10 Worksheet 10: Digging into the underlying time series concepts, Part 3
Before proceeding, make sure that you have finished the preceding worksheets.
So far, you have encountered stationary and nonstationary stochastic processes. You will now be exploring how they can be applied to a classic problem in finance – can we predict returns? You will also be exploring the impact of nonstationarity on the use of linear regression with time series data.
10.1 Why bother with the ACF?
Since the sample ACF contains information about the extent and strength of dependence, it can be used to summarize how predictable returns are from their past.
Returns do not have to be limited to stock returns. Any asset or economic time series which returns or growth rates are metrics for performance or rewards could be used as inputs for the application of ACFs.
There is another use of ACFs, which we will downplay in this workbook. The ACF is sometimes used to discriminate among different time series models. You have seen the differences in the theoretical ACFs of the AR(1) process and the MA(1) process in the previous worksheet. In a way, you can use the sample ACF to somehow help you identify which time series linear process can be suitable. This is sometimes called the Box-Jenkins methodology. To give a bit more detail, arima.sim() allows you to generate observed paths for linear processes called AutoRegressive Integrated Moving Average \((p,d,q)\) or ARIMA\((p,d,q)\) processes. An AR(1) process is an ARIMA(1,0,0) process, an MA(1) process is an ARIMA(0,0,1) process, and a random walk is an ARIMA(0,1,0) process. The Box-Jenkins methodology is a procedure to figure out what would be the “best” \(p, d, q\) according to some criterion usually associated with good forecasting ability or even good fit. Fortunately, the auto.arima() command from the forecast package by Hyndman et al. (2026) can automatically select the “best” \(p, d, q\). But as with all things automatic, there are choices to be made and they are laid out in Hyndman and Khandakar (2008).
Unfortunately, this is a difficult iterative process, as you may have seen in the previous worksheet. We are typically using the limited data to identify or discriminate across a broad range of stochastic processes. But it can be useful as a starting point to know what kind of process could have generated the data you see and it could be useful for quick forecasting tasks.
For our purposes, we will ditch Box-Jenkins and focus on the summaries provided by the ACF to gain a sense of whether stock returns are predictable. Intuitively, if stock returns exhibit dependence, then stock returns can be predicted from its past.
10.2 Exercise A
In this exercise, you will be inspecting the sample ACFs of stock returns.
I will demonstrate for the Philippine Stock Exchange, Inc. (PSEi). We will examine what the ACFs look like for returns at different frequencies. So far you have encountered daily returns. We will also examine transformations of returns. So far you have encountered the stock returns as they are.
Take note that in what follows we only get to see one observed path of stock returns.