## Monday, November 30, 2015

A month ago,  I attended the SF Fed/Bank of Canada conference on fixed income. I had the chance to comment on Michael Bauer and Jim Hamilton's "Robust Bond Risk Premia.My comments here.

As usual when faced with a really nice paper, I used most of my discussion time to survey the field and give my views on current facts and challenges, which is why my comments might be interesting to blog readers.

Some highlights: I reran regressions of bond returns in the style of Joslin, Priebsch, and Singleton, forecasting returns with the first  three principal components of yields, and growth and inflation. Here are the results:

First row: the slope factor forecasts returns with the usual 18% R2. Second row: Inflation and growth do not forecast returns at all. Third row: in combination with the first three principal components, the R2 rises to 0.26 by adding growth and inflation. Inflation now becomes a significant predictor, and its presence raises the coefficient and t statistic on the level and slope factors. This is an interesting OLS puzzle.

If you plot inflation, you see it is mostly a downward trend in this sample period. So, it occurred to me, what if I used a trend instead? The last two rows of the table add a trend. Indeed, with the trend, growth and inflation disappear. In fact, we can drop growth, inflation, and the third principal component, forecast returns with amazing t statistics and an R2 of 0.62, which must be an all time high.

What's going on here? Is the trend just picking up a trend in returns? Here is a plot of expected returns (a + b x_t) and actual returns (r_t+1) for four of the models in Table 1.

The point: the trend is not just picking up a trend in returns. And the 62% R2 is not a pathology of one big outlier, a trend, or something else.  Instead, the trend serves to filter the level factor, and to a lesser extent the slope factor. The message is not "a trend seems to forecast a trend in returns" but "the cyclical variations picked up by detrended level and slope factors seem to forecast returns."

So what does this all mean? Is this proof growth and inflation don't work because they are driven out by trends? No, the trend is after all a proxy for something economic.  (This is roughly Cieslak and Povala's point, who get over 50% R2 in a longer sample with smoothed inflation.) Is this all a big econometric goof, because serially correlated right hand variables are a mistake? No, and my comments go into this at length. Bauer and Hamilton's point is this econometric problem, but they don't get close to t statistics of 10. OLS cares about serial correlation of the residuals, but not of the right hand variables.  In the end, it's a interpretation issue, not an econometric one.

The biggest point of my comments: It's time to get past forecasting returns one at a time. Classic finance got past "is AT&T a good investment?" in the 1960s, after all, and moved on to portfolios and covariances. Here, the more interesting outstanding question is the factor structure of expected returns  -- do expected returns on all bonds move together over time? -- and the risk premium question -- what are the factors, covariance with which drives that variation in expected returns?

To this question, perhaps we should take a lesson from the VAR literature of the 1980s, and stop worrying tremendously about equation by equation parsimony in forecasting. Instead, accept that forecasting regressions will be a somewhat overfit, but put our attention in the cross-equation structure of forecasts.

To be specific, the next graph shows the expected returns of bonds with maturity 1-10 years -- the fitted value of each bond's return-forecasting regression. The graph is clear: these are not 10 different series. The expected returns on all bonds move in lockstep. There is a strong one-factor structure in expected returns.

Finance 101: Expected return = covariance of return with something, times risk premium. What's that something? In this context, the bonds whose expected return moves most over time should have returns that covary proportionally more with some factor. What is it? The next picture plots how much each bond moves with the common factor shown in Figure 11 against the covariance of the 10 bond returns with innovations in the bond principal components, growth, and inflation.

Again, the pattern is pretty clear: time-varying expected return corresponds completely with covariances with the level factor. Covariances with the other factors are all about zero, and do not vary in the same way as expected returns.

In sum, this simple exploration shows a pretty strong pattern: 1) There is a strong one-factor model of expected returns -- expected returns on bonds of all maturity move together over time. 2) There is a strong one-factor model of risk: the single time-varying risk premium in all bonds corresponds to covariance with a single factor, innovations to the level of interest rates.

This is all very simplified of course. The point: This kind of characterization of the joint behavior of bonds of various maturities -- and later of bonds, stocks, and foreign exchange -- seems like a more interesting unanswered question than the precise identity of forecasting variables for each security, taken in isolation.

These points are a bit of a rehash of older papers, Decomposing the yield curve and more generally  Discount Rates. But they are also an extension --- the "Decomposing the yield curve" point holds using the JPS forecasters and factors, and updated data.  This kind of inquiry needs a lot more work.

1. This would explain why the information equilibrium model does fairly well -- it describes that trend as log r = a log NGDP/MB + b.

http://informationtransfereconomics.blogspot.com/2015/11/principal-component-information.html

It also captures the trend of rising rates from the 1960s to the 1980s (since NGDP/MB rises until 1980, then falls, and finally falls precipitously in 2009):

http://informationtransfereconomics.blogspot.com/2015/08/comparison-of-interest-rate-predictions.html

2. John, my results for average excess returns on 2 - 5 year Treasuries regressed on yield curve and macroeconomic factors inflation and real activity (~Ang Piazzesi 2003) from 1969 through 2008. Slope coefficient only agreement: Panel A Restricted regressions

constant curvature slope level R2
-0.025 0.102 2.595 5.711 0.210
(0.018) (0.088) (0.699) (2.185)

Constant Inflation Real
1.10 -1.72 1.06 0.48
(0.37) (0.27) (0.23)

Panel B Unrestricted regressions
constant Curvature Slope Level Inflation Real
Coef 0.15 8.91 2.05 0.01 -1.24 1.42
Std error (1.15) (1.90) (0.38) (0.06) (0.24) (0.19)

3. I wanted to see how yield curve and macroeconomic factors behaved doing recessions: plots of excess returns and macro factors, excess returns and yield curve factors, and yield curve and macro factors https://nancyahammond.wordpress.com/2015/12/07/104/

4. That's pretty neat to see my name on a paper on your blog, even if it's just as an RA :)

Comments are welcome. Keep it short, polite, and on topic.

Thanks to a few abusers I am now moderating comments. I welcome thoughtful disagreement. I will block comments with insulting or abusive language. I'm also blocking totally inane comments. Try to make some sense. I am much more likely to allow critical comments if you have the honesty and courage to use your real name.