# VGLTSMs Family Functions: Order--q Moving Average Model with covariates Estimates the intercept, standard deviation (or variance) of the random noise (not necessarily constant), and the conditional--mean model coefficients of an order--q moving average (MA) process with covariates (MAX(q)) by maximum likelihood estimation using Fisher scoring. ```r MAXff(order = 1, zero = c(if (nomean) NULL else "Mean", "MAcoeff"), xLag = 0, type.EIM = c("exact", "approximate")[1], var.arg = TRUE, nomean = FALSE, noChecks = FALSE, lmean = "identitylink", lsd = "loglink", lvar = "loglink", lMAcoeff = "identitylink", imean = NULL, isd = NULL, ivar = NULL, iMAcoeff = NULL) ``` ## Arguments - `order`: The order 'q' of the MA model, recycled if needed. By default `q = 1`. - `zero`: Integer or character--string vector. Same as `ARIMAXff`. Details at `zero`. - `xLag`: Same as `ARIMAXff`. - `type.EIM, var.arg`: Same as `ARIMAXff`. - `nomean`: Logical. `nomean = TRUE` supresses estimation of the **mean** (intercept of the conditional--mean model). - `noChecks`: Logical. Same as `ARIMAXff`. - `lmean, lsd, lvar, lMAcoeff`: Link functions applied to the mean (intercept), the standard deviation or variance of the random noise, and the MA coefficients (conditional--mean model). Note, `lmean` plays the role of `ldrift`. - `imean, isd, ivar, iMAcoeff`: Same as `ARIMAXff`. Note, `imean` plays the role of `idrift`. ## Details Similar to `ARXff`, this family function fits an order--$q$ moving average model with covariates (MAX(q)), another special case of the class VGLM--ARIMA (Miranda and Yee, 2018). Observations, $Y[t]$, are modelled as $$ Y_t | \Phi_{t - 1} = \mu_t + \phi_{1} \varepsilon_{t - 1} +\ldots + \phi_q \varepsilon_{t - q} + \varepsilon_t,%Y[t] | \Phi[t - 1] = \mu[t] + \phi[1] e[t - 1] + \ldots +\phi[q] e[t - p] + e[t], $$ where $\mu[t]$ is the (possibly time--dependent) intercept, modelled as c("$\n$", "$mu[t] = \\mu + \\beta^T x[t]$"), and the errors are mean--zero Gaussian: c("$\n$", "$ e[t] | \\Phi[t - 1] ~ N(0 , \\sigma[e]^2 | \\Phi [t - 1] ) $"). The symbol $\Phi[t]$ denotes the history of the joint process c("$\n$", "$(Y[t], X[t + 1])\n$"), at time $t$ for a time--varying covariate vector $x[t]$. At each step of Fisher scoring, the **exact** log-likelihood based on model above is computed through `dMAq`. The linear predictor by default is $$ \boldsymbol{\eta} = \left(\mu_t, \log \sigma^{2}_{\varepsilon_{t | \Phi_{t - 1}}},\phi_1, \ldots, \phi_q\right)^T.\eta = (\mu[t], log \sigma[e]^2 | \Phi[t - 1],\phi[1], \ldots, \phi[q] )^T. $$ The unconditional mean of the process is simply c("$%$", "$\n$", "$E(Y[t]) = \\mu $"), provided no covariates added. This family function is not restricted to the noise to be strictly white noise (in the sense of **constant variance**). That is, covariates may be incorporated in the linear predictor c("$\n$", "$log \\sigma[e]^2 | \\Phi[t - 1].\n$") Also, it handles **multiple responses** so that a matrix can be used as the response. For further details on VGLM/VGAM--link functions, such as `logitlink`, refer to `Links`. ## Returns An object of class `"vglmff"` (see `vglmff-class`) to be used by VGLM/VGAM modelling functions, e.g., `vglm` or `vgam`. ## Note Further choices for the random noise, besides Gaussian, will be implemented over time. `zero` can be either an **integer** vector or a vector of character strings specifying either the position(s) or name(s) (partially or not) of the parameter(s) modeled as intercept-only. For `MAXff`, the parameters are placed and named as follows (by convention): `c("Mean", "noiseVar" or "noiseSD", "MAcoeff")`. Users can modify the `zero` argument accordingly. For simplicity, the second choice recommended. See `CommonVGAMffArguments` for further details on `zero`. If