BECE-142 Applied Econometrics SOLVED ASSIGNMENT 2024-25
Q 1. (a) A research study involves examining the impact of Pradhan Mantri Jan Dhan Yojana initiative on the economically weaker section in the state of Madhya Pradesh. Suggest an appropriate research design (in terms of quantitative and qualitative research designs) to undertake such a study. Give reasons.
ANS.
Research Design for Pradhan Mantri Jan Dhan Yojana Impact Study
- Quantitative Research Design:
- Approach: Employ statistical methods to measure the program's impact on financial inclusion.
- Methods:
- Large-scale surveys to gather data on financial indicators (account ownership, usage, savings).
- Econometric modeling (regression, difference-in-differences) to quantify the relationship between program participation and economic outcomes.
- Reasons:
- Provides statistically valid and generalizable evidence.
- Enables precise measurement of the program's effects.
- Allows for controlling confounding variables.
- Qualitative Research Design:
- Approach: Explore the lived experiences and perceptions of beneficiaries.
- Methods:
- In-depth interviews to understand individual experiences.
- Focus groups to gather collective insights.
- Case studies to provide detailed narratives.
- Reasons:
- Provides rich, contextual understanding.
- Captures nuances and complexities that quantitative data may miss.
- Reveals the "why" behind observed outcomes.
- Mixed-Methods Approach (Recommended):
- Combine both quantitative and qualitative methods.
1 - Reasons:
- Provides a comprehensive and robust understanding of the program's impact.
- Triangulates findings, enhancing validity.
- Allows for deeper insights by integrating statistical evidence with qualitative narratives.
- Combine both quantitative and qualitative methods.
Q1 (b) Discuss the difference between Univariate, Bivariate and Multivariate analysis?
Ans.
- Univariate Analysis:
- Examines a single variable.
- Describes its distribution (mean, median, standard deviation).
- Example: Analyzing the average age of respondents.
- Bivariate Analysis:
- Examines the relationship between two variables.
- Determines the strength and direction of the relationship.
- Example: Analyzing the relationship between education level and income.
- Multivariate Analysis:
- Examines the relationships among three or more variables.
- Explores complex interactions.
- Example: Analyzing the combined effects of education, age, and gender on income.
Ans.
No, correlation does not imply causation. Just because a study finds a relationship between obesity and cancer does not mean that obesity directly causes cancer. Several possibilities could explain the correlation:
- Common Risk Factors: Obesity and cancer may share underlying risk factors, such as poor diet, lack of exercise, or genetic predisposition.
- Indirect Effects: Obesity might contribute to conditions (like inflammation or hormonal imbalances) that increase cancer risk, but other factors could also play a role.
- Reverse Causation: In some cases, an illness (like cancer) could lead to weight gain or changes in metabolism, making it appear as if obesity caused cancer.
- Confounding Variables: Other lifestyle factors, such as smoking, alcohol consumption, or environmental exposures, might influence both obesity and cancer risk.
The Akaike Information Criterion (AIC) and the Adjusted
R² criterion are both used for model selection, but they serve different
purposes and have their own strengths. Whether AIC is superior to Adjusted R²
depends on the context of model selection.
Comparing AIC and Adjusted R²
|
Criterion |
Akaike Information Criterion (AIC) |
Adjusted R² |
|
Purpose |
Measures model fit while penalizing complexity (avoids
overfitting). |
Adjusts R² for the number of predictors to avoid
inflation. |
|
Penalty for Extra Variables |
Stronger penalty for additional parameters. |
Adjusts for the number of predictors but does not penalize
as strongly as AIC. |
|
Interpretation |
Lower AIC is better (compares models, not absolute fit). |
Higher Adjusted R² is better (indicates goodness of fit). |
|
Use Case |
Best for comparing models with different numbers of
predictors. |
Best for explaining variance in a single model. |
- Penalty
for Complexity:
- AIC
discourages overfitting by penalizing excessive predictors more
effectively than Adjusted R².
- Adjusted
R² still increases when new variables improve model fit, even if they are
not truly necessary.
- Model
Comparison Across Different Models:
- AIC
is useful when comparing non-nested models (models that don’t simply add
or remove variables but are fundamentally different).
- Adjusted
R² is mostly useful for nested models (adding/removing predictors in the
same framework).
- Likelihood-Based
Approach:
- AIC
is derived from likelihood estimation and is grounded in information
theory, making it more generalizable across different types of
statistical models.
Illustration
Example: Choosing a Model for Predicting House Prices
Suppose we build two regression models:
- Model
1 (Simple model): Predicts house prices using square footage.
- Model
2 (Complex model): Predicts house prices using square footage, number
of bedrooms, crime rate, and distance to the city center.
- Adjusted
R² may favor Model 2 because adding more variables improves the fit
slightly.
- AIC might prefer Model 1 if the additional variables do not significantly improve the model’s predictive power, penalizing unnecessary complexity.
For a variable XX, the natural logarithm is given by:
Y=ln(X)
where Y represents the logged value of X.
