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\u0646\u0638\u0631\u060c \u0633\u0633\u062a\u06d2 \u0645\u0627\u0688\u0644 \u06a9\u06cc \u0637\u0631\u0641 \u0627\u0633\u062a\u0641\u0633\u0627\u0631\u0627\u062a \u06a9\u0627 \u0631\u0627\u0633\u062a\u06c1 \u0628\u062f\u0644 \u062f\u06cc\u062a\u0627 \u06c1\u06d2\u06d4 \u0634\u0631\u062d \u06a9\u06cc \u062d\u062f \u0628\u0646\u062f\u06cc \u0628\u0646\u06cc\u0627\u062f\u06cc \u0688\u06be\u0627\u0646\u0686\u06d2 \u06a9\u06cc \u0648\u062c\u0648\u06c1\u0627\u062a \u06a9\u06cc \u0628\u0646\u0627\u0621 \u067e\u0631 \u0646\u0627\u0641\u0630 \u06a9\u06cc \u062c\u0627\u062a\u06cc \u06c1\u06d2\u060c \u0642\u0637\u0639 \u0646\u0638\u0631 \u0627\u0633 \u06a9\u06d2 \u06a9\u06c1 \u0635\u0627\u0631\u0641 \u0627\u0635\u0644 \u0645\u06cc\u06ba \u06a9\u06cc\u0627 \u062f\u0631\u062e\u0648\u0627\u0633\u062a \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u06cc\u06c1 \u0628\u06d2 \u062a\u0631\u062a\u06cc\u0628\u06cc \u0627\u06c1\u0645 \u06c1\u06d2\u060c \u0627\u0648\u0631 \u062f\u0648 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u06d2 \u06a9\u0645 \u0633\u06d2 \u06a9\u0645 \u0645\u0631\u0628\u0639 (2SLS) \u06c1\u0645\u06cc\u06ba \u0627\u0633 \u0633\u06d2 \u063a\u06cc\u0631 \u0645\u0628\u06c1\u0645 \u0648\u062c\u06c1 \u06a9\u0627 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u0646\u06a9\u0627\u0644\u0646\u06d2 \u06a9\u06cc \u0627\u062c\u0627\u0632\u062a \u062f\u06cc\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u06cc\u06c1 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u0622\u067e \u06a9\u0648 \u0627\u0632\u06af\u0631 \u0645\u06cc\u06ba \u062a\u0634\u062e\u06cc\u0635\u06cc \u062f\u0631\u0633\u062a\u06af\u06cc \u06a9\u06cc \u067e\u0648\u0631\u06cc \u062a\u0631\u062a\u06cc\u0628 \u0633\u06d2 \u0622\u06af\u0627\u06c1 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0631\u0648\u0679\u0646\u06af \u06a9\u0646\u0641\u0627\u0624\u0646\u0688\u0631\u0632 OLS \u06a9\u0627 \u062a\u0639\u0635\u0628 \u06a9\u06cc\u0648\u06ba \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062f\u0648 \u0645\u0646\u0633\u0644\u06a9 \u0631\u06cc\u06af\u0631\u06cc\u0634\u0646\u0632 \u0645\u06cc\u06ba \u0634\u0631\u0648\u0639 \u0633\u06d2 2SLS \u06a9\u06cc\u0633\u06d2 \u0628\u0646\u0627\u06cc\u0627 \u062c\u0627\u0626\u06d2\u060c \u067e\u06c1\u0644\u06d2 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u06d2 F-statistic \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0622\u0644\u06d2 \u06a9\u06cc \u0637\u0627\u0642\u062a \u06a9\u0648 \u06a9\u06cc\u0633\u06d2 \u0686\u06cc\u06a9 \u06a9\u06cc\u0627 \u062c\u0627\u0626\u06d2\u060c \u0627\u0648\u0631 \u0645\u0642\u0627\u0645\u06cc \u0627\u0648\u0633\u0637 \u0639\u0644\u0627\u062c \u06a9\u06d2 \u0627\u062b\u0631 \u06a9\u0648 \u06a9\u06cc\u0633\u06d2 \u0628\u062d\u0627\u0644 \u06a9\u06cc\u0627 \u062c\u0627\u0626\u06d2 \u062c\u0633 \u06a9\u0627 2SLS \u0627\u0635\u0644 \u0645\u06cc\u06ba \u0627\u0648\u0633\u0637 \u0639\u0644\u0627\u062c \u06a9\u06d2 \u0627\u062b\u0631 \u06a9\u0648 \u063a\u0644\u0637 \u0633\u0645\u062c\u06be\u0646\u06d2 \u06a9\u06d2 \u0628\u062c\u0627\u0626\u06d2 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u0644\u06af\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u0622\u062e\u0631 \u062a\u06a9\u060c \u0622\u067e \u062c\u0627\u0646 \u0644\u06cc\u06ba \u06af\u06d2 \u06a9\u06c1 \u0627\u067e\u0646\u06d2 \u0644\u0627\u06af\u0632 \u0645\u06cc\u06ba \u06a9\u0646\u0641\u06cc\u0648\u0632\u0646\u06af \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u0641\u06cc\u0635\u0644\u0648\u06ba \u06a9\u0648 \u06a9\u06cc\u0633\u06d2 \u067e\u06c1\u0686\u0627\u0646\u0627 \u062c\u0627\u0626\u06d2\u060c \u0627\u0646\u0641\u0631\u0627\u0633\u0679\u0631\u06a9\u0686\u0631 \u0633\u06af\u0646\u0644\u0632 \u0633\u06d2 \u062f\u0631\u0633\u062a \u0679\u0648\u0644\u0632 \u062c\u06cc\u0633\u06d2 \u0631\u06cc\u0679 \u06a9\u0648 \u0645\u062d\u062f\u0648\u062f \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0641\u0627\u0644 \u0628\u06cc\u06a9\u0633 \u06a9\u0648 \u06a9\u06cc\u0633\u06d2 \u0628\u0646\u0627\u06cc\u0627 \u062c\u0627\u0626\u06d2\u060c \u0627\u0648\u0631 \u062d\u062f \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0627\u0639\u062a\u0645\u0627\u062f \u0648\u0627\u0644\u06d2 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0648\u0642\u0641\u0648\u06ba \u06a9\u06cc \u0628\u062c\u0627\u0626\u06d2 \u0645\u0646\u0627\u0633\u0628 \u0633\u0627\u0626\u0632 \u06a9\u06d2 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0648\u0642\u0641\u0648\u06ba \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u06a9\u0627\u0632\u0644 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u062a\u06cc\u0627\u0631 \u06a9\u0631\u06cc\u06ba \u062c\u0648 \u0645\u06cc\u0646\u0648\u0626\u0644 2SLS \u0628\u0637\u0648\u0631 \u0688\u06cc\u0641\u0627\u0644\u0679 \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p><strong>\u0633\u0627\u062a\u06be \u0648\u0627\u0644\u06d2 \u0646\u0648\u0679<\/strong>: \u0627\u0633 \u0645\u0636\u0645\u0648\u0646 \u06a9\u06d2 \u062a\u0645\u0627\u0645 \u06a9\u0648\u0688 \u0628\u0644\u0627\u06a9\u0633 \u06a9\u0648 \u0622\u062e\u0631 \u0633\u06d2 \u0622\u062e\u0631 \u062a\u06a9 \u0639\u0645\u0644 \u0645\u06cc\u06ba \u0644\u0627\u06cc\u0627 \u06af\u06cc\u0627 \u06c1\u06d2\u06d4 <code>iv_demo.ipynb<\/code> \u0633\u0627\u062a\u06be\u06cc \u0645\u062e\u0632\u0646 \u0633\u06d2 <code>github.com\/RudrenduPaul\/product-experimentation-causal-inference-genai-llm\/tree\/main\/11_instrumental_variables\/<\/code>.<\/p>\n<h2 id=\"heading-table-of-contents\">\u0627\u0646\u0688\u06cc\u06a9\u0633<\/h2>\n<h2 id=\"heading-why-routing-confounds-regression\">\u06a9\u06cc\u0648\u06ba \u0631\u0648\u0679\u0646\u06af \u0631\u062c\u0639\u062a \u06a9\u0648 \u0627\u0644\u062c\u06be\u0627\u062a\u06cc \u06c1\u06d2\u06d4<\/h2>\n<p>\u0631\u0648\u0679\u0646\u06af \u0633\u0633\u0679\u0645 \u0628\u0627\u06c1\u0645 \u0645\u0631\u0628\u0648\u0637 \u0641\u06cc\u0635\u0644\u06d2 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u06a9\u0645 \u0627\u0639\u062a\u0645\u0627\u062f \u0627\u0633\u06a9\u0648\u0631\u0632\u060c \u0637\u0648\u06cc\u0644 \u0679\u0648\u06a9\u0646 \u06a9\u0627\u0624\u0646\u0679\u060c \u06cc\u0627 \u067e\u06cc\u0686\u06cc\u062f\u06c1 \u0645\u0644\u0679\u06cc \u0633\u0679\u06cc\u067e \u0627\u0646\u0679\u06cc\u0646\u0679 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0622\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u0648 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u067e\u0631 \u0631\u0648\u0679 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u0633\u0648\u0627\u0644\u0627\u062a \u062c\u0648 \u0645\u062e\u062a\u0635\u0631\u060c \u0648\u0627\u0636\u062d\u060c \u0627\u0648\u0631 \u0627\u06cc\u06a9 \u0633\u0633\u062a\u06d2 \u0645\u0627\u0688\u0644 \u06a9\u06cc \u0635\u0644\u0627\u062d\u06cc\u062a\u0648\u06ba \u06a9\u06d2 \u0627\u0646\u062f\u0631 \u06c1\u06cc\u06ba \u0633\u0633\u062a\u06d2 \u0637\u0631\u06cc\u0642\u06d2 \u0633\u06d2 \u0631\u0648\u0679 \u06a9\u06cc\u06d2 \u062c\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u06cc\u06c1 \u0627\u0631\u062a\u0628\u0627\u0637 \u0631\u0648\u0679\u0646\u06af \u067e\u0631\u062a \u06a9\u0627 \u0645\u0631\u06a9\u0632 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p>\u0645\u0633\u0626\u0644\u06c1 \u06cc\u06c1 \u06c1\u06d2 \u06a9\u06c1 \u0648\u06c1\u06cc \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u062c\u0648 \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u0641\u06cc\u0635\u0644\u0648\u06ba \u06a9\u0648 \u0622\u06af\u06d2 \u0628\u0691\u06be\u0627\u062a\u06cc \u06c1\u06cc\u06ba \u062f\u0644\u0686\u0633\u067e\u06cc \u06a9\u06d2 \u0646\u062a\u0627\u0626\u062c \u06a9\u0648 \u0628\u06be\u06cc \u0645\u062a\u0627\u062b\u0631 \u06a9\u0631\u062a\u06cc \u06c1\u06cc\u06ba\u06d4<\/p>\n<p>\u067e\u06cc\u0686\u06cc\u062f\u06c1 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u0648 \u0645\u06a9\u0645\u0644 \u06a9\u0631\u0646\u0627 \u0632\u06cc\u0627\u062f\u06c1 \u0645\u0634\u06a9\u0644 \u06c1\u06d2\u060c \u0642\u0637\u0639 \u0646\u0638\u0631 \u0627\u0633 \u06a9\u06d2 \u06a9\u06c1 \u06a9\u0648\u0646 \u0633\u0627 \u0645\u0627\u0688\u0644 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u067e\u0631 \u06a9\u0627\u0631\u0631\u0648\u0627\u0626\u06cc \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u062c\u0628 \u0622\u067e \u0644\u06a9\u06be\u062a\u06d2 \u06c1\u06cc\u06ba <code>task_completed ~ routed_to_premium + controls<\/code>, <code>controls<\/code> \u0634\u0631\u0627\u0626\u0637 \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u06a9\u06d2 \u0642\u0627\u0628\u0644 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u062c\u06c1\u062a\u0648\u06ba \u06a9\u0648 \u062c\u0630\u0628 \u06a9\u0631 \u0633\u06a9\u062a\u06cc \u06c1\u06cc\u06ba\u060c \u062c\u06cc\u0633\u06d2 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06cc \u0644\u0645\u0628\u0627\u0626\u06cc\u060c \u0635\u0627\u0631\u0641 \u06a9\u06cc \u0645\u0635\u0631\u0648\u0641\u06cc\u062a \u06a9\u0627 \u062f\u0631\u062c\u06c1 \u0628\u0646\u062f\u06cc\u060c \u0627\u0648\u0631 \u06c1\u0631 \u0686\u06cc\u0632 \u062c\u0648 \u0644\u0627\u06af \u0627\u0646 \u06c1\u06d2\u06d4<\/p>\n<p>\u0648\u06c1 \u062c\u06c1\u062a \u062c\u0633 \u06a9\u0627 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0646\u06c1\u06cc\u06ba \u06a9\u06cc\u0627 \u062c\u0627 \u0633\u06a9\u062a\u0627 <code>routed_to_premium<\/code> \u06c1\u0645 \u06af\u062a\u0627\u0646\u06a9 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631\u060c \u062e\u0644\u0644 \u06a9\u06cc \u0633\u0645\u062a \u06a9\u06d2 \u0644\u062d\u0627\u0638 \u0633\u06d2\u060c \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06a9\u0648 \u0646\u06cc\u0686\u06d2 \u06a9\u06cc \u0637\u0631\u0641 \u0645\u062a\u0639\u0635\u0628 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba (\u067e\u06cc\u0686\u06cc\u062f\u06c1 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u0628\u06be\u06cc\u062c\u06d2 \u062c\u0627\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u06a9\u0645 \u06a9\u062b\u0631\u062a \u0633\u06d2 \u0645\u06a9\u0645\u0644 \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062c\u0633 \u0633\u06d2 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0648\u0627\u0642\u0639\u06cc \u0627\u0646 \u0633\u06d2 \u0628\u062f\u062a\u0631 \u0646\u0638\u0631 \u0622\u062a\u06d2 \u06c1\u06cc\u06ba) \u06cc\u0627 \u0627\u0648\u067e\u0631 \u06a9\u06cc \u0637\u0631\u0641\u06d4<\/p>\n<p>\u0627\u0633 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u0645\u06cc\u06ba \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u06cc\u06d2 \u06af\u0626\u06d2 \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627 \u0633\u06cc\u0679 \u0645\u06cc\u06ba\u060c OLS \u06a9\u0627 \u062a\u062e\u0645\u06cc\u0646\u06c1 +3.3 \u0641\u06cc\u0635\u062f \u067e\u0648\u0627\u0626\u0646\u0679\u0633 \u062a\u06a9 \u067e\u06c1\u0646\u0686\u062a\u0627 \u06c1\u06d2\u060c \u062d\u0627\u0644\u0627\u0646\u06a9\u06c1 \u0627\u0635\u0644 \u06a9\u0627\u0632\u0644 \u0627\u062b\u0631 +6 \u0641\u06cc\u0635\u062f \u067e\u0648\u0627\u0626\u0646\u0679\u0633 \u06c1\u06d2\u06d4 \u06cc\u06c1 2.7 \u067e\u06cc \u067e\u06cc \u06a9\u0627 \u0646\u06cc\u0686\u06d2 \u06a9\u06cc \u0637\u0631\u0641 \u062a\u0639\u0635\u0628 \u06c1\u06d2\u060c \u062c\u0648 \u0645\u06a9\u0645\u0644 \u0637\u0648\u0631 \u067e\u0631 \u063a\u06cc\u0631 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0634\u062f\u06c1 \u0633\u0648\u0627\u0644 \u06a9\u06cc \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u06a9\u06cc \u0648\u062c\u06c1 \u0633\u06d2 \u06c1\u06d2\u06d4<\/p>\n<p>\u0631\u062c\u0639\u062a \u0679\u06be\u0648\u0633 \u0646\u0638\u0631 \u0622\u062a\u06cc \u06c1\u06d2\u060c p-value \u0646\u0645\u0627\u06cc\u0627\u06ba \u0646\u0638\u0631 \u0622\u062a\u06cc \u06c1\u06d2\u060c \u0627\u0648\u0631 \u0645\u0639\u06cc\u0627\u0631\u06cc OLS \u0622\u0624\u0679 \u067e\u0679 \u0645\u06cc\u06ba \u06a9\u0648\u0626\u06cc \u0627\u06cc\u0633\u06cc \u0686\u06cc\u0632 \u0646\u06c1\u06cc\u06ba \u06c1\u06d2 \u062c\u0648 \u06a9\u0633\u06cc \u0645\u0633\u0626\u0644\u06d2 \u06a9\u06cc \u0646\u0634\u0627\u0646\u062f\u06c1\u06cc \u06a9\u0631\u062a\u06cc \u06c1\u0648\u06d4 \u06cc\u06c1\u06cc \u0686\u06cc\u0632 \u0627\u0633 \u0646\u0627\u06a9\u0627\u0645\u06cc \u06a9\u06d2 \u0645\u0648\u0688 \u06a9\u0648 \u062e\u0637\u0631\u0646\u0627\u06a9 \u0628\u0646\u0627\u062a\u06cc \u06c1\u06d2\u06d4 \u06cc\u06c1 \u0645\u0639\u06cc\u0627\u0631\u06cc \u0631\u062c\u0639\u062a \u06a9\u06cc \u062a\u0634\u062e\u06cc\u0635 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0646\u0638\u0631 \u0646\u06c1\u06cc\u06ba \u0622\u062a\u0627\u06d4<\/p>\n<p>2SLS \u0628\u0627\u0644\u06a9\u0644 \u0627\u0633\u06cc \u0688\u06be\u0627\u0646\u0686\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0628\u0646\u0627\u06cc\u0627 \u06af\u06cc\u0627 \u062a\u06be\u0627\u06d4 \u0631\u0648\u0679\u0646\u06af \u06a9\u06cc \u0645\u062e\u062a\u0644\u0641 \u062d\u0627\u0644\u062a\u0648\u06ba \u06a9\u06d2 \u0627\u06cc\u06a9 \u0628\u06cc\u0631\u0648\u0646\u06cc \u0630\u0631\u06cc\u0639\u06c1 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u06d2 \u062c\u0648 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06d2 \u0645\u0639\u06cc\u0627\u0631 \u0633\u06d2 \u063a\u06cc\u0631 \u0645\u0631\u0628\u0648\u0637 \u06c1\u06d2\u06d4 \u0634\u0631\u062d \u06a9\u0648 \u0645\u062d\u062f\u0648\u062f \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0648\u062c\u06c1 \u0633\u06d2 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u06cc\u06c1 \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p><em>\u0634\u06a9\u0644 1: IV \u06a9\u0627\u0632\u0644 \u0688\u06be\u0627\u0646\u0686\u06c1\u06d4 \u0622\u0644\u06c1 (Z = \u0634\u0631\u062d \u06a9\u06cc \u062d\u062f \u06a9\u0627 \u0645\u062a\u0628\u0627\u062f\u0644) \u0645\u0637\u0627\u0628\u0642\u062a \u06a9\u0648 \u067e\u0648\u0631\u0627 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 (Z \u0627\u06cc\u06a9 \u0631\u0648\u0679\u0646\u06af \u067e\u06cc\u0634\u06cc\u0646 \u06af\u0648\u0626\u06cc \u06c1\u06d2)\u060c \u0627\u062e\u0631\u0627\u062c (\u0646\u062a\u06cc\u062c\u06d2 \u0645\u06cc\u06ba \u06a9\u0648\u0626\u06cc \u0628\u0631\u0627\u06c1 \u0631\u0627\u0633\u062a Z \u0646\u06c1\u06cc\u06ba \u06c1\u06d2)\u060c \u0627\u0648\u0631 \u0622\u0632\u0627\u062f\u06cc (Z \u063a\u06cc\u0631 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0634\u062f\u06c1 \u0633\u0648\u0627\u0644 \u06a9\u06cc \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u06a9\u06d2 \u0633\u0627\u062a\u06be \u063a\u06cc\u0631 \u0645\u0646\u0633\u0644\u06a9 \u06c1\u06d2)\u06d4 \u0633\u0631\u062e \u0688\u06cc\u0634\u0688 \u062a\u06cc\u0631 \u0627\u0644\u062c\u06be\u0646 \u06a9\u0627 \u0631\u0627\u0633\u062a\u06c1 \u062f\u06a9\u06be\u0627\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u0628\u0648\u0644\u06cc OLS \u06a9\u0627 \u062a\u0639\u0635\u0628 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/em><\/p>\n<h2 id=\"heading-what-an-instrumental-variable-is\">\u0627\u06cc\u06a9 \u0622\u0644\u06c1 \u0645\u062a\u063a\u06cc\u0631 \u06a9\u06cc\u0627 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\u062f\u0631\u062d\u0642\u06cc\u0642\u062a \u0627\u06cc\u0646\u0688\u0648\u062c\u06cc\u0646\u0633 \u0645\u062a\u063a\u06cc\u0631 \u06a9\u0648 \u0645\u062a\u0627\u062b\u0631 \u06a9\u0631\u0646\u0627 \u0686\u0627\u06c1\u06cc\u06d2\u06d4 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u0627\u0646\u0688\u06cc\u06a9\u06cc\u0679\u0631 \u06a9\u0648 \u0645\u062d\u062f\u0648\u062f \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u0627 \u0631\u06cc\u0679 \u062c\u0648 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0627\u06c1\u0644 \u0627\u0633\u062a\u0641\u0633\u0627\u0631\u0627\u062a \u06a9\u06d2 15% \u067e\u0631 \u0686\u0644\u062a\u0627 \u06c1\u06d2 \u0627\u0633 \u067e\u0631 \u0645\u0639\u0646\u06cc \u062e\u06cc\u0632 \u0627\u062b\u0631 \u0688\u0627\u0644\u06d2 \u06af\u0627 \u06a9\u06c1 \u0622\u06cc\u0627 \u0627\u0646 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u0648 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u06a9\u06cc \u0637\u0631\u0641 \u0631\u0648\u0679 \u06a9\u06cc\u0627 \u06af\u06cc\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u06cc\u06c1 \u0642\u06cc\u0627\u0633 \u0642\u0627\u0628\u0644 \u0627\u0645\u062a\u062d\u0627\u0646 \u06c1\u06d2\u06d4 \u067e\u06c1\u0644\u06d2 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u06d2 F-statistic \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0686\u06cc\u06a9 \u06a9\u0631\u06cc\u06ba\u06d4 \u0627\u0633\u0679\u06cc\u06af\u0631 \u0627\u06cc\u0646\u0688 \u0627\u0633\u0679\u0627\u06a9 (1997) \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06c1 \u0631\u0648\u0627\u06cc\u062a\u06cc \u062d\u062f F> 10 \u0645\u0642\u0631\u0631 \u06a9\u06cc \u06af\u0626\u06cc \u06c1\u06d2\u060c \u062c\u0648 \u06a9\u06c1 \u0628\u062f\u062a\u0631\u06cc\u0646 \u0635\u0648\u0631\u062a \u0645\u06cc\u06ba OLS \u06a9\u06d2 \u0645\u0642\u0627\u0628\u0644\u06d2 \u0645\u06cc\u06ba 2SLS \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06a9\u0627\u0631 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062a\u0642\u0631\u06cc\u0628\u0627\u064b 10% \u06a9\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u062a\u0639\u0635\u0628 \u06a9\u06d2 \u0645\u0633\u0627\u0648\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u06cc\u0646\u0688\u0631\u06cc\u0648\u0632\u060c \u0627\u0633\u0679\u0627\u06a9\u060c \u0627\u0648\u0631 \u0633\u0646 (2019) \u06a9\u06cc \u062d\u0627\u0644\u06cc\u06c1 \u062a\u062d\u0642\u06cc\u0642 \u0633\u06d2 \u067e\u062a\u06c1 \u0686\u0644\u062a\u0627 \u06c1\u06d2 \u06a9\u06c1 \u0686\u06be\u0648\u0679\u06d2 \u0646\u0645\u0648\u0646\u0648\u06ba \u06cc\u0627 \u0627\u06cc\u06a9 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0622\u0644\u0627\u062a \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u062a\u0631\u062a\u06cc\u0628\u0627\u062a \u0645\u06cc\u06ba \u06cc\u06c1 \u062d\u062f \u0628\u06c1\u062a \u0632\u06cc\u0627\u062f\u06c1 \u062c\u0627\u0626\u0632 \u06c1\u0648 \u0633\u06a9\u062a\u06cc \u06c1\u06d2\u06d4 \u0645\u062d\u062f\u0648\u062f \u0627\u0639\u062f\u0627\u062f \u0648 \u0634\u0645\u0627\u0631 \u06a9\u06d2 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id=\"heading-prerequisites\">\u0634\u0631\u0627\u0626\u0637<\/h2>\n<p>\u0622\u067e \u06a9\u0648 \u062f\u0631\u062c \u0630\u06cc\u0644 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u06af\u06cc:<\/p>\n<p>\u0627\u0633 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u06a9\u06d2 \u0644\u06cc\u06d2 \u067e\u06cc\u06a9\u062c\u0632 \u0627\u0646\u0633\u0679\u0627\u0644 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<pre><code class=\"language-bash\">pip install numpy pandas statsmodels scipy\n<\/code><\/pre>\n<p>\u06cc\u06c1 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u0645\u06cc\u06ba \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06c1\u0648\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0686\u0627\u0631 \u067e\u06cc\u06a9\u062c\u0648\u06ba \u06a9\u0648 \u0627\u0646\u0633\u0679\u0627\u0644 \u06a9\u0631\u06d2 \u06af\u0627\u06d4 <code>statsmodels<\/code> OLS \u0627\u0648\u0631 \u0641\u0627\u0631\u0645\u0648\u0644\u0627 API \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u060c <code>scipy<\/code> \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e \u0642\u062f\u0645 \u0645\u06cc\u06ba \u0634\u0645\u0627\u0631\u06cc\u0627\u062a\u06cc \u062d\u0633\u0627\u0628 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u062d\u0627\u0635\u0644 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u0633\u0627\u062a\u06be\u06cc \u0631\u06cc\u067e\u0648\u0632\u0679\u0631\u06cc \u06a9\u0648 \u06a9\u0644\u0648\u0646 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<pre><code class=\"language-bash\">git clone https:\/\/github.com\/RudrenduPaul\/product-experimentation-causal-inference-genai-llm.git\ncd product-experimentation-causal-inference-genai-llm\npython data\/generate_data.py --seed 42 --n-users 50000 --out data\/synthetic_llm_logs.csv\n<\/code><\/pre>\n<p>\u0627\u0633 \u0628\u0627\u062a \u06a9\u0648 \u06cc\u0642\u06cc\u0646\u06cc \u0628\u0646\u0627\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u06a9\u06c1 \u0646\u062a\u0627\u0626\u062c \u0627\u0633 \u06a9\u0627\u063a\u0630 \u06a9\u06d2 \u0645\u062a\u0648\u0642\u0639 \u0622\u0624\u0679 \u067e\u0679 \u0633\u06d2 \u0645\u0645\u0627\u062b\u0644 \u06c1\u0648\u06ba\u060c \u06c1\u0645 \u0633\u0627\u062a\u06be\u06cc \u0631\u06cc\u067e\u0648\u0632\u0679\u0631\u06cc \u06a9\u0648 \u06a9\u0644\u0648\u0646 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u0641\u06a9\u0633\u0688 \u0633\u06cc\u0688\u0632 \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u0645\u0634\u062a\u0631\u06a9\u06c1 50,000 \u0635\u0627\u0631\u0641 \u06a9\u06d2 \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u06a9\u0648 \u062f\u0648\u0628\u0627\u0631\u06c1 \u062a\u062e\u0644\u06cc\u0642 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<h2 id=\"heading-setting-up-the-working-example\">\u0648\u0631\u06a9\u0646\u06af \u0645\u062b\u0627\u0644 \u0633\u06cc\u0679 \u0627\u067e<\/h2>\n<p>\u0627\u0633 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u0645\u06cc\u06ba\u060c \u06c1\u0645 \u0627\u06cc\u06a9 \u0645\u0634\u062a\u0631\u06a9\u06c1 \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u0633\u06d2 \u0635\u0627\u0631\u0641 \u06a9\u0648\u0648\u06cc\u0631\u06cc\u0679\u0633 \u06a9\u06d2 \u0627\u0648\u067e\u0631 \u062a\u06cc\u0646 \u0646\u0642\u0644\u06cc \u0645\u062a\u063a\u06cc\u0631\u0627\u062a \u0634\u0627\u0645\u0644 \u06a9\u0631\u06a9\u06d2 \u06a9\u0648\u0688 \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u0645\u06a9\u0645\u0644 IV \u06a9\u0627\u0632\u0644 \u06af\u0631\u0627\u0641 \u0628\u0646\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<ul>\n<li>\n<p><code>rate_limit_fallback<\/code>: \u0622\u0644\u06c1 Z. \u062e\u0627\u0644\u0635 \u0628\u0631\u0646\u0648\u0644\u06cc (0.15) \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0646\u0645\u0648\u0646\u06c1\u060c \u062a\u0645\u0627\u0645 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u06cc \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u0633\u06d2 \u0645\u06a9\u0645\u0644 \u0637\u0648\u0631 \u067e\u0631 \u0622\u0632\u0627\u062f\u06d4<\/p>\n<\/li>\n<li>\n<p><code>routed_to_premium_actual<\/code>: \u0627\u06cc\u0646\u0688\u0648\u062c\u06cc\u0646\u0633 \u067e\u0631\u0648\u0633\u06cc\u0633\u0646\u06af D. \u0631\u0648\u0679\u0646\u06af \u062f\u0648\u0646\u0648\u06ba \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u0686\u0644\u062a\u06cc \u06c1\u06d2\u06d4 <code>query_confidence<\/code> (\u0645\u0634\u0627\u06c1\u062f\u06c1) \u0627\u0648\u0631 <code>query_complexity<\/code> (\u0646\u0627\u0642\u0627\u0628\u0644 \u0645\u0634\u0627\u06c1\u062f\u06c1)\u060c \u0644\u06c1\u0630\u0627 OLS \u0645\u062a\u0639\u0635\u0628 \u06c1\u06d2\u06d4<\/p>\n<\/li>\n<li>\n<p><code>task_completed_iv<\/code>: \u0646\u062a\u06cc\u062c\u06c1 Y. \u06a9\u0648 \u0645\u0639\u0644\u0648\u0645 +6pp \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u0627\u062b\u0631 \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 IV causal \u06af\u0631\u0627\u0641 \u067e\u0631 \u062f\u0648\u0628\u0627\u0631\u06c1 \u0646\u0642\u0644 \u06a9\u06cc\u0627 \u062c\u0627 \u0633\u06a9\u062a\u0627 \u06c1\u06d2 \u062a\u0627\u06a9\u06c1 \u06cc\u06c1 \u062f\u06cc\u06a9\u06be\u0627 \u062c\u0627 \u0633\u06a9\u06d2 \u06a9\u06c1 \u0622\u06cc\u0627 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u0644\u06af\u0627\u0646\u06d2 \u0648\u0627\u0644\u0627 \u0632\u0645\u06cc\u0646\u06cc \u0633\u0686\u0627\u0626\u06cc \u06a9\u0648 \u0628\u0627\u0632\u06cc\u0627\u0641\u062a \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<\/li>\n<\/ul>\n<p>\u0634\u0641\u0627\u0641\u06cc\u062a \u06a9\u0627 \u0646\u0648\u0679: \u062d\u0642\u06cc\u0642\u06cc \u067e\u06cc\u062f\u0627\u0648\u0627\u0631 \u06a9\u06d2 \u062a\u062c\u0632\u06cc\u0627\u062a \u0645\u06cc\u06ba\u060c \u0634\u0631\u062d \u06a9\u0648 \u0645\u062d\u062f\u0648\u062f \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u0648\u0627\u0642\u0639\u0627\u062a API \u06af\u06cc\u0679 \u0648\u06d2 \u0644\u0627\u06af\u0632 \u0645\u06cc\u06ba \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0635\u0627\u0631\u0641 \u06a9\u06d2 \u0679\u06cc\u0644\u06cc \u0645\u06cc\u0679\u0631\u06cc \u0679\u06cc\u0628\u0644 \u0645\u06cc\u06ba \u0627\u06cc\u0633\u06cc \u06a9\u0648\u0626\u06cc \u0627\u0646\u062f\u0631\u0627\u062c \u0646\u06c1\u06cc\u06ba \u06c1\u06d2\u06d4 \u0627\u0646 \u062f\u0648 \u0630\u0631\u0627\u0626\u0639 \u06a9\u0648 \u0645\u0644\u0627 \u06a9\u0631\u060c \u0622\u067e \u0627\u06cc\u06a9 \u0622\u0644\u06c1 \u0628\u0646\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p>\u06cc\u06c1\u0627\u06ba\u060c \u0646\u0642\u0644\u06cc \u0627\u0635\u0644\u06cc \u0622\u0644\u06d2 \u06a9\u06cc \u0633\u0627\u062e\u062a\u06cc \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u06a9\u0648 \u0645\u062d\u0641\u0648\u0638 \u0631\u06a9\u06be\u062a\u06cc \u06c1\u06d2\u06d4 \u06cc\u0639\u0646\u06cc\u060c \u06cc\u06c1 \u067e\u0631\u0648\u0688\u06a9\u0634\u0646 \u06af\u06cc\u0679 \u0648\u06d2 \u0644\u0627\u06af \u06a9\u06d2 \u0628\u063a\u06cc\u0631 \u0627\u0648\u0631 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06d2 \u0645\u0639\u06cc\u0627\u0631 \u0633\u06d2 \u0642\u0637\u0639 \u0646\u0638\u0631 \u0628\u0646\u06cc\u0627\u062f\u06cc \u0688\u06be\u0627\u0646\u0686\u06d2 \u06a9\u06cc \u0648\u062c\u0648\u06c1\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u0686\u0644\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">import numpy as np\nimport pandas as pd\nimport statsmodels.formula.api as smf\n\nnp.random.seed(42)\n\ndf = pd.read_csv(\"data\/synthetic_llm_logs.csv\")\nrng = np.random.default_rng(99)\nn = len(df)\n\n# Unobserved confounder: complex queries route premium AND complete less often\nquery_complexity = rng.normal(0, 1, n)\n\n# Endogenous routing: depends on query_confidence (observable)\n# and query_complexity (unobserved): this is the confounding structure\nlog_odds = -2.0 + 4.0 * (1.0 - df[\"query_confidence\"]) + 0.6 * query_complexity\npremium_prob = 1.0 \/ (1.0 + np.exp(-log_odds))\ndf[\"routed_to_premium_iv\"] = rng.binomial(1, premium_prob).astype(int)\n\n# Instrument: pure Bernoulli(0.15), independent of all query characteristics\ndf[\"rate_limit_fallback\"] = rng.binomial(1, 0.15, n)\n\n# Actual routing: premium if intended, unless fallback overrides\ndf[\"routed_to_premium_actual\"] = (\n    df[\"routed_to_premium_iv\"] * (1 - df[\"rate_limit_fallback\"])\n).astype(int)\n\n# Outcome: known causal structure with +0.06 premium effect\nengagement_base = np.where(df.engagement_tier == \"heavy\", 0.70,\n                  np.where(df.engagement_tier == \"medium\", 0.55, 0.35))\ncompletion_prob = np.clip(\n    engagement_base\n    + 0.06 * df[\"routed_to_premium_actual\"]  # true causal effect\n    - 0.04 * query_complexity                 # unobserved confounder\n    + rng.normal(0, 0.02, n),\n    0.01, 0.99\n)\ndf[\"task_completed_iv\"] = rng.binomial(1, completion_prob).astype(int)\n\n# Encode engagement tier as dummies\ndf = pd.get_dummies(df, columns=[\"engagement_tier\"], drop_first=True)\ntier_dummies = [c for c in df.columns if c.startswith(\"engagement_tier_\")]\ncovariate_str = \" + \".join([\"query_confidence\"] + tier_dummies)\n\nprint(f\"Rate-limit fallback rate:      {df.rate_limit_fallback.mean():.3f}\")\nprint(f\"Premium routing rate (actual): {df.routed_to_premium_actual.mean():.3f}\")\nprint(f\"Mean confidence | fallback=1:  {df[df.rate_limit_fallback==1].query_confidence.mean():.3f}\")\nprint(f\"Mean confidence | fallback=0:  {df[df.rate_limit_fallback==0].query_confidence.mean():.3f}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">Rate-limit fallback rate:      0.151\nPremium routing rate (actual): 0.271\nMean confidence | fallback=1:  0.716\nMean confidence | fallback=0:  0.715\n<\/code><\/pre>\n<p>\u0627\u0648\u067e\u0631 \u0648\u0627\u0644\u06d2 \u06a9\u0648\u0688 \u0645\u06cc\u06ba\u060c \u0641\u0627\u0644 \u0628\u06cc\u06a9=1 \u0627\u0648\u0631 \u0641\u0627\u0644 \u0628\u06cc\u06a9=0 \u06af\u0631\u0648\u067e\u0633 \u06a9\u06d2 \u062f\u0631\u0645\u06cc\u0627\u0646 \u062a\u0642\u0631\u06cc\u0628\u0627\u064b \u0627\u06cc\u06a9 \u062c\u06cc\u0633\u06d2 \u0627\u0648\u0633\u0637 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0627\u0633\u06a9\u0648\u0631 \u0627\u0633 \u0628\u0627\u062a \u06a9\u06cc \u062a\u0635\u062f\u06cc\u0642 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u06a9\u06c1 \u0622\u0644\u06c1 \u0642\u0627\u0628\u0644 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0631\u0648\u0679\u0646\u06af \u0633\u06af\u0646\u0644 \u0633\u06d2 \u0622\u0632\u0627\u062f \u06c1\u06d2\u06d4 \u06cc\u06c1 \u0622\u0632\u0627\u062f\u06cc \u06a9\u06d2 \u0645\u0641\u0631\u0648\u0636\u06d2 \u06a9\u06cc \u062c\u0627\u0646\u0686 \u06c1\u06d2 \u062c\u0648 \u06a9\u0633\u06cc \u0628\u06be\u06cc \u0645\u062c\u0648\u0632\u06c1 \u0622\u0644\u06d2 \u067e\u0631 \u0686\u0644\u0627\u0626\u06cc \u062c\u0627 \u0633\u06a9\u062a\u06cc \u06c1\u06d2\u06d4 <code>query_complexity<\/code> \u06cc\u06c1 \u0627\u0633 \u062a\u062e\u0631\u0648\u067e\u0646 \u0645\u06cc\u06ba \u0642\u0627\u0628\u0644 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06c1\u06d2 \u0644\u06cc\u06a9\u0646 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631 \u0645\u06cc\u06ba \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0646\u06c1\u06cc\u06ba \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u0631\u062c\u0639\u062a \u062a\u062c\u0632\u06cc\u06c1 \u06cc\u06c1 \u062d\u0627\u0635\u0644 \u0646\u06c1\u06cc\u06ba \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-step-1-naive-ols-biased-baseline\">\u0645\u0631\u062d\u0644\u06c1 1: \u0628\u0648\u0644\u06cc OLS (\u0645\u062a\u0639\u0635\u0628 \u0628\u06cc\u0633 \u0644\u0627\u0626\u0646)<\/h2>\n<p>\u0645\u0639\u06cc\u0627\u0631\u06cc \u0631\u062c\u0639\u062a \u06a9\u0648 \u0686\u0644\u0627\u0646\u06d2 \u0633\u06d2 \u067e\u06c1\u0644\u06d2 \u0627\u06cc\u06a9 \u0645\u062a\u0639\u0635\u0628 \u0628\u0646\u06cc\u0627\u062f\u06cc \u0644\u0627\u0626\u0646 \u0642\u0627\u0626\u0645 \u06c1\u0648\u062a\u06cc \u06c1\u06d2 \u062c\u0633 \u06a9\u0627 \u0633\u0627\u0645\u0646\u0627 \u0645\u062a\u0636\u0627\u062f \u0688\u06be\u0627\u0646\u0686\u06d2 \u06a9\u0648 \u0645\u062f\u0646\u0638\u0631 \u0631\u06a9\u06be\u06d2 \u0628\u063a\u06cc\u0631 \u06a9\u06cc\u0627 \u062c\u0627 \u0633\u06a9\u062a\u0627 \u06c1\u06d2\u06d4 \u0632\u06cc\u0627\u062f\u06c1 \u062a\u0631 \u0627\u0646\u062c\u06cc\u0646\u0626\u0631\u0646\u06af \u0679\u06cc\u0645\u06cc\u06ba \u0627\u0646 \u0646\u0645\u0628\u0631\u0648\u06ba \u06a9\u0648 \u06cc\u06c1 \u0633\u0645\u062c\u06be\u06d2 \u0628\u063a\u06cc\u0631 \u0627\u0637\u0644\u0627\u0639 \u062f\u06cc\u062a\u06cc \u06c1\u06cc\u06ba \u06a9\u06c1 \u06cc\u06c1 \u0631\u06cc\u0627\u0636\u06cc \u06a9\u06d2 \u0644\u062d\u0627\u0638 \u0633\u06d2 \u0679\u0648\u0679\u06d2 \u06c1\u0648\u0626\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<pre><code class=\"language-python\">ols_formula = f\"task_completed_iv ~ routed_to_premium_actual + {covariate_str}\"\nols_model = smf.ols(ols_formula, data=df).fit(cov_type=\"HC3\")\n\nols_coef = ols_model.params[\"routed_to_premium_actual\"]\nols_se   = ols_model.bse[\"routed_to_premium_actual\"]\nols_pval = ols_model.pvalues[\"routed_to_premium_actual\"]\nprint(f\"OLS estimate of premium routing effect: {ols_coef:+.4f}\")\nprint(f\"HC3 standard error:                      {ols_se:.4f}\")\nprint(f\"p-value:                                 {ols_pval:.4f}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">OLS estimate of premium routing effect: +0.0327\nHC3 standard error:                      0.0050\np-value:                                 0.0000\n<\/code><\/pre>\n<p>\u06cc\u06c1\u0627\u06ba \u06a9\u06cc\u0627 \u06c1\u0648 \u0631\u06c1\u0627 \u06c1\u06d2: OLS +3.3 \u0641\u06cc\u0635\u062f \u067e\u0648\u0627\u0626\u0646\u0679\u0633 (\u0627\u0645\u06a9\u0627\u0646\u06cc \u0627\u06a9\u0627\u0626\u06cc\u0627\u06ba\u060c \u0627\u0633 \u06a9\u06d2 \u0628\u0639\u062f) \u0628\u0627\u0632\u06cc\u0627\u0641\u062a \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 <code>task_completed_iv<\/code> \u0627\u06cc\u06a9 \u0644\u06a9\u06cc\u0631\u06cc \u0627\u0645\u06a9\u0627\u0646\u06cc \u0645\u0627\u0688\u0644 \u0645\u06cc\u06ba\u060c 0\/1 \u0627\u06cc\u06a9 \u0628\u0627\u0626\u0646\u0631\u06cc \u0646\u062a\u06cc\u062c\u06c1 \u06c1\u06d2)\u06d4 \u0627\u0635\u0644 \u0648\u062c\u06c1 \u0627\u062b\u0631 +6.0pp \u06c1\u06d2\u06d4 2.7pp \u062a\u0639\u0635\u0628 \u063a\u06cc\u0631 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0634\u062f\u06c1 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06cc \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0631\u0648\u0679\u0646\u06af \u0633\u06d2 \u0622\u062a\u0627 \u06c1\u06d2\u06d4 \u0632\u06cc\u0627\u062f\u06c1 \u0645\u0634\u06a9\u0644 \u0633\u0648\u0627\u0644\u0627\u062a \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679 \u06a9\u06cc\u06d2 \u062c\u0627\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u06a9\u0645 \u06a9\u062b\u0631\u062a \u0633\u06d2 \u0645\u06a9\u0645\u0644 \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u06cc\u06c1 \u0646\u06cc\u0686\u06d2 \u06a9\u06cc \u0637\u0631\u0641 \u062e\u0644\u0644 \u06a9\u0627 \u0637\u0631\u06cc\u0642\u06c1 \u06a9\u0627\u0631 \u06c1\u06d2\u06d4 p-value \u0627\u06c1\u0645 \u0646\u0638\u0631 \u0622\u062a\u06cc \u06c1\u06d2 \u0627\u0648\u0631 \u0645\u0639\u06cc\u0627\u0631\u06cc \u063a\u0644\u0637\u06cc \u062f\u0631\u0633\u062a \u062f\u06a9\u06be\u0627\u0626\u06cc \u062f\u06cc\u062a\u06cc \u06c1\u06d2\u06d4 \u0627\u0633 \u0622\u0624\u0679 \u067e\u0679 \u0645\u06cc\u06ba \u0627\u06cc\u0633\u0627 \u06a9\u0686\u06be \u0628\u06be\u06cc \u0646\u06c1\u06cc\u06ba \u06c1\u06d2 \u062c\u0648 \u0627\u0633 \u0628\u0627\u062a \u06a9\u06cc \u0646\u0634\u0627\u0646\u062f\u06c1\u06cc \u06a9\u0631\u06d2 \u06a9\u06c1 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u063a\u0644\u0637 \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u0646 \u0646\u0645\u0628\u0631\u0648\u06ba \u06a9\u0648 \u0630\u06c1\u0646 \u0645\u06cc\u06ba \u0631\u06a9\u06be\u06cc\u06ba\u06d4 \u0627\u0633\u0679\u06cc\u062c 2 \u06a9\u06d2 2SLS \u0646\u062a\u0627\u0626\u062c \u0627\u0633 \u0641\u0631\u0642 \u06a9\u0648 \u0638\u0627\u06c1\u0631 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<h2 id=\"heading-step-2-two-stage-least-squares-2sls-from-scratch\">\u0645\u0631\u062d\u0644\u06c1 2: \u0634\u0631\u0648\u0639 \u0633\u06d2 \u062f\u0648 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u06d2 \u0633\u0628 \u0633\u06d2 \u06a9\u0645 \u0686\u0648\u06a9\u0648\u0631 (2SLS)<\/h2>\n<p>\u062f\u0648 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u06d2 \u06a9\u0645 \u0633\u06d2 \u06a9\u0645 \u0645\u0631\u0628\u0639 \u0631\u06cc\u0679 \u06a9\u06cc \u062d\u062f \u06a9\u06d2 \u0645\u062a\u0628\u0627\u062f\u0644 \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06c1 \u062a\u06cc\u0627\u0631 \u06a9\u0631\u062f\u06c1 \u062e\u0627\u0631\u062c\u06cc \u0631\u0648\u0679\u0646\u06af \u062a\u063a\u06cc\u0631 \u06a9\u0648 \u0627\u0644\u06af \u06a9\u0631 \u06a9\u06d2 \u0627\u0648\u0631 \u0635\u0631\u0641 \u0627\u0633 \u062a\u063a\u06cc\u0631 \u06a9\u0648 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u0648\u062c\u06c1 \u0627\u062b\u0631 \u06a9\u0627 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u0644\u06af\u0627 \u06a9\u0631 \u062a\u0639\u0635\u0628 \u06a9\u0648 \u062f\u0631\u0633\u062a \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<h3 id=\"heading-stage-1-predict-routing-from-the-instrument-and-covariates\">\u0645\u0631\u062d\u0644\u06c1 1: \u0622\u0644\u0627\u062a \u0627\u0648\u0631 covariates \u0633\u06d2 \u0631\u0648\u0679\u0646\u06af \u06a9\u06cc \u067e\u06cc\u0634 \u06af\u0648\u0626\u06cc \u06a9\u0631\u06cc\u06ba\u06d4<\/h3>\n<pre><code class=\"language-python\">stage1_formula = f\"routed_to_premium_actual ~ rate_limit_fallback + {covariate_str}\"\nstage1 = smf.ols(stage1_formula, data=df).fit(cov_type=\"HC3\")\n\nprint(f\"Stage 1 instrument coefficient: {stage1.params['rate_limit_fallback']:+.4f}\")\nprint(f\"p-value:                         {stage1.pvalues['rate_limit_fallback']:.4f}\")\n\ndf[\"rtp_hat\"] = stage1.fittedvalues\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">Stage 1 instrument coefficient: -0.3190\np-value:                         0.0000\n<\/code><\/pre>\n<p>\u06cc\u06c1 \u06a9\u0648\u0688 \u0688\u06cc\u0648\u0627\u0626\u0633 \u06a9\u06d2 \u0627\u06cc\u0646\u0688\u0648\u062c\u06cc\u0646\u0633 \u0631\u0648\u0679\u0646\u06af \u0645\u062a\u063a\u06cc\u0631\u0627\u062a \u06a9\u0648 \u0627\u0633\u06cc \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0634\u062f\u06c1 \u06a9\u0648\u0648\u06cc\u0631\u06cc\u0679\u0633 \u067e\u0631 \u0648\u0627\u067e\u0633 \u0644\u06d2 \u062c\u0627\u062a\u0627 \u06c1\u06d2 \u062c\u0633\u06d2 \u06c1\u0645 \u0645\u0631\u062d\u0644\u06c1 2 \u0645\u06cc\u06ba \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u06cc\u06ba \u06af\u06d2\u06d4<\/p>\n<p>\u0645\u0646\u0627\u0633\u0628 \u0642\u06cc\u0645\u062a <code>rtp_hat<\/code> \u0627\u0633 \u0645\u06cc\u06ba \u062f\u0648 \u0627\u062c\u0632\u0627\u0621 \u06c1\u06cc\u06ba: \u062e\u0627\u0631\u062c\u06cc \u062a\u063a\u06cc\u0631 \u062c\u0633 \u06a9\u06cc \u0648\u0636\u0627\u062d\u062a \u0622\u0644\u06c1 \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06c1 \u06a9\u06cc \u06af\u0626\u06cc \u06c1\u06d2 \u0627\u0648\u0631 \u062e\u0627\u0631\u062c\u06cc \u062a\u063a\u06cc\u0631 \u062c\u0633 \u06a9\u06cc \u0648\u0636\u0627\u062d\u062a covariates \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06c1 \u06a9\u06cc \u06af\u0626\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>Endogenous \u0627\u062c\u0632\u0627\u0621 (\u0645\u062a\u063a\u06cc\u0631\u0627\u062a \u062c\u0648 \u063a\u06cc\u0631 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0634\u062f\u06c1 \u0633\u0648\u0627\u0644 \u06a9\u06cc \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0645\u0646\u0633\u0644\u06a9 \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba) \u0628\u0627\u0642\u06cc\u0627\u062a \u0645\u06cc\u06ba \u0631\u06c1\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 <code>rtp_hat<\/code>. \u0645\u0646\u0641\u06cc \u06af\u062a\u0627\u0646\u06a9 <code>rate_limit_fallback<\/code> \u0645\u0637\u0627\u0628\u0642\u062a \u06a9\u06d2 \u0645\u0641\u0631\u0648\u0636\u0648\u06ba \u06a9\u0648 \u0686\u06cc\u06a9 \u06a9\u0631\u06cc\u06ba\u06d4 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u0644\u0627\u06af\u0648 \u06c1\u0648\u0646\u06d2 \u06a9\u06d2 \u0628\u0639\u062f\u060c \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u06a9\u0627 \u0627\u0645\u06a9\u0627\u0646 \u062a\u0642\u0631\u06cc\u0628\u0627\u064b 32 \u0641\u06cc\u0635\u062f \u067e\u0648\u0627\u0626\u0646\u0679\u0633 \u06a9\u0645 \u06c1\u0648 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<h3 id=\"heading-stage-2-regress-outcome-on-the-predicted-routing\">\u0645\u0631\u062d\u0644\u06c1 2: \u067e\u06cc\u0634\u0646 \u06af\u0648\u0626\u06cc \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u0631\u062c\u0639\u062a \u06a9\u06d2 \u0646\u062a\u0627\u0626\u062c\u06d4<\/h3>\n<pre><code class=\"language-python\">stage2_formula = f\"task_completed_iv ~ rtp_hat + {covariate_str}\"\nstage2 = smf.ols(stage2_formula, data=df).fit(cov_type=\"HC3\")\n\ntsls_coef = stage2.params[\"rtp_hat\"]\ntsls_se   = stage2.bse[\"rtp_hat\"]\nprint(f\"2SLS estimate:              {tsls_coef:+.4f}\")\nprint(f\"Stage-2 SE (underestimate): {tsls_se:.4f}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">2SLS estimate:              +0.0599\nStage-2 SE (underestimate): 0.0188\n<\/code><\/pre>\n<p>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2: \u0634\u0641\u0679 <code>routed_to_premium_actual<\/code> \u06a9\u06d2 \u0633\u0627\u062a\u06be <code>rtp_hat<\/code> \u06cc\u06c1 \u0631\u0648\u0679\u0646\u06af \u062a\u063a\u06cc\u0631 \u06a9\u06d2 \u0627\u06cc\u0646\u0688\u0648\u062c\u06cc\u0646\u0633 \u062d\u0635\u06d2 \u06a9\u0648 \u06c1\u0679\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u062f\u0648\u0633\u0631\u06d2 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u0627 \u06af\u062a\u0627\u0646\u06a9 (+0.0599) \u06a9\u0627\u0645 \u06a9\u06cc \u062a\u06a9\u0645\u06cc\u0644 \u067e\u0631 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u06a9\u0627\u0632\u0644 \u0627\u062b\u0631 \u06a9\u0627 2SLS \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06c1\u06d2\u060c \u062c\u0648 \u0632\u0645\u06cc\u0646\u06cc \u0633\u0686\u0627\u0626\u06cc \u0633\u06d2 \u062a\u0642\u0631\u06cc\u0628\u0627\u064b \u0628\u0627\u0644\u06a9\u0644 +0.06 \u0627\u0648\u067e\u0631 \u06c1\u06d2\u06d4<\/p>\n<p>\u0645\u0639\u06cc\u0627\u0631\u06cc \u063a\u0644\u0637\u06cc\u0648\u06ba \u06a9\u06d2 \u0628\u0627\u0631\u06d2 \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u0627\u06c1\u0645 \u0646\u0648\u0679: \u062f\u0633\u062a\u06cc 2SLS 2-\u0627\u0633\u0679\u06cc\u062c SE \u062a\u06cc\u0627\u0631 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u0628\u06c1\u062a \u0686\u06be\u0648\u0679\u0627 \u06c1\u06d2\u06d4 \u0627\u0633\u0679\u06cc\u062c 2 OLS \u06a9\u0627 \u0639\u0644\u0627\u062c <code>rtp_hat<\/code> \u06cc\u06c1 \u062f\u0631\u062d\u0642\u06cc\u0642\u062a \u0627\u06cc\u06a9 \u0645\u0642\u0631\u0631\u06c1 \u0645\u0639\u0644\u0648\u0645 \u0631\u06cc\u06af\u0631\u06cc\u0633\u0631 \u06c1\u06d2 \u062c\u0633 \u06a9\u0627 \u0627\u0646\u062f\u0627\u0632\u06c1 \u0645\u0631\u062d\u0644\u06c1 1 \u0645\u06cc\u06ba \u0688\u06cc\u0679\u0627 \u0633\u06d2 \u0644\u06af\u0627\u06cc\u0627 \u06af\u06cc\u0627 \u06c1\u06d2\u06d4 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06a9\u06cc \u062e\u0631\u0627\u0628\u06cc \u0627\u06cc\u06a9 \u0645\u062a\u063a\u06cc\u0631 \u062c\u0632\u0648 \u06a9\u0627 \u0627\u0636\u0627\u0641\u06c1 \u06a9\u0631\u062a\u06cc \u06c1\u06d2 \u062c\u0648 \u0645\u0631\u062d\u0644\u06c1 2 \u0633\u06d2 \u0628\u0627\u0642\u06cc\u0627\u062a \u0645\u06cc\u06ba \u0646\u0638\u0631 \u0646\u06c1\u06cc\u06ba \u0622\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u0633\u0679\u06cc\u06a9 \u06c1\u0648\u0644\u0688\u0631\u0632 \u06a9\u0648 \u0646\u062a\u0627\u0626\u062c \u06a9\u06cc \u0627\u0637\u0644\u0627\u0639 \u062f\u06cc\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u06cc\u06ba: <code>linearmodels.IV2SLS<\/code> (\u062f\u06cc\u06a9\u06be\u06cc\u06ba &quot;\u0622\u06af\u06d2 \u06a9\u06cc\u0627 \u06a9\u0631\u0646\u0627 \u06c1\u06d2\u06d4&#8221;) \u06cc\u06c1 \u0635\u062d\u06cc\u062d \u0633\u06cc\u0646\u0688\u0648\u0686 \u062a\u063a\u06cc\u0631 \u06a9\u0627 \u062d\u0633\u0627\u0628 \u0644\u06af\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<h3 id=\"heading-compare-ols-and-2sls-side-by-side\">OLS \u0627\u0648\u0631 2SLS \u06a9\u0627 \u0633\u0627\u062a\u06be \u0633\u0627\u062a\u06be \u0645\u0648\u0627\u0632\u0646\u06c1 \u06a9\u0631\u06cc\u06ba:<\/h3>\n<pre><code class=\"language-python\">print(f\"OLS estimate (biased):  {ols_coef:+.4f}\")\nprint(f\"2SLS estimate (IV):     {tsls_coef:+.4f}\")\nprint(f\"True premium effect:   +0.0600\")\nprint(f\"OLS bias:               {ols_coef - 0.06:+.4f}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">OLS estimate (biased):  +0.0327\n2SLS estimate (IV):     +0.0599\nTrue premium effect:   +0.0600\nOLS bias:               -0.0273\n<\/code><\/pre>\n<p>\u06cc\u06c1\u0627\u06ba OLS 2.7pp \u062a\u06a9 \u062d\u0642\u06cc\u0642\u06cc \u0627\u062b\u0631 \u0633\u06d2 \u0645\u062d\u0631\u0648\u0645 \u06c1\u06d2\u060c \u062c\u0648 \u06a9\u06c1 45% \u06a9\u0645 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06c1\u06d2\u06d4 2SLS \u0627\u0633\u06d2 0.01pp \u06a9\u06d2 \u0627\u0646\u062f\u0631 \u0628\u062d\u0627\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u062e\u0644\u0627 \u06a9\u06cc \u0633\u0645\u062a \u0627\u0636\u0637\u0631\u0627\u0628 \u06a9\u06d2 \u0637\u0631\u06cc\u0642\u06c1 \u06a9\u0627\u0631 \u06a9\u06d2 \u0645\u0637\u0627\u0628\u0642 \u06c1\u06d2\u06d4 \u063a\u06cc\u0631 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0634\u062f\u06c1 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06cc \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u06a9\u06cc \u0648\u062c\u06c1 \u0633\u06d2 OLS \u06a9\u0648\u0641\u06cc\u0634\u06cc\u0646\u0679\u0633 \u06a9\u0648 \u0627\u06cc\u06a9 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u067e\u0631 \u0645\u0634\u06a9\u0644 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u0648 \u0631\u0648\u0679 \u06a9\u0631\u06a9\u06d2 \u0627\u0648\u0631 \u0645\u06a9\u0645\u0644 \u06a9\u0648 \u06a9\u0645 \u06a9\u0631\u06a9\u06d2 \u0646\u06cc\u0686\u06d2 \u06a9\u06cc \u0637\u0631\u0641 \u0627\u06cc\u0688\u062c\u0633\u0679 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/umang.pk\/wp-content\/uploads\/2026\/07\/1785378685_31_\u0622\u0644\u0627\u062a-\u06a9\u06d2-\u0645\u062a\u063a\u06cc\u0631\u0627\u062a-\u06a9\u06d2-\u0633\u0627\u062a\u06be-\u067e\u0631\u0648\u0688\u06a9\u0679-\u06a9\u06d2-\u062a\u062c\u0631\u0628\u0627\u062a-\u0627\u0632\u06af\u0631-\u0645\u06cc\u06ba.png\" alt=\"dc39aa79-c955-4c4e-9962-f5a648d6e383\" style=\"display:block;margin:0 