Statistical Time Series Models: AR, MA and ARIMA Explained

Learn Statistical Time Series Models in detail, including Auto Regression (AR), Moving Average (MA), ARIMA, formulas, examples, differences, FAQs and MCQs.

Statistical Time Series Models

Auto Regression (AR), Moving Average (MA) and ARIMA Model

Statistical Time Series Models are important techniques used to analyze the relationship between observations collected over time and to forecast future values. These models use the historical behavior of a time series, its previous observations, and error terms to understand patterns and make predictions.

The three important models covered in this topic are:

Auto Regression (AR) – Basic Concept
Moving Average (MA) – Basic Concept
ARIMA Model – Introduction

Introduction to Statistical Time Series Models

What is a Statistical Time Series Model?

A Statistical Time Series Model is a mathematical or statistical model used to describe the behavior and relationships present in observations recorded sequentially over time.

For example, consider the monthly sales of a company:

Month	Sales
January	100
February	110
March	115
April	125
May	130
June	140

The current month’s sales may be related to sales in previous months. Statistical time series models help identify such relationships.

They can be used to:

Understand historical patterns.
Identify dependencies between observations.
Analyze random fluctuations.
Study trends and stationarity.
Model relationships between current and past observations.
Forecast future values.

Need for Statistical Time Series Models

Basic forecasting techniques such as Naïve Forecasting and Simple Moving Average are useful for simple situations. However, many real-world time series contain more complex relationships.

Statistical models can help to:

Analyze relationships between past and current observations.
Identify autocorrelation.
Model the effect of previous observations.
Model the effect of previous forecast errors.
Handle non-stationary data through differencing.
Develop mathematical forecasting models.
Improve understanding of the structure of time series data.
Generate forecasts for future periods.

Major Statistical Time Series Models

The important models introduced in this topic are:

Auto Regression (AR)

An AR model explains the current value of a time series using its previous values.

Moving Average (MA)

An MA model explains the current value using current and previous error terms.

ARIMA

ARIMA combines:

Auto Regression
Differencing
Moving Average

The general notation is:

$A R I M A (p, d, q)$

Auto Regression (AR) – Basic Concept

What is Auto Regression?

Auto Regression (AR) is a statistical time series model in which the current value of a variable depends on its own previous values.

The word “Auto” means self, while “Regression” refers to modeling a variable based on other values.

Therefore:

Auto Regression models the current value of a time series using one or more of its previous values.

Basic Idea of AR

Consider temperature data.

Suppose:

Yesterday’s temperature = 30°C
Today’s temperature = 31°C

Today’s temperature may be related to yesterday’s temperature.

Similarly:

Current sales may depend on previous sales.
Current demand may depend on previous demand.
Current electricity consumption may depend on previous consumption.
Current production may depend on previous production.

An AR model attempts to mathematically represent these relationships.

AR(1) Model

The simplest Auto Regression model is AR(1).

The number 1 indicates that one lagged observation is used.

The general formula is:

$Y_{t} = c + ϕ_{1} Y_{t - 1} + ϵ_{t}$

Where:

$Y_{t}$ = Current value
$c$ = Constant
$ϕ_{1}$ = AR coefficient
$Y_{t - 1}$ = Previous observation
$ϵ_{t}$ = Random error term

Example of AR(1)

Suppose the model is:

$Y_{t} = 5 + 0.8 Y_{t - 1} + ϵ_{t}$

Suppose:

$Y_{t - 1} = 100$

and assume:

$ϵ_{t} = 0$

Then:

$Y_{t} = 5 + (0.8) (100)$

$Y_{t} = 5 + 80$

$Y_{t} = 85$

Therefore, the estimated current value is: $85$

AR(2) Model

If the current value depends on the previous two observations, the model is called AR(2).

