# Mean reverting (time series models)

In the context of mean reversion in time series model, what is the mean? Is it the trendline?

long term mean … just means over time that you would not stray very far from a long term trend.

got it. I was unsure if the mean was static or changing over time. I couldn’t figure how it could be static and I just wanted to be certain.

Yes it is static, otherwise, the time series is not covariance stationary, and you would use least squares method.

What about an autoregressive time series model? How can a mean be static?

Autoregressive time (AR) series is an alternative trend model, but it must be covariance stationary. If it isn’t, then use some other trend model but make sure no serial correlation, HSK, etc. If you plot the price of a stock, that would not be covariance stationary because it just keeps growing (hopefully). However, to make it covariance stationary, you can use the *change* in stock price from day to day, or the log of that if necessary. If IBM is trading at \$100 today, for tomorrow it might go up a little, go down a little, or stay at \$100. If it was a random walk, our best guess is that the price for tomorrow will be same as today! That’s because the change will revert to a mean of zero, unchanged. A random walk is an AR(1) series which has an intercept of zero (b_o=0) and slope of 1 (b_1 = 1). For example, AR(1): Xt = b0 + b1 X_t-1 + e. Setting b0=0 and b1=1, gets you a random walk: Xt=X_t-1 + e (i.e., today’s value is same as yesterday’s plus some random error whose expected value is zero). Having said that, I’m no statistician, but that’s how I see it, be warned!

Further to the above… then it would seem easy to check whether a time series is covariance stationary by simply checking whether b1=1. If b1=1 then there is unit root and the series is not covariance stationary. However, you can’t do that directly like that, you have to use a test developed by Dickey and Fuller (I think that’s the name). It is a test similar to a -t-stat test but you use values developed specifically for this test.

I’m not really following you here. I’m asking about mean reversion in the context of an AR model where, by definition, the value is expected to trend upward over time. If the value is said to be mean reverting, at any given point in time, is the mean going to be that trending line, or is it the average of all observations, in which case the mean is a horizontal line.

Robert A Wrote: ------------------------------------------------------- > I’m not really following you here. I’m asking > about mean reversion in the context of an AR model > where, by definition, the value is expected to > trend upward over time. The value does not trend up, that’s what I corrected earlier. The value is a fixed average.

If you want to model the consumer price index over time, how would you define the mean, or calculate mean reversion? Say for example the index was 100, 103, 106, 109.1, 112, 114.9, 118, etc. How are we approaching the calculation of the mean?

This series does not look like it is covariance stationary…we don’t know, but a graph of it and past experience indicates that inflation keeps rising. If that’s the case, then you can’t use an AR model for that. But you can try and define a new series as +3%, +2.91%, +2.92%, +2.66%, +1.79%, etc. Here you have a series that has been going up for the last 5 years. It is not expected to do that very often as we expect CPI to rise by, probably, 2.5% annually on average. So, there may be some lower positive or even negative growth to get the average 2.5%. The point is that if the graph shows a steadily rising line, it’s no good. Draw a graph of the original index and compare with the pernatage index, and you’ll see what covariance stationary means.

Okay, I just went back and re-read part of Reading 13. This isn’t sinking in too readily, but with each pass, I get a bit more of it. If we take my example of the CPI (and let’s set aside whether this is C/S), we need to be evaluating the change in price from one period to the next, not the cumulative price index itself. The price index, which is expected to rise, cannot have a mean, and that is what has been throwing me off. Am I correct with this? Two last questions. On p459, Table 5: 1. The R-squared is .3808. What is inflation being correlated with? 2. The standard errors in the top and middle panels are 3.4239, .2803 and .0417. How do they compute these? thank you!

R sqrd shows that prev month’s inflation can explain 38% variation of next month’s inflation. Top panel std. error is SEE, the deviation of predeicted value from mean, the other two are calculated in a different way, beyond the scope of CFA.