Gaussian Maximum Likelihood Estimation method for Reduced-Rank Regression. This method is not robust in the sense that it assumes a Gaussian distribution for the innovations which does not take into account the heavy-tailedness of the true distribution and outliers.
RRR(y, x, z = NULL, mu = TRUE, r = 1)Matrix of dimension N*P. The matrix for the response variables. See Detail.
Matrix of dimension N*Q. The matrix for the explanatory variables to be projected. See Detail.
Matrix of dimension N*R. The matrix for the explanatory variables not to be projected. See Detail.
Logical. Indicating if a constant term is included.
Integer. The rank for the reduced-rank matrix \(AB'\). See Detail.
A list of the estimated parameters of class RRR.
The input specifications. \(N\) is the sample size.
The estimated constant vector. Can be NULL.
The estimated exposure matrix.
The estimated factor matrix.
The estimated coefficient matrix of z. Can be NULL.
The estimated covariance matrix of the innovation distribution.
The formulation of the reduced-rank regression is as follow: $$y = \mu +AB' x + D z+innov,$$ where for each realization \(y\) is a vector of dimension \(P\) for the \(P\) response variables, \(x\) is a vector of dimension \(Q\) for the \(Q\) explanatory variables that will be projected to reduce the rank, \(z\) is a vector of dimension \(R\) for the \(R\) explanatory variables that will not be projected, \(\mu\) is the constant vector of dimension \(P\), \(innov\) is the innovation vector of dimension \(P\), \(D\) is a coefficient matrix for \(z\) with dimension \(P*R\), \(A\) is the so called exposure matrix with dimension \(P*r\), and \(B\) is the so called factor matrix with dimension \(Q*r\). The matrix resulted from \(AB'\) will be a reduced rank coefficient matrix with rank of \(r\). The function estimates parameters \(\mu\), \(A\), \(B\), \(D\), and \(Sigma\), the covariance matrix of the innovation's distribution, assuming the innovation has a Gaussian distribution.
S. Johansen, "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models,"Econometrica, vol. 59,p. 1551, Nov. 1991.
For robust reduced-rank regression estimation see function RRRR.
set.seed(2222)
data <- RRR_sim()
res <- RRR(y=data$y, x=data$x, z = data$z)
res
#> Reduced-Rank Regression
#> ------------
#> Specifications:
#> N P Q R r
#> 1000 3 3 1 1
#>
#> Coefficients:
#> mu A B D Sigma1 Sigma2 Sigma3
#> 1 0.099542 -0.308282 0.790166 0.268161 2.508671 -0.186934 0.231774
#> 2 0.185923 0.859776 0.467165 1.163374 -0.186934 2.325132 -0.104148
#> 3 0.132058 1.678437 -0.296864 1.959463 0.231774 -0.104148 2.943445