Robust and Unbiased Variance of GLM Coefficients for Misspecified Autocorrelation and Hemodynamic Response Models in fMRI

Table 1

The four methods of estimation and their corresponding variance.

Type	Mean	Variance
𝑊	𝛽𝑂=(𝑋^′𝑋)^−1𝑋^′𝑌	𝑉𝑊=1𝑛(𝑋^′𝑋)^−1𝑋^′𝑊𝑋(𝑋^′𝑋)^−1
OLS	𝛽𝑂=(𝑋^′𝑋)^−1𝑋^′𝑌	𝑉𝑂=𝜎^2𝑂(𝑋^′𝑋)^−1
GLS	𝛽𝐺=(𝑋^′𝑅𝑋)^−1𝑋^′𝑅𝑌	𝑉𝐺=𝜎^2𝐺(𝑋^′𝑅𝑋)^−1
𝑆	𝛽𝑆=(𝑋^′𝑆^′𝑆𝑋)^−1𝑋^′𝑆^′𝑆𝑌	𝑉𝑆=𝜎^2𝑆(𝑋^′𝑆^′𝑆𝑋)^−1𝑋^′𝑆^′𝑆𝑅𝑆^′𝑆𝑋(𝑋^′𝑆^′𝑆𝑋)^−1