# The Laplace Likelihood Ratio Test for Heteroscedasticity

Authors: J. Martin van Zyl
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

It is shown that the likelihood ratio test for heteroscedasticity,
assuming the Laplace distribution, gives good
results for Gaussian and fat-tailed data. The
likelihood ratio test, assuming normality, is
very sensitive to any deviation from normality,
especially when the observations are from a
distribution with fat tails.  Such a likelihood
test can also be used as a robust test for a
constant variance in residuals or a time series
if the data is partitioned into
groups.


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## Body

## 1. Introduction

The likelihood ratio test for equal variances will be derived under the assumption of Laplace or double exponential distributed observations or residuals. The excess kurtosis of this distribution is three and it is leptokurtic. It is shown that the likelihood ratio test for the equality of variances when assuming the Laplace distribution for the residuals is more robust than the normal one. The distributional properties of this ratio are very similar to that when normality is assumed, but with a better approximation in the asymptotic chi-square approximation of the log-likelihood than the normal case.One of the factors to consider when checking the fit of a model in time series is to see if the residuals are white noise. The use of volatility models for log returns attracted a lot of attention in the last few years, and ARCH and GARCH models are fitted when heteroscedasticity is present. A test for a constant variance after partitioning the residuals, suitable for observations from a distribution with fatter tails than a normal, can also be used to check for white noise.The tests of Levene [1], Brown and Forsythe [2], and Gastwirth et al. [3] are often used as more robust tests than the normal likelihood ratio test for the equality of variances in general statistical tests for example, ANOVA’s. Bootstrap methods can be helpful to investigate the distribution of test statistics in these problems. A review and some suggestions are given in the paper of Boos and Brownie [4]. A simulation study comparing the methods shows much more robust performance of the Laplace likelihood ratio than the normal likelihood ratio and the Levene test.Assume, that a total ofn independent observations {ut}, t=1,…,n, from a normal distribution, are available. There are k groups with sample sizes  n1,…,nk. When equal sample sizes are under consideration, it will be assumed that n=kn0.  Let njσ̂j2=∑j=1nj(uj-u̅j)2, j=1,…,k denote the estimated variances for each partition, where u̅j denotes the sample mean of the jth group or partition. The term njσ̂j2/σ2~χnj-12 is a gamma variable with parameters (nj-1)/2, 1/2.In the case of equal variances, theσ2’s cancel in the likelihood ratio. The likelihood ratio λN [5] for the hypothesis of the equality of variances, H0:σ12=⋯=σk2=σ2,  for normal data is(1.1)λN=∏j=1k(σ̂j2)nj/2(∑j=1knjσ̂j2/n)n/2.A weak point of the statistic is that it is very sensitive to deviations from normality. The statistic and its asymptotic chi-square approximation,-2log(λN), was studied widely and many corrections were suggested to improve the approximation.The ideas and results of Bartlett [6] and Box [7, 8] were the basis for many of the asymptotic corrections later derived for the statistic. They considered the statistic M1: (1.2)M1=(n-k)log⁡(σ2)-∑j=1k(nj-1)log⁡(σj2),σ̂2=∑j=1k(nj-1)σ̂j2(n-k),(nj-1)σ̂j2=∑jnj(uj-u̅j)2,j=1,…,k.Letνj=nj-1, j=1,…,k.  M1 can be denoted in the normal case as M1(0;ν1,…,νk), where the zero indicates excess kurtosis of zero. The statistic can be generalized to M1(γ21,…,γ2k;ν1,…,νk), where each of the k variance estimates is from a population with a different kurtosis, γ2j, j=1,…,k. It is shown [8] that if the kurtosis is equal to γ2 for the k samples, the statistic is distributed as δ-1M1(0;δν1,…,δνk), δ=(1+(1/2)γ2)-1,  in large samples, for any distribution having finite cumulants. Or that the statistic is distributed as (1+(1/2)γ2)χk-12  in large samples.To find the moments of the log likelihood ratio was no problem, but the exact distributions ofλN  and also log(λN) are both extremely complex and not practical to use. The multivariate version of the normal likelihood tests concerning covariance matrices is covered in detail in the book of Muirhead [9]. The equal-sized partitioned series can be put in a sphericity test setting where the ellipticity statistic, which has a very similar form, is used.The likelihood ratioλN is an interesting statistic, and  λN2/n  can be viewed as the ratio of the geometric mean of the estimated variances to the arithmetic mean of gamma variables, which is equal to one, only when the individual terms are independent and equal. The ratio of the geometric mean to the arithmetic mean of gamma variables was studied by Glaser [10]. Another way to look at λN is to notice that it can be written as the product of Dirichlet random variables, or in this work it will be considered as the product of beta random variables. Let(1.3)wj=njσ̂j2∑j=1k(njσ̂j2)=σ̂j2∑j=1kσ̂j2forequalsamplesizes.
This ratio has a beta distribution with parameters νj=(nj-1)/2, ν=(n-k)/2. The likelihood ratio can be expressed in terms of the product of beta random variables and
(1.4)λN2=∏j=1k(σ̂j2)nj(∑j=1knjσ̂j2/n)n=(nn/∏j=1knjnj)∏j=1k(njσ̂j2)nj(∑j=1knjσ̂j2)n=kn∏j=1k(σ̂j2)nj(∑j=1kσ̂j2)n=kn∏j=1kwjn0.
The product of beta and Dirichlet random variables was studied by Springer and Thompson [11] and Rogers and Young [12]. The resulting density is complicated and expressed in terms of the Meijer’s G and H-functions [13].It will be shown that the likelihood test derived from residuals which have the Laplace distribution can also be expressed as a product of beta random variables for large sample sizes.