auto\" width=\"1633\" height=\"763\" loading=\"lazy\" title=\"\"><\/p>\n<p><em>\u0634\u06a9\u0644 2: 50,000 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u0633\u06d2 \u0688\u06cc\u0679\u0627 \u067e\u0631 \u0645\u0628\u0646\u06cc \u0646\u062a\u0627\u0626\u062c\u06d4 \u0628\u0627\u0626\u06cc\u06ba \u067e\u06cc\u0646\u0644: \u0645\u062a\u0628\u0627\u062f\u0644 \u06af\u0631\u0648\u067e\u0648\u06ba \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06c1 \u0631\u0648\u0679\u0646\u06af \u06a9\u0627 \u062a\u0646\u0627\u0633\u0628 \u0627\u06cc\u06a9 \u0633\u0637\u062d\u06cc \u062a\u0639\u0644\u0642\u0627\u062a \u06a9\u06cc \u062a\u0635\u062f\u06cc\u0642 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0641\u0627\u0644 \u0628\u06cc\u06a9=0 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u0627 \u067e\u0627\u062a\u06be \u067e\u0631\u06cc\u0645\u06cc\u0645 39.1% \u06c1\u06d2\u060c \u0641\u0627\u0644 \u0628\u06cc\u06a9=1 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u0627 \u067e\u0627\u062a\u06be \u067e\u0631\u06cc\u0645\u06cc\u0645 0% \u06c1\u06d2 (\u0645\u06a9\u0645\u0644 \u0627\u0648\u0648\u0631 \u0631\u0627\u0626\u0688)\u06d4 \u062f\u0627\u0626\u06cc\u06ba \u067e\u06cc\u0646\u0644: OLS CI (\u0633\u0631\u062e) \u0627\u0635\u0644 +0.06pp \u0627\u062b\u0631 \u06a9\u0648 \u0645\u06a9\u0645\u0644 \u0637\u0648\u0631 \u067e\u0631 \u06a9\u06be\u0648 \u062f\u06cc\u062a\u0627 \u06c1\u06d2\u06d4 2SLS CI (\u0633\u0628\u0632) \u0627\u0633 \u06a9\u0627 \u0627\u062d\u0627\u0637\u06c1 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0648\u0633\u06cc\u0639 \u062a\u0631 2SLS \u0648\u0642\u0641\u06d2 \u0628\u0627\u0632\u06cc \u06a9\u06d2 \u0627\u062e\u0631\u0627\u062c\u0627\u062a \u06a9\u06cc \u0639\u06a9\u0627\u0633\u06cc \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u062c\u0648 \u0635\u0631\u0641 \u0622\u0644\u06c1 \u0645\u06cc\u06ba \u062e\u0627\u0631\u062c\u06cc \u062a\u063a\u06cc\u0631\u0627\u062a \u067e\u0631 \u0645\u0646\u062d\u0635\u0631 \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/em><\/p>\n<h2 id=\"heading-step-3-weak-instrument-diagnostics\">\u0645\u0631\u062d\u0644\u06c1 3: \u06a9\u0645\u0632\u0648\u0631 \u0622\u0644\u0627\u062a \u06a9\u06cc \u062a\u0634\u062e\u06cc\u0635 \u06a9\u0631\u06cc\u06ba\u06d4<\/h2>\n<p>\u0627\u06cc\u06a9 \u062f\u0631\u0633\u062a \u0679\u0648\u0644 \u062c\u0633 \u06a9\u0627 \u0646\u062a\u0627\u0626\u062c \u067e\u0631 \u0628\u06c1\u062a \u06a9\u0645 \u0627\u062b\u0631 \u06c1\u0648\u062a\u0627 \u06c1\u06d2 \u0648\u06c1 \u0627\u06cc\u06a9 \u06a9\u0645\u0632\u0648\u0631 \u0679\u0648\u0644 \u06c1\u06d2\u06d4 \u0627\u06cc\u06a9 \u06a9\u0645\u0632\u0648\u0631 \u0622\u0644\u06c1 2SLS \u062a\u062e\u0645\u06cc\u0646\u06d2 \u062a\u06cc\u0627\u0631 \u06a9\u0631\u06d2 \u06af\u0627 \u062c\u0633 \u0645\u06cc\u06ba \u0628\u06c1\u062a \u0632\u06cc\u0627\u062f\u06c1 \u062a\u063a\u06cc\u0631 \u06c1\u0648 \u06af\u0627 \u062c\u0648 \u0686\u06be\u0648\u0679\u06d2 \u0646\u0645\u0648\u0646\u0648\u06ba \u0645\u06cc\u06ba OLS \u0627\u0646\u062f\u0627\u0632\u0648\u06ba \u06a9\u06cc \u0637\u0631\u0641 \u0628\u0691\u06be\u06d2 \u06af\u0627\u060c \u0627\u0633 \u0637\u0631\u062d \u0645\u0642\u0635\u062f \u06a9\u0648 \u0634\u06a9\u0633\u062a \u062f\u06d2 \u06af\u0627\u06d4 \u0645\u0639\u06cc\u0627\u0631\u06cc \u062a\u0634\u062e\u06cc\u0635\u06cc \u0627\u06cc\u06a9 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u0627 F-statistic \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">stage1_restricted = smf.ols(\n    f\"routed_to_premium_actual ~ {covariate_str}\", data=df\n).fit()\n\nf_stat, f_pval, _ = stage1.compare_f_test(stage1_restricted)\nprint(f\"First-stage F-statistic (instrument): {f_stat:.2f}\")\nprint(f\"p-value:                               {f_pval:.4f}\")\n\nif f_stat > 10:\n    print(\"Instrument is STRONG (F > 10). 2SLS estimates are reliable.\")\nelif f_stat > 4:\n    print(\"Instrument is BORDERLINE WEAK (4 < F < 10). Interpret with caution.\")\nelse:\n    print(\"Instrument is WEAK (F < 4). 2SLS estimates are unreliable.\")\n\nprint(f\"nFirst-stage coefficient on instrument: \"\n      f\"{stage1.params['rate_limit_fallback']:+.4f}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">First-stage F-statistic (instrument): 3780.94\np-value:                               0.0000\nInstrument is STRONG (F > 10). 2SLS estimates are reliable.\n\nFirst-stage coefficient on instrument: -0.3190\n<\/code><\/pre>\n<p>\u0627\u0648\u067e\u0631 \u0648\u0627\u0644\u0627 \u06a9\u0648\u0688 F-\u0679\u06cc\u0633\u0679 \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062a\u0627\u06a9\u06c1 \u067e\u06c1\u0644\u06d2 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u06d2 \u0645\u06a9\u0645\u0644 \u0645\u0627\u0688\u0644 (\u0622\u0644\u0627\u062a \u06a9\u06d2 \u0633\u0627\u062a\u06be) \u06a9\u0627 \u0645\u062d\u062f\u0648\u062f \u0645\u0627\u0688\u0644 (\u0628\u063a\u06cc\u0631 \u0622\u0644\u0627\u062a) \u0633\u06d2 \u0645\u0648\u0627\u0632\u0646\u06c1 \u06a9\u06cc\u0627 \u062c\u0627 \u0633\u06a9\u06d2\u06d4 3780 \u06a9\u0627 F \u0627\u0646\u06af\u0648\u0679\u06be\u06d2 \u06a9\u06d2 Staiger-Stock \u06a9\u06d2 \u0627\u0635\u0648\u0644 \u0633\u06d2 \u0628\u06c1\u062a \u0632\u06cc\u0627\u062f\u06c1 \u06c1\u06d2\u06d4 50,000 \u0645\u0634\u0627\u06c1\u062f\u0627\u062a \u067e\u0631 \u0644\u0627\u06af\u0648 \u06c1\u0648\u0646\u06d2 \u0648\u0627\u0644\u06cc 15% \u062a\u0639\u0645\u06cc\u0644 \u06a9\u06cc \u0634\u0631\u062d \u0628\u0691\u06d2 \u0627\u0648\u0631 \u062f\u0631\u0633\u062a \u0627\u0646\u062f\u0627\u0632\u06d2 \u06a9\u06d2 \u0645\u0637\u0627\u0628\u0642 \u067e\u06c1\u0644\u06d2 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u06d2 \u0627\u062b\u0631\u0627\u062a \u067e\u06cc\u062f\u0627 \u06a9\u0631\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>F-statistic \u062d\u0642\u06cc\u0642\u06cc \u067e\u0631\u0648\u0688\u06a9\u0634\u0646 \u0688\u06cc\u0679\u0627 \u0633\u06cc\u0679\u0633 \u0645\u06cc\u06ba \u06a9\u0645 \u06c1\u06d2 \u062c\u0633 \u0645\u06cc\u06ba \u06a9\u0645 \u0644\u0627\u06af\u062a \u06a9\u06cc \u0634\u0631\u062d \u06cc\u0627 \u0686\u06be\u0648\u0679\u06d2 \u0688\u06cc\u0679\u0627 \u0633\u06cc\u0679 \u06c1\u06cc\u06ba\u06d4 \u0627\u06af\u0631 F-statistic 10 \u0633\u06d2 \u06a9\u0645 \u06c1\u06d2\u060c \u062a\u0648 \u0632\u06cc\u0627\u062f\u06c1 \u0637\u0627\u0642\u062a\u0648\u0631 \u0679\u0648\u0644 \u062a\u0644\u0627\u0634 \u06a9\u0631\u06cc\u06ba \u06cc\u0627 \u0646\u062a\u06cc\u062c\u06c1 \u0627\u062e\u0630 \u06a9\u0631\u0646\u06d2 \u0633\u06d2 \u067e\u06c1\u0644\u06d2 \u0645\u0632\u06cc\u062f \u0627\u0645\u067e\u06cc\u0679\u06cc\u0634\u0646 \u0688\u06cc\u0679\u0627 \u0634\u0627\u0645\u0644 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<p>\u0622\u0644\u06d2 \u06a9\u06cc \u0637\u0627\u0642\u062a \u0627\u0648\u0631 \u0627\u062e\u0631\u0627\u062c \u06a9\u06cc \u062f\u0631\u0633\u062a\u06af\u06cc \u06a9\u06d2 \u062f\u0631\u0645\u06cc\u0627\u0646 \u062a\u062c\u0627\u0631\u062a \u06c1\u06d2 \u062c\u0648 \u0648\u0627\u0636\u062d \u0637\u0648\u0631 \u067e\u0631 \u062c\u06be\u0646\u0688\u0627 \u0644\u06af\u0627\u0646\u06d2 \u06a9\u06d2 \u0642\u0627\u0628\u0644 \u06c1\u06d2\u06d4 \u0622\u067e \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u06a9\u06cc \u0634\u0631\u062d \u0645\u06cc\u06ba \u0627\u0636\u0627\u0641\u06c1 \u06a9\u0631 \u06a9\u06d2 \u0627\u067e\u0646\u06d2 \u0622\u0644\u06d2 \u06a9\u0648 \u0645\u0632\u06cc\u062f \u0645\u0636\u0628\u0648\u0637 \u0628\u0646\u0627 \u0633\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u0644\u06cc\u06a9\u0646 \u0627\u06af\u0631 \u0622\u067e \u0635\u0627\u0631\u0641 \u06a9\u06d2 \u062a\u062c\u0631\u0628\u06d2 \u06a9\u0648 \u0645\u062a\u0627\u062b\u0631 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u0633\u06d2 \u06a9\u0627\u0641\u06cc \u0627\u0648\u0646\u0686\u0627 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062a\u0648 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u0627\u0637\u0645\u06cc\u0646\u0627\u0646 \u0627\u0648\u0631 \u062f\u0648\u0628\u0627\u0631\u06c1 \u06a9\u0648\u0634\u0634 \u06a9\u06d2 \u0631\u0648\u06cc\u06d2 \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u06a9\u0627\u0645 \u06a9\u06cc \u062a\u06a9\u0645\u06cc\u0644 \u06a9\u0648 \u0628\u0631\u0627\u06c1 \u0631\u0627\u0633\u062a \u0645\u062a\u0627\u062b\u0631 \u06a9\u0631\u0646\u0627 \u0634\u0631\u0648\u0639 \u06a9\u0631 \u062f\u06cc\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062c\u0648 \u0627\u062e\u0631\u0627\u062c \u06a9\u06cc \u067e\u0627\u0628\u0646\u062f\u06cc \u06a9\u06cc \u062e\u0644\u0627\u0641 \u0648\u0631\u0632\u06cc \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0645\u0642\u0635\u062f \u0627\u06cc\u06a9 \u0637\u0627\u0642\u062a\u0648\u0631 \u0679\u0648\u0644 \u06c1\u06d2 \u062c\u0648 \u0645\u0637\u0627\u0628\u0642\u062a \u0627\u0648\u0631 \u0627\u062e\u0631\u0627\u062c \u062f\u0648\u0646\u0648\u06ba \u06a9\u0648 \u067e\u0648\u0631\u0627 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>endogeneity \u06a9\u06cc \u0633\u0645\u062a \u06a9\u0648 \u0686\u06cc\u06a9 \u06a9\u0631\u06cc\u06ba:<\/p>\n<pre><code class=\"language-python\">gap = ols_coef - tsls_coef\nprint(f\"OLS minus 2SLS gap: {gap:+.4f}\")\nif abs(gap) > 0.005:\n    print(\"Gap suggests endogeneity bias is present in OLS.\")\nelse:\n    print(\"Small gap: OLS and 2SLS broadly agree.\")\nprint(\"For a formal Hausman endogeneity test, use linearmodels IV2SLS.\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">OLS minus 2SLS gap: -0.0272\nGap suggests endogeneity bias is present in OLS.\nFor a formal Hausman endogeneity test, use linearmodels IV2SLS.