Please humanize this assignment please note that the equations and appendixes that some portions of the assignment refer to hasn't been included as I haven't finished with it yet all I'm looking for is a rewrite so it doesn't get flagged for AI usage please keep the contents of the assignment the same and please don't go above 2100 words thank you in advance.

Q1. Difference-in-Differences Estimation

1. Parallel Trend Assumption

The difference-in-differences (DiD) framework relies heavily on the parallel trends assumption, which posits that in the absence of treatment, the average outcomes of treated and untreated groups would have evolved in a similar fashion over time. In other words, the untreated precincts provide a credible counterfactual for the treated ones.

Applied here, this assumption implies that if no facial recognition bans had been introduced, both treated and untreated precincts would have displayed similar time paths in stop-and-frisk searches (SAF), clearance rates (CLEAR), and arrests (ARR). This is intuitive because precincts operate under the same broad institutional, cultural, and federal conditions, with the primary difference being the presence or absence of the ban.

One limitation is that the dataset only includes a short pre-treatment window (2017 before bans, 2019 after), which constrains our ability to statistically test parallelism. Ideally, we would use multiple pre-treatment periods to conduct placebo regressions or trend tests. In criminology research, longer horizons are common to validate pre-trends. Nonetheless, graphical comparisons and regression baselines (Appendix A1) suggest that initial levels of policing outcomes were broadly similar, providing reasonablethough imperfectsupport for the assumption.

2. Why DiD is Preferred

Compared with a nave before-and-after comparison, DiD is preferred because it removes time-invariant differences between groups and accounts for common shocks. For instance, if both treated and untreated precincts faced a nationwide decline in crime due to demographic shifts, DiD subtracts this out.

From a methodological perspective, DiD exploits the interaction between time and treatment, allowing us to isolate the incremental effect of the ban. This helps mitigate omitted variable bias that would otherwise confound results. Wooldridge (2016) emphasises that this makes DiD particularly suitable for policy evaluation where randomisation is infeasible.

3. Manual DiD Estimate (Stop-and-Frisk)

The manual DiD estimator is:

Plugging in averages:

This implies bans reduced stop-and-frisk rates by 16.7 percentage points relative to untreated precincts. This substantial magnitude highlights the potential civil liberties impact of bans.

4. Regression-Based DiD Estimate

We next estimate:

The coefficient of interest, 3\beta_33, is the DiD estimate. Results (Appendix A3) give 3=0.245, which matches the manual calculation in direction and size but is statistically insignificant (p=0.615).

This means while bans reduce stop-and-frisk, the evidence is statistically weak given the small number of treated precincts and the limited post-ban window.

5. DID with Controls and Multiple Outcomes

Adding controls (income, unemployment, police numbers, population growth) and extending to all outcomes provides a fuller picture (Appendices A4-A6):

• Stop-and-frisk (SAF): Surprisingly, with controls, the DiD coefficient turns positive (+2.12) and significant. This suggests bans may have led to compensatory policingperhaps police increased stop-and-frisk in response to losing surveillance technology.
• Arrests (ARR): The effect remains negative but insignificant, implying no robust evidence that bans reduced arrests.
• Clearance (CLEAR): Negative but insignificant, pointing to a weak decline in investigative capacity.

Overall, the policy effect is mixed: bans may shift policing strategies (raising stop-and-frisk) without improving outcomes like clearance.

6. Diagnostics and Model Validity

Diagnostics strengthen confidence in the findings:

• Control coefficients align with theory (e.g., more police more arrests).
• R ranges from 0.55-0.68, showing moderate explanatory power.
• HAC and robust SEs confirm stability though widen confidence intervals.
• Residuals pass basic checks, though the limited time frame weakens power to test trends.

Policy Implication: Bans trade off civil liberty protection against ambiguous impacts on policing effectiveness. Legislators must weigh whether symbolic gains in privacy justify potential declines in clearance and compensatory increases in intrusive practices.

Q2. Instrumental Variable Estimation - Psychologists and Mental Health

1. OLS Estimation

OLS results (Appendix Q2.1) show a negative, insignificant coefficient on psy (-0.457, p = 0.821). This counterintuitive finding is consistent with selection bias: those seeking treatment are disproportionately those already struggling.

