Solution to Exercise 2.5.

Problem Statement: Under the supposition that $\rho_j = 0$ for j > 2,

(a) Obtain approximate standard errors for r₁, r₂, and r_j, j > 2.

(b) Obtain the approximate correlation between r₄ and r₅.

Solution: For part (a), we apply (2.1.11), p. 32:

Here, (10) is right out of the book, and (11) is just a different version. Of course, the standard error is just the square root of this. There will only be finitely many terms in the sum since $\rho_k = 0$ for k > 2.

For k = 1,

Here, (12) is obtained by eliminating the summands which we know are from the assumption $\rho_k = 0$, k > 2, (13) is just all the summations expanded out, and (14) was obtained by ``simplifying'' the expression in Maple:

> s:=(1+2*r1^2)*(1+2*r1^2+2*r2^2)+(r1^2+2*r2)-4*r1*(2*r1*r2+2*r1);
                 2           2       2      2
   s := (1 + 2 r1 ) (1 + 2 r1  + 2 r2 ) + r1  + 2 r2 - 4 r1 (2 r1 r2 + 2 r1)

> simplify(s);
                    2       2       4       2   2              2
            1 - 3 r1  + 2 r2  + 4 r1  + 4 r1  r2  + 2 r2 - 8 r1  r2

Now, for k=2,

The steps are all the same as before. Here is the Maple code for the simplification:

> t:=(1+2*r1^2)*(1+2*r1^2+2*r2^2)+r2^2-4*r2*(2*r2+r1^2);
                      2           2       2      2           2
        t := (1 + 2 r1 ) (1 + 2 r1  + 2 r2 ) + r2  - 4 r2 (r1  + 2 r2)

> simplify(t);
                       2       2       4       2   2       2
               1 + 4 r1  - 5 r2  + 4 r1  + 4 r1  r2  - 4 r1  r2

Finally, the last part of (a) is k > 2. By supposition, $\rho_k = 0$, so we can immediately drop the terms $2 \rho_k^2 \sum_{v=-\infty}^{\infty} \rho_v^2$ and $4 \rho_k \sum_{v=-\infty}^{\infty} \rho_v \rho_{v-k}$ from (11). Also, for any integer v, at least one of v+k or v-k is outside the interval [-2,2], so the term $\sum_{v=-\infty}^{\infty} \rho_{v+k}\rho_{v-k}$ is also . Thus,

Finally, for part (b), we use (2.1.14), p. 34. Since 4 and 5 are bigger than 2, this formula applies (note that it only applies once we are into the region where k > q and q is the lag of the last nonzero autocorrelation). We have

Remarks: The takehome message is that these formulae exist. If you are like me and aren't satisfied until you know where they come from, you can take our advanced time series class. The chances of these formulae ever coming up on an exam are two: slim and none.

Remember: these results are valid for Gaussian processes. Under a ``long range'' independence assumption, one can derive formulae for the approximate variances (variances of asymptotic normal distributions) which depend on the first four moments, basically. The particular algebraic forms are highly dependent on the moments of multivariate normal distributions.