Chapter 7 — Correlation

Overview

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🔌1

You exceeded your current quota

Fig 1 — Educational Diagram: You exceeded your current quota

⚡ PHYSICAL LAW / FORMULA

You exceeded your current quota

Key Point: Pearson correlation coefficient: r = Σ (x_i - x̄)(y_i - ȳ) / sqrt[Σ (x_i - x̄)^2 * Σ (y_i - ȳ)^2]

Meaning in economics (quota): In trade and resource management, a quota is a fixed limit on the quantity of a good that can be imported, produced or used. The message "You exceeded your current quota" means the limit has been reached and further quantity cannot be legally or technically consumed/imported without special permission.

Relevance to Chapter: Correlation: Correlation studies the direction and strength of association between two variables. When a quota is in effect the relationship between variables (for example, imports and domestic price, or production and market supply) can change. Studying correlation helps us understand how strongly quota size is associated with outcomes like price level, domestic production, shortage incidence or black‑market activity.

How to analyze empirically: Collect paired observations (e.g., quota size and domestic price across countries or years). Plot a scatter diagram to see the pattern, compute correlation coefficients (Pearson for linear association, Spearman for monotonic/rank association) and test significance. Compare correlations before and after imposing a quota to see whether the association becomes stronger, weaker or changes sign. Remember correlation ≠ causation: a high correlation only indicates association; causality requires further analysis (experiments, instrumental variables, policy controls).

Practical implications: If exceeding a quota is associated with steep price increases (positive correlation between quota tightness and price), policymakers know tightening quotas raises prices and may cause shortages. If quota exceedance correlates with increased smuggling or black market size, enforcement and alternative policies (tariffs, subsidies) might be preferred.

📌 Examples

• International trade: A country sets an import quota on sugar. Data on annual quota limit and domestic sugar price show a negative correlation between quota limit and price (smaller quota → higher price). If importers report "You exceeded your current quota" frequently, shortages and higher prices follow.
• Fisheries management: A catch quota for a fish species. Observations of quota remaining and market catch volume can be correlated with fish prices. Exceeding quota often precedes temporary closures and price spikes.
• Telecom/data: A mobile operator sets a monthly data cap. When many users see "You exceeded your current quota", correlation between data cap tightness and customer complaints/incremental revenue can be measured. Tighter caps may correlate with higher overage charges.
• Carbon emissions trading: A firm’s quota (allowance) for emissions and its marginal abatement cost. If firms repeatedly exceed allowances, correlation between allowance tightness and compliance costs or purchase of extra permits can be analyzed.

🧮 Formulas

1. \[Pearson correlation coefficient: r = Σ (x_i - x̄)(y_i - ȳ) / sqrt[Σ (x_i - x̄)^2 * Σ (y_i - ȳ)^2]\]
2. \[Spearman rank correlation: r_s = 1 - (6 Σ d_i^2) / [n(n^2 - 1)]\]

   \[where d_i = difference in ranks\]
3. \[Covariance: cov(X,Y) = Σ (x_i - x̄)(y_i - ȳ) / (n - 1)\]
4. \[Coefficient of determination: R^2 = r^2 (fraction of variance in Y explained by linear association with X)\]
5. \[Significance test for Pearson r: t = r * sqrt((n - 2) / (1 - r^2))\]

   \[with df = n - 2\]
6. \[Simple linear regression (to model effect of quota on price): Price = a + b*(Quota) + ε\]

   \[where b measures change in price per unit change in quota\]

1. Define correlation and state what its sign and magnitude indicate. / सहसंबंध को परिभाषित कीजिए और बताइए कि इसका चिह्न तथा परिमाण क्या दर्शाते हैं।
   Show answer

   Correlation measures the direction and strength of association between two variables; the sign (+/-) shows whether they move together or in opposite directions, while the magnitude (closeness to 1) shows the strength of the relationship. / सहसंबंध दो चरों के बीच संबंध की दिशा और सामर्थ्य को मापता है; चिह्न (+/-) दर्शाता है कि वे एक साथ या विपरीत दिशा में चलते हैं, और परिमाण (1 के निकटता) संबंध की सामर्थ्य दर्शाता है।

2. Distinguish between Pearson's correlation coefficient and Spearman's rank correlation, stating when each is used. / पियर्सन सहसंबंध गुणांक और स्पीयरमैन कोटि सहसंबंध में अंतर बताइए तथा प्रत्येक का प्रयोग कब होता है।
   Show answer

   Pearson's r measures linear association between quantitative variables, while Spearman's rank correlation measures monotonic association using ranks and is used for ordinal data or when the relationship is non-linear but ordered. / पियर्सन का r मात्रात्मक चरों के बीच रैखिक संबंध मापता है, जबकि स्पीयरमैन कोटि सहसंबंध कोटियों का उपयोग कर एकदिष्ट संबंध मापता है और कोटिक्रमिक आंकड़ों या अरैखिक परंतु क्रमबद्ध संबंध के लिए प्रयोग होता है।

