# VGLTSM family function for the Two--dimensional Error--Correction Model (Engle and Granger, 1987) for $I(1)$--variables Estimates a bidimensional error-correction model of order--($K$, $L$), as proposed by Engle--Granger (Two step--approach; 1987), with bivariate normal errors by maximum likelihood estimation using Fisher scoring. ```r ECM.EngleGran(ecm.order = c(1, 1), zero = c("var", "cov"), resids.pattern = c("intercept", "trend", "neither", "both")[1], lag.res = 1, lmean = "identitylink", lvar = "loglink", lcov = "identitylink", ordtsDyn = 0) ``` ## Arguments - `ecm.order`: Length--2 (positive) integer vector. The order of the ECM model. - `zero`: Integer or character--string vector. Details at `zero`. - `resids.pattern`: Character. How the static linear regression $y[2, t] ~ y[1, t]$ must be settle to estimate the residuals $zhat[t]$. The default is a linear model with intercept, and no trend term. See below for details. - `lag.res`: Numeric, single positive integer. The error term for the long--run equilibrium path is lagged up to order `lag.res`. See below for further details. - `lmean, lvar, lcov`: Same as `MVNcov`. - `ordtsDyn`: Positive integer. Allows to compare the estimated coefficients with those provided by the package 'tsDyn'. See below for further details. ## Details This is an implementation of the two--step approach as proposed by Engle--Granger [1987] to estimate an order--($K$, $L$) bidimensional error correction model (ECM) with bivariate normal errors. This ECM class models the dynamic behaviour of two cointegrated $I(1)$-variables, say $y[1, t]$ and $y[2, t]$ with, probably, $y[2, t]$ a function of $y[1, t]$. Note, the response must be a two--column matrix, where the first entry is the regressor, i.e, $y[1, t]$ above, and the regressand in the second colum. See Example 2 below. The general specification of the ECM class described by this family function is $$ \Delta y_{1, t} |\Phi_{t - 1} =~\phi_{0, 1} + \gamma_1 \widehat{z}_{t - k} +\sum_{i = 1}^K \phi_{1, i} \Delta y_{2, t - i} +\sum_{j = 1}^L \phi_{2, j} \Delta y_{1, t - j} +\varepsilon_{1, t},\Deltay[1, t] |\Phi[t - 1] = \phi[0, 1] + \gamma[1] zhat[t - k] +\sum_i^K \phi[1, i] \Delta y[2, t - i] +\sum_j^L \phi[2, j] \Delta y[1, t - j] + e[1, t], $$ $$ \Delta y_{2, t} |\Phi_{t - 1}=~\psi_{0, 1} + \gamma_2 \widehat{z}_{t - k} +\sum_{i = 1}^K \psi_{1, i} \Delta y_{1, t - i} +\sum_{j = 1}^L \psi_{2, j} \Delta y_{2, t - j} +\varepsilon_{2, t}.\Deltay[2, t] |\Phi [t - 1] = \psi[0, 1] + \gamma[2] zhat[t - k] +\sum_i^K \psi[1, i] \Delta y[1, t - i] +\sum_j^L \psi[2, j] \Delta y[2, t - j] + e[1, t]. $$ Under the binormality assumption on the errors c("$\n$", "$(e[1, t], e[2, t])^T\n$") with covariance matrix $V$, model above can be seen as a VGLM fitting linear models over the conditional means, c("$\n$", "$\\mu\\Deltay[1, t] = E(\\Deltay[1, t] | \\Phi[t - 1])$") and c("$\n$", "$\\mu\\Deltay[2, t] = E(\\Deltay[2, t] | \\Phi[t - 1])$"), producing $$ (\Delta y_{1, t} |\Phi_{t - 1} ,\Delta y_{2, t} |\Phi_{t - 1} )^T \sim N_{2}(\mu_{\Delta y_{1, t}} ,\mu_{\Delta y_{2, t}}, \boldsymbol{\textrm{V}})(\Deltay[1, t] | \Phi[t - 1], \Deltay[2, t] | \Phi[t - 1])^T ~N2(\mu \Deltay[1, t], \mu \Deltay[2, t] , V). $$ The covariance matrix is assumed to have elements c("$\n$", "$\\sigma^2[1], \\sigma^2[2],$") and $Cov[12].