Abstract
Allias and Nichèle (Eur Rev Agric Econ, 34(4):517–538, 2007) proposed a Markov-switching almost ideal demand system (MS-AIDS) model by extending the idea of Hamilton (Econometrica, 57(2):357–384, 1989). In this paper, we propose a Bayesian estimation for MS-AIDS model and illustrate applicability of our proposed method. We then run two sets of simulation studies to confirm the validity of the proposed method. In the empirical study on the Japanese meat market, our Bayesian estimation improves the MSEs for all meat products over the ML estimation, while successfully capturing the regime shifts of meat demand coinciding with the timing of bovine spongiform encephalopathy cases in Japan and US.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Hamilton (1989) proposed the Markov-switching model to date the timing of recessions and booms with real gross national product (GNP) data in US. He found that the regime shift from positive to negative growth rate has a recurrent feature of the US business cycle.
“Adding up” guarantees that the total expenditure is equal to the sum of expenditures on the category of products under consideration. “Homogeneity” guarantees that if prices of products increase to \(\tau p_{1 t}, \dots , \tau p_{Nt}\) for a scalar \(\tau > 0\), representative consumer has to increase his expenditure from \(m_{0 t}\) to \(\tau m_{0 t}\) to keep his utility level. “Symmetry” guarantees that the substitution effect in the Slutsky equation is symmetric.
That is, the prior of \(\varvec{\theta }\) and the prior of \(\varvec{\pi }\) are independent.
These number are set to mimic Japanese meat (beef, pork, chicken, and fish) market from 1998 to 2006 to be analyzed in this paper, so that the prices and the expenditures are in Japanese yen. In the actual analysis, the number of observations are only available monthly with \(108\) observations.
Geweke (1992) proposed a convergence diagnostic for Markov chains based on a test for equality of the means of the first and last part of a Markov chain after a burn-in period. Given the samples of parameter \(\{ \theta ^{(j)} \}_{j=1}^{n}\), we consider the two sample averages \(\bar{\theta }_{A}\) and \(\bar{\theta }_{B}\) such that
$$\begin{aligned} \bar{\theta }_{A} = \frac{1}{n_{A}} \sum _{j = 1}^{n_{A}} \theta ^{(j)} \quad \text {and} \quad \bar{\theta }_{B} = \frac{1}{n_{B}} \sum _{j = n^{*}}^{n} \theta ^{(j)}, \end{aligned}$$(12)where \(n^{*} = n - n_{B} + 1\). From these values in (12), Geweke (1992) proposes the following statistic called convergence diagnostic (CD):
$$\begin{aligned} \text {CD} = \frac{\bar{\theta }_{A} - \bar{\theta }_{B} }{\sqrt{ \mathbf {Var}(\bar{\theta }_{A}) + \mathbf {Var}(\bar{\theta }_{B})} } \sim \mathcal {N}( 0, 1 ), \end{aligned}$$(13)where \(\mathbf {Var}(\bar{\theta }_{A})\) and \(\mathbf {Var}(\bar{\theta }_{B})\) are variances of \(\bar{\theta }_{A}\) and \(\bar{\theta }_{B}\).
Japanese native beef cattle.
The initial value of lagged budget share \(\bar{w}_{j, t-1}\) is observed budget share data on December 1997.
Given the parameter \(\theta \in \Theta \), marginal likelihood for data \(\varvec{y}\) \(\equiv \{ y_{1}, y_{2}, \dots , y_{T} \}\) conditional on model \(M\) is defined as
$$\begin{aligned} p( \varvec{y} | M) = \int \limits _{\Theta } \mathcal {L}( \theta | \varvec{y}, M ) p(\theta | M) d\theta \end{aligned}$$(15)Given the samples \(\{ \theta ^{(j)} \}_{j=1}^{n}\) from the posterior distribution \(p(\theta | \varvec{y}, M)\), Newton and Raftery (1994) estimates the marginal likelihood \(p(\varvec{y} |M )\) as
$$\begin{aligned} \hat{p}(\varvec{y} |M ) = \left[ \frac{1}{n}\sum _{j = 1}^{n} \frac{1}{\mathcal {L}( \theta ^{(j)} | \varvec{y}, M ) } \right] ^{-1} \end{aligned}$$(16)and the Bayes factor for model \(i\) against model \(j\) is obtained as
$$\begin{aligned} \text {BF}_{ij} = \frac{\hat{p}(\varvec{y} |M_{i} )}{\hat{p}(\varvec{y} |M_{j} )}. \end{aligned}$$(17)Table 7 Log-marginal likelihood and log-bayes factor The Dirichlet distribution for \(\varvec{\pi }_{i}\) is defined as
$$\begin{aligned} p(\varvec{\pi }_{i}| u_{i 1}, u_{i 2}, \dots , u_{i K} ) = \frac{\Gamma (u_{i 0})}{\Gamma (u_{i1}) \cdots \Gamma (u_{i K})} \pi _{i1}^{u_{i1} - 1} \cdots \pi _{i K}^{u_{iK} - 1}, \end{aligned}$$where \(0 \le \pi _{i j} \le 1\), \(\sum _{j = 1}^{K} \pi _{i j} = 1\), \(u_{i j} > 0\), and \(u_{i 0} = \sum _{j = 1}^{K} u_{i j}\).