The formula is:

$Y_{t} = c + ϕ_{1} Y_{t - 1} + ϕ_{2} Y_{t - 2} + ϵ_{t}$

Where:

$Y_{t - 1}$ = Previous observation
$Y_{t - 2}$ = Observation from two periods earlier

Example

Suppose:

$Y_{t} = 10 + 0.5 Y_{t - 1} + 0.3 Y_{t - 2} + ϵ_{t}$

Given:

$Y_{t - 1} = 100$ $Y_{t - 2} = 80$

and:

$ϵ_{t} = 0$

Then:

$Y_{t} = 10 + (0.5) (100) + (0.3) (80)$

$= 10 + 50 + 24$

$= 84$

Therefore:

$Y^{t} = 84$

AR(p) Model

The general Auto Regression model is called AR(p).

Here, $p$ represents the number of lagged observations included in the model.

The general formula is:

$Y_{t} = c + ϕ_{1} Y_{t - 1} + ϕ_{2} Y_{t - 2} + \dots + ϕ_{p} Y_{t - p} + ϵ_{t}$

Where:

$p$ = Number of lags
$ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}$ = AR coefficients
$ϵ_{t}$ = Error term

Understanding Lag

A lag represents a previous observation in a time series.

Consider:

Time	Value
$t - 3$	100
$t - 2$	110
$t - 1$	120
$t$	130

Therefore:

$Y_{t - 1} = 120$

$Y_{t - 2} = 110$

$Y_{t - 3} = 100$

If a model uses the previous three observations, it is an AR(3) model.

Advantages of AR Models

Easy to understand conceptually.
Uses historical observations directly.
Captures dependency between current and previous values.
Useful for forecasting.
Can model autocorrelation.
Can be implemented using statistical software.
Useful for stationary time series.

Limitations of AR Models

Appropriate lag selection is important.
Basic AR models generally require stationarity.
A model with too many lags may become unnecessarily complex.
Strong seasonality may require additional modeling.
Sudden structural changes may reduce model performance.
Incorrect model specification can produce poor forecasts.

Moving Average (MA) – Basic Concept

What is an MA Model?

In statistical time series analysis, a Moving Average (MA) model is a model in which the current value depends on the current and previous error terms.

An MA model represents a time series using current and past random shocks or error terms.

Important Difference: SMA vs Statistical MA

A very important point is that Simple Moving Average forecasting and the Moving Average statistical model are not the same.

Simple Moving Average

Uses the average of previous actual observations.

Statistical MA Model

Uses current and previous error terms.

Therefore:

Simple Moving Average ≠ Statistical Moving Average Model

This distinction is particularly important in examinations.

MA(1) Model

The simplest Moving Average model is MA(1).

The general formula is:

$Y_{t} = μ + ϵ_{t} + θ_{1} ϵ_{t - 1}$

Where:

$Y_{t}$ = Current observation
$μ$ = Mean of the series
$ϵ_{t}$ = Current error term
$ϵ_{t - 1}$ = Previous error term
$θ_{1}$ = MA coefficient

Example of MA(1)

Suppose:

$Y_{t} = 100 + ϵ_{t} + 0.5 ϵ_{t - 1}$

Given:

$ϵ_{t} = 2$

and:

$ϵ_{t - 1} = 4$

Then:

$Y_{t} = 100 + 2 + (0.5) (4)$

$Y_{t} = 100 + 2 + 2$

$Y_{t} = 104$

Therefore:

$Y^{t} = 104$

MA(2) Model

If the current observation depends on the previous two error terms, it is called an MA(2) model.

The formula is:

$Y_{t} = μ + ϵ_{t} + θ_{1} ϵ_{t - 1} + θ_{2} ϵ_{t - 2}$

Here:

$ϵ_{t - 1}$ = Previous error
$ϵ_{t - 2}$ = Error from two periods earlier

MA(q) Model

The general Moving Average model is called MA(q).

