Fixed-Income Comments
A calendar month ago, I attended the SF Fed/Bank of Canada conference on fixed income. I had the direct chances to comment on Michael Bauer as well as Jim Hamilton's "Robust Bond Risk Premia.” My comments here.
As green when faced amongst a actually squeamish paper, I used most of my give-and-take fourth dimension to survey the champaign as well as plough over my views on electrical flow facts as well as challenges, which is why my comments mightiness move interesting to weblog readers.
Some highlights: I reran regressions of bond returns inwards the style of Joslin, Priebsch, as well as Singleton, forecasting returns amongst the kickoff three main components of yields, as well as increase as well as inflation. Here are the results:
First row: the gradient gene forecasts returns amongst the green 18% R2. Second row: Inflation as well as increase create non forecast returns at all. Third row: in combination with the kickoff 3 main components, the R2 rises to 0.26 past times adding increase as well as inflation. Inflation straightaway becomes a pregnant predictor, as well as its presence raises the coefficient as well as t statistic on the marker as well as gradient factors. This is an interesting OLS puzzle.
If yous plot inflation, yous run into it is mostly a downward tendency inwards this sample period. So, it occurred to me, what if I used a tendency instead? The final 2 rows of the tabular array add together a trend. Indeed, amongst the trend, increase as well as inflation disappear. In fact, nosotros tin drib growth, inflation, as well as the 3rd main component, forecast returns amongst amazing t statistics as well as an R2 of 0.62, which must move an all fourth dimension high.
What's going on here? Is the tendency simply picking upward a tendency inwards returns? Here is a plot of expected returns (a + b x_t) as well as actual returns (r_t+1) for 4 of the models inwards Table 1.
The point: the tendency is non simply picking upward a tendency inwards returns. And the 62% R2 is non a pathology of i large outlier, a trend, or something else. Instead, the tendency serves to filter the marker factor, as well as to a lesser extent the gradient factor. The message is non "a tendency seems to forecast a tendency inwards returns" but "the cyclical variations picked upward past times detrended marker as well as gradient factors seem to forecast returns."
So what does this all mean? Is this proof increase as well as inflation don't move because they are driven out past times trends? No, the tendency is after all a proxy for something economic. (This is roughly Cieslak as well as Povala's point, who larn over 50% R2 inwards a longer sample amongst smoothed inflation.) Is this all a large econometric goof, because serially correlated correct mitt variables are a mistake? No, as well as my comments cash inwards one's chips into this at length. Bauer as well as Hamilton's indicate is this econometric problem, but they don't larn around t statistics of 10. OLS cares nigh series correlation of the residuals, but non of the correct mitt variables. In the end, it's a interpretation issue, non an econometric one.
The biggest indicate of my comments: It's fourth dimension to larn past times forecasting returns i at a time. Classic finance got past times "is AT&T a practiced investment?" inwards the 1960s, after all, as well as moved on to portfolios as well as covariances. Here, the to a greater extent than interesting outstanding enquiry is the factor construction of expected returns -- create expected returns on all bonds movement together over time? -- as well as the hazard premium enquiry -- what are the factors, covariance amongst which drives that variation inwards expected returns?
To this question, peradventure nosotros should convey a lesson from the VAR literature of the 1980s, as well as halt worrying tremendously nigh equation past times equation parsimony inwards forecasting. Instead, convey that forecasting regressions volition move a somewhat overfit, but set our attending inwards the cross-equation construction of forecasts.
To move specific, the adjacent graph shows the expected returns of bonds amongst maturity 1-10 years -- the fitted value of each bond's return-forecasting regression. The graph is clear: these are non 10 dissimilar series. The expected returns on all bonds movement inwards lockstep. There is a rigid one-factor construction inwards expected returns.
Finance 101: Expected furnish = covariance of furnish amongst something, times hazard premium. What's that something? In this context, the bonds whose expected furnish moves most over fourth dimension should receive got returns that covary proportionally to a greater extent than amongst some factor. What is it? The adjacent moving painting plots how much each bond moves amongst the mutual gene shown inwards Figure eleven against the covariance of the 10 bond returns amongst innovations inwards the bond main components, growth, as well as inflation.
Again, the designing is pretty clear: time-varying expected furnish corresponds completely amongst covariances amongst the marker factor. Covariances amongst the other factors are all nigh zero, as well as create non vary inwards the same agency every bit expected returns.
In sum, this uncomplicated exploration shows a pretty rigid pattern: 1) There is a rigid one-factor model of expected returns -- expected returns on bonds of all maturity movement together over time. 2) There is a rigid one-factor model of risk: the unmarried time-varying hazard premium inwards all bonds corresponds to covariance amongst a unmarried factor, innovations to the marker of involvement rates.
This is all really simplified of course. The point: This form of characterization of the articulation behaviour of bonds of diverse maturities -- as well as afterward of bonds, stocks, as well as unusual telephone substitution -- seems similar a to a greater extent than interesting unanswered enquiry than the precise identity of forecasting variables for each security, taken inwards isolation.
These points are a chip of a rehash of older papers, Decomposing the yield curve as well as to a greater extent than to a greater extent frequently than non Discount Rates. But they are besides an extension --- the "Decomposing the yield curve" indicate holds using the JPS forecasters as well as factors, as well as updated data. This form of research needs a lot to a greater extent than work.
