Ogura (n.d.) MCMC Methods for the Demand Systems: the Monte Carlo Results

Summary

Ogura (n.d.) applies the conjugate Gibbs sampler for seemingly unrelated regression (SUR; Percy 1992; Chib-Greenberg 1995b) to an Almost Ideal Demand System (AIDS, Deaton-Muellbauer 1980) estimated on Japanese household expenditure data (1965–1995, 5 goods). The paper first verifies via Monte Carlo ($T=31$, 1,000 iterations) that Markov chain Monte Carlo (MCMC) achieves lower root mean squared error (RMSE) than traditional SUR in 80% of parameters, then estimates the demand system using Gibbs sampling and tests homogeneity via the Newton-Raftery (1994) harmonic mean Bayes factor, obtaining $B_{01}=1.002$ in favour of the homogeneity restriction.

Key Claims

• AIDS as SUR: The linear approximate AIDS (LA/AIDS) budget-share equations $w_{it}=a_i+\sum_j b_{ij}\ln p_{jt}+c_i\ln(x_t/P_t)+u_{it}$ satisfy the SUR structure with identical regressors in each equation (Zellner 1962 same-regressor case); the Stone price index linearization avoids the nonlinear price index at the cost of inconsistency when the sample is small.
• Gibbs sampler: Two conjugate blocks — $\boldsymbol{\beta}|\Sigma\sim N(b_1,B_1)$ with $B_1=[(Z'\Sigma^{-1}Z)^{-1}+B_0^{-1}]^{-1}$, $b_1=B_1[B_0^{-1}b_0+Z'\Sigma^{-1}\mathbf{y}]$; and $\Sigma|\boldsymbol{\beta}\sim IW(v_0+T,\;R_0+(\mathbf{y}-\mathbf{Z}\boldsymbol{\beta})'(\mathbf{y}-\mathbf{Z}\boldsymbol{\beta}))$. Prior: $N(0,10^6 I)$ for $\boldsymbol{\beta}$, $IW(10,0.5I)$ for $\Sigma$; 10,000 draws after 1,000 burn-in; initial values from traditional SUR.
• Monte Carlo results: At $T=31$, MCMC RMSE beats SUR RMSE for 80% of parameters. Bias is smaller for MCMC in most cases; the intercept of equation 5 has the largest bias in both methods because the adding-up condition accumulates errors from all other equations.
• Demand theory: Adding-up is automatic (use $n-1$ equations). Homogeneity ($\sum_j b_{ij}=0$) and symmetry ($b_{ij}=b_{ji}$) are separate restrictions; symmetry is not imposed in this paper.
• Japanese application: Food and Fuel are necessities (income elasticities 0.60, 0.78); Housing, Clothing, Miscellaneous are luxuries ($\eta>1$). Own-price elasticities all negative. Geweke (1992) convergence test: $p>0.05$ for all parameters.
• Homogeneity test: likelihood-ratio (LR) statistic $=0.522$ vs. $\chi^2(4)=9.488$ $\to$ homogeneity not rejected by SUR. Bayes factor via Newton-Raftery harmonic mean: $B_{01}=1.002$ — "strong" on Jeffreys (1961) scale ($1<B_{01}<2$). Note: the harmonic mean estimator has poor numerical properties; a $B_{01}$ this close to 1 is probably numerical noise rather than genuine evidence.
• MCMC credible intervals: Unlike frequentist SUR, the Bayesian approach directly yields 95% credible intervals for price and income elasticities, enabling straightforward significance testing of the elasticity values themselves.

Concepts Introduced or Extended

• Almost Ideal Demand System
• Seemingly Unrelated Regression
• Gibbs Sampler
• Bayes Factor

Entities Mentioned

• Hisashi Tanizaki

Quotes

"Our results show that the MCMC estimates are more reliable than the traditional SUR estimates." (p. 7)

"In the Bayes estimation, it is possible to directly test the hypothesis that the values of parameters (or elasticities) fall in a particular range." (p. 9)

My Take

A modest working paper from a Kobe graduate student — essentially a textbook application of the Percy (1992) / Chib-Greenberg (1995b) conjugate SUR Gibbs sampler to an AIDS demand system. The methodological contribution is minimal: the Gibbs sampler and its application to SUR were already fully documented in sources already in this wiki. The Monte Carlo evidence ($T=31$, only a partial coefficient table shown) is limited. The harmonic mean Bayes factor should be treated with skepticism — the Newton-Raftery estimator is known to have infinite variance in some settings, and $B_{01}=1.002$ is essentially uninformative. The main value to the wiki is as a concrete application of the SUR Gibbs sampler and an entry point into AIDS demand systems. The companion paper (Ogura-Ohtani, Applied Economics Letters, forthcoming) contains the actual theoretical contribution on homogeneity testing under elliptically symmetric errors.