$Y_{t} = μ + ϵ_{t} + θ_{1} ϵ_{t - 1} + θ_{2} ϵ_{t - 2} + \dots + θ_{q} ϵ_{t - q}$

Where:

$q$ = Number of lagged error terms
$θ_{1}, θ_{2}, \dots, θ_{q}$ = MA coefficients
$ϵ_{t}$ = Error term

Basic Idea of MA Model

The basic idea can be summarized as:

Current Value → Current Error + Previous Errors

Whereas:

AR → Previous Values

This is the key conceptual difference between AR and MA.

Advantages of MA Models

Models the effect of random shocks.
Useful for short-term dependencies.
Helps model error structures.
Forms an important part of ARIMA models.
Useful in statistical time series analysis.
Can capture patterns that cannot be adequately represented by an AR model alone.

Limitations of MA Models

Error terms are not directly observed and must be estimated.
Choosing an appropriate $q$ is important.
Model estimation can be more complex than simple forecasting methods.
Basic MA models are generally designed for stationary series.
Incorrect model specification can reduce forecasting performance.

Difference Between AR and MA

Feature	AR Model	MA Model
Full Form	Auto Regression	Moving Average
Main idea	Uses previous observations	Uses previous error terms
Parameter	$p$	$q$
Main coefficient	$ϕ$	$θ$
Example	AR(1)	MA(1)
Main dependency	Past values	Past errors
Main use	Modeling value dependency	Modeling shock/error dependency

Easy way to remember:

AR → Past Values

MA → Past Errors

ARIMA Model – Introduction

What is ARIMA?

ARIMA stands for:

AutoRegressive Integrated Moving Average

ARIMA is one of the most important statistical models used for time series forecasting.

It combines three major ideas:

AR – AutoRegressive
I – Integrated
MA – Moving Average

The general notation is:

$A R I M A (p, d, q)$

Meaning of p, d and q

The three parameters of ARIMA are:

$p$ – AutoRegressive Order

It represents the number of lagged observations used by the AR component.

$d$ – Degree of Differencing

It represents the number of times differencing is applied to make the series stationary.

$q$ – Moving Average Order

It represents the number of lagged error terms used by the MA component.

Therefore:

ARIMA(p,d,q) = AR order + Differencing order + MA order

Components of ARIMA

AR – AutoRegressive Component

The AR component uses previous observations.

$Y_{t} \leftarrow Y_{t - 1}, Y_{t - 2}, \dots$

I – Integrated Component

The Integrated component refers to differencing.

Differencing is used to reduce or remove non-stationarity from the series.

MA – Moving Average Component

The MA component uses previous error terms.

$Y_{t} \leftarrow ϵ_{t - 1}, ϵ_{t - 2}, \dots$

What is Differencing?

Differencing is the process of calculating the difference between consecutive observations.

The first difference is:

$Y_{t'} = Y_{t} - Y_{t - 1}$

Example

Consider:

Year	Sales
2022	100
2023	120
2024	135
2025	150

First differences:

$120 - 100 = 20$

$135 - 120 = 15$

$150 - 135 = 15$

Therefore, the differenced series is: $20, 15, 15$

Purpose of Differencing

The main purpose of differencing is:

To help transform a non-stationary time series into a stationary time series.

For example, if a series has a strong upward trend, first-order differencing may reduce that trend.

What is Stationarity?

A stationary time series is a series whose important statistical properties remain relatively stable over time.

These properties commonly include:

Mean
Variance
Autocovariance structure

In simple terms:

A stationary series has a statistical behavior that does not change substantially over time.

Stationarity is an important concept in ARIMA modeling.

Non-Stationary Time Series

A time series may be non-stationary when it contains:

Strong trend
Changing mean
Changing variance
Certain forms of persistent dependence
Seasonal behavior that has not been modeled appropriately

Differencing is one of the common methods used to address non-stationarity.

ARIMA(1,0,0)

Consider:

$A R I M A (1, 0, 0)$

Here:

$p = 1$
$d = 0$
$q = 0$

This means:

One AR lag
No differencing
No MA component

It is closely related to an AR(1) model.