Use Logs in Economic Data
- Convert Non-Linear Relationships into Linear Forms
- Interpret Economic Elasticities Directly
- Reduce Skewness and Stabilize Variance (Heteroscedasticity)
- Ease
of Interpretation in Growth Models
Factors Contributing to the ‘Log Effect’
- Exponential Growth Phenomena
- Diminishing Returns and Scale Effects
- Proportional Changes Over Absolute Changes
- Financial and Market Data Processing
Both the Logit and Probit models are used for
binary outcome variables (e.g., success/failure, yes/no, employed/unemployed).
They model the probability that an event occurs as a function of independent
variables.
|
Feature |
Logit Model |
Probit Model |
|
Function Used |
Logistic function |
Normal cumulative distribution function (CDF) |
|
Formula |
 |  |
|
Interpretation of Coefficients |
Log-odds ratio |
Marginal effects based on standard normal distribution |
|
Tail Behavior |
Longer tails, more sensitive to extreme values |
Shorter tails, less sensitive to outliers |
|
Usage |
More common in economics, machine learning |
Preferred in social sciences where normality assumption
holds |
Scenario:
A bank wants to predict whether a loan application will be approved
(1) or denied (0) based on the applicant’s income (X1) and credit
score (X2).
The estimated logit model:
- If Income
(X1X_1X1)
increases by $1,000, the log-odds of approval increase by 0.03.
- If Credit
Score (X2X_2X2)
increases by 10 points, the log-odds increase by 8%.
A multiple regression model extends simple regression
by including multiple independent variables to predict a dependent variable.
The key assumptions remain largely the same as in simple regression, but
multiple regression introduces additional considerations due to the presence of
multiple predictors.
1. Linearity
- The
relationship between the dependent variable (YYY) and the independent
variables (X1,X2,…,XnX_1, X_2, \dots, X_nX1,X2,…,Xn) should be linear.
- Violation:
Non-linear relationships can lead to biased estimates.
- Solution:
Transform variables (e.g., log transformation) or use non-linear
regression models.
2. Independence (No Autocorrelation)
- Observations
should be independent of each other.
- In time
series data, autocorrelation (correlation of residuals over time) is a
concern.
- Violation:
Leads to inefficient estimates and unreliable hypothesis testing.
- Solution:
Use Durbin-Watson test to detect autocorrelation and apply remedies
like differencing or autoregressive models.
3. No Perfect Multicollinearity
- Independent
variables should not be highly correlated with each other.
- Violation:
High multicollinearity inflates standard errors, making coefficient
estimates unreliable.
- Solution:
- Check
Variance Inflation Factor (VIF); if VIF > 10, remove or combine
correlated predictors.
- Use
Principal Component Analysis (PCA) or Ridge Regression if
necessary.
4. Homoscedasticity (Constant Variance of Errors)
- The
variance of residuals (errors) should be constant across all values
of XXX.
- Violation
(Heteroscedasticity): Unequal variance leads to inefficient estimates.
- Solution:
- Use
Breusch-Pagan Test or White’s Test to detect
heteroscedasticity.
- Apply
log transformation or use robust standard errors.
5. Normality of Residuals
- The
residuals (errors) should be normally distributed for valid
hypothesis testing and confidence intervals.
- Violation:
Impacts the reliability of statistical tests (t-tests, F-tests).
- Solution:
- Use
histograms, Q-Q plots, or the Shapiro-Wilk test to check
normality.
- Apply
log transformation or use non-parametric methods.
6. No Omitted Variable Bias
- The
model should include all relevant predictors; omitting important
variables biases estimates.
- Violation:
Leads to underfitting, making the model unreliable.
- Solution:
Use theory-driven model selection and statistical tests like Ramsey’s
RESET test.
1. Fixed Effects (FE) Model
- Assumes
that individual-specific effects (αi\alpha_iαi) are correlated with
explanatory variables.
- Controls
for time-invariant unobserved factors within each entity (e.g.,
person, company, or country).
- Suitable
when analyzing the impact of variables within an entity over time.
- Differences
across entities are absorbed into the intercept.
2. Random Effects (RE) Model
- Assumes
individual-specific effects (αi\alpha_iαi) are uncorrelated with explanatory
variables.
- More
efficient than FE if the assumption holds.
- Allows
for both within-group and between-group variation.
Assumptions of Fixed Effects Model
- Linearity
- The
relationship between independent and dependent variables is linear.
- Strict
Exogeneity
- The independent variables (Xit) should not be correlated with the error term (ϵit)
- Time-Invariant
Individual Effects (αi)
- Each
entity has its own fixed effect, which does not change over time.
- No
Perfect Multicollinearity
- Independent
variables should not be perfectly correlated.
- Homoscedasticity
and No Serial Correlation
- The
residuals should have constant variance (no heteroscedasticity).
- No autocorrelation
(errors should not be correlated across time for the same entity).
Ordinary Least Squares (OLS) is a widely used method for
estimating linear regression models, but it is not appropriate
for estimating binary dependent variable models (e.g., Yes/No, 0/1
outcomes). Here’s why OLS is problematic and why alternative methods like Logit
or Probit models are preferred.
Violation of the Assumption of Linearity
- OLS
assumes that the relationship between the independent variables (XX) and
the dependent variable (YY) is linear.
