5.4.1 Visual Assessment
Quantile-quantile plots and density plots are powerful tools for the visual assessment of the goodness of fit for an empirical distribution. A quantile-quantile plot of an ordered sample y = (y1 ≤ . . . ≤ yn) plotsyj (that is, the empirical(j−1/2)/n-quantile of the data) against the(j−1/2)/n-quantile of the fitted distribution, which we assume to have a continuous distribution function. If the fit is good, then the points(xj, yj),j = 1, . . . , n, should lie close to the liney =x. Figure 5.6 shows the strong deviation from normality of the log return distribution for 5-year bonds. The fact that the points lie far below the line x = y for small quantiles and far above this line for large quantiles shows that the empirical distribution has fatter tails than the fitted normal distribution.3
In density plots, the empirical density of the sample, that is, a Gaussian kernel estimation of the den- sity from the sample, is compared with the density of the fitted distribution. Figure 5.7 shows the two densities. It can be clearly seen that the empirical distribution is leptokurtic, that is, it puts more mass around the origin and in the tails than a normal distribution with the same mean and standard deviation.
In terms of bond prices, this means that relatively small daily price changes and relatively large daily price changes take place more frequently than the Gaussian HJM model predicts. On the other hand, price changes of medium size are observed less frequently than in the Gaussian model. Choosing the log scale for they-axis allows us to study the tail behavior of the distributions. Figure 5.8 compares the log densities of the empirical distribution and the log density of the fitted normal distribution. The log den- sity of the normal distribution is a parabola, while the log of the empirical density resembles a hyperbola,
3The normal distribution was fitted by choosing the sample mean and the sample standard deviation.
quantiles of fitted normal distribution
empirical quantiles of log return dataset
-0.006 -0.004 -0.002 0.0 0.002 0.004 0.006
-0.015-0.010-0.0050.00.0050.010
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Figure 5.6: Quantile-quantile plot: Empirical quantiles of log returns on 5-year bonds against quantiles of fitted normal distribution. January 1986 to May 1995.
log return x of zero-bond
density( x )
-0.010 -0.005 0.0 0.005 0.010
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empirical normal
Figure 5.7: Density plot: Empirical density of log returns on 5-year bonds and density of fitted normal distribution. January 1986 to May 1995.
log return x of zero-bond
log( density( x ) )
-0.010 -0.005 0.0 0.005 0.010
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Figure 5.8: Log-density plot: Logarithm of empirical density of log returns on 5-year bonds against log density of fitted normal distribution. German bond market, January 1986 to May 1995.
at least in the central region. We see that the empirical distribution puts considerably more mass into the tails than one would expect if log returns were normally distributed. As a conclusion, we can say that, judged by the visual measures of quantile-quantile plot and density plot, the Gaussian model performs poorly as a description of the empirically observed daily movements of bond prices.
5.4.2 Quantitative Assessment
In this subsection, we apply two common goodness-of-fit tests to test the null hypothesis that the dis- counted log returns on zero-coupon bonds are normally distributed. The Kolmogorov distance of two probability distributions on(IR,B)(given by their distribution functionsF andG) is defined by
dK(F, G) := sup
x∈IR|F(x)−G(x)|.
IfGis the empirical distributionGxof a samplex= (x1, . . . , xn)of sizen, this distance can be written as follows.
dK(F, Gx) = max
1≤k≤n
n
F(xk−)−k−1 n ,k
n−F(xk) o
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where F(x−) denotes the left limit of the distribution function F at the point x, i. e. F(x−) is the measure assigned to the open interval (−∞, x). For a distribution F without point masses we have F(x−) =F(x), and consequently
dK(F, Gx) = max
1≤k≤n
n
F(xk)−k−1 n ,k
n−F(xk) o
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α 20% 10% 5% 2% 1%
λ1−α 1.08 1.23 1.36 1.52 1.63
Table 5.2: Critical values for the Kolmogorov-Smirnov test ifn >40. (From: Hartung (1986))
1−α 85% 90% 95% 97.5% 99%
Q1−α 0.775 0.819 0.895 0.995 1.035
Table 5.3: Approximate quantiles of the distribution of the modified Kolmogorov-Smirnov statisticDmod for the test against a normal distribution whose parameters are estimated from the sample via the standard estimators. (From: Stephens (1986))
The Kolmogorov-Smirnov test uses the Kolmogorov distance of the empirical distribution function Gx and a given continuous distribution functionF to test whetherxwas sampled from the distributionF. It rejects this hypothesis if the Kolmogorov distance is too large, that is, if
Dn:=√
ndK(F, Gx)≥λ1−α, with a valueλ1−αthat depends on the significance levelα.