ARIMA(0,0,1)

Consider:

$A R I M A (0, 0, 1)$

Here:

$p = 0$
$d = 0$
$q = 1$

This corresponds to an MA(1) model.

ARIMA(1,1,1)

Consider:

$A R I M A (1, 1, 1)$

This means:

$p = 1$ → One AR lag
$d = 1$ → First-order differencing
$q = 1$ → One MA error lag

Thus:

ARIMA(1,1,1) uses first-order differencing and combines one AR term with one MA term.

General Steps for Building an ARIMA Model

A typical ARIMA modeling process includes the following steps.

Step 1: Collect Time Series Data

Obtain historical observations in chronological order.

Examples:

Monthly sales
Daily temperature
Quarterly revenue
Annual enrollment

Step 2: Visualize the Data

Plot the time series to identify:

Trend
Seasonality
Fluctuations
Outliers
Structural changes

Step 3: Check Stationarity

Determine whether the time series is stationary.

This can involve:

Visual inspection
Statistical tests
ACF/PACF analysis

Step 4: Apply Differencing

If the series is non-stationary, appropriate differencing may be applied.

Step 5: Select $p, d, q$

Determine appropriate ARIMA parameters:

$(p, d, q)$

Model identification may use:

ACF
PACF
Information criteria
Domain knowledge
Model comparison

Step 6: Fit the Model

Estimate the ARIMA model parameters using the available historical data.

Step 7: Evaluate the Model

Check:

Residuals
Forecast errors
Model fit
Information criteria
Residual autocorrelation

Step 8: Forecast Future Values

Once an appropriate model has been selected and validated, it can be used to generate future forecasts.

Advantages of ARIMA

It is a well-established statistical forecasting framework.
It can model autocorrelation.
Differencing can help handle non-stationary series.
It includes both AR and MA components.
It is useful for many time series forecasting problems.
It provides a systematic modeling framework.
It can produce short-term forecasts.
It is supported by many statistical and programming tools.

Limitations of ARIMA

Statistical knowledge is required.
Selection of $p, d, q$ can be challenging.
Data preprocessing may be necessary.
Basic ARIMA does not explicitly model seasonality.
Strong nonlinear relationships may not be captured effectively.
Outliers can affect model estimation.
Structural changes can reduce forecasting performance.
Forecast accuracy depends on the quality and characteristics of the historical data.

Applications of ARIMA

ARIMA can be applied in many areas.

Business

Sales forecasting
Demand forecasting
Revenue forecasting
Inventory planning

Finance and Economics

Economic indicators
Financial time series
Inflation-related analysis
Business forecasting

Education

Student enrollment forecasting
Attendance trend analysis
Academic performance trends
Course demand forecasting

Healthcare

Patient demand forecasting
Hospital resource planning
Healthcare service demand

Energy

Electricity demand
Energy consumption
Power-load forecasting

Government

Population-related forecasting
Economic planning
Resource demand estimation

AR vs MA vs ARIMA

Feature	AR	MA	ARIMA
Full Form	AutoRegressive	Moving Average	AutoRegressive Integrated Moving Average
Main concept	Previous values	Previous errors	AR + Differencing + MA
Parameters	$p$	$q$	$p, d, q$
Differencing	Not a component	Not a component	Included
Main use	Past-value dependency	Error dependency	Time series forecasting
Example	AR(1)	MA(1)	ARIMA(1,1,1)

Simple Conceptual Example

Suppose a company’s monthly sales are being forecast.

