# Estimate a SUR Model Use this function to estimate a Seemingly Unrelated Regression model. ```r estim.sur( data, searchSigMaxIter = 0, searchSigMaxProb = 0.1, restriction = NULL, pcaOptionsY = NULL, pcaOptionsX = NULL, simFixSize = 0, simTrainFixSize = 0, simTrainRatio = 0.75, simSeed = 0, simMaxConditionNumber = Inf ) ``` ## Arguments - `data`: A list that determines data and other required information for the search process. Use `get.data()` function to generate it from a `matrix` or a `data.frame`. - `searchSigMaxIter`: An integer for the maximum number of iterations in searching for significant coefficients. Use 0 to disable the search. - `searchSigMaxProb`: A number for the maximum value of type I error to be used in searching for significant coefficients. If p-value is less than this, it is interpreted as significant. - `restriction`: A `km x q` matrix of restrictions where `m` is the number of endogenous data, `k` is the number of exogenous data, and `q` is the number of unrestricted coefficients. - `pcaOptionsY`: A list of options to use principal components of the endogenous data, instead of the actual values. Set `NULL` to disable. Use `get.options.pca()` for initialization. - `pcaOptionsX`: A list of options to use principal components of the exogenous data, instead of the actual values. Set `NULL` to disable. Use `get.options.pca()` for initialization. - `simFixSize`: An integer that determines the number of out-of-sample simulations. Use zero to disable the simulation. - `simTrainFixSize`: An integer representing the number of data points in the training sample in the out-of-sample simulation. If zero, `trainRatio` will be used. - `simTrainRatio`: A number representing the size of the training sample relative to the available size, in the out-of-sample simulation. It is effective if `trainFixSize` is zero. - `simSeed`: A seed for the random number generator. Use zero for a random value. - `simMaxConditionNumber`: A number for the maximum value for the condition number in the simulation. ## Details As described in section 10.2 in if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="greene2020econometric;textual",package="ldt",cached_env=.Rdpack.currefs) , this type of statistical model consists of multiple regression equations, where each equation may have a different set of exogenous variables and the disturbances between the equations are assumed to be correlated. The general form with $m$ equations can be written as $y_i=z_i'\gamma_i+v_i$ and $E(v_i v_j)=\sigma_{ij}^2$ for $i=1,\ldots m$. Assuming that a sample of $N$ independent observations is available, we can stack the observations and use the following system for estimation: $$ Y = X B + V, \quad \mathrm{vec}B = R\gamma, $$ where the columns of $Y:N \times m$ contain the endogenous variables for each equation and the columns of $X: N\times k$ contain the explanatory variables, with $k$ being the number of unique explanatory variables in all equations. Note that $X$ combines the $z_i$ variables, and the restrictions imposed by $R:mk\times q$ and $\gamma:q\times 1$ determine a set of zero constraints on $B: k \times m$, resulting in a system of equations with different sets of exogenous variables. Imposing restrictions on the model using the $R$ matrix is not user-friendly, but it is suitable for use in this package, as users are not expected to specify such restrictions, but only to provide a list of potential regressors. Note that in this package, most procedures, including significance search, are supposed to be automated. The unrestricted estimators (i.e., $\hat{B}=(X'X)^{-1}X'Y$, and $\hat{\Sigma}=(\hat{V}'\hat{V})/N$ where $\hat{V}=Y-X\hat{B}$) are used to initialize the feasible GLS estimators: $$ \tilde{B} = RW^{-1}R'[\hat{V}-1 \otimes x']\mathrm{vec}Y, \quad \tilde{\Sigma}=(\tilde{V}'\tilde{V})/N, $$ where $W = R'[\hat{V}^{-1} \otimes X'X]R$ and $\tilde{V}=Y-X\tilde{B}$. The properties of these estimators are discussed in proposition 5.3 in if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="lutkepohl2005new;textual",package="ldt",cached_env=.Rdpack.currefs) . See also section 10.2 in if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="greene2020econometric;textual",package="ldt",cached_env=.Rdpack.currefs) . The maximum likelihood value is calculated by $-\frac{N}{2}(m(\ln 2\pi+1)+\ln|\tilde{\Sigma}|)$. The condition number is calculated by multiplying 1-norm of $W$ and its inverse (e.g., see page 94 in if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="trefethen1997numerical;textual",package="ldt",cached_env=.Rdpack.currefs) ). Furthermore, given an out-of-sample observation such as $x:k\times 1$, the prediction is $y^f = \tilde{B}'x$, and its variance is estimated by the following formula: $$ \mathrm{var}y^f = \tilde{V} + (x' \otimes I_m)R W^{-1}R'(x \otimes I_m). $$ Note that the focus in `ldt` is model uncertainty and for more sophisticated implementations of the FGLS estimator, you may consider using other packages such as `systemfit`. Finally, note that the main purpose of exporting this method is to show the inner calculations of the search process in search.sur function. ## Examples ```r # Example 1 (simulation, small model): set.seed(123) sample <- sim.sur(sigma = 2L, coef = 3L, nObs = 100) print(sample$coef) print(sample$sigma) data <- data.frame(sample$y, sample$x) # Use systemfit to estimate: exp_names <- paste0(colnames(sample$x), collapse = " + ") fmla <- lapply(1:ncol(sample$y), function(i) as.formula(paste0("Y", i, " ~ -1 +", exp_names))) fit <- systemfit::systemfit(fmla, data = data, method = "SUR") print(fit) # Use 'ldt::estim.sur' function fit <- estim.sur(data = get.data(cbind(sample$y, sample$x), endogenous = ncol(sample$y), addIntercept = FALSE)) # or, by using formula list: fit <- estim.sur(data = get.data(data = data, equations = fmla, addIntercept = FALSE)) print(fit) print(fit$estimations$sigma) plot_data <- plot(fit, equation = 1) # Example 2 (simulation, large model with significancy search): num_obs <- 100 sample <- sim.sur(sigma = 2L, coef = 3L, nObs = num_obs) print(sample$coef) # create irrelevant data: num_x_ir <- 20 x_ir <- matrix(rnorm(num_obs * num_x_ir), ncol = num_x_ir) data_x <- data.frame(sample$x, x_ir) colnames(data_x) <- c(colnames(sample$x), paste0("z", 1:num_x_ir)) fit <- estim.sur(data = get.data(cbind(sample$y, data_x), endogenous = ncol(sample$y), addIntercept = FALSE), searchSigMaxIter = 100, searchSigMaxProb = 0.05) print(fit$estimations$coefs) # coefficient matrix, with lots of zero restrictions # Example 3 (simulation, large model with PCA): # by using data of the previous example fit <- estim.sur(data = get.data(cbind(sample$y, data_x), endogenous = ncol(sample$y), addIntercept = FALSE), pcaOptionsX = get.options.pca(2,4)) print(fit$estimations$coefs) # coefficients are: intercept and the first exogenous variable and 4 PCs ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also search.sur

estim.sur function