## MA Backcasting in Eviews vs. Box/Jenkins

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jpfeifer
Posts: 3
Joined: Wed Apr 08, 2015 11:47 am

### MA Backcasting in Eviews vs. Box/Jenkins

According to the Eviews help, MA terms are backcasted by running the forward model backwards in time:

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\tilde \epsilon_t=u_t-\theta_1*\tilde \epsilon_{t+1}-...-\theta_q*\tilde \epsilon_{t+q}

with \tilde \epsilon_{T+i}= for i>0. This allows computing

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{\tilde \epsilon_{0},...,\tilde \epsilon_{-(q-1)}
. The help seems to suggest that the \tilde \epsilon are then used in the backward model

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\hat \epsilon_t=u_t-\theta_1*\hat \epsilon_{t-1}-...-\theta_q*\hat \epsilon_{t-q}

i.e. the \tilde \epsilon_{i}, i<1 are used as the \hat \epsilon_{i}. But the reference is the the Box/Jenkins (1970) book. In its 2016 edition, Chapter 7.1.4, the procedure seems to be quite different as the forward and backward model deliver two different, distinct sets of innovations. They therefore do not allow using the innovations from the forward model in the backward model. To solve this problem, they backcast the u_t for t={0,..,q-1} using the backcasts for \tilde \epsilon_t and employ that \tilde \epsilon_{i}=0,i<1 due to them being independent from u_t. These u_t, t<1 are then used in the backward model to run a forward recursion with \hat \epsilon_{i}=0 for i<q-1 due to independence. This then allows computing {\hat \epsilon_{0},...,\hat \epsilon_{-(q-1)}.

So what exactly does Eviews do here?

EViews Glenn
EViews Developer
Posts: 2598
Joined: Wed Oct 15, 2008 9:17 am

### Re: MA Backcasting in Eviews vs. Box/Jenkins

I don't quite understand everything you wrote in your comment (in particular the last couple of sentences) so my apologies if my answer doesn't address all of your issues.

That said, I believe that the biggest issue is that you are misreading the discussion of backcasting my our documentation.

The discussion of MA initialization in our manual is divided into three parts: forward recursion, backward recursion, and no backcasting.

The first part discusses forward recursion. It notes that if we have (1) initial estimate of the MA coefficients, (2) values for the unconditional residuals u_t, and importantly (3) estimates of the pre-sample values of the innovations \epsilon, we may use forward recursion to obtain the remaining values of the innovations, which may then be used in conditional least squares estimation. Left out of this section is how to obtain (3).

The second part of the discussion addresses the task of obtaining (3), the estimates of the pre-sample values of the innovations, using backward recursion. We note that given (a) actual values of the unconditional residuals, and after (b) initializing the innovations beyond the estimation sample to zero, and (c) initializing the pre-sample values of the unconditional residuals to zero, we may use the backward recursion to obtain estimates of the pre-sample innovations.

You are correct in noting that the backcasting should not employ the forward recursion results and our notation here reflects the fact that the backward recursion estimates of the innovation depend only on the unconditional residuals, are completely separate from the forward recursion, and are only used to obtain the pre-sample values to initialize the forward recursion.

The third section notes that you may instead obtain (3) by setting the pre-sample values to zero.

This backcasting procedure is as outlined in Box and Jenkins (1976) and appears to be referred to as the first cycle in Box, Jenkins, Reinsel (2008).