What is Volatility?

the investor gauge of fear”

A measurement of risk.
It is a random variable that follows a stochastic process.
It is the variability of the asset price changes over a particular period of time.
The magnitude of fluctuations in the return of an asset is called its volatility.
It is usually measured by the standard deviation of returns

Volatility Definition

Volatility refers to the fluctuations observed in some phenomenon over time. Within econom
In financial economics, volatility is often defined as the standard deviation (or “sigma”) o

Broad Classification of Volatility Estimation

qTraditional Volatility Estimators — Based on either squared returns or standard deviation

Ø Extreme Value Volatility Estimators —These also incorporate high and low prices observe
ØConditional Volatility Models — These models (ARCH/GARCH type models) take into accou
ØImplied Volatility — In case of options, most of the parameters relevant for their valuation

Characteristics of Volatility

—Persistence: This refers to the persistence of shocks to the variance. Shocks persist for a l
—Mean reverting: There is normal level of volatility to which volatility will eventually return.
—Models generally
—Linear in parameters
—Structural models (& some time series models (univariate)) unable to explain
ØLeptokurtosis – Fat tails and excessive peakedness
ØVolatility clustering or volatility pooling – volatility appearing in bunches
—Large returns follow large returns (say in big uptrends)
—Small returns follow small returns (normal mkt condns)
ØLeverage effects – Volatility that rises more during a fall than volatility that is present dur

—A non-linear model of the error terms u_t

—Should be non-linear in the mean i.e g(.)
—Should be non-linear in the variance i.e σ^2(.)
—General tests will not be able to detect non-linearity
—BDS test to detect non-linearity
—In general
—CLRM (Classical Linear Regression Model) à Linear in mean & Linear in variance
—But generally time series will be non-linear in Variance
—GARCH models – Linear in mean & non-linear in variance
—Bicorrelation models – Non-Linear in mean & linear in variance
—Hybrid threshold model with GARCH – Non- linear in mean & non-linear in variance

—Recollect that normal linear reg. models

—Assumption is Variance of error is constant ( Homoscedastic)
—If it is not constant then it is heteroscedastic and
—The resultant regression model would not be correct
—Time series – unlikely the errors would have constant variance
—Further time series would have “VOLATILITY CLUSTERING”

Characteristics of Volatility
—Innovations may have asymmetric impact on volatility: Positive and negative shocks do no
—Exogeneous variables may influence volatility: Deterministic events may also have an imp
iod of time.
s volatility.

over time. Within economics, it is used slightly more formally to describe, without a specific implied metric, the
deviation (or “sigma”) of the random Wiener-driven component

ns or standard deviation of returns over a period.

gh and low prices observed unlike traditional estimators which are based on closing prices of the asset.
e models) take into account the time-varying nature of volatility. There have been quite a few extensions of the
elevant for their valuation can be directly observed or estimated, except volatility. Volatility is, therefore, backed

nce. Shocks persist for a long time and they explain the large fluctuations observed in stock market.
ity will eventually return. Very long run forecast of volatility should converge to the same normal level, no matt

le to explain

atility that is present during an uptrend

ear in variance

-linear in variance
nd negative shocks do not have same impact on volatility called leverage effect
nts may also have an impact on the volatility. Such things as announcements, macroeconomic news release eff
pecific implied metric, the variability of the random (unforeseen) component of a time series.

ces of the asset.

e a few extensions of the basic conditional volatility models to incorporate ‘observed’ characteristics of asset/ s
tility is, therefore, backed out from the observed option values and is used as volatility forecast.

stock market.
me normal level, no matter when they are made
conomic news release effect can influence the volatility.
characteristics of asset/ stock returns.
y forecast.
•Under standard assumptions, the conditional mean is non-constant, conditional variance i
• However, in some situations, the basic assumption of constant conditional variance may
• A time series is said to be heteroscedastic if its variance changes over time otherwise it i

Testing for ARCH Disturbances in Linear Regression Model

—ARCH disturbances can be tested by any of the following methods (preferably all).
—Method 1. If the normality assumption is used to describe the conditional error distribution
—Therefore testing for non-normality of the residuals can be considered.
—Method 2. The autocorrelation structure of residuals and the squared residuals can be exa
An indication of ARCH is that the residuals will be uncorrelated but the squared residuals w

ARCH Models

—ARCH (p): Engle (1982) introduced the ARCH (q) model assuming that the conditional vari
—εt is an independent and identically distributed sequence with zero mean and unit varianc
—The application of the linear ARCH model has problems concerning the need of a long lag

Testing for ARCH effects

—Test for ARCH effects

—Run any linear equation and save the residuals
—Take residuals à square it and regress on own lags to test for ARCH
—Null and alternate hypotheses are
—ARCH & GARCH exercise for given data set –
—Jan 2020 to Dec 2020

GARCH models

—GARCH (p,q) Models, Bollerslev (1986): The extension of the ARCH model to the GARCH m

—Unconditional variance is given by

Problems with GARCH models

1)Non-negativity conditions may be violated.

üArtificial constraints induced to avoid this non – negativity
2)GARCH models cannot account for LEVERAGE effects but account for vol. clustering and l
3)Volatility i.e both positive and negative volatility treated same way in GARCH
4)But Leverage effects –
2)Downsides – Volatility is more
3)Upsides – Volatility is lesser
4)Hence, ASYMMETRIC GARCH is used to account for this
GJR Model (Glosten Jagannathan Runkle)

GARCH models

EGARCH model: An alternative for asymmetric volatilities is the Exponential GARCH, or EGA
üLog ensures that values are positivity overcoming non-negativity constraint
üAsymmetries also allowed under EGARCH
Multivariate GARCH models
ØGARCH models are univariate in nature i.e single variable
ØBut if many variables are there Multivariate GARCH model is used
ØDiagonal VECH & BEKK models are widely used to determine volatility spill ove

Exponentially Weighted Moving Average model (EWMA)

ØEWMA – another model to model volatility

ØGives more weightage to latest observations over other observations
nt, conditional variance is constant and the conditional distribution is normal.
onditional variance may not be true. This behavior is normally referred as heteroscedasticity.
s over time otherwise it is called homoscedastic

s (preferably all).
nditional error distribution, then a property of ARCH is that the unconditional error distribution will be non-norma

ared residuals can be examined.

t the squared residuals will show autocorrelation

g that the conditional variance depends on past volatility measured as a linear function of past squared values o
ro mean and unit variance.
ng the need of a long lag length q and the non-negativity conditions imposed in parameters.

H model to the GARCH model is similar to the extension of the standard time series AR model to the general AR

nt for vol. clustering and leptokurtosis

way in GARCH

xponential GARCH, or EGARCH model, introduced by Nelson (1991). The EGARCH (p, q) model is defined by
constraint

odel is used
ermine volatility spill over between various assets/markets
 ibution will be non-normal.

of past squared values of the process εt i.e

R model to the general ARMA model.

model is defined by
vA cointegrating relationship is a long term or equilibrium phenomenon.
vPossible it may deviate from their long term relationship in the short run but wo
Spot and Futures prices of a given commodity – oil, gold, etc.
Equity prices and dividends etc.
No cointegration – they would wander apart from their mean
Spot and futures are expected to be cointegrated – because they are prices for t

Non stationary items must be differenced “d” times to make it stationary – Integ
Yt ~ I (d)
∆d = Yt ~ I (0)
∆ is the difference operator and applying d times make it a stationary process
I (2) – contains two unit roots and would require differencing two times to make
I(1) – single unit root and single differencing would make it stationary
Most financial time series are I(1)

ØIn general, if there is a series of I(1) a linear combination of I(1) is also I(1).
üDesirable to obtain the residuals that are I(0)
ØCointegration – “ A set of variables if the linear combination of these is STATIO
ØCointegrating relationship is a long-term equilibrium phenomenon
ØThough it may deviate in the short run their association would return to equilib
ØMany time series are non-stationary but move together over time
ØBut in some special case it could be I(0)
ØThe logic mostly comes from economic theory –
ØSpot and Futures
Purchasing power parity , etc

ØMultiple cointegrating relationships

ØIf only y_t & x_t à 2 variables there can be only1 linear combination or one coin
ØIf there are k variables then the max cointegrating relationship
r≤ k -1
Say k =2, then k-1 = 2-1 =1 i.e r =1
i.e max of one cointegrating relationship

vError Correction or Equilibrium correction models are used to model cointegrati

Non-stationary variables I(1) cointegrated then VECM can be used to investigat
Z is Error Correction term. OLS residuals from the following
•Speed of adjustment is phi – Measures the speed at which Y returns to equilibr

vCointegration example between Niftybees ETF and Nifty

vTime period (1 Jan 2020 to 31 Dec 2020)
vADF tests – Unit root for Niftybees and Nifty
vJohansen’s co-integration test
vVECM – application Eviews
vGranger Causality test
 Variable
Interpret
Pair Order in
ation
EViews
How spot
A) QAU
QAU, then gold
ETF and
Spot affects
Spot Gold
QAU ETF
How
B) QAU
futures
ETF and QAU, then
prices
Gold Futures
affect
Futures
QAU ETF
How
futures
C) Spot
Spot, then prices
Gold and
Futures affect
Futures
spot
prices
um phenomenon.
hip in the short run but would return to their association in the long run

ause they are prices for the same asset

make it stationary – Integrated of order “d”. Written as

it a stationary process
encing two times to make it stationary
ke it stationary

ion of I(1) is also I(1).

nation of these is STATIONARY”

phenomenon
on would return to equilibrium in the long run
er over time

r combination or one cointegrating relationship “ r”

ationship

used to model cointegration

can be used to investigate short run and long term relationship

which Y returns to equilibrium

ØVector Autoregression (VAR) model is an extension of univariate autoregression model to
ØVAR model is a multi-equation system where all the variables are treated as endogenous
ØThere is one equation for each variable as dependent variable. Right-hand side of each eq

ØVAR is a hybrid of univariate time series and simultaneous equation model

ØSimultaneous equation model is

üy1t = Nifty index

üy2t = Volatility Index (VIX)
üu = Stochastic disturbance term
üt= time

VAR uses

ØIn accounting, predict different accounting variables such as sales, earnings, and accruals
ØIn marketing, VAR can be used to evaluate the impact of different factors on consumer be
Ø1) Psychological factors- motivation, perception, learning & attitude & beliefs
Ø2) Social factors –Family, reference groups, roles & status
Ø3) Cultural factors – culture, sub-culture, social class
Ø4) Personal factors – Age,income, lifestyle, occupation
Ø5) Economic factors – personal income, family income, savings, etc
ØIn economics, VAR is used to forecast macroeconomic variables, such as GDP, money sup
ØIn finance, predict spot prices and future prices of securities; foreign exchange rates acro
ØVAR is better than some structural models to forecast.
ØMcNees has shown that forecasting of GDP, unemployment , etc is more accurate using V
ØAnalysis of system response to different shocks/impacts.

