A calendar month ago, I attended the SF Fed/Bank of Canada conference on fixed income. I had the jeopardy to comment on Michael Bauer together with Jim Hamilton's "Robust Bond Risk Premia.” My comments here.
As park when faced amongst a actually squeamish paper, I used most of my give-and-take fourth dimension to survey the plain together with plough over my views on electrical flow facts together with challenges, which is why my comments mightiness hold upwards interesting to spider web log readers.
Some highlights: I reran regressions of bond returns inwards the style of Joslin, Priebsch, together with Singleton, forecasting returns amongst the starting fourth dimension three nous components of yields, together with increment together with inflation. Here are the results:
First row: the gradient gene forecasts returns amongst the park 18% R2. Second row: Inflation together with increment create non forecast returns at all. Third row: in combination with the starting fourth dimension 3 nous components, the R2 rises to 0.26 past times adding increment together with inflation. Inflation at 1 time becomes a pregnant predictor, together with its presence raises the coefficient together with t statistic on the score together with gradient factors. This is an interesting OLS puzzle.
If yous plot inflation, yous encounter it is mostly a downward tendency inwards this sample period. So, it occurred to me, what if I used a tendency instead? The terminal 2 rows of the tabular array add together a trend. Indeed, amongst the trend, increment together with inflation disappear. In fact, nosotros tin drib growth, inflation, together with the tertiary nous component, forecast returns amongst amazing t statistics together with an R2 of 0.62, which must hold upwards an all fourth dimension high.
What's going on here? Is the tendency merely picking upwards a tendency inwards returns? Here is a plot of expected returns (a + b x_t) together with actual returns (r_t+1) for 4 of the models inwards Table 1.
The point: the tendency is non merely picking upwards a tendency inwards returns. And the 62% R2 is non a pathology of 1 large outlier, a trend, or something else. Instead, the tendency serves to filter the score factor, together with 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 upwards past times detrended score together with gradient factors seem to forecast returns."
So what does this all mean? Is this proof increment together with inflation don't operate because they are driven out past times trends? No, the tendency is after all a proxy for something economic. (This is roughly Cieslak together with 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, together with my comments popular off into this at length. Bauer together with Hamilton's betoken is this econometric problem, but they don't larn about t statistics of 10. OLS cares close 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 betoken of my comments: It's fourth dimension to larn past times forecasting returns 1 at a time. Classic finance got past times "is AT&T a practiced investment?" inwards the 1960s, after all, together with moved on to portfolios together with covariances. Here, the to a greater extent than interesting outstanding inquiry is the factor construction of expected returns -- create expected returns on all bonds motion together over time? -- together with the jeopardy premium inquiry -- what are the factors, covariance amongst which drives that variation inwards expected returns?
To this question, mayhap nosotros should bring a lesson from the VAR literature of the 1980s, together with halt worrying tremendously close equation past times equation parsimony inwards forecasting. Instead, conduct keep that forecasting regressions volition hold upwards a somewhat overfit, but seat our attending inwards the cross-equation construction of forecasts.
To hold upwards specific, the side past times side 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 unlike series. The expected returns on all bonds motion inwards lockstep. There is a rigid one-factor construction inwards expected returns.
Finance 101: Expected provide = covariance of provide amongst something, times jeopardy premium. What's that something? In this context, the bonds whose expected provide moves most over fourth dimension should conduct keep returns that covary proportionally to a greater extent than amongst some factor. What is it? The side past times side flick plots how much each bond moves amongst the mutual gene shown inwards Figure xi against the covariance of the 10 bond returns amongst innovations inwards the bond nous components, growth, together with inflation.
Again, the designing is pretty clear: time-varying expected provide corresponds completely amongst covariances amongst the score factor. Covariances amongst the other factors are all close zero, together with create non vary inwards the same means equally expected returns.
In sum, this elementary exploration shows a pretty rigid pattern: 1) There is a rigid one-factor model of expected returns -- expected returns on bonds of all maturity motion together over time. 2) There is a rigid one-factor model of risk: the unmarried time-varying jeopardy premium inwards all bonds corresponds to covariance amongst a unmarried factor, innovations to the score of involvement rates.
This is all really simplified of course. The point: This sort of characterization of the articulation conduct of bonds of diverse maturities -- together with afterwards of bonds, stocks, together with unusual central -- seems similar a to a greater extent than interesting unanswered inquiry 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 together with to a greater extent than mostly Discount Rates. But they are too an extension --- the "Decomposing the yield curve" betoken holds using the JPS forecasters together with factors, together with updated data. This sort of research needs a lot to a greater extent than work.
