BECE-142 Free Solved Assignment

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.

Answer: Research Design for Examining the Impact of Pradhan Mantri Jan Dhan Yojana (PMJDY) on Economically Weaker Sections in Madhya Pradesh

1. Quantitative Research Design

Sampling:

• A stratified random sampling technique could be used, where households in different income strata (low, middle, and very low-income groups) across various districts of Madhya Pradesh are selected. Stratified sampling ensures that different segments of the economically weaker sections are represented adequately.

Data Collection:

• Data can be gathered through structured surveys and questionnaires administered to beneficiaries of PMJDY. These surveys should capture details on:
  • Number of bank accounts opened before and after PMJDY.
  • Changes in income levels, savings patterns, and access to loans.
  • Changes in household spending behavior and financial literacy.

Analysis:

• Statistical techniques like regression analysis could be applied to assess the relationship between PMJDY participation and the economic indicators of the beneficiaries. Key variables such as income, savings, credit access, and financial stability can be considered in the regression model to quantify the direct impact of the initiative.

Control Group:

• A control group of non-beneficiaries (those who are eligible but did not participate in the scheme) can be used to compare the differences in financial behavior, providing insights into the program’s true impact.

2. Qualitative Research Design

Sampling:

• A purposive sampling approach could be used to select individuals from various socio-economic backgrounds. This would ensure a diverse range of experiences from beneficiaries, non-beneficiaries, and local community leaders.

Data Collection:

• In-depth interviews and focus group discussions (FGDs) can be conducted with PMJDY beneficiaries to understand:
  • Their personal experiences of using the bank accounts.
  • The challenges faced in opening and maintaining accounts (e.g., accessibility, digital literacy, documentation).
  • The social and psychological impacts of financial inclusion, such as empowerment and social status.

Analysis:

• Thematic analysis can be used to identify common themes from the interviews and discussions. This approach will help in understanding how the PMJDY has affected social cohesion, financial literacy, and community-level economic changes.

Conclusion

Discuss the difference between Univariate, Bivariate and Multivariate analysis?

Answer: Difference Between Univariate, Bivariate, and Multivariate Analysis

Key Characteristics:

• It involves one variable, making it the most basic form of statistical analysis.
• Common measures include mean, median, mode, variance, standard deviation, and frequency distributions.
• Graphs commonly used in univariate analysis include histograms, box plots, and bar charts.

Example:

Key Characteristics:

• It examines the relationship between two variables.
• The analysis often involves correlation, regression, or cross-tabulation (contingency tables).
• Graphs used include scatter plots, line graphs, or bar charts.

Example:

Key Characteristics:

• It involves three or more variables, often simultaneously.
• It can handle both dependent and independent variables.
• Common methods include multiple regression, factor analysis, discriminant analysis, principal component analysis (PCA), and multivariate analysis of variance (MANOVA).
• Graphs include 3D scatter plots or cluster maps.

Example:

A study is conducted to find the relationship between city temperature on a single day and the elevation of the city. A random sample of various cities is presented in the following table:

Answer: To analyze the relationship between city temperature and elevation, we can go through each part of the problem as follows:

(i) Finding the Regression Equation

• $a$$a$aa$a$ is the intercept (temperature when elevation is zero),
• $b$$b$bb$b$ is the slope (the change in temperature for a one-unit change in elevation).

(ii) Finding the Residuals and Creating a Residual Plot

(iii) Estimating City Temperature for an Elevation of 5500 feet

(iv) Finding the Correlation Coefficient and Coefficient of Determination

• Interpretation of $r$$r$rr$r$: If $r$$r$rr$r$ is close to -1, it indicates a strong negative correlation (as elevation increases, temperature decreases).
• Interpretation of $R^2$$R^2$R^(2)R^2$R^2$: A high $R^2$$R^2$R^(2)R^2$R^2$ value, closer to 1, indicates that a large portion of the temperature variance is explained by elevation.

Conclusion

A study shows that there is a correlation between people who are obese and those that have cancer. Does that mean being obese causes cancer?

Answer: A correlation between obesity and cancer does not necessarily imply causation. Correlation simply indicates a relationship or association between two variables, where changes in one variable tend to be accompanied by changes in the other. However, it does not confirm that one variable directly causes the other. In the case of obesity and cancer, the observed correlation suggests that obese individuals may have a higher incidence of cancer, but this relationship does not automatically mean that obesity is the direct cause of cancer.