Advantages of VAR

ØNeed not specify which variable is endogenous or exogenous – all are endogenous.
ØVAR allows values to depend on its own lags or combination of white noise termsà more fl
ØAR model in fact is a restricted VAR model
ØVAR is better than some structural models to forecast.
ØSims 1980 –proved structural models performed badly in out of sample forecasting
ØMcNees has shown that forecasting of GDP, unemployment , etc. is more accurate using V

Difficulties whiles using VAR

•VAR- “a-theoretical” models- little theoretical info usedà on relationships b/w variables
•Lesser application to policy regulations
•Possible to obtain a spurious relationship also.
•Finding the appropriate lag length
•But there are workarounds to find them
•Many parameters to be estimated , eg 3 variables with 3 lags then parameters to be estim

VAR types
•Primitive VAR
•Includes the contemporaneous variable in the equation

•Include contemporaneous term now i.e current “t” term

•The above represents VARX model or VAR in primitive form

•VAR with exogenous variables or VARX

•Includes the exogenous variable in the equation

•Where X is the exogenous variables or external regressors

 autoregression model to multivariate time series data.
e treated as endogenous but exogenous variables are also included if needed. In that case, it becomes VARX.
ight-hand side of each equation includes lagged values of all dependent variables in the system, no contempor

ion model

es, earnings, and accruals.

nt factors on consumer behavior and forecast its future change. Consumer behavior depends on
ude & beliefs

such as GDP, money supply, and unemployment

eign exchange rates across markets

is more accurate using VAR

ll are endogenous.
hite noise termsà more flexible than univariate AR models
sample forecasting
. is more accurate using VAR

onships b/w variables

en parameters to be estimated will be 30

 ase, it becomes VARX.
he system, no contemporaneous variables.
•Is a statistical test to understand whether one time series is useful in forecasting another
üX Granger causes Y based on the past values of Y and X
üIncludes the past lags of the variables themselves

•Time series variables have to be stationary before performing Granger causality test
ul in forecasting another

anger causality test

—Forecasting – Determine the values that a series is likely to take.
—Why forecasting?
—Useful to make financial decisions
—More accurate the forecast more the utility
—Examples in finance where forecasting is useful
—Forecasting stock returns
—Forecasting the price of a house
—Forecasting the riskiness of portfolio
—Forecasting the volatility of bond returns
—Forecasting number of likely defaults on home loans

Forecasting in Econometrics

vTime series forecasting

—Involves forecasting the future values given its previous values and error terms
vBasic concepts
ü In sample
ü Out sample
vDetermining In sample & Outsample period
ü Left to the reseracher’s discretion
üUsually a 80 to 20 combo used

vSome more concepts used in forecasting

vRolling windows – The length of the period remains same
vRecursive windows – Length of the period changes due to addition
vSample size higher the better but in case it is lesser – rolling windows method is better as

Forecasting accuracy
vForecasts produced for out of sample
vCompared with actual values & diff. is aggregated in various ways
vMain types

types:
—In sample
—Out sample
—One step forecast – Forecasting done only for the next step
Multistep forecast – done for the next ‘s’ steps i.e 1,2,3,… s steps
—Point forecast – predicts single value
—Interval forecasts – gives a range of values
nd error terms

dows method is better as it considers more data points by rolling over method
—Developed by G. P. E Box and G. M Jenkins, “Time Series Analysis: Forecasting and Contro
—Popularly known as Box-Jenkins (BJ) methodology, but technically known as ARIMA method
—A time series Yt is explained by past or lagged values of Y itself and error terms.
—ARIMA models are not derived from any economic theory.

—One can think of a time series as having structure related to:

• long run trend
• Its’ past values
— AutoRegressive (AR) structure
• Its’ past error terms
— Moving Average (MA) structure

ü1) Identification (Model Building)

Ø Plotting time series data
Ø Pattern of the ACF and PACF - Within +/- 1.96/sqrt.n – significant (i.e. +/- 2 Standard erro
ü2) Estimation
Ø By Least squares or
ØMaximum Likelihood

—) Diagnostic Checking
—How is model fit estimated for linear regressions???
—For time series Information criteria (widely used)
—Model validity tests
ØResiduals plot – Correlogram
ØBreusch Godfrey LM test F sig. > .05 – no serial correlation in residuals
ØDurbin Watson statistic for residuals = 2, no serial correlation , d=0 – positive correlation
Test normality of residuals – Jarque Bera test, Shapiro Wilks test, etc
s: Forecasting and Control”, 1976.
y known as ARIMA methodology.
nd error terms.

t (i.e. +/- 2 Standard error)

 residuals
d=0 – positive correlation, d=4 negative correlation
—Components
—Trend —Indicates long term behaviour of the time series
—Seasonal —Regular pattern – visible. Eg: Due to weather
—Can be modelled using X13 – ARIMA SEATS
—Irregular —Random fluctuation
Cyclical —Gradual up and down movements – Due to economic factors
—The autocorrelation measures the correlation between some value of a series (e.g., Y) and
e.g., autocorrelation at a lag of 1 is the pairwise correlation between Yt and Yt-1 for all n-1

For all n-k pairs of observations

—The autocorrelation function (ACF) (Correlogram) maps the ACF’s for various lags.
—Significance @ 95% confidence level – if it falls outside the region the null of no autocorre

—The Partial Auto Correlation measures the direct correlation (for example say) between Y(
ue of a series (e.g., Y) and the value of that series at some lag.
en Yt and Yt-1 for all n-1 pairs.

s for various lags.

n the null of no autocorrelation is rejected -

example say) between Y(t) and Y (t-2) after removing the intermediary effect of Y(t-1)
§Many financial and economic time series exhibit non stationarity:
•Stock market indices, exchange rates, interest rates, commodity prices
§Leads to spurious regressions
§What are the consequences of spurious regressions?
§Properties of Stationary Time Series
ü Constant mean over time (i. e., no trend)
ü Constant variance over time
ü Correlation over time depends on relationship between Yt and Y at some lag k, (Yt-k) doe

ØStationarity, is defined as a quality of a process in which the statistical parameters (mean

ØA weakly stationary process has a constant mean, constant variance and constant autova
ØA truly stationary (or strongly(strictly) stationary) process has all higher-order moments c

vHow to detect stationarity of series?

qVisual inspection of time series
ü Can evaluate the Autocorrelation Function (ACF)
q Can also conduct a formal test:
ü Augmented Dickey-Fuller test (ADF test)
ü Phillip Perron’s test (PP test)

—A non stationary series is said to have a unit root.

—e.g. Yt= ρYt-1+ut
—If ρ=1 then series Yt has a unit root
—Null Hypothesis:
— Ho: Series has unit root and is non stationary
— HA: Series is stationary
—Here, if you reject the null, you have stationarity and thus, don’t need to difference.
 at some lag k, (Yt-k) doesn’t depend on any other variable.

istical parameters (mean and standard deviation) of the process do not change with time. (Challis and Kitney,1

ance and constant autovariance structure

higher-order moments constant including the variance and mean. i.e the distribution of values remain the sam

need to difference.
me. (Challis and Kitney,1991)

of values remain the same as time progresses

—Structural models – Multivariate in nature
—Eg: Linear regression analysis
—Stock returns = F ( GDP, Inflation, Unemployment…)
ØSometimes structural models are not viable
ØAll variables are not observable or measurable- Unemployment rate in India
ØLower frequency –Eg: Stock prices are daily, macro economic variables are quarterly (GDP
ØHow to model ??? Can we model variables with different time frequencies – Stocks (daily
ØStructural models are not suitable for out of sample forecasting
—Univariate models
ØBased only on their past values and also based on their error term
•Time Series models -
ØAlso known as “ a-theoretical” models and not based on underlying theory
ØARMA and ARIMA (p,d,q) – important class of time series models
ØAR- AutoRegressive & MA Moving Average

qRegression analysis example

—The regression model

Significance of error term

Vagueness of theory
ØUnsure of other variables in the above e.g. u_i captures all other terms missing
2) Unavailability of data
ØSome data may be unavailable .
ØEg: Family wealth. Difficult to get the wealth details.
3) Core variables vs peripheral variables
ØOther variables like family size, age, sex, education, etc could also play a role.
ØAlso interaction of these variables could play a role
ØFor sake of brevity and cost purpose – they may be ignored and U I captures those terms
4) Intrinsic randomness
ØEven after capturing all variables some intrinsic randomness may be missing Ui captures
5)Poor proxy variables
ØMilton Firedman Permanent consumption is dependent on permanent income
ØProxy variables used - current consumption vs current income
ØThese proxies may not capture them accurately
6) Principle of parsimony
ØRegression is explained by 2 or 3 variables and other variables – unsure whether they can
7) Wrong functional form
ØLinear form and non-linear form – not chosen properly
ØUi will capture those errors

qRegression analysis assumptions

—The regression model is linear in the parameters

—2) X values are fixed in repeated sampling
—) Zero mean value of disturbance
—E(ui ) = 0
—Homoscedasticity or equal variance of (ui )

—No autocorrelation between the disturbances

•Cov (u_i, u_j )=0
— 6) zero or no covariance between u_i, X_i
•Cov (u_i, X_i )=0
—7) No. of observations n must be greater than explanatory variables
—Cannot estimate the parameters from lesser no. of observations
—8) Variability in X values
—Cannot estimate the parameters
—9) Correctly specify the functional form to avoid specification error or bias
—10) There is no perfect multicollinearity among the explanatory variables
 ate in India
riables are quarterly (GDP)
quencies – Stocks (daily prices), GDP (quarterly), IIP (monthly)??

ng theory

er terms missing

so play a role.

U I captures those terms

y be missing Ui captures that

anent income

unsure whether they can have an effect – better to use lesser no. of variables
•Exponential Smoothing
Ø Not based on ARIMA approach
ØUses a linear combination of previous values for modelling and forecasting
ØGives more weight on recent observations than the past values
orecasting
Regression analysis is a way to understand the relationship between variables, but for it to

1. Linearity – The relationship should be straight

The dependent variable (what you're predicting) should have a straight-line relationship w

Imagine plotting points on a graph—if they roughly form a straight line, that’s good. If the

2. Independence – No pattern in errors

The errors (differences between actual and predicted values) should not be related to eac

If you're predicting sales based on advertising, the error in one month's prediction should

3. Homoscedasticity – Equal spread of errors

The variability of errors should be roughly the same across all levels of the independent v

If the errors grow larger as values increase (like a funnel shape), then the assumption is v

4. Normality of Errors – Errors should follow a bell curve

If you make a histogram of the prediction errors, they should roughly form a normal (bell

This is important when making statistical inferences from the model.

5. No Perfect Multicollinearity – Independent variables shouldn’t b

If you're using multiple predictors, they shouldn’t be highly correlated.

Example: If you're predicting house prices, using both "size in square feet" and "number

These assumptions help ensure that regression gives meaningful and reliable results. If the

4o

You said:

explain time series data, cross sectional data, pooled data and longitudinal data with two e

ChatGPT said:

Let's break down these different types of data with simple explanations and two examples
1. Time Series Data

Time series data is collected over time at regular intervals (e.g., daily, monthly, yearly). It

Examples:

✅ Financial Example: Stock prices of Apple from 2015 to 2025 (collected daily or monthly
✅ Layman Example: Daily temperature recorded in New York for the past 10 years.

2. Cross-Sectional Data

Cross-sectional data is collected at a single point in time across different entities (comp

Examples:

✅ Financial Example: Net profits of 50 companies in 2023 (data is for one year, across diff
✅ Layman Example: The heights of 100 students in a school measured in January 2024.

3. Pooled Data

Pooled data combines both time series and cross-sectional data, but it is not necessar

Examples:

✅ Financial Example: Quarterly profits of 20 banks from 2018 to 2023 (mix of cross-sectio
✅ Layman Example: Exam scores of students from 5 different schools over the past 3 yea

4. Longitudinal Data (Panel Data)

Longitudinal (or panel) data tracks the same entities over multiple time periods, making

Examples:

✅ Financial Example: Annual revenue of the same 30 companies from 2010 to 2023 (sam
✅ Layman Example: Weight of 50 people tracked every year for 10 years (same individua

Key Differences:
 Time series → One entity, multiple time points.

Cross-sectional → Many entities, one time point.

Pooled data → Combination of both but not necessarily structured.

Longitudinal data → Many entities tracked consistently over time.

Hope this makes it clear! 😊

4o

You said:

structural model means

ChatGPT said:

What is a Structural Model?

A structural model is a conceptual or mathematical representation of how different varia

Think of it as a blueprint that shows why and how certain factors influence an outcome.

Key Features of a Structural Model:

1. Causal Relationships → Shows which variables affect others and how strongly.

2. Multiple Equations → Often involves several interconnected equations instead of a s

3. Latent Variables → Can include factors that aren’t directly measurable (e.g., custom

4. Hypothesis Testing → Used to test theories and validate relationships.

Examples of Structural Models

✅ Financial Example:

A model that explains how interest rates, inflation, and consumer spending

It would use equations to represent how these economic factors interact and impact over
✅ Layman Example:

A model that shows how diet, exercise, and sleep affect body weight and overall h

It explains which factor has the biggest impact and how they work together.

Where are Structural Models Used?

Economics: Explaining how policies affect unemployment or inflation.

Marketing: Understanding how advertising, product quality, and pricing affect sales.

Social Sciences: Studying how education and family background influence income.

