<p>It is well known that Jorda(2005) proposed the following model called local projection:
<span class="math-container">$$y_{t+h} - y_{t-1} = \beta_h shock_{t} + \gamma_h ctr_{t-1} + \epsilon_{t,h}, h = 0,1,2,\dots,H.$$</span>
I am trying to calculate AIC for such a model. To do this, I have to calculate the likelihood function, which in turn relies on the joint distribution of <span class="math-container">$\vec{\epsilon_t} = (\epsilon_{t,0},\epsilon_{t,1},\dots,\epsilon_{t,H})$</span>. I guess that <span class="math-container">$\vec{\epsilon_t}$</span> has some <span class="math-container">$H+1$</span>-dimensional Gaussian distribution since we usually estimate this model by OLS. But I have no idea how to determine the variance-covariance matrix of the distribution.</p>
<p>It seems reasonable for me to calculate <span class="math-container">$Var(\epsilon_{t,h})$</span> by Newey-West correction (Newey, West, 1987, 1994), though I am still not very sure about that.</p>
<p>What bothers me most is whether it is appropiote to assume <span class="math-container">$Cov(\epsilon_{t,h}, \epsilon_{t, j}) = 0$</span>, since there may be autocorrelations between the residuals given the time-series context. <strong>I also do not know how to calculate these possibly-existing covariances.</strong> Can anyone please help me?</p>
<p><strong>Note:</strong> I know there is a R-package with which one can estimate an LP and calculate the AIC. But I prefer to realise it by myself.</p>

CC BY-SA 4.0
2024-07-19T09:35:56.280Z
Have you estimated the model? So you have residuals? Then you have vectors $\hat \epsilon_t$ is not enough to calculate variance-covariance matrix estimate based on these? For example using the formula $\frac{1}{T}\sum_t \hat \epsilon_t\hat \epsilon_t^\top$ being outer product.
Jesper Hybel
CC BY-SA 4.0
2024-07-19T09:53:37.393Z
@JesperHybel It seems to me that a direct calculation of variance may be flawed due to potential heteroskedasticity and autocorrelation, as is described above.
zyy
CC BY-SA 4.0
2024-07-19T09:55:40.003Z
@JesperHybel For example, it will be incorrect to calculate confidential interval using the covariance matrix in your comment, one has to adjust for heteroskedasticity by Newey-West corrections or so. See Jorda (2005, 2023).
zyy
CC BY-SA 4.0
2024-07-19T23:51:17.213Z
You can estimate the vector model single equation instead of as VAR. You add lags until you get rid of serial correlation. Formula for HAC s.e.s is here https://www.stata.com/manuals14/tsnewey.pdf and here is a similar question: https://stats.stackexchange.com/questions/390139/implement-var-model-in-r-with-hac-corrected-standard-errors. It is explained on page 166 of the paper you refer to how to calculate stds for impulse response function elements.
Jesper Hybel
CC BY-SA 4.0
2024-07-22T16:00:48.833Z
Cross-posted on Cross Validated [here](https://stats.stackexchange.com/questions/651389).
Richard Hardy
<macroeconomics><econometrics><time-series><causality>
On assumptions of local projection method
zyy
# On assumptions of local projection method

(Asked by: username_0 on 2024-07-19)

It is well known that Jorda(2005) proposed the following model called local projection:
$$y\_{t+h} - y\_{t-1} = \beta\_h shock\_{t} + \gamma\_h ctr\_{t-1} + \epsilon\_{t,h}, h = 0,1,2,\dots,H.$$
I am trying to calculate AIC for such a model. To do this, I have to calculate the likelihood function, which in turn relies on the joint distribution of $\vec{\epsilon\_t} = (\epsilon\_{t,0},\epsilon\_{t,1},\dots,\epsilon\_{t,H})$. I guess that $\vec{\epsilon\_t}$ has some $H+1$-dimensional Gaussian distribution since we usually estimate this model by OLS. But I have no idea how to determine the variance-covariance matrix of the distribution.

It seems reasonable for me to calculate $Var(\epsilon\_{t,h})$ by Newey-West correction (Newey, West, 1987, 1994), though I am still not very sure about that.

What bothers me most is whether it is appropiote to assume $Cov(\epsilon\_{t,h}, \epsilon\_{t, j}) = 0$, since there may be autocorrelations between the residuals given the time-series context. **I also do not know how to calculate these possibly-existing covariances.** Can anyone please help me?

**Note:** I know there is a R-package with which one can estimate an LP and calculate the AIC. But I prefer to realise it by myself.

**Tags:** macroeconomics, econometrics, time-series, causality
> **Comments:**
> **username_1**: Have you estimated the model? So you have residuals? Then you have vectors $\hat \epsilon_t$ is not enough to calculate variance-covariance matrix estimate based on these? For example using the formula $\frac{1}{T}\sum_t \hat \epsilon_t\hat \epsilon_t^\top$ being outer product.  
> **username_0**: @JesperHybel It seems to me that a direct calculation of variance may be flawed due to potential heteroskedasticity and autocorrelation, as is described above.  
> **username_0**: @JesperHybel For example, it will be incorrect to calculate confidential interval using the covariance matrix in your comment, one has to adjust for heteroskedasticity by Newey-West corrections or so. See Jorda (2005, 2023).  
> **username_1**: You can estimate the vector model single equation instead of as VAR. You add lags until you get rid of serial correlation. Formula for HAC s.e.s is here https://www.stata.com/manuals14/tsnewey.pdf and here is a similar question: https://stats.stackexchange.com/questions/390139/implement-var-model-in-r-with-hac-corrected-standard-errors. It is explained on page 166 of the paper you refer to how to calculate stds for impulse response function elements.  
> **username_2**: Cross-posted on Cross Validated [here](https://stats.stackexchange.com/questions/651389).  

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