Iqtisodiy taraqqiyot va tahlil, 2024-yil, may
www.e-itt.uz
247
OLS CONFIDENCE INTERVALS IN NON-LINEAR MODELS:
BOOTSTRAP APPROACH
PhD Rakhimov Zarrukh Aminovich
Westminster International University in Tashkent
ORCID: 0009-0001-0583-4819
Abstract.
Linear models has been a powerful econometric tool used to show the relationship
between two or more variables. Many studies also use linear approximation for nonlinear cases
as it still might show valid results. OLS method requires the relationship of dependent and
independent variables to be linear, although many studies employ OLS approximation even for
nonlinear cases. In this study, we are introducing alternative method of intervals estimation,
bootstrap, in linear regressions when the relationship is nonlinear. We compare the traditional
and bootstrap confidence intervals when data has nonlinear relationship. As we need to know the
true parameters, we carry out a simulation study. Our research findings indicate that when error
term has non-normal shape, bootstrap interval will outperform the traditional method due to no
distributional assumption and wider interval width.
Key words:
OLS, nonlinear model, sample size, confidence Interval, bootstrap, accuracy,
interval size, variance.
CHIZIQLI BO'LMAGAN MODELLARDA OLS ISHONCH INTERVALLARI:
BOOTSTRAP YONDASHUV
PhD Raximov Zarrux Aminovich
Toshkent Xalqaro Vestminster Universiteti
Аннотация.
Чизиқли моделлар икки ёки ундан ортиқ ўзгарувчилар ўртасидаги
муносабатни кўрсатиш учун ишлатиладиган кучли економетрик восита бўлган. Кўпгина
тадқиқотлар, шунингдек, чизиқли бўлмаган ҳолатлар учун чизиқли яқинлашувдан
фойдаланади, чунки у ҳали ҳам ҳақиқий натижаларни кўрсатиши мумкин. ОЛС усули
боғлиқ ва мустақил ўзгарувчилар муносабатларининг чизиқли бўлишини талаб қилади,
гарчи кўплаб тадқиқотлар ҳатто чизиқли бўлмаган ҳолатлар учун ҳам ОЛС
яқинлашувидан фойдаланади. Ушбу тадқиқотда биз чизиқли регрессияларда, агар
муносабатлар чизиқли бўлмаса, интервалларни баҳолашнинг муқобил усули, юклаш
усулини киритамиз. Маълумотлар чизиқли бўлмаган муносабатларга ега бўлса, биз
анъанавий ва юклаш ишонч оралиқларини солиштирамиз. Ҳақиқий параметрларни
билишимиз кераклиги сабабли, биз симуляция тадқиқотини ўтказамиз. Тадқиқот
натижаларимиз шуни кўрсатадики, агар хато атамаси ноодатий шаклга ега бўлса,
юклаш оралиғи тақсимот тахмини ва кенгроқ интервалли кенглиги туфайли анъанавий
усулдан устун бўлади.
Калит сўзлар:
ОЛС, чизиқли бўлмаган модел, намуна ҳажми, ишонч оралиғи, юклаш
чизиғи, аниқлик, интервал ўлчами, дисперсия.
UO‘K: 330.43
V SON - MAY, 2024
247-255
Iqtisodiy taraqqiyot va tahlil, 2024-yil, may
www.e-itt.uz
248
ДОВЕРИТЕЛЬНЫЕ ИНТЕРВАЛЫ OLS В НЕЛИНЕЙНЫХ МОДЕЛЯХ:
МЕТОД БУТСТРАПА
PhD Рахимов Заррух Аминович
Международный Вестминстерский Университет в Ташкенте
Аннотация.
Линейные модели стали мощным эконометрическим инструментом,
используемым для демонстрации взаимосвязи между двумя или более переменными.
Многие исследования также используют линейную аппроксимацию для нелинейных
случаев, поскольку она все еще может дать достоверные результаты. Метод МНК
требует, чтобы отношения зависимых и независимых переменных были линейными,
хотя во многих исследованиях используется приближение МНК даже для нелинейных
случаев. В этом исследовании мы представляем альтернативный метод оценки
интервалов, бутстрап, в линейных регрессиях, когда взаимосвязь нелинейна. Мы
сравниваем традиционные доверительные интервалы и доверительные интервалы
начальной загрузки, когда данные имеют нелинейную зависимость. Поскольку нам
необходимо знать истинные параметры, мы проводим моделирование. Результаты
нашего исследования показывают, что, когда член ошибки имеет ненормальную форму,
бутстрап-интервал
превосходит
традиционный
метод
из-за
отсутствия
предположения о распределении и более широкой ширины интервала.
