Modelling Inflation using Generalized Additive Mixed Models (GAMM)

International journal of Chemistry, Mathematics and Physics (IJCMP) [Vol-1, Issue-1, May-Jun, 2017]
AI Publications ISSN: 2456-866X
www.aipublications.com Page | 73
Modelling Inflation using Generalized Additive
Mixed Models (GAMM)
Jamilatuz Zahro1
, Rezzy Eko Caraka2,3
1
Magister Aktuaria, Institut Teknologi Bandung, Indonesia
2
School of Mathematics, Faculty of Science and Technology, the National University of Malaysia, Malaysia
3
Bioinformatics and Data Science Research Center, Bina Nusantara University, Indonesia
Abstract— Inflation becomes an important thing to become
a benchmark for economic growth, investor considerations
factor in choosing the type of investment, as well as
determining factors for the government in formulating fiscal
policy, monetary or non-monetary to be run. Inflation
calculations carried out using the Consumer Price Index,
known as CPI as an indicator to measure the cost of
consumption of goods and services markets. Based on an
analysis using GAMM was concluded R2
value of 0.996 or
can be interpreted that the inflation amounted to 99.6 %
can be explained by the variables used in this study and 0.4
% is explained by other factors
Keywords— Inflation ; General Additive Mixed Models
;CPI ; Economic Growth.
I. INTRODUCTION
The Government has set the inflation target for the period
2016, 2017 and 2018 through the issuance of the Finance
Minister Regulation Number 93.PMK.011 / 2014 on
Inflation Target Year 2016, Year 2017 and Year 2018
inflation target type in this rule inflation is the Consumer
Price Index (CPI) annual (year on year). For 2016, the
inflation target is set at 4.0 percent. For 2017 by 4.0 percent,
and in 2018 by 3.5 percent. All three levels of 1 percent
deviation. In carrying out these policies, BI given various
authorities to ensure the independence, transparency, and
accountability of monetary policy are made. One of the
main tasks of BI functions and indicators of success in
managing its monetary policy is controlled by targeted
inflation rate. Inflation targeting policy has become a best
practice central banks in the world, including in Indonesia
in the last decade. In simple terms defined inflation as rising
prices in general and continuously. The price increase of
one or two items cannot be called inflation unless the
increase was widespread (or result in higher prices) on other
goods.
Indicators are often used to measure the rate of inflation is
the Consumer Price Index (CPI). CPI changes over time
show the price movement of a package of goods and
services consumed by society. Since July 2008, a package
of goods and services in the CPI basket has been done on
the basis of Cost of Living Survey (SBH) 2007 conducted
by the Central Bureau of Statistics Indonesia (BPS). Then,
the BPS will monitor the development of prices of goods
and services on a monthly basis in several cities, in
traditional and modern markets to some types of products /
services in each city.
Inflation, as measured by the CPI in Indonesia, are grouped
into 7 categories of expenditure (based on the Classification
of individual consumption by purpose - COICOP), namely:
Group Material, Food, Beverages and Tobacco, Housing,
Clothing, Health, Education and Sports, Transport, and
Communications. Modeling food price inflation conducted
by Prahutama and Caraka (2015) Based on multivariable
spline model of the variables change in the price of rice,
chicken, chili and vegetable crops contributed to the
inflation rate amounted to 93.94%.
In order to support the economy in Indonesia, the
government takes the role in formulating fiscal policy,
monetary or non-monetary. In addition, it is necessary also
a deep concern related to inflation. This is because when
inflation is high, the price of goods and services exports
become relatively more expensive and lead to domestic
products and services cannot compete with goods and
services from abroad. Exports will also tend to decrease
followed by an increase in imports from other countries are
likely to increase Caraka et al (2016). In a certain area,
inflation to it is an important that he had made the standard-
bearer of economic well-being of society, the factors
Directors investors in selecting a kind of investment, and
the determining factor for the government to formulate
policy fiscal, monetary, as well as non-monetary that will be
applied Suparti et al (2016).
Generalized additive models (GAM) is an extension of the
usual linear regression to replace linear function into
functional additives so that these models can be used even
though relations response variable and several predictor

