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A random walk Metropolis-Hastings algorithm has been widely used in sampling the parameter of spatial interaction in spatial autoregressive model from a Bayesian point of view. In addition, as an alternative approach, the griddy Gibbs sampler is proposed by [1] and utilized by [2]. This paper proposes an acceptance-rejection Metropolis-Hastings algorithm as a third approach, and compares these three algorithms through Monte Carlo experiments. The experimental results show that the griddy Gibbs sampler is the most efficient algorithm among the algorithms whether the number of observations is small or not in terms of the computation time and the inefficiency factors. Moreover, it seems to work well when the size of grid is 100.

Spatial models have been widely used in various research fields such as physical, environmental, biological science and so on. Recently, a lot of researches are also emerging in econometrics (e.g., [

Although there are a lot of works using spatial models in a Bayesian framework, previous literature has rarely examined sampling methods for the parameter of spatial correlation. [

In this paper, we examine the efficiency of the existing Markov chain Monte Carlo methods for the spatial autoregressive (hereafter SAR) model which is the simplest and most commonly used model in the spatial models. Moreover, we propose an acceptance-rejection Metropolis-Hastings (hereafter ARMH) algorithm as an alternative MH algorithm, which is proposed by [

We illustrate the properties of these algorithms using simulated data set given the three number of observations and the seven values of spatial correlation. From the results, we find that the GGS is the most efficient method whether the number of observations is small or not in terms of both the computation time and the inefficiency factors. Furthermore, we show that it is efficient when the number of grid in the GGS sampler is one hundred. These results give a benchmark of sampling the spatial correlation parameter of the models.

The rest of this paper is organized as follows. Section 2 summarizes the SAR model. Section 3 discusses the computational strategies of the MCMC methods, and reviews three sampling methods for spatial correlation parameter. Section 4 gives the Monte Carlo experiments using simulated data set and discusses the results. Finally, we summarize the results and provide concluding remarks.

Spatial autoregressive model explains the spatial spillover using a weight matrix (see [

Let

and we define

Next, let

where

Then the likelihood function of the model (1) is given as follows:

where

Since we adopt the Bayesian approach, we complete the model by specifying the prior distribution over the parameters. We use the following independent prior distribution:

Given a prior density

Finally, we assume the following prior distributions:

where

Since the joint posterior distribution is given by (3), we can now adopt the MCMC method. The Markov chain sampling scheme can be constructed from the full conditional distributions of

From (3), the full conditional distribution of

As it is difficult to sample from the standard distribution, we examine three approaches for sampling

The GGS was proposed by [

and

Thus, we select the grid

Finally, we sample

The RMH method is a simple algorithm because it needs the previous value and a random walk process such as

where

And finally set

An acceptance-rejection Metropolis-Hastings (ARMH) algorithm method was proposed by [

Next, suppose the candidate

In this step, the candidate draw is accepted with probability

The advantage of this method is that it is free to functional form which differs from the GGS and RMH. In this paper, in order to construct the candidate function, we utilize the result of [

where

The full conditional distributions of

where

In this section, we examine the properties of three MCMC methods by simulated data sets. Desirable properties for sampling methods in MCMC are efficiency and well mixing, which yield fast convergence. [

The inefficiency factor is defined as

lated from the sampled values. It is used to measure how well the chain mixes and is the ratio of the numerical variance of the sample posterior mean to the variance of the sample mean from the hypothetical uncorrelated draws (see [

We now explain the simulated data for an experiment. First, we give the weight matrix as an exogenous variable. We construct the spatial weight matrix

Given

where the

The prior distributions are as follows:

We perform the MCMC procedure by generating 35,000 draws in a single sample path and discard the first 20,000 draws as the initial burn-in. For the GGS, we consider the number of grid,

Observation: | |||||
---|---|---|---|---|---|

Parameter | RMH | GGS | ARMH | ||

−0.9 | 7.2 | 3.2 | 3.4 | 3.4 | 2.8 |

−0.6 | 27.6 | 4.4 | 4.4 | 4.7 | 3.7 |

−0.3 | 15.4 | 23.7 | 6.6 | 6.9 | 4.3 |

0 | 41.6 | 9.0 | 10.1 | 11.5 | 6.6 |

0.3 | 79.8 | 24.6 | 19.6 | 20.9 | 13.1 |

0.6 | 117.0 | 46.3 | 45.2 | 52.3 | 44.6 |

0.9 | 806.1 | 312.9 | 223.1 | 327.2 | 324.9 |

Observation: | |||||

Parameter | RMH | GGS | ARMH | ||

−0.9 | 10.7 | 4.8 | 5.0 | 5.3 | 4.7 |

−0.6 | 17.0 | 6.7 | 7.3 | 7.5 | 4.6 |

−0.3 | 34.7 | 9.0 | 10.2 | 10.9 | 5.0 |

0 | 72.4 | 15.4 | 16.2 | 17.8 | 9.5 |

0.3 | 85.1 | 24.5 | 25.5 | 32.6 | 19.9 |

0.6 | 202.3 | 36.1 | 56.6 | 65.8 | 51.3 |

0.9 | 609.1 | 379.3 | 338.0 | 342.1 | 338.9 |

Observation: | |||||

Parameter | RMH | GGS | ARMH | ||

−0.9 | 22.2 | 7.1 | 8.3 | 5.7 | 7.8 |

−0.6 | 31.0 | 11.5 | 12.4 | 13.5 | 9.0 |

−0.3 | 64.8 | 17.6 | 17.6 | 19.1 | 13.8 |

0 | 75.7 | 23.5 | 26.4 | 33.6 | 23.7 |

0.3 | 163.6 | 57.4 | 67.3 | 65.6 | 50.5 |

0.6 | 697.3 | 164.2 | 117.5 | 163.3 | 159.3 |

0.9 | 860.4 | 695.1 | 628.7 | 694.0 | 780.6 |

have similar shapes but that the sample paths (top of the figure) and autocorrelation functions (bottom of the figure) are different. From the sample paths, we can find that the ARMH and GGS mix better than the RMH. As same as the sample paths, autocorrelation functions shows the same tendency. The figure of autocorrelation indicates that both GGS and ARMH perform similarly in the autocorrelation disappear. On the contrary, the result for the RMH indicates that serious autocorrelation for parameter at large lag length.

