Bayesian Tests of Mean-Variance Efficiency

Matthew Pollard

10th November 2007

Abstract

This paper presents two Bayesian tests for the mean-variance efficiency of a port-folio. The first uses generalization of confidence intervals into RN , the second usesBayes factors. The tests are applied in testing mean-variance efficiency of the marketcapitalization-weighted US stock market.

1 Introduction

Mean-variance efficiency is an important concept in modern finance. Both Markowitz’s(1956) portfolio selection model and Sharpe-Lintner (1965) capital asset pricing model(CAPM) use mean-variance efficient portfolios. In portfolio selection, mean-variance ef-ficient portfolios are held by rational investors; in the CAPM, all assets are priced inrelation to the market portfolio, which is assumed to be mean-variance efficient. Testingportfolio efficiency is therefore an important in verifying these models: do investors holdmean-variance efficient portfolios; and is the market mean-variance efficient?

There has been considerable research focus deriving efficiency tests. Roll (1977) provedthat portfolio mean-variance efficiency is equivalent to excess returns on all assets beinglinearly related to the excess portfolio returns with zero intercept. The testable con-sequence under this equivalent representation is thus whether the estimated interceptfor each asset is zero. The influential paper of Gibbons, Ross and Shanken (1989) usedordinary-least squares regression theory to construct a generalized t-test of zero-intercepts.Importantly, they assume that returns drawn i.i.d. from the multivariate-normal distri-bution. There has been considerable discussion regarding the power of their test whenreturns do not conform to this distributional assumption. On daily and weekly time-frames, asset returns are non-normal and not identically distributed (Cont, 2003), anddisplay volatility clustering and excess kurtosis.

The General Method of Moments (Hansen, 1982) and Bayesian inference are twoalternative methods that allow weakened distributional assumptions. GMM doe not makeexplicit distributional assumptions for returns and tests zero-intercepts by a goodness-of-fit statistic for moment conditions. Bayesian methods require parametric specification ofthe return distribution, however yield exact in-sample inference and enjoy several powerfultests based on the posterior joint distribution of the asset intercepts.

We construct two Bayesian tests of mean variance efficiency using the joint distributionof estimated intercepts. The first test uses the multivariate posterior distribution of

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the intercept parameters tests the zero-intercept hypothesis by constructing Bayesiancredibility sets. The second test uses Bayes factors to compare the posterior likelihood ofthe null linear regression with zero intercept to the alternative with free intercept. Thetwo tests are applied to the value-weighted CRSP index using 10 size decile portfolios asproxies for stock returns. Mean-variance efficiency is accepted at 95% confidence usingboth tests.

The paper is organized as follows. Section 2 defines mean-variance efficiency, dis-cusses Roll’s spanning result and presents the test for mean-variance efficiency. Section3 discusses three statistical methods for testing the null: OLS, GMM and Bayesian, andsection 4 presents the two Bayesian tests. Section 5 specifies the two models for returnsused and specifies prior distributions. Section 6 presents the results.

2 Mean-Variance Efficiency

Let xp denote weights for a portfolio over N assets. Let µ and Σ respectively denote thevector of mean returns and the N ×N matrix of variance-covariances. Given a set levelof expected return, E = x

′pµ, xp is a mean-variance efficient portfolio if it solves

xp = arg minx(x′Σx), such that x

′µ = E. (1)

This condition is equivalent to the following spanning result due to Roll (1977): xp isefficient iff for all assets i,

E(ri − rzp) = βi(Erp − Erzp) (2)

where βi = Cov(ri,rp)V ar(rp) and rzp is the return on the (unique) portfolio satisfying Cov(rp, rzp) =

0. If a risk-free asset does not exist in the set of assets, (2) is the statement of the BlackCAPM with arbitrary benchmark portfolio. If a risk-free rate rf exists, rzp = rf and (2)is the Sharpe-Litnier CAPM.

Testing (1) directly is difficult since the distribution of the efficient frontier curve ishard to derive; see Kan and Smith (2006). Testing (2) is straightforward since it specifiesa linear regression with no intercept. An immediate testable implication is whether theintercept is zero. If we run the vector regression,

rt − rzp,t = α + β(rp,t − rzp,t) + εt,

where rt = (r1,t, r2,t, ..., rN,t)T , then testing the null hypothesis

H0 : {α1 = ... = αN = 0} (3)

is a test of mean-variance efficiency. The general alternative hypothesis is that one ormore αi 6= 0.

