moment

Purpose

Computes a cross-product matrix. This is the same as x’x.

Format

y = moment(x, d)

Parameters:

• x (NxK matrix or M-dimensional array where the last two dimensions are NxK) – data

• d (scalar) –

  controls handling of missing values.

  0	missing values will not be checked for. This is the fastest option.
  1	”listwise deletion” is used. Any row that contains a missing value in any of its elements is excluded from the computation of the moment matrix. If every row in x contains missing values, then moment(x,1) will return a scalar zero.
  2	”pairwise deletion” is used. Any element of x that is missing is excluded from the computation of the moment matrix. Note that this is seldom a satisfactory method of handling missing values, and special care must be taken in computing the relevant number of observations and degrees of freedom.

Returns:

y (KxK matrix or M-dimensional array) – where the last two dimensions are KxK, the cross-product of x.

Examples

rndseed 129070;

// Create data
x = ones(100, 3)*rndn(100, 3);
b_true = {1.3 0.5 0.75 -1.9};

y = x*b_true' + 0.5*rndn(100,3);

// Create moment matrix
xx = moment(x, 2);

// Find inverse of moment matrix
ixx = invpd(xx);

// Find coefficients
b = ixx*missrv(x, 0)'y;

print "b_true~b_est";
b_true'~b_est;

1.3000000        1.2808949
0.50000000       0.49703322
0.75000000       0.73297298
-1.9000000       -1.8087071

In this example, the regression of y on x is computed. The moment matrix (xx) is formed using the moment() command (with pairwise deletion, since the second parameter is 2). Then xx is inverted using the invpd() function. Finally, the ols coefficients are computed. missrv() is used to emulate pairwise deletion by setting missing values to 0.

Remarks

The fact that the moment matrix is symmetric is taken into account to cut execution time almost in half.

If x is an array, the result will be an array containing the cross-products of each 2-dimensional array described by the two trailing dimensions of x. In other words, for a 10x4x4 array x, the resulting array y will contain the cross-products of each of the 10 4x4 arrays contained in x, so \(y[n,.,.]=x[n,.,.]'x[n,.,.]\) for \(1 <= n <= 10\).

If there is no missing data then d = 0 should be used because it will be faster.

The / operator (matrix division) will automatically form a moment matrix (performing pairwise deletions if trap 2 is set) and will compute the ols coefficients of a regression. However, it can only be used for data sets that are small enough to fit into a single matrix. In addition, the moment matrix and its inverse cannot be recovered if the / operator is used.