Dependent errors leading to artificially small p values?

Question

Why do dependent regression errors in a Multiple Linear Regression model (a violation of the assumptions of the MLR model) lead to underestimated standard errors and artificially small p-values? What is the connection between these quantities?

Answer 1

Okey so p-value is a quantity that shows the relation between the expected value of a coefficient and the standard error (the uncertainty of the value).

If the errors are dependent then the coefficients are still unbiased, as the derivation for this does not change. However the variance-covariance matrix of the errors is no longer a diagonal matrix. But OLS will assume it is diagonal.

This is how it should be calculated: \begin{equation} \begin{aligned} V\left[b_{\text {OLS }}\right] & =E\left[\left(X^{\prime} X\right)^{-1} X^{\prime} \varepsilon \varepsilon^{\prime} X\left(X^{\prime} X\right)^{-1}\right] \\ & =\left(X^{\prime} X\right)^{-1} X^{\prime} E\left[\varepsilon \varepsilon^{\prime}\right] X\left(X^{\prime} X\right)^{-1} \\ & =\left(X^{\prime} X\right)^{-1} X^{\prime} \Omega X\left(X^{\prime} X\right)^{-1} \end{aligned} \end{equation}

But it will apply the standard OLS equations where: \begin{equation} \Omega=\sigma^2 I \end{equation}

hence it applies: \begin{equation} V\left[b_{\text {OLS }}\right] = \sigma^2\left(X^{\prime} X\right)^{-1} \end{equation}

This quantity is smaller than the actual $V\left[b_{\text {OLS }}\right]$. Thus the standard errors of the coefficients are artificially lowered, hence also smaller p-values