I am looking for a way to characterise the frequency response of the slope from a linear regression. We are exploring the effect of window length of the regression to the magnitude of the slope of the regression and find that for our data (discrete time series data), the slope is positively related to the window length.
The slope can be thought of as the expected change of our output variable for a given input variable over a time period equal to the window length. We find that for a short window length, this slope is very low whereas for a long window length the slope is high. We interpret this to mean that the system responds to changes in the input in the long term but not in the short term. To me, that sounds a lot like a low pass filter killing off the fast changes in the input but passing the slower (or rather, more sustained) changes.
The slope, or beta, of the regression for a rolling window is, in a way, similar to a moving average and it is possible to characterise the frequency response of a moving average filter as a function of its window length as shown here. I was wondering if anyone knew of a similar way to characterise the frequency response of the beta of a rolling regression as a function of its window length in a similar manner?
Note that I have read this paper which explores using regression beta as a filter but I was unable to get an intuitive understanding of why the larger the window length is, the lower the cutoff frequency appears to be.
Here are a few descriptive charts of our data that might show why we are asking this question:
In the chart above, the thick grey line is the regression over all the data (i.e. a long window length) and has a slope of 1.79. The coloured lines (and points) show the regressions for single years of data (the betas for these lines are in the brackets in the legend of the chart). The beta for any individual year is significantly lower than the overall beta. In other words the grey line is much steeper than the other lines. This is true for any arbitrary 1 year window on the data (i.e. there is no year-long window of data that has a slope as steep as the overall data).
This next chart shows the rolling betas for different window lengths:
So the question we are trying to answer (and hoping to do so by characterising regression as a LPF) is why do the lines in this chart have higher biases (higher on average) as the window length increases (e.g. the yellow line is higher than the green line). How can we interpret this phenomenon?