Calculate the least squares estimates

This function fits the model using least squares. It takes an optional pattern matrix P as in (6.51), which specifies which \(\beta _{ij}\)'s are zero.

bothsidesmodel(x, y, z = diag(qq), pattern = matrix(1, nrow = p, ncol = l))

Arguments

x

An \(N \times P\) design matrix.

y

The \(N \times Q\) matrix of observations.

z

A \(Q \times L\) design matrix

pattern

An optional \(N \times P\) matrix of 0's and 1's indicating which elements of \(\beta\) are allowed to be nonzero.

Value

A list with the following components:

Beta

The least-squares estimate of \(\beta\).

SE

The \(P \times L\) matrix with the \(ij\)th element being the standard error of \(\hat{\beta}_ij\).

T

The \(P \times L\) matrix with the \(ij\)th element being the \(t\)-statistic based on \(\hat{\beta}_{ij}\).

Covbeta

The estimated covariance matrix of the \(\hat{\beta}_{ij}\)'s.

df

A \(p\)-dimensional vector of the degrees of freedom for the \(t\)-statistics, where the \(j\)th component contains the degrees of freedom for the \(j\)th column of \(\hat{\beta}\).

Sigmaz

The \(Q \times Q\) matrix \(\hat{\Sigma}_z\).

Cx

The \(Q \times Q\) residual sum of squares and crossproducts matrix.

See also

bothsidesmodel.chisquare, bothsidesmodel.df, bothsidesmodel.hotelling, bothsidesmodel.lrt, and bothsidesmodel.mle.

Examples

# Mouth Size Example from 6.4.1
data(mouths)
x <- cbind(1, mouths[, 5])
y <- mouths[, 1:4]
z <- cbind(c(1, 1, 1, 1), c(-3, -1, 1, 3), c(1, -1, -1, 1), c(-1, 3, -3, 1))
bothsidesmodel(x, y, z)
#> $Beta
#>           [,1]       [,2]       [,3]        [,4]
#> [1,] 24.968750  0.7843750  0.2031250 -0.05625000
#> [2,] -2.321023 -0.3048295 -0.2144886  0.07215909
#> 
#> $SE
#>           [,1]       [,2]      [,3]       [,4]
#> [1,] 0.4860008 0.08599951 0.1275764 0.08868435
#> [2,] 0.7614168 0.13473534 0.1998739 0.13894168
#> 
#> $T
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 51.375949  9.120691  1.592183 -0.634272
#> [2,] -3.048294 -2.262432 -1.073120  0.519348
#> 
#> $Covbeta
#>              [,1]          [,2]          [,3]          [,4]         [,5]
#> [1,]  0.236196733  0.0042588778 -0.0026296165 -0.0097208807 -0.236196733
#> [2,]  0.004258878  0.0073959162 -0.0031372514  0.0005659801 -0.004258878
#> [3,] -0.002629616 -0.0031372514  0.0162757457 -0.0003583097  0.002629616
#> [4,] -0.009720881  0.0005659801 -0.0003583097  0.0078649148  0.009720881
#> [5,] -0.236196733 -0.0042588778  0.0026296165  0.0097208807  0.579755617
#> [6,] -0.004258878 -0.0073959162  0.0031372514 -0.0005659801  0.010453609
#> [7,]  0.002629616  0.0031372514 -0.0162757457  0.0003583097 -0.006454513
#> [8,]  0.009720881 -0.0005659801  0.0003583097 -0.0078649148 -0.023860343
#>               [,6]          [,7]          [,8]
#> [1,] -0.0042588778  0.0026296165  0.0097208807
#> [2,] -0.0073959162  0.0031372514 -0.0005659801
#> [3,]  0.0031372514 -0.0162757457  0.0003583097
#> [4,] -0.0005659801  0.0003583097 -0.0078649148
#> [5,]  0.0104536092 -0.0064545132 -0.0238603435
#> [6,]  0.0181536125 -0.0077005262  0.0013892239
#> [7,] -0.0077005262  0.0399495577 -0.0008794873
#> [8,]  0.0013892239 -0.0008794873  0.0193047908
#> 
#> $df
#> [1] 25 25 25 25
#> 
#> $Sigmaz
#>             z1           z2           z3           z4
#> z1  3.77914773  0.068142045 -0.042073864 -0.155534091
#> z2  0.06814205  0.118334659 -0.050196023  0.009055682
#> z3 -0.04207386 -0.050196023  0.260411932 -0.005732955
#> z4 -0.15553409  0.009055682 -0.005732955  0.125838636
#> 
#> $ResidSS
#> [1] 80
#> 
#> $Dim
#> [1] 18
#> 
#> $Cp
#> [1] 116
#>