# The Unrestricted estimator This function calculates the unrestricted estimator as [REMOVE_ME]$$ \hat{\beta}^{U} = (X^{\top} X)^{-1} X^{\top} y [REMOVE_ME_2]$$ where $^{\top}$ denotes the transpose of a matrix. It is important to note that the input matrices $X$ and $y$ should be standardized, for example, by using `scale`. Alternatively, the user can employ `lm` to obtain this estimator, but it is crucial to remember to set `intercept = FALSE`. ```r unrReg(X, y) ``` ## Arguments - `X`: Matrix with input observations, of dimension `n` x `p`, where each row is an observation vector; - `y`: Vector with response observations of size `n`. ## Returns An object of class `unrestricted` is a list containing at least the following components: - **`coef`**: A named vector of coefficients. - **`residuals`**: The residuals, that is, the response values minus fitted values. - **`s2`**: The estimated variance. - **`fitted.values`**: The fitted values. ## Description This function calculates the unrestricted estimator as $$ \hat{\beta}^{U} = (X^{\top} X)^{-1} X^{\top} y $$ where $^{\top}$ denotes the transpose of a matrix. It is important to note that the input matrices $X$ and $y$ should be standardized, for example, by using `scale`. Alternatively, the user can employ `lm` to obtain this estimator, but it is crucial to remember to set `intercept = FALSE`. ## Details The corresponding unrestricted estimator of $\sigma^2$ is $$ s^2 = \frac{1}{n-p}(y-X\hat{\beta}^{U})^{\top}(y - X\hat{\beta}^{U}). $$ ## Examples ```r data(cement) n_obs <- 100 p_vars <- 5 beta <- c(2, 1, 3, 0, 5) simulated_data <- simdata(n = n_obs, p = p_vars, beta) X <- simulated_data$X y <- simulated_data$y unrReg(X, y) data(cement) X <- as.matrix(cbind(1, cement[, 1:4])) y <- cement$y # Based on Kaciranlar et al. (1999) H <- matrix(c(0, 1, -1, 1, 0), nrow = 1, ncol = 5, byrow = TRUE) h <- rep(0, nrow(H)) unrReg(X, y) H <- matrix(c(0, 1, -1, 1, 0, 0, 0, 1, -1, -1, 0, 1, -1, 0, -1), nrow = 3, ncol = 5, byrow = TRUE) h <- rep(0, nrow(H)) unrReg(X, y) ``` ## References Saleh, A. K. Md. Ehsanes. (2006). **Theory of Preliminary Test and Stein‐Type Estimation With Applications**, Wiley.

unrReg function