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Parameter Errors (uncertainties)

Posted 10 years ago


How does Mathematica compute the Standard Errors for parameters?

To give an example, let's take the stock prices for IBM and S&P500, and transform them in continuous-time returns:

data = Import["", "Table"];
{IBM, SP500} = Log[data[[2 ;;, 2 ;;]] + 1]\[Transpose];

I want to regress current returns of IBM on current S&500 and 1-period lagged returns of these assets. So, I define the dependent variable Y and the independent variable X as follows:

 Y = Drop[IBM, 1];
 X = {ConstantArray[1, Length[Y]], Drop[SP500, 1], Drop[IBM, -1], Drop[SP500, -1]}\[Transpose];

The parameters of the regression is very simple to get:

\[Beta] = Inverse[X\[Transpose].X].(X\[Transpose].Y)

And they correspond to the ones obtained from the LinearModelFit:

lm = LinearModelFit[Append[X[[All, 2 ;;]]\[Transpose], Y]\[Transpose], {x1, x2, x3}, {x1, x2, x3}]
lm["BestFitParameters"] == \[Beta]
Out[]= True

However my Standard Errors are slightly different.

Letting epsilon to denote the residuals, i.e.

\[Epsilon] = Y - X.\[Beta];

standard errors can be obtained in the following way:

\[Epsilon].\[Epsilon]/(Length[Y] - 1) Inverse[X\[Transpose].X] // Diagonal // Sqrt
Out[]= {0.00172569, 0.0308209, 0.0316489, 0.0403108}

LinearModelFit standard errors on the other hand are:

Out[]= {0.00172916, 0.0308831, 0.0317127, 0.0403921}

I could not find the source of this difference.

POSTED BY: Sandu Ursu
Posted 10 years ago

Just found my mistake:

I've forgotten to subtract the number of columns in X. Now it works :)

\[Epsilon].\[Epsilon]/(Length[Y] - Dimensions[X][[2]])Inverse[X\[Transpose].X] == lm["CovarianceMatrix"]
Out[]=  True
POSTED BY: Sandu Ursu
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