2

3 Tests of H0

There are several statistical methods for testing H0 : ordinary-least squares regressiontheory, generalized method of moments (GMM) and Bayesian. A short overview of eachis provided.

3.1 Ordinary Least Squares

Several standard regression tests exist for H0 if returns satisfy usual ordinary least-squares(OLS) assumptions: normality, serial independence and homoscedasticity. Under theseassumptions, t− and F−tests for significance may be used.

The t-test for an individual αi is performed by calculating the t statistic,

ti =αi√

ε′εT−2/(X ′X)−1.

,

and comparing each ti to the 1 − α quantile of the Student-t distribution with T − 2degrees of freedom. Each t−test is carried out separately do not constitute a joint test ofthe null. Bonferonni adjustments are one way to correct this. Consider the hypothesis

H∗0 :

N⋂i=1

{αi = 0}.

Under this construction, H∗0 is rejected if one (or more) individual t-tests rejects αi = 0.

Clearly, the probability of rejecting H∗0 is higher than rejecting an individual null {αi = 0}.

Let the probability of a incorrectly rejecting the hypothesis {αi = 0} equal to x, that is

p(reject {αi = 0}|H0) = x.

If the test-statistics are independent, then the the probability of incorrectly H∗0 is equal

top(reject H∗

0 |H0) = 1− (1− x)N > x

Bonferroni adjustment is the process of scaling down the significance level x to x∗ so thatp(reject H∗

0 |H0) = x; solving for x∗ yields

x∗ = 1− (1− x)1/N .

Now testing H∗0 with x∗ significance has equal type-1 error as H0. H∗

0 still tests if each nullis simultaneously true, and is consistent with H0 under the OLS assumptions and inde-pendence. While this procedure seems reasonable, Bonferroni adjustment are associatedwith several inferential problems; see Perneger, 1998.

A joint test of H0 can be constructed using the OLS General Linear Hypothesis test. Let R = [1, ..., 1︸ ︷︷ ︸

N

, 0, ..., 0] andX be the T × N matrix of repeated (rmt ) rows, β =

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[α1, ..., αN , β1, ..., βN ]T and r = [0, ..., 0]. The test statistic

f =(Rβ − r)T

[R(XT X)−1RT

]−1 (Rβ − r)/(2N − 1)εT ε/(T − 2N)

has an F distribution with (2N − 1, T − 2N) degrees of freedom. H0 is rejected iff >F (x, 2N − 1, T − 2N). This test is similar in construction to the Gibbons-Ross-Shanken (1989) T 2 test.

If normality, independence, homoscedasticity assumptions are not met, OLS tests per-form poorly (incorrect coverage probability and low power to reject alternatives). It is welldocumented that daily and monthly returns data is highly non-normal and heteroscedas-tic (for instance, Cont, 2001; Harris & Kucukozmen, 2001). Two statistical methods thatare robust to these statistical properties are General Method of Moments and Bayesian.A short summary follows of both methods.

3.2 General Method of Moments

GMM tests whether the moment restrictions specified by a model hold. Generally, a modeland set of distibutions it specifies (X, Θ) can be characterized by the entire (infinite) setof the moment restrictions:

E[g1(X, Θ)− x1] = 0

E[g2(X,Θ)− x2] = 0...

......

where X is the vector of random variables and Θ is the set of unknown parameters. GMMtruncates this list of moment restrictions and estimates the parameters Θ to minimize theweighted squared error in the moment restrictions,

Θ = argminΘ

[1T

T∑t=1

gk(Θ, xt)

]′

k

W

[1T

T∑t=1

gk(Θ, xt)

]k

where W is a suitably chosen weighting matrix.If the number of estimated parameters, M , is less than the number of moment re-

strictions, N , the system is“over-identified” and a test statistic for the sample momentgoodness-of-fit can be constructed. The test statistic is the weighted vector of samplemoment errors,

J = g(Θ)′Wg(Θ),

and under the null H0 : E[gk(Θ, X)] = 0, ∀k, T × J has an asymptotic chi-squareddistribution with N−M degrees of freedom. A p-value test is used to test the null model:if J exceeds the 1 − α quantile χ2

N−M,1−α of the chi-squared distribution, the model isrejected.

