LLS

Syntax: LLS(A, Y)

A = a range
Y = a range

LLS generates:

1. the linear least squares solution, X, to the overdetermined system of equations AX=Y

2. the standard errors of the least square estimates of each element of the parameter vector X, the T-statistics that compare each parameter to zero, and the significance level of each.

3. the variance of the model, the R-squared correlation for the model, and its corresponding T-statistic and significance level. (Note: These values are only valid if A contains a column of 1's, which corresponds to the y-intercept. Omitting the column of 1's is equivalent to forcing the y-intercept of the model to be equal to zero.)

If any columns of A are linearly dependent, the function produces an error.

The output of LLS is a a table with four rows and n+1 columns, where n is the number of independent variables in the model (i.e., the number of columns in the A matrix). The table is presented in the following format:

$$\begin{array}{ccccc} \hat{x}_1 & \hat{x}_2 & \cdots & \hat{x... ...hat{x}_2}) & \cdots & p(t_{\hat{x}_n}) & p(t_R) \\ \end{array}$$

where

$\hat{x}_i$	=	the least squares estimate of the ith coefficient, corresponding to the independent variable in the ith column of A.
$\mbox{\it SE}(\hat{x}_i)$	=	the standard error of $\hat{x}_i$.
$t_{\hat{x}_i}$	=	the t-statistic for testing whether $x_i$ is significantly different from zero.
$p(t_{\hat{x}_i})$	=	the probability of error in rejecting the null hypothesis that $x_i=0$, based on two-sided t-test.
$R^2$	=	the mean squared error of the model.
MSE	=	the model coefficient of determination (the square of the model correlation coefficient, R).
$t_R$	=	the t-statistic for testing whether $R$ is significantly different from zero.
$p(t_R)$	=	the probability of error in rejecting the null hypothesis that $R=0$, based on a two-sided t-test.

Examples:

Matrix D3..E5 =

	D	E
3	1	3
4	2	4
5	7	4

Matrix G3..G5 =

	G
3	9
4	5
5	4

LLS(D3..E5, G3..G5) =

-0.76479076	2.2640693	15.308802
0.9516909	1.0921967	0.54349619
-0.80361256	2.0729501	0.49703082
0.56904671	0.2861429	0.60906673

LLS(A1..A2, B1..C2) = Error - LLS, improper dimensions

Excel function: N/A