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1. Estimate the model with OLS, which gives you usual $SE(\beta_1)$, the RMSE $\hat \sigma$, and the residuals $\hat u_1,...,\hat u_T$.
2. Get the ${\hat r_1,...,\hat r_T}$ residuals from the auxiliary OLS regression of $x_t$ on a constant and calculate $\hat a_t = \hat u_t \cdot \hat r_t$ for $t=1,...,T$. If you had more regressors, you would include them as additional covariates.
3. Assuming that the serial correlation lasts up to one period, calculate $$\hat v(1)=\sum_{t=1}^T \hat a_t^2+ \sum_{t=2}^T \hat a_t \cdot \hat a_{t-1}.$$ The first term in $\hat v$ is what gets you the het-consistent standard error. The second term is the autocorrelation part. Assuming the autocorrelation is positive, this is why your standard errors blow up: you have less information. If you autocorrelation lasted longer, you would have weighted additional terms for each each lag. You might want to use a finite sample correction multiplier of $\frac{T}{T-k}$ here, where we count the constant as one of the $k$ covariates.
4. Calculate N-W standard error of $\beta_1$ as $$\left[ \frac{SE(\beta_1)}{\hat \sigma} \right]^2 \cdot \sqrt{ \hat v(1)}.$$

1. Estimate the model with OLS, which gives you usual $SE(\beta_1)$, the RMSE $\hat \sigma$, and the residuals $\hat u_1,...,\hat u_T$.
2. Get the ${\hat r_1,...,\hat r_T}$ residuals from the auxiliary OLS regression of $x_t$ on a constant and calculate $\hat a_t = \hat u_t \cdot \hat r_t$ for $t=1,...,T$. If you had more regressors, you would include them as additional covariates.
3. Assuming that the serial correlation lasts up to one period, calculate $$\hat v(1)=\sum_{t=1}^T \hat a_t^2+ \sum_{t=2}^T \hat a_t \cdot \hat a_{t-1}.$$ The first term in $\hat v$ is what gets you the het-consistent standard error. The second term is the autocorrelation part. Assuming the autocorrelation is positive, this is why your standard errors blow up: you have less information. If you autocorrelation lasted longer, you would have additional weighted terms for each each lag. You might want to use a finite sample correction multiplier of $\frac{T}{T-k}$ here, where we count the constant as one of the $k$ covariates.
4. Calculate N-W standard error of $\beta_1$ as $$\left[ \frac{SE(\beta_1)}{\hat \sigma} \right]^2 \cdot \sqrt{ \hat v(1)}.$$

1. Estimate the model with OLS, which gives you usual $SE(\beta_1)$, the RMSE $\hat \sigma$, and the residuals $\hat u_1,...,\hat u_T$.
2. Get the ${\hat r_1,...,\hat r_T}$ residuals from the auxiliary OLS regression of $x_t$ on a constant and calculate $\hat a_t = \hat u_t \cdot \hat r_t$ for $t=1,...,T$. If you had more regressors, you would include them here.
3. Assuming that the serial correlation lasts up to one period, calculate $$\hat v(1)=\sum_{t=1}^T \hat a_t^2+ \sum_{t=2}^T \hat a_t \cdot \hat a_{t-1}.$$ The first term in $\hat v$ is what gets you the het-consistent standard error. The second term is the autocorrelation part. Assuming the autocorrelation is positive, this is why your standard errors blow up: you have less information. If you autocorrelation lasted longer, you would have weighted additional terms for each each lag. You might want to use a finite sample correction multiplier of $\frac{T}{T-k}$ here, where we count the constant as one of the $k$ covariates.
4. Calculate N-W standard error of $\beta_1$ as $$\left[ \frac{SE(\beta_1)}{\hat \sigma} \right]^2 \cdot \sqrt{ \hat v(1)}.$$

1. Estimate the model with OLS, which gives you usual $SE(\beta_1)$, the RMSE $\hat \sigma$, and the residuals $\hat u_1,...,\hat u_T$.
2. Get the ${\hat r_1,...,\hat r_T}$ residuals from the auxiliary OLS regression of $x_t$ on a constant and calculate $\hat a_t = \hat u_t \cdot \hat r_t$ for $t=1,...,T$. If you had more regressors, you would include them as additional covariates.
3. Assuming that the serial correlation lasts up to one period, calculate $$\hat v(1)=\sum_{t=1}^T \hat a_t^2+ \sum_{t=2}^T \hat a_t \cdot \hat a_{t-1}.$$ The first term in $\hat v$ is what gets you the het-consistent standard error. The second term is the autocorrelation part. Assuming the autocorrelation is positive, this is why your standard errors blow up: you have less information. If you autocorrelation lasted longer, you would have weighted additional terms for each each lag. You might want to use a finite sample correction multiplier of $\frac{T}{T-k}$ here, where we count the constant as one of the $k$ covariates.
4. Calculate N-W standard error of $\beta_1$ as $$\left[ \frac{SE(\beta_1)}{\hat \sigma} \right]^2 \cdot \sqrt{ \hat v(1)}.$$

This is based on Wooldridge, Jeffrey M. "A computationally simple heteroskedasticity and serial correlation robust standard error for the linear regression model." Economics Letters 31.3 (1989): 239-243.

Take your model of $$y_t=\beta_0+\beta_1x_t+u_t,$$ where $t=1,...,T$. We will assume there are no other regressors and that the serial correlation only last up to one period.