\n<\/code><\/pre>\n<p>\u06cc\u06c1\u0627\u06ba \u06a9\u06cc\u0627 \u06c1\u0648 \u0631\u06c1\u0627 \u06c1\u06d2: OLS \u0627\u0648\u0631 2SLS \u06a9\u06d2 \u062f\u0631\u0645\u06cc\u0627\u0646 \u0641\u0631\u0642 endogeneity \u06a9\u06cc \u062a\u0634\u062e\u06cc\u0635 \u06c1\u06d2\u06d4 2.7pp \u06a9\u0627 \u0641\u0631\u0642 \u0627\u0633 \u0628\u0627\u062a \u06a9\u06cc \u062a\u0635\u062f\u06cc\u0642 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u06a9\u06c1 \u0631\u0648\u0679\u0646\u06af \u0645\u062a\u063a\u06cc\u0631 \u0648\u0627\u0642\u0639\u06cc \u063a\u06cc\u0631 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0634\u062f\u06c1 \u0627\u0644\u062c\u06be\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0645\u062a\u063a\u06cc\u0631 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0645\u0646\u0633\u0644\u06a9 \u06c1\u06d2 \u0627\u0648\u0631 \u06cc\u06c1 \u06a9\u06c1 OLS \u06a9\u0686\u06be \u0627\u0644\u062c\u06be\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0645\u062a\u063a\u06cc\u0631 \u0627\u062b\u0631 \u06a9\u0648 \u062c\u0630\u0628 \u06a9\u0631 \u0631\u06c1\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0628\u0627\u0636\u0627\u0628\u0637\u06c1 \u0637\u0648\u0631 \u067e\u0631 \u062f\u0631\u0633\u062a \u06c1\u0627\u0624\u0633\u0645\u06cc\u0646 \u0679\u06cc\u0633\u0679 \u06a9\u06d2 \u0644\u06cc\u06d2 (\u0627\u06cc\u06a9 \u0627\u06cc\u0633\u0627 \u0679\u06cc\u0633\u0679 \u062c\u0648 \u06a9\u06c1 null \u06a9\u06d2 \u062a\u062d\u062a \u0645\u0639\u0644\u0648\u0645 \u062a\u0642\u0633\u06cc\u0645 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0627\u06cc\u06a9 chi-square statistic \u067e\u06cc\u062f\u0627 \u06a9\u0631\u062a\u0627 \u06c1\u06d2)\u060c \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u06cc\u06ba: <code>linearmodels.IV2SLS<\/code>\u06cc\u06c1 \u0627\u06cc\u06a9 \u0628\u0644\u0679 \u0627\u0646 \u0679\u06cc\u0633\u0679 \u06c1\u06d2\u06d4 \u0645\u0646\u062f\u0631\u062c\u06c1 \u0628\u0627\u0644\u0627 \u0648\u0627\u0642\u0641\u06cc\u062a \u06a9\u06cc \u062c\u0627\u0646\u0686 \u0635\u0631\u0641 \u0627\u06cc\u06a9 \u0641\u0648\u0631\u06cc \u062a\u0634\u062e\u06cc\u0635 \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-step-4-the-late-is-the-quantity-you-actually-care-about\">\u0645\u0631\u062d\u0644\u06c1 4: \u062a\u0627\u062e\u06cc\u0631 \u0648\u06c1 \u0645\u0642\u062f\u0627\u0631 \u06c1\u06d2 \u062c\u0633 \u06a9\u0627 \u0622\u067e \u0627\u0635\u0644 \u0645\u06cc\u06ba \u062e\u06cc\u0627\u0644 \u0631\u06a9\u06be\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/h2>\n<p>2SLS \u0645\u0642\u0627\u0645\u06cc \u0627\u0648\u0633\u0637 \u0639\u0644\u0627\u062c \u06a9\u06d2 \u0627\u062b\u0631 (LATE) \u06a9\u0627 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u0644\u06af\u0627\u062a\u0627 \u06c1\u06d2\u060c \u062c\u0633\u06d2 Complier Average Causal Effect (CACE) \u0628\u06be\u06cc \u06a9\u06c1\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u0644\u06cc\u0679 \u0635\u0631\u0641 \u06a9\u0645\u067e\u0627\u0626\u0644\u0631\u0632 \u067e\u0631 \u0644\u0627\u06af\u0648 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u06d4 \u06cc\u0639\u0646\u06cc\u060c \u06cc\u06c1 \u0627\u0633\u062a\u0641\u0633\u0627\u0631\u0627\u062a \u06a9\u0627 \u0627\u06cc\u06a9 \u0645\u062e\u0635\u0648\u0635 \u0630\u06cc\u0644\u06cc \u0633\u06cc\u0679 \u06c1\u06d2 \u062c\u0633 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0631\u0648\u0679\u0646\u06af \u062f\u0631\u0627\u0635\u0644 \u062a\u0628\u062f\u06cc\u0644 \u06c1\u0648\u062a\u06cc \u06c1\u06d2 \u062c\u0628 \u0622\u0644\u06c1 \u0686\u0644\u062a\u0627 \u06c1\u06d2\u06d4 \u0634\u0631\u062d \u06a9\u0648 \u0645\u062d\u062f\u0648\u062f \u06a9\u0631\u0646\u0627 \u0635\u0631\u0641 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0627\u06c1\u0644 \u0627\u0633\u062a\u0641\u0633\u0627\u0631\u0627\u062a \u06a9\u0648 \u0645\u062a\u0627\u062b\u0631 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0646 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0645\u0648\u0627\u062e\u0630\u06c1 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u060c \u0627\u0633 \u0644\u06cc\u06d2 \u0644\u06cc\u0679 \u0635\u0631\u0641 \u0627\u0633 \u0630\u06cc\u0644\u06cc \u0622\u0628\u0627\u062f\u06cc \u067e\u0631 \u0644\u0627\u06af\u0648 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">compliers_mask = df[\"rate_limit_fallback\"] == 1\ncomplier_count = compliers_mask.sum()\ncomplier_pct   = complier_count \/ n * 100\n\nprint(f\"Approximate complier population: {complier_count:,} ({complier_pct:.1f}% of queries)\")\nprint(f\"nComplier mean confidence:     {df[compliers_mask]['query_confidence'].mean():.3f}\")\nprint(f\"Non-complier mean confidence: {df[~compliers_mask]['query_confidence'].mean():.3f}\")\nprint(f\"n2SLS LATE estimate: {tsls_coef:+.4f}\")\nprint(\"This is the causal effect of premium routing for queries rerouted\")\nprint(\"by rate-limit fallbacks, not all queries in the dataset.\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">Approximate complier population: 7,575 (15.2% of queries)\n\nComplier mean confidence:     0.716\nNon-complier mean confidence: 0.715\n\n2SLS LATE estimate: +0.0599\nThis is the causal effect of premium routing for queries rerouted\nby rate-limit fallbacks, not all queries in the dataset.\n<\/code><\/pre>\n<p>\u0627\u0633 \u06a9\u0648\u0688 \u0645\u06cc\u06ba\u060c \u0645\u0631\u062a\u0628 \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u06a9\u06cc \u0622\u0628\u0627\u062f\u06cc 7,575 \u0633\u0648\u0627\u0644\u0627\u062a \u06c1\u06cc\u06ba (\u0627\u06cc\u0633\u06d2 \u0633\u0648\u0627\u0644\u0627\u062a \u062c\u0648 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u06a9\u06cc \u0634\u0631\u062d \u06a9\u0648 \u0645\u062d\u062f\u0648\u062f \u06a9\u0631\u0646\u06d2 \u06a9\u0627 \u062a\u062c\u0631\u0628\u06c1 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u0627\u0646\u06c1\u06cc\u06ba \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0633\u06d2 \u06a9\u0645 \u0644\u0627\u06af\u062a \u0645\u06cc\u06ba \u062a\u0628\u062f\u06cc\u0644 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2)\u06d4 \u0627\u0646 \u06a9\u0627 \u0627\u0648\u0633\u0637 \u0627\u0639\u062a\u0645\u0627\u062f (0.716) \u062a\u0642\u0631\u06cc\u0628\u0627\u064b \u063a\u06cc\u0631 \u062a\u0639\u0645\u06cc\u0644 \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u06af\u0631\u0648\u067e (0.715) \u0633\u06d2 \u0645\u0645\u0627\u062b\u0644 \u06c1\u06d2\u060c \u0627\u0633 \u0628\u0627\u062a \u06a9\u06cc \u062a\u0635\u062f\u06cc\u0642 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u06a9\u06c1 \u0633\u0648\u0627\u0644 \u06a9\u06cc \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u0633\u06d2 \u0642\u0637\u0639 \u0646\u0638\u0631 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u067e\u0631 \u0639\u0645\u0644 \u062f\u0631\u0622\u0645\u062f \u06a9\u06cc\u0627 \u06af\u06cc\u0627 \u062a\u06be\u0627\u06d4<\/p>\n<p>\u062c\u0628 \u0645\u0631\u062a\u0628 \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u0627 \u06a9\u0633\u06cc \u0642\u0627\u0628\u0644 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u067e\u0631 \u062a\u0645\u0627\u0645 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u06d2 \u0646\u0645\u0627\u0626\u0646\u062f\u06c1 \u0679\u06a9\u0691\u0648\u06ba \u06a9\u0648 \u062f\u06cc\u06a9\u06be\u062a\u0627 \u06c1\u06d2\u060c \u062a\u0648 LATE \u0627\u06a9\u062b\u0631 \u0627\u0648\u0633\u0637 \u067e\u0631\u0648\u0633\u06cc\u0633\u0646\u06af \u0627\u06cc\u0641\u06cc\u06a9\u0679\u06cc\u0648\u0646\u0633 (ATE) \u06a9\u0627 \u0627\u06cc\u06a9 \u0645\u0639\u0642\u0648\u0644 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0642\u0627\u0628\u0644 \u0645\u0634\u0627\u06c1\u062f\u06c1 \u0646\u0645\u0627\u0626\u0646\u062f\u06af\u06cc \u06a9\u0645 \u0627\u0632 \u06a9\u0645 \u062a\u0634\u062e\u06cc\u0635\u06cc \u0645\u0639\u06cc\u0627\u0631 \u067e\u0631 \u067e\u0648\u0631\u0627 \u0627\u062a\u0631\u062a\u06cc \u06c1\u06d2\u06d4 \u0631\u0633\u0645\u06cc \u062f\u06cc\u0631 \u0633\u06d2 ATE \u06a9\u06d2 \u062d\u0627\u0644\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u06cc\u0627 \u062a\u0648 \u062a\u0645\u0627\u0645 \u0627\u06a9\u0627\u0626\u06cc\u0648\u06ba \u06a9\u06d2 \u0644\u06cc\u06d2 \u06cc\u06a9\u0633\u0627\u06ba \u0639\u0644\u0627\u062c \u06a9\u06d2 \u0627\u062b\u0631 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2 \u06cc\u0627 \u067e\u06be\u0631 \u0622\u0628\u0627\u062f\u06cc \u0645\u06cc\u06ba \u0645\u0648\u062c\u0648\u062f \u062a\u0645\u0627\u0645 \u0627\u06a9\u0627\u0626\u06cc\u0648\u06ba \u06a9\u06d2 \u0644\u06cc\u06d2 \u0645\u0648\u0632\u0648\u06ba \u0622\u0644\u06c1\u06d4 \u0627\u06af\u0631 \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u0627\u062b\u0631\u0627\u062a \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06cc \u0627\u0642\u0633\u0627\u0645 \u0645\u06cc\u06ba \u0645\u062a\u0636\u0627\u062f \u06c1\u06cc\u06ba (\u0645\u062b\u0627\u0644 \u06a9\u06d2 \u0637\u0648\u0631 \u067e\u0631\u060c \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u067e\u06cc\u0686\u06cc\u062f\u06c1 \u0633\u0648\u0627\u0644\u0627\u062a \u0645\u06cc\u06ba \u0622\u0633\u0627\u0646 \u0633\u0648\u0627\u0644\u0627\u062a \u0633\u06d2 \u06a9\u06c1\u06cc\u06ba \u0632\u06cc\u0627\u062f\u06c1 \u0645\u062f\u062f \u06a9\u0631\u062a\u06cc \u06c1\u06d2)\u060c \u0644\u06cc\u0679 ATE \u0633\u06d2 \u0646\u0645\u0627\u06cc\u0627\u06ba \u0637\u0648\u0631 \u067e\u0631 \u0645\u062e\u062a\u0644\u0641 \u06c1\u0648 \u0633\u06a9\u062a\u0627 \u06c1\u06d2 \u06cc\u06c1\u0627\u06ba \u062a\u06a9 \u06a9\u06c1 \u062c\u0628 \u06a9\u0645\u067e\u0644\u0627\u0626\u0646\u0679 \u06a9\u0627 \u0627\u0648\u0633\u0637 \u0627\u0639\u062a\u0645\u0627\u062f \u0646\u0627\u0646 \u06a9\u0645\u067e\u0644\u0627\u0626\u0646\u0679 \u06af\u0631\u0648\u067e \u0633\u06d2 \u0645\u0644\u062a\u0627 \u062c\u0644\u062a\u0627 \u06c1\u0648\u06d4<\/p>\n<p>\u062a\u0632\u0648\u06cc\u0631\u0627\u062a\u06cc \u0635\u0644\u0627\u062d\u06cc\u062a \u06a9\u06cc \u0645\u0646\u0635\u0648\u0628\u06c1 \u0628\u0646\u062f\u06cc \u06a9\u06d2 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\u06a9\u0648\u0646 \u0633\u06d2 \u0633\u0648\u0627\u0644\u0627\u062a \u0645\u062d\u062f\u0648\u062f \u06c1\u06cc\u06ba\u06d4 2SLS \u0627\u0633 \u0633\u0648\u0627\u0644 \u06a9\u0627 \u0628\u0631\u0627\u06c1 \u0631\u0627\u0633\u062a \u062c\u0648\u0627\u0628 \u062f\u06cc\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-step-5-bootstrap-confidence-intervals\">\u0645\u0631\u062d\u0644\u06c1 5: \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u0627 \u0648\u0642\u0641\u06c1<\/h2>\n<p>\u062f\u0633\u062a\u06cc 2SLS \u062f\u0648\u0633\u0631\u06d2 \u062f\u0631\u062c\u06d2 \u06a9\u06cc \u0645\u0639\u06cc\u0627\u0631\u06cc \u063a\u0644\u0637\u06cc\u0627\u06ba \u067e\u06cc\u062f\u0627 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u0628\u06c1\u062a \u0686\u06be\u0648\u0679\u06cc \u06c1\u06cc\u06ba\u060c \u062c\u06cc\u0633\u0627 \u06a9\u06c1 \u0645\u0631\u062d\u0644\u06c1 2 \u0645\u06cc\u06ba \u0628\u06cc\u0627\u0646 \u06a9\u06cc\u0627 \u06af\u06cc\u0627 \u06c1\u06d2\u06d4 \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e CI \u062f\u0631\u0633\u062a \u062a\u062c\u0632\u06cc\u0627\u062a\u06cc \u062a\u063a\u06cc\u0631 \u0641\u0627\u0631\u0645\u0648\u0644\u06c1 \u0627\u062e\u0630 \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06a9\u06d2 \u0628\u063a\u06cc\u0631 \u063a\u06cc\u0631 \u06cc\u0642\u06cc\u0646\u06cc \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u06d2 \u0642\u0627\u0628\u0644 \u0627\u0639\u062a\u0645\u0627\u062f \u062a\u062e\u0645\u06cc\u0646\u06c1 \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e \u067e\u0648\u0631\u06d2 \u062f\u0648 \u0642\u062f\u0645\u06cc \u0637\u0631\u06cc\u0642\u06c1 \u06a9\u0627\u0631 \u06a9\u0648 \u0627\u06cc\u06a9 \u0633\u0627\u062a\u06be \u062f\u0648\u0628\u0627\u0631\u06c1 \u0646\u0645\u0648\u0646\u06c1 \u0628\u0646\u0627\u062a\u0627 \u06c1\u06d2\u060c \u062f\u0648\u0646\u0648\u06ba \u0645\u0631\u0627\u062d\u0644 \u0645\u06cc\u06ba \u0646\u0645\u0648\u0646\u06d2 \u0644\u06cc\u0646\u06d2 \u06a9\u06d2 \u062a\u063a\u06cc\u0631 \u06a9\u0648 \u067e\u06a9\u0691\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">rng_boot = np.random.default_rng(7)\nols_boot, tsls_boot = [], []\n\nfor _ in range(500):\n    samp = df.sample(len(df), replace=True,\n                     random_state=int(rng_boot.integers(1_000_000_000)))\n\n    # OLS bootstrap\n    ols_b = smf.ols(\n        f\"task_completed_iv ~ routed_to_premium_actual + {covariate_str}\",\n        data=samp\n    ).fit()\n    ols_boot.append(ols_b.params[\"routed_to_premium_actual\"])\n\n    # 2SLS bootstrap (two stages together)\n    s1b = smf.ols(\n        