Common mistake 1 with Yield curve (Nelson–Siegel / Svensson)

Reading the level factor as "the interest rate". It is the asymptote of the fitted curve, which no traded bond is quoted at.

Common mistake 2 with Yield curve (Nelson–Siegel / Svensson)

Forecasting far past the data. Three AR(1) factors revert to their means, so at long horizons this predicts the average curve of the sample period - which is a statement about your sample, not about the future.

Common mistake 3 with Yield curve (Nelson–Siegel / Svensson)

Fitting it to a curve that is not a curve. Corporate or emerging-market yields with liquidity gaps at some maturities will still produce three factors and a respectable-looking picture.

Common mistake 4 with Yield curve (Nelson–Siegel / Svensson)

Trusting the band too far. The curve’s interval here maps the factor uncertainties through the loadings treating the factors as independent, which understates it where they move together.

EconStatsCraft › Yield curve (Nelson–Siegel / Svensson)

Yield curve (Nelson–Siegel / Svensson)

A yield curve is a shape, not a series. Three common factors - level, slope and curvature - describe almost all of it, so forecasting the curve means forecasting three numbers instead of thirteen, and the forecast is a curve rather than lines that might cross.

What it needs

Yields by maturity - One column per maturity. Names like m3, m120 or “10 Yr” are read automatically.

The intuition

A yield curve is a dozen or more maturities that move almost in lockstep, so describing it with a dozen numbers is wasteful. Nelson-Siegel replaces them with three that have direct readings: a level that sets where the whole curve sits, a slope that bends the short end away from the long, and a curvature that lifts or dips the middle. Almost every curve ever observed is well described by those three.

When to use it

Any term-structure question: where rates are going across maturities, whether an inversion is expected to persist, what a bond portfolio’s duration is exposed to. Also the standard benchmark a macro-augmented yield model has to beat, and a hard one to beat.

Where the estimator comes from

The Nelson-Siegel form for the yield at maturity \tau is y(\tau)=\beta_0+\beta_1\frac{1-e^{-\lambda\tau}}{\lambda\tau}+\beta_2\left(\frac{1-e^{-\lambda\tau}}{\lambda\tau}-e^{-\lambda\tau}\right). The first loading is 1 at every maturity, so \beta_0 is the LEVEL. The second decays from 1 at \tau\to0 to 0 at the long end, so \beta_1 is the SLOPE and \beta_0+\beta_1 is the instantaneous short rate. The third loading is zero at both ends and humped in between, so \beta_2 is CURVATURE and \lambda sets where the hump sits. Given \lambda the model is linear in the betas and fits by OLS, which is why \lambda is usually fixed or profiled over a grid rather than estimated jointly.

The model

y_t(\tau) = L_t + S_t\frac{1-e^{-\lambda\tau}}{\lambda\tau} + C_t\left(\frac{1-e^{-\lambda\tau}}{\lambda\tau} - e^{-\lambda\tau}\right)

What it assumes

That three factors are enough. They usually explain over 99% of the variation in a government curve - the fit table shows where they do not, and it is almost always the very short end, where policy and money-market frictions live. Svensson relaxes this with a fourth factor, and because a freer model always fits better in sample, the report compares it against the three-factor fit rather than presenting the improvement on its own.
A fixed decay λ, which is what makes this the two-step estimator rather than the one-step Kalman version. λ sets where the curvature loading peaks; 0.0609 puts it near 30 months, which is the conventional choice for monthly maturities.
For Svensson, that the two humps can be told apart. They cannot fully: the two curvature loadings are similar in shape by construction, so the split between C and C2 is less well determined than either is alone. λ₂ = 0.0199 puts the second hump at 90 months, which is where the design matrix on a 1-month-to-30-year grid is best conditioned - chosen by measuring, not by taste. The report prints the condition number so the cost is visible.
That the factors are forecastable from their own past. There are no macro variables in here at all, which is deliberate: this is the benchmark.

