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Richard Hardy
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Uncorrelated errors with the regressor in a reduced form VAR

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MAOC
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Uncorrelated errors in a reduced form VAR

I have a reduced form VAR

$$\begin{equation} y_t = c_o + A y_{t-1} + \epsilon_t \end{equation}$$

Where, $y_t \in \mathbb{R}^2$, $A$ is a $2$X$2$ matrix and

$$\begin{equation} E(\epsilon_t \epsilon_\tau ')=\left\{ \begin{array}{@{}ll@{}} \Omega, & \text{if}\ t=\tau \\ 0, & \text{otherwise} \end{array}\right. \end{equation} $$

$\Omega$ is not necessarily digonal. I have to show that I can use Least squares method. But, for this, I have to show that the errors $\epsilon_t$ are not correlated with the regressor $y_{t-1}$, i.e.

$$E(y_{t-1} \epsilon_t' ) = 0$$

I tried to use the Wald decomposition:

$$y_{t-1} = \mu + \sum_{i=0}^{\infty} A^{i} \epsilon_{t-1 - i} $$ But

$$E[ y_{t-1} \epsilon_t'] = E[\mu \epsilon_t'] + \sum_{i=0}^{\infty} A^{i} E[\epsilon_{t-1 - i} \epsilon_t'] =^{i =1} A \Omega$$

With this, I can not reach my goal. Some ideias?