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st: Constructing confidence intervals for a sum of forecasts


From   Ian Sue Wing <isw@bu.edu>
To   statalist@hsphsun2.harvard.edu
Subject   st: Constructing confidence intervals for a sum of forecasts
Date   Mon, 26 Jul 2010 13:00:36 -0400

Dear StataListers,

I am generating a dynamic forecast of two variables, x and y, after running a vector autoregression, and I want to construct the forecast confidence interval for the sum of their first differences, D.x + D.y. My question is what is the correct way to do this.

I can easily generate the se's of the first differences of individual variables using -fcast compute, diff-. Let's call these se_Dx_hat and se_Dy_hat. It is simple to compute the se of their sum if I assume independence:

se(Dx_hat+Dy_hat) = sqrt( se_Dx_hat^2 + se_Dy_hat^2 )

However, I am not clear about how to correctly handle potential correlation in the first-differences, which this expression omits.

From first principles,

D.x + D.y = x(t) - x(t-1) + y(t) - y(t-1)

var(D.x + D.y) = var(D.x) + var(D.y) + 2 cov(D.x, D.y)

Now,

var(D.x) = var(x(t)) + var(x(t-1)) - 2 cov(x(t), x(t-1)) = se_Dx_hat^2,

with a similar expression for y. What I don't have is the covariance term:

2 * [ cov(x(t),y(t)) - cov(x(t),y(t-1)) - cov(x(t-1),y(t)) + cov(x(t-1),y(t-1)) ] = 2 * [ 2 * cov(x(t),y(t)) - cov(x(t),y(t-1)) - cov(x(t-1),y(t)) ]

Should I just assume that each of the terms in square brackets is a constant, given by, first, the data, and second, the appropriate elements in e(V) generated by my var? If not, can anyone recommend an alternative way of doing this calculation?

Thanks,

-i

--
Ian Sue Wing                      675 Commonwealth Ave., Boston MA 02215
Associate Professor               Tel: (617) 353-5741
Dept. of Geography&  Environment  Fax: (617) 353-5986
Boston University                 Web: http://people.bu.edu/isw

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