A GMM test of H0 uses the moment restriction E{ri − rzp − βi(rp − rf )} = 0. Three

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unknown parameters are Erzp, Erp and βi (for all assets i). This introduces two additionalrestrictions for estimation. The full sample moments restrictions are:

g =1T

T∑t=1

ri,t − rzp − βi(rp,t − rzp) ∀i(r2

m,t − [Erp,t]2)βi − (rp,t − Erpri)rp,t − Erp

The second condition estimates βi and is derived by rearranging βi = Cov(ri,rp)

V ar(rp) . The thirdestimates Erm. For N assets, There are 2N + 1 moment restrictions and N + 2 unknownparameters. The system is over-identified for N ≥ 2 and the test statistic has distribution

TJ = Tg′SWg ∼ χ2N−1.

3.3 Bayesian Inference and MCMC

Bayesian inference is a formal system for turning prior beliefs and new evidence into pos-terior beliefs. Beliefs are specified as probability distributions over plausible but unknownstates, S. Suppose data Y are observed. Bayes theorem provides the formula for posteriorbeliefs,

p(S|Y ) =P (Y |S)p(S)∫p(Y |S)p(S)dS

.

or simply, p(S|Y ) ∝ p(Y |S)p(S). The posterior belief is a weighted average of the priorbelief, p(S), with weights equal to the likelihood of the data, p(Y |S).

An advantage for Bayesian methods is that in-sample inference is exact if the modelis properly specified and the posterior distribution is known. Bayesians do not rely on as-ymptotic results (central limit theorem, law of large numbers) for inference. These resultsare used in classical statistics to approximate the distribution of a sample statistic. In theBayesian inference, the exact posterior distribution of the state is obtained and inferenceusing this distribution follows immediately. The downside is that deriving closed form,analytical expressions for posterior distributions is extremely difficult; most models yieldintractable posterior distributions. Due to this technical reason, Bayesian methods wererarely used until the development of fast PCs and Markov Chain Monte Carlo (MCMC).

MCMC is a method of drawing samples from the posterior distribution by numericallysimulating the path of a Markov chain. The chain is specially constructed to have anequilibrium distribution equal to the posterior distribution. If the conditional distributionof each unknown parameter in the model is known, a Gibbs sampler is used to drawsamples. If not, the slower Metropolis-Hastings sampler is used. Until recently, MCMCwas not computationally feasible and researchers had to program up their own samplingalgorithms from scratch, which demands considerable effort. The purpose built MCMCsoftware BUGS/WinBUGS (Spiegelhalter et. al, 1986) solves this problem. Sampling istreated as a black-box and the researcher needs to only specify the model as a sequenceof conditional distributions and specify prior distributions. WinBUGS is used to performthe two Bayesian tests.

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4 Two Bayesian Tests

The first test is based on constructing N -th dimensional confidence intervals, or “credibil-ity sets” in Bayesian terminology, over the joint posterior distribution of intercepts. If thevector 0 = (0, ..., 0)lies within this set, H0 is accepted and the portfolio is mean-varianceefficient.

The second test uses Bayes Factors, a tool used in Bayesian model selection. Bayesfactor are the ratio of posterior model likelihoods for two models. Here, the comparison isbetween the model with free intercept (α ∈ RN ) and the model with constrained intercept(α = 0). If the likelihood for the free-intercept model is considerably higher, we rejectH0.

4.1 Credibility Set Test

Let r denote the matrix of observed returns for each asset, and p(α|r) denote the posteriordistribution of α = (α1, ..., αN ). The test of H0 at (1− x)% confidence is

Accept H0 if (0, ..., 0) ∈ Minimal Credibility Setx(α),

where Minimal Credibility Setx(α) is defined as the N -dimensional ball satisfying

(1) For all α∗ drawn from p(α1, ..., αN |r), Pr[α∗ ∈ Minimal Credibility Setx(α)] = x

(2) The volume the set is minimal.