AR Model

Current sales depend on previous sales:

$S a l e s_{t} = f (S a l e s_{t - 1}, S a l e s_{t - 2})$

MA Model

Current sales depend on previous forecast errors:

$S a l e s_{t} = f (ϵ_{t - 1}, ϵ_{t - 2})$

ARIMA Model

If the sales series has a trend:

Differencing → AR component + MA component → Forecast

Important Formula Summary

AR(p)

$Y^{t} = c + ϕ^{1} Y^{t - 1} + ϕ^{2} Y^{t - 2} + \dots + ϕ^{p} Y^{t - p} + ϵ^{t}$

MA(q)

$Y^{t} = μ + ϵ^{t} + θ^{1} ϵ^{t - 1} + \dots + θ^{q} ϵ^{t - q}$

First Difference

$Y^{t'} = Y^{t} - Y^{t - 1}$

ARIMA

$A R I M A (p, d, q)$

Frequently Asked Questions (FAQ)

Q1. What is a Statistical Time Series Model?

Answer: A Statistical Time Series Model is a mathematical/statistical model used to analyze relationships and dependencies among observations recorded over time and to forecast future values.

Q2. What is Auto Regression?

Answer: Auto Regression is a model in which the current value of a time series depends on one or more of its previous values.

Q3. What does AR(1) mean?

Answer: AR(1) means that the current observation is modeled using one previous observation or one lag.

Q4. What does $p$ represent in AR(p)?

Answer: $p$ represents the number of lagged observations included in the AR model.

Q5. What is a lag?

Answer: A lag represents a previous observation in a time series. For example, $Y_{t - 1}$ is the first lag.

Q6. What is a Moving Average model?

Answer: A statistical Moving Average model represents the current observation using current and previous error terms.

Q7. What does MA(1) mean?

Answer: MA(1) means that the model uses one previous error term.

Q8. What does $q$ represent in MA(q)?

Answer: $q$ represents the number of lagged error terms used in the MA model.

Q9. What is the main difference between AR and MA?

Answer: AR uses previous observations, whereas MA uses previous error terms.

Q10. Is Simple Moving Average the same as the statistical MA model?

Answer: No. Simple Moving Average uses averages of actual observations, while the statistical MA model uses error terms.

Q11. What is ARIMA?

Answer: ARIMA stands for AutoRegressive Integrated Moving Average. It combines AR, differencing, and MA components.

Q12. What is the notation of ARIMA?

Answer:

$A R I M A (p, d, q)$

Q13. What does $p$ represent in ARIMA?

Answer: $p$ represents the order of the AutoRegressive component.

Q14. What does $d$ represent in ARIMA?

Answer: $d$ represents the degree or order of differencing.

Q15. What does $q$ represent in ARIMA?

Answer: $q$ represents the order of the Moving Average component.

Q16. What is differencing?

Answer: Differencing is the process of subtracting a previous observation from the current observation.

$Y_{t'} = Y_{t} - Y_{t - 1}$

Q17. Why is differencing used?

Answer: Differencing is used to help transform a non-stationary time series into a stationary series.

Q18. What does ARIMA(1,1,1) mean?

Answer: It means one AR term, first-order differencing, and one MA term.

Q19. What is stationarity?

Answer: Stationarity means that important statistical properties of a time series remain relatively stable over time.

Q20. What is ARIMA mainly used for?

Answer: ARIMA is mainly used for time series modeling and forecasting.

Q21. Can basic ARIMA handle seasonality directly?

Answer: Basic ARIMA does not explicitly model seasonal patterns. For strong seasonality, SARIMA (Seasonal ARIMA) can be used.

Q22. What is ARIMA(1,0,0)?

Answer: It is essentially an AR(1)-type model with no differencing and no MA component.

Q23. What is ARIMA(0,0,1)?

Answer: It is essentially an MA(1)-type model with no differencing and no AR component.

Q24. Why is parameter selection important in ARIMA?

Answer: Appropriate selection of $p, d, q$ helps the model represent the time series structure accurately and can improve forecasting performance.

Q25. What should be checked before developing an ARIMA model?

Answer: The analyst should examine data quality, trend, seasonality, stationarity, autocorrelation, outliers, and other relevant characteristics.