This paper reviewed the MCMC estimation procedures for sampling the spatial correlation of SAR model, and proposed the ARMH algorithm as more efficient than the RMH in order to show the property of the GGS proposed by [

Observation: | |||||
---|---|---|---|---|---|

Parameter | RMH | GGS | ARMH | ||

−0.9 | 22.17 | 11.12 | 22.68 | 1:05.96 | 24.24 |

−0.6 | 23.22 | 11.57 | 23.06 | 1:05.99 | 23.95 |

−0.3 | 23.31 | 11.71 | 23.21 | 1:07.18 | 23.99 |

0 | 23.27 | 11.83 | 23.27 | 1:07.87 | 24.01 |

0.3 | 23.20 | 12.26 | 23.10 | 1:09.49 | 23.99 |

0.6 | 24.16 | 12.07 | 22.70 | 1:08.36 | 24 |

0.9 | 23.17 | 12.06 | 23.64 | 1:08.66 | 24.02 |

Observation: | |||||

Parameter | RMH | GGS | ARMH | ||

−0.9 | 1:31.73 | 18.67 | 35.49 | 1:37.61 | 1:44.90 |

−0.6 | 1:41.36 | 17.25 | 35.75 | 1:37.83 | 1:42.99 |

−0.3 | 1:43.81 | 19.93 | 36.64 | 1:39.25 | 1:43.22 |

0 | 1:40.10 | 18.30 | 40.15 | 1:38.47 | 1:43.53 |

0.3 | 1:40.90 | 19.90 | 40.04 | 1:41.40 | 1:43.55 |

0.6 | 1:41.73 | 18.91 | 37.35 | 1:43.97 | 1:43.13 |

0.9 | 1:43.36 | 17.93 | 37.89 | 1:40.33 | 1:42.27 |

Observation: | |||||

Parameter | RMH | GGS | ARMH | ||

−0.9 | 8:40.79 | 26.88 | 56.62 | 2:43.63 | 9:05.58 |

−0.6 | 8:43.81 | 26.66 | 56.76 | 2:45.73 | 9:07.63 |

−0.3 | 8:59.71 | 26.74 | 58.84 | 2:44.22 | 9:08.03 |

0 | 8:57.87 | 26.92 | 57.48 | 2:46.64 | 8:56.41 |

0.3 | 9:03.95 | 27.02 | 58.49 | 2:45.99 | 8:51.45 |

0.6 | 9:12.82 | 28.24 | 58.13 | 2:48.13 | 9:01.35 |

0.9 | 9:22.86 | 27.10 | 57.84 | 2:48.15 | 8:59.61 |

Note: Time denotes CPU time on a Pentium Core2 Duo, including discarded and rejected draws.

Parameter | ||||||
---|---|---|---|---|---|---|

AR step | MH step | AR step | MH step | AR step | MH step | |

−0.9 | 0.9866 | 0.9116 | 0.9578 | 0.8975 | 0.9881 | 0.9505 |

−0.6 | 0.9999 | 0.9500 | 0.9999 | 0.9438 | 1.0000 | 0.9724 |

−0.3 | 1.0000 | 0.9848 | 1.0000 | 0.9805 | 1.0000 | 0.9906 |

0 | 1.0000 | 0.9849 | 1.0000 | 0.9787 | 1.0000 | 0.9949 |

0.3 | 1.0000 | 0.9670 | 1.0000 | 0.9544 | 1.0000 | 0.9861 |

0.6 | 0.9991 | 0.9553 | 0.9958 | 0.9375 | 1.0000 | 0.9802 |

0.9 | 0.9997 | 0.9716 | 0.9977 | 0.9649 | 0.9997 | 0.9821 |

−0.050 | |||||

0.087 | 0.106 | ||||

−0.059 | 0.205 | 0.183 | |||

0.161 | −0.007 | 0.002 | −0.025 | ||

−0.995 | 0.054 | −0.075 | 0.078 | −0.160 |

Note: True parameter is 0.9. The number of observation set to be 100.

the results of selecting

Finally, we will state our remaining issues. In this paper, we found that the GGS was the most efficient algorithm in sampling the intensity of spatial interaction. On the other hand, we showed the problem of the SAR model such that the spatial correlation and constant term was weakly identified. Thus, we have to construct the model which is identified, or appropriate algorithm to sample the intensity of spatial interaction. Furthermore, we found that the number of grids is appropriate when

We gratefully acknowledge helpful discussions and suggestions with Toshiaki Watanabe on several points in the paper. Mototsugu Fukushige gave insightful comments and suggestions when we made a presentation at Japan Association for Applied Economics in June, 2009. This research was partially supported by KAKENHI # 25245035 for K. Kakamu.