no constraints are entered in the fitting process, (e.g., via `cm.ARMA`) this family function internally verifies by default whether the estimated series is invertible (since `noChecks = FALSE`). To ignore this step, set `noChecks = TRUE`. If the estimated MA process is non-invertible, MLE coefficients will conform with the corresponding invertible MA model. Further details about these checks are shown within the `summary()` output. ## References Miranda, V. and Yee, T.W. (2018) Vector Generalized Linear Time Series Models. **In preparation**. Madsen, H. (2008) **Time Series Analysis** Florida, USA: **Chapman & Hall**. (Sections 5.3 to 5.5). Tsay, R. (2013) An Introduction to Analysis of Financial data with `R`. New Jersey, USA: **Wiley** Sections 2.2 to 2.4. ## Author(s) Victor Miranda and Thomas W. Yee. ## Warning By default, a moving-average model of order-$1$ is fitted. If different, the `order` is recycled up to the number of responses entered in the `vglm \ vgam` call has been matched. Successful convergence depends on reasonably setting initial values. If initial values computed by the algorithm are not adequate, make use of the the optional initial values (`imean`, `isd`, etc.) For constraints on the paramaters see `cm.ARMA`. ## See Also `ARIMAXff`, `ARXff`, `checkTS.VGAMextra`, `CommonVGAMffArguments`, `Links`, `vglm`, `vgam`. ## Examples ```r set.seed(2) nn <- 130 ### Coefficients phi1 <- 0.34; phi2 <- -1.19; phi3 <- 0.26 ### Intercept mu <- c(-1.4, 2.3) ### Noise standar deviations (Two responses) sdMA <- c(sqrt(6.5), sqrt(4.0)) ### A single covariate. Xcov <- runif(nn) # Generating two MA processes, TS1 and TS2, Gaussian noise. # Note, the SD noise for TS2 is function of Xcov. y1 <- mu[1] + arima.sim(nn, model = list( ma = c(phi1, phi1^2)), rand.gen = rnorm, sd = exp(sdMA[1]) ) y2 <- mu[2] + arima.sim(nn, model = list( ma = c(phi1, phi2, phi3) ), rand.gen = rnorm, sd = exp(Xcov + sdMA[2]) ) # OUR DATA tsdata <- data.frame(x2 = Xcov , TS1 = y1, TS2 = y2) #------------------------------------------------------------------------# # 1. A simple MA(3) to compare with 'arima()'. myfit0 <- vglm(TS1 ~ 1, MAXff(order = 3, type.EIM = "exact", var.arg = FALSE), #constraints = cm.ARMA(Model = ~ 1, # lags.cm = 2, # Resp = 1), data = tsdata, trace = TRUE) Coef(myfit0)[c(3, 4, 1)] fitArima <- arima(tsdata$TS1, order = c(0, 0, 2)) coef(fitArima) AIC(myfit0); BIC(myfit0) # ------------------------------------------------------------------------# # 2. Estimate an MA(3), intercept-only, using initial values. myfit <- vglm(TS2 ~ 1, MAXff(order = 3, type.EIM = c("exact", "approx")[1], # Optional initial values. imean = 3, iMAcoeff = c(0.3, -0.2, 0.25), var.arg = TRUE), data = tsdata, trace = TRUE) Coef(myfit) summary(myfit) constraints(myfit) #----------------------------------------# # Same model fitted using arima() #----------------------------------------# fitArima <- arima(tsdata$TS2, order = c(0, 0, 3)) coef(fitArima) #------------------------------------------------------------------------# # 3. An MAX(3) with one covariate, testing its effect over the # standard deviation of the Gaussian noise. Note the 'zero' argument. myfit1 <- vglm(TS2 ~ x2, # Or Multiple responses! # cbind(TS1, TS2) ~ 1, MAXff(order = 3, type.EIM = "exact", xLag = 1, # Optional initial values: # idev.mean = 1.4, # iMAcoeff = c(2.3, -1.2, 0.25), isd = 1.6, # NOTE THE ZERO ARGUMENT: zero = c("Mean", "MAcoeff"), var.arg = TRUE), data = tsdata, trace = TRUE) coef(myfit1, matrix = TRUE) summary(myfit1) vcov(myfit1) constraints(myfit1) #------------------------------------------------------------------------# # Model above CANNOT be fitted using arima() #------------------------------------------------------------------------# ```

MAXff function