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** R Programming Sample Assignment**

*Implement reduced rank regression as a simple R function. **SOLUTION **The R code is displayed below.*

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ReducedRankRegression <- function(Xraw,Y,Rank) {

# This code implements a two-stage reduced-rank estimation.

# First, we estimate the covariance matrix Sigma of the residuals

# via OLS. Second, we engage reduced-rank regression supplying Sigma

# as an input.

Xraw <- as.matrix(Xraw);

Y <- as.matrix(Y);

# STAGE 1

N <- dim(Y)[1];

q <- dim(Y)[2];

X <- cbind(rep(1,N), Xraw);

OlsRank <- dim(X)[2];

EffectiveRank <- Rank + 1;

Bols <- t( solve( t(X)%*%X ) %*% t(X)%*%Y );

Sigma <- matrix(0,q,q);

Sxx <- matrix(0,q+1,q+1);

Sxy <- matrix(0,q+1,q);

Syy <- matrix(0,q,q);

for(t in 1:N) {

Sxx <- Sxx + X[t,]%*%t(X[t,]);

Sxy <- Sxy + X[t,]%*%t(Y[t,]);

Syy <- Syy + Y[t,]%*%t(Y[t,]);

Resid <- Y[t,] - Bols%*%X[t,];

Sigma <- Sigma + Resid%*%t(Resid);

}

Sxx <- Sxx / N;

Sxy <- Sxy / N;

Syx <- t(Sxy);

Syy <- Syy / N;

Sigma <- Sigma / (N-OlsRank);

# STAGE 2

Gamma <- solve(Sigma);

HalfGamma <- chol(Gamma);

H <- HalfGamma%*%Syx%*%solve(Sxx)%*%Sxy%*%HalfGamma;

SVDoutput <- svd(H);

RelevEigenVectors <- SVDoutput$u[,1:EffectiveRank];

MiddleSum <- matrix(0,q,q);

for(j in 1:EffectiveRank) {

MiddleSum <- MiddleSum +

RelevEigenVectors[,j]%*%t(RelevEigenVectors[,j]);

} B <- solve(HalfGamma)%*%MiddleSum%*%HalfGamma %*%

Syx%*%solve(Sxx);

list(B=B, Sigma=Sigma, Bols=Bols);

}