Reasons Why Correlation Does Not Equal Causation

1. Confounding Variables: There may be other underlying factors, known as confounders, that influence both obesity and cancer risk. For example, lifestyle factors such as poor diet, lack of physical activity, and smoking can contribute to both obesity and cancer. These confounders may explain the association without obesity being a direct cause of cancer.
2. Biological Complexity: While obesity is linked to certain health risks, cancer is a complex disease with multiple contributing factors, including genetics, environment, lifestyle, and infections. Obesity may be one of the risk factors that increase the likelihood of developing cancer, but it is rarely the sole cause. Other biological mechanisms, such as chronic inflammation associated with obesity, could contribute to cancer risk.
3. Reverse Causation: In some cases, the relationship may work in the opposite direction, where pre-existing conditions or factors that lead to cancer could also contribute to weight gain. Although this is less likely with cancer and obesity, it demonstrates why causation cannot be assumed from correlation alone.
4. Need for Further Research: Establishing causation requires rigorous experimental studies, such as longitudinal studies or randomized controlled trials, which can help determine whether obesity directly leads to an increased risk of cancer. Observational studies alone are insufficient to prove causation.

Conclusion

A standard error of the estimate is a measure of the accuracy of predictions. Elucidate.

Answer: Standard Error of the Estimate: Understanding Prediction Accuracy

Calculation and Interpretation

• $Y_i$$Y_i$Y_(i)Y_i$Y_i$ is the actual observed value,
• ${\hat{Y}}_i$${\widehat{Y}}_i$hat(Y)_(i)\hat{Y}_i${\hat{Y}}_i$ is the predicted value from the regression model,
• $n$$n$nn$n$ is the number of observations.

Significance of SEE in Predictions

1. Evaluating Model Fit: SEE provides insight into how well the model fits the data. A lower SEE suggests that the model captures the underlying pattern in the data more effectively, with fewer discrepancies between predicted and observed values.
2. Comparing Models: When comparing multiple regression models, SEE can be a useful criterion for selecting the model with higher predictive accuracy. A model with a lower SEE is generally preferred as it indicates greater reliability in predictions.
3. Confidence Intervals for Predictions: SEE is essential for constructing confidence intervals around predicted values. A smaller SEE results in narrower confidence intervals, indicating more precise predictions, whereas a larger SEE widens the intervals, reflecting higher uncertainty in predictions.

Conclusion

Do you think that Akaike information criterion (AIC) is superior to adjustment ${\overline{R}}^2$${\overline{R}}^2$bar(R)^(2)\bar{R}^2${\overline{R}}^2$ criterion in determining the choice of Model? Give reasons and illustration in support of your answer.

Answer: Yes, the Akaike Information Criterion (AIC) is often considered superior to the adjusted ${\overline{R}}^2$${\overline{R}}^2$bar(R)^(2)\bar{R}^2${\overline{R}}^2$ criterion for model selection, especially in complex models with multiple predictors. Both metrics are used to evaluate model quality, but they serve different purposes and have unique advantages.

Reasons Why AIC is Often Preferred

1. Focus on Predictive Accuracy:
   • AIC measures the relative quality of a statistical model by balancing goodness of fit with model complexity. It penalizes models for having more parameters, thus discouraging overfitting. The formula for AIC is:$$AIC=2k-2\ln (L)$$$$AIC=2k-2\ln (L)$$AIC=2k-2ln(L)AIC = 2k – 2\ln(L)$$AIC=2k-2\ln (L)$$where $k$$k$kk$k$ is the number of parameters and $L$$L$LL$L$ is the maximum likelihood of the model. Lower AIC values indicate a better model.
   • In contrast, adjusted ${\overline{R}}^2$${\overline{R}}^2$bar(R)^(2)\bar{R}^2${\overline{R}}^2$ focuses on the proportion of variance explained by the model, adjusted for the number of predictors. It’s useful for assessing fit within the dataset but doesn’t penalize complexity as strongly as AIC.
2. Comparative Nature:
   • AIC is particularly useful when comparing non-nested models (models that are not simple extensions of one another) and models with different numbers of predictors. Adjusted ${\overline{R}}^2$${\overline{R}}^2$bar(R)^(2)\bar{R}^2${\overline{R}}^2$ is more appropriate for nested model comparisons.
3. Flexibility Across Different Types of Models:
   • AIC applies to a broad range of models, including linear regression, logistic regression, and time series models. It is effective in handling models where assumptions like normality may not hold, making it a versatile tool across different statistical frameworks.