Ключевые слова:
МНК, нелинейная модель, размер выборки, доверительный
интервал, бутстрап, точность, размер интервала, дисперсия.
Introduction.
Ordinary Least Squares (OLS) regression is a cornerstone of statistical analysis. It
establishes a linear relationship between a dependent variable and one or more independent
variables, allowing us to estimate the effect of changes in the independent variables on the
dependent variable. However, OLS relies on several key assumptions to ensure the validity of
its results. One of the most critical assumptions is linearity in the relationship between the
independent and dependent variables. This implies that the change in the dependent variable
is constant for each unit change in the independent variable.
In many real-world scenarios, however, the relationship between variables is not
perfectly linear. This limitation can lead to biased estimates and unreliable conclusions when
using OLS for non-linear models. For instance, imagine analyzing the effect of fertilizer on crop
yield. Initially, as fertilizer increases, crop yield might rise proportionally. But beyond a certain
point, adding more fertilizer might have diminishing returns or even negative effects. A linear
model would not capture this complexity.
This paper explores the use of the bootstrap approach to construct confidence intervals
for parameters estimated in non-linear models. Confidence intervals provide a range of
plausible values for a population parameter, with a certain level of confidence (e.g., 95%). While
traditional methods for constructing confidence intervals in non-linear models can be complex
and rely on specific assumptions about the error distribution, the bootstrap offers a more
robust and flexible alternative.
The bootstrap method is a resampling technique that utilizes the data you already have
to create new datasets (called bootstrap samples). By analyzing these bootstrap samples, we
can estimate the sampling distribution of the parameter estimates, which allows us to construct
reliable confidence intervals even for non-linear models.
This chapter outlines the limitations of OLS for non-linear models, introduces the concept
of confidence intervals, and emphasizes the importance of the linearity assumption in OLS. It
then highlights the advantages of the bootstrap approach for constructing confidence intervals
in scenarios where the linearity assumption is violated. The subsequent chapters will delve
Iqtisodiy taraqqiyot va tahlil, 2024-yil, may
www.e-itt.uz
249
deeper into the details of the bootstrap method, its application to non-linear models, and the
interpretation of the resulting confidence intervals.
Literature review.
Limitations of OLS for Non-Linear Models: Ordinary Least Squares (OLS) regression
remains a fundamental tool for statistical analysis. However, its core assumption of linearity
between the independent and dependent variables can lead to significant drawbacks when
applied to non-linear models (Weisberg, 2014). When the true relationship is non-linear, OLS
estimates become biased and unreliable (Fox, 2016). This bias arises because the linear model
fails to capture the true curvature or non-linear trend in the data (Montgomery and Chatterjee,
2015).
Consequently, relying solely on OLS for non-linear models can lead to misleading
interpretations of the relationships between variables and inaccurate predictions (Pindyck and
Rubinfeld, 2013). For example, imagine analyzing the effect of temperature on ice cream sales.
An OLS model might suggest a constant increase in sales with rising temperature. However, in
reality, sales might peak at a certain temperature as consumers turn to other options in extreme
heat.
Confidence Intervals and Their Importance: Confidence intervals are a crucial aspect of
statistical inference. They provide a range of plausible values for a population parameter, with
a specific level of confidence (e.g., 95%). In regression analysis, confidence intervals are
typically constructed around the estimated coefficients of the model (Faraway, 2014). These
intervals help us assess the precision of our estimates and the degree of uncertainty associated
with them.
Narrower confidence intervals indicate more precise estimates, while wider intervals
suggest greater uncertainty. By interpreting confidence intervals alongside the estimated
coefficients, we can gain valuable insights into the statistical significance of the relationships
between variables (Gamerman and Lopes, 2006).
Challenges of Constructing Confidence Intervals in Non-Linear Model: Traditional methods
for constructing confidence intervals in non-linear models can be complex and rely on specific
assumptions about the error distribution (Wu, 2004). For instance, delta methods or likelihood-
ratio tests often require normality of the error terms, which might not always hold true in non-
linear models (Harvey, 2013). Additionally, these methods can be computationally intensive,
especially for intricate non-linear relationships.
