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Sunday, February 24, 2019

Comments On Using Regression Models On Treasury Yields

Ernie Tedeschi recently published "Deficits are Raising Interest Rates. But Other Factors are Lowering Them.", in which  he discusses the results of some regression models that attempt to model the 10-year Treasury yield based on a number of factors, including fiscal deficits or debt. His work is a well-documented return to a genre of models that are (or at least were) popular among economists. I have always had reservations about this class of models, which I will outline here.


Tedeschi's article is well-documented and straightforward, and should be easy to replicate with standard statistical software and publicly-available data. In interests of brevity, I will let the reader consult the article to see its conclusions. Tedeschi's work has replicated the previous results once extended to a longer set of data. As the title suggests, he finds that the models suggest that deficits or debt (each appears in a different model) are associated with raising Treasury yields, but other factors are lowering them.

In this article, I will focus on the discussion of the use of fiscal deficits in such models. The debt level results are less convincing, as I will only briefly touch on them in the appendix. Furthermore, there is a discussion of the relation to the term premium in affine term structure models. From my perspective, affine term structure models are only adding value analytically if there is a way to make money systematically based on their outputs, and I see little evidence of such an ability.

Deficits and Rates

The results suggest that a 1% increase in the cyclically-adjusted deficit raises the 10-year Treasury yield by 18 basis points (0.18%). Since the model already incorporates growth and inflation forecasts (from the Survey of Professional Forecasters), this effect is supposed to be interpreted as being on top of the expected stimulative effect of an increase in the fiscal deficit.

From my perspective, the 18 basis point value for the coefficient is the most annoying possible outcome. If it were something like 10 basis points, it would be economically insignificant. Or if it were 50 basis points, it would be easy to demonstrate that the result is incorrect by looking at the data. At 18 basis points, it is on the boundary of being measurable.

From a policy perspective, a sensible person would realise that the effects are insignificant. We need to move the cyclically-adjusted fiscal deficit by more than 5% in order to raise the 10-year yield by 100 basis points. Although any rise in interest rates causes anguished hand-wringing in the business press, 100 basis points is still a nothingburger in the grand scheme of things; Treasurys often have trading ranges that are at least that wide. Meanwhile, a stimulus package of 5% of GDP is massive, considering that this is a cyclically-adjusted measure. To put it in terms that the business press commonly uses, THIS IS OVER ONE TRILLION DOLLARS! And if we wanted to raise interest rates over the 10-year life of the bond, we would follow the business press and quote the total amount: OVER TEN TRILLION DOLLARS!

Unless the increase in the deficit is the result of a policy that has a multiplier close to zero -- for example, handing tax cuts to billionaires -- one would expect that the growth effects would raise the 10-year Treasury yield by way more than 100 basis points

Feedback Loops

From a theoretical perspective, we should expect that fiscal deficits should raise interest rates if we accept some conventional assumptions about interest rates and growth. If we believe that a higher term rate of interest reduces growth courtesy of the reduction in demand for term borrowing, a rise in rates should lower growth and inflation expectations. There is a theoretical debate around the effects of interest rates on the economy, but I would note that recent decades featured very large housing bubble effects (including mortgage equity withdrawal) from mortgage rates. If we go further back into the data sets, unhinged Fed rate hikes were correlated with recessions -- a correlation that can be plausibly linked to punitive rates hammering the housing market.

Furthermore, the model uses consensus economic survey data, not actual growth and inflation rates. There is no doubt that the consensus believes that higher interest rates inhibit growth and inflation.

As a result, we end up close to the neoclassical arguments about interest rates offsetting fiscal policy: if there is a fiscal stimulus, if inflation expectations remain anchored, that should have the effect of raising nominal rates while keeping inflation and growth expectations unchanged. Therefore, the regression model should exhibit the effect that changes in the fiscal stance raises nominal interest rates, even though there may be zero "supply" effects on interest rates.

For some reason, the conventional view is that anything that raises interest rates is a disastrous drag on growth. This shows the limitations of the "all else equal" thinking pushed in economics departments: the inability to solve models in their entirety results in warped thinking about outcomes. Even if the accept the premise that increasing the deficit raises interest rates, and that higher interest rates lower growth, the net result of the increased deficit is still that growth is faster. The change in interest rates is just an endogenous automatic stabiliser (under usual assumptions).