- However,
in a binary model, the true relationship is often non-linear.
- OLS
forces a linear probability model (LPM), which may predict
probabilities less than 0 or greater than 1, which is nonsensical.
Heteroscedasticity in Residuals
- In
OLS, residuals (ϵi\epsilon_i) should have constant variance
(homoscedasticity).
- In
binary models, residual variance depends on XX, leading to heteroscedasticity.
- This
violates an OLS assumption, making standard errors biased and
inefficient.
Inefficiency of OLS Estimates
- OLS
does not maximize the likelihood for binary models.
- Instead,
Maximum Likelihood Estimation (MLE) is preferred (as used in Logit
and Probit models), leading to more efficient parameter
estimates.
- OLS fails
for binary models due to non-linearity, heteroscedasticity, and invalid
probabilities.
- Logit
and Probit models provide better, more efficient estimates
using Maximum Likelihood Estimation (MLE).
- OLS may
be used as an approximation, but it is not theoretically sound
for binary outcomes.
Ans.
Identification refers to the ability to uniquely estimate
the true parameters of an economic model using observed data. If
a model is not identified, it means multiple sets of parameter values
could explain the data equally well, making estimation impossible or
unreliable.
Identification is crucial in simultaneous equations
models (SEM) and causal inference, where distinguishing between correlation
and causation is essential.
The Problem of Identification
The identification problem arises when the parameters of an
economic model cannot be uniquely determined because the available data
does not contain enough variation or information. This issue is common in:
- Simultaneous
Equations Models (SEM)
- Causal
Inference and Instrumental Variables (IV)
- Structural
vs. Reduced Form Models
Conditions for Identification
A system of equations is identified if we can determine unique
estimates for its parameters. There are three possible cases:
1. Under-Identification (Not Identified)
- The
number of unknown parameters exceeds the number of independent
equations.
- The
model cannot be estimated because there is not enough information.
- Example:
A demand-supply system with the same variables in both equations.
2. Exact Identification (Just Identified)
- The
number of independent equations matches the number of unknown
parameters.
- The
model can be estimated with a unique solution.
3. Over-Identification (Overidentified)
- More
equations than parameters exist, allowing estimation through techniques
like instrumental variables (IV).
- The
model can be estimated, but statistical methods like the Sargan test
are needed to check validity.
In hypothesis testing, errors occur when we make
incorrect conclusions about a population based on sample data. The two types of
errors are:
1. Type I Error (False Positive)
- Occurs
when we reject a true null hypothesis (H0).
- It
is denoted by α (alpha), which represents the significance level
of the test.
- Example:
- A court
case where an innocent person is wrongly convicted.
- A medical
test where a healthy person is diagnosed with a disease.
Illustration:
If H0 = "The patient is healthy" and H1"The patient has a disease":
- A Type
I error occurs if the test wrongly detects a disease in a healthy
patient.
2. Type II Error (False Negative)
- Occurs
when we fail to reject a false null hypothesis (H0).
- It
is denoted by β (beta), and 1 - β is the test’s power.
- Example:
- A court
case where a guilty person is wrongly acquitted.
- A medical
test failing to detect a disease in a sick patient.
Illustration:
If H0 = "The patient is healthy" and H1 =
"The patient has a disease":
- A Type
II error occurs if the test fails to detect a disease in a sick
patient.
1. Research Methodology
- Definition:
Research methodology is the philosophical framework and overall
approach to conducting research.
- It
explains the why and how of research.
- It
includes the research design, sampling techniques, data collection
methods, and data analysis strategies.
Key Aspects of Research Methodology:
- Research
Paradigms (Qualitative, Quantitative, Mixed Methods)
- Research
Design (Descriptive, Experimental, Case Study, etc.)
- Sampling
Methods (Random Sampling, Stratified Sampling, etc.)
- Data
Collection Techniques (Surveys, Interviews, Observations)
Research Methods
- Definition:
Research methods are the specific techniques and procedures used to
collect and analyze data.
- It
focuses on the tools and techniques of research.
- Research methods are part of research methodology.
Sampling Design
- Definition:
Sampling design refers to the plan or strategy used to select a
subset (sample) from a larger population for research or analysis.
- The
goal is to ensure the sample is representative of the population to
make accurate inferences.
Key Aspects of Sampling Design:
- Target
Population – The group from which the sample is drawn.
- Sampling
Frame – A list or database of individuals in the population.
- Sample
Size – The number of units selected for the study.
- Sampling
Technique – The method used to select the sample.
Statistical Design
- Definition:
Statistical design refers to the mathematical and analytical framework
used to organize, analyze, and interpret data.
- It
ensures that the study is structured correctly for valid
statistical inferences.
Key Components of Statistical Design:
- Choice
of Variables – Selecting independent and dependent variables.
- Control
of Confounding Factors – Ensuring extraneous variables don’t affect
results.
- Choice
of Statistical Tests – Selecting appropriate tests like t-tests,
ANOVA, regression, or chi-square tests.
- Design
of Experiments (DOE) – Structuring the study to minimize bias and
maximize accuracy.
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