Difference Between Structural and Non-Structural Models

Non-
Structur
Feature Structur
al Model
al Model

Cause-
Purely
and-effect
Focus statistical
relationsh
patterns
ips

Single
Multiple, equation
Equation
interconn or direct
s
ected correlatio
n

Predicting
Economic stock
Example
growth prices
s
models using past
trends

Would you like a deeper example with equations? 😊

4o

You said:

structural models are not suitable for out of sample forecasting. explain
ChatGPT said:

Why Are Structural Models Not Suitable for Out-of-Sample Foreca

Structural models focus on explaining the causal relationships between variables rather

1. Assumption of Stable Relationships

Structural models assume that the relationships between variables remain

In reality, economic, financial, or behavioral relationships can change due to

Example: A model explaining inflation based on interest rates and unemployment might

2. Overfitting to Historical Data

Structural models are often built based on historical data and theoretical assumptio

When applied to new, unseen data, they may not capture emerging patterns or unexpect

Example: A model explaining stock market performance using interest rates and corpora

3. Complexity and High Sensitivity

Structural models often involve multiple equations and interdependencies

A small error in one variable can propagate and lead to incorrect forecasts

Example: A model predicting GDP growth might fail if consumer confidence drops unexp

4. Difficulty in Capturing External Shocks

Structural models struggle with unexpected events like financial crises, pandemics, or

They assume a predictable world, which is not always the case.

Example: A model explaining oil prices using supply and demand might fail if a war disru
Alternative for Out-of-Sample Forecasting?

For better forecasting, data-driven models like:

✅ Time Series Models (ARIMA, Exponential Smoothing) → Work well for trend-based forec
✅ Machine Learning Models (Neural Networks, Random Forests) → Adapt to changing da

Would you like an example comparing structural vs. data-driven forecasting? 😊

4o

You said:

univariate model and multi variate model explain

ChatGPT said:

Univariate vs. Multivariate Models

These two terms refer to the number of variables being analyzed in a model.

Multivari
Univariat
Feature ate
e Model
Model

A model
with one A model
depende with one
nt depende
variable nt
and one variable
Definitio
indepen but
n
dent multiple
variable indepen
(or just dent
one variables
variable .
in total).
 Understan
Understan ding
ding or relationsh
predicting ips
a single between
Focus variable multiple
over time factors
or across affecting
observati the
ons. dependen
t variable.

Complexi More
Simple
ty complex

Stock House
price price
Example trends, prediction
Use temperat ,
Cases ure economic
forecastin forecastin
g g

1. Univariate Model (One Variable)

A univariate model focuses on analyzing a single variable. If forecasting, it predicts base

Examples:

✅ Financial Example:

Predicting the future stock price of Tesla using only its past stock prices

✅ Layman Example:

Tracking the daily temperature in New York over 10 years to predict tomorrow's tempe

2. Multivariate Model (Multiple Variables)

A multivariate model analyzes the relationship between a dependent variable

Examples:

✅ Financial Example:
 Predicting Tesla’s stock price using interest rates, inflation, and trading volume

✅ Layman Example:

Predicting a student’s exam score based on study hours, sleep duration, and past

Key Difference in Modeling Approach

Univariate models are simpler but may miss important influencing factors.

Multivariate models provide a richer analysis but require more data and careful selec

Would you like an example equation for both? 😊

4o

You said:

exponential smoothening, AR, MA, ARMA, ARIMA explain

ChatGPT said:

Time Series Forecasting Models Explained Simply

These models help in forecasting future values based on past observations. Let’s break the

1. Exponential Smoothing (ES)

📌 Idea: More recent data is more important than older data.
📌 How it works: It assigns more weight to recent observations and less weight to old

Formula:

St=αYt+(1−α)St−1S_t = \alpha Y_t + (1 - \alpha) S_{t-1}St​=αYt​+(1−α)St−1​

where:

StS_tSt​= smoothed value at time ttt

YtY_tYt​= actual observation at time ttt

 α\alphaα = smoothing factor (0 < α < 1)

✅ Example: Predicting tomorrow’s gold price using past prices, giving more importance t

2. Autoregressive Model (AR)

📌 Idea: Future values depend on past values of the same variable.
📌 How it works: Uses a regression of the variable on its own past values.

Formula (AR(1) model example):

Yt=c+ϕYt−1+ϵtY_t = c + \phi Y_{t-1} + \epsilon_tYt​=c+ϕYt−1​+ϵt​

where:

YtY_tYt​= value at time ttt

Yt−1Y_{t-1}Yt−1​= value at previous time step

ϕ\phiϕ = coefficient showing how past values influence current values

ϵt\epsilon_tϵt​= error term

✅ Example: Predicting monthly sales based only on previous months’ sales.

3. Moving Average Model (MA)

📌 Idea: Future values depend on past error terms (random shocks).
📌 How it works: The model corrects for previous prediction errors rather than relying on p

Formula (MA(1) model example):

Yt=c+θϵt−1+ϵtY_t = c + \theta \epsilon_{t-1} + \epsilon_tYt​=c+θϵt−1​+ϵt​

where:

ϵt−1\epsilon_{t-1}ϵt−1​= previous period’s error term

θ\thetaθ = coefficient of the error term

✅ Example: Predicting stock market movements where today’s price is affected by unexpe
4. Autoregressive Moving Average (ARMA) = AR +
📌 Idea: Combines AR and MA models to account for both past values and past errors
📌 How it works: It captures both trends and noise in the data.

Formula (ARMA(1,1) model example):

Yt=c+ϕYt−1+θϵt−1+ϵtY_t = c + \phi Y_{t-1} + \theta \epsilon_{t-1} + \epsilon_tYt​=c+ϕY

✅ Example: Predicting exchange rates where both previous rates and external shocks m

5. Autoregressive Integrated Moving Average (AR

📌 Idea: Extends ARMA by making the data stationary (removing trends or seasonality).
📌 How it works:

AR (p) → Uses past values.

I (d) → Differencing to make data stationary.

MA (q) → Uses past error terms.

Formula (ARIMA(p,d,q)):

(1−∑ϕiLi)Yt=(1+∑θjLj)ϵt(1 - \sum \phi_i L^i) Y_t = (1 + \sum \theta_j L^j) \epsilon_t(1−∑ϕi​L

where LLL is the lag operator.

✅ Example: Predicting GDP growth where raw data needs to be differenced to remove lo

Quick Summary of Differences

Uses Uses
Removes
Model Past Past
Trends?
Values? Errors?
ES ❌ ❌ ✅
AR ✅ ❌ ❌
MA ❌ ✅ ❌
ARMA ✅ ✅ ❌
ARIMA ✅ ✅ ✅
Would you like a practical example with Python or Excel formulas? 😊

4o

You said:

stationary timeseris meaning

ChatGPT said:

What is a Stationary Time Series?

A stationary time series is a time series whose statistical properties do not change
1️⃣Constant Mean – The average value remains the same over time.
2️⃣Constant Variance – The spread of the data (volatility) does not change.
3️⃣No Seasonality/Trends – Patterns do not change over time.

Why is Stationarity Important?

Many forecasting models (like ARMA, ARIMA) assume that the data is stationary

If a time series is non-stationary, we need to transform it (e.g., differencing) before ap

Examples of Stationary and Non-Stationary Time Series

✅ Stationary Example (Good for Forecasting)

Stock Market Returns (not stock prices!) – The percentage change in stock prices often

Daily Temperature Deviations – If we measure how much today’s temperature differs

❌ Non-Stationary Example (Needs Transformation)

Stock Prices – They often follow an upward or downward trend over time.

GDP Growth – GDP usually grows over time, meaning the mean is not constant.

How to Check for Stationarity?

📌 Visual Inspection – Plot the time series and see if it has a trend or changing variance.
📌 Statistical Tests:
1️⃣Augmented Dickey-Fuller (ADF) Test – If p-value < 0.05, the series is stationary.
2️⃣KPSS Test – If p-value < 0.05, the series is not stationary.

How to Make a Time Series Stationary?

1️⃣Differencing – Subtract the previous value from the current value (Yt−Yt−1Y_t - Y_{t-1}
2️⃣Log Transformation – Taking the log of values can stabilize variance.
3️⃣Seasonal Adjustments – Remove seasonal patterns.

Would you like an example of differencing to make a series stationary? 😊

4o

You said:

moments order

ChatGPT said:

Order of Moments in Statistics

Moments are statistical measures that describe the shape and distribution

1st Order Moment: Mean (Central Tendency)

📌 What it tells us: The average value of the dataset.

📌 Formula:

μ=1n∑i=1nXi\mu = \frac{1}{n} \sum_{i=1}^{n} X_iμ=n1​i=1∑n​Xi​

✅ Example: The average closing price of a stock over the last year.

2nd Order Moment: Variance (Spread of Data)

📌 What it tells us: How much the data points deviate from the mean.
📌 Formula:

σ2=1n∑i=1n(Xi−μ)2\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (X_i - \mu)^2σ2=n1​i=1∑n​(

✅ Example: The volatility of a stock (higher variance means more risk).

3rd Order Moment: Skewness (Symmetry of Distribution)

📌 What it tells us: Whether the data is skewed to the left (-) or right (+).
📌 Formula:

Skewness=1n∑i=1n(Xi−μσ)3\text{Skewness} = \frac{1}{n} \sum_{i=1}^{n} \left(\frac{X

✅ Example: If stock returns are negatively skewed, it means there are more large negative

4th Order Moment: Kurtosis (Tails of Distribution)

📌 What it tells us: The "peakedness" and whether the data has fat tails (more extreme va
📌 Formula:

Kurtosis=1n∑i=1n(Xi−μσ)4\text{Kurtosis} = \frac{1}{n} \sum_{i=1}^{n} \left(\frac{X_i -

✅ Example: A stock with high kurtosis has more extreme gains or losses, making it riskier.

Summary Table

Moment Name Meaning Example

Average
Central temperat
1st Mean
Value ure of a
city
Stock
2nd Variance Spread price
volatility
Salary
Asymmetr distributio
3rd Skewness
y n in a
company
Risk of
Extreme stock
4th Kurtosis
values market
crashes

Would you like a visualization of these moments? 😊

4o
You said:

causation vs correlation

ChatGPT said:

Causation vs. Correlation – What’s the Difference?

Many people confuse correlation and causation, but they are very different concepts in s

1️⃣Correlation (Association)

📌 Definition: Correlation means that two variables move together but one does not nec

📌 Example:
✅ Ice cream sales & Shark attacks – Both increase in summer, but ice cream does not c
✅ Stock Market & Google Searches – A rising stock market might correlate with more se

📌 Key Fact: Correlation is measured using the correlation coefficient (rrr)

r>0r > 0r>0 → Positive correlation (both increase/decrease together)

r<0r < 0r<0 → Negative correlation (one increases, the other decreases)

r=0r = 0r=0 → No correlation

2️⃣Causation (Cause & Effect)

📌 Definition: Causation means that a change in one variable directly causes a chang

📌 Example:
✅ Smoking & Lung Cancer – Smoking causes lung cancer, proven through medical studi
✅ High Interest Rates & Low Borrowing – When interest rates increase, borrowing decr

📌 Key Fact: Causation requires experimental evidence or strong statistical methods

Key Differences

Correlati Causatio
Feature
on n
 One
Two
variable
Definitio variables
causes a
n move
change in
together
another

Implies
cause- ❌ No ✅ Yes
effect?
Correlatio Statistical
Measure n tests,
ment coefficient experime
(rrr) nts

Ice cream
Smoking
sales &
Example & lung
shark
cancer
attacks

Common Mistake: "Correlation does not imply causation"

Just because two things are related does not mean one causes the other
2️⃣Reverse causation – Example: Does stress cause poor sleep, or does poor sleep cause
3️⃣Pure coincidence – Some correlations happen by chance.

Want a fun example?

There's a strong correlation between the number of movies Nicolas Cage appears in
Does this mean Nicolas Cage causes drowning? No! It’s just a random correlation. 😆

Would you like a real-world dataset example to analyze correlation vs. causation? 😊

4o

You said:

lagging the variable in time series. why do we do that

ChatGPT said:

Lagging a Variable in Time Series – Why Do We Do That?

In time series analysis, lagging a variable means shifting it backward in time by a certa
For example, if we have a dataset of monthly sales, a lag of 1 month (Yt−1Y_{t-1}Yt−1

Reasons for Lagging Variables

🔹 1️⃣Capturing Past Effects (Autoregression)

Many time series processes have dependencies on past values.