Illustration

Conclusion

What do you mean by the term ‘Logs’ in the context of economic data? Give an account of the factors contributing to ‘Log effect’. Give illustration in support of your answer.

Answer: Logs in the Context of Economic Data

Reasons for Using Logs and the ‘Log Effect’

1. Handling Skewed Data:
   • Many economic variables, such as income, expenditure, and GDP, are right-skewed, meaning they have a long tail on the right side. Taking the log of these variables can make the distribution more symmetric, which is often desirable in regression analysis for meeting the assumption of normally distributed residuals.
2. Interpreting Elasticities and Growth Rates:
   • Log transformations are useful for interpreting relationships in terms of percentage changes. In a log-linear model, the coefficient of a predictor represents the percentage change in the dependent variable for a 1% change in the predictor. This makes it easier to understand how sensitive the dependent variable is to changes in explanatory variables.
3. Stabilizing Variance:
   • Economic data often have heteroscedasticity, where variance increases with the level of a variable. Log transformation can stabilize variance, making the variable more suitable for linear regression models.
4. Modeling Non-linear Relationships:
   • Many economic relationships are non-linear (e.g., diminishing returns to scale). Taking logs allows us to model these relationships within a linear regression framework, enabling simpler analysis and interpretation.

Illustration of Log Effect

Conclusion

Distinguish between Logit model and Probit model. Explain with illustration the process involved in estimation of Logit model.

Answer: Distinction Between Logit and Probit Models

1. Link Function and Distribution:
   • Logit Model: The Logit model uses the logistic function as its link function. It assumes a logistic distribution for the error term. The probability of the outcome variable is given by:$$P(Y=1|X)=\frac{1}{1+e^{-(\alpha +\beta X)}}$$$$P(Y=1|X)=\frac{1}{1+e^{-(\alpha +\beta X)}}$$P(Y=1|X)=(1)/(1+e^(-(alpha+beta X)))P(Y=1|X) = \frac{1}{1 + e^{-(\alpha + \beta X)}}$$P(Y=1|X)=\frac{1}{1+e^{-(\alpha +\beta X)}}$$
   • Probit Model: The Probit model uses the cumulative distribution function (CDF) of the standard normal distribution as its link function. It assumes a normal distribution for the error term. The probability of the outcome variable is given by:$$P(Y=1|X)=\mathrm{\Phi }(\alpha +\beta X)$$$$P(Y=1|X)=\mathrm{\Phi }(\alpha +\beta X)$$P(Y=1|X)=Phi(alpha+beta X)P(Y=1|X) = \Phi(\alpha + \beta X)$$P(Y=1|X)=\mathrm{\Phi }(\alpha +\beta X)$$where $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi$\mathrm{\Phi }$ represents the CDF of the standard normal distribution.
2. Interpretation of Coefficients:
   • In the Logit model, coefficients are interpreted in terms of odds ratios (the odds of the outcome occurring).
   • In the Probit model, coefficients are interpreted in terms of z-scores.
3. Application: Both models are widely used in econometrics and social sciences. The Logit model is often preferred for its interpretability in terms of odds ratios, while the Probit model is useful when a normal distribution of the error term is assumed.