Weaknesses of Regression Models

To be very clear, Ernie Tedeschi notes that regression models have limitations; he is just reporting what happens when we apply a popular existing methodology. I want to explain why I do not spend any time developing such models. (I have seen dozens of variants of these models proposed in my time in the financial industry.)

From a high level perspective, these models are purely descriptive econometric results without any underlying model structure. The economists that developed them give them a backstory, but a backstory is not the same thing as a model. Model identification was not the area of theory I looked at in control systems, but I believe it was safe to say that picking variables in a semi-random fashion and regressing them against each other was an accepted methodology to deal with complex systems.

What is a ordinary least-squares (OLS) regression? One way to approach the task is to find a variable that best explains the target variables. (In this case, the most likely candidate is the short-term nominal rate, or the 3-month Treasury bill rate.)

The regression will find the a, b coefficients to minimise the error in the following equation:

a (short rate) + b = (10-year yield) + (error).

That is, we apply a scaling and add a constant to the input series to best approximate the target series.

We can then see how "data mining" work (as the term is used in economics, not by data scientists.) Once we calculate the error, we see that it a time series. If it is really small, our work is done. If not, we just need to find a variable that is correlated to the error. We throw it in the mix, and re-run the results. The new variable will be able to notch out a portion of the error that is correlated to itself. We can then repeat the process. All we need to do is reduce the error is to keep searching for new series that are correlated with the residual. So long as not all economic and financial time series are correlated -- which they are not -- we may be able to reduce the error.

To be clear, most of the variables chosen appear plausible, such as taking real growth and inflation expectations (conforming to neoclassical views about interest rates). So the overall exercise does not appear to be data mining. However, the inclusion of foreign official holdings and information on the senior population look somewhat bolted on. For the United States, the usual growth and inflation series regressions did a good job historically of fitting nominal rates. (One suspects that this reflects how earlier bond market participants priced bonds.) However, regression models started to blow up in the 2000s (which I had to deal with in my professional life). The models were over-predicting interest rates. However, foreign official purchases of Treasurys took off (mainly Chinese purchases). Therefore, the folklore took hold that foreign reserve purchases were suppressing interest rates. The fact that bonds had an epic bull market from 2005-2015 did nothing to shake the consensus from the view that bonds were in some form of reserve manager-driven bubble.

The reality is that 100% of bonds have to be owned by somebody. Even if we accept that there was a surge of price insensitive buying by green reserve managers more than a decade ago, market makers and relative value traders will have long adjusted to their habits.

Since there is only one run of history, and there no model dynamics to analyse to find other criteria to judge the model, all we can do is wait decades to see whether the models blow up or not -- if we confine our analysis to the United States. (Based on the historical record, it would not be surprising to see new variables inserted into the regression models to preserve the fit.) The only useful alternative is to turn to other countries -- does the methodology work there?

The methodology historically did not adapt to other countries well. One problem is that one typically had much shorter runs of data to work with. Furthermore, there were a number of large shocks in quite a few countries, like the ERM fiasco in the United Kingdom. It may be that the increasing length of data runs since when I attempted to build such models (mainly from 2000-2002) would make the models more plausible.


In case it is not obvious, I have distaste for such models. The problem is straightforward: they suggest that the 10-year bond is floating in space, detached from the rest of the yield curve. This completely ignores the fundamentals for fixed income pricing. We need to model the curve as a cohesive whole. Modelling the entire curve is obviously very complex, but perhaps we can step in that direction.

At the minimum, we should realise that the front end is anchored by the policy rate. There is literally nothing bond market participants can do to the very front end (modulo default scenarios). If we go a bit further along the curve, only soon-to-be-bankrupt traders believe that the 2-year note trades very far from the expected average for the overnight rate over the next two years. (Yes, there is a small term premium and various basis spreads, but the uncertainty in those factors is an order of magnitude smaller than rate expectations.Very simply, if your time series for market participant consensus rate expectations for the 2-year is markedly different from the observed 2-year yield, your expectations series is just plain wrong. (One typical way of achieving that feat is to use economist consensus rate forecasts.)

