# Fitting orthogonal panel models `opm` is used to fit orthogonal panel models. ```r opm(x, ...) ## Default S3 method: opm(x, y, n.samp, add.time.indicators = FALSE, ...) ## S3 method for class 'formula' opm(x, data = environment(x), subset = NULL, index = 1:2, n.samp, ...) ``` ## Arguments - `x`: a formula (see description of parameter `formula` below) or an array of dimension `time x variable x case` of terms. - `...`: further arguments passed to other methods. - `y`: a matrix of dimensions `time x case` of responses. - `n.samp`: number of samples to use to estimate the parameters. - `add.time.indicators`: (logical) if `TRUE`, adds dummy variables for time. - `data`: an optional data frame, list, or environment containing the variables in the model. If not found in `data`, the variables are taken from `environment(x)`, typically the environment from which `opm` is called. - `subset`: an optional vector specifying a subset of observations to be used in the fitting process. - `index`: a two-element vector containing the index of the case and time variables, respectively. Variable indices can be specifed by name or position. This argument is ignored if the model is not specified by the formula, because the index is implicit in the organization of the terms and response arrays. ## Returns An object of class `opm` with the following elements: - **`samples`**: parameter samples used to estimate the model, as a list with following elements: - **`call`**: the matched call - **`index`**: the index variables, when using the formula interface - **`time.indicators`**: `TRUE` if dummy time variables are used (see Notes), `FALSE` otherwise - **`terms`**: the `terms` object used The function `summary` (i.e., `summary.opm`) can be used to obtain or print a summary of the results. The generic accessor functions `coefficients`, `fitted.values`, `residuals`, `logLik`, and `df.residual` can be used to extract various useful features of the value returned by `opm`. ## Details The model can be either specified symbolically with the formula `response ~ term1 + term2 ...` or with the terms and response given as a pair of 3- and 2-dimensional arrays, `x` and `y` respectively. The arrays have to be in the format `time x variable x case` for terms and `time x case` for the response. The lagged dependent variable does not need to be included in the formula or data, as it is included automatically. ## Note Dummy time variables exist as an additional column for each wave of data, excluding the first and second wave (i.e., at $t=0$ and $t=1$ using the terminology from Lancaster (2000)). The new variables are named `tind.`$t$, where $t = 2, ...$, and appear as such as elements of the estimated `beta` coefficient. ## Examples ```r set.seed(123) N <- 5 T <- 2 beta <- .5 rho <- .5 v <- 1 f <- runif(N, -2, 2) K <- length(beta) beta <- matrix(beta, K, 1) ## $x_i = 0.75 f + N(0, 1)$: x <- array(.75*f, dim=c(N, K, (T+1))) + rnorm(N*K*(T+1)) ## $y_{i,t} = \rho y_{i,t-1} + \beta x_{i,t} + f_i + N(0,1)$: y <- matrix(0, N, T+1) for (t in seq_len(T+1)) { yy <- if (t>1) y[,t-1] else 0 y[,t] <- rho * yy + f + x[,,t] %*% beta + rnorm(N, sd = sqrt(1/v)) } d <- data.frame(i = rep(seq(N), T+1), t = rep(seq(T+1), each = N), as.data.frame(matrix(aperm(x, c(1, 3, 2)), N*(T+1), K, dimnames = list(NULL, paste0('x', seq(K))))), y = c(y)) opm(y~x1, d, n.samp = 10) ```

opm function