Moving Average Auto correlation

From CFAI book:

EQ 12: xt = εt + θεt−1, E(εt) = 0, E(εt2) = σ2, Cov(εts) = E(εtεs) = 0 for t ≠ s

This equation is called a moving-average model of order 1, or simply an MA(1) model. Theta (θ) is the parameter of the MA(1) model.33

Equation 12 is a moving-average model because in each period, xt is a moving average of ε_t_ and ε_t–1, two uncorrelated random variables that each have an expected value of zero. Unlike the simple moving-average model of Equation 11, this moving-average model places different weights on the two terms in the moving average (1 on εt, and θ on εt_–1).

“First, we examine the variance of xt in Equation 12 and its first two autocorrelations. Because the expected value of xt is 0 in all periods and ε_t_ is uncorrelated with its own past values, the first autocorrelation is not equal to 0, _ but the second and higher autocorrelations are equal to 0 _. Further analysis shows that all autocorrelations except for the first will be equal to 0 in an MA(1) model. Thus for an MA(1) process, any value xt is correlated with xt−1 and xt+1 but with no other time-series values; we could say that an MA(1) model has a memory of one period.”

Can someone show mathematically why second or higher autocorrelation are equal to zero. Thanks

It’s because of equation. Particularly this part:

Cov(εts) = E(εtεs) = 0 for ts,

For the first lag, t = s so the expected value of E(εtεs) is not 0, which means the first autocorrelation is also not 0.

But once we get to Lag 2 (and higher), ts so the equation applies where Cov(εts) = E(εtεs) = 0, which means the autocorrelation(s) equal 0.

Hope this helps.

This is just not correct. The first lag t and s are not equal, it’s the same as saying corr(εt(t-1)) which by definition in a MA(1) process is nonzero for order 1 and zero for greater than order 1 (really, it just drops off “sufficiently quickly” in most cases). Again, this is by definition of the process. If t=s, as you claim, the correlation is 1 and the covariance equals the variance of εs. This isn’t really about the subscripts (except the part where you misrepresented the idea).

I actually read this from another source since the CFAI doesn’t go into too much detail but from my understanding, the autocorrelation function would not be 1 since θ (theta) is the parameter for εt-1

So for an MA(1) model, ρ equals:

ρ1 = θ1/1 + θ12

ρs = 0 for s > 2

Here is the article if you were interested in reading it:

https://newonlinecourses.science.psu.edu/stat510/node/48/

The article uses different subscripts so it may get confusing but I’ve converted it below.

Looking further in MA(1) model, xt = μ + ε_t_ + θ1ε_t-1_

Mean: E(xt)

= E(μ + ε_t + θ1εt-1_)

= μ + 0 + (θ1)(0)

= μ

Variance: Var(xt)

= Var(μ+εt1εt−1)

= 0 + Var(εt) + Var(θ1εt−1)

= σ2ε + θ22ε

= (1 + θ122ε

As for the covariance between xt and xt-s. This is E(xt - μ)(xt-s - μ), which equals:

= E[(εt + θ1εt−1)(εt−s + θ1εt−s−1)]

= E[εtεt−s + θ1εt−1εt−s + θ1εtεt−s−1 + θ12εt−1εt−s−1]

When s = 1, the previous expression = θ1σε2

I was saying that if t=s (your original post) that implies the error in period t correlated with itself, which is a correlation of 1 (or a covariance equal to the variance of the error, again assuming t=s). I wasn’t referring to the parameter theta or the ACF for adjacent periods (which would mean t not equal to s, against what you wrote earlier). I agree the CFA curriculum is not great and the PSU resources are usually pretty good- I think I’ve listed them in here previously as a supplement!

The correlation would not be 1. The MA model puts different weights on the two terms. Let’s say θ1 = 0.7. Since ε_t _ = 1 and εt−1 = θ1, the correlation would be:

= 0.7/1+0.72

= 0.4698

You’re not understanding what I’m saying: you claimed when t=s. This literally means the correlation between one period and itself. By definition, correlation is 1 in that case. For example, say we have 12 periods 1,2,…11,12. The correlation of ε t=1 with ε_s=1 _(so t=s as you specified in your first post), is by definition 1. The formula for the ACF isn’t applicable in this case. Run an example on your computer if you don’t believe it.

If periods are 1 period apart (therefore t does not = s , against your first specification) then the formula applies and is consistent with a MA(1) process as you wrote (which is for one period and a different period); again this is when t doesn’t equal s. In your original post you said t equals s which implies the correlation of the series in a given period with itself.

For some reason, you’re hung up on something I’m not saying.

Once more, t doesn’t equal s for the first lag.