1. Estimate the model with OLS, which gives you usual $SE(\beta_1)$, the RMSE $\hat \sigma$, and the residuals $\hat u_1,...,\hat u_T$.
2. Get the ${\hat r_1,...,\hat r_T}$ residuals from the auxiliary OLS regression of $x_t$ on a constant and calculate $\hat a_t = \hat u_t \cdot \hat r_t$ for $t=1,...,T$
3. Assuming that the serial correlation lasts one period, calculate $$\hat v(1)=\sum_{t=1}^T \hat a_t^2+ \sum_{t=2}^T \hat a_t \cdot \hat a_{t-1}.$$ The first term in $\hat v$ is what gets you the het-consistent standard error. The second term is the autocorrelation part. Assuming the autocorrelation is positive, this is why your standard errors blow up: you have less information. You might want to use a finite sample correction multiplier of $\frac{T}{T-k}$ here, where we count the constant as one of the $k$ covariates.
4. Calculate N-W standard error of $\beta_1$ as $$\left[ \frac{SE(\beta_1)}{\hat \sigma} \right]^2 \cdot \sqrt{ \hat v(1)}.$$

/* N-W Standard Errors With One Regressor and Serial Correlation That Dies Down After 1 Period */
webuse idle2, clear
tsset time
list time usr idle, clean noobs
/* Step 1 */
reg usr idle
scalar se_beta1 = _se[idle]
scalar sigmahat = e(rmse)
predict double uhat, resid
/* Step 2 */
reg idle
predict double rhat, resid
gen double ahat = uhat*rhat
/* Step 3 */
gen double v = ahat^2
replace v    = v + ahat*L1.ahat in 2/L
sum v
scalar v1     = r(sum) 
scalar v1_fsc = r(sum)*(30/28) // Stata uses a finite sample correction of T/(T-k)
/* Step 4 */
di "Usual N-W SE = " sqrt(scalar(v1))*[scalar(se_beta1)/scalar(sigmahat)]^2
di "Stata's N-W SE  = " sqrt(scalar(v1_fsc))*[scalar(se_beta1)/scalar(sigmahat)]^2
/* Compare to newey command */
newey usr idle, lag(1)
di _se[idle]
/* Compare To Robust Sandwich SE */ 
sum v
scalar v0     = r(sum)*(30/28)
di "Stata's Sandwich SE  = " sqrt(scalar(v0))*[scalar(se_beta1)/scalar(sigmahat)]^2
reg usr idle, robust
di _se[idle]

Take your model of $$y_t=\beta_0+\beta_1x_t+u_t,$$ where $t=1,...,T$. We will assume there are no other regressors and that the serial correlation only lasts up to one period (so shocks do not persist for very long).

1. Estimate the model with OLS, which gives you usual $SE(\beta_1)$, the RMSE $\hat \sigma$, and the residuals $\hat u_1,...,\hat u_T$.
2. Get the ${\hat r_1,...,\hat r_T}$ residuals from the auxiliary OLS regression of $x_t$ on a constant and calculate $\hat a_t = \hat u_t \cdot \hat r_t$ for $t=1,...,T$. If you had more regressors, you would include them here.
3. Assuming that the serial correlation lasts up to one period, calculate $$\hat v(1)=\sum_{t=1}^T \hat a_t^2+ \sum_{t=2}^T \hat a_t \cdot \hat a_{t-1}.$$ The first term in $\hat v$ is what gets you the het-consistent standard error. The second term is the autocorrelation part. Assuming the autocorrelation is positive, this is why your standard errors blow up: you have less information. If you autocorrelation lasted longer, you would have weighted additional terms for each each lag. You might want to use a finite sample correction multiplier of $\frac{T}{T-k}$ here, where we count the constant as one of the $k$ covariates.
4. Calculate N-W standard error of $\beta_1$ as $$\left[ \frac{SE(\beta_1)}{\hat \sigma} \right]^2 \cdot \sqrt{ \hat v(1)}.$$

/* N-W Standard Errors With One Regressor and Serial Correlation That Dies Down After 1 Period */
webuse idle2, clear
tsset time
list time usr idle, clean noobs
/* Step 1 */
reg usr idle
scalar se_beta1 = _se[idle]
scalar sigmahat = e(rmse)
predict double uhat, resid
/* Step 2 */
reg idle
predict double rhat, resid
gen double ahat = uhat*rhat
/* Step 3 */
gen double v = ahat^2
replace v    = v + ahat*L1.ahat in 2/L
sum v
scalar v1     = r(sum) 
scalar v1_fsc = r(sum)*(30/28) // Stata uses a finite sample correction of T/(T-k)
/* Step 4 */
di "Usual N-W SE = " sqrt(scalar(v1))*[scalar(se_beta1)/scalar(sigmahat)]^2
di "Stata's N-W SE  = " sqrt(scalar(v1_fsc))*[scalar(se_beta1)/scalar(sigmahat)]^2
/* Compare to newey command */
newey usr idle, lag(1)
di _se[idle]
/* Compare To Smaller Robust Sandwich SE */ 
sum v
scalar v0     = r(sum)*(30/28)
di "Stata's Sandwich SE  = " sqrt(scalar(v0))*[scalar(se_beta1)/scalar(sigmahat)]^2
reg usr idle, robust
di _se[idle]