## 2. The Likelihood Ratio for Laplace Distributed Variables

The Laplace or double exponential density is given by(2.1)pX(x)=12ϕexp⁡(-|x-θ|ϕ),-∞<x<∞,ϕ>0.The variance is2ϕ and the median of the observations is the maximum likelihood estimate of the mean θ. The maximum likelihood estimate of ϕ is ∑j=1n|xj-θ̂|/n for a sample of size n, where θ̂ denotes the estimated median.  For θ known, ∑j=1n|xj-θ|/n is distributed as a (2n)-1ϕχ2n2 variable.  The properties of the Laplace distribution are reviewed in the book of Johnson et al. [14]. The variance of the median is O(n-1), and the absolute deviations |xj-θ̂|,  j=1,…,n, are asymptotically independent [15, page 335].For the seriesu1,…,un partitioned into k parts, the likelihood ratio λL, for the test  H0:ϕ1=⋯=ϕk=ϕ, is(2.2)λL=∏j=1kϕ̂jnjϕ̂n,
with  ϕ̂j=∑j=1k|xj-θ̂j|/nj and ϕ̂=(1/n)(n1ϕ̂1+⋯+nkϕ̂k).For equal sample sizes,n0, the ratio is simplified to(2.3)λL=kn∏j=1k(ϕ̂j∑j=1kϕ̂j)n0,
and λ1/n  is proportional to the geometric mean of the ratios. If θ was known, the ratio ϕ̂j/∑j=1kϕ̂j has a beta distribution with parameters nj, n-nj. This variance is approximately half that of the beta variable for the normal case.Terms involving the distribution of the sum of the log of powers of beta random variables are found in the normal and Laplace likelihood ratio. The moment-generating function of the log of a beta variable with parametersnj and n-nj to a power is
(2.4)φ(t)=E(log⁡(etlog⁡xh))=E(xth)=Γ(n)Γ(nj+ht)Γ(n+ht)Γ(nj),
and the log of the moment-generating function of the sum of k such variables is
(2.5)E(log⁡(φ(t)))=klog⁡(Γ(n))-klog⁡(Γ(nj))+∑j=1klog⁡(Γ(nj+ht))-klog⁡(Γ(n+ht)),
showing that the expected value of -2log(λL) found from the cumulant generating function is
(2.6)E(-2log⁡(λL))=-2nlog⁡(k)-2∑j=1kE(log⁡(xjnj))=-2nlog⁡(k)-2∑j=1k(njψ(nj)-njψ(n)),
where xj denotes a beta variable with parameters nj,  n-nj, and ψ is the digamma function, the derivative of the log of the gamma function. By making use of the approximation of ψ [13],  ψ(n)≈log(n)-(1/n)(1/2+1/12n):(2.7)E(-2log⁡(λL))=k-1-2nlog⁡(k)+2∑j=1k(njlog⁡(n)-njlog⁡(nj))+∑j=1k1(1+6nj)-1(1+6n)≈k-1,
for n=kn0, n0=n1=⋯=nk, and large sample sizes, the expected value -2log(λL) is equal to that of a χk-12  random variable. For the normal ratio, it would be νj=(nj-1)/2 in place of the nj’s, showing that the large sample  χ2 approximation for the Laplace likelihood ratio would be a better approximation for the same sample size n, assuming the distributional assumption concerning θ.The assumption of an expected value of zero for log returns is often made in financial time series of returns, but in most problems the expected value would be unknown, and the median of the observations of a specific partition would be used. The median is a maximum likelihood estimator and good approximations can be expected for reasonable large sample sizes.In Figure1, the histogram of simulated and expected frequencies of 1000 ratios is shown. The ratios were calculated using Gaussian white noise series. The ϕ̂’s are estimated using the median as the estimator of θ. A sample of 200 was partitioned into k=5 equal parts and a histogram of 1000 ratios, ϕ̂1/∑j=1kϕ̂j, is shown in Figure 1. The expected values are from a beta distribution with parameters n1=40, n-n1=160. The estimated mean was 0.1995 compared with the theoretical value of 0.2, and the estimated variance 7.9835e-004 compared with the theoretical value of 7.9602e-004.Figure 1