variables are not linear. And like GLM, GAM's response on
the distribution not only on the normal distribution but also
the distribution of which is included in the exponential
family can be analyzed by this method. The additive model
theory is comprehensive in revealing things that are more
complex, especially with regard to the influence of random
components and a variety of variables that form the data
distribution is not normal. Furthermore, the model GAMM
is expected to be more efficient in identifying the spread of
the influence of random components so that they can more
precisely explain the influence of random components in a
model.
II. LITTERATURE REVIEW
2.1 Additive Model
Generalized additive mixed models are used when there is
no linear relationship between the variables in response to
some of the predictor variables. Generalized linear model in
linear mixed models changed to the additive model.
Additive model is a development of linear models where the
predictor component in the form of the sum smoothing
function (Hastie and Thibshirani, 1999). The relationship
between the predictor variables in the additive model are
independent, and each of the predictor variables contributes
to the response variable. Suppose we have a set of data
{yi, xi1, xi2, … , xip}
i=1
n
with n is the number of observations.
Then the additive model can be written as follows:
Yi = f0 + ∑ fj(Xij) + εi
p
j=1
(1)
fj(∙) = single function possessed by each predictor
p is the number of independent variables and E(ε) =
0,var(ε) = σ2
.
2.2 Smoothing Spline
The smoothing function is a tool to summarize the trend in
the response variable Y as a function of one or more
predictor variablesX1 , … , Xp. Smoothing is used to
summarize the trend is referred to as scatterplot smoother.
Usefulness of the smoothing function is easier to see the
trend in the scatterplot smoother generated between the
response variable and the predictor variable X. Y
Resurfacing in response Y can be done by calculating the
average value Y of each category of data that is worth
categorical predictor. While smoothing techniques for non-
categorical data is to do with the running mean smoothing
techniques or spline kernel. In the additive model are the
sum i function that is a sum sole function of each predictor
variable. The equation that has a large number of
observations that often produces a form of regression curves
were not in accordance with actual conditions. Thus, the
curve cannot describe the tendency of the curve in certain
parts. The concept used in solving the problem by dividing
the data into several sections and then connect each part, in
order to obtain a precise estimate. This concept is called a
piecewise of a regression equation. The method used in the
estimation approach is the smoothing spline.Hastie and
Tibshirani (1990) discussing the various smoothing a scatter
diagram. One of smoothing the scatter diagram is
smoothing spline which is a solution:
S(x) = ∑(Yi − f(xi))2
n
i=1
+  ∫(f′′
(x))2
dx (2)
With  is the smoothing parameter in the interval
10   and great value will produce a smooth curve,
while the small  will produce the rough curve. The first
term in the above equation is used to measure the density of
the data, while the second term shows the curve of a
function.
Fig.1: Illustration Smoothing Spline

Figure 1 shows a scatter diagram left guise of a plot against
the response variable predictor variable X. The right image,
smoothing the scatter diagram has been added to describe
the tendency (trend) in response to the variable predictor
variable X (Hastie and Tibshirani, 2004).
2.3 Selection of Parameter Smoothing
Smoothing spline estimator is highly dependent on the
smoothing parameter, so the selection of smoothing
parameter (smoothingparameter) is essential in finding the
most appropriate spline estimator. If the parameter value is
very small smoothing spline estimator will give you a very
rough. Conversely, if the value of a smoothing parameter is
very large it will produce a very smooth spline estimator.
As a result, need to have parameters in order to obtain
optimal smoothing spline estimator is most appropriate for
the data. One of the criteria in the selection of smoothing
parameter in nonparametric model of the generalized cross
validation (GCV) is expressed as:
∑ (
yi − fλ
̂(xi)
1 −
tr(Sλ)
n
)
2
(3)
n
i=1
2.4 Generalized Additive Mixed Models
Generalized additive mixed models (GAMM) is an
extension of the generalized linear mixed model (GLMM) ,
namely by replacing the linear function becomes a function
Additive GLMM (Lin and Zhang , 1999) .Generalized
additive mixed models are defined as follows :
g(μi) = 𝐗i
T
𝛃 + ∑ fj(Xij)
p
j=1
+ 𝐙i
𝐓
𝐛i (4)
g(μi) = function circuit that will connect the mean
observation, i = 1, n, and predictors of all –j, j=1,
p.
𝐗i
T
= transpose of matrix effects remain p x
1,observation of the i-unit
𝛃 = Coefficient vector p x 1.
fj(∙) = Single function possessed by each predictor
𝐙i
T
= transpose of a matrix of random effects q x
1,observation of the –i th unit
𝐛i = vector of random effects q x 1, observation of
the –i th unit
𝐛i~ N 𝐦 (𝟎, 𝐐), Qis the covariance matrix Q for random
effects
2.5 Parameter Estimation of Generalized Additive Mixed
Models
Estimation is the prediction of the values of the population
parameters based on the existing data, to estimate the
parameters of generalized additive mixed models, first
described function probability density (pdf) of exponential
family as the response variable, as follows:
f(y; θ) = exp[a(y)b(θ) + c(θ) + d(y)] (4)
The likelihood function for a family of exponential estimate
𝛃based n independent samples of Yi :
l𝐢 = y𝐢b(θi) + c (θi) + d(yi) (5)
Which is known to the expected value and variance of the
response variable is
E(Yi) = μi = −c′
(θi)/b′
(θi)
var(Yi) =
[b′′(θi)c′(θi) − c′′
(θi)b′
(θi)]
[b′(θi)]3
g(μi) = 𝐗i
T
𝛃 + ∑ fj(Xitj)
p
j=1
+ 𝐙i
T
𝐛𝐢 = η𝐢
While the function of log-likelihood obtained by calculating
the natural logarithm of the function likelihood for
generalized additive mixed models are:
l = ∑ li
N
i=1
= ∑ yib(θi) + ∑ c(θi) + ∑ d(yi) (6)
In generalized additive mixed models used maximum
likelihood estimation (MLE) to search parameter β andb
Value estimator generalized additive mixed models
obtained by maximizing the log-likelihood function.
To obtain the value β̂ that maximizes the log-
likelihood to be lowered by a step
Values βj
̂ obtained from the first derivative
∂l
∂βj
= 0
The first step is to find the first derivative of the log-
likelihood function of β the first conducted using the chain
rule. Can be written as:
∂l
∂βj
= ∑ [
∂li
∂βj
]
N
i=1
= ∑ [
∂l
∂θi
∂θi
∂μi
∂μi
∂βj
]
N
i=1
Rule-based chain that has been written above, the following
translation of the decrease in variable l to θ
l = ∑ li
N
i=1
= ∑ yib(θi) + ∑ c(θi) + ∑ d(yi)
∂l
∂θi
= yib′(θi) + c′(θi) = b′(θi)(yi − μi)

