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I found a peculiar feature in some data that I am analyzing and was wondering whether there was a technical term for this type of phenomenon and whether anyone has come across it before.

I am doing a ordinary least squares linear regression of one financial time series on another, both of which appear stationary (bond yields and dividend yields for a specific sector). I am not using shrinkage. If I run the regression over small samples of the data (e.g. rolling 1-year periods), the regression slope (the estimated coefficient to the independent variable) is quite low and the intercept (relatively high. However, the more the sample size increases, the higher the slope becomes and the lower the intercept (i.e. the regression line steepens). This increase in slope (and corresponding decrease in intercept) appears surprisingly monotonic for increases in sample size.

Let me illustrate what is going on more precisely. I am using weekly observations, across 14 years. If I group the data by year, and run the regression separately for each year, the highest year with the highest regression slope records a slope of 1.07. The slope for the entire data set is however much higher at 1.79. It seems to me like the relationship is much weaker over small data sets than over the entire data set, or in this case weaker in the long-term than in the short-term. I.e. the one variable influences the other more in the long-term than short-term.

A colleague of mine thinks that the problem can be framed in terms of signal processing, and has posted the following question: https://dsp.stackexchange.com/questions/25323/frequency-response-of-a-rolling-linear-regression. I was wondering whether there was a purely statistical answer, and would greatly appreciate any help. Specifically: 1) is there a technical term for this type of phenomenon, 2) is there a better way to detect it and test its significance and 3) are there any places in nature or elsewhere where this typically occurs?

Here is a sample of the data illustrating the issue: bond yields and dividend yields

The thick grey line is the regression over all data; the colored lines/points are for selected individual years.

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    $\begingroup$ "Beta" normally could mean one of three different quantities in this context: an estimated coefficient; an estimated coefficient when the two series are standardized; and an estimate of volatility (or relative volatility). Furthermore, how the estimates behave will depend on your estimation procedure. Could you clarify these points? $\endgroup$ – whuber Aug 19 '15 at 12:19
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    $\begingroup$ A couple other questions in addition to whuber's. You say "simple regression", but I'd still like to ask: are you using shrinkage (like ridge regression)? What happens to the intercept in your model as your "beta" increases? $\endgroup$ – Matthew Drury Aug 19 '15 at 14:37
  • $\begingroup$ @whuber thank you for your comments, please see my edited question. Hope that clarifies it. $\endgroup$ – Rene Aug 20 '15 at 13:24
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    $\begingroup$ It would be interesting to learn more about how you have inferred that these two series are stationary. Your regression results suggest they are not, but that the lack of stationarity may be second order: if the volatility of one (or both) series has been gradually changing over time (which indeed has happened to many financial time series), that alone would explain this phenomenon. $\endgroup$ – whuber Aug 20 '15 at 14:00
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    $\begingroup$ At the risk of asking the obvious, have you tried visualising the data? It should be quite simple, seeing as it's a single variable. PS: apologies for not adding this as a comment. I don't have enough reputation points yet. $\endgroup$ – user3353185 Aug 26 '15 at 8:16
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From the plot, it's not clear how these are "stationary" time series, as the mean values of both clearly change from year to year. Both these time series seem to have significant and correlated overall trends, in this case presumably due to underlying macroeconomic phenomena that affect both dividend and bond yields.

The high regression coefficients over long time periods represent the joint responses of both variables to those underlying phenomena, which account for much of the variance in either type of yield over those periods of time. Over shorter time periods where the macroeconomic influences might be relatively constant, you are tending more to examine possible (and presumably weaker) inter-relations between the 2 variables.

Proper analysis of time series typically starts with identifying and removing these overall trends and any seasonal (or similar) components before examining the intra-series and inter-series correlations. If you are going to be analyzing econometric time series, draw on the decades of literature and take advantage of well-vetted tools like those provided in R.

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    $\begingroup$ +1, I don't know why this answer did not receive any attention. $\endgroup$ – forecaster Sep 26 '15 at 18:13

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