1. Specify the Model: Define the dependent variable $Y$$Y$YY$Y$ (e.g., success = 1, failure = 0) and the independent variable(s) $X$$X$XX$X$.
2. Log-Likelihood Function: The likelihood function for a binary outcome in the Logit model is:
   $$L=\prod _{i=1}^{n}P(Y_i|X_i{)}^{Y_i}[1-P(Y_i|X_i){]}^{1-Y_i}$$$$L=\prod _{i=1}^n P(Y_i|X_i{)}^{Y_i}[1-P(Y_i|X_i){]}^{1-Y_i}$$L=prod_(i=1)^(n)P(Y_(i)|X_(i))^(Y_(i))[1-P(Y_(i)|X_(i))]^(1-Y_(i))L = \prod_{i=1}^{n} P(Y_i|X_i)^{Y_i} [1 – P(Y_i|X_i)]^{1 – Y_i}$$L=\prod _{i=1}^{n}P(Y_i|X_i{)}^{Y_i}[1-P(Y_i|X_i){]}^{1-Y_i}$$
   where $P(Y_i|X_i)=\frac{1}{1+e^{-(\alpha +\beta X_i)}}$$P(Y_i|X_i)=\frac{1}{1+e^{-(\alpha +\beta X_i)}}$P(Y_(i)|X_(i))=(1)/(1+e^(-(alpha+betaX_(i))))P(Y_i|X_i) = \frac{1}{1 + e^{-(\alpha + \beta X_i)}}$P(Y_i|X_i)=\frac{1}{1+e^{-(\alpha +\beta X_i)}}$.
3. Maximization of the Likelihood: Using numerical optimization techniques, MLE maximizes the log-likelihood function to estimate the values of $\alpha$$\alpha$alpha\alpha$\alpha$ and $\beta$$\beta$beta\beta$\beta$ that make the observed data most probable.
4. Interpretation of Results: The estimated coefficient $\beta$$\beta$beta\beta$\beta$ in the Logit model represents the change in the log odds of the outcome for a one-unit increase in $X$$X$XX$X$. The odds ratio, calculated as $e^{\beta }$$e^{\beta }$e^(beta)e^{\beta}$e^{\beta }$, indicates the factor by which the odds of the outcome change for a one-unit change in $X$$X$XX$X$.

Illustration

What are the various assumptions considered for running a multiple regression model? Are these assumptions different from the ones considered under simple regression model?

Answer: Assumptions for Running a Multiple Regression Model

1. Linearity: The relationship between the dependent variable and each independent variable should be linear.
2. Independence of Errors: Observations should be independent of each other, with errors not correlated across observations.
3. Homoscedasticity: The variance of the error terms should be constant across all levels of the independent variables.
4. No Perfect Multicollinearity: Independent variables should not be perfectly correlated, as this would make it difficult to isolate their individual effects.
5. Normality of Errors: The residuals (errors) should follow a normal distribution, especially for small sample sizes.

Comparison with Simple Regression Assumptions

What is the difference between random effects approach and fixed effects approach for estimation of parameters? State the assumptions of fixed effects model.

Answer: Difference Between Random Effects and Fixed Effects Approaches

• Fixed Effects (FE) Approach: The fixed effects approach controls for all time-invariant characteristics of the entities by allowing each entity to have its own intercept. It assumes that individual characteristics are correlated with the predictor variables. This method removes entity-specific effects by "demeaning" the data, focusing on within-entity variation.
• Random Effects (RE) Approach: The random effects approach assumes that entity-specific characteristics are uncorrelated with the predictor variables. It considers both within- and between-entity variation, making it more efficient when the RE assumption holds.

Assumptions of Fixed Effects Model

1. Time-Invariant Characteristics: Assumes entity-specific characteristics (e.g., culture, policies) are constant over time.
2. No Correlation Among Error Terms: Error terms are uncorrelated over time within an entity.
3. No Perfect Multicollinearity: Predictor variables must not be perfectly correlated.
4. Strict Exogeneity: Assumes no correlation between error terms and predictor variables within each time period.

"OLS is appropriate method for estimating the parameters of Binary dependent variable model" Comment.

Answer: Ordinary Least Squares (OLS) is generally not appropriate for estimating the parameters of a binary dependent variable model. In cases where the dependent variable is binary (e.g., 0 or 1), OLS has several limitations:

1. Non-Linearity of Probability: OLS assumes a linear relationship between independent and dependent variables, but probabilities are inherently non-linear. For binary outcomes, OLS may predict probabilities outside the [0,1] range, which is meaningless in a probabilistic context.
2. Homoscedasticity Violation: OLS assumes constant variance (homoscedasticity) of errors. In binary models, the variance of the error term depends on the predicted probability, leading to heteroscedasticity.
3. Non-Normality of Errors: OLS assumes normally distributed errors, but binary outcomes create non-normal residuals, violating this assumption and compromising hypothesis testing accuracy.
4. Efficiency and Interpretability: The coefficients estimated by OLS lack a clear interpretation in terms of probabilities or odds. Models like Logit and Probit are specifically designed for binary outcomes and estimate parameters based on maximum likelihood estimation, providing probabilities bounded within [0,1] and interpretable coefficients.

What is meant by the problem of identification? Explain the conditions for identification.

