_form <- bvec_to_bvar(bvec_est) ``` The output of `bvec_to_bvar` is an object of class `bvar` for which summary statistics, forecasts, impulse responses and variance decompositions can be obtained in the usual manner using `summary`, `predict`, `irf` and `fevd`, respectively. ```{r} summary(bvar_form) ``` ### Forecasts Forecasts with credible bands can be obtained with the function `predict`. If the model contains deterministic terms, new values can be provided in the argument `new_d`. If no values are provided, the function sets them to zero. For the current model, seasonal dummies need to be provided. They are taken from the original series. The number of rows of `new_d` must be the same as the argument `n.ahead`. ```{r forecasts, fig.width=5.5, fig.height=5.5} # Generate deterministc terms for function predict new_d <- data$data$X[3 + 1:10, c("const", "season.1", "season.2", "season.3")] # Genrate forecasts bvar_pred <- predict(bvar_form, n.ahead = 10, new_d = new_d) # Plot forecasts plot(bvar_pred) ``` ### Forecast error impulse response ```{r feir, fig.width=5.5, fig.height=4.5} FEIR <- irf(bvar_form, impulse = "R", response = "Dp", n.ahead = 20) plot(FEIR, main = "Forecast Error Impulse Response", xlab = "Period", ylab = "Response") ``` ## Forecast error variance decomposition ```{r fevd-oir, fig.width=5.5, fig.height=4.5} bvar_fevd_oir <- fevd(bvar_form, response = "Dp", n.ahead = 20) plot(bvar_fevd_oir, main = "OIR-based FEVD of inflation") ``` ## References Koop, G., León-González, R., & Strachan R. W. (2010). Efficient posterior simulation for cointegrated models with priors on the cointegration space. *Econometric Reviews, 29*(2), 224-242. Lütkepohl, H. (2006). *New introduction to multiple time series analysis* (2nd ed.). Berlin: Springer. Pesaran, H. H., & Shin, Y. (1998). Generalized impulse response analysis in linear multivariate models, *Economics Letters, 58*, 17-29.