$ Hence, the parameter vector is $$ \boldsymbol{\theta} =(\phi_{0, 1}, \gamma_1, \phi_{1, i}, \phi_{2, j},\psi_{0, 1}, \gamma_2, \psi_{1, i}, \psi_{2, j},\sigma_1^2, \sigma_2^2, \textrm{Cov}_{12})^T,\theta = (\phi[0, 1], \gamma[1], \phi[1, i], \phi[2, j],\psi[0, 1], \gamma[2], \psi[1, i], \psi[2, j],\sigma^2[1], \sigma^[2], Cov[12])^T, $$ for $i = 1, \ldots, K$ and $j = 1, \ldots, L$. The linear predictor is $$ \boldsymbol{\eta} = (\mu_{\Delta y_{1, t} },\mu_{\Delta y_{2, t} },{\color{blue}\texttt{loglink}}~\sigma_1^2,{\color{blue}\texttt{loglink}}~\sigma_2^2,\textrm{Cov}_{12})^T.\eta = (\mu \Deltay[1],\mu \Deltay[2],log \sigma^2[1], log \sigma^2[2],Cov[12] )^T. $$ The estimated cointegrated vector, c("$\n$", "$\\betahat*$") = c("$\n$", "$(1, -\\betahat)^T$") is obtained by linear regression depending upon `resids.pattern`, as follows: 1) c("$\n$", "$y[2, t] = \\beta[0] + \\beta[1] * y[1, t] + z[t]$"), if `resids.pattern = "intercept"`, 2) c("$\n$", "$y[2, t] = \\beta[1] * y[1, t] + \\beta[2] * t + z[t]$"), if `resids.pattern = "trend"`, 3) c("$\n$", "$y[2, t] = \\beta[1] * y[1, t] + z[t]$"), if `resids.pattern = "neither"`, or else, 4) c("$\n$", "$y[2, t] = \\beta[0] + \\beta[1] * y[1, t] + \\beta[2] * t + z[t]$"), if `resids.pattern = "both"`, where c("$\n$", "$\\betahat* = (\\beta[0], \\beta[1], \\beta[2])^T,\n$") and $z[t]$ assigns the error term. Note, the **estimated residuals**, $zhat[t]$ are (internally) computed from any of the linear models 1) -- 4) selected, and then lagged up to order `alg.res`, and embedded as explanatories in models $\Deltay[1, t] |\Phi[t - 1]$ and $\Deltay[2, t] |\Phi[t - 1]$ above. By default, $zhat[t - 1]$ are considered (as `lag.res` = 1), although it may be any lag $zhat[t - k]$, for $k > 0$. Change this through argument `lag.res`. ## Returns An object of class `"vglmff"` (see `vglmff-class`) to be used by VGLM/VGAM modelling functions, e.g., `vglm` or `vgam`. ## Note Reduced--Rank VGLMs (RR-VGLMs) can be utilized to aid the increasing number of parameters as $K$ and $L$ grows. See `rrvglm`. By default, c("$\n$", "$\\sigma^2[1], \\sigma^[2]$") and $Cov[12]$ are intercept--only. Set argument `zero` accordingly to change this. Package `tsDyn` also has routines to fit ECMs. However, the bivariate--ECM handled (similar to that one above) differs in their parametrization: `tsDyn` considers the **current** estimated residual, $zhat[t]$ instead of $zhat[t - 1]$ in models $\Deltay[1, t] |\Phi[t - 1]$ and $\Deltay[2, t] |\Phi[t - 1]$. See Example 3 below which compares ECMs fitted with `VGAMextra` and `tsDyn`. ## References Engle, R.F. and Granger C.W.J. (1987) Co-integration and error correction: Representation, estimation and testing. **Econometrica**, `55`(2), 251--276. Pfaff, B. (2011) **Analysis of Integrated and Cointegrated Time Series with `R`.