Helper function to determine \(\beta\) estimates for MLE regression.

Generates \(\beta\) estimates for MLE using a conditioning approach.

bsm.simple(x, y, z)

Arguments

x

An \(N \times (P + F)\) design matrix, where \(F\) is the number of columns conditioned on. This is equivalent to the multiplication of \(xyzb\).

y

The \(N \times (Q - F)\) matrix of observations, where \(F\) is the number of columns conditioned on. This is equivalent to the multiplication of \(Yz_a\).

z

A \((Q - F) \times L\) design matrix, where \(F\) is the number of columns conditioned on.

Value

A list with the following components:

Beta

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

SE

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

T

The \((P + F) \times L\) 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 \((Q - F) \times (Q - F)\) matrix \(\hat{\Sigma}_z\).

Cx

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

Details

The technique used to calculate the estimates is described in section 9.3.3.

See also

bothsidesmodel.mle and bsm.fit

Examples

# Taken from section 9.3.3 to show equivalence to methods.
data(mouths)
x <- cbind(1, mouths[, 5])
y <- mouths[, 1:4]
z <- cbind(1, c(-3, -1, 1, 3), c(-1, 1, 1, -1), c(-1, 3, -3, 1))
yz <- y %*% solve(t(z))
yza <- yz[, 1:2]
xyzb <- cbind(x, yz[, 3:4])
lm(yza ~ xyzb - 1)
#> 
#> Call:
#> lm(formula = yza ~ xyzb - 1)
#> 
#> Coefficients:
#>        [,1]      [,2]    
#> xyzb1  24.93713   0.82680
#> xyzb2  -2.27175  -0.35044
#> xyzb3   0.18897   0.19136
#> xyzb4  -1.24459   0.06324
#> 
bsm.simple(xyzb, yza, diag(2))
#> $Beta
#>            [,1]       [,2]
#> [1,] 24.9371256  0.8268033
#> [2,] -2.2717454 -0.3504386
#> [3,]  0.1889661  0.1913639
#> [4,] -1.2445893  0.0632445
#> 
#> $SE
#>           [,1]       [,2]
#> [1,] 0.5205837 0.09051471
#> [2,] 0.7935186 0.13797033
#> [3,] 0.7732514 0.13444644
#> [4,] 1.1123569 0.19340725
#> 
#> $T
#>            [,1]       [,2]
#> [1,] 47.9022397  9.1344632
#> [2,] -2.8628761 -2.5399562
#> [3,]  0.2443787  1.4233468
#> [4,] -1.1188759  0.3270017
#> 
#> $Covbeta
#>              [,1]         [,2]         [,3]         [,4]          [,5]
#> [1,]  0.271007398  0.005297044 -0.273389373 -0.005343601  0.1199197919
#> [2,]  0.005297044  0.008192913 -0.005343601 -0.008264923  0.0023439227
#> [3,] -0.273389373 -0.005343601  0.629671752  0.012307409 -0.1262809490
#> [4,] -0.005343601 -0.008264923  0.012307409  0.019035812 -0.0024682563
#> [5,]  0.119919792  0.002343923 -0.126280949 -0.002468256  0.5979177242
#> [6,]  0.002343923  0.003625334 -0.002468256 -0.003817641  0.0116867527
#> [7,]  0.064067145  0.001252241 -0.083442521 -0.001630947 -0.0272399259
#> [8,]  0.001252241  0.001936835 -0.001630947 -0.002522578 -0.0005324249
#>               [,6]          [,7]          [,8]
#> [1,]  0.0023439227  0.0640671449  0.0012522406
#> [2,]  0.0036253343  0.0012522406  0.0019368347
#> [3,] -0.0024682563 -0.0834425209 -0.0016309470
#> [4,] -0.0038176405 -0.0016309470 -0.0025225780
#> [5,]  0.0116867527 -0.0272399259 -0.0005324249
#> [6,]  0.0180758456 -0.0005324249 -0.0008234991
#> [7,] -0.0005324249  1.2373378647  0.0241847013
#> [8,] -0.0008234991  0.0241847013  0.0374063642
#> 
#> $df
#> [1] 23
#> 
#> $Sigmaz
#>            [,1]       [,2]
#> [1,] 3.88871861 0.07600794
#> [2,] 0.07600794 0.11756112
#> 
#> $Cx
#>             [,1]        [,2]         [,3]         [,4]
#> [1,]  0.06969067 -0.07030320  0.030837868  0.016475130
#> [2,] -0.07030320  0.16192268 -0.032473666 -0.021457588
#> [3,]  0.03083787 -0.03247367  0.153757004 -0.007004859
#> [4,]  0.01647513 -0.02145759 -0.007004859  0.318186526
#>