2. Causal Concerns

OLS is biased because of:

• Reverse causality: poor mental health higher likelihood of treatment.
• Unobserved heterogeneity: socioeconomic status, workplace culture, family support.
• Measurement error: self-reported health introduces noise.

Thus, OLS cannot be interpreted causally.

3. IV Estimation with CS and Rebate

We use cs and rebate as instruments for therapy participation. Both strongly predict access (first-stage F > 10), and plausibly affect mental health only through treatment.

Results (Appendix Q2.2):

• Coefficient on psy rises to 9.009 (p = 0.036).
• This suggests therapy substantially improves mental health once endogeneity is addressed.

This is a textbook case where IV reveals the "hidden" treatment effect that OLS obscured.

4. Instrument Relevance and Validity

• First-stage regressions show both instruments are relevant predictors.
• J-statistic (p = 0.821) confirms validity: instruments are exogenous.

5. Instruments Separately

• cs alone: Positive (9.98), marginally insignificant.
• rebate alone: Positive (7.37), also insignificant.
• Combined: Strongest and most precise.

This illustrates the econometric principle that multiple valid instruments improve efficiency.

6. Robustness

Robust SEs confirm results are stable: effect remains ~9 and significant.

Policy Implication: Evidence supports public investment in therapy access. The results suggest under OLS, benefits appear negligible, but once selection is corrected, therapy produces large health gains. This justifies subsidies and employer programs.

Q3. Time Series Modelling - Brent Crude and WTI

1. Stationarity

ADF tests reject unit roots for both Brent and WTI returns (p

2. Model Estimation

Several specifications were compared. Simpler models (ARDL(1,0), (1,1)) underfit, while higher-order models (3,1), (2,3)) risk overfitting.

Chosen model: ARDL(2,2)

3. Interpretation

• Own lags: Negative, significant mean reversion in Brent returns.
• WTI returns: Contemporaneous effect large (0.88), with persistent lag effects Brent reacts immediately and continues to adjust.

This reflects financial market integration, where WTI shocks transmit quickly due to US liquidity.

4. Diagnostics

• Adjusted R = 0.83.
• AIC and SIC lowest among tested models.
• DW 2 no serial correlation.
• LM test p = 0.097 residuals white noise.

5. Why ARDL(2,2)?

• Parsimonious yet flexible.
• Best fit among alternatives.
• Aligns with economic intuition: Brent is influenced both contemporaneously and with lags.

Implication: For trading strategies, monitoring WTI is criticalBrent is essentially "led" by US market movements.

Q4. Regression on Extramarital Affairs

Introduction

The analysis of extramarital affairs provides a useful case study for econometric modelling of social behaviour. Unlike physical markets, relationship dynamics are influenced by psychological, cultural, and moral factors, many of which are unobservable in the data. This makes the econometric exercise challenging, as our model will inevitably omit key determinants such as personality traits, peer networks, or workplace dynamics. Nonetheless, with 601 observations from a confidential survey, the dataset provides valuable insights into measurable correlates of affairs.

The dependent variable is number of affairs per year (Naffairs). Because this variable is skewed, many applied researchers transform it into a log form to normalise distribution, though our core results hold with the raw variable as well.

(a) Baseline Model

The baseline specification regresses Naffairs on age, years married, religiosity, education, and marital satisfaction. This mirrors the lecture emphasis on beginning with a simple linear model before experimenting with complexity.

Results show:

• Satisfaction (Ratemarr): Strongly negative and significant. This aligns perfectly with economic and psychological expectations: when marital satisfaction is high, the "utility" of seeking external relationships declines. Each additional unit of satisfaction (on a 1-5 scale) reduces expected affairs by a sizeable margin, making this the most robust predictor.
• Religiosity (Relig): Negative and significant. More religious individuals are less likely to have affairs. This reflects the role of social norms and moral costs religion imposes sanctions (both internal and communal) that discourage infidelity.
• Age: Weakly negative, suggesting older respondents are somewhat less likely to engage in affairs, though the effect is not large.
• Years married (Yrsmarr): Positive and significant. Each additional year of marriage increases risk slightly, consistent with theories of relationship "fatigue" or the idea that opportunities to stray accumulate over time.
• Education (Educ): Statistically insignificant. Schooling does not appear to influence infidelity once other controls are included.

