In the previous section, we explored a random process, which serves as a good starting point in the study of time series. However, most time series we encounter, like the Amazon sales series, will not be random. In such cases, our job will be to apply statistical tools to model the nonrandom behavior by formulating a mathematical summary of the behavior. In the next section, we learn that regression can serve as one of those statistical tools.

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But before we move on to formal modeling, we consider here a very special non-random process called a random walk. We give it special attention because it is prominently discussed in finance applications as it often well describes stock price movements over time. A random walk process can be represented by the following equation:

This equation is different than the one given for the random process (page 646) in that there is the term of found on the right side of the equation. If the variable represents a price of a stock, then the random walk model says that the observed price for period is equal to a constant plus the previous period’s price plus a random deviation. As defined earlier, these deviations are independent, with mean 0 and standard deviation .

Random walks exhibit curious behavior as can be seen with our next example.

It should be clear from Example 13.8 that a random walk is anything but random. If we look closely at the random walk model equation, we can gain insights about the random walk process by rearranging terms:

Recognize that represents the change in from one period to the next. In the area of time series, period-to-period changes of a time series are known as first differences. For convenience, define as . This gives us the following relationship:

Compare this equation with the equation for a random process (page 646). Other than the variable name being here, you should find the equation to be the same. The equation tells us that differences of a random walk follow a random process, with the underlying mean change of the random walk observations being .

If the mean change in the random walk is zero, then the random walk is said to have no drift. With no drift, there is no tendency of the random walk to go in any particular direction, as was seen with the random walk model simulated in Example 13.8. If is not 0, then the random walk is said to have drift. Random walks with will tend to drift upward, while random walks with will tend to drift downward.

To make a forecast of a future price, we write again the random walk model but shift the subscript one period into the future:

The parameter represents the mean price change. For the Honda series, we find that the sample mean of the price changes is 0.0029. Testing this sample mean against a null hypothesis of , we would find a -value of 0.968; thus, there is not sufficient evidence to reject the null hypothesis. In that light, we proceed with our forecast based on a zero mean price change value. This implies that we are proceeding with a random walk model with no drift. The term represents the future random deviation, which is unknown. If the random deviations have mean 0, our “best” guess for is simply 0. In the end, the forecast equation for predicting Honda’s closing price for time is given by:

In other words, the best forecast of the next period’s price is simply the current period’s price. Using the current observation as the forecast of the next period is known as a naive forecast.

Price changes versus returns

Our discussion of the random walk model centered around the ups and downs of prices. Earlier in Section 13.1, we looked at the ups and downs of Disney prices in terms of percentages. Percentage changes in prices are referred to as returns. The advantage of using returns over price changes is made evident with our next example.

The phenomenon we are witnessing with the increasing variability of price changes is common with economic time series. In particular, many economic time series change as a rate or percentage. Suppose that prices change on average by 1% whether up or down. This would mean that if the price level is $10, the expected price change would be $0.10. But if the price level is at $100, then the expected price change would be $1.00, which is 10 times greater than when the price level was $10.

We can compute returns using the standard formula:

In finance, there can be some ambiguity with the use of the term “return.” It can be defined as the change in price (or value), which, in the preceding formula, is given by the numerator term. But, it is also common practice to have return represent the change in value as a percentage. When return is calculated as a percentage, we could refer to the computed quantity as a “rate of return.” We, however, will go with the standard practice of simply calling the computed values “returns.”

For example, the weekly closing price on July 21, 2014, was $34.99, while the weekly closing price on July 28, 2014, was $34.60. This means the return was

Consider now the following calculation:

The closeness of these values is not accidental. It turns out that the difference of natural logged values is approximately a percent change. We have seen the benefits of the log transformation in many examples. Given these benefits, along with its nice interpretation, it is no wonder we find that so many data analyses in economics and finance are routinely done in logged units.

The stabilizing effect of the log transformation on the variance of a time series can be quite useful in simplifying modeling efforts. Take a look back at Figure 13.4 (page 648), which shows the time plot of Amazon sales. You should not be surprised that our eventual modeling of this series will involve the log transformation.