Simulated ratios and the expected frequencies of the beta density.The sample size ofn=200 is not very large for a time series. It can be seen that the ratios of the estimated ϕj’s to the sum of the ϕj’s are approximately beta distributed for this sample size.A simulation study was conducted to check and compare the asymptoticχ2  approximations of λN, λL of  the likelihood ratios and the Brown-Forsythe test. Series of n=200 observations, partitioned into k=5, groups were simulated 1000 times and tested at the α=0.05 level. The median was used in the Brown-Forsythe variation of the Levene test. The expected values for the normal and Laplace χ2  approximation are k-1=4, and for the Levene statistic, the expected value of a F4;196 variable is 1.0103.The following data was generated: normal white noise, white noise from the Laplace distribution, and independent values from the stable distribution with indexα=1.9 and 1.5.These series were partitioned into 5 groups. Time series were generated to check the results when the test is used for checking a constant variance in residuals. The series generated were an AR(1) and a Garch series. The disturbance terms are normally distributed. The Garch series is the IGARCH(1,1) fitted by Tsay [16] to the excess returns of the SP&500.  The results are shown in Table 1.Table 1
Results of simulation study comparing the likelihood ratio tests and the Levene test.
LaplaceNormalLeveneProportion rejectedMean of-2log(λL)Proportion rejectedMean of-2log(λN)Proportion rejectedMean of LeveneWGaussian White Noise0.00202.32770.05004.07830.04300.9455Laplace White Noise0.05704.06060.42509.76040.08401.0904t-distribution (df=4) White Noise0.05604.02530.524013.89470.11201.2181Stable (α=1.5)0.488019.29670.9370107.94990.34103.5335Stable (α=1.9)0.05304.30120.395023.55640.11901.5190AR(1)ρ1=0.10.00402.37100.06304.12580.04500.9491AR(2)ρ1=0.50.02903.59890.21406.47600.15901.4636IGARCH(1,1)0.789034.67040.941083.80570.69707.9725Both tests are sensitive to heteroskedasticity, but the Laplace test is less sensitive when testing for a constant variance for non-Gaussian white noise. It is interesting to note that both tests are sensitive to large autocorrelation in the series and also when the variance is infinite for stable data withα<2.00. The stable noise with index α=1.9 is close to Gaussian, but theoretically only E(xα) is finite. All the tests detect that the series with α=1.5 is not second-order stationary.The tests are not very sensitive when autocorrelation is present as in the AR(1) models, but the sensitivity increases as the first-order autocorrelation increases. All the tests easily detect the heteroscedasticity in the IGARCH(1,1) model.


## 3. Conclusions and Suggestions

The Laplace likelihood test performs much better on data with heavier tails, and better than the Brown-Forsythe variation of the Levene test for example, in the case of thet-distributed series. The normal likelihood ratio test is only effective in the normal data case.It seems that these tests are sensitive for serial correlation and heteroskedasticity in series and can be used as a check for white noise in residuals. The use of filtering [17] can be applied to investigate and improve results when testing for heteroscedasticity where serial correlation is present in time series models. An investigation into the use of size-adjustment can improve the effectiveness of this test, especially when working with GARCH type series.



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*Source: 249564-2011-06-15.xml*