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Empirical Economics

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01.12.2014

Estimating the Markov-switching almost ideal demand systems: a Bayesian approach

verfasst von: Satoshi Kabe, Yuichiro Kanazawa

Erschienen in: Empirical Economics | Ausgabe 4/2014

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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.

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Nächster Artikel On estimation of almost ideal demand system using moving blocks bootstrap and pairs bootstrap methods

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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

	Model 1	Model 2	Model 3	Model 4
Model 1	1098.416	–	–	–
Model 2	18.927	1117.343	–	–
Model 3	62.157	43.230	1160.573	–
Model 4	63.452	44.525	1.295	1161.868

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.

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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

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Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Assoc 90(430):773–795CrossRef

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Titel: Estimating the Markov-switching almost ideal demand systems: a Bayesian approach
verfasst von: Satoshi Kabe
Yuichiro Kanazawa
Publikationsdatum: 01.12.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 4/2014
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-013-0777-3

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