MCQs – Statistical Time Series Models

1. AR stands for:

A) Average Regression
B) Auto Regression
C) Automatic Ratio
D) Average Reduction

Answer: B) Auto Regression

2. An AR model primarily uses:

A) Future values
B) Previous observations
C) Random numbers
D) Maximum observations

Answer: B) Previous observations

3. AR(1) uses:

A) One variable
B) One lag
C) One future value
D) One moving average

Answer: B) One lag

4. In AR(p), $p$ represents:

A) Number of variables
B) Number of lagged observations
C) Number of errors
D) Number of forecasts

Answer: B) Number of lagged observations

5. Which is the general form of an AR(1) model?

A) $Y_{t} = c + ϕ_{1} Y_{t - 1} + ϵ_{t}$

B) $Y_{t} = c + ϵ_{t}$

C) $Y_{t} = Y_{t} + 1$

D) $Y_{t} = c + θ ϵ_{t}$

Answer: A)

6. MA stands for:

A) Mathematical Average
B) Moving Average
C) Multiple Analysis
D) Mean Autocorrelation

Answer: B) Moving Average

7. A statistical MA model primarily uses:

A) Previous actual observations
B) Previous error terms
C) Future observations
D) Maximum values

Answer: B) Previous error terms

8. MA(1) contains:

A) Zero lagged errors
B) One lagged error
C) Two lagged errors
D) Three lagged errors

Answer: B) One lagged error

9. In MA(q), $q$ represents:

A) AR order
B) Differencing order
C) Number of lagged error terms
D) Number of variables

Answer: C) Number of lagged error terms

10. What is the main difference between AR and MA?

A) AR uses past values; MA uses past errors
B) AR uses future values; MA uses past values
C) They are exactly identical
D) AR uses only averages

Answer: A) AR uses past values; MA uses past errors

11. ARIMA stands for:

A) Auto Regression Integrated Moving Average
B) AutoRegressive Integrated Moving Average
C) Average Regression Integrated Model Analysis
D) Automatic Regression Integrated Model

Answer: B) AutoRegressive Integrated Moving Average

12. The standard notation for ARIMA is:

A) ARIMA(p,q)
B) ARIMA(p,d,q)
C) ARIMA(d,q)
D) ARIMA(p,d)

Answer: B) ARIMA(p,d,q)

13. In ARIMA, $p$ represents:

A) Differencing order
B) AR order
C) MA order
D) Error value

Answer: B) AR order

14. In ARIMA, $d$ represents:

A) AR order
B) MA order
C) Differencing order
D) Number of observations

Answer: C) Differencing order

15. In ARIMA, $q$ represents:

A) AR order
B) MA order
C) Differencing order
D) Mean

Answer: B) MA order

16. The primary purpose of differencing is to:

A) Increase the dataset size
B) Help achieve stationarity
C) Delete observations
D) Increase error

Answer: B) Help achieve stationarity

17. First-order differencing is represented by:

A) $Y_{t} + Y_{t - 1}$

B) $Y_{t} - Y_{t - 1}$

C) $Y_{t} / Y_{t - 1}$

D) $Y_{t} Y_{t - 1}$

Answer: B) $Y_{t} - Y_{t - 1}$

18. In ARIMA(1,1,1), $d = 1$ means:

A) No differencing
B) First-order differencing
C) Second-order differencing
D) One MA term

Answer: B) First-order differencing

19. ARIMA(1,0,0) is closely related to:

A) MA(1)
B) AR(1)
C) MA(2)
D) ARIMA(0,1,1)

Answer: B) AR(1)

20. ARIMA(0,0,1) is closely related to:

A) AR(1)
B) MA(1)
C) AR(2)
D) ARIMA(1,1,1)

Answer: B) MA(1)

21. ARIMA is mainly used for:

A) Image classification
B) Time series forecasting
C) Database normalization
D) Web development

Answer: B) Time series forecasting

22. A stationary time series generally has:

A) Stable statistical properties
B) Increasing file size
C) Constant number of columns
D) Constant computer memory