Example: Stock Prices – Today’s stock price is influenced by yesterday’s stock price.

Model:

Yt=β0+β1Yt−1+ϵtY_t = \beta_0 + \beta_1 Y_{t-1} + \epsilon_tYt​=β0​+β1​Yt−1​+ϵt​

where Yt−1Y_{t-1}Yt−1​(lagged value) helps predict YtY_tYt​.

🔹 2️⃣Dealing with Delayed Effects

Some events take time to show their impact.

Example: Interest Rates & Inflation – If the central bank raises interest rates today, its

By using lagged interest rates, we account for this delay.

🔹 3️⃣Removing Serial Correlation (Avoiding Overfitting)

Many time series models assume that errors (residuals) are uncorrelated.

If errors are correlated (autocorrelated), lagging variables can help capture the missing in

🔹 4️⃣Creating Moving Averages

Lagging variables help calculate moving averages, used in forecasting and smoothing d

Example: 5-day moving average of stock prices is computed using the past 5 days’ p

🔹 5️⃣Avoiding Look-Ahead Bias

In financial models, using future data to predict past data is a mistake (not realistic).

Lagging ensures we only use information available at that time.

Example: Lagging in Stock Market Prediction

📊 Stock Price at Time ttt

Stock
Day Lag 1 ($) Lag 2 ($)
Price ($)
1 100 — —
2 102 100 —
3 105 102 100
4 103 105 102

Here, the Lag 1 column shifts the stock price back by 1 period, and Lag 2

How Much Lag Should We Use?

📌 Use statistical tests like:

✅ Autocorrelation Function (ACF) – Checks how past values relate to the current one.
✅ Partial Autocorrelation Function (PACF) – Identifies the most significant lags.

Conclusion

✅ Lagging a variable helps account for past influences, delayed effects, and depe
✅ It improves forecasting accuracy, removes biases, and avoids data leakage.

Would you like an example using Python or Excel to calculate lagged variables? 😊

2-Feb
4o

You said:

vector auto regression model

ChatGPT said:

Vector AutoRegression (VAR) Model – Explained Simply

A Vector AutoRegression (VAR) model is a statistical model used in time series analy

1️⃣What is a VAR Model?

📌 The VAR model is an extension of the autoregressive (AR) model, but it includes
📌 Each variable in the system is modeled as a linear combination of its own past valu
📌 It captures the dynamic relationships between multiple time series.

Example: Suppose we want to model the relationship between:

✅ GDP growth (Y1)
✅ Inflation rate (Y2)
✅ Interest rate (Y3)

In a VAR(1) model (one lag), the equations would look like this:

Y1t=a1+b11Y1t−1+b12Y2t−1+b13Y3t−1+e1tY1_t = a_1 + b_{11} Y1_{t-1} + b_{12} Y2

Each variable is predicted using its own lagged values and the lagged values of all o

2️⃣When to Use a VAR Model?

✅ When multiple time series are interdependent (i.e., they influence each other).
✅ When we want to analyze relationships between economic or financial variables
✅ When we need to forecast multiple time series based on their historical data.

3️⃣Examples of VAR Model Usage

📌 Financial Example (Macroeconomic Indicators):

Stock market index (Nifty 50), interest rate, and inflation – These variables influen

📌 Simple Example (Layman’s Understanding):

Temperature, electricity consumption, and ice cream sales – If temperature increa

4️⃣Advantages of VAR Model

✅ Handles multiple variables at once instead of analyzing them separately.

✅ Captures interdependencies – Unlike single-variable models, it shows how different va
✅ Good for forecasting when relationships exist between variables.

5️⃣Disadvantages of VAR Model

❌ Needs a lot of data – Since multiple variables are used, more observations are required
❌ Computationally complex – More variables mean more equations to solve.
❌ Hard to interpret – The model gives correlations, but not necessarily causal relation

6️⃣VAR vs. AR Models – Key Difference

VAR
AR
Model
Model
Feature (Vector
(Autoreg
AutoReg
ressive)
ression)
Number
of Single Multiple
Variable variable variables
s

Uses past
Uses past values of
Equation
values of itself AND
Form
itself other
variables

Analyzing
relationsh
Forecastin
ips
g one
Usage between
time
multiple
series
time
series

7️⃣How to Choose the Number of Lags in VAR?

Akaike Information Criterion (AIC)

Bayesian Information Criterion (BIC)

Likelihood Ratio Test

These statistical tests help determine the optimal number of lags to use.

Conclusion

✅ VAR is a powerful model for analyzing multiple time series that influence each
✅ It’s widely used in finance, economics, and business forecasting.
✅ But it does not establish causality—just relationships between variables.
Would you like an example in Python or Excel for practical understanding? 😊

4o

You said:

acf. pacf , aic

ChatGPT said:

ACF, PACF, and AIC – Explained Simply

In time series analysis, ACF, PACF, and AIC help us understand patterns in the data and c

1️⃣AutoCorrelation Function (ACF)

📌 What is ACF?
The AutoCorrelation Function (ACF) measures the correlation between a time series an

📌 How to Read an ACF Plot?

The x-axis shows different lag values (e.g., Lag 1, Lag 2, etc.).

The y-axis shows the correlation coefficient (ranging from -1 to 1).

If a lag has a high positive ACF, it means the past value influences the present value.

📌 Example (Stock Market Returns)

If today's stock return (RtR_tRt​) is highly correlated with yesterday's return (Rt−1R_{t-1}

If there’s no correlation, the ACF values will be close to zero for all lags.

📌 Use of ACF:
✅ Detects patterns like seasonality and trends.
✅ Helps choose the right model (e.g., MA, ARMA, or ARIMA).

2️⃣Partial AutoCorrelation Function (PACF)

📌 What is PACF?
The Partial AutoCorrelation Function (PACF) measures the direct relationship between
📌 How is PACF Different from ACF?

ACF shows the total correlation of a time series with its lags (including indirect effects).

PACF removes the influence of other lags and shows the pure correlation

📌 Example (Inflation Rates)

Suppose inflation today (ItI_tIt​) depends on both last month (It−1I_{t-1}It−1​) and two mo

The ACF will show correlation for both Lags 1 and 2.

But PACF at Lag 2 will be low if Lag 1 already explains most of the variation

📌 Use of PACF:
✅ Helps decide how many AR terms to include in a model.
✅ If PACF cuts off sharply after a certain lag, we might use an AR model.

3️⃣Akaike Information Criterion (AIC)

📌 What is AIC?
The Akaike Information Criterion (AIC) is a measure of how well a statistical model fits

📌 Formula for AIC:

AIC=−2ln⁡(L)+2kAIC = -2 \ln(L) + 2kAIC=−2ln(L)+2k

where:

LLL = Likelihood of the model (how well it fits the data)

kkk = Number of parameters in the model

📌 Why is AIC Important?

Lower AIC values indicate a better model (better balance of fit and simplicity).

Helps compare different models (e.g., ARIMA(1,1,0) vs. ARIMA(2,1,1)).

📌 Example (Choosing an ARIMA Model)

Suppose we test three different models:

ARIMA(1,1,0) → AIC = 100

 ARIMA(2,1,1) → AIC = 120

ARIMA(1,1,1) → AIC = 90

Since ARIMA(1,1,1) has the lowest AIC (90), we choose this as the best model.

📌 Use of AIC:
✅ Helps select the best AR, MA, ARMA, or ARIMA model.
✅ Avoids overfitting by penalizing unnecessary complexity.

Summary Table: ACF vs. PACF vs. AIC

AIC
PACF (Akaike
ACF
(Partial Informat
Feature (AutoCor
AutoCorr ion
relation)
elation) Criterion
)

Measures
direct Measures
Measures
correlatio model fit
total
n after while
Purpose correlatio
removing penalizing
n with
intermedi complexit
past lags
ate y
effects

Helps Helps
Identifies
determine choose
Usage seasonalit
number of the best
y & trends
AR terms model

Slowly Cuts off

declines for AR
Lower
for AR models,
Graph value =
models, slowly
Behavior Better
cuts off declines
model
for MA for MA
models models

Conclusion

✅ ACF helps detect overall dependencies and moving averages (MA).

✅ PACF helps determine the number of autoregressive (AR) terms.
✅ AIC helps pick the best model with the right balance of fit and simplicity.
Would you like an example with Python or Excel to see these concepts in action? 😊

4o

You said:

VECF

ChatGPT said:

Vector Error Correction Model (VECM) – Explained Simply

📌 What is VECM?

A Vector Error Correction Model (VECM) is a special type of Vector AutoRegressio

📌 When Do We Use VECM?

We use VECM when:

1. Two or more time series are non-stationary (i.e., they have trends).

2. There is a long-run equilibrium relationship between them (cointegration).

3. We want to model both short-term fluctuations and long-term relationships

📌 How Does VECM Work?

VECM corrects deviations from long-term equilibrium by adding an error correctio

The ECT measures how far the system is from equilibrium and adjusts the next-period va

📌 VECM Equation (Simplified)

For two variables YtY_tYt​and XtX_tXt​, a basic VECM can be written as:

ΔYt=α(βXt−1−Yt−1)+γ1ΔYt−1+γ2ΔXt−1+ϵt\Delta Y_t = \alpha (\beta X_{t-1} - Y_{t-1}) +

where:

ΔYt\Delta Y_tΔYt​and ΔXt\Delta X_tΔXt​are first differences (removing trends).

βXt−1−Yt−1\beta X_{t-1} - Y_{t-1}βXt−1​−Yt−1​is the error correction term

α\alphaα is the adjustment coefficient (how quickly the system corrects itself).
 γ1\gamma_1γ1​and γ2\gamma_2γ2​capture short-run dynamics.

ϵt\epsilon_tϵt​is the error term.

📌 Example 1: Financial Data (Stock Prices & Interest Rates)

Let’s say we study the relationship between:

Stock Market Index (NIFTY 50) YtY_tYt​

Interest Rate (Repo Rate) XtX_tXt​

Since stock prices and interest rates move together in the long run (lower interest rate
✅ Capture short-term deviations from equilibrium.
✅ Predict how quickly stock prices adjust to interest rate changes.

📌 Example 2: Everyday Concept (Temperature & Ice Cream Sales)

YtY_tYt​= Ice cream sales

XtX_tXt​= Temperature

Even if ice cream sales and temperature fluctuate daily, they follow a long-term relati
✅ A VECM can model short-term deviations (e.g., rainy days reducing sales) while keeping t

📌 VECM vs. VAR – What’s the Difference?

VECM
VAR
(Vector
(Vector
Feature Error
AutoReg
Correctio
ression)
n Model)
For non-
For
stationary
When stationary
but
Used? time
cointegrat
series
ed series
 Both
short-
Only
term
short-
fluctuatio
Captures term
ns & long-
relationsh
term
ips
equilibriu
m

Error
Correctio
❌ No ✅ Yes
n Term
(ECT)?

📌 Conclusion

✅ VECM is useful when two or more time series have a long-term relationship
✅ It ensures that the system corrects itself when there are deviations from equilibrium.
✅ Used widely in finance, economics, and forecasting real-world relationships.

Would you like a Python example to see VECM in action? 🚀

4o

You said:

how to interpret the results of VAR model in eviews

ChatGPT said:

Interpreting VAR Model Results in EViews

When you estimate a Vector AutoRegression (VAR) model in EViews, the output provid

📌 Step 1: Understanding the VAR Equation Output

EViews presents the estimated VAR equations for each dependent variable. For example, in

Yt=c1+α11Yt−1+α12Xt−1+α13Yt−2+α14Xt−2+ϵY,tY_t = c_1 + \alpha_{11} Y_{t-1} + \a

Each equation explains how the current value of a variable depends on its past values and

🔹 Key Components to Analyze

 1. Coefficient Values:

These tell us the effect of past values on the current variable.

If α12=0.5\alpha_{12} = 0.5α12​=0.5, it means a 1-unit increase in Xt−1X_{t-1}

A significant positive coefficient suggests a positive relationship, while a

2. Standard Errors & t-Statistics:

If the t-statistic is large (typically greater than 2 in absolute value) and the

This means the past value significantly influences the current value.

3. R-Squared (R2R^2R2):

Shows how well the past values explain the variability in the dependent variable.

