Case-Shiller Housing Index: Log vs. Linear Scale

The S&P/Case-Shiller Housing Index is one of the most watched indices for the U.S. housing market. For a number of reasons, I think it is probably the best (in terms of methodology). This post starts a series of posts where I will examine the Case-Shiller and eventually culminate in my time series model for the Case-Shiller and show what the model forecasts going forward.

Linear Scaling

Typically, the Case-Shiller is presented as a long term time series showing housing prices going back to 1987. It usually is shown with linear scaling as seen in the Jan-09 S&P/Case-Shiller press release on page 2. Here is my plot of the 10-city composite going back to 1987. The y-axis here is a linear scale. The linear scale presents equal absolute changes in the index value equally. While that might make sense initially, what it means is that a 10 point move in the index back in '87 when it was under 100 will look the same as a 10 point move in the index when it was over 200 in '06. So one's caveman instinct looking at the chart would lead to these conclusions graphically (also pointed out in the chart):

1. Housing prices look like they were increasing more than twice as fast in '05 as they were in '88.
   
2. Starting in the late '90s, prices increased faster and faster till the peak in '06. The appeared to be going parabolic.
   

Both of those graphically based intuitive conclusions are completely wrong.

Log Scaling

The preferred alternative is to display the y-axis in a log scale. There are a couple of ways to do that but the easiest is if one's graphing software allows the option to change the y-axis to log scaling. Here is a chart of the same Case-Shiller 10-city composite but in log scale. A log scale puts similar % moves in the index in similar proportion. Take a look for a second and you will come to a very different intuitive conclusion about what was going on with housing prices in the different time periods. Here is how I would restate the above conclusions based on the chart:

1. Housing prices were increasing only slightly faster '05 than they were in '88.
   
2. The increase that started in the late 90's and went through '06 was fairly linear. With maybe only a slight increase in rate of gains in the last few years.
   

In this case, log scaling is much more appropriate in my opinion. It is not really a right or wrong perspective, just that the linear scale on this series misleads one's instinct. As it turns out, this is often the recommended perspective for data series in which % changes are the important factor, not absolute changes. Sometime in the future I will touch on this for stock charts but the conclusion is the same, log scaling is preferred. There are in fact other more statistical reasons that I will try to touch on as I get to doing more statistical analysis and modeling on the Case-Shiller.

One additional Case-Shiller example. CalculatedRisk recently had a good post comparing the different tiers of houses and how their prices had appreciated and crashed at different rates. Although I won't disagree at all with the conclusions drawn in that post, I think the linear scaling overstates the discrepancy between tiers. Here is a comparison for this data:

(click on any of the charts for a larger view) The linear scaling on the right overstates how fast prices appreciated and more importantly it leads one to think that higher tier homes increased even faster compared to the other tiers than they really did. The other thing you notice, is that much more detail is evident in the early '90s on the log scale which the linear scaling washes out. That is one big issue with linear scaling for data that increases alot over time: lower values in the past get compressed and become flat and hard to see. Again, the bottom line here doesn't necessarily change, but how bad the situation really was is exaggerated in the linear scale chart.

Enjoy and thanks for reading! I will continue in the following days with further analysis of the Case-Shiller Housing Index.