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Geographically weighted regression models with the measurement error are a modeling method that combines the global regression models with the measurement error and the weighted regression model. The assumptions used in this model are a normally distributed error with that the expectation value is zero and the variance is constant. The purpose of this study is to estimate the parameters of the model and find the properties of these estimators. Estimation is done by using the Weighted Least Squares (WLS) which gives different weighting to each location. The variance of the measurement error is known. Estimators obtained are

The measurement error is the error appeared when a recorded value isn’t exactly equal to the true value in terms of a measurement process, so that the true value of the explanatory variables is represented by a value that is obtained through a measurement process that isn’t necessarily correspond to the true value.

[

There are many researches that have been discussed about parametric regression models with the measurement error, including [2,4]. Nonparametric regression models with the measurement errors have been developed by [

Some researches have been done on the nonlinear regression model with the measurement errors such as [

According to [

Beside CAR, there are several ways to analyze the spatial data. One of them is the geographically weighted regression model [

Based on the problems and the development of the research before, in this case, we are interested in examining the GWR model with the measurement error. So that, the purpose of this study is to determine the parameter estimators β and to examine the statistical properties of the resulting on the GWR model with the measurement error.

GWR Model is a regression model of global development of the basic idea which is taken from the nonparametric regression [

GWR Model can be written as follows [

with states the point of coordinates (latitude, longitude) region is the value of a random variable and x_{i} is the value of a fixed variable which is known and does not contain errors. This means that x_{i} can be observed directly.

If x_{i} can’t be observed directly, it will be observed variables z_{i}. In this case, there has been a measurement error of the x_{i}. Measurement error is referred to as an error on the variables x_{i}. Measurement error models are:

where is a random variables. The observed random variables called indicator variables, and unobserved variables referred to as latent variables.

So that from Equations (2.1) and (2.2) the regression model becomes:

The first step from this model approach is forming a weighting matrix for each observation (location). Weighting matrix is used to estimate the parameters in the location. Suppose the weight for each location

is, , then the location parameter allegedly by adding the element weighting in Equation (2.1).

Suppose is the element of the i-th row of the matrix 𝑿. Then the value of y at the location of the observation can be written as follows:

We let then

If can’t be directly observed or experienced measurement error, then the Equation (3.1) becomes:

In Equation (3.2) it is assumed that has mean 0 and constant variance.

Means, so that

where

From Equation (3.3) with

^{,}^{}

obtained

so

If Equation (3.4) derived to and the result equated to zero then obtain parameter estimators

The parameter estimation of geographically weighted regression models with the measurement error for each location is

After obtaining the estimators it will look for the properties of the estimator in Equation (3.5). From Equation (2.1), weighted regression with measurement errors obtained:

which can also be written as

where

In matrix notation, the regression Equation (4.1) is

Estimating the parameters obtained from the weighted

From the above Equation we can see that

so that the estimator is a biased estimator for.

[

To obtain the parameter estimates, we create a “quasi” normal Equation by pre-multiplying both sides of Equation (4.2) with, and hence, we obtain

Since, Equation (4.3) above becomes

Therefore, the unbiased estimates obtained by the instrumental variable technique are

Despite the fact that it is sometimes difficult to find variables serving purely as instrumental variables [

To prove whether is an efficient estimator is

with

should be as small as possible so that efficient estimator.

In this paper, we assume that the variance of the measurement error is known. Assumptions used in this model are the errors normally distributed with that expected value is zero and the variance is constant. Geographically weighted regression models with the measurement error use the instrumental variable method. gives the unbiased estimation in the linear model, with terms. and are independent. The properties of estimators of geographically weighted regression models with the measurement error are an efficient estimator if should be as small as possible.