The Naive Forecast
The naive forecast is very simple: the next value will be the same as the last value.
What will be the high temperature for tomorrow? It will be the same as the high temperature today.
It's called naive for a reason. You don't need to have any more information or knowledge about the "system" you're working with than a measurement or description of the present.
While the naive forecast is rarely used to produce a final forecast, it is used to tell us about the "system" under investigation. Since I work with finance and economics, my "systems" are the stock markets, bond markets, currency markets, commodity markets, and national economies. Although I do look at climate and weather, I don't do it professionally.
The naive forecast is one of the simplest "models" about a system that can be built. A model is an abstraction, usually mathematical, of a physical system that is used to describe the system and, in my case, predict what the system will do.
Since it is such a simple model, the naive forecast is also used as a benchmark. Can a more complicated or complex model reliably beat the naive model.
Here I will use the Standard and Poor's 500 stock index as a measure of the stock market. I will pick on the S&P 500 a lot throughout my posts for two reasons: 1) it is a measure of a real world, open market system; and 2) it deviates significantly from simple statistical measures.
So, how would a naive forecast of the S&P 500 be produced. Simple. The average of the S&P 500 for next month will be the same as the average of the S&P 500 for this month. Or, if you don't like averages, the value at the end of next month will be the same as the value at the end of this month.
Or if you have an aversion to values, you can say that if the S&P 500 was "up" last month, it will be "up" this month.
The S&P 500 closed at 3259.21 in December 2019. Therefore, the S&P 500 will close at 3259 for January 2020.
The stock market was "up" in December 2019. Therefore the stock market will be "up" in January 2020.
If I used the naive forecast of the S&P 500, how would it actually perform?
First, let's define "error" as the difference between the actual values and the forecast values:
error = actual - predicted
So for the naive forecast, the error will simply be the difference in the S&P 500 from one month to prior month.
Below is a plot of the naive forecast error over time from 1980 through 2019. The y-axis is the difference in the average S&P 500 from one month to the prior month.
Another way to visualize the error of our naive forecast is with a histogram. A histogram shows the range of error values along the bottom and the count of error values along the y-axis.
The histogram above shows how the errors of the naive model cluster around one point and how the errors spread out. The histogram also shows that as the errors become very large (+/- 200), the count of these extreme values is very low.
This explaination of the naive forecast and the errors from the naive model is just one of the building blocks I need to delve deeper into the financial markets and economics.
In further posts, I will use the S&P 500 errors above to explore how the stock markets work.