This set exists and is unique provided that the posterior distribution p(α|r) is smooth.The credibility set is a generalization of confidence intervals, which exist in R1, to spaceRN . The condition that the volume is minimal is equivalent to the boundary of the setoccupying the highest probability density region, which is a desirably property for anyconfidence interval. With N = 2, the set is an oval; with N = 3, it is an ovoid.

The posterior joint distribution p(α|Y ) is the distribution of intercepts implied fromdata Y and prior information p(α), By Bayes rule, the posterior joint distribution of α is

p(α|Y ) = 1C

∫αN

...

∫α1

p(Y |α1, ..., αN )p(α1, ..., αN )d(α1, ..., αN)

where C is an integration constant ensuring that p(α|Y ) integrates to 1.

4.2 Bayes Factor Test

The Bayes Factor is a Bayesian model selection statistic used to compare evidence fortwo competing models, M0 and MA. It is defined as the ratio of posterior likelihoods fordata Y :

BF (M0,MA) =p(Y |MA)p(Y |M0)

.

6

Here, M0 is the constrained model with α = 0 and MA is the unconstrained model withα ∈ RN .

Bayesian likelihoods are not the same as classical (“frequentist”) likelihoods. A classicallikelihood is calculated by substituting estimates (α, β) of parameters into the conditionalformula p(Y |α, β), that is, it measures the likelihood at single point. Estimated parame-ters (α, β) are commonly picked to maximise this likelihood. A Bayesian likelihood is theweighted average of point likelihoods corresponding to different parameter values. Theweights equal the posterior distribution of each parameter. Formally,

Frequentist Likelihood = p(Y |α, β),

Bayesian Likelihood =∫

β

∫α

p(Y |α, β)p(α, β|Y )dαdβ

The Bayesian likelihood automatically penalizes models with uncertain parameters byaveraging p(Y |α, β) over the whole posterior space p(α, β|Y ).

If BF ≥1, this is evidence for MA over M0, or equivalently rejecting H0 in favour ofHA. When Y accords to H0, we should expect a Bayes factor of slightly less than one: M0

nests into MA; and thus there should be little difference between the fitted regressionsunder H0. Using classical likelihoods, this ratio has expectation of exactly 1. However,since MA contains an additional parameter for each asset, the Bayesian likelihood for thismodel should be lower due to greater parameter uncertainty.

In deciding whether to accept MA, a threshold Bayes Factor value is needed. Ingeneral, there is no one-to-one correspondence between Bayes-factors and classical p-values, i.e. “accept MA with 95% confidence if BF > threshold(0.95)”. Rules of thumbhave instead been adopted. Kass and Rafterly (1995) suggest BF > 10 is “substantial”evidence forMA, and BF > 20 is“decisive”evidence. A critical threshold of 20 is adoptedin this paper.

5 Models

The regression model is specified by:

rt − rzp,t = α + β(rp,t − rzp,t) + εt,

where rt is the vector of N asset returns at time t, and α, β are vectors with elements αi,

βi.

We consider two models for the distribution of asset returns: i.i.d. multivariate normaland i.i.d multivariate student-t.

(1) : rt ∼ i.i.d MV N(µ,Σ)

(2) : rt ∼ i.i.d. MV T (µ,Σ,k)

7

where µ is the vector of means, Σ is the variance-covariance matrix and k is the vectorof degrees of freedom for each student-t.

The multivariate normal model is used by Gibbons, Ross and Shanken (1989). Thestudent-t distribution allows excess-kurtosis in returns to be modelled. More complicatedspecifications for error structure, such as multivariate-GARCH(1,1), are possible to runWinBUGS. For simplicity, we focus on these two i.i.d models.

Prior distributions must be specified for each parameter in the model. We use highlydiffuse priors that carry little prior information. Conjugate distributions are also used sothat the fast Gibbs algorithm can be used for sampling. The priors are:

αi ∼ N (0, 100), βi ∼ N (1, 100), Σi,j ∼ Wishart(100× Ii=j , 10), ki ∼ χ2(5)

for each asset i. The variance-covariance matrix has a prior Wishart distribution, withdensity function

f(Σ) = |R|k/2|Σ|(k−p−1)/2exp(−1

2(RΣ)

)where k = 10 and R is a 10×10 matrix with i − jth element {100 × Ii=j}, that is, theprior variance for each asset is 100 and the prior covariances are zero.

