# Summary methods for Vector Generalized Time Series Models S4 summary methods for models fitted with time series family functions from `VGAMextra`. These function are all `methods` for objects of class `vglm` with signature `vgltsmff-class`. methods ## Details Implementation of vector generalized time series (TS) family functions (**vgltsmff**) in `VGAMextra` is entirely based on the structure of family functions of class `vglmff-class` from `VGAM`. More precisely, **vgltsmff** family functions can be created by calls of the form `new("vgltsmff",...)`, following the structure `vglmff-class`. See `vglmff-class` for additional details. In this line, specific S4 dispatching methods are currently implemented at `VGAM` to show (or plot) essential statistical information about the model fitted. For the generic `summary`, specifically, S4 methods for objects with signature `vgtsff` are incorporated in `VGAMextra` to display supplementary analyses commonly required by TS practicioners. That is, additional information to the default output shown by `summaryvglm` for family functions at `VGAM`, as follows: a) The standard errors, which are computed from the asymptotic distribution of the MLE estimates, unlike the asymptotic approach (z-value) from `VGAM`. b) Checks on stationarity and/or invertibility for autoregressive (AR), moving average (MA), and autoregressive moving-average (ARMA) models via the polynomial roots. c) The AIC, AICC and BIC criteria for model identification. Notice that, for intercept-only models in the 'vglm' context, the asypmtotic distribution of the estimates, either conditional or unconditional, will coincide with the theoretical distributions as long as $n$ increases. In particular, for the AR($p$) process, the MLEs and the Yule-Walker estimates will concur asymptotically. Where covariates or parameter constraints are involved, the standard errors for the estimates from time series family functions at `VGAMextra` are calculated from the predicted values allocated in the slot `@predictors`, when `summary(...)` is called. In this case, the **conditional** mean, c("$\n$", "$E[\\eta(j) |$") x $]$, is considered as the estimate, where: $$ \eta_j = \sum_{k = 1}^{p} \beta_{(j)k} \times x_{k},\eta[ j ] = \sum \beta[ (j) k] * x[ k ], $$ for $j = 1, \ldots, M$. ## Note As for the intercept, notice that this is called **drift-term** at `ARXff` and `ARMAXff`, whilst it is refered as **intercept** in `MAXff`. This parameter is also estimated by TS family functions in `VGAMextra`. In the MA model, particularly, it is the mean of the process. The drift-term, denoted as $\mu^{*} $, is linearly linked to the mean, $\mu$, of the AR and ARMA processes in turn, as follows: $$ \mu \to \frac{\mu^{*} }{1 - \sum \theta_i}.\mu -> \mu^* /(1 - \sum \theta[i]). $$ Here, $\theta[i]$ are the AR coefficients. Hence, the standard error for the **drift-term** is accordingly computed based on the asymptotic distribution of the mean. More precisely, the relation $$ V(\mu^{*}) = (1 - \sum \theta_i)^{-2} \times\frac{\sigma_{\varepsilon}^2 }{n},(1 - \sum \theta[i])^(-2) * \sigma[e]^2 / n, $$ is considered, where $\sigma[e]^2$ is the variance of the random errors. Finally, the AIC, AICC and BIC criteria are computed of the well-known expressions $$ AIC = (-2) \times Log-likelihood + 2 \times kAIC = (-2) * Log-likelihood + 2 * k, $$ $$ AICC = AIC + \frac{2~k~(k + 1)}{n - k - 1}AICC = AIC + 2 * k * (k + 1) /( n - k - 1), $$ and $$ BIC = (-2) \times Log-likelihood + k~\times~ln(n)BIC = (-2) * Log-likelihood + k * ln(n), $$ with $k$ denoting the number of parameters. ## Returns An object of class `summary.vglm` printed by specific methods defined at `VGAMextra` for objects with signature vgltsff-class. ## See Also `vgtsff-class`, `summaryvlgm`, `ARXff`, `MAXff`, `ARMAXff`, `vglm`. ## References Woodward, H., Gray, H. and Elliot A. (2012) **Applied Time Series Analysis**. Taylor & Francis/CRC, Florida, USA. ## Author(s) V. Miranda and T.W. Yee. ## Examples ```r #------------------------------------------------------------------------# # Fitting a simple Moving Average model to compare with arima(). #------------------------------------------------------------------------# set.seed(0628) nn <- 300 theta <- c(0.2, -0.37) # Autoregressive coefficients phi <- c(0.25) # MA coefficients. mu <- c(1.5, 0.85) # Mean (not drift) of the MA process. x2 <- runif(nn) tsd1 <- mu[1]/(1 - sum(theta)) + arima.sim(n = nn, model = list(order = c(2, 0, 0), ar = theta), sd = exp(1.5)) tsd2 <- mu[2]/(1 - sum(theta)) + arima.sim(n = nn, model = list(order = c(2, 0, 1), ar = theta, ma = phi), sd = exp(1 + 2 * x2)) tsdata <- data.frame(TS1 = tsd1, TS2 = tsd2, x2 = x2) head(tsdata) ### An ARIMA(2, 0, 0) model, that is an AR(2) model ### #fit1 <- vglm(TS1 ~ 1, # ARIMAXff(order = c(2, 0, 0), var.arg = FALSE, type.EIM = "exact"), # data = tsdata, crit = "log", trace = TRUE) fit1 <- vglm(TS1 ~ 1, ARXff(order = 2, var.arg = FALSE, type.EIM = "exact"), data = tsdata, crit = "log", trace = TRUE) m.coe <- Coef(fit1) ## Using arima to compare to summary(vgtsff) summary(fit1) arima(tsdata$TS1, order = c(2, 0, 0)) ## Similar SE's than VGAMextra. m.coe[1] / (1 - sum(m.coe[-(1:2)])) # THIS IS SIMILAR TO THE INTERCEPT # ESTIMATED BY arima(): 1.1898 ### An ARIMA(2, 0, 1) models, that is an ARMA(2, 1) ### ### The errors standard deviation is a function of 'x2' ### ### NOTICE: ARIMA and ARMA use the "identitylink" for coefficients ### #fit2 <- vglm(TS2 ~ x2, # ARMAXff(order = c(2, 1), var.arg = FALSE, type.EIM = "exact", # zero = NULL), # # constraints = list('x2' = rbind(0, 1, 0, 0, 0)), # data = tsdata, crit = "loglikelihood", trace = TRUE) #m.coe <- coef(fit2) #coef(fit2, matrix = TRUE) ## Compare summary(vglm) to arima(). #summary(fit2) #arima(tsdata$TS2, order = c(2, 0, 1)) ```

summaryS4VGAMextra function