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Panel data combine cross-section data and time series data. If the cross-section is locations, there is a need to check the correlation among locations.
*ρ* and
*λ* are parameters in generalized spatial model to cover effect of correlation between locations. Value of
*ρ* or
*λ* will influence the goodness of fit model, so it is important to make parameter estimation. The effect of another location is covered by making contiguity matrix until it gets spatial weighted matrix (
*W*). There are some types of
*W*—uniform
*W*, binary
*W*, kernel Gaussian
*W* and some
*W* from real case of economics condition or transportation condition from locations. This study is aimed to compare uniform
*W* and kernel Gaussian
*W* in spatial panel data model using RMSE value. The result of analysis showed that uniform weight had RMSE value less than kernel Gaussian model. Uniform W had stabil value for all the combinations.

Panel data analysis combines cross-section data and time series data, in sampling when the data are taken from different locations. It’s commonly found that the observation value at one location depends on observation value in another location. In the other name, there is spatial correlation between the observations, which is spatial dependence. Spatial dependence in this study is covered by generalized spatial model which is focussed on dependence between locations and errors [

Some recent literature of spatial cross-section data is Spatial Ordinal Logistic Regression by Aidi and Purwaningsih [

Data used in the panel data modelisa combination of cross section and time-series data. Crossection data is data collected at one time of many units of observation, then time-series data is data collected over time to an observation. If each unit has a number of observations a cross individuals in the same period of time series, it is calleda balanced panel data. Conversely, if each individual unit has a number of observations a cross different period of time series, it is called an unbalanced panel data (unbalanced panel data).

In general, panel data regression model is expressed as follows:

with

Residual components of the direction of the regression model in Equation (1) can be defined as follows:

where

Spatial weighted matrix is basically a matrix that describes the relationship between regions and obtained by distance or neighbourhood information. Diagonal of the matrix is generally filled with zero value. Since the weighting matrix shows the relationship between the overall observation, the dimension of this matrix is N × N [

After determining the spatial weighting matrix to be used, further normalization in the spatial weighting matrix. In general, the matrix used for normalization normalization row (row-normalize). This means that the matrix is transformed so that the sum of each row of the matrix becomes equal to one. There are other alternatives in the normalization of this matrix is to normalize the columns of the matrix so that the sum of each column in the weighting matrix be equal to one. Also, it can also perform normalization by dividing the elements of the weighting matrix with the largest characteristic root of the matrix ( [

There are several types of Spatial Weight

with

Generalized spatial model expressed in the following equation:

where

Data used in this study was gotten from simulation using generalized spatial panel data model as Equation (5) with initiation of some parameter. Simulation was done use R program. The following step is used to generate the spatial data panel which is consist of index n and t. In dexnindicates the number of locations and indextindicates the number of period in each locations. Here is the proccess:

1) Determining the number of locations to be simulated is

2) Makes 3 types of map location on step 1.

3) Creating a binary spatial weighted matrix based on the concept of queen contiguity of each type of map locations. In this step, to map the 3 locations it will form a 3 × 3 matrix, 9 locations will form a 9 × 9 matrix and 25 locations form a 25 × 25 matrix.

4) Creating spatial uniform weighted matrix based on the concept of queen contiguity of each type of map locations.

5) Making weighted matrix kernel Gaussian based on the concept of distance. To make this matrix, previously researchers randomize the centroid points of each location. After setting centroid points, then measure the distance between centroids and used it as a reference to build kernel Gaussian W. Gaussian kernel W as follows:

6) Specifies the number of time periods to be simulated is

7) Generating the data

8) Cronecker multiplication between matrix identtity of time periods and W, then get new matrix named IW.

9) Multiply matrix IW and

10) Build a spatial panel data models and get the value of RMSE.

11) Repeat steps 7)-9) until 1000 replications for each combination on types of

Types of W: W binary, W uniform and Gaussian kernel W;

Types of

Types of

Types of

12) Get the RMSE value for all of 1000 replicationsoh each combination between W,

13) Determine the best W based on the smallest RMSE for all combinations.

Simulation generate data for vector Y as dependent variable and X matrix as independent variable. Y and X is generate with parameter initiation. After doing simulation, we can get RMSE for each combinations and proccessing it, then we can calculate RMSE for each W, N, T,

W types | Location types | Periods types | Generalized spatial panel data model | Average RMSE | Average RMSE | ||
---|---|---|---|---|---|---|---|

ρ = 0.3, λ = 0.3 | ρ = 0.5, λ = 0.5 | ρ = 0.8, λ = 0.8 | |||||

Uniform W | N = 3 | T = 3 | 1.076 | 1.23 | 2.06 | 1.771 | 1.634 |

T = 6 | 1.223 | 1.387 | 2.684 | ||||

T = 12 | 1.251 | 1.464 | 2.957 | ||||

T = 36 | 1.296 | 1.524 | 3.099 | ||||

Average | 1.211 | 1.401 | 2.7 | ||||

N = 9 | T = 3 | 1.293 | 1.365 | 1.775 | 1.578 | ||

T = 6 | 1.341 | 1.401 | 1.976 | ||||

T = 12 | 1.357 | 1.429 | 2.054 | ||||

T = 36 | 1.362 | 1.448 | 2.139 | ||||

Average | 1.338 | 1.411 | 1.986 | ||||

N = 25 | T = 3 | 1.383 | 1.433 | 1.755 | 1.553 | ||

T = 6 | 1.397 | 1.446 | 1.812 | ||||

T = 12 | 1.403 | 1.467 | 1.843 | ||||

T = 36 | 1.407 | 1.409 | 1.877 | ||||

Average | 1.398 | 1.439 | 1.822 | ||||

Kernel Gaussian W | N = 3 | T = 3 | 1.137 | 1.137 | 1.137 | 1.748 | 1.809 |

T = 6 | 1.352 | 1.352 | 1.806 | ||||

T = 12 | 1.405 | 2.971 | 2.014 | ||||

T = 36 | 1.461 | 3.098 | 2.11 | ||||

Average | 1.339 | 2.14 | 1.767 | ||||

N = 9 | T = 3 | 2.101 | 1.115 | 1.056 | 1.243 | ||

T = 6 | 1.353 | 1.138 | 1.097 | ||||

T = 12 | 1.255 | 1.15 | 1.106 | ||||

T = 36 | 1.261 | 1.161 | 1.119 | ||||

Average | 1.493 | 1.141 | 1.095 | ||||

N = 25 | T = 3 | 1.49 | 1.282 | 1.168 | 2.436 | ||

T = 6 | 5.705 | 1.286 | 1.169 | ||||

T = 12 | 6.004 | 1.293 | 1.177 | ||||

T = 36 | 6.19 | 1.294 | 1.179 | ||||

Average | 4.847 | 1.289 | 1.173 |

Based on

Based on

After looking at the result, it can be concluded that uniform W is better than kernel Gaussian W almost for all combinations of N and T. Then uniform W is better in

The first, authors would like to thankful to Allah SWT, my parents, lecturer and all of friends. This research was supported by private funds.