The beta distribution for \(\pi _{11}\) is defined as
$$\begin{aligned} p( \pi _{11} | u_{11}, u_{12}) = \frac{\Gamma (u_{11} + u_{12})}{\Gamma (u_{11})\Gamma (u_{12})} \pi _{11}^{u_{11} - 1} (1 - \pi _{11})^{u_{12} - 1}, \end{aligned}$$where \(0 \le \pi _{11} \le 1\) and \(u_{11}\), \(u_{12} > 0\).
In other words, we assume that the prior distribution’s location parameters \(\left\{ \varvec{\theta }_{j} \right\} _{j=0}^{K}\) and scale-like parameters \(\left\{ \varvec{\Sigma }_{j} \right\} _{j = 1}^{K}\) can be freely moved and form a \(K\)-dimensional rectangular parameter space.
That is, when \(s_{t} = k\), the matrix \(\varvec{X}_{t}\) consists of \(k - 1\) of matrices of size \((N- 1) \times [ 3 (N- 1) + N(N- 1 ) / 2 ]\) whose elements are all 0, \(\varvec{X}_{t}^{(1)}\), and \(K - k\) of matrices of size \((N- 1) \times [ 3 (N- 1) + N(N- 1 ) / 2 ]\) whose elements are all 0, and \(\varvec{X}_{t}^{(0)}\), all aligned from left to right.
References
Allais O, Nichèle V (2007) Capturing structural changes in french meat and fish demand over the period 1991–2002. Eur Rev Agric Econ 34(4):517–538
Banks J, Blundell R, Lewbel A (1997) Quadratic engel curves and consumer demand. Rev Econ Stat 79(4):527–539
Carter C, Kohn R (1994) On Gibbs sampling for state space models. Biometrika 81(3):541–553
Chib S (1996) Calculating posterior distributions and modal estimates in Markov mixture models. J Econom 75(1):79–97
Christensen LR, Jorgenson DW, Lau LJ (1975) Transcendental logarithmic utility functions. Am Econ Rev 65(3):367–383
Cooper RJ, McLaren KR (1992) An empirically oriented demand system with improved regularity properties. Can J Econ 25(3):652–668
Deaton A, Muellbauer J (1980) An almost ideal demand system. Am Econ Rev 70(3):312–326
Diebold FXJ, Lee H, Weinbach G (1994) Regime switching with time-varying transition probabilities. In: Hargreaves C (ed) Nonstationary time series analysis and cointegration (Advanced Texts in Econometrics, CWJ Granger and G. Mizon, eds.). Oxford University Press, Oxford, pp 283–302
Geweke J (1992) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: Dawid AP, Bernado JM, Berger JO, Smith AFM (eds) Bayesian statistics, 4th edn. Oxford University Press, Oxford
Hamilton JD (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2):357–384
Hamilton JD (1990) Analysis of time series subject to changes in regime. J Econ 45(1–2):39–70
Hamilton JD (1991) A quasi-Bayesian approach to estimating parameters for mixtures of normal distributions. J Bus Econ Stat 9:27–39
Ishida T, Ishikawa N, Fukushige M (2006) Impact of BSE and bird flu on consumers’ meat demand in Japan. Discussion Paper Series in Osaka University, Suita.
Ishida T, Ishikawa N, Fukushige M (2010) Impact of BSE and bird flu on consumers’ meat demand in Japan. Appl Econ 42(1):49–56
Jin HJ, Koo WW (2003) The effect of the BSE outbreak in Japan on consumers’ preferences. Eur Rev Agric Econ 30(2):173–192
Kabe S, Kanazawa Y (2012) Another view of impact of BSE crisis in Japanese meat market through the almost ideal demand system model with Markov switching. Appl Econ Lett 19(16):1643–1647
Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Assoc 90(430):773–795
Kim CJ, Piger J, Startz R (2008) Estimation of Markov regime-switching regression models with endogenous switching. J Econom 143(2):263–273
Moosa IA, Baxter JL (2002) Modeling the trend and seasonal within an aids model of demand for alcoholic beverage in the United Kingdom. J Appl Econom 17(2):95–106
Newton MA, Raftery AE (1994) Approximate Bayesian inference with the weighted likelihood bootstrap. J R Stat Soc B 56(1):3–48
Ohtani K, Katayama S (1986) A gradual switching regression model with autocorrelated errors. Econ Lett 21(2):169–172
Peterson HH, Chen YJK (2005) The impact of BSE on Japanese retail meat demand. Agribusiness 21(3):313–327
Rickertsen K (1996) Structural change and the demand for meat and fish in Norway. Eur Rev Agric Econ 23(3):316–330
Scott SL (2002) Bayesian methods for hidden Markov models. J Am Stat Assoc 97(457):337–351
Theil H (1965) The information approach to demand analysis. Econometrica 33(1):67–87
Acknowledgments
We thank two referees for many constructive comments on earlier versions of the article for helpful discussions and comments. We are grateful for financial support from the Japan Society for the Promotion of Science under Grant-in-Aid for Scientific Research (B) 20310081 and Challenging Exploratory Research 25590051.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kabe, S., Kanazawa, Y. Estimating the Markov-switching almost ideal demand systems: a Bayesian approach. Empir Econ 47, 1193–1220 (2014). https://doi.org/10.1007/s00181-013-0777-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00181-013-0777-3