The model's R is modest (~0.13). This is typical in behavioural regressions: human relationships are complex, and many drivers are unobservable. Low R does not mean the model is useless, only that predictive precision is limited.

(b) Extended Model

The second specification adds nonlinear terms (age, yrs), children (Kids), gender, and religiosity squared to capture possible curvature or threshold effects.

Findings:

• Age and yrs: Both insignificant. There is no evidence of nonlinear lifecycle patterns beyond the linear effects already captured.
• Children: Insignificant, which may seem surprising. Intuitively, one might expect children to reduce infidelity (higher costs of family breakdown). But once satisfaction and religiosity are controlled for, children provide no additional explanatory power. This may reflect two offsetting forces: children increase the cost of divorce but also increase stress, which may heighten risk.
• Gender: Also insignificant. This is notable, as sociological literature often posits men are more likely to have affairs. At least in this dataset, once satisfaction and religiosity are included, the gender gap disappears.
• Religiosity squared: Not significant. The negative relationship between religiosity and affairs appears linear.

The adjusted R does not improve materially, and many added coefficients are uninterpretable. This illustrates a common econometric lesson: adding variables does not guarantee stronger models. In fact, it risks overfitting, reducing clarity, and inflating standard errors.

(c) Parsimonious Model

Given the limited contribution of additional variables, the parsimonious model retains only the strongest predictors: age, years married, satisfaction, and religiosity.

Advantages:

• Adjusted R improves slightly, showing the model is as efficient as more complex versions.
• Coefficients are consistent, stable, and interpretable.
• Results align with both economic theory (cost-benefit of infidelity) and sociological frameworks (moral norms and relationship quality).

This demonstrates the parsimony principle taught in class: models should be as simple as possible while still capturing the essence of the phenomenon.

(d) Strengths and Weaknesses of the Models

Strengths:

• Coefficients on satisfaction and religiosity are robust across specifications.
• Signs generally align with theoretical expectations.
• Parsimonious model avoids noise from weak predictors.

Weaknesses:

• Low explanatory power: Only ~13% of variation explained. This reflects the inherently unobservable nature of personal choices.
• Endogeneity: Marital satisfaction may be endogenous dissatisfaction could both cause and result from affairs. This reciprocal relationship means OLS may understate true effects.
• Measurement error: Respondents may underreport affairs due to social desirability bias, biasing coefficients downward.
• Omitted variables: Income, cultural background, personality traits, and partner's characteristics are absent. These could be strong drivers of infidelity but are unmeasured.

(e) Diagnostics

Residuals appear normally distributed without severe heteroskedasticity. Variance Inflation Factors (VIF) indicate low collinearity (

(f) Economic and Social Rationale

Despite its limits, the model provides important insights. Marital satisfaction emerges as the central determinant: dissatisfaction sharply increases the likelihood of infidelity. Economically, this fits a rational-choice model of marriage individuals weigh costs and benefits, and when satisfaction is low, the relative "return" on extramarital activity rises.

Religiosity functions as a constraint, lowering the perceived benefit of affairs by raising moral and social costs. This suggests that social institutions, such as religious or community norms, play a disciplining role in behaviour.

Years married increases the risk of affairs, consistent with sociological theories of relationship fatigue and declining novelty. This highlights the need for sustained investment in long-term relationships.

The lack of gender effect challenges popular stereotypes. It suggests that once we control for satisfaction and religiosity, men and women behave similarly. This insight can shift the framing of infidelity from being "male-driven" to being fundamentally linked to relationship quality and social values.

Policy Implication: While not a traditional policy area, the results offer guidance for counselling and family support services. Interventions that raise relationship satisfaction such as communication workshops, marital education, or conflict resolution programs may reduce infidelity. Likewise, values-based programs (religious or secular) can reinforce fidelity norms.

The affairs regression highlights the value and limitations of econometric models in social science. The modest R shows predictive weakness, but the robustness of satisfaction and religiosity underlines their centrality. The lesson is not that econometrics can perfectly predict behaviour, but that it can identify consistent correlates which inform theory and policy.