3. 'Correlation does not imply causation.' Explain this statement with reference to a quota and price example. / 'सहसंबंध कारणता नहीं दर्शाता।' इस कथन को कोटा और कीमत के उदाहरण से समझाइए।
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   A high correlation between quota size and price only shows they are associated, not that one causes the other; both could be driven by a third factor, so establishing causality needs further analysis like controlled study or instrumental variables. / कोटा आकार और कीमत के बीच उच्च सहसंबंध केवल यह दर्शाता है कि वे संबंधित हैं, यह नहीं कि एक दूसरे का कारण है; दोनों किसी तीसरे कारक से प्रभावित हो सकते हैं, अतः कारणता सिद्ध करने हेतु नियंत्रित अध्ययन जैसी अतिरिक्त जांच आवश्यक है।

4. What is a scatter diagram and how does its pattern help interpret correlation? / प्रकीर्ण आरेख क्या है और इसका प्रतिरूप सहसंबंध की व्याख्या में कैसे सहायक है?
   Show answer

   A scatter diagram plots paired observations of two variables; an upward pattern suggests positive correlation, a downward pattern negative correlation, and a scattered cloud with no trend suggests little or no correlation. / प्रकीर्ण आरेख दो चरों के युग्मित प्रेक्षणों को अंकित करता है; ऊपर की ओर प्रवृत्ति धनात्मक सहसंबंध, नीचे की ओर ऋणात्मक सहसंबंध, और बिना प्रवृत्ति का बिखरा समूह बहुत कम या कोई सहसंबंध नहीं दर्शाता है।

5. If the correlation between import quota size and domestic sugar price is r = -0.85, interpret this value. / यदि आयात कोटा आकार और घरेलू चीनी कीमत के बीच सहसंबंध r = -0.85 है, तो इस मान की व्याख्या कीजिए।
   Show answer

   r = -0.85 indicates a strong negative linear correlation, meaning a larger quota is strongly associated with a lower domestic price (and a smaller quota with a higher price). / r = -0.85 एक प्रबल ऋणात्मक रैखिक सहसंबंध दर्शाता है, अर्थात बड़ा कोटा घरेलू कीमत के घटने से और छोटा कोटा कीमत के बढ़ने से प्रबलता से संबद्ध है।

6. What does the coefficient of determination (R^2 = r^2) tell us? / निर्धारण गुणांक (R^2 = r^2) हमें क्या बताता है?
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   R^2 gives the fraction of the variation in the dependent variable that is explained by its linear association with the other variable; for example r = 0.8 gives R^2 = 0.64, meaning 64% of variation is explained. / R^2 आश्रित चर की कुल विचरणशीलता का वह अंश देता है जो दूसरे चर के साथ रैखिक संबंध द्वारा समझाया जाता है; उदाहरण के लिए r = 0.8 पर R^2 = 0.64, अर्थात 64% विचरण समझाया जाता है।

7. Calculate Spearman's rank correlation for 5 observations where the sum of squared rank differences (Σd^2) is 10. / 5 प्रेक्षणों के लिए स्पीयरमैन कोटि सहसंबंध की गणना कीजिए जहाँ कोटि अंतरों के वर्गों का योग (Σd^2) 10 है।
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   Using r_s = 1 - 6Σd^2 / [n(n^2-1)] = 1 - (6×10)/[5(25-1)] = 1 - 60/120 = 0.5, so there is a moderate positive rank correlation. / सूत्र r_s = 1 - 6Σd^2 / [n(n^2-1)] = 1 - (6×10)/[5(25-1)] = 1 - 60/120 = 0.5 का उपयोग करने पर, मध्यम धनात्मक कोटि सहसंबंध प्राप्त होता है।

8. Why is comparing correlation 'before and after' a policy such as imposing a quota analytically useful? / कोटा लगाने जैसी नीति से 'पहले और बाद' के सहसंबंध की तुलना विश्लेषणात्मक रूप से क्यों उपयोगी है?
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   Comparing correlations before and after the quota shows whether the association between variables (e.g., imports and price) strengthens, weakens, or changes sign, helping evaluate the policy's effect on outcomes like prices or shortages. / कोटा से पहले और बाद के सहसंबंधों की तुलना दर्शाती है कि चरों (जैसे आयात और कीमत) के बीच संबंध प्रबल, क्षीण या चिह्न-परिवर्तित होता है, जिससे कीमतों या कमी जैसे परिणामों पर नीति के प्रभाव का मूल्यांकन होता है।

Foundational laws & principles behind this chapter. Each one opens a full page — what it says, why it matters, five practice questions and the mistakes to avoid.