Lagged Interest Rates and Plotting Changes Against Past Values
Summary
The document explains the meaning of a lagged interest rate in a time-series plot. A lag is a past observation aligned with a later date: with a one-period lag, the rate from the previous observation is paired with the current observation. For example, if annual data are used, the prior year's rate is the lagged value for the current year. A two-period lag would use the rate from two observations earlier. The interval therefore depends on the frequency of the series, not on a fixed calendar unit.
In the referenced statistical example, the plot places the change in the rate against its lagged level to examine whether changes become more dispersed when the prior rate is larger, a pattern associated with heteroskedasticity. The answer describes forming a lag by shifting the series so each current date is matched to an earlier rate; the first observation has no prior value, and the final observation may have no subsequent change depending on how the change series is defined. The discussion offers a basic interpretation, but no plot reconstruction, statistical test, or evidence that the relationship is present in other interest rate data.
Key ideas
- A lagged rate is a past observation aligned with the current date.
- A one-period lag refers to the previous observation, so its calendar span depends on the data frequency.
- A plot of rate changes against lagged rates can be used to inspect whether variability changes with the prior rate level.
- Constructing a lag requires shifting the series and aligning observations by date.
- The explanation defines the axes conceptually but does not validate the pattern statistically.
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# What is lagged interest rate? # What is lagged interest rate? I am trying to reproduce a plot in "Statistics and Data engineering for Financial Engineering" by D. Ruppert. The author uses the risk free returns data available in the Ecdat package in R. Specifically the rf variable in the Capm data set of this package. He wants to demonstrate that changes in the rate are more variable for large values of the rate (heteroskedacity). So in figure 4.19 (a) he plots the change in the rate vs the lagged rate, and indeed for larger values the points in the scatter plot are more dispersed. I want to reproduce this plot. But I have no idea what the "lagged rate" here means. Any ideas? I have attached the plot below. I would like to reconstruct figure 4.19 (a), but I'm not exactly sure what to plot on the x-axis. ## Answer by jessica (score 3, accepted) https://quant.stackexchange.com/a/8093 Lagged means past values. The lag can be by as long as you want. If Interest Rates today are 0% and yesterday they were 0.25%. Yesterdays value is what we call the lagged value. Let's say its now 2012 and we are looking at IR in yearly frequency. IR is 0.1%. To lag IR we simply look back at the last value. So what was IR last year? It was 0.3%. Notice how 0.3% is in IR(-1) for 2012 (which is the present). If the concept isn't clear just reread this more carefully. When your lagging a time series all your doing is making todays value, yesterday. How do you that, you shift the array down. Notice you can make the lag whaterver value you feel. IR(-2) means, your looking at IR 2 years ago. In the graph above, the author is simply trying to analyze the effect of yesterday's (past values of IR) on today's IR value. How much does the past values influence today's value? That's what the graphs are trying to delineate. (Can someone help arrange the data matrix I wrote below, when its displayed it not formatted correctly. Thank you!) Date IR IR(-1)<--Lagged Interest Rate 2008 1.5% -- 2009 1.1% 1.5% 2010 0.9% 1.1% 2011 0.3% 0.9% 2012 0.1% 0.3% 2013 -- 0.1%
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