The Bootstrap Approach: A Viable Alternative: The bootstrap method offers a robust and
flexible alternative for constructing confidence intervals in non-linear models (Efron and
Tibshirani, 1993). It is a resampling technique that leverages the data you already have to
create new datasets, called bootstrap samples. These samples are generated by drawing
observations from the original data with replacement, meaning an observation can be selected
multiple times in a single bootstrap sample (Efron and Tibshirani, 1994).
By repeatedly fitting the non-linear model to these bootstrap samples and obtaining the
corresponding parameter estimates, the bootstrap approach allows us to estimate the sampling
distribution of the parameter estimates. This sampling distribution reflects the variability of
the estimates had we collected different samples from the same population (Davison and
Hinkley, 1997).
Once the sampling distribution is obtained, we can calculate percentiles to construct
confidence intervals. For instance, the 2.5th and 97.5th percentiles of the bootstrap distribution
would provide a 95% confidence interval for the parameter estimate (Efron and Tibshirani,
1998).
The beauty of the bootstrap lies in its ability to bypass the need for specific assumptions
about the error distribution. It relies solely on the data itself, making it a more generalizable
approach for constructing confidence intervals in non-linear models (Leek, 2010).
Iqtisodiy taraqqiyot va tahlil, 2024-yil, may
www.e-itt.uz
250
OLS Confidence Intervals and Limitations in Non-Linear Models
The Power of OLS Regression: Ordinary Least Squares (OLS) regression remains a
workhorse in statistical analysis. It establishes a linear relationship between a dependent
variable (Y) and one or more independent variables (X), allowing us to estimate how changes
in X impact Y. The core of OLS lies in minimizing the squared deviations between the actual Y
values and the Y values predicted by the fitted line. This minimization process yields estimates
for the slope (β₁) and intercept (β₀) coefficients of the linear model:
Y = β₀ + β₁X + ε
Here, ε represents the random error term, capturing the unexplained variation in Y. OLS
assumes ε follows a normal distribution with a mean of zero and constant variance
(homoscedasticity).
Confidence Intervals: Quantifying Uncertainty: A crucial aspect of regression analysis is
constructing confidence intervals (CIs) for the estimated coefficients. These CIs provide a range
of plausible values for the population coefficient, with a specific level of confidence (e.g., 95%).
In OLS, CIs are typically constructed around the estimated slope (β₁) and intercept (β₀).
The width of the CI reflects the precision of the estimates. Narrower intervals indicate
more precise estimates, with a higher degree of confidence that the true population coefficient
falls within that range. Conversely, wider CIs suggest greater uncertainty about the true
coefficient value. Interpreting CIs alongside the estimated coefficients provides valuable
insights into the statistical significance of the relationships between variables.
Non-Linear Relationships: While OLS delivers powerful insights in linear scenarios, its
core assumption of linearity can become a significant limitation when the true relationship
between X and Y is non-linear. In non-linear models, the linear model fails to capture the true
curvature or non-linear trend in the data. This can lead to several issues:
Biased Estimates: When the relationship is non-linear, the OLS estimates (β₀ and β₁)
become biased. This means the estimates are systematically skewed away from the true
population values. Imagine analyzing the effect of advertising spending on sales. A non-linear
relationship might exist where initial advertising has a high impact, but the effect plateaus or
even diminishes with further spending. An OLS model would likely underestimate the initial
impact and overestimate the effect of higher spending levels.
Unreliable Inferences: Biased estimates lead to unreliable inferences about the
relationships between variables. Significance tests based on OLS might incorrectly suggest a
statistically significant relationship when none exists, or vice versa. This can lead to misleading
conclusions about the true impact of X on Y.
Limited Generalizability: OLS predictions based on a non-linear relationship are only
reliable for the range of X values observed in the data. Extrapolating beyond this observed
range can lead to inaccurate predictions, as the linear model doesn't capture the true
underlying trend.
Visualizing the Challenges: A Worked Example: Consider the scenario where we want to
analyze the effect of fertilizer application (X) on crop yield (Y). Imagine the true relationship is
a quadratic curve, where initially yield increases with fertilizer application, but eventually
reaches a peak and starts to decline due to over-fertilization.
An OLS model would fit a straight line to this data. This line might intersect the true curve
at two points, potentially leading to misleading interpretations. The estimated slope might
suggest a positive relationship throughout the observed range, even though the true effect
plateaus and then becomes negative.
Furthermore, OLS CIs constructed around the slope estimate would not accurately reflect
the true uncertainty in the non-linear relationship.
