AF NS2−S, C
In theAF NS2−S, Cthe inverse ofSis given by
S−1=
⎛
⎝1 −σσ1222 −σσ1333 0 σ122 0 0 0 σ133
⎞
⎠
and the determinant is given by
det(S) =σ22σ33
wherey(t, T) is the observed yield, with maturityT−t.
Using the probability integral transform, we obtain the realized quantiles in the conditional distribution. Consider a sequence of probability integral transforms,{ut}Tt=1, then two conditions must hold for the model to be well specified.
First, the probability integral transforms must follow aUniformdistribution over the unit interval, [0,1]. Second, the sequence must be independent -the past values of -the sequence cannot be used to forecast -the next value.
To conduct a formal test, we consider the method of Berkowitz (2001).
Consider the sequence{zt}Tt=1, where zt= Φ−1(ut)
and Φ−1(u) is the inverse of the standard normal distribution function.
If{ut}Tt=1form a uniform and independent sequence, then{zt}Tt=1form a standard Normal and independent sequence.
Next step in the test of Berkowitz (2001) is to consider theAR(1) model:18 zt+1−μ=ρ(zt−μ) +t+1, t+1∼ N(0, σ2)
so that under the hypothesis of independent and standard Normal data, we have thatμ= 0, ρ= 0 andσ= 1. The Likelihood-ratio test size is given by
LRPIT=−2 (l(z|0,0,1)−l(z|μ, ρ, σ)) where (ignoring constant terms)
l(z|μ, ρ, σ) =−1 2
log σ2
1−ρ2+(z1−μ/(1−ρ))2 σ2/(1−ρ2) + (T−1) log(σ2) +
T t=2
(zt−(1−μ)ρ−ρZt−1)2 σ2
Likelihood ratio tests for independence only (ρ= 0) and for standard Nor-mality only (μ= 0 andσ= 1) can be defined in an analogous fashion.
18More lags and possibly time variation could be included in this specification. We however, choose this simple specification as it is intuitive and should be sufficient to test against a standard normal distribution.
Results from Probability Integral Transform tests
Tables 4.18 to 4.22 present the results from the PIT-tests.
First, the tables show the likelihood ratio-test for zero mean, unit variance and no autocorrelation. It is evident that most of the models fail this test;
in fact it is only for the 3-month maturity and for theAF NS2−LC-model that the hypothesis of zero mean, unit variance and no autocorrelation is not rejected.
Second to assess to which extent, the results are driven by autocorrelation in the PITs, we consider the test, where the autocorrelation can be estimated freely, but still with zero mean and unit variance.
Interestingly, theAF NS3andAF NS3−CA, perform quite well for most maturities, when ignoring autocorrelation in the PIT time series. This im-plies that, when just considering matching the distribution, i.e. mainly mean and variance, theAF NS3andAF NS3−CAperform well in short term forecasts. Recall that for most models, the one-month forecasts are unbi-ased, as measured by thet-statistics, and coupled with the high correlations with the E-GARCH volatility, these results are not a big surprise.
Evaluating density forecasts 1 and 5 years ahead
When considering density forecasts 1 and 5 years ahead, we consider fore-casting quantiles, rather than the PIT-test above.
Consider a quantile on levelα, and lety(t, t+τ) be the yield at timet with maturityτ, and let Q(α|Ωt−s) be the yield at the quantile levelα conditional on the information up to timet−s, in one of the considered models. Then define ahit function as
I(t, τ, α) =
1 if y(t, t+τ)≤Q(α|Ωt−s) 0 if y(t, t+τ)> Q(α|Ωt−s)
in this manner the hit function forms a sequence of zeros and ones, e.g.
(0,0,1,0, . . . ,1), which gives a history of whether the quantile has been exceeded or not. Christoffersen (1998) points out that the problem of de-termining the accuracy of a VaR model (or in this case quantiles), is to test if the sequence{I(t, τ, α)}Tt=1satisfies two properties.
First, theUnconditional Coverage Property, which implies that the proba-bility of exceedingQ(α|Ωt−s) should be equal toα, or more in the previous notation Pr (I(t, τ, α) = 1) =α. Obviously if this is not the case, the model would under- or overestimate the actual risk.
Second, theIndependence Property, implies that there is a restriction on how often violations can occur. For instanceI(t1, τ, α) must be independent of I(t2, τ, α), such thatI(t1, τ, α) cannot be used for forecastingI(t2, τ, α) (for t1< t2). If a model cannot adopt to changing market conditions, then the model could suffer from violations for multiple periods in a row, making it less useful for risk management.
In the present setting with a downward trend in the interest-rate level, we do not expect the independence property to hold. This is also indicated by the PIT-tests, where all model fail the full test, which includes autocorrelation.
Visual inspection of our time series also confirms that the independence property does not hold. Instead we only consider the unconditional coverage property.
Results from the quantile forecasts
Table 4.23 to 4.32 present the results from the quantile forecasts. For each model, the tables show the estimated frequency of quantile exceedences (here termed quantile prediction probabilities(QPD)):
QP D≡αˆ= 1 T
T t=1
I(t, τ, α)
whereI(t, τ, α) is the hit function defined above, andTis the sample size.
Each table presents the QPDs, along witht-statistics based standard er-rors from the estimation of the QPDs. Standard erer-rors are calculated as
SD=
ˆ
α(1−α)/T, where ˆˆ αis the estimated QPD. Thus we do not take autocorrelation in the exceedences into account, however the results in this section are quite clear so the biased standard deviations are not likely to affect the results greatly.
For the one year forecasts (Tables 4.23 to 4.27) it is mainly theAF NS1−L andAF NS1−C-models that perform well, by matching the quantiles for the majority of the yields. On the other hand, theAF NS2−SC model performs the worst, forecasting quantiles that are far too low, compared to the data. For the CIR-based models,CIR−2, AF NS3andAF NS3−CA, the results are quite interesting. For the lower quantiles, the models forecast quantiles that are too high compared to the data, i.e. putting to little risk in the lower tail of the distribution. For the higher quantiles the picture is a bit mixed; for the shorter maturities, the Nelson-Siegel-based models forecast quantiles that are too low, i.e. understating the upward risk to interest-rate changes. For longer-term interest-rates all the CIR-based models forecast
the upper quantiles quite well, i.e. there is no statistical difference between the observed and model-based quantiles.
For the five-year forecast (Tables 4.28 to 4.32), there is an interesting finding for the non-CIR-based models (AF NS1 and AF NS2-models). They all (expect for the 3-month yield) forecast lower quantiles that are far too low. For instance for the 10 year yield, there are no realized observations below the 10 percent quantile. In terms of the upper quantiles, the picture is almost the same, there are no or only a few observations above the 90 percent quantile. All in all, this implies that these models may provide a good forecast of the mean, but distributions are generally too wide.
In terms of the CIR-based models (theCIR−2, AF NS3andAF NS3−CA), they do not perform very well either. In term of the lower quantiles, the models typically have a large portion of the observed yields below the 5 percent quantile (i.e. around 40-70 percent). In terms of the upper quantiles these models perform slightly worse than the non-CIR-based models. We believe the poor performance of the CIR-based models to be an artifact of the downward trend in the level of the interest-rates, coupled with a model structure that ensures strict positivity of interest-rates. This is especially a weakness when one is mostly exposed to risk of falling interest-rates.