As park when faced amongst a actually squeamish paper, I used most of my give-and-take fourth dimension to survey the plain together with plough over my views on electrical flow facts together with challenges, which is why my comments mightiness hold upwards interesting to spider web log readers.
Some highlights: I reran regressions of bond returns inwards the style of Joslin, Priebsch, together with Singleton, forecasting returns amongst the starting fourth dimension three nous components of yields, together with increment together with inflation. Here are the results:
First row: the gradient gene forecasts returns amongst the park 18% R2. Second row: Inflation together with increment create non forecast returns at all. Third row: in combination with the starting fourth dimension 3 nous components, the R2 rises to 0.26 past times adding increment together with inflation. Inflation at 1 time becomes a pregnant predictor, together with its presence raises the coefficient together with t statistic on the score together with gradient factors. This is an interesting OLS puzzle.
If yous plot inflation, yous encounter it is mostly a downward tendency inwards this sample period. So, it occurred to me, what if I used a tendency instead? The terminal 2 rows of the tabular array add together a trend. Indeed, amongst the trend, increment together with inflation disappear. In fact, nosotros tin drib growth, inflation, together with the tertiary nous component, forecast returns amongst amazing t statistics together with an R2 of 0.62, which must hold upwards an all fourth dimension high.
What's going on here? Is the tendency merely picking upwards a tendency inwards returns? Here is a plot of expected returns (a + b x_t) together with actual returns (r_t+1) for 4 of the models inwards Table 1.
The point: the tendency is non merely picking upwards a tendency inwards returns. And the 62% R2 is non a pathology of 1 large outlier, a trend, or something else. Instead, the tendency serves to filter the score factor, together with 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 upwards past times detrended score together with gradient factors seem to forecast returns."
So what does this all mean? Is this proof increment together with inflation don't operate because they are driven out past times trends? No, the tendency is after all a proxy for something economic. (This is roughly Cieslak together with 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, together with my comments popular off into this at length. Bauer together with Hamilton's betoken is this econometric problem, but they don't larn about t statistics of 10. OLS cares close 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 betoken of my comments: It's fourth dimension to larn past times forecasting returns 1 at a time. Classic finance got past times "is AT&T a practiced investment?" inwards the 1960s, after all, together with moved on to portfolios together with covariances. Here, the to a greater extent than interesting outstanding inquiry is the factor construction of expected returns -- create expected returns on all bonds motion together over time? -- together with the jeopardy premium inquiry -- what are the factors, covariance amongst which drives that variation inwards expected returns?
To this question, mayhap nosotros should bring a lesson from the VAR literature of the 1980s, together with halt worrying tremendously close equation past times equation parsimony inwards forecasting. Instead, conduct keep that forecasting regressions volition hold upwards a somewhat overfit, but seat our attending inwards the cross-equation construction of forecasts.
To hold upwards specific, the side past times side 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 unlike series. The expected returns on all bonds motion inwards lockstep. There is a rigid one-factor construction inwards expected returns.
Finance 101: Expected provide = covariance of provide amongst something, times jeopardy premium. What's that something? In this context, the bonds whose expected provide moves most over fourth dimension should conduct keep returns that covary proportionally to a greater extent than amongst some factor. What is it? The side past times side flick plots how much each bond moves amongst the mutual gene shown inwards Figure xi against the covariance of the 10 bond returns amongst innovations inwards the bond nous components, growth, together with inflation.
Again, the designing is pretty clear: time-varying expected provide corresponds completely amongst covariances amongst the score factor. Covariances amongst the other factors are all close zero, together with create non vary inwards the same means equally expected returns.
In sum, this elementary exploration shows a pretty rigid pattern: 1) There is a rigid one-factor model of expected returns -- expected returns on bonds of all maturity motion together over time. 2) There is a rigid one-factor model of risk: the unmarried time-varying jeopardy premium inwards all bonds corresponds to covariance amongst a unmarried factor, innovations to the score of involvement rates.
This is all really simplified of course. The point: This sort of characterization of the articulation conduct of bonds of diverse maturities -- together with afterwards of bonds, stocks, together with unusual central -- seems similar a to a greater extent than interesting unanswered inquiry 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 together with to a greater extent than mostly Discount Rates. But they are too an extension --- the "Decomposing the yield curve" betoken holds using the JPS forecasters together with factors, together with updated data. This sort of research needs a lot to a greater extent than work.