Statistical properties

The three factors correspond closely to the first three principal components of yield changes, which empirically explain about 99% of the variation in a curve. Nelson-Siegel is a parametric version of that finding with interpretable loadings.
Fixing lambda makes the model linear and the fit instant; estimating it jointly makes the objective non-convex with local minima, which is why Diebold and Li fix it at the value maximising the curvature loading at two to three years.
The Svensson extension adds a second hump for curves that need it, at the cost of a second lambda and much less stable estimation.
Nelson-Siegel does not guarantee absence of arbitrage. The Christensen-Diebold-Rudebusch arbitrage-free version adds a correction term and is the right choice for pricing rather than description.
Extrapolation beyond the longest observed maturity converges to beta0, which is a modelling assumption rather than information from the data.
Treating the fitted betas as a three-variable time series and forecasting them with a VAR is the Diebold-Li approach, and it forecasts the curve better than a random walk at long horizons.

How to read the output

Check the factor-interpretation table FIRST. Level should correlate near 1 with the longest yield, slope with short minus long, curvature with the empirical hump. If they do not, λ is wrong for these maturities and everything after is suspect.
The cross-section RMSE is in basis points and is the honest summary of fit. Around 10 bp is what this model achieves on US Treasuries.
The forecast is a curve by construction, which is the point: forecasting thirteen maturities separately can produce a curve that crosses itself, and this cannot.
The slope factor is the recession indicator people watch. Negative long-minus-short - an inverted curve - has preceded most US recessions.

A worked reading

13 maturities from 1 to 360 months over 120 dates, fitted with a decay of λ = 0.0609 - which places the curvature loading's peak at 24 months of those available. Three factors reproduce the whole cross-section to a root-mean-square error of 6.8 basis points, which is the fact the entire literature rests on: a curve with thirteen degrees of freedom is really moving in three. The factors also mean what their names claim, and the note checks rather than asserts it - Level correlates 0.9964 with the 360-month yield, Slope 0.9947 with the one-month-minus-360-month spread, and Curvature 0.9311 with 2×m24 − m1 − m360. Forecasting is then done on the three factors rather than the thirteen yields: a VAR(1) takes Level from 3.2746 to 3.5975 and Curvature from 1.2422 to 0.5610 over six periods, and the curve is rebuilt from those. Note the interval on the reconstructed one-month yield runs from 0.62 to 3.80 - a 300 basis point band around a 17 basis point predicted move, which is the honest uncertainty of a yield forecast and is usually omitted from the chart.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Poor fit at the very short end	Money-market instruments behave differently	Consider excluding maturities under three months
Curvature coefficient huge	lambda misplaced, so the hump is in the wrong place	Profile lambda over a grid
Betas jumping between days	Joint estimation hitting different local minima	Fix lambda across the sample
A curve needing two humps	Nelson-Siegel is too rigid	Use Svensson, and expect less stable betas
Fitted forward rates going negative at the long end	No arbitrage-free constraint	Use the CDR arbitrage-free form
Fewer than four maturities	Three parameters plus lambda is not identified	The model needs more points than parameters

Common mistakes

Reading the level factor as "the interest rate". It is the asymptote of the fitted curve, which no traded bond is quoted at.
Forecasting far past the data. Three AR(1) factors revert to their means, so at long horizons this predicts the average curve of the sample period - which is a statement about your sample, not about the future.
Fitting it to a curve that is not a curve. Corporate or emerging-market yields with liquidity gaps at some maturities will still produce three factors and a respectable-looking picture.
Trusting the band too far. The curve’s interval here maps the factor uncertainties through the loadings treating the factors as independent, which understates it where they move together.

Where to read this properly

Nelson and Siegel (1987), "Parsimonious Modeling of Yield Curves", Journal of Business 60(4).
Diebold and Li (2006), "Forecasting the Term Structure of Government Bond Yields", Journal of Econometrics 130(2).
Svensson (1994), IMF Working Paper 94/114.
Christensen, Diebold and Rudebusch (2011), Journal of Econometrics 164(1), for the arbitrage-free version.

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Yield curve (Nelson–Siegel / Svensson)

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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