Answer: Problem of Identification

Conditions for Identification

1. Order Condition: A necessary condition for identification is that the number of instruments (exogenous variables) must be at least as large as the number of endogenous variables in each equation. Specifically, for a model with $K$$K$KK$K$ endogenous variables, an equation is identified if the number of excluded exogenous variables is at least $K-1$$K-1$K-1K – 1$K-1$.
2. Rank Condition: The rank condition is a sufficient condition for identification. It states that the matrix of coefficients on the excluded exogenous variables must have full column rank. This condition ensures that each endogenous variable has a unique relationship with the exogenous variables, allowing distinct parameter values.

Distinguish between Type I and Type II errors.

Answer: Type I and Type II Errors: Definitions and Differences

1. Type I Error (False Positive):
   • Occurs when the null hypothesis is incorrectly rejected even though it is actually true.
   • It is also called an alpha error, and its probability is denoted by $\alpha$$\alpha$alpha\alpha$\alpha$, typically set at 0.05 or 5%.
   • Example: Concluding a drug is effective when it isn’t.
2. Type II Error (False Negative):
   • Occurs when the null hypothesis is incorrectly accepted even though it is actually false.
   • Known as a beta error, with its probability denoted by $\beta$$\beta$beta\beta$\beta$.
   • Example: Failing to detect an effective drug’s impact.

Key Differences:

• Impact: Type I error is considered more serious in contexts where false positives carry high risks, while Type II error may be more concerning in tests needing high sensitivity.
• Control: Decreasing Type I error often increases the likelihood of Type II error and vice versa, making a balance essential based on study goals.

Distinguish between Research methodology and research methods.

Answer: Research Methodology vs. Research Methods

1. Research Methodology:
   • Research methodology refers to the overall approach, principles, and rationale guiding a research study. It encompasses the philosophical basis of research (e.g., qualitative vs. quantitative), the logic of how research is conducted, and why certain techniques are chosen.
   • Methodology involves the theoretical analysis of research processes, helping researchers determine the best path for conducting a study. It includes decisions on sampling, data collection, and analysis techniques based on the research question.
   • Example: A researcher may choose an experimental methodology to test causal relationships or a survey methodology for descriptive analysis.
2. Research Methods:
   • Research methods are the specific techniques and procedures used to gather and analyze data. These are practical tools that researchers employ to collect evidence and generate findings.
   • Methods are the "how-to" steps in a study, such as surveys, interviews, experiments, or observations.
   • Example: A researcher conducting a survey could use a questionnaire as the research method to collect responses.

Distinguish between Sampling design and statistical design.

Answer: Sampling Design vs. Statistical Design

1. Sampling Design:
   • Sampling design refers to the process of selecting a subset of individuals or observations from a larger population to participate in a study. Its primary purpose is to ensure that the sample accurately represents the population, enabling generalizable findings.
   • Key elements in sampling design include determining the sampling technique (e.g., random, stratified, cluster) and sample size.
   • Example: A researcher choosing a stratified sampling design to ensure representation of different age groups within a population.
2. Statistical Design:
   • Statistical design relates to the structure of the experiment or study and specifies how data will be collected, measured, and analyzed. It involves planning the statistical methods to ensure valid, reliable, and interpretable results.
   • Elements of statistical design include defining variables, setting up controls, establishing treatments, and choosing appropriate statistical tests.
   • Example: A researcher planning an ANOVA design to compare the effectiveness of different teaching methods.

Distinguish between Pooled cross section data and panel data.

Answer: Pooled Cross-Section Data vs. Panel Data

1. Pooled Cross-Section Data:
   • Pooled cross-section data combines multiple cross-sectional datasets collected at different points in time. However, each cross-section represents a new sample, so the individuals or units are not necessarily the same across time periods.
   • This data type is often used to examine changes or trends over time without tracking the same subjects, making it useful for analyzing broad shifts in populations.
   • Example: A study of household income levels in 2000 and 2010 using a different sample of households each time.
2. Panel Data:
   • Panel data, also known as longitudinal data, consists of repeated observations of the same individuals or units over multiple time periods. This structure allows researchers to analyze both cross-sectional (differences among units) and time-series (changes over time) aspects.
   • Panel data is valuable for understanding individual-level dynamics, such as how a specific person’s income changes over time.
   • Example: Tracking the same set of companies’ revenues and expenses annually from 2010 to 2020.