Could someone explain these ACF and PACF plots for what I think is a lag1 series?

Question

I am trying to get my head around simple time series analysis. I think this R code shows my confusion.

### simulating lag1 autoregressive process
library(dplyr)
library(ggplot2)
library(GGally)
set.seed(12345)
n <- 1000
divisor <- 2 # how much less variance to give the lag1 variance contribution to tmp3
tmp1 <- rnorm(n)  # base data
tmp2 <- dplyr::lag(tmp1, n = 1) # lag1 component
tmp <- tmp1 + tmp2/divisor
tmp <- na.omit(tmp)
tmpDF <- as.data.frame(cbind(tmp,
                             dplyr::lag(tmp, n = 1),
                             dplyr::lag(tmp, n = 2),
                             dplyr::lag(tmp, n = 3),
                             dplyr::lag(tmp, n = 4)))
colnames(tmpDF) <- c("tmp","lag1","lag2","lag3","lag4")
ggpairs(tmpDF)
acf(tmp, lag.max = 10)
pacf(tmp, lag.max = 10)

That generates this ACF plot:

Which is pretty much what I'd expect. However it gives this PACF:

Thinking about that ACF plot made me realise that what I had there wasn't a typical autocorrelation process but the simple addition of a lag1 component to data, not the same thing! Oops. OK. I think this, though probably clumsy is a true lag1 autocorrelation process:

tmpDF <- as.data.frame(cbind(tmp,
                             dplyr::lag(tmp, n = 1),
                             dplyr::lag(tmp, n = 2),
                             dplyr::lag(tmp, n = 3),
                             dplyr::lag(tmp, n = 4)))
colnames(tmpDF) <- c("tmp","lag1","lag2","lag3","lag4")
ggpairs(tmpDF)
acf(tmp, lag.max = 10)
pacf(tmp, lag.max = 10)

And that generates this ACF plot with the decreasing later correlations I had expected.

and this PACF plot.

OK, that's what I expected so I've answered part of my own question but can anyone explain to me why I get those sequential and "significant" partial autocorrelations in the first example?

TIA,

Chris

Answer 1

One way to interpret what's going on here is the equivalence of an MA(1) to an AR($\infty$). Reflecting the MA representation, the ACF goes to 0 after 1 lag. Reflecting the AR representation, PACF (speaking loosely) has non-zero terms out to $\infty$.

Moving average order 1 is equivalent to an auto-regressive order $\infty$

You have created an MA(1) model which is equivalent to an AR($\infty$) model. Your model is:

$$y_t = \epsilon_t + \frac{1}{2} \epsilon_{t-1} $$

Using the lag operator, you can write this as:

$$ y_t = \left( 1 + \frac{1}{2} L \right) \epsilon_t $$

You can show this is invertible since $\left|\frac{1}{2}\right| < 1$ hence:

$$ \left(1 + \frac{1}{2} L \right)^{-1} y_t = \epsilon_t $$

As I discuss in this answer, this gives you $\sum_{j=0}^\infty \left(-\frac{1}{2} \right)^j y_{t-j} = \epsilon_t $.

\begin{align*} y_t - \frac{1}{2} y_{t-1} + \frac{1}{4} y_{t-2} - \frac{1}{8} y_{t-3} + \ldots &= \epsilon_t \end{align*}

From the MA representation, you can easily compute the autocorrelation function:

$$ \rho(k) = \left\{ \begin{array}{rr} 1 & \text{ for }k = 0 \\ \frac{1}{2} & \text{ for } k = 1 \\ 0 & \text{otherwise}\end{array}\right. $$

The the slowly tailing off partial autocorrelation reflects the infinite order AR representation. Based on answers here and here, you'll find the PACF for $k\geq 1$ is:

$$ \alpha(k) = \frac{\left( \frac{1}{2}\right)^k(-1)^{k+1}}{\sum_{j=0}^k \left(\frac{1}{2}\right)^{2j}} $$