The situation is somewhat different if one wants to test whether the samplexwas drawn from a distribu- tion from a parameterized classF ={Fθ :θ∈Θ}, whereΘ⊂IRdfor some dimensiond. Then usually one first estimates the unknown parameterθfrom the samplex, say, by maximum likelihood. Then one calculates the Kolmogorov distance between the empirical distributionGxand the estimated distribution Fθ. However, since one has used the samplexto determine the distributionFθ, the distribution of the Kolmogorov distance is not known in general. For the Kolmogorov-Smirnov test on normal distribution, a formula for the tail probability was derived in Tyurin (1985). Another approach by Stephens (1974) (see Stephens (1986)) uses the fact that the modified Kolmogorov-Smirnov statistic
Dnmod :=Dnã(√
n−0.01 + 0.85/√ n) (5.4)
has a distribution that exhibits a very weak dependence onn. Approximate quantiles of this distribution are given in Stephens (1986), Table 4.7. We reproduce them in Table 5.3.
We analyze the log return data for different maturities. The values ofDnthat we get ifF is a normal distribution fitted to the respective samplexare displayed in Table 5.4. (There is virtually no difference between the values ofDandDmodhere because the additional term−0.01+0.85/√
nis close to zero for n= 2342.) Comparison with the critical values given in Table 5.3 yields that the Kolmogorov-Smirnov
Zero coupon bonds with time to maturity [in years]
1 2 3 4 5 6 7 8 9 10
D 4.37 3.86 4.2 4.82 4.56 4.08 4.3 4.15 4.01 3.87 Dmod 4.37 3.86 4.2 4.82 4.56 4.08 4.3 4.15 4.01 3.87
Table 5.4: Values of Kolmogorov-Smirnov test statistic: Test of the normal fit of the log-return distribu- tion for zero bonds with maturities of up to ten years. January 1986 to May 1995.
Zero coupon bonds with time to maturity [in years]
1 2 3 4 5 6 7 8 9 10
normal fit 391 316 366 518 465 351 386 375 348 331
Table 5.5: Values of χ2 test statistic: Test of the normal fit of the log-return distribution for zero bonds with maturities of up to ten years. The number of classes was 45; the 90%-, 95%-, 98%, and 99%- quantiles of theχ2(44)-distribution are 56.4, 60.5, 65.3, 68.7, respectively.
test clearly rejects the assumption of normality.
The χ2 test for goodness of fit counts the number of sample points falling into certain intervals and compares these counts with the expected number in these intervals under the null hypothesis. Following the recommendation in Moore (1986), Section 3.2.4, we choose a numberM :=d2n2/5eof equiprobable classes.4
The log-return datasets under consideration have length n = 2342, soM = 45. We choose the j/M- quantiles (j= 1, . . . , M−1) of the fitted distribution as the boundary points of the classes.
Table 5.5 shows the values of theχ2-test statistic for the null hypothesis of normality. As the number of degrees of freedom, we have chosen44, which is the number of classes minus one. The exact distribution of the test statistic under the null hypothesis is now known. However, the correct quantiles lie between those of χ2(44) and χ2(42)if the distribution has two unknown parameters that are estimated via the maximum likelihood. (See Moore (1986), Section 3.2.2.) Choosing the quantiles ofχ2(44)thus yields a test that is too conservative. But even this conservative test rejects the assumption of normality.