Higher R2R^2R2 (closer to 1) suggests a good fit.

📌 Step 2: Checking Stability of the VAR Model

EViews provides a Roots of the Characteristic Polynomial table.

✅ If all roots lie inside the unit circle, the model is stable.
❌ If any root is outside, the VAR model is unstable, meaning predictions might be unrelia

📌 Step 3: Examining Granger Causality Tests

Does one variable "cause" another?

If XtX_tXt​Granger-causes YtY_tYt​, then past values of XtX_tXt​help predict YtY_tYt​.

p-value < 0.05 → Reject the null hypothesis (X does influence Y).

p-value > 0.05 → Fail to reject (no significant causality).

📌 Step 4: Impulse Response Functions (IRF)
These show how a shock to one variable affects another over time.

Example: A positive shock to interest rates might negatively impact stock prices over sev

Look at the direction and duration of the impact (how long the effect lasts).

📌 Step 5: Variance Decomposition

Tells us how much of the forecast error variance of a variable is explained by shocks to it

If 90% of the variance in GDP is explained by GDP itself → it is largely self-driven.

If 30% of the variance in stock prices is due to interest rates, interest rates play a

📌 Conclusion

✅ Step 1: Look at coefficient signs & significance (do past values matter?).
✅ Step 2: Ensure stability (roots inside the unit circle).
✅ Step 3: Use Granger Causality to check predictive power.
✅ Step 4: Interpret Impulse Response Functions (how shocks affect the system).
✅ Step 5: Use Variance Decomposition for importance of variables.

Want me to explain a specific part in more detail? 🚀

4o

You said:

in need to build ar 1, ar 2 , ar 3, ar4 and ma 1, ma2 ma 3 ma4 and write the aic value in a m

ChatGPT said:

Building AR(1) to AR(4) and MA(1) to MA(4) Models in EViews and

To determine the best-fit model, we need to estimate Autoregressive (AR) models of or

📌 Step 1: Writing a General Equation to Estimate A
Instead of estimating each model separately, you can use loops in EViews

Method 1: Using a General Equation (One-by-One Estimation)

To estimate AR and MA models one by one, you can use the following single equation sp

scss
Copy code
Y C AR(n) MA(n)

where:

Y is your dependent variable (time series).

C is the constant term.

AR(n) specifies an autoregressive model of order n (AR(1) to AR(4)).

MA(n) specifies a moving average model of order n (MA(1) to MA(4)).

👉 Example commands in EViews:

AR(1): Y C AR(1)

AR(2): Y C AR(2)

AR(3): Y C AR(3)

AR(4): Y C AR(4)

MA(1): Y C MA(1)

MA(2): Y C MA(2)

MA(3): Y C MA(3)

MA(4): Y C MA(4)

Steps to Estimate the Model in EViews:

1. Go to Quick → Estimate Equation.

2. Enter the model specification (e.g., Y C AR(1)).

 3. Click OK, and note down the AIC value from the output.

4. Repeat for AR(2) to AR(4) and MA(1) to MA(4).

📌 Step 2: Automating Model Selection (Loop Appr

If you want to automate this process and collect AIC values in a matrix, use the followin

eviews
Copy code
!max_order = 4
matrix(5,5) AIC_Matrix ' Create a 5x5 matrix to store AIC values (1st row & col for labels)

for !i = 1 to !max_order
equation ar_model.ls y c ar(!i) ' Estimate AR model
scalar ar_aic = @aic ' Store AIC value
AIC_Matrix(1,!i+1) = ar_aic ' Store in matrix

equation ma_model.ls y c ma(!i) ' Estimate MA model

scalar ma_aic = @aic ' Store AIC value
AIC_Matrix(2,!i+1) = ma_aic ' Store in matrix
next

' Label the rows and columns for better readability

AIC_Matrix(1,1) = "AR Model"
AIC_Matrix(2,1) = "MA Model"
AIC_Matrix(3,1) = ""
AIC_Matrix(4,1) = ""
AIC_Matrix(5,1) = ""

for !j = 1 to !max_order
AIC_Matrix(3,!j+1) = !j
AIC_Matrix(4,!j+1) = ""
AIC_Matrix(5,!j+1) = ""
next

' Display the matrix

show AIC_Matrix

📌 Step 3: Interpreting the Results

The AIC matrix will look like this in EViews:
 Model 1 2 3 4
AR X1 X2 X3 X4
MA Y1 Y2 Y3 Y4

Where:

X1, X2, X3, X4 are the AIC values for AR(1) to AR(4).

Y1, Y2, Y3, Y4 are the AIC values for MA(1) to MA(4).

The model with the lowest AIC value is the best fit.

📌 Step 4: Selecting the Best Model

1. Identify the model with the lowest AIC from the matrix.

2. If AR models have lower AIC than MA models, use an AR model.

3. If MA models have lower AIC than AR models, use an MA model.

4. If neither is optimal, consider ARMA or ARIMA models.

📌 Conclusion

✅ You can manually estimate AR(n) and MA(n) models in EViews using Y C AR(n)
✅ To automate AIC comparison, use EViews script loops to store values in a matrix.
✅ Choose the model with the lowest AIC for the best predictive accuracy.

Would you like additional help with ARMA or ARIMA models? 🚀

4o

You said:

Closeret c ar(1) ma(1) Closeret c ar(1) ar(2) ma(1) like this write me the equations

ChatGPT said:

Here are the equations for different AR (Autoregressive) and MA (Moving Average) m
📌 AR and MA Model Equations in EViews

Each equation follows the pattern:

eviews
Copy code
DependentVariable C AR(p) MA(q)

where:

Closeret = Dependent variable (replace with your actual series).

C = Constant term.

AR(p) = Autoregressive term of order p.

MA(q) = Moving Average term of order q.

📌 Equations for AR(1) to AR(4) & MA(1) to MA(4)

1️⃣AR(1), MA(1):

eviews
Copy code
Closeret C AR(1) MA(1)

2️⃣AR(2), MA(1):

eviews
Copy code
Closeret C AR(1) AR(2) MA(1)

3️⃣AR(3), MA(1):

eviews
Copy code
Closeret C AR(1) AR(2) AR(3) MA(1)

4️⃣AR(4), MA(1):

eviews
Copy code
Closeret C AR(1) AR(2) AR(3) AR(4) MA(1)

5️⃣AR(1), MA(2):
eviews
Copy code
Closeret C AR(1) MA(1) MA(2)

6️⃣AR(2), MA(2):

eviews
Copy code
Closeret C AR(1) AR(2) MA(1) MA(2)

7️⃣AR(3), MA(2):

eviews
Copy code
Closeret C AR(1) AR(2) AR(3) MA(1) MA(2)

8️⃣AR(4), MA(2):

eviews
Copy code
Closeret C AR(1) AR(2) AR(3) AR(4) MA(1) MA(2)

9️⃣AR(1), MA(3):

eviews
Copy code
Closeret C AR(1) MA(1) MA(2) MA(3)

🔟 AR(2), MA(3):

eviews
Copy code
Closeret C AR(1) AR(2) MA(1) MA(2) MA(3)

1️⃣1️⃣AR(3), MA(3):

eviews
Copy code
Closeret C AR(1) AR(2) AR(3) MA(1) MA(2) MA(3)

1️⃣2️⃣AR(4), MA(3):

eviews
Copy code
Closeret C AR(1) AR(2) AR(3) AR(4) MA(1) MA(2) MA(3)

1️⃣3️⃣AR(1), MA(4):
eviews
Copy code
Closeret C AR(1) MA(1) MA(2) MA(3) MA(4)

1️⃣4️⃣AR(2), MA(4):

eviews
Copy code
Closeret C AR(1) AR(2) MA(1) MA(2) MA(3) MA(4)

1️⃣5️⃣AR(3), MA(4):

eviews
Copy code
Closeret C AR(1) AR(2) AR(3) MA(1) MA(2) MA(3) MA(4)

1️⃣6️⃣AR(4), MA(4) (Full Model):

eviews
Copy code
Closeret C AR(1) AR(2) AR(3) AR(4) MA(1) MA(2) MA(3) MA(4)

📌 Next Steps

Run these equations in EViews one by one or use a loop script to automate the estim

Compare AIC values to determine the best model.

If none of these models fit well, consider ARMA(p, q) or ARIMA(p, d, q)

Would you like a script to run all these automatically and extract the best-fit model? 🚀

4o

You said:

can you give me the loop script

ChatGPT said:

Yes! Below is an EViews loop script that will automatically estimate different ARMA mode
📌 EViews Loop Script for ARMA Model Selection

eviews
Copy code
!max_ar = 4 ' Maximum AR order
!max_ma = 4 ' Maximum MA order
matrix(!max_ar+1, !max_ma+1) AIC_matrix ' Matrix to store AIC values

' Loop through AR orders

for !p = 0 to !max_ar
' Loop through MA orders
for !q = 0 to !max_ma
if !p = 0 and !q = 0 then
' Skip the model without AR and MA terms
else
' Generate equation name dynamically
equation eq{!p}{!q}.ls Closeret C
if !p > 0 then
for !i = 1 to !p
eq{!p}{!q}.append AR(!i)
next
endif
if !q > 0 then
for !j = 1 to !q
eq{!p}{!q}.append MA(!j)
next
endif
' Store AIC in matrix
AIC_matrix(!p+1, !q+1) = eq{!p}{!q}.@aic
endif
next
next

' Display the AIC matrix

show AIC_matrix

📌 How This Script Works

1. Loops over AR(1) to AR(4) and MA(1) to MA(4) to estimate models.

2. Dynamically constructs model equations for each combination.

3. Stores the AIC values in a matrix to help identify the best model.

4. Displays the AIC matrix so you can compare values easily.

📌 How to Use It

Open EViews.

Copy & paste the script into the command window.

Press Enter.

Check the AIC matrix to find the model with the lowest AIC (best fit).

🚀 Would you like modifications to include BIC or HQIC as well?

4o

You said:

can you compile the individual equations into a single downloadable excel file

ChatGPT said:

Yes! I'll generate an Excel file containing all the ARMA model equations along with plac

I have created an Excel file containing all the ARMA model equations with placeholders for

Download ARMA Models Excel

Great question! The ARCH and GARCH models are both used in time series analysis

🔹 What is ARCH?

ARCH = Autoregressive Conditional Heteroskedasticity

Introduced by Robert Engle (who later won the Nobel Prize for it).

It models the current period's volatility (variance) based on past squared error term

⚙ ARCH(1) equation:

σt2=α0+α1εt−12\sigma_t^2 = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2σt2​=α0​+α1​εt−12

Where:
 σt2\sigma_t^2σt2​= forecast variance at time t

εt−12\varepsilon_{t-1}^2εt−12​= squared residual from previous time step

α0>0\alpha_0 > 0α0​>0, α1≥0\alpha_1 \geq 0α1​≥0

👉 So, if you had a big surprise (shock) in the last period, you expect more volatility now.

🔹 What is GARCH?

GARCH = Generalized ARCH

Developed by Tim Bollerslev as an extension of ARCH.

GARCH includes both past shocks (squared errors) and past variances

It handles volatility clustering (i.e., high-volatility periods tend to follow high-volatility

⚙ GARCH(1,1) equation:

σt2=α0+α1εt−12+β1σt−12\sigma_t^2 = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \beta

Where:

β1σt−12\beta_1 \sigma_{t-1}^2β1​σt−12​= influence of past variance

This extra term allows the model to "remember" volatility longer

🆚 Key Differences

Feature ARCH GARCH

Past
✔️Used ✔️Used
Errors
Past ❌ Not
✔️Used
Variance Used
More
Model Less
flexible &
Flexibility flexible
realistic

Volatility Weakly Strongly

Clustering captured captured
Real- Mostly Widely
World theoretica used in
Usage l or base finance

🧠 Simple Analogy:

ARCH is like saying: "If I tripped badly yesterday, I'm nervous today."

GARCH is like saying: "If I tripped badly yesterday and I’ve been nervous all week, I'm
en variables, but for it to work well, we have to make some assumptions. Think of these as "rules of the game"

straight-line relationship with the independent variable(s).

ht line, that’s good. If they curve, regression may not be the best choice.

ould not be related to each other.

month's prediction shouldn't depend on the error in another month.

vels of the independent variable.