** Seattle, Washington, USA: **Springer**. ## Author(s) Victor Miranda ## See Also `MVNcov`, `rrvglm`, `CommonVGAMffArguments`, `Links`, `vglm`. ## Examples ```r ## Example 1. Comparing the Engle -- Granger procedure carried oud by two procedures. ## ECM.EngleGran() makes easier the fitting process. ## Here, we will use: ## A) The R code 4.2, in Chapter 4, Pfaff (2011). ## This code 1) generates artificial data and 2) fits an ECM, following ## the Engle --Granger procedure. ## B) The ECM.EngleGran() family function to fit the same model assuming ## bivariate normal innovations. ## The downside in the R code 4.2 is the assumption of no--correlation among ## the errors. These are generated indenpendently. ## A) ## STEP 1. Set up the data (R code as in Pfaff (2011)). nn <- 100 set.seed(123456) e1 <- rnorm(nn) # Independent of e2 e2 <- rnorm(nn) y1 <- cumsum(e1) y2 <- 0.6 * y1 + e2 lr.reg <- lm(y2 ~ y1) error <- residuals(lr.reg) error.lagged <- error[-c(nn - 1, nn)] dy1 <- diff(y1) dy2 <- diff(y2) diff.dat <- data.frame(embed(cbind(dy1, dy2), 2)) colnames(diff.dat) <- c('dy1', 'dy2', 'dy1.1', 'dy2.1') ## STEP 2. Fit the ECM model, using lm(), R code as in Pfaff (2011). ecm.reg <- lm(dy2 ~ error.lagged + dy1.1 + dy2.1, data = diff.dat) summary(ecm.reg) ## B) Now, using ECM.EngleGran() and VGLMs, the steps at A) can be skipped. ## Enter the I(1)--variables in the response vector only, putting down the ## the dependent variable from the I(1) set, i.e. y2, in the second column. coint.data <- data.frame(y1 = y1, y2 = y2) fit.ECM <- vglm(cbind(y1, y2) ~ 1, ECM.EngleGran, data = coint.data, trace = TRUE) ## Check coefficients ## coef(fit.ECM, matrix = TRUE) ## Compare 'Diff2' with summary(ecm.reg) coef(summary(ecm.reg)) head(depvar(fit.ECM)) # The estimated differences (first order) vcov(fit.ECM) constraints(fit.ECM, matrix = TRUE) ## Not run: ### Example 2. Here, we compare ECM.EngleGran() from VGAMextra with VECM() from ## package "tsDyn" when fitting an ECM(1, 1). We will make use of ## the argument 'ordtsDyn' so that the outcomes can be compared. library("tsDyn") # Need to be installed first. fit.tsDyn1 <- with(coint.data, VECM(cbind(y2, y1), lag = 1, estim = "2OLS")) # MODEL 1 summary(fit.tsDyn1) ### Fit same model using ECM.EngleGran(). NOTE: Set ordtsDyn = 1 !! # MODEL 2 fit.ECM.2 <- vglm(cbind(y1, y2) ~ 1, ECM.EngleGran(ecm.order = c(1, 1), resids.pattern = "neither", ordtsDyn = 1), data = coint.data, trace = TRUE) coef.ECM.2 <- coef(fit.ECM.2, matrix = TRUE) fit.tsDyn1$coefficients ## From pakage 'tsDyn'. t(coef.ECM.2[, 1:2][c(2, 1, 4, 3), ][, 2:1]) ## FROM VGAMextra ### Example 3. An ECM(2, 2), with residuals estimated by OLS, with NO intercept ### and NO trend term. The data set is 'zeroyld', from package tsDyn. ### ECM.EngleGran() and with VECM() willbe compared again. data(zeroyld, package = "tsDyn") # Fit a VECM with Engle-Granger 2OLS estimator: vecm.eg <- VECM(zeroyld, lag=2, estim = "2OLS") summary(vecm.eg) # For the same data, fit a VECM with ECM.EngleGran(), from VGAMextra. # Set ordtsDyn = 1 for compatibility! fit.ECM.3 <- vglm(cbind(long.run, short.run) ~ 1, ECM.EngleGran(ecm.order = c(2, 2), resids.pattern = "neither", ordtsDyn = 1), data = zeroyld, trace = TRUE) coef.ECM.3 <- coef(fit.ECM.3, matrix = TRUE) #### Compare results vecm.eg$coefficients # From tsDyn t(coef.ECM.3[, 1:2][c(2, 1, 5, 3, 6, 4 ),][, 2:1]) # FROM VGAMextra ## End(Not run) ```

ECM.EngleGran function