Answer: A) Stable statistical properties

23. Which is NOT a component of ARIMA?

A) AR
B) Integrated
C) MA
D) Classification

Answer: D) Classification

24. The “I” in ARIMA is associated with:

A) Integration/Differencing
B) Information
C) Independence
D) Input

Answer: A) Integration/Differencing

25. Which model is commonly considered for strong seasonal patterns?

A) SARIMA
B) AR(1) only
C) MA(1) only
D) Naïve model only

Answer: A) SARIMA

26. In an AR model, $ϕ$ generally represents:

A) AR coefficient
B) Error term
C) Mean
D) Differencing order

Answer: A) AR coefficient

27. In an MA model, $θ$ generally represents:

A) AR coefficient
B) MA coefficient
C) Mean
D) Time period

Answer: B) MA coefficient

28. In ARIMA(2,1,1), the AR order is:

A) 1
B) 2
C) 3
D) 4

Answer: B) 2

29. In ARIMA(2,1,1), the differencing order is:

A) 0
B) 1
C) 2
D) 3

Answer: B) 1

30. In ARIMA(2,1,1), the MA order is:

A) 0
B) 1
C) 2
D) 3

Answer: B) 1

Important Examination Points

Auto Regression (AR)

AR → Previous Values

$Y_{t} = c + ϕ_{1} Y_{t - 1} + \dots + ϕ_{p} Y_{t - p} + ϵ_{t}$

Moving Average (MA)

MA → Previous Errors

$Y_{t} = μ + ϵ_{t} + θ_{1} ϵ_{t - 1} + \dots + θ_{q} ϵ_{t - q}$

ARIMA

ARIMA → AR + Differencing + MA

$A R I M A (p, d, q)$

Parameters

p → AR order
d → Differencing order
q → MA order

Most Important Difference

AR uses past observations, whereas MA uses past error terms.

Easy Memory Trick

AR → Past Values
I → Difference
MA → Past Errors

Therefore:

ARIMA = Past Values + Differencing + Past Errors

Statistical Time Series Models

Statistical Time Series Models

Auto Regression (AR), Moving Average (MA) and ARIMA Model

Introduction to Statistical Time Series Models

What is a Statistical Time Series Model?

Need for Statistical Time Series Models

Major Statistical Time Series Models

Auto Regression (AR) – Basic Concept

What is Auto Regression?

Basic Idea of AR

AR(1) Model

Example of AR(1)

AR(2) Model

Example

AR(p) Model

Understanding Lag

Advantages of AR Models

Limitations of AR Models

Moving Average (MA) – Basic Concept

What is an MA Model?

Important Difference: SMA vs Statistical MA

MA(1) Model

Example of MA(1)

MA(2) Model

MA(q) Model

Basic Idea of MA Model

Advantages of MA Models

Limitations of MA Models

Difference Between AR and MA

Easy way to remember:

ARIMA Model – Introduction

What is ARIMA?

Meaning of p, d and q

Components of ARIMA

AR – AutoRegressive Component

What is Differencing?

Purpose of Differencing

What is Stationarity?

Non-Stationary Time Series

ARIMA(1,0,0)

ARIMA(0,0,1)

ARIMA(1,1,1)

General Steps for Building an ARIMA Model

Step 7: Evaluate the Model

Advantages of ARIMA

Limitations of ARIMA

Applications of ARIMA

AR vs MA vs ARIMA

Simple Conceptual Example

Important Formula Summary

Frequently Asked Questions (FAQ)

Q7. What does MA(1) mean?

Q14. What does d represent in ARIMA?

Q22. What is ARIMA(1,0,0)?

MCQs – Statistical Time Series Models

6. MA stands for:

11. ARIMA stands for:

16. The primary purpose of differencing is to:

23. Which is NOT a component of ARIMA?

30. In ARIMA(2,1,1), the MA order is:

Important Examination Points

By Mr. Santosh Nalawade

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Q14. What does $d$ represent in ARIMA?