"This again?"

Yup, this one again:

The big one

When I first put the trend line on it, I was imitating what I've seen many times. Straight line thru scatterplot. But then I got thinking about it. I wonder what a curved trend line would look like. I was thinking: Maybe it shows a high that suggests the P2P value we'd need to maximize economic growth.

I don't really have a reason for picking a "linear" trendline over one of the curved ones. I picked linear because that's the one I always see. Don't remember many curved trendlines on scatterplots. The only one I can remember is the Phillips curve, and things didn't work out so well for that one.

I've long felt there must be an optimum range for the P2P ratio, one that best promotes economic growth. And a curved trendline might show such a range, where the linear cannot. So I decided to look at some trendlines. Note: I can't tell you anything technical about trendlines. (Maybe you can tell me?)

LINEAR All these "little" graphs are the same except for the type of trend line they show. Click them to see them bigger, should the need arise. This first one is the linear, same as I've been using since Monday. It shows a general trend -- the "gist" of the numbers, as I said the other day. Of course, I don't believe for a moment that all those dots in all those spots could ever meld together to produce a perfectly straight trend, except in the world of numbers.

LOGARITHMIC The "exponential" trendline option was not available  for this particular graph. This little graph shows the "logarithmic" line. Looks similar to the linear line and sits in about the same position --high on the left, low on the right. But this line has a slight sag to it. Told ya the line shouldn't be perfectly straight!

POLYNOMIAL 2 This one is a second order polynomial trend. Like the logarithmic line, it looks similar to the linear, again higher on the left than on the right. But this line instead of being straight or sagging has a bit of an arch to it.

POLYNOMIAL 3 Third order polynomial. Finally, a trend line with some shape to it! Something of an S-curve. Higher on the left and lower on the right. But this time the ends of the line are not the high and low points. Now that is something I can believe! The high point of the curve looks to be somewhere between 2.0 and 3.0 on the horizontal axis. So, a private-to-public ratio somewhere around 2.5. Sounds right to me.

POLYNOMIAL 4 Fourth order polynomial. An S-curve, like the previous trendline. A little less curve on the left, a little more on the right. I think these trendlines are overly influenced by the absence of data beyond the endpoints -- as people say of the Hodrick-Prescott calc, and as I suggested recently about starting a dataset in the midst of wartime conditions. Anyway, the high point of the curve again appears to be between 2 and 3. A policy target?

POLYNOMIAL 5 Fifth order polynomial. The shape is a little more complex now. A little too complex, I think. I don't think you're going to get a high-low-high-low-high-low response from growth by repeatedly increasing the level of private debt relative to public debt. It just doesn't seem reasonable. The sixth order polynomial looks very much like this one.

I wonder what curve calculation old Bill Phillips used...