Iqtisodiy taraqqiyot va tahlil, 2024-yil, may
www.e-itt.uz
251
The Need for Alternative Approaches: The limitations of OLS in non-linear models highlight
the need for alternative approaches that can provide more reliable estimates and CIs. The next
chapter will explore the bootstrap method, a robust and flexible technique for constructing CIs
in non-linear models, even when the specific form of the non-linearity is unknown. The
bootstrap approach offers a valuable tool for researchers and analysts to gain deeper insights
from data exhibiting non-linear relationships.
The Bootstrap: A Resampling Rescue: The limitations of OLS confidence intervals (CIs) in
non-linear models necessitate alternative approaches. The bootstrap method emerges as a
powerful and flexible technique for constructing reliable CIs in these scenarios. Unlike
traditional methods, the bootstrap does not rely on specific assumptions about the error
distribution or the form of the non-linearity (Efron and Tibshirani, 1993).
The core idea of the bootstrap revolves around resampling the data you already have.
Here's how it works:
Sample with Replacement: We create new datasets, called bootstrap samples, by drawing
observations from the original data with replacement. This means an observation can be
selected multiple times in a single bootstrap sample. The size of the bootstrap sample is usually
equal to the size of the original data.
Repeat and Estimate: This resampling process is repeated a large number of times (e.g.,
1000 times). For each bootstrap sample, we fit the non-linear model and obtain the
corresponding estimates for the model coefficients.
Distribution of Estimates: By repeating step 2 numerous times, we generate a distribution
of the estimates for each coefficient (e.g., the slope and intercept). This distribution reflects the
variability of the estimates had we collected different samples from the same population.
Constructing Bootstrap CI: Once we have the distribution of the estimates from the
bootstrap samples, we can construct CIs for the non-linear model coefficients. Here's a common
approach:
Percentile Method: We identify the percentiles of the bootstrap distribution that
correspond to the desired confidence level (e.g., 2.5th and 97.5th percentiles for a 95% CI).
These percentiles represent the lower and upper bounds of the CI for the coefficient.
For instance, the 2.5th percentile of the bootstrap distribution for the slope estimate (β₁)
would be the value below which 2.5% of the bootstrap estimates lie. Similarly, the 97.5th
percentile would be the value above which only 2.5% of the estimates fall. The resulting interval
represents the range of values within which we are confident (e.g., 95%) the true population
value for β₁ lies.
Advantages of Bootstrap CIs in Non-Linear Models: The bootstrap approach offers several
advantages for constructing CIs in non-linear models:
Robustness: Unlike traditional methods, the bootstrap doesn't require specific
assumptions about the error distribution or the form of the non-linearity. It relies solely on the
data itself, making it a more generalizable approach. (Efron and Tibshirani, 1994)
Flexibility: The bootstrap can be applied to various non-linear models, regardless of their
complexity. This makes it a versatile tool for researchers working with diverse datasets.
(Davison & Hinkley, 1997)
Interpretability: Bootstrap CIs can be easily interpreted alongside the estimated
coefficients, providing a clear picture of the uncertainty associated with the estimates in a non-
linear context. (Wasserman, 2004)
Applying Bootstrap CIs to Our Fertilizer Example: Recall our example where the true
relationship between fertilizer application (X) and crop yield (Y) is non-linear (quadratic). OLS
CIs would be unreliable in this scenario. However, the bootstrap method can be applied as
follows: We would draw bootstrap samples from the original data on fertilizer application and
crop yield. For each bootstrap sample, we would fit a non-linear model (e.g., a quadratic
function) to estimate the coefficients. By repeating this process a large number of times, we
Iqtisodiy taraqqiyot va tahlil, 2024-yil, may
www.e-itt.uz
252
would obtain a distribution of the estimated coefficients for the non-linear model (e.g., the
intercept and the coefficient for the quadratic term). Using percentiles from this bootstrap
distribution, we could construct CIs for each coefficient. These CIs would reflect the uncertainty
in the estimates due to the non-linear relationship.
By interpreting the bootstrap CIs alongside the estimated coefficients, we can gain a more
accurate understanding of the effect of fertilizer application on crop yield. For instance, a wide
CI for the coefficient of the quadratic term might suggest that the exact peak yield and the point
of diminishing returns are uncertain due to the non-linearity in the data.
Methodology: simulation.