The conditional likelihood of data r used to calculate Bayes Factors are as follow. Forthe normal model, the conditional likelihood under MA is

p(r|, α, β,Σ) =T∏

t=1

(2π)−10/2|Σ|1/2exp(−1

2(rt − α− βrm,t)′T (rt − α− βrm,t)

). (4)

For the student-t model, the conditional likelihood under MA is:

p(r|, α, β,Σ,k) =Γ[(k + 10)/2]

Γ(k/2)k10/2π10/2|Σ|1/2

[1 +

1k(rt − α− βrm)′Σ(rt − α− βrm)

]−(k+10)/2

(5)Under M0, the formulae are the same after setting α = 0.

6 Data

The test portfolio for mean-variance efficiency is the US market weighted by market capi-talization. This is proxied by the monthly CRSP1 value-weighted aggregated index, whichis a weighted sum of the NYSE, AMEX & NASDAQ indices. Ten CRSP stock portfoliosare used as proxies for asset returns, which are sorted by capitalization decile. The datawindow is January 1926 to December 2006. Gibbons, Ross and Shanken (1989) the samedataset but with the equally-weighted CRSP index. The zero-correlation portfolio returnrzp is proxied by the 1-month T-bill rates supplied by the CRSP risk-free rate index. Thet-bill rate is treated as risk-free, that is, it is assumed that Cov(rp, rzp) = 0.

1Center for Research in Security Prices. Data available through Wharton Research Data Services,wrds.wharton.edu.

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7 Results

7.1 Posterior Distribution Test

The posterior distribution p(α1, ..., α10|Y ) is sampled from using the Gibbs algorithm10,000 times for the multivariate-normal model, and a Metropolis-Hastings algorithm forthe multivariate-t model. 2000 initial observations are discarded to ensure stationarity ofthe chain, leaving 8000 observations pairs, {α(k)

1 , ..., α(k)10 }K=8000

k=1 . The minimal credibilityset is fitted by constructing a smoothed kernel density estimate for each αi, joining densi-ties into an array and sliding a hyperplane perpendicular to the density until the volumeabove the intersection is equal to probability x. The method is outlined in Hyndman(1996), who has written an R library2 of credibility set tools.

Multivariate-Normal The joint distribution p(α1, ..., α10|r) is shown in figure 1 throughcross-sectional distribution p(αk, α10|r) for ascending k = 1 to 10. There is strong neg-ative correlation between α1 and α10, which represent the lowest and highest deciles ofcapitalization respectively. Cross-sections of the R10 minimum credibility set are drawnat 1% (outer circle) and 5% (inner circle). The point 0 = (0, ..., 0) is not an element ofthis set at 1% or 5%, as seen in the top-left panel which plots p(α1, α10|r) : the point(0, 0) lies out of the set. Consequently, mean-variance efficiency is rejected at 1% and 5%.

Multivariate student-t Figure 2 shows the cross sectional plots and minimum cred-ibility sets at 1% and 5%. A similar correlation structure is present. The credibility sethas slightly wider volume and have slightly different positioning. The point 0 is containedin the set at 1% and marginally at 5%: the point (0, 0) is only just contained in the 5%set in p(α4, α10|r). Mean-variance efficiency is accepted at both levels.

7.2 Bayes Factor Test

A random sample of 8000 estimated parameters is generated for the unrestricted modelMA (α ∈ RN ) and the restricted model M0 (α = 0). The pairs (α(k)

A , β(k)A ,Σ(k)

A ) and(β(k)

0 ,Σ(k)0 ) of parameter values for the two respective models are substituted into the

conditional likelihood formulae given in (4) and (5). The Bayes factor is estimated by

BF (M0,MA) =∑K

i=1 p(r|α(k)A , β

(k)A ,Σ(k)

A )∑Ki=1 p(r|β(k)

0 ,Σ(k)0 )

.

Multivariate Normal The estimated Bayes factor is

BF (M0,MA) = 51.25

which indicates decisive evidence for the unconstrained model, α ∈ RN over the con-strained model α = 0. The Bayes factor exceeds the adopted “decisive” threshold of 20.Mean-variance efficiency is rejected.

