Timeline for calculate R² as a function of μ,σ
Current License: CC BY-SA 4.0
11 events
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| Mar 14, 2020 at 17:09 | comment | added | elemolotiv | @everyone thanks for your feedbacks. The concept of $R^2$ does not make sense in the context of a random walk, but for the records... I simulated it and there is a numerical relationship $E[R^2] = f(\frac{\mu}{\sigma})$ which looks like this: i.imgur.com/0fLVYEW.png - intuitively when $\mu \gg \sigma$ the random walks look more like a straight line so $E[R^2] \approx 1$ and viceversa. | |
| Mar 12, 2020 at 10:38 | answer | added | ReneBt | timeline score: 2 | |
| Mar 12, 2020 at 5:26 | comment | added | ReneBt | THere is a discussion here with external links that show what I refer to stats.stackexchange.com/questions/330857/… | |
| Mar 11, 2020 at 22:30 | comment | added | mlofton | @ReneBt: Could you explain what you mean by a series of eigenvectors whose features differ in frequency of features. Or, maybe an example if possible ? Thanks. | |
| Mar 11, 2020 at 22:28 | comment | added | mlofton | Renet and Whuber sound like they know more about this but the "true" $R^2$ between your random walk and the straight line is probably zero. One way to test this is to run thousands of simulatons so that you get thousands of realizations between the straight line and the random walk. Then calculate the average $R^2$ of all of them. I'm pretty sure it will be close to zero. There is a possibility of spurious correlation ( see Granger and Newbold for that ) but, if you simulate enough times, I would think you'd wash that effect away. | |
| Mar 11, 2020 at 16:47 | comment | added | whuber♦ | $R^2$ is not a useful statistic in this context for many reasons, not least of which is because the regression is invalid: it relies on assuming the responses $y_i$ are independent but they are (by construction) very strongly correlated. | |
| Mar 11, 2020 at 11:56 | comment | added | ReneBt | Dispersion is going to depend on the timescale you look at. The longer the interval, the bigger the possible random walks from the straight line. If you do a PCA of random walk in one dimension it decomposes into a series of eigenvectors whose features differ in freqency of features. | |
| Mar 11, 2020 at 11:13 | comment | added | elemolotiv | @mlofton thank you. I appreciate it might be inappropriate, but I thought I could use $R^2$ to measure the "average dispersion" of the random walk of $y_t$ compared the "ideal trajectory", that is a straight line from $y_0=0$ to $y_t=\mu t$. So I supposed there is a relationship between $R^2$ and $\mu$, $\sigma$. | |
| Mar 11, 2020 at 7:45 | comment | added | mlofton | Hi: There's no analytic relationship between your model for $y_{t}$ and your model for $f_{t}$ because $y_{t}$ is difference stationary and $f_t$ is trend stationary. These are two different types of non-stationarity in time series and I think that you're mistake is that you're comparing them. | |
| Mar 11, 2020 at 7:28 | history | edited | elemolotiv | CC BY-SA 4.0 |
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| Mar 11, 2020 at 7:18 | history | asked | elemolotiv | CC BY-SA 4.0 |
