A great many people are confused on how best to measure trend. This is understandable because in most cases, nobody has explained to them that trend is a complex thing that requires more than one measurement. There are two primary categories of trend measurement. Trend strength is a measure of how much change in the data takes place during a trend. The greater the rate of change, the greater the strength of the trend. Trend strength is often referred to as "momentum" when applied to financial data. Trend quality is a measure of what fraction of a given interval of a time series can be considered as trend. Most financial trends are mixed: they are combinations of trend and noise (aka "choppiness"). It is important to know how much actual trend is present before good trading decisions can be made. There are only a few trend quality indicators in the public domain. The two most prominent are the Kaufman Efficiency Ratio and the True Strength Index (ironically, not a measure of trend strength). I honestly can't tell you what ADX measures about trend. I have always lost interest halfway through the description. Some technical indicators are way too kludgey for me to take them seriously. Linear regression is a simultaneous measurement of trend strength and trend quality. IMO it fails at both because (a) a pure trend can be nonlinear, which linear regression cannot differentiate from choppiness, and (b) the linear regression gives false levels of the actual overall momentum.
I want to say 2 things to this toppic. 1. trend = is a longer time price move in one direction, definied by the timeframe. Usually generall definied by (for an up trend) higher high and higher low. thats the first sign of an uptrend. 2. trend = is a pattern/situation that appears in the markets over and over again, and thereby, called as an "trend". There are a lot of ways to measure both kind of trends here. Indicator work very well for me, you just know how to use them, and this is very difficult to learn. But more important is point 2 of trend definition here. So trade with the trend. trendtrader.
Trend is tricky. First, the components that make up the trend must be strictly defined. Then and only then can you then strictly define what makes up "your" trend. (Remember "your" trend is only relevant to you and the chart you are trading.) Then strictly define your triggers and exits that define your decision points in "your" trend. Finally, remember to "not" trade your trend but trade your triggers as they play out inside your defined trend. If you can do this without distraction, you have a high probability of consistent success and profits.
Regarding (a), on the contrary, just try doing linear regression in Excel. Various goodness of fit statistics are generated. It is easy to tell a good fit from an attempt to fit chop.
Various? Aside from R-squared, what other GoF is there? My point is that a noiseless nonlinear trend (e.g., parabolic, cubic) will "appear" as a noisy linear trend to linear regression because you're forcing a square peg into a round hole, regardless of how much actual choppiness there is.
Agreed. Maybe linear regression can be seen as a first order approximation. The only benefit is that you basically only estimate one parameter (the slope) which makes it sort of robust (which of course is of little value if it cannot be used for other reasons). One problem with a non-linear modelling of trends is that you will have to fit more parameters to the data with the obvious risk of over-fitting. As always it is hard to balance model complexity/descriptive power against the risk of over-fitting. So as you say, this is neither as easy or straight-forward as one might first think.
R-squared and Adjusted R-squared Correlation between y and y-hat F-test and significance level of F You are right that R-squared is goodness of fit. I mean to say, in layman's terms there is more than one way to tell the regression is trying to fit chop. If the residuals and measures of error are high, this indicates that the data is not very linear. I agree with the other poster that overfitting becomes a concern. Mentioning "noiseless" seems to indicate that noise is low enough to not be a problem. There is typically alot of noise, in equity markets, especially in these times of high volatility. You'd have to give a concrete example where parabolic or cubic fit works well on real-world data. Lots of times linear is "good enough" because attempting to model non-linear trend plus non-linear noise requires many more data points than are available for usable significance/confidence. Complex fits can work in the currency market, because the volatility is low, and even there the fits are linear trend plus non-linear noise, not non-linear trend plus non-linear noise.
Analysts often run to LR because it's easy, not because it's truly appropriate or even "good enough" lots of time. In measuring financial trend, you have three options: Option #1 - Account for the curvature of data somehow. Option #2 - Make irrelevant the curvature of the data. Option #3 - Just use a linear regression and hope for the best. The third option is the worst by far. The KER uses the second option and works better than LR as a result.