How does MATLAB calculate standard error in 'fitlm'?

I would like to know how MATLAB calculates the standard error in the 'fitlm' function both when an intercept is set to 'false' and when an intercept is included in the model.  

Answer 1

[MathWorks Support Team]

The following code can be used to calculate the standard error on mock data both in the case that the intercept is set to 'false' and when an intercept is present:

%%Sample Data Definition
X = [1 2 3 4 5 6 7 8 9 10];
Y = [1 1.5 2.1 4 5 6.3 7.3 7.8 9 9.9];
%%Calculation of Standard Error Without Intercept
n = length(X);                                       %  Number of observations
Slope = sum(X.*Y)/sum(X.^2);                         %  Calculates slope
yfit=X*Slope;                                        %  Fitted response values based on the slope
r = Y - yfit;                                        %  r is the residuals, which is the observed minus fitted values

SSE = sum(r.^2);                                     %  SSE is the sum of squared errors
MSE=SSE/(n-1);                                       %  Mean Squared Error 
Code_NoIntercept_SE=sqrt(MSE/sum(X.^2))              %  Standard error of the slope

model1 = fitlm(X,Y,'Intercept',false);               %  Calls 'fitlm'
MATLAB_NoIntercept_SE=model1.Coefficients.SE         %  Queries the standard error of MATLAB's model

%%Clear Variables then Redefine Data
clear all
X = [1 2 3 4 5 6 7 8 9 10];
Y = [1 1.5 2.1 4 5 6.3 7.3 7.8 9 9.9];

%%Calculation of Standard Error With Intercept
n = length(X);                                      %  Number of observations
XBar=mean(X);                                       %  Calculates mean of X
YBar=mean(Y);                                       %  Calculates mean of Y
Sxx=sum((X-XBar).^2);  
Sxy=sum((X-XBar).*(Y-YBar));

Slope = Sxy/Sxx;                                    %  Calculates Slope
Intercept= YBar-Slope*XBar;                         %  Calculates Intercept
yfit=Intercept + X*Slope;                           %  Fitted response values based on the slope
r = Y - yfit;                                       %  r is the residuals, which is the observed minus fitted values

SSE = sum(r.^2);                                    %  SSE is the sum of squared errors
MSE=SSE/(n-2);                                      %  Mean Squared Error 
Code_Intercept_SE=[                                 %  Standard Error of the regression coefficients
    sqrt(MSE*sum(X.^2)/(n*Sxx));                    %  Standard Error of the intercept coefficient
    sqrt(MSE/Sxx)]                                  %  Standard Error of the slope coefficient

model2 = fitlm(X,Y,'Intercept',true);               %  Calls 'fitlm'
MATLAB_Intercept_SE=model2.Coefficients.SE          %  Queries the standard error of MATLAB's model