# The Nobel Memorial Prize for Robert F. Engle

## Summary (2 min read)

### 1. Introduction

- And the authors use neither the no-arbitrage approach nor the equilibrium approach.
- Instead, the authors use the Nelson-Siegel (1987) exponential components framework to distill the entire yield curve, period-by-period, into a threedimensional parameter that evolves dynamically.

### 2. Modeling and Forecasting the Term Structure I: Methods

- Here the authors introduce the framework that they use for fitting and forecasting the yield curve.
- The authors argue that the well-known Nelson-Siegel (1987) curve is well-suited to their ultimate forecasting purposes, and they introduce a novel twist of interpretation, showing that the three coefficients in the Nelson-Siegel curve may be interpreted as latent level, slope and curvature factors.
- The authors also argue that the nature of the factors and factor loadings implicit in the Nelson-Siegel model facilitate consistency with various empirical properties of the yield curve that have been cataloged over the years.
- Finally, motivated by their interpretation of the Nelson-Siegel model as a three-factor model of level, slope and curvature, the authors contrast it to various multi-factor models that have appeared in the literature.

### Constructing "Raw" Yields

- Let us first fix ideas and establish notation by introducing three key theoretical constructs and the relationships among them: the discount curve, the forward curve, and the yield curve.
- A second discount-curve approach to yield construction is due to Vasicek and Fong (1982) , who fit exponential splines to the discount curve, using a negative transformation of maturity instead of maturity itself, which ensures that the forward rates and zero-coupon yields converge to a fixed limit as maturity increases.
- In their approach, the unit loading on the first factor is imposed from the beginning, which potentially enables us to estimate the other factors more efficiently.
- The long-term factor , for example, governs the yield curve level.
- In closing this sub-section, it is worth noting that what the authors have called the "Nelson-Siegel curve" is actually a different factorization than the one originally advocated by Nelson and Siegel (1987) , who used .

### The Data

- The authors use end-of-month price quotes (bid-ask average) for U.S. Treasuries, from January 1985 through December 2000, taken from the CRSP government bonds files.
- Fama-Bliss forward rates into unsmoothed Fama-Bliss zero yields.
- The variation in slope and curvature is less strong, but nevertheless apparent.

### Fitting Yield Curves

- As discussed above, the authors fit the yield curve using the three-factor model, 9 Other weightings and loss functions have been explored by Bliss (1997b) , Soderlind and Svensson (1997), and Bates (1999) .
- In Figure 5 the authors dig deeper by plotting the raw yield curve and the threefactor fitted yield curve for some selected dates.
- As noted in Bliss (1997b) , regardless of the term structure estimation method used, there is a persistent discrepancy between actual bond prices and prices estimated from term structure models.
- The authors include the VAR forecasts for completeness, although one might expect them to be inferior to the AR forecasts for at least two reasons.
- In Figure 8 (right column) the authors provide some evidence on the goodness of fit of the AR(1) models fit to the estimated level, slope and curvature factors, showing residual autocorrelation functions.

### Out-of-Sample Forecasting Performance of the Three-Factor Model

- A good approximation to yield-curve dynamics should not only fit well in-sample, but also forecast well out-of-sample.
- The authors model's 6-monthahead forecasting results, reported in Table 5 , are noticeably improved, and their model's 12-month-ahead forecasting results, reported in Table 6 , are much improved.
- The strong yield curve forecastability at the 12-month-ahead horizon is, for example, very attractive from the vantage point of active bond trading and the vantage point of credit portfolio risk management.
- A 17 We note, however, that their enthusiasm must be tempered by the fact that their in-sample and out-of-sample periods are not identical to Duffee's, so definitive comparisons can not be made.the authors.the authors.
- 17 Finally, the authors note that although their approach is closely related to direct principal components regression, neither their approach nor their results are identical.

### Notes to Table 4:

- The authors present the results of out-of-sample 1-month-ahead forecasting using eight models, as described in detail in the text.
- The authors estimate all models recursively from 1985:1 to the time that the forecast is made, beginning in 1994:1 and extending through 2000:12.

### Notes to Table 5:

- The authors present the results of out-of-sample 6-month-ahead forecasting using eight models, as described in detail in the text.
- The authors estimate all models recursively from 1985:1 to the time that the forecast is made, beginning in 1994:1 and extending through 2000:12.

### Notes to Table 6:

- The authors present the results of out-of-sample 12-month-ahead forecasting using twelve models, as described in detail in the text.
- The authors estimate all models recursively from 1985:1 to the time that the forecast is made, beginning in 1994:1 and extending through 2000:12.

### Notes to Table 7:

- The authors present Diebold-Mariano forecast accuracy comparison tests of their three-factor model forecasts (using univariate AR(1) factor dynamics) against those of the Random Walk model (RW) and the Fama-Bliss forward rate regression model (FB).
- The null hypothesis is that the two forecasts have the same mean squared error.
- - Notes to Figure 7 : We define the level as the 10-year yield, the slope as the difference between the 10-year and 3-month yields, and the curvature as the twice the 2-year yield minus the sum of the 3-month and 10year yields.the authors.the authors.

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### Cites background or methods from "The Nobel Memorial Prize for Robert..."

...F-ARCH (Factor ARCH) The multivariate factor ARCH model developed by Diebold and Nerlove (1989) (see also Latent GARCH) and the factor GARCH model of Engle, Ng and Rothschild (1990) assumes that the temporal variation in the N×N conditional covariance matrix for a set of N returns can be described by univariate GARCH models for smaller set of K<N portfolios,...

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...A leading example is the N-dimensional factor ARCH model of Diebold and Nerlove (1989), , where and denote N×1 vectors of factor loadings and i....

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...…Laurent and Rombouts (2006), Bera and Higgins (1993), Bollerslev, Chou and Kroner (1992), Bollerslev, Engle and Nelson (1994), Degiannakis and Xekalaki (2004), Diebold (2004), Diebold and Lopez (1995), Engle (2001, 2004), Engle and Patton (2001), Pagan (1996), Palm (1996), and Shephard (1996)....

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...A leading example is the N-dimensional factor ARCH model of Diebold and Nerlove (1989), , where and denote N×1 vectors of factor loadings and i.i.d. innovations respectively, and the conditional variance of is determined by an ARCH model in lagged squared values of the latent factor (see also F-ARCH). Models in which the innovations are subject to censoring is another example (see Tobit-GARCH). In contrast to standard ARCH and GARCH models, for which the likelihood functions are readily available through a prediction error decomposition type argument (see ARCH), the likelihood functions for latent GARCH models are generally not available in closed form. General estimation and inference procedures for latent GARCH models based on Markov Chain Monte Carlo methods have been developed by Fiorentini, Sentana and Shephard (2004) (see also SV)....

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...Progress continues to be made, as for example in Engle (2002a, 2002b)....

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...9 For interesting commentary on the furious pace of development, both retrospective and prospective, see Engle (2002b)....

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