As green when faced amongst a actually squeamish paper, I used most of my give-and-take fourth dimension to survey the champaign as well as plough over my views on electrical flow facts as well as challenges, which is why my comments mightiness move interesting to weblog readers.
Some highlights: I reran regressions of bond returns inwards the style of Joslin, Priebsch, as well as Singleton, forecasting returns amongst the kickoff three main components of yields, as well as increase as well as inflation. Here are the results:
First row: the gradient gene forecasts returns amongst the green 18% R2. Second row: Inflation as well as increase create non forecast returns at all. Third row: in combination with the kickoff 3 main components, the R2 rises to 0.26 past times adding increase as well as inflation. Inflation straightaway becomes a pregnant predictor, as well as its presence raises the coefficient as well as t statistic on the marker as well as gradient factors. This is an interesting OLS puzzle.
If yous plot inflation, yous run into it is mostly a downward tendency inwards this sample period. So, it occurred to me, what if I used a tendency instead? The final 2 rows of the tabular array add together a trend. Indeed, amongst the trend, increase as well as inflation disappear. In fact, nosotros tin drib growth, inflation, as well as the 3rd main component, forecast returns amongst amazing t statistics as well as an R2 of 0.62, which must move an all fourth dimension high.
What's going on here? Is the tendency simply picking upward a tendency inwards returns? Here is a plot of expected returns (a + b x_t) as well as actual returns (r_t+1) for 4 of the models inwards Table 1.
The point: the tendency is non simply picking upward a tendency inwards returns. And the 62% R2 is non a pathology of i large outlier, a trend, or something else. Instead, the tendency serves to filter the marker factor, as well as to a lesser extent the gradient factor. The message is non "a tendency seems to forecast a tendency inwards returns" but "the cyclical variations picked upward past times detrended marker as well as gradient factors seem to forecast returns."
So what does this all mean? Is this proof increase as well as inflation don't move because they are driven out past times trends? No, the tendency is after all a proxy for something economic. (This is roughly Cieslak as well as Povala's point, who larn over 50% R2 inwards a longer sample amongst smoothed inflation.) Is this all a large econometric goof, because serially correlated correct mitt variables are a mistake? No, as well as my comments cash inwards one's chips into this at length. Bauer as well as Hamilton's indicate is this econometric problem, but they don't larn around t statistics of 10. OLS cares nigh series correlation of the residuals, but non of the correct mitt variables. In the end, it's a interpretation issue, non an econometric one.
The biggest indicate of my comments: It's fourth dimension to larn past times forecasting returns i at a time. Classic finance got past times "is AT&T a practiced investment?" inwards the 1960s, after all, as well as moved on to portfolios as well as covariances. Here, the to a greater extent than interesting outstanding enquiry is the factor construction of expected returns -- create expected returns on all bonds movement together over time? -- as well as the hazard premium enquiry -- what are the factors, covariance amongst which drives that variation inwards expected returns?
To this question, peradventure nosotros should convey a lesson from the VAR literature of the 1980s, as well as halt worrying tremendously nigh equation past times equation parsimony inwards forecasting. Instead, convey that forecasting regressions volition move a somewhat overfit, but set our attending inwards the cross-equation construction of forecasts.
To move specific, the adjacent graph shows the expected returns of bonds amongst maturity 1-10 years -- the fitted value of each bond's return-forecasting regression. The graph is clear: these are non 10 dissimilar series. The expected returns on all bonds movement inwards lockstep. There is a rigid one-factor construction inwards expected returns.
Finance 101: Expected furnish = covariance of furnish amongst something, times hazard premium. What's that something? In this context, the bonds whose expected furnish moves most over fourth dimension should receive got returns that covary proportionally to a greater extent than amongst some factor. What is it? The adjacent moving painting plots how much each bond moves amongst the mutual gene shown inwards Figure eleven against the covariance of the 10 bond returns amongst innovations inwards the bond main components, growth, as well as inflation.
Again, the designing is pretty clear: time-varying expected furnish corresponds completely amongst covariances amongst the marker factor. Covariances amongst the other factors are all nigh zero, as well as create non vary inwards the same agency every bit expected returns.
In sum, this uncomplicated exploration shows a pretty rigid pattern: 1) There is a rigid one-factor model of expected returns -- expected returns on bonds of all maturity movement together over time. 2) There is a rigid one-factor model of risk: the unmarried time-varying hazard premium inwards all bonds corresponds to covariance amongst a unmarried factor, innovations to the marker of involvement rates.
This is all really simplified of course. The point: This form of characterization of the articulation behaviour of bonds of diverse maturities -- as well as afterward of bonds, stocks, as well as unusual telephone substitution -- seems similar a to a greater extent than interesting unanswered enquiry than the precise identity of forecasting variables for each security, taken inwards isolation.
These points are a chip of a rehash of older papers, Decomposing the yield curve as well as to a greater extent than to a greater extent frequently than non Discount Rates. But they are besides an extension --- the "Decomposing the yield curve" indicate holds using the JPS forecasters as well as factors, as well as updated data. This form of research needs a lot to a greater extent than work.
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