Further reduction in variable θagainstμ. Based on the
expected value μ of the known
∂θi
∂μi
=
1
∂μi
∂θi
μi = −c′
(θi)/b′
(θi)
∂μi
∂θi
=
−c′′
(θi)
b′(θi)
+
c′(θi)b′′
(θi)
[b′(θi)]2
=
−c′′
(θi)b′
(θi)
[b′(θi)]2
+
c′(θi)b′′
(θi)
[b′(θi)]2
=
c′(θi)b′′
(θi) − c′′
(θi)b′
(θi)
[b′(θi)]2
x
b′
(θi)
b′(θi)
= b′
(θi)
[b′′(θi)c′(θi) − c′′
(θi)b′
(θi)]
[b′(θi)]3
= b′(θi)var(Yi)
∂θi
∂μi
=
1
∂μi
∂θi
=
1
b′(θi)var(Yi)
Generalized additive mixed modelsg(μi) = 𝐗i
T
𝛃 +
∑ fj(Xitj)
p
j=1 + 𝐙i
T
𝐛𝐢 = η𝐢, described as follows
∂μi
∂βj
=
∂μi
∂ηj
∂ηi
∂βj
=
∂μi
∂ηj
xij
Then obtained for the first derivative of the function log-
lihood against βj
∂l
∂βj
= ∑ [
∂li
∂βj
]
N
i=1
= ∑ [
∂l
∂θi
∂θi
∂μi
∂μi
∂βj
]
N
i=1
= ∑ [b′(θi)(yi − μi)
1
b′(θi)var(Yi)
∂μi
∂ηj
xij]
N
i=1
= ∑ [
(yi − μi)
var(Yi)
xij
∂μi
∂ηj
]
N
i=1
This form is not closed-form so it does not provide a
solution for the log-likelihood function equation toβj still
intertwine with each other. Closed-form shape not have
resulted in the value of the parameter estimates cannot be
obtained analytically. The estimated value parameter
generalized additive mixed models using iterative numerical
method called Newton-Raphson method.βj
2.6 Inference Generalized Additive Mixed
Models
Inference parameters need to be conducted to determine
whether the parameters in the model of generalized additive
mixed models, significant or not. The test statistic used is
Test T. The main hypothesis to be tested is
 Ho : βi = 0
H1:βi ≠ 0
 Significance level: α
 ttest =
rxiy√n−2
√1−rxiy
2
rxiy = correlation y andxifor parametersβ for
all-i
n = Number of observations
 The rejections:
Ho will be rejected if the t-test is less than the t-
tablettest < ttabel(α ,n)
Nakagawa and Schielzeth (2013) describes the marginal R2
to measure variant, described by a fixed factor. Fixed-
effects in variants is as numerator. Total variance explained
by the model as the denominator includes random variants,
components disperse additives (for non-normal models) and
a special distribution variant expressed by the following:
RGLMM(m)
2
=
σf
2
σf
2
+ ∑ σl
2
+ σe
2 + σd
2u
l=1
2.7 Prediction Based on Mixed Generalized
Additive Models (poison)
Predictions for new observations, can be done by evaluating
the values of the new observation into a function that has
been formed.
g(μi) = 𝐗i
T
𝛃 + ∑ fj(Xij)
p
j=1
+ 𝐙i
𝐓
𝐛i
suppose has owned generalized additive model with a
mixture of 2 predictor variables as fixed effects linear
relationship X1 = a, X2 = b, and one fixed effects that do
not have a linear relationship X3 = c , was Z = d is the
predictor variable random effects, can be done by entering
these values on a model that has been formed so as to obtain
the value of the response.
g(μi) = β1(X1 = a) + β2(X2 = a) + f3
̂(X3 = c)
+ b1(Z1 = d)
III. RESEARCH METHODOLOGY
The data used in this paper is secondary data obtained on
the website of Bank Indonesia (bi.go.id) as for the steps
outlined as follows:
1. Analysis of Variable Response
2. Testing liniearity of predictor variables
3. Smoothing Variable Response
4. Generalized Additive Model Mixed Model