2hdrcde, available through cran.r-project.org

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Figure 1: Multivariate Normal. Cross sections of the 1% and 5% minimum credibilityset (α1, ..., α10). The set lies in R10 and each cross-section shows the joint posteriorp(αk, α10|r), k = 1 to 9. The set does not contain 0 = (0, ..., 0) at 1% or 5%, which isvisible in the top-left panel; consequently efficiency is rejected at 1% and 5%.

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Figure 2: Multivariate Student-t. Cross sections of the 1% and 5% minimum credibil-ity set for (α1, ..., α10). The contains 0 = (0, ..., 0) both 1% or 5%, consequently efficiencyis accepted.

11

Multivariate student-t The estimated Bayes factor is

BF (M0,MA) = 21.10

which indicates significance evidence for the unconstrained model, α ∈ RN over the con-strained model α = 0. Using the threshold of 20, mean-variance efficiency is marginallyrejected.

8 Discussion

Results from the tests illustrate the importance of distributional assumptions in paramet-ric test methods. Under the i.i.d multivariate-normal model, mean-variance efficiency isrejected “decisively” by the Bayes factor test, and rejected at the 1% and 5% level in thecredibility set test. However, under the i.i.d multivariate-t model, the Bayes factor isconsiderably lower is only marginally decisiveness. Mean-variance efficiency is rejected at5% but not 1% in the credibility set test.

In conclusion, this paper presents two new Bayesian tests of mean-variance efficiency;the first uses the the joint distribution of intercept parameters to construct credibilitysets; the second

References

[1] Cont, R., “Empirical Properties of Asset Returns: Stylized Facts and Statistical Is-sues,” Quantitative Finance, 1, 2, 223-236, 2001.

[2] Gibbons, M., Ross, S., and Shanken, J., “A test of the efficiency of a given portfolio,”Econometrica, 57, 5, 1121-1152, 1989.

[3] Harris, R. & Kucukozmen, C., “The Empirical Distribution of UK and US StockReturns,” Journal of Business Finance & Accounting, 28, 5-6, 715-740, 2001.

[4] Hyndman, R.J., “Computing and graphing highest density regions,” American Statis-tician, 50, 120-126, 1996.

[5] Kan, R., & Smith, D., “The Distribution of the Sample Minimum-Variance Frontier,”working paper, Simon Fraser University, 2006.

[6] Perneger, T., “What’s wrong with Bonferroni adjustments,” British Medical Journal,1998.

[7] Roll, R., “A critique of the asset pricing theory’s test, Part I: On past and potentialtestability of the theory,” Journal of Financial Economics, 4, 2, 1977.

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A WinBUGS Model

For brevity, the two models that follow estimate the system with 2 assets; the k − thadditional asset can be added by repeated the equations for meank, priors alphak andbetak. An extrw rows and columns in the Wishart distribution matrix R with 0.01diagonal must be added.

A.1 Normal Model

model{for (t in 1:N) {mu[t,1]<-alpha1+beta1*m[t]mu[t,2]<-alpha2+beta2*m[t]y[t, 1:2]˜dmnorm(mu[t,],tau[,])}tau[1 : 2,1 : 2] ˜ dwish(R[ , ], 2)

R[1, 1] <- 0.001R[1, 2] <- 0R[2, 1] <- 0;R[2, 2] <- 0.001

beta1˜dnorm(0,0.01)beta2˜dnorm(0,0.01)alpha1˜dnorm(0,0.01)alpha2˜dnorm(0,0.01)}

A.2 Student-t Model

model{for (t in 1:N) {mu[t,1]<-alpha1+beta1*m[t]mu[t,2]<-alpha2+beta2*m[t]y[t, 1:2]˜dmt(mu[t,],tau[,],k)}tau[1 : 2,1 : 2] ˜ dwish(R[ , ], 2)

R[1, 1] <- 0.001R[1, 2] <- 0R[2, 1] <- 0;R[2, 2] <- 0.001

beta1˜dnorm(0,0.01)beta2˜dnorm(0,0.01)alpha1˜dnorm(0,0.01)alpha2˜dnorm(0,0.01)k˜dchisqr(5)I(2,)}

13