You are left with modelling the 2- to 10-year part of the instantaneous forward curve, which you can approximate with averages. If you do that in a reasonable fashion, you can greatly pin down a fair value estimate that is plausibly close to observed 10-year yields. You can then try to look for factors that explain as much as possible of the deviation from fair value. Perhaps fiscal deficits will work, perhaps they will not.

If we look at raw fiscal deficits, we should not be surprised if they add value. Both the fiscal deficit and the slope (which is what we are modelling) are cyclical variables (with a negative correlation with each other). That is, the raw fiscal deficit will work, but so would any other cyclical variable.

The use of a cyclically-adjusted budget balance is supposed to eliminate these shenanigans -- but that relies on trusting the mainstream economists' methodologies for cyclical adjustment. There is a very large body of heterodox economic literature critiquing the mainstream techniques for cyclical adjustments.*

Concluding Remarks

I am skeptical about the effects of fiscal variables on bond yields as a result of supply effects, however, I think this skepticism needs to be somewhat nuanced. The results presented by Ernie Tedeschi are consistent with a view along the lines of "fiscal deficits have a barely measurable effect on interest rates." Why we can view the effects as minor are as follows.
  • In order to generate a movement in interest rates that is larger than the trading ranges seen on multi-month perspectives, we need relatively large deficit changes based on the estimated coefficients. Those deficit changes should have a growth impact larger than this fiscal effect.
  • The normal reaction function of markets and central banks suggests that an increasing deficit will be associated with higher rates, even if there is no supply effect. The estimation methodology combines both the normal reaction function feedback as well as the supply effect into a single value.
  • To salvage the model fits, we need to insert variables that conveniently rise from small levels just when the previous regression models started to fail.
  • We need to be suspicious of the cyclically-adjusted budget deficit: what is it, really?
  • We should be modelling the slope.

Technical Appendix: Debt Levels and Yields

Figure: Debt-to-GDP versus yield scatter plot

I have resurrected one of my favourite charts: the bond yield versus the government debt-to-GDP ratio scatter plot. If one looks at the scales, one rapidly notes that the "model" fit suggests that a higher debt-to-GDP ratio lowers bond yields. The fit is not perfect, but at the same time, it is not bad for a single economic time series versus a financial one. (We can obviously get good fits from one interest rate to another.) 

The usual reaction of economists to this chart is akin to horror; a greater supply of bonds raises the price (price up, yield down). Since we cannot contradict our assumptions, we need to a lot of statistical work to get the coefficient to be the "correct" sign.

If we looked at regression models that use debt without the demographics/foreign official holding variables, they blow sky post-2010. We need those variables to save the models. 

If we look elsewhere, Japan really crushes any such modelling attempts. The only way to save the models is possibly the demographics dodge. Otherwise, any non-zero coefficient on the government debt ratio would suggest that rates would have otherwise been highly negative, which is an implausible outcome. 

Debt levels are accumulated flows, and a high debt-to-GDP ratio mechanically reflects previously low nominal GDP growth rates. (Sorry, R&R.) It is largely wishful thinking that we can somehow disentangle that reality from observed yields to get a supply effect estimate.


* Needless to say, these critiques are never cited in mainstream work. What did you think would happen?

(c) Brian Romanchuk 2019


  1. I agree that deficits tend to increase interest rates. But how about this for a more precise statement of the relationship (which I think is MMT compliant).

    So long as the economy is not at capacity, there is no need to pay any interest at all to those holding state liabilities (i.e. base money / goveernment debt). As soon as the economy reaches capacity, and assuming the deficit rises further, then it will become increasingly necessary to pay interest in order to persuade holders of state liabilities not to spend their money. Ergo if you have "capacity" on the horizongtal axis (with full capacity at the right hand side) and interest on the vertical axis, there should be a horizontal line which then slopes upward at the "full capacity" point.

    It would be nice to tease that relationship out of the welter of background noise and show that it exists, but maybe that's impossible.

    1. (This is Brian.)
      The analysis used here can only tease out historical relationships, and even then, it leaves open a lot of questions. No real way to handle what you are asking.

  2. holders of state liabilities can only spend the "money" if they get someone else to buy it, and they become the holder?

    1. (This is Brian. Blogger is not letting me log on in my iPad. Sigh.)
      Government-issued money is a government liability. So you are looking for anyone willing to sell things for money.

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