, then the assumption is violated.

ell curve

ughly form a normal (bell-shaped) distribution.

riables shouldn’t be too similar

quare feet" and "number of rooms" might be redundant because they convey similar information.

and reliable results. If they are violated, we might need to use other statistical techniques or adjust the model.

gitudinal data with two examples foe each - one should be finalcial data and concepts related and the other exa

ations and two examples for each—one related to finance and one that’s easy to understand.
daily, monthly, yearly). It helps track trends, patterns, and forecasts.

collected daily or monthly).

the past 10 years.

s different entities (companies, individuals, cities, etc.). It helps in comparing variables across different subjects

is for one year, across different companies).

asured in January 2024.

ta, but it is not necessarily structured (i.e., different entities might not be observed at all time points).

2023 (mix of cross-sectional and time series).

hools over the past 3 years (not all students are tracked every year).

ple time periods, making it more structured than pooled data.

s from 2010 to 2023 (same companies tracked over time).

10 years (same individuals observed over time).
on of how different variables in a system are related to each other. It goes beyond simple correlations and aims

influence an outcome.

s and how strongly.

d equations instead of a single formula.

measurable (e.g., customer satisfaction).

ationships.

sumer spending influence GDP growth.

interact and impact overall growth.

y weight and overall health.

rk together.

d pricing affect sales.

nd influence income.

ral Models
-of-Sample Forecasting?

between variables rather than just predicting future values. While they are useful for understanding how variabl

les remain constant over time.

ange due to policy shifts, market crashes, or new consumer trends.

and unemployment might fail when a new economic policy is introduced.

theoretical assumptions.

ging patterns or unexpected changes.

nterest rates and corporate profits may not account for sudden geopolitical events or technological disruptions

dependencies, making them complex.

ct forecasts.

r confidence drops unexpectedly due to a global crisis.

ial crises, pandemics, or political instability.

nd might fail if a war disrupts oil production.

well for trend-based forecasts.
) → Adapt to changing data patterns.

orecasting? 😊

ed in a model.
recasting, it predicts based on past values of the same variable.

tock prices (e.g., using an ARIMA model).

predict tomorrow's temperature.

endent variable and multiple independent variables.

, and trading volume.

eep duration, and past grades.

cing factors.

ore data and careful selection of variables.

ervations. Let’s break them down step by step in simple terms.

ns and less weight to older ones using a smoothing factor (α).

(1−α)St−1​
giving more importance to today’s price than last week’s price.

onths’ sales.

s rather than relying on past values of the variable itself.

θϵt−1​+ϵt​

ice is affected by unexpected shocks from previous days.

ARMA) = AR + MA
alues and past errors.

-1} + \epsilon_tYt​=c+ϕYt−1​+θϵt−1​+ϵt​

es and external shocks matter.

g Average (ARIMA) = ARMA + Differencing

trends or seasonality).

a_j L^j) \epsilon_t(1−∑ϕi​Li)Yt​=(1+∑θj​Lj)ϵt​

differenced to remove long-term trends.

perties do not change over time. This means:

ot change.

data is stationary for accurate predictions.

g., differencing) before applying these models.

e Series

ange in stock prices often fluctuates around a constant mean.

day’s temperature differs from the yearly average, it remains relatively stable.

over time.

n is not constant.

d or changing variance.
e series is stationary.

ue (Yt−Yt−1Y_t - Y_{t-1}Yt​−Yt−1​).

stribution of a dataset. They are used in probability and statistics to summarize data characteristics.

_i - \mu)^2σ2=n1​i=1∑n​(Xi​−μ)2
tribution)

m_{i=1}^{n} \left(\frac{X_i - \mu}{\sigma}\right)^3Skewness=n1​i=1∑n​(σXi​−μ​)3

e are more large negative returns (riskier).

at tails (more extreme values).

=1}^{n} \left(\frac{X_i - \mu}{\sigma}\right)^4Kurtosis=n1​i=1∑n​(σXi​−μ​)4

losses, making it riskier.

 ery different concepts in statistics and data analysis.

r but one does not necessarily cause the other.

but ice cream does not cause shark attacks. 🌊🍦

ght correlate with more searches for "best investments," but searches do not cause the market to rise.

fficient (rrr), ranging from -1 to 1:

irectly causes a change in another.

en through medical studies. 🚬

increase, borrowing decreases because loans become more expensive.

ng statistical methods like controlled experiments or causal inference models.

ausation"

the other. There could be: 1️⃣A third variable (confounding factor) – Example: Hot weather increases both
r does poor sleep cause stress?

olas Cage appears in and the number of people who drown in pools each year. 🎥🏊
a random correlation. 😆

n vs. causation? 😊

Do That?

kward in time by a certain number of periods. This creates a "lagged variable" that represents past values of
month (Yt−1Y_{t-1}Yt−1​) means using last month's sales as a new variable in the model.

esterday’s stock price.

Yt​=β0​+β1​Yt−1​+ϵt​

es interest rates today, its full effect on inflation may be seen after several months.

correlated.

elp capture the missing information.

ecasting and smoothing data.

d using the past 5 days’ prices.

istake (not realistic).

 and Lag 2 shifts it back by 2 periods.

ate to the current one.

st significant lags.

ayed effects, and dependencies in time series data.

voids data leakage.

ed variables? 😊

Simply

sed in time series analysis when there are multiple interdependent variables. Unlike simple autoregressi

model, but it includes multiple variables instead of just one.

 on of its own past values AND the past values of all other variables in the system.

1} Y1_{t-1} + b_{12} Y2_{t-1} + b_{13} Y3_{t-1} + e1_tY1t​=a1​+b11​Y1t−1​+b12​Y2t−1​+b13​Y3t−1​+e1t​Y2t=a

e lagged values of all other variables.

fluence each other).

or financial variables over time.
historical data.

n – These variables influence each other, and a VAR model can help understand their relationships.

es – If temperature increases, both electricity consumption (AC usage) and ice cream sales might rise. A VAR m

m separately.
it shows how different variables affect each other.

observations are required.

ions to solve.
cessarily causal relationships (i.e., one variable affecting another directly).

es that influence each other over time.

ween variables.
tanding? 😊

patterns in the data and choose the best forecasting model.

between a time series and its lagged values. It tells us how strongly past values influence future values

nces the present value.

ay's return (Rt−1R_{t-1}Rt−1​), the ACF at Lag 1 will be high.

PACF)

rect relationship between a time series and its past values after removing the effects of intermediate lags
cluding indirect effects).

correlation of each lag with the current value.

1I_{t-1}It−1​) and two months ago (It−2I_{t-2}It−2​).

ost of the variation.

ell a statistical model fits the data while balancing complexity and accuracy.

f fit and simplicity).

s the best model.
epts in action? 😊

ed Simply

Vector AutoRegression (VAR) model used when time series variables are cointegrated—meaning they ha

ave trends).

hem (cointegration).

g-term relationships.

dding an error correction term (ECT) to a Vector AutoRegression (VAR) model.

djusts the next-period values accordingly.

beta X_{t-1} - Y_{t-1}) + \gamma_1 \Delta Y_{t-1} + \gamma_2 \Delta X_{t-1} + \epsilon_tΔYt​=α(βXt−1​−Yt−1

moving trends).

rrection term (long-run equilibrium).

m corrects itself).
rest Rates)

run (lower interest rates often boost stock prices), we check for cointegration. If cointegrated, a VECM can:

Ice Cream Sales)

llow a long-term relationship (hotter days increase sales).

ng sales) while keeping the long-run link intact.
 -term relationship but may fluctuate in the short term.
ons from equilibrium.
lationships.

Views, the output provides key information, including coefficients, standard errors, t-statistics, R-square

uation Output
variable. For example, in a VAR(2) model with two variables (YtY_tYt​and XtX_tXt​), you might see:

\alpha_{11} Y_{t-1} + \alpha_{12} X_{t-1} + \alpha_{13} Y_{t-2} + \alpha_{14} X_{t-2} + \epsilon_{Y,t}Yt​=c

ds on its past values and the past values of other variables.

ncrease in Xt−1X_{t-1}Xt−1​leads to a 0.5-unit increase in YtY_tYt​, holding other factors constant.

onship, while a negative coefficient indicates an inverse relationship.

value) and the p-value is below 0.05, the coefficient is statistically significant.

ependent variable.

VAR Model

dictions might be unreliable.

ity Tests

elp predict YtY_tYt​.

ns (IRF)

pact stock prices over several periods.

the effect lasts).

explained by shocks to itself vs. other variables.

it is largely self-driven.

tes, interest rates play a key role.

ct the system).

d write the aic value in a matrix form to understnd which model is the best fit. now give me way by which i can g

dels in EViews and Comparing AIC Values

ssive (AR) models of order 1 to 4 and Moving Average (MA) models of order 1 to 4, and then compare
n to Estimate AR and MA Models in EViews
in EViews or estimate them in a structured way.

e Estimation)

wing single equation specification in the Equation Estimation Window in EViews:

on (Loop Approach in EViews Programming)
a matrix, use the following EViews script:

w & col for labels)

 sing Y C AR(n) and Y C MA(n).
e values in a matrix.
e accuracy.

me the equations

MA (Moving Average) models in EViews:

pt to automate the estimation.

A(p, d, q) models.

he best-fit model? 🚀

mate different ARMA models (AR(1) to AR(4) and MA(1) to MA(4)) and store their AIC values in a matrix
ate models.
ble excel file

quations along with placeholders for AIC values. You'll be able to download it and use it to fill in AIC values afte

ons with placeholders for AIC values. You can download it using the link below:

ime series analysis, especially for modeling volatility (variance) over time, which is very common in

past squared error terms (i.e., how wrong past predictions were).

t-1}^2σt2​=α0​+α1​εt−12​
us time step

ct more volatility now.

t variances in the equation.

d to follow high-volatility periods).

repsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2σt2​=α0​+α1​εt−12​+β1​σt−12​

n nervous all week, I'm definitely nervous today."
e as "rules of the game" that need to be followed to get reliable results. Here’s a simple breakdown of these as

formation.

ues or adjust the model.

related and the other example with any easily understandable concepts by a lay man
across different subjects.

all time points).

ple correlations and aims to explain cause-and-effect relationships between variables.
nderstanding how variables interact, they often fail in out-of-sample forecasting (predicting future data poin

technological disruptions.
characteristics.
market to rise.
t weather increases both ice cream sales and shark attacks.

represents past values of the original variable.

nlike simple autoregressive (AR) models that handle just one variable, VAR models allow us to analyze
−1​+b13​Y3t−1​+e1t​Y2t=a2+b21Y1t−1+b22Y2t−1+b23Y3t−1+e2tY2_t = a_2 + b_{21} Y1_{t-1} + b_{22} Y2_

elationships.

ales might rise. A VAR model can help understand how these three variables are interlinked.
nfluence future values.

ts of intermediate lags.
rated—meaning they have a long-term relationship but may deviate in the short term.

ilon_tΔYt​=α(βXt−1​−Yt−1​)+γ1​ΔYt−1​+γ2​ΔXt−1​+ϵt​
ntegrated, a VECM can:
s, t-statistics, R-squared values, and impulse response functions. Here's how you can interpret them:

ou might see:

t-2} + \epsilon_{Y,t}Yt​=c1​+α11​Yt−1​+α12​Xt−1​+α13​Yt−2​+α14​Xt−2​+ϵY,t​Xt=c2+β11Yt−1+β12Xt−1+β13Yt−
er factors constant.
e me way by which i can generate all this model using one equation in evies

to 4, and then compare their Akaike Information Criterion (AIC) values in a matrix format.
alues in a matrix.
it to fill in AIC values after running the models in EViews. Let me create and share the file now. 🚀

h is very common in financial data like stock returns. Let’s break it down simply:
e breakdown of these assumptions:
edicting future data points that were not part of the model's training set). Here’s why:
low us to analyze multiple time series that influence each other over time.
} Y1_{t-1} + b_{22} Y2_{t-1} + b_{23} Y3_{t-1} + e2_tY2t​=a2​+b21​Y1t−1​+b22​Y2t−1​+b23​Y3t−1​+e2t​Y3t=a
ou can interpret them:

