# mvrnormSeries: Generate data under the Dynamic or Rao-Yu Time Series Models In sae2: Small Area Estimation: Time-Series Models

## Description

Function to generate data under a Rao-Yu time series model, a dynamic model, or a mixture of both. The function can produce either univariate or multivariate observations. All components of the returned random variable have unconditional mean zero. The function calls `mvrnorm` in MASS.

## Usage

 ```1 2 3 4``` ```mvrnormSeries(NV=1, D, T, sigma.e, rho.dyn, sigma.v.dyn, sigma.u.dyn, rho.u.dyn, rho.RY, sigma.v.RY, sigma.u.RY, rho.u.RY, tol=1e-6, conditional.mean=FALSE) ```

## Arguments

 `NV` The number of variables. `D` The number of domains. `T` The number of time instances (constant for all domains). `sigma.e` The covariance matrix for the variation due to sampling, specified either as a single square matrix with `NV*T` rows and columns, or as a list of `D` matrices, each with `NV*T` rows and columns. The rows should vary over `T` more quickly than over `NV`. Sampling covariances between domains are assumed to be zero. `rho.dyn` The temporal correlation parameter in the dynamic model. This parameter is not a true correlation, however, and it may exceed 1. `sigma.v.dyn` A vector of length `NV` with the v (nu) component of the variance under the dynamic model. This component measures the variance of the random effect at the beginning of the series. `sigma.u.dyn` A vector of length `NV` with the u component of the variance under the dynamic model. `rho.u.dyn` For `NV>1`, the cross-sectional correlation(s) under the model. The vector should specify `(NV*(NV-1))/2` correlations, in lower triangular form. For example, for `NV=3`, the correlations should be specified in the order (2,1), (3,1), (3,2). `rho.RY` The temporal correlation parameter in the Rao-Yu model. This is a true correlation, unlike the parameter in the dynamic model. `sigma.v.RY` A vector of length `NV` with the v (nu) component of the variance under the Rao-Yu model. This component reflects a constant random effect for each domain unchanging over time. `sigma.u.RY` A vector of length `NV` with the u component of the variance under the Rao-Yu model. `rho.u.RY` For `NV>1`, the cross-sectional correlation under the model. The vector should specify `(NV*(NV-1))/2` correlations, in lower triangular form. For example, for `NV=3`, the correlations should be specified in the order (2,1), (3,1), (3,2). `tol` A tolerance parameter used by `mvrnorm` in MASS to determine if the covariance matrix is non-singular. `conditional.mean` If true, the function will also return the generated values of the random effects.

## Details

The function assembles the covariance matrix from the covariance matrix under the dynamic model (if specified), the Rao-Yu model (if specified) and a required sampling covariance matrix.

## Value

If `conditional.mean=FALSE`, then for `NV=1`, a multivariate normal random vector with mean zero and length `D*T`. For `NV>1`, a matrix with `D*T` rows and `NV` columns.

If `conditional.mean=TRUE`, a list with the first element as above and a second element that is the sum of the random effects without the sampling error. Simulation studies can evaluate the small area estimates using the first element of the list as input against the second element of the list, which is the target of the small area estimation.

## Author(s)

Robert E. Fay

`mvrnorm`
 ```1 2 3 4 5``` ```set.seed(7) mvrnormSeries(D=2, T=5, sigma.e=diag(5), rho.dyn=.8, sigma.v.dyn=2, sigma.u.dyn=.72, conditional.mean=TRUE) mvrnormSeries(NV=2, D=2, T=5, sigma.e=diag(10), rho.dyn=.8, sigma.v.dyn=2, sigma.u.dyn=.72, rho.u.dyn=.8) ```