In this part, we will look into simulation of a regression model with non-normal error
term. Simulation is necessary in our study, as we need to know the true parameter in the first
place. Using real live data will almost never allow us to know the true population coefficients.
Secondly, we need to control the form of non-normality we are introducing in our case. Our
simulation starts with the simplest form of linear model with one explanatory variable as given
below
Y =
𝛃
0
+
𝛃
1
*
𝑋1
+
Ɛ
where
𝑋1 ~ 𝑁(5,
4)
Ɛ ~ 𝑁(0, 𝑋1/2
)
where intercept (
𝛽
0
) and
𝛽
1
are defined by us. Independent variables (X
1
) come from
normal distribution with mean of 5 and standard deviation of 4. Error term has mean of 0 and
variance dependent of X
1
variable. This way we simulate non normal distribution of error term
known as heteroscedasticity.
Afterwards, we construction confidence intervals using both approaches, traditional and
bootstrap ones. In order to evaluate the performance at difference sample size, first we start
with sample size of 30 and then we increase it by 10 observations up to 200 observations. All
of the simulations are carried out in R software.
We take the following steps for simulation of linear model with heteroscedasticity
with different sample sizes
Step 1: set intercept
𝛃
0
= 4 and coefficient
𝛃
1
=5
Step 2: Set sample size to n=30
Step 3: generate X1 ~ N(5, 4) starting with sample size n
Step 4: generate
Ɛ ~ 𝑁(0, 𝑋1/2
) starting with sample size n
Step 5: generate Y with Y
=
𝛃
0
+
𝛃
1
*
𝑋1
+
Ɛ
Step 6: Estimate confidence intervals using traditional and bootstrap methods in repeated
simulations (1000 times). Here we construction 95 percent confidence intervals
Step 7: evaluate how many times (out of 1000), true parameters were within estimated
OLS and bootstrap confidence intervals
Step 8: repeat step 2 to step 8 by adding 10 observations to sample size (n=n+10). Finish
when sample size reaches 200 observations
Traditional and bootstrap confidence intervals estimations are discussed in above
sections. For traditional intervals, we use the following formula which is estimated in any
statistical package when we construct our linear model.
𝛽̂
1
± 𝑡
𝛼
2
∗ 𝑠𝑒(𝛽̂
1
)
In case of bootstrap confidence intervals, we get our bootstrap 95 per cent intervals by
taking 2.5
th
and 97.5
th
percentiles from 1000 estimated bootstrap coefficients. This kind of
bootstrap approach is known in literature as bootstrapping pairs.
Iqtisodiy taraqqiyot va tahlil, 2024-yil, may
www.e-itt.uz
253
Results.
In this section, we will be looking at the results of our simulation carried out in R. In Figure
1, one can see non-normal behavior of the error term with respect to X1 explanatory variable.
This gives a visual image of the violation of the one of the assumptions of the linear model, i.e.
constant variance of the error term.
Figure1: non-normal variance of the error term in the linear model
We now look at the performance of the confidence intervals constructed using traditional
and bootstrap approaches. In order to evaluate the performance, we calculate how often the
true coefficient from our simulation was falling within the given interval. Ideally, the true
coefficient must fall in at least 95 per cent of simulated cases. We will be terming frequency of
true coefficient falling within our estimated intervals as accuracy.
The results in Figure 2 indicate that accuracy of traditional confidence intervals are below
our benchmark of 95 per cent across all sample sizes (from 30 to 200). In other works,
traditional approach is suffering a lot from the violations of the OLS assumption. In contrast,
bootstrap confidence intervals at above 95 per cent benchmark, meaning that bootstrap
intervals are including true coefficients in at least 95 per cent of cases.
Figure 2. Accuracy of Confidence intervals: case of nonlinear model
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
20
40
60
80
100
120
140
160
180
200
Accuracy of Confidence intervals: case of nonlinear model
accuracy_simple
accuracy_boot
benchmark
Iqtisodiy taraqqiyot va tahlil, 2024-yil, may
www.e-itt.uz
254
One of the explanations of better performance of bootstrap method lies in the fact that
this approach does not have any distributional or OLS assumptions in contrast to traditional
method. Second reason of stronger performance is explained with Figure 3. One can see that
the size of bootstrap intervals are simply larger which increases changes of including true
coefficients.
