Comprehensive Exploration of Time Series Models: Theory and Practice

This report provides a comprehensive overview of various time series models, offering insights into their theoretical foundations and practical applications. It begins with an introduction to time series data and its significance, followed by detailed explanations of key models such as Autoregressive Integrated Moving Average (ARIMA), Moving Average model, Vector Autoregression (VAR), Nonlinear Autoregressive Exogenous model, Distributed Lag model, and Autoregressive Fractionally Integrated Moving Average. Each model's characteristics, including its underlying assumptions and practical applications, are discussed. The report highlights the importance of these models in forecasting, data analysis, and understanding patterns in time-dependent data across various fields, such as economics, finance, and environmental science. The document also underscores the importance of model assumptions and their implications for the validity of the results. The report concludes by summarizing the effectiveness of time series models in estimating future values and highlights the importance of selecting the appropriate model based on data characteristics and the research objectives.

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Table of Contents
Introduction................................................................................................................................2
Time series models.....................................................................................................................2
Autoregressive integrated moving average............................................................................3
Moving average model...........................................................................................................3
Vector auto regression............................................................................................................3
Nonlinear autoregressive exogenous model...........................................................................3
Distributed lag model.............................................................................................................3
Autoregressive fractionally integrated moving average.........................................................3
Application and assumption.......................................................................................................3
Autoregressive integrated moving average............................................................................4
Moving average model...........................................................................................................4
Vector auto regression............................................................................................................4
Nonlinear autoregressive exogenous model...........................................................................4
Distributed lag model.............................................................................................................5
Autoregressive fractionally integrated moving average.........................................................5
Conclusion..................................................................................................................................5
Bibliography...............................................................................................................................6

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Introduction
A primary time series is a pattern of the points of data which graph in the time order.
Mainly it is concerned with the sequence pointed at the successive equally points on time.
Thus, it is an arrangement of the discrete data.
Primary time series are very rare plotted via line graphs. The use of the times series is
in figures, signal handling, pattern response, econometrics, calculated economics, weather
predicting (Rao, 2012).
Primary time series enquiry consists of the approaches, which are, used for the
analysing time series statistics in reference to extract the features of the data and other
meaningful data. Time series forecasting can be used as the model of the time series for
predicting the future based on the value which is based on the previously observed value.
This paper will give the theoretical knowledge about time series models and their
applicability with assumptions.
Time series models
These models are very valuable models when anyone have serially connected data.
Most the corporate houses effort on time series data to examine sales amount for the next
year, website traffic, competition place and much more. Though, it is also one of the parts,
which many forecasters do not comprehend.

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In addition to this, the times series model makes use of the one method ordering of
time so that the morals can be derived from the past values instead of deriving the values
from the future values (Box, 2015).
There are various types of the primary time model such which are stated below:
Autoregressive integrated moving average
This model is the simplification of an autoregressive moving average (ARMA) model.
These model are fixed to the time series data for well understanding of the data or to predict
future facts (Lee, 2011).
Moving average model
This model is called as the moving average process, which is the common approach
for model related to the time series. This model can also be termed as the moving average
process. This specifies that the variable of the output depend linearly on the past as well as
current values.
Vector auto regression
This is the stochastic model which is used to capture the dependency among the
various times series.
Nonlinear autoregressive exogenous model
In the primary time series modeling, this model is a nonlinear autoregressive model
that has exogenous efforts (Pham, 2010).
Distributed lag model
This is that model of the primary time series in which a regression equation is used to
estimate the current values of a variable, which is dependent on the current values a d, the
lagged values.

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Autoregressive fractionally integrated moving average
This model is the types of the time series model which simplify ARIMA models by
slowing the non-integer values of the differencing parameters.
Application and assumption
Autoregressive integrated moving average
The main presentation is in the area of the short term predicting where at least 40
historical data opinions are needed. It mechanism is best when the data displays a stable
design over time.
This model works with the assumption of stationarity which means that they must
have constant variance and mean.
Moving average model
It is based on the notion of a perpetual mean and the instance has a linear style in the
mean during the learning.
This applicability of this model is in technical inquiry that has smooth out value
action by sifting from the price fluctuations. It has one of the drawbacks that it is centered on
past values (Li, 2012).
Vector auto regression
In the innovation are assumed to be zero mean random variables with having constant
variance, correlated with one another and have probability functions.
This model is used in capturing the linear interdependencies between the numerous
multiple time series.

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Nonlinear autoregressive exogenous model
This model is used in predicting the hysteretic behaviour of passive control system.
The assumption connected with this model is that it assumed that the variables are
correlated with each other.
Distributed lag model
This model concept is taken from the demand analysis, this model is used in the
analysis of consumer demand problem. The wide application of this model is in
econometrics.
In this model, OLS yield is the consistent estimators of the β1, β2, β3….. βr. In a big
sample, β1 is considered normal (Gasparrini, 2010).
Autoregressive fractionally integrated moving average
The use of this model is in predicting upcoming cost in the time series of data Xt.
Box-Jenkins technique is used to estimate the ARMA model.
This model is used on the assumption of time series, this series includes normally
distributed variance and it is assumed that the mean and variance are remaining content for a
longer period.
Conclusion
Therefore, it can be summarized that the time series model is very effective in
estimating the future value on a fixed time basis. To predict the value various model are used
which are stated above.

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Bibliography
Box, G. E. (2015). Time series analysis: forecasting and control. . John Wiley & Sons.
Gasparrini, A. A. (2010). Distributed lag non‐linear models. Statistics in medicine. Statistics
in medicine, 2224-2234.
Lee, Y. S. (2011). Forecasting time series using a methodology based on autoregressive
integrated moving average and genetic programming. Knowledge-Based Systems, 66-
72.
Li, Q. G. (2012). Application of an autoregressive integrated moving average model for
predicting the incidence of hemorrhagic fever with renal syndrome. The American
journal of tropical medicine and hygiene, 364-370.
Pham, H. T. (2010). A hybrid of nonlinear autoregressive model with exogenous input and
autoregressive moving average model for long-term machine state forecasting. .
Expert Systems with Applications., 3310-3317.
Rao, T. S. (2012). An introduction to bispectral analysis and bilinear time series models.
Springer Science & Business Media.