Yt−1+β12Xt−1+β13Yt−2+β14Xt−2+ϵX,tX_t = c_2 + \beta_{11} Y_{t-1} + \beta_{12} X_{t-1} + \beta_{13}

ix format.
file now. 🚀
1​+b23​Y3t−1​+e2t​Y3t=a3+b31Y1t−1+b32Y2t−1+b33Y3t−1+e3tY3_t = a_3 + b_{31} Y1_{t-1} + b_{32} Y2_{
} X_{t-1} + \beta_{13} Y_{t-2} + \beta_{14} X_{t-2} + \epsilon_{X,t}Xt​=c2​+β11​Yt−1​+β12​Xt−1​+β13​Yt−2​+β
} Y1_{t-1} + b_{32} Y2_{t-1} + b_{33} Y3_{t-1} + e3_tY3t​=a3​+b31​Y1t−1​+b32​Y2t−1​+b33​Y3t−1​+e3t​
−1​+β12​Xt−1​+β13​Yt−2​+β14​Xt−2​+ϵX,t​
1​+b33​Y3t−1​+e3t​
This output presents the results of the

🔍 1. Breusch-Godfrey Seria
Null Hypothesis (H₀):

There is no serial correlation in the residuals

Interpret
Metric Value
ation

Statisticall
F-statistic 5.200445 y
significant

Reject H₀
Prob(F-
0.0001 at 1%
statistic)
level

Statisticall
Obs*R-
24.06877 y
squared
significant

Prob(Chi-
0.0002 Reject H₀
Square)

➡️Conclusion: There is significant serial cor

🔁 2. Test Equation Details

This regression identifies which specific lags of r

Coefficie t-
Lag
nt Statistic

RESID(-
-0.1481 -2.3835
1)

RESID(-
0.0479 0.7628
2)
RESID(-
0.0995 1.5904
3)

RESID(-
0.0585 0.9301
4)

RESID(-
0.2485 3.9926
5)

➡️Key Findings:

Significant autocorrelation exists at

Other lags are not individually significant but

⚠️Implications & Recomme

1. ARCH(1) model is insufficient

2. Consider fitting a GARCH(1,1)

3. Alternatively, include autoregressive (AR) te

✅ Summary:
There is statistically significant autocorre

This suggests model misspecification

Significant autocorrelation violates key model

he results of the Breusch-Godfrey Serial Correlation LM Test and the underlying regression used to detect

-Godfrey Serial Correlation LM Test:

relation in the residuals up to 5 lags.

s significant serial correlation in the residuals up to lag 5. This violates a key assumption and suggests that

uation Details (Regression of RESID on its lags):

es which specific lags of residuals are contributing to the autocorrelation.

Interpret
p-Value
ation
Significan
t — lag 1
shows
0.0179
negative
autocorrel
ation.
Not
0.4463 significant
.
 Not
0.1131 significant
.
Not
0.3532 significant
.
Highly
significa
nt —
strong
0.0001
positive
autocorrel
ation at
lag 5.

lation exists at lag 1 and lag 5.

dividually significant but collectively contribute to overall serial correlation.

ns & Recommendations:
s insufficient — it doesn't fully eliminate serial correlation in residuals.

GARCH(1,1) or higher-order ARCH(q) or ARMA-GARCH model to better capture the time series dynamics and

de autoregressive (AR) terms in the mean equation if you haven't already.

y significant autocorrelation in the residuals of your ARCH(1) model (especially at lag 5).

misspecification, and improvements are needed (e.g., switching to GARCH or increasing lag orders).

lation violates key model assumptions and may affect forecast reliability.
egression used to detect autocorrelation (serial correlation) in the residuals of your ARCH model. Here's a clea

mption and suggests that the ARCH(1) model may not have fully captured all autocorrelations
 ime series dynamics and volatility structure.

asing lag orders).

RCH model. Here's a clear interpretation of each section:

correlations in the data.

This output shows the results from estimating an ARCH(1) model on the NIFTYRET

🧪 Model Specification: ARCH(1)

The variance equation:

σt2=α0+α1⋅ϵt−12\sigma_t^2 = \alpha_0 + \alpha_1 \cdot \epsilon_{t-1}^2σt2​=α0​+α1​⋅ϵt−

This is the basic ARCH(1) form where:

α0\alpha_0α0​(C) is the constant term

α1\alpha_1α1​(RESID(-1)2^22) captures how much past shocks (squared residuals) impa

🔍 Mean Equation:
Coefficie z- Interpret
Variable Prob.
nt Statistic ation
Not
statisticall
y
significant
C 0.001113 1.1927 0.233
→ mean
return is
close to
zero.

📌 Hypothesis for Mean Equation:

H₀ (Null Hypothesis): Coefficient = 0 → Mean return is not significantly different from z

Conclusion: Fail to reject H₀ (p = 0.2330) → No significant trend in returns.

📈 Variance Equation (ARCH Effects):

Coefficie z- Interpret
Variable Prob.
nt Statistic ation
 Statisticall
y
C (α₀) 0.000195 19.8479 0
significant
.

Highly
significant
→
confirms
RESID(-
0.5943 7.0898 0 ARCH
1)^2 (α₁)
effect
(volatility
clustering
).

📌 Hypotheses for ARCH(1):

H₀: α₁ = 0 (no ARCH effect; no volatility clustering)

H₁: α₁ ≠ 0 (presence of ARCH effect)

➡️Conclusion: Reject H₀ since p < 0.01 → Significant ARCH effect exists.

📊 Goodness of Fit & Model Diagnostics:

Interpret
Metric Value
ation
Not
meaningf
R-squared -0.000779 ul for
volatility
models.
S.E. of Standard
regressio 0.020085 error of
n residuals.
Used for
Log model
659.4321
likelihood compariso
n.
 All
negative
Akaike, → useful
Schwarz, for
Hannan- comparin
Quinn g models;
lower is
better.
Close to 2
→ no
autocorr
Durbin-
elation
Watson 2.2811
in
stat
residuals
, which is
good.

✅ Final Interpretation:
The ARCH(1) model is appropriate for modeling time-varying volatility in NIFTY return

The significant coefficient on RESID(−1)2\text{RESID}(-1)^2RESID(−1)2 confirms

The mean return is not significant, indicating that returns are centered around zero.

Residuals show no autocorrelation (Durbin-Watson ≈ 2.28), which is a good diagnost

📌 Recommendations:
1. Consider extending the model to GARCH(1,1) to capture both ARCH and GARCH effec

2. You may check for normality of residuals and ARCH effects in standardized residuals to
 on the NIFTYRET (Nifty Returns) series. The model captures time-varying volatility, commonly found in financ

n_{t-1}^2σt2​=α0​+α1​⋅ϵt−12​

(squared residuals) impact current volatility

nificantly different from zero.

d in returns.
ect exists.
g volatility in NIFTY returns.

SID(−1)2 confirms volatility clustering, i.e., large shocks are followed by large shocks (of either sign), and sm

e centered around zero.

which is a good diagnostic outcome.

h ARCH and GARCH effects (i.e., persistence in volatility).

standardized residuals to assess model adequacy.

commonly found in financial return series. Let's interpret this with hypotheses.
s (of either sign), and small by small.
This output shows the results of an ARCH LM test (Lagrange Multiplier test) for

🔍 Purpose of the Test:

To check if the residuals from your model exhibit ARCH effects (i.e., time-varying volatility

✅ Test Summary:
Interpret
Test Value p-value
ation

F- Not
0.379823 0.5383
statistic significant

Obs*R-
Not
squared 0.38232 0.5364
significant
(LM Stat)

📌 Hypotheses of the ARCH LM Test:

H₀ (Null Hypothesis): No ARCH effect (i.e., constant variance of residuals).

H₁ (Alternative Hypothesis): Presence of ARCH effect (i.e., residual variance depends

➡️Decision: Since p-values > 0.05, we fail to reject H₀.

📊 Test Equation Summary (Auxiliary Regression):

This is the regression of the squared residuals on their lag:

WGT_RESIDt2=α0+α1⋅WGT_RESIDt−12+ϵt\text{WGT\_RESID}^2_t = \alpha_0 + \alpha_1

Coefficie Interpret
Variable p-value
nt ation

Significan
C (α₀) 1.042662 0 t intercept
term.
 Not
significant
WGT_RE
→ no
SID²(-1) -0.039266 0.5383
ARCH
(α₁)
effect
detected.

📌 Hypotheses for α₁:

H₀: α₁ = 0 → No ARCH effect

H₁: α₁ ≠ 0 → Presence of ARCH effect

➡️Conclusion: Since p = 0.5383, fail to reject H₀ → No significant ARCH effect.

🧾 Overall Interpretation:
Your residuals do not exhibit time-varying volatility.

The test does not support the use of an ARCH model for this series (WGT_RESID).

Volatility appears to be homoskedastic (constant) over time in this case.

A simple OLS model might be sufficient for modeling this variable.

✍️Final Summary in Plain English:

The ARCH LM test was performed on the residuals of the WGT return series to check for v
tiplier test) for heteroskedasticity in the residuals from your estimated model, specifically testing whether

.e., time-varying volatility).

of residuals).

sidual variance depends on past squared residuals).

Regression):

_t = \alpha_0 + \alpha_1 \cdot \text{WGT\_RESID}^2_{t-1} + \epsilon_tWGT_RESIDt2​=α0​+α1​⋅WGT_RESIDt−12

ant ARCH effect.

is series (WGT_RESID).

n this case.

eturn series to check for volatility clustering. The test yielded high p-values, indicating that there is
cifically testing whether ARCH effects are present in the residuals.

=α0​+α1​⋅WGT_RESIDt−12​+ϵt​
that there is no significant evidence of ARCH effects in the residuals. Therefore, heteroskedasticity is no
eteroskedasticity is not present, and there is no need to model volatility using ARCH/GARCH
g ARCH/GARCH for this particular time period and variable.
Vector Autoregression (VAR) Model Interpretation

This output shows a VAR(2) model estimation involving two return series:

SPOT_RET — returns of the spot market

ETF_RET — returns of the ETF

🔁 Model Structure (Lag Order = 2):

Two equations (one for each variable):

1️⃣SPOT_RET Equation:

SPOT_RETt=−0.1851⋅SPOT_RETt−1−0.0476⋅SPOT_RETt−2+0.1515⋅ETF_RETt−1+0.0274⋅ET

Interpretation:

The current spot return is negatively impacted by its own lags, especially lag 1.

ETF_RET(-1) has a positive and relatively stronger impact, suggesting that

ETF_RET(-2) has a small effect.

The constant term is very close to zero (essentially negligible).

2️⃣ETF_RET Equation:

ETF_RETt=0.2381⋅SPOT_RETt−1+0.1206⋅SPOT_RETt−2−0.1474⋅ETF_RETt−1−0.0531⋅ETF_

Interpretation:

The ETF return is positively influenced by past spot returns — notably SPOT_RET(-1)

Past ETF returns negatively influence current ETF returns, especially the 1-period lag.

This suggests mean-reversion in ETF returns but positive spillover from spot retu

🔗 Key Takeaways:
There is strong evidence of mutual influence between spot and ETF returns:

Lagged ETF returns help predict spot returns

Lagged spot returns help predict ETF returns

This indicates a bi-directional short-run relationship.

The stronger positive coefficients on ETF_RET(-1) in the SPOT_RET equation and SPOT_RE

📌 Strategic Implications (e.g., for trading or risk m

Traders might exploit lead-lag dynamics between spot and ETF returns.

Risk managers should consider spillover effects when modeling volatility or risk in port
5⋅ETF_RETt−1+0.0274⋅ETF_RETt−2−0.00016\text{SPOT\_RET}_t = -0.1851 \cdot \text{SPOT\_RET}_{t-1} - 0.0

s, especially lag 1.

, suggesting that ETF returns may help predict spot returns in the short run.
TF_RETt−1−0.0531⋅ETF_RETt−2−0.00024\text{ETF\_RET}_t = 0.2381 \cdot \text{SPOT\_RET}_{t-1} + 0.1206 \

— notably SPOT_RET(-1), which has a large and positive coefficient (0.2381).

pecially the 1-period lag.

pillover from spot returns to ETF returns.

and ETF returns:

ET equation and SPOT_RET(-1) in the ETF_RET equation suggest feedback effects in both directions.

ading or risk management):

F returns.

g volatility or risk in portfolios containing both spot and ETF instruments.