Figure 3. Size of confidence intervals
We should note here that bootstrap method might not always outperform traditional
methods even in case of nonlinear models or violations of OLS assumptions. Our study show
that specifically when error term has non-constant variance, bootstrap method is showing a
more promising outcome. Thus, researchers are highly encouraged to consult with related
papers that is suitable in their specific case before applying traditional or bootstrap methods.
Conclusion.
This paper looked into cases when data is nonlinear distributed in error term and
investigated at two ways of confidence interval estimations of coefficients of OLS regression. In
the first chapter, we gave an introductory guide to our topic explaining the relevance and
applicability of our study. Afterwards, we revised related literature on topic of using OLS
approach with nonlinear data and looked into possible pitfalls. Revision of existing papers
indicate that there is limited literature on application of bootstrap approach in case of nonlinear
data. Following this chapter, we looked into theory of ordinary least squares and traditional
approach of constructing confidence intervals. In the same chapter, we also presented how
bootstrap confidence intervals are applicable in this case and how they are estimated. We have
employed bootstrapping pair approaches that does not have any distributional assumptions. In
order to evaluate the performance, we need to know the true parameters. Therefore, we carried
out a simulation of a simple linear model with one explanatory variable. In order to evaluate
performance of both approaches we simulated our regression with error term that has different
variance across independent variable. We simulated our model with different sample size,
spanning from 30 to 200 observations. Our simulation indicates that traditional approach
suffers from introduced nonlinearity as its 95 percent intervals include true coefficient in less
than 95 per cent of the cases. In contrast, bootstrap method has been steadily performing at
above 95 per cent accuracy across all sample sizes (from 30 to 200 observations). This indicates
that bootstrap approach is performing better than traditional approach due to two core
reasons. Firstly, bootstrap method has no distributional assumption in contrast to OLS method.
0
0,2
0,4
0,6
0,8
1
1,2
1,4
20
40
60
80
100
120
140
160
180
200
Size of confidence intervals
mean_conf_interval_simple
mean_conf_interval_boot
Iqtisodiy taraqqiyot va tahlil, 2024-yil, may
www.e-itt.uz
255
Secondly, our simulation showed that bootstrap intervals are usually wider in size increasing
the chances of including the true coefficient.
We need to mention that future researcher should use this method with care, as it might
not be suitable for all nonlinear cases. Thus, they are recommended to look into studies that are
related to their case.
Reference:
Davison , A. C. , and Hinkley , D. V. (1997). Bootstrap Methods and Their Applications.
Cambridge University Press, Cambridge .
Efron , B., and Tibshirani , R. (1986). Bootstrap methods for standard errors, confidence
intervals and other measures of statistical accuracy. Statistical Science. Vol. 1 , 54 – 77
Efron, B. (1979). Bootstrap methods: Another look at the jackknife. The Annals of Statistics,
7(1), 1-26.
Efron, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans. SIAM,
Philadelphia
Efron, B., and Tibshirani, R. J. (1993). An introduction to the bootstrap. Chapman and
Hall/CRC.
Efron, B., and Tibshirani, R. J. (1994). An introduction to the bootstrap (Vol. 57). Chapman
and Hall/CRC.
Efron, B., and Tibshirani, R. J. (1998). The art of statistical learning (Vol. 1). Springer.
Faraway, J. J. (2006). Extending the linear model with R: Generalized linear, mixed effects
and nonparametric regression models. Chapman and Hall/CRC.
Faraway, J. J. (2014). Linear models and extensions (Vol. 14). Springer.
Fox, J. (2016). An R companion to applied regression (3rd ed.). Sage Publications.
Freedman , D. A. (1981). Bootstrapping regression models. Annals of Statistics, 9, 1218 –
1228
Gamerman, J. A., and Lopes, H. F. (2006). Markov chain Monte Carlo: Stochastic simulation
for Bayesian inference (2nd ed.). Chapman and Hall/CRC.
Harvey, D. I. (2013). Linear regression analysis for count data (2nd ed.). John Wiley and Sons.
Montgomery, D. C.,and Chatterjee, S. (2015). Design and analysis of experiments (8th ed.).
John Wiley and Sons.
Pindyck, R. S., and Rubinfeld, D. L. (2013). Econometric models and economic forecasts (5th
ed.). Pearson Education.
Wasserman, L. (2004). All of statistics: A concise course with applications. Springer.
Weisberg, S. (2014). Applied linear regression (4th ed.). John Wiley and Sons.
Wu, C. F. J. (2004). Asymptotic theory for nonlinear regression. Chapman and Hall/CRC.
