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Question: By mapping the OLS regression into the GMM framework, write the formula for the standard error of the OLS regression coefficients that corrects for autocorrelation but not heteroskedasticity. Furthermore, show that in this case, the conventional standard errors are OK if the $x$'s are uncorrelated over time, even if the errors $\varepsilon$ are correlated over time.

Attempt: So the general model is $y_t = \beta' x_t + \varepsilon_t$. OLS picks parameters $\beta$ to minimize the variance of the residual: $$\min_{\beta} E_T[(y_t-\beta' x_t)^2] $$ where the notation $E_t(\cdot) = \frac{1}{T} \sum_{t=1}^T( \cdot )$ denotes the sample mean. We find $\widehat{\beta}$ from the first-order condition, which states that: $$g_T(\beta) = E_T[x_t(y_t - x_t' \beta)] =0$$ In the GMM context, here, the number of moments equals the number of parameters. Thus, we set the sample moments exactly to zero and solve for the estimate analytically: $$\widehat{\beta} = [E_T(x_tx_t')]^{-1} E_T(x_t y_t)$$ Using the known result from GMM theory that $$Var(\widehat{b}) = \frac{1}{T} (ad)^{-1} aSa^{\prime} (ad)^{-1 \prime}$$ where in this case $a = I$ (the identity matrix), $d = -E[x_t x_t']$, and $S = \sum_{j=-\infty}^{\infty} E[f(x_t, b), f(x_{t-j}, b)']$ with $f(x_t, \beta) = x_t(y_t - x_t'\beta) = x_t \varepsilon_t$.

So the general formula for the standard error of OLS is $$Var(\widehat{\beta}) = \frac{1}{T}E(x_t x_t')^{-1} \left[\sum_{j=-\infty}^{\infty} E(\varepsilon_t x_t x_{t-j}' \varepsilon_{t-j})\right]E(x_t x_t')^{-1}$$

Now I know from the OLS assumptions:

(i) No autocorrelation: $E(\varepsilon_t \mid x_t, x_{t-1}, \cdots, \varepsilon_{t-1}, \varepsilon_{t-2}, \cdots) =0$

(ii) No heteroskedasticity: $E(\varepsilon_t^2 \mid x_t, x_{t-1}, \cdots, \varepsilon_{t-1}, \cdots) = constant = \sigma_{\varepsilon}^2$


What would the OLS standard error become if I correct for autocorrelation but not heteroskedasticity? Also how do I show that the conventional standard errors are OK if the $x$'s are uncorrelated over time, even if the errors $\varepsilon$ are correlated over time?

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