{SPOT\_RET}_{t-1} - 0.0476 \cdot \text{SPOT\_RET}_{t-2} + 0.1515 \cdot \text{ETF\_RET}_{t-1} + 0.0274 \cd
T\_RET}_{t-1} + 0.1206 \cdot \text{SPOT\_RET}_{t-2} - 0.1474 \cdot \text{ETF\_RET}_{t-1} - 0.0531 \cdot \tex

both directions.
RET}_{t-1} + 0.0274 \cdot \text{ETF\_RET}_{t-2} - 0.00016SPOT_RETt ​=−0.1851⋅SPOT_RETt−1​−0.0476⋅SPOT_
_{t-1} - 0.0531 \cdot \text{ETF\_RET}_{t-2} - 0.00024ETF_RETt ​=0.2381⋅SPOT_RETt−1​+0.1206⋅SPOT_RETt−2​−0
T_RETt−1​−0.0476⋅SPOT_RETt−2​+0.1515⋅ETF_RETt−1​+0.0274⋅ETF_RETt−2​−0.00016
​+0.1206⋅SPOT_RETt−2​−0.1474⋅ETF_RETt−1​−0.0531⋅ETF_RETt−2​−0.00024
Date: 04/16/25 Time: 03:37
Sample (adjusted): 1/08/2020 12/30/2020
Included observations: 245 after adjustments
Trend assumption: Linear deterministic trend
Series: NIFTYBEES NIFTY
Lags interval (in first differences): 1 to 4

Unrestricted Cointegration Rank Test (Trace)

Hypothesized Trace 0.05

No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None * 0.094153 24.37846 15.49471 0.0018

At most 1 0.000619 0.151616 3.841465 0.6970

Trace test indicates 1 cointegrating eqn(s) at the 0.05 level

* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

Hypothesized Max-Eigen 0.05

No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None * 0.094153 24.22685 14.26460 0.0010

At most 1 0.000619 0.151616 3.841465 0.6970

Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level

* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegrating Coefficients (normalized by b'*S11*b=I):

NIFTYBEES NIFTY
-1.950963 0.021002
0.816668 -0.007985

Unrestricted Adjustment Coefficients (alpha):

D(NIFTYBEES) 0.075589 -0.041121

D(NIFTY) -4.333123 -4.501244

1 Cointegrating Equation(s): Log likelihood -1716.686

Normalized cointegrating coefficients (standard error in parentheses)

NIFTYBEES NIFTY
1.000000 -0.010765
(7.7E-05)

Adjustment coefficients (standard error in parentheses)

D(NIFTYBEES) -0.147472
(0.21251)
D(NIFTY) 8.453765
(23.0951)
This output presents the results of a Johansen Cointegration Test performed on two tim

🧪 1. Johansen Cointegration Test (Trace & Max Eig

✅ Trace and Maximum Eigenvalue Results:

Critical
Hypothe Trace Conclusi
Value p-value
sis Statistic on
(5%)
❗Reject
null → 1
None (r = cointegr
24.38 15.49 0.0018
0) ating
equation
exists

✅ Fail to
reject null
At most 1
0.15 3.84 0.697 → only 1
(r ≤ 1)
relations
hip

Same conclusion from Max Eigenvalue test:

Max statistic: 24.23 > 14.26 (critical) → reject null

p-value: 0.0010

✅ Conclusion:

There exists one cointegrating relationship between NIFTYBEES and NIFTY —

🔗 2. Cointegrating Vector (β):

From the normalized cointegration coefficients:

NIFTYBEES=0.010765⋅NIFTY\text{NIFTYBEES} = 0.010765 \cdot \text{NIFTY}NIFTYBEES=0

Or more precisely, the cointegrating equation (normalized on NIFTYBEES):

NIFTYBEESt−0.010765⋅NIFTYt=Stationary\text{NIFTYBEES}_t - 0.010765 \cdot \text{NIFTY

This implies:

There's a long-run equilibrium relationship.

If the difference between NIFTYBEES and 0.010765 × NIFTY deviates from the norm, it wi

⚙️3. Adjustment Coefficients (α):

These reflect how each variable responds to deviations from the long-run equilibrium

Adjustm
ent Interpret
Variable
Coefficie ation
nt (α)

Weak
response
→ small,
negative
adjustm
D(NIFTY
-0.147472 ent
BEES)
toward
equilibriu
m. ETF
adjusts
slowly.

Strong,
positive
adjustm
ent — but
high
standard
D(NIFTY) 8.453765
error
(23.09),
suggests
statistical
insignific
ance.

📌 Interpretation:

NIFTYBEES shows modest but significant error correction, adjusting toward long-term
 NIFTY (the index) does not appear to adjust, indicating it is weakly exogenous

🔄 4. Overall Interpretation:
Cointegration exists → NIFTYBEES and NIFTY share a long-term relationship

NIFTYBEES follows NIFTY closely in the long run.

Short-run disequilibrium is mostly corrected by changes in NIFTYBEES

Hence, NIFTY drives NIFTYBEES, and not vice versa.

📊 Use Case Implications:

✅ ETF Tracking Efficiency: NIFTYBEES effectively tracks the NIFTY index over the long r

🔁 Pair Trading / Mean Reversion: You can build a strategy based on the mean-revertin

🔍 Risk Management: Strong cointegration implies that portfolio diversification benefits

est performed on two time series: NIFTYBEES (an ETF tracking the NIFTY index) and NIFTY (the underlyi

ace & Max Eigenvalue Tests)

ES and NIFTY — they are cointegrated.

ext{NIFTY}NIFTYBEES=0.010765⋅NIFTY

NIFTYBEES):

010765 \cdot \text{NIFTY}_t = \text{Stationary}NIFTYBEESt​−0.010765⋅NIFTYt​=Stationary

 ates from the norm, it will correct over time.

e long-run equilibrium.

adjusting toward long-term equilibrium.

weakly exogenous — it drives the relationship.

erm relationship.

n NIFTYBEES, not NIFTY.

FTY index over the long run.

sed on the mean-reverting behavior of the spread.

o diversification benefits may be limited.

nd NIFTY (the underlying index itself). Here's a breakdown of the key interpretations:
Vector Error Correction Estimates
Date: 04/16/25 Time: 04:02
Sample (adjusted): 1/06/2020 12/30/2020
Included observations: 247 after adjustments
Standard errors in ( ) & t-statistics in [ ]

Cointegrating Eq: CointEq1

NIFTYBEES(-1) 1.000000

NIFTY(-1) -0.010671
(5.8E-05)
[-184.732]

C 0.264439

Error Correction: D(NIFTYB... D(NIFTY)

CointEq1 -0.408138 -10.05350

(0.19772) (21.2531)
[-2.06425] [-0.47304]

D(NIFTYBEES(-1)) -0.247023 -22.25373

(0.27321) (29.3681)
[-0.90415] [-0.75775]

D(NIFTYBEES(-2)) 0.965724 95.82725

(0.24264) (26.0825)
[ 3.97999] [ 3.67400]

D(NIFTY(-1)) 0.001665 0.103992

(0.00269) (0.28899)
[ 0.61917] [ 0.35985]

D(NIFTY(-2)) -0.008063 -0.747452

(0.00236) (0.25403)
[-3.41176] [-2.94232]

C 0.070105 6.083351
(0.10991) (11.8147)
[ 0.63784] [ 0.51490]

R-squared 0.100894 0.087314

Adj. R-squared 0.082240 0.068379
Sum sq. resids 715.7218 8269947.
S.E. equation 1.723311 185.2434
F-statistic 5.408785 4.611175
Log likelihood -481.8699 -1637.193
Akaike AIC 3.950363 13.30521
Schwarz SC 4.035612 13.39046
Mean dependent 0.079190 7.106478
S.D. dependent 1.798867 191.9213

Determinant resid covariance (dof adj.) 5345.178

Determinant resid covariance 5088.647
Log likelihood -1754.999
Akaike information criterion 14.32388
Schwarz criterion 14.52279
Number of coefficients 14
This output presents the results from a Vector Error Correction Model (VECM) using the two varia

Let’s interpret this VECM result step by step:

🔁 1. Cointegrating Equation (Long-Run Relationship)

The cointegrating equation is:

NIFTYBEESt−1=0.010671⋅NIFTYt−1+0.264439\text{NIFTYBEES}_{t-1} = 0.010671 \cdot \text{NIFTY}

Or,

NIFTYBEESt−1−0.010671⋅NIFTYt−1−0.264439=0\text{NIFTYBEES}_{t-1} - 0.010671 \cdot \text{NIFT

This represents the long-run equilibrium relationship between the two. The small coefficient (0.01

⚖️2. Error Correction Term (ECT or CoinEq1)

This measures the speed of adjustment toward the long-run equilibrium.

Depende ECT
t- Interpret
nt Coefficie
Statistic ation
Variable nt

✅
Significa
nt:
D(NIFTY NIFTYBEE
-0.408138 -2.06
BEES) S adjusts
to restore
equilibriu
m

❌ Not
significant
D(NIFTY) -10.0535 -0.473 : NIFTY
does not
adjust

✅ Conclusion:

Only NIFTYBEES adjusts to eliminate disequilibrium → NIFTY drives the relationship

 The negative sign of -0.4081 means about 40.8% of the disequilibrium is corrected in the next p

🔄 3. Short-run Dynamics (Lagged Differences)

For D(NIFTYBEES):

Interpret
Variable Coeff. t-Stat
ation

D(NIFTYB ❌ Not
-0.247 -0.9
EES(-1)) significant

✅
D(NIFTYB Significan
0.9657 3.98
EES(-2)) t positive
effect

D(NIFTY(- ❌ Not
0.0017 0.62
1)) significant

✅
D(NIFTY(- Significan
-0.0081 -3.41
2)) t negative
effect

Interpretation: Past changes in NIFTYBEES (lag 2) and past changes in NIFTY (lag 2)

For D(NIFTY):

Interpret
Variable Coeff. t-Stat
ation

D(NIFTYB ❌ Not
-22.25 -0.76
EES(-1)) significant

✅
D(NIFTYB
95.83 3.67 Significan
EES(-2))
t

D(NIFTY(- ❌ Not
0.1039 0.36
1)) significant
 ✅
D(NIFTY(-
-0.747 -2.94 Significan
2))
t

Interpretation: Lagged changes in NIFTYBEES (lag 2) and NIFTY (lag 2) significantly affect sho

However, since the error correction term for NIFTY is not significant, it doesn't respond to long-

📊 4. Model Fit
D(NIFTY
Metric D(NIFTY)
BEES)

R-squared 10.10% 8.73%

Adj. R-
8.22% 6.84%
squared
Akaike
Informatio 3.95 13.31
n Crit.
Schwarz
4.03 13.39
Criterion

These values are modest — typical for financial time series — but acceptable given the cointegratio

✅ Final Summary
NIFTYBEES and NIFTY are cointegrated, confirming a long-run equilibrium relationship.

NIFTY drives NIFTYBEES, as NIFTY doesn't adjust to disequilibrium.

The error correction term is significant for NIFTYBEES, which restores equilibrium by adjusting

The short-run dynamics show significant effects from past lags, especially at lag 2 for both series.

The VECM confirms that NIFTYBEES is a follower, and NIFTY is the leader in the long-run relatio
ECM) using the two variables: NIFTYBEES and NIFTY.

lationship)

10671 \cdot \text{NIFTY}_{t-1} + 0.264439NIFTYBEESt−1​=0.010671⋅NIFTYt−1​+0.264439

0.010671 \cdot \text{NIFTY}_{t-1} - 0.264439 = 0NIFTYBEESt−1​−0.010671⋅NIFTYt−1​−0.264439=0

he small coefficient (0.010671) reflects the scale difference between the index and ETF price.

relationship.
is corrected in the next period.

n NIFTY (lag 2) influence short-run changes in NIFTYBEES.

2) significantly affect short-run changes in NIFTY.

doesn't respond to long-run disequilibrium.

ble given the cointegration and error correction dynamics.

brium relationship.

s equilibrium by adjusting its prices.

y at lag 2 for both series.

der in the long-run relationship.

−0.264439=0