Impacts of Tourism and Fiscal Expenditure on Remote Islands in Japan: A Panel Data Analysis

values. We show the list of islands in Table A1. Note that we conducted a panel data analysis of
data from 1975 to 2001 (26 years).

In this paper, we adopt a partial adjustment model and its modification to analyze the data.
First, we set up the following partial adjustment model (note that the subscript i denotes individual
islands and the subscript t denotes time):

Yit^-Yit-1 ^=θ(Yit^*-Yit-1)⁺uit                                          (1)

where Y is the per capita taxable income, θ is the partial adjustment coefficient, and Y^* is the
ideal per capita taxable income given by Y_it^* = α+ βG_it + γT_it + δN_it where G is the per capita
fiscal expenditure, T is the per capita number of tourists, and N is the population size.
Substituting this formula into (1) and adding an error term, we have the following equation for an
empirical analysis in this study:

Y_it =(1-θ)Y_it-1 +θα+θβG_it +θγT_it+θδN_it +u_it.

If we assume Y^* is given by a distributed lag model with M lags⁸, we have

Yit ⁼ α^{+ β}0Git ^{+ β}1Git -1 ^{+ β}2Git -2 ^{L + β}MGit-M ⁺ γ0Tit ⁺ γ1Tit -1 ⁺γ2Tit-2 ^{L +}γMTit-M

^+δ0Nit ^{+ δ}1Nit-1 ^+δ2Nit-2 ^{L +δ}MNit-M

If we introduce a parameter that represents the difference between a variable at time t and that at
time t-1, that is, ΔG_it = G_it - G_{it -1} , ΔT_it = T_it - T_{it -1} and ΔN_it = N_it - N_it-1 , we then rewrite the
above equation as follows:

^Yit^*=α+βGit^+φ0^ΔGit^+φ1^ΔGit-1^+φ2^ΔGit-2^L+φM-1^ΔGit-M+1

+γT_it +ψ₀ΔT_it +ψ₁ΔT_it-1 +ψ₂ΔT_it-2 L +ψ_{M -1}ΔT_it

-M+1

^+δNit ^+ξ0^ΔNit ^+ξ1^ΔNit-1^+ξ2^ΔNit-2^L+ξM^ΔNit-M+1.

Thus, the fiscal and tourism multipliers and the effect of population size in the long run are
denoted by β, γ, and δ, respectively⁹. Using this formula, (1) is then modified as:

Yit=(1-θ)Yit-1+θα+θβGit+θφ0ΔGit+θφ1ΔGit-1L+θφM-1ΔGit-M+1
+θγTit+θψ0ΔTit+θψ1ΔTit-1L+θψM-1ΔTit-M+1
+θδNit+θξ0ΔNit+θξ1ΔNit-1L+θξM-1ΔNit-M+1+uit.

Applying this model to panel data allows us to relax the constancy of the intercept α . In
particular, the variable α can be divided into three separate components as follows:

α_it = μ+η_i +τ_t

where μ is a constant over all time and islands, η_i is a variable depending only on the individual
island i, τ_t is a variable depending only on the time t. We have two popular models for this
specification. One is a fixed effects model that treats both η_i and τ_t as nonstochastic variables,
and the other is a random effects model that assumes both η_i and τ_t are random variables with

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