5. Conduct an analysis in the model
Response variable used in this paper is the inflation rate in
Indonesia (Y). While the predictor variables food prices
(X1), food, beverages cigarettes and tobacco (X2), housing,
water electricity, gas and fuel (X3), clothing (X4), health
(X5), education recreation and sports (X6), transport
communications and services finance (X7). In this study,
the analysis conducted by generalized additive mixed
models, first analyzed the response variable is entered into
the distribution of exponential families. Furthermore, the
smoothing to variable nonlinear predictor, the best model
building.
IV. RESULTS
The first step in modeling GAMM is to check at the
distribution of the response data. Based on the analysis it
can be seen that the normal distribution of data which are
included in the distribution of exponential family.
Fig.1: Distribution Variable Response
(a)
(b)
Fig.2: Linearity Test (a) Variable after Smoothing (b)
Figure 2.b explains that four predictor variables namely;
clothing, health, Education, Recreation and Sport and
transport, communications and financial services, has made
smoothing fine. The next stage is to determine Inference
parameters need to be conducted to determine whether the
parameters in the model of generalized additive mixed
models. Based interference Variable test can be seen in
tabel.1
Tabel.1: Interference Variable Test
Parameter Estimate t-value P_Value Results
intercept 0.111562 14.80691 0000 Significant*
Food material 0.233736 91.52170 0000 Significant*
Food, Beverages, Cigarettes and
Tobacco
0.173329 13.31899 0000 Significant*

Housing, Water, Electricity, Gas and
Fuels
0.258612 25.34501 0000 Significant*
Clothing 1,000 16.48502 0000 Significant*
Health 1,000 3.21606 0017 Significant*
Education, Recreation and Sports 1,000 13.65569 0000 Significant*
Transport, Communications and
Financial Services
2,258 30.18056 0000 Significant*
* Significant at significance level α = 5%
From Tabel.1 can be explained estimate the value of each
variable used in this study. Significant value of each
variable below 0.05 means that all significant variables and
can be used in a Generalized Additive Model Mixed. After
that we must to check Inference feasibility of this model
with hypothesis:
 Ho: Yi = 0 (Model improperly used)
H1:Yi ≠ 0 (Model fit for use)
 Significance level: α
 Calculate statistics:
Ftest =
R2(K − 1)
(1 − R2)(n − K)
R2
= Coefficient of determination
n = Number of observations
k = Number of regression coefficients
 The rejection:
Ho will be rejected ifFtest < Ftabel(α ,n)
Tabel.2: Statistics Test
Model Sum of Squares Df Mean Square F Sig.
Regression 407.618 7 58.231 3.679E4 .000a
Residual .203 128 0.002
Total 407.821 135
a
Significant at significance level α = 5%
From Table 2 significant value of our model predictor 0.00
it means that all variables have a significant impact on the
response variable in the model Mixed Generalized Additive
Model. In the same time it can be interpreted the value of
R2
0.996. The inflation amounted to 99.6% can be explained
byfood prices (X1), food, beverages cigarettes and tobacco
(X2), housing, water electricity, gas and fuel (X3), clothing
(X4), health (X5), education recreation and sports (X6),
transport communications and services finance (X7) and
0.4% is explained by other factors beyond the research. Bias
taken estimated values in Table 1 was formed the following
models:
Inflation = 0.111562 + 0.233736 (prepared food) + 0.173329 (food, drinks, tobacco and cigarette)
+ 0.258612( housing, water, electricity, gas and fuel constitute) + 1 (clothing) + 1 (health)
+ 1( education, recreation dan sport) + 2.258 (Transport, communication, and financial services)
V. CONCLUSION
Bank Indonesia has the objective to achieve and maintain
rupiah stability. The stability of the rupiah among others
include the stability of prices of goods and services
reflected in Inflation. Stability inflation is essential for
sustainable economic development and improve the
welfare. Model generalized additive mixed models
considered appropriate in the modeling of inflation,
inflationary factors do not linear smoothing and response
variables have a scope wider distribution, ie distribution
entered into an exponential family. Additive model in
GAMM is comprehensive in revealing things that are more
complex, especially with regard to mixed effect models are

random and fixed effect, components of varieties and forms
of distribution data.
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Modelling Inflation using Generalized Additive Mixed Models (GAMM)

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