f\"routed_to_premium_actual ~ rate_limit_fallback + {covariate_str}\",\n        data=samp\n    ).fit()\n    samp = samp.copy()\n    samp[\"rtp_hat\"] = s1b.fittedvalues\n    s2b = smf.ols(\n        f\"task_completed_iv ~ rtp_hat + {covariate_str}\",\n        data=samp\n    ).fit()\n    tsls_boot.append(s2b.params[\"rtp_hat\"])\n\nols_ci  = (np.percentile(ols_boot, 2.5),  np.percentile(ols_boot, 97.5))\ntsls_ci = (np.percentile(tsls_boot, 2.5), np.percentile(tsls_boot, 97.5))\ntrue_eff = 0.0600\n\nprint(f\"OLS  95% CI: [{ols_ci[0]:+.4f}, {ols_ci[1]:+.4f}]\")\nprint(f\"2SLS 95% CI: [{tsls_ci[0]:+.4f}, {tsls_ci[1]:+.4f}]\")\nprint(f\"Ground truth: +{true_eff:.4f}\")\nprint(f\"OLS CI covers ground truth:  {ols_ci[0] <= true_eff <= ols_ci[1]}\")\nprint(f\"2SLS CI covers ground truth: {tsls_ci[0] <= true_eff <= tsls_ci[1]}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">OLS  95% CI: [+0.0227, +0.0426]\n2SLS 95% CI: [+0.0247, +0.0969]\nGround truth: +0.0600\nOLS CI covers ground truth:  False\n2SLS CI covers ground truth: True\n<\/code><\/pre>\n<p>\u0627\u0633 \u06a9\u0648\u0688 \u06a9\u06d2 \u0644\u06cc\u06d2\u060c OLS 95% CI ([+0.023, +0.043]) \u0645\u06a9\u0645\u0644 \u0637\u0648\u0631 \u067e\u0631 \u0627\u0635\u0644 +0.06 \u0627\u062b\u0631 \u0633\u06d2 \u0645\u062d\u0631\u0648\u0645 \u06c1\u0648\u06af\u06cc\u0627\u06d4 \u0627\u0633 \u0648\u0642\u0641\u06c1 \u0645\u06cc\u06ba \u062a\u0645\u0627\u0645 \u0627\u0642\u062f\u0627\u0631 \u0627\u0635\u0644 \u0633\u0686\u0627\u0626\u06cc \u0633\u06d2 \u06a9\u0645 \u06c1\u06cc\u06ba\u06d4 OLS \u0648\u0627\u0636\u062d \u0637\u0648\u0631 \u067e\u0631 \u063a\u0644\u0637 \u06c1\u06d2\u06d4 2SLS CI([+0.025, +0.097]) \u0627\u0635\u0644 \u062d\u0642\u0627\u0626\u0642 \u0633\u06d2 \u0646\u0645\u0679\u062a\u0627 \u06c1\u06d2\u06d4 \u06cc\u06c1 OLS \u0648\u0642\u0641\u06c1 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0648\u0633\u06cc\u0639 \u06c1\u06d2\u060c IV \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06a9\u06cc \u0628\u0627\u0632\u06cc \u0644\u0627\u06af\u062a \u06a9\u0648 \u0638\u0627\u06c1\u0631 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0627\u0633 \u06a9\u0627 \u0645\u0637\u0644\u0628 \u06c1\u06d2 \u06a9\u06c1 \u062a\u0627\u062b\u06cc\u0631 \u062d\u0627\u0635\u0644 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0622\u067e \u06a9\u0648 \u0627\u0633 \u06a9\u06cc \u062f\u0631\u0633\u062a\u06af\u06cc \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0627\u062f\u0627\u0626\u06cc\u06af\u06cc \u06a9\u0631\u0646\u06cc \u06c1\u0648\u06af\u06cc\u06d4<\/p>\n<p>\u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e \u06c1\u0631 \u062a\u06a9\u0631\u0627\u0631 \u067e\u0631 \u062f\u0648\u0646\u0648\u06ba \u0645\u0631\u0627\u062d\u0644 \u06a9\u0648 \u062f\u0648\u0628\u0627\u0631\u06c1 \u0646\u0645\u0648\u0646\u06c1 \u0628\u0646\u0627\u062a\u0627 \u06c1\u06d2\u060c \u0644\u06c1\u0630\u0627 \u063a\u06cc\u0631 \u06cc\u0642\u06cc\u0646\u06cc \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u062f\u0648 \u0645\u0631\u0627\u062d\u0644 \u06a9\u06cc \u0633\u0627\u062e\u062a \u06a9\u0648 \u062f\u0631\u0633\u062a \u0637\u0631\u06cc\u0642\u06d2 \u0633\u06d2 \u0628\u06cc\u0627\u0646 \u06a9\u0631\u062a\u06cc \u06c1\u06d2\u06d4 \u062f\u0633\u062a\u06cc \u0646\u0641\u0627\u0630 \u0633\u06d2 2SLS \u0646\u062a\u0627\u0626\u062c \u06a9\u06cc \u0627\u0637\u0644\u0627\u0639 \u062f\u06cc\u062a\u06d2 \u0648\u0642\u062a \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e CI \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u06cc\u06ba\u06d4 \u0627\u0633 \u06a9\u06cc \u0648\u062c\u06c1 \u06cc\u06c1 \u06c1\u06d2 \u06a9\u06c1 \u06cc\u06c1 \u062f\u0648\u0633\u0631\u06d2 \u062f\u0631\u062c\u06d2 \u06a9\u06d2 \u067e\u06cc\u0631\u0627\u0645\u06cc\u0679\u0631 SE \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0645\u0633\u062a\u062d\u06a9\u0645 \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-when-instrumental-variables-fail\">\u062c\u0628 \u0627\u0646\u0633\u0679\u0631\u0648\u0645\u06cc\u0646\u0679\u0644 \u0645\u062a\u063a\u06cc\u0631\u0627\u062a \u0646\u0627\u06a9\u0627\u0645 \u06c1\u0648\u062c\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/h2>\n<p>IV \u062a\u062c\u0632\u06cc\u06c1 \u0645\u06cc\u06ba \u067e\u0631\u0648\u067e\u06cc\u0646\u0633\u06cc\u0679\u06cc \u0633\u06a9\u0648\u0631 \u06cc\u0627 \u0631\u06cc\u06af\u0631\u06cc\u0634\u0646 \u0688\u0633\u06a9\u0648\u0646\u0679\u06cc\u0648\u0679\u06cc \u0641\u06cc\u0644 \u0645\u0648\u0688\u0632 \u06a9\u06d2 \u0645\u0642\u0627\u0628\u0644\u06d2 \u0645\u06cc\u06ba \u0632\u06cc\u0627\u062f\u06c1 \u06a9\u067e\u0679\u06cc \u0646\u0627\u06a9\u0627\u0645\u06cc \u06a9\u06d2 \u0637\u0631\u06cc\u0642\u06d2 \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0627\u0633 \u06a9\u06cc \u0648\u062c\u06c1 \u06cc\u06c1 \u06c1\u06d2 \u06a9\u06c1 \u0686\u0627\u0631 \u0645\u06cc\u06ba \u0633\u06d2 \u062f\u0648 \u0645\u0641\u0631\u0648\u0636\u0648\u06ba \u06a9\u0648 \u0627\u06a9\u06cc\u0644\u06d2 \u0688\u06cc\u0679\u0627 \u0633\u06d2 \u0646\u06c1\u06cc\u06ba \u062c\u0627\u0646\u0686\u0627 \u062c\u0627 \u0633\u06a9\u062a\u0627\u06d4<\/p>\n<h3 id=\"heading-weak-instruments\">\u06a9\u0645\u0632\u0648\u0631 \u0622\u0644\u06c1<\/h3>\n<p>10 \u0633\u06d2 \u06a9\u0645 \u06a9\u0627 \u067e\u06c1\u0644\u0627 \u062f\u0631\u062c\u06c1 F \u0627\u06cc\u06a9 \u0633\u0646\u06af\u06cc\u0646 \u0634\u0646\u0627\u062e\u062a\u06cc \u0645\u0633\u0626\u0644\u06c1 \u06a9\u06cc \u0646\u0634\u0627\u0646\u062f\u06c1\u06cc \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u06a9\u0645\u0632\u0648\u0631 \u0622\u0644\u0627\u062a \u06a9\u06cc \u0648\u062c\u06c1 \u0633\u06d2 2SLS \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06a9\u0646\u0646\u062f\u06c1 \u0645\u062d\u062f\u0648\u062f \u0646\u0645\u0648\u0646\u0648\u06ba \u0645\u06cc\u06ba \u0628\u0691\u06d2 \u062a\u063a\u06cc\u0631 \u06a9\u06cc \u0637\u0631\u0641 \u0628\u0691\u06be\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 OLS \u0645\u062a\u0639\u0635\u0628 \u0628\u06cc\u0633 \u0644\u0627\u0626\u0646 \u06a9\u0648 \u0646\u0642\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0646\u0641\u06cc\u0633 \u06a9\u0627\u0645 \u0627\u0646\u062c\u0627\u0645 \u062f\u06cc\u062a\u0627 \u06c1\u06d2\u06d4 IV \u06a9\u06d2 \u0646\u062a\u0627\u0626\u062c \u06a9\u06cc \u062a\u0634\u0631\u06cc\u062d \u06a9\u0631\u0646\u06d2 \u0633\u06d2 \u067e\u06c1\u0644\u06d2 F-statistic \u0686\u06cc\u06a9 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<p>\u0627\u06af\u0631 F 10 \u0633\u06d2 \u06a9\u0645 \u06c1\u06d2\u060c \u062a\u0648 \u0627\u06cc\u06a9 \u0645\u0636\u0628\u0648\u0637 \u067e\u0631\u0648\u0688\u06a9\u0679 \u062a\u0644\u0627\u0634 \u06a9\u0631\u06cc\u06ba \u06cc\u0627 \u0648\u0627\u0636\u062d \u06a9\u0645\u0632\u0648\u0631 \u0645\u0635\u0646\u0648\u0639\u0627\u062a \u06a9\u06cc \u0648\u0627\u0631\u0646\u0646\u06af \u06a9\u06d2 \u0633\u0627\u062a\u06be \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06a9\u06cc \u0627\u0637\u0644\u0627\u0639 \u062f\u06cc\u06ba\u06d4 \u06cc\u06c1\u0627\u06ba \u06a9\u0627 \u0622\u0644\u06c1 \u0645\u0636\u0628\u0648\u0637 \u06c1\u06d2\u060c \u06a9\u06cc\u0648\u0646\u06a9\u06c1 50,000 \u0633\u0648\u0627\u0644\u0627\u062a \u067e\u0631 \u0644\u0627\u06af\u0648 15% \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u06a9\u06cc \u0634\u0631\u062d 7,500 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0631\u0648\u0679\u0646\u06af \u062a\u0628\u062f\u06cc\u0644\u06cc\u0627\u06ba (F = 3780) \u067e\u06cc\u062f\u0627 \u06a9\u0631\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<h3 id=\"heading-exclusion-restriction-violations\">\u0627\u062e\u0631\u0627\u062c \u06a9\u06cc \u067e\u0627\u0628\u0646\u062f\u06cc\u0648\u06ba \u06a9\u06cc \u062e\u0644\u0627\u0641 \u0648\u0631\u0632\u06cc<\/h3>\n<p>\u0627\u06af\u0631 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u06a9\u0648 \u0645\u062d\u062f\u0648\u062f \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0634\u0631\u062d \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u0641\u06cc\u0635\u0644\u0648\u06ba \u06a9\u06d2 \u0639\u0644\u0627\u0648\u06c1 \u062f\u06cc\u06af\u0631 \u0686\u06cc\u0646\u0644\u0632 \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u06a9\u0627\u0645 \u06a9\u06cc \u062a\u06a9\u0645\u06cc\u0644 \u06a9\u0648 \u0645\u062a\u0627\u062b\u0631 \u06a9\u0631\u062a\u06cc \u06c1\u06d2\u060c \u062a\u0648 \u0627\u062e\u0631\u0627\u062c \u0646\u0627\u06a9\u0627\u0645 \u06c1\u0648 \u062c\u0627\u0626\u06d2 \u06af\u0627\u06d4<\/p>\n<p>\u062f\u0648 \u0645\u0645\u06a9\u0646\u06c1 \u062e\u0644\u0627\u0641 \u0648\u0631\u0632\u06cc\u0627\u06ba \u06c1\u06cc\u06ba\u06d4 \u0627\u0633 \u06a9\u0627 \u0645\u0637\u0644\u0628 \u06cc\u06c1 \u06c1\u06d2 \u06a9\u06c1 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u0627\u06cc\u0648\u0646\u0679\u0633 \u0632\u06cc\u0627\u062f\u06c1 \u0679\u0631\u06cc\u0641\u06a9 \u06a9\u06d2 \u062f\u0648\u0631\u0627\u0646\u06cc\u06d2 \u0645\u06cc\u06ba \u06a9\u0644\u0633\u0679\u0631 \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba \u062c\u0628 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u06d2 \u067e\u06cc\u0686\u06cc\u062f\u06c1 \u0628\u06cc\u0686 \u0622\u067e\u0631\u06cc\u0634\u0646\u0632 \u06a9\u0631\u0646\u06d2 \u06a9\u0627 \u0632\u06cc\u0627\u062f\u06c1 \u0627\u0645\u06a9\u0627\u0646 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u060c \u062c\u0633 \u06a9\u06d2 \u0646\u062a\u06cc\u062c\u06d2 \u0645\u06cc\u06ba \u0622\u0644\u06c1 \u06a9\u0648 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06cc \u062f\u0634\u0648\u0627\u0631\u06cc \u0633\u06d2 \u062c\u0648\u0691\u062a\u0627 \u06c1\u06d2\u06d4 \u0645\u062a\u0628\u0627\u062f\u0644 \u0637\u0648\u0631 \u067e\u0631\u060c \u0648\u06c1 \u0635\u0627\u0631\u0641\u06cc\u0646 \u062c\u0648 \u0645\u062a\u0628\u0627\u062f\u0644 \u06a9\u0627 \u062a\u062c\u0631\u0628\u06c1 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u0648\u06c1 \u0627\u0646\u062d\u0637\u0627\u0637 \u067e\u0630\u06cc\u0631 \u0631\u062f\u0639\u0645\u0644 \u06a9\u0648 \u062f\u06cc\u06a9\u06be \u0633\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u0633\u06cc\u0634\u0646 \u06a9\u0648 \u062a\u0631\u06a9 \u06a9\u0631 \u0633\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062c\u0633 \u0633\u06d2 \u0635\u0627\u0631\u0641 \u06a9\u06cc \u0645\u0627\u06cc\u0648\u0633\u06cc \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u0628\u0631\u0627\u06c1 \u0631\u0627\u0633\u062a ZY \u0631\u0627\u0633\u062a\u06c1 \u0628\u0646\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0645\u0637\u0627\u0628\u0642\u062a \u06a9\u0648 \u0628\u0631\u0642\u0631\u0627\u0631 \u0631\u06a9\u06be\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u062f\u0648\u0646\u0648\u06ba \u06c1\u06cc \u0627\u062e\u0631\u0627\u062c \u06a9\u06cc \u062e\u0644\u0627\u0641 \u0648\u0631\u0632\u06cc \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0622\u067e \u0688\u06cc\u0679\u0627 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0627\u0633 \u06a9\u06cc \u062c\u0627\u0646\u0686 \u0646\u06c1\u06cc\u06ba \u06a9\u0631 \u0633\u06a9\u062a\u06d2\u06d4 \u062f\u0644\u0627\u0626\u0644 \u06a9\u06cc \u0628\u0646\u06cc\u0627\u062f \u0646\u0638\u0627\u0645 \u06a9\u06d2 \u0639\u0644\u0645 \u067e\u0631 \u06c1\u0648\u0646\u06cc \u0686\u0627\u06c1\u06cc\u06d2\u06d4<\/p>\n<h3 id=\"heading-late-vs-ate-confusion\">\u062f\u06cc\u0631 \u0628\u0645\u0642\u0627\u0628\u0644\u06c1 ATE \u0627\u0644\u062c\u06be\u0646<\/h3>\n<p>\u0627\u06af\u0631 \u06a9\u0645\u067e\u0627\u0626\u0644\u0631 \u063a\u06cc\u0631 \u0645\u0639\u0645\u0648\u0644\u06cc \u06c1\u06d2 \u062a\u0648\u060c \u0648\u0633\u06cc\u0639 \u0631\u0648\u0679\u0646\u06af \u067e\u0627\u0644\u06cc\u0633\u06cc \u06a9\u06cc \u062a\u0628\u062f\u06cc\u0644\u06cc\u0648\u06ba \u06a9\u0627 \u062c\u0648\u0627\u0632 \u067e\u06cc\u0634 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062f\u06cc\u0631 \u0633\u06d2 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u0646\u0627 \u063a\u0644\u0637 \u06c1\u0648\u06af\u0627\u06d4 \u0627\u06af\u0631 \u0634\u0631\u062d \u06a9\u06cc \u062d\u062f \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u063a\u06cc\u0631 \u0645\u062a\u0646\u0627\u0633\u0628 \u0637\u0648\u0631 \u067e\u0631 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\u062f\u0648\u0631 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u0633 \u06a9\u06cc \u0627\u0637\u0644\u0627\u0639 \u062f\u06cc\u0646\u0627 \u06af\u0648\u06cc\u0627 \u06cc\u06c1 \u0627\u06cc\u06a9 ATE \u06c1\u06d2 \u062a\u0645\u0627\u0645 \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06d2 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u06a9\u0648 \u0631\u0648\u0679 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0641\u0627\u0626\u062f\u06d2 \u06a9\u0648 \u0628\u0691\u06be\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u0645\u0631\u062d\u0644\u06c1 4 \u0645\u06cc\u06ba \u06a9\u0645\u067e\u0627\u0626\u0644\u0631 \u067e\u0631\u0627\u067e\u0631\u0679\u06cc\u0632 \u0679\u06cc\u0628\u0644 \u062a\u0634\u062e\u06cc\u0635\u06cc \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0645\u0648\u0627\u0641\u0642\u062a \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0627\u0648\u0631 \u063a\u06cc\u0631 \u062a\u0639\u0645\u06cc\u0644 \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0627\u067e\u0646\u06d2 \u0645\u0634\u0627\u06c1\u062f\u0627\u062a \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u062c\u06cc\u0633\u06d2 \u0646\u0638\u0631 \u0622\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062a\u0648 LATE ATE \u06a9\u0627 \u0627\u06cc\u06a9 \u0642\u0627\u0628\u0644 \u0627\u0639\u062a\u0645\u0627\u062f \u062a\u062e\u0645\u06cc\u0646\u06c1 \u06c1\u06d2\u06d4<\/p>\n<h3 id=\"heading-defiers-and-the-monotonicity-assumption\">\u0628\u0627\u063a\u06cc \u0627\u0648\u0631 \u06cc\u06a9\u062c\u06c1\u062a\u06cc \u06a9\u06d2 \u0645\u0641\u0631\u0648\u0636\u06d2\u06d4<\/h3>\n<p>\u062f\u06cc\u0631 \u0633\u06d2 \u062a\u062c\u0632\u06cc\u06c1 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u06cc\u06a9\u062c\u06c1\u062a\u06cc \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u06d4 \u06cc\u0639\u0646\u06cc\u060c \u0622\u0644\u06c1 \u062a\u0645\u0627\u0645 \u0645\u062a\u0627\u062b\u0631\u06c1 \u06cc\u0648\u0646\u0679\u0648\u06ba \u06a9\u0648 \u0627\u06cc\u06a9 \u06c1\u06cc \u0633\u0645\u062a \u0645\u06cc\u06ba \u0644\u06d2 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u0641\u0627\u0644 \u0628\u06cc\u06a9\u0633 \u06a9\u06cc \u0634\u0631\u062d \u06a9\u0648 \u0645\u062d\u062f\u0648\u062f \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0635\u0648\u0631\u062a \u0645\u06cc\u06ba\u060c \u06cc\u06c1 \u062a\u0642\u0631\u06cc\u0628\u0627\u064b \u06cc\u0642\u06cc\u0646\u06cc \u0637\u0648\u0631 \u067e\u0631 \u067e\u0648\u0631\u0627 \u06c1\u0648 \u062c\u0627\u0626\u06d2 \u06af\u0627 \u06a9\u06cc\u0648\u0646\u06a9\u06c1 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u06c1\u0645\u06cc\u0634\u06c1 \u0645\u062a\u0627\u062b\u0631\u06c1 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u0627\u0645\u06a9\u0627\u0646\u0627\u062a \u06a9\u0648 \u06a9\u0645 \u06a9\u0631 \u062f\u06d2 \u06af\u0627\u06d4<\/p>\n<p>\u0627\u06af\u0631 \u0645\u0639\u0627\u0648\u0636\u06d2 \u06a9\u0627 \u06a9\u0686\u06be \u0637\u0631\u06cc\u0642\u06c1 \u06a9\u0627\u0631 \u0645\u0648\u062c\u0648\u062f \u06c1\u06d2 (\u0645\u062b\u0627\u0644 \u06a9\u06d2 \u0637\u0648\u0631 \u067e\u0631 \u0641\u0627\u0644 \u0628\u06cc\u06a9: \u0627\u06cc\u06a9 \u0633\u0648\u0627\u0644 \u0627\u06af\u0644\u06cc \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06cc \u062a\u0631\u062c\u06cc\u062d\u06cc \u0628\u06c1\u062a\u0631\u06cc \u06a9\u0648 \u0645\u062a\u062d\u0631\u06a9 \u06a9\u0631\u062a\u0627 \u06c1\u06d2)\u060c \u0645\u0633\u062a\u0631\u062f \u06c1\u0648\u062c\u0627\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 \u06cc\u06a9\u062c\u06c1\u062a\u06cc \u06a9\u0627 \u0645\u0641\u0631\u0648\u0636\u06c1 \u0679\u0648\u0679 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u062f\u06cc\u0631 \u0633\u06d2 \u0628\u06be\u0631\u0648\u0633\u06c1 \u06a9\u0631\u0646\u06d2 \u0633\u06d2 \u067e\u06c1\u0644\u06d2 \u0633\u0645\u062a \u06a9\u06cc \u0645\u0633\u062a\u0642\u0644 \u0645\u0632\u0627\u062c\u06cc \u06a9\u0648 \u0686\u06cc\u06a9 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<h2 id=\"heading-what-to-do-next\">\u0622\u06af\u06d2 \u06a9\u06cc\u0627 \u06a9\u0631\u0646\u0627 \u06c1\u06d2\u06d4<\/h2>\n<p>\u0627\u0633 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u0645\u06cc\u06ba \u062f\u0633\u062a\u06cc 2SLS \u06a9\u0627 \u0646\u0641\u0627\u0630 \u0645\u06cc\u06a9\u0627\u0646\u0632\u0645 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0634\u0641\u0627\u0641 \u06c1\u06d2\u060c \u0644\u06cc\u06a9\u0646 \u063a\u0644\u0637 \u0645\u0639\u06cc\u0627\u0631\u06cc \u062e\u0631\u0627\u0628\u06cc \u067e\u06cc\u062f\u0627 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0627\u0646 \u0646\u062a\u0627\u0626\u062c \u06a9\u06d2 \u0644\u06cc\u06d2 \u062c\u0648 \u0622\u067e \u0627\u0633\u0679\u06cc\u06a9 \u06c1\u0648\u0644\u0688\u0631\u0632 \u06a9\u0648 \u0631\u067e\u0648\u0631\u0679 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u06cc\u0627 \u0634\u0627\u0626\u0639 \u0634\u062f\u06c1 \u062a\u062c\u0632\u06cc\u0648\u06ba \u0645\u06cc\u06ba \u0634\u0627\u0645\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u06cc\u06ba: <code>linearmodels.IV2SLS<\/code>:<\/p>\n<pre><code class=\"language-python\"># Production-grade 2SLS with correct standard errors\n# pip install linearmodels\nfrom linearmodels.iv import IV2SLS\n\nexog_vars = [\"query_confidence\"] + tier_dummies\niv_model = IV2SLS.from_formula(\n    f\"task_completed_iv ~ 1 + {' + '.join(exog_vars)} \"\n    f\"[routed_to_premium_actual ~ rate_limit_fallback]\",\n    data=df\n).fit(cov_type=\"robust\")\n\nprint(iv_model.summary)\n<\/code><\/pre>\n<p>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2: <code>linearmodels<\/code> \u06cc\u06c1 \u062f\u0631\u0633\u062a 2SLS \u062a\u063a\u06cc\u0631\u0627\u062a \u06a9\u06cc \u06af\u0646\u062a\u06cc \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u062f\u0648 \u0645\u0631\u0627\u062d\u0644 \u06a9\u06d2 \u0688\u06be\u0627\u0646\u0686\u06d2 \u06a9\u0648 \u0628\u06cc\u0627\u0646 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u060c \u0645\u0646\u0627\u0633\u0628 \u067e\u06c1\u0644\u06d2 \u0645\u0631\u062d\u0644\u06d2 \u06a9\u06d2 \u062a\u0634\u062e\u06cc\u0635\u06cc \u062e\u0644\u0627\u0635\u06d2 \u0686\u0644\u0627\u062a\u0627 \u06c1\u06d2\u060c \u0627\u0648\u0631 \u0627\u06cc\u06a9 \u0631\u0633\u0645\u06cc Hausman endogeneity \u0679\u06cc\u0633\u0679 \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0646\u062d\u0648 \u0642\u0648\u0633\u06cc\u0646 \u0645\u06cc\u06ba endogenous \u0645\u062a\u063a\u06cc\u0631\u0627\u062a \u0627\u0648\u0631 \u0622\u0644\u0627\u062a \u06a9\u0648 \u0628\u0646\u062f \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 <code>[D ~ Z]<\/code>.<\/p>\n<p>\u0645\u06a9\u0645\u0644 \u0639\u0645\u0644 \u062f\u0631\u0622\u0645\u062f (\u0628\u0634\u0645\u0648\u0644 \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0648\u0642\u0641\u06d2 \u0627\u0648\u0631 \u0634\u06a9\u0644 2 \u0645\u06cc\u06ba \u062a\u0635\u0648\u0631\u0627\u062a) github.com\/RudrenduPaul\/product-experimentation-causal-inference-genai-llm\/tree\/main\/11_instrumental_variables\/ \u067e\u0631 \u0633\u0627\u062a\u06be\u06cc \u0646\u0648\u0679 \u0628\u06a9 \u0645\u06cc\u06ba \u06c1\u06d2\u06d4 \u0631\u06cc\u067e\u0648\u0632\u0679\u0631\u06cc \u06a9\u0648 \u06a9\u0644\u0648\u0646 \u06a9\u0631\u06cc\u06ba\u060c \u0627\u06cc\u06a9 \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u0628\u0646\u0627\u0626\u06cc\u06ba\u060c \u0627\u0648\u0631 \u0627\u0633\u06d2 \u0686\u0644\u0627\u0626\u06cc\u06ba\u06d4 <code>iv_demo.ipynb<\/code> \u0634\u0631\u0648\u0639 \u0633\u06d2 \u062e\u062a\u0645 \u062a\u06a9 \u06c1\u0631 \u06a9\u0648\u0688 \u0628\u0644\u0627\u06a9 \u06a9\u0648 \u062f\u0648\u0628\u0627\u0631\u06c1 \u067e\u06cc\u0634 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<p>IV \u062a\u06a9 \u06a9\u0628 \u067e\u06c1\u0646\u0686\u0646\u0627 \u0686\u0627\u06c1\u06cc\u06d2 \u0627\u0633 \u06a9\u06d2 \u0628\u0627\u0631\u06d2 \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u062d\u062a\u0645\u06cc \u0646\u0648\u0679: \u0627\u06af\u0631 \u0622\u067e \u06a9\u0627 \u0633\u0633\u0679\u0645 \u062c\u0628\u0631\u06cc \u0631\u0648\u0679\u0646\u06af \u0631\u06cc\u0646\u0688\u0645\u0627\u0626\u0632\u06cc\u0634\u0646 \u06a9\u0648 \u0633\u067e\u0648\u0631\u0679 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 (\u0628\u06d2 \u062a\u0631\u062a\u06cc\u0628 \u0637\u0648\u0631 \u067e\u0631 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u0627 \u0627\u06cc\u06a9 \u062d\u0635\u06c1 \u067e\u0631\u06cc\u0645\u06cc\u0645 \u06a9\u0648 \u062a\u0641\u0648\u06cc\u0636 \u06a9\u0631\u0646\u0627 \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0627\u0633\u06a9\u0648\u0631 \u0633\u06d2 \u0642\u0637\u0639 \u0646\u0638\u0631)\u060c \u0645\u0639\u06cc\u0627\u0631\u06cc A\/B \u0679\u06cc\u0633\u0679\u0646\u06af \u0622\u0633\u0627\u0646 \u06c1\u06d2 \u0627\u0648\u0631 \u0645\u062c\u0645\u0648\u0639\u06cc \u0637\u0648\u0631 \u067e\u0631 ATE \u062a\u062e\u0645\u06cc\u0646\u06c1 \u062a\u06cc\u0627\u0631 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>IV \u0627\u06cc\u06a9 \u0627\u0686\u06be\u0627 \u0679\u0648\u0644 \u06c1\u06d2 \u062c\u0628 \u0631\u06cc\u0646\u0688\u0645\u0627\u0626\u0632\u06cc\u0634\u0646 \u0645\u0645\u06a9\u0646 \u0646\u06c1 \u06c1\u0648\u060c \u06cc\u0639\u0646\u06cc \u062c\u0628 \u067e\u0631\u0648\u0688\u06a9\u0634\u0646 \u0645\u0646\u0637\u0642 \u0645\u06cc\u06ba \u0631\u0648\u0679\u0646\u06af \u06a9\u06d2 \u0627\u0635\u0648\u0644 \u0644\u0627\u06af\u0648 \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062c\u0628 \u0622\u067e \u062c\u0627\u0646 \u0628\u0648\u062c\u06be \u06a9\u0631 \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u0648 \u0633\u0628 \u0633\u06d2 \u0628\u06c1\u062a\u0631 \u0637\u0631\u06cc\u0642\u06d2 \u0633\u06d2 \u0631\u0648\u0679 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0645\u062a\u062d\u0645\u0644 \u0646\u06c1\u06cc\u06ba \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u06cc\u0627 \u062c\u0628 \u0622\u067e \u06a9\u0648 \u062a\u0627\u0631\u06cc\u062e\u06cc \u0645\u0634\u0627\u06c1\u062f\u0627\u062a\u06cc \u0688\u06cc\u0679\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0622\u067e \u0627\u06cc\u06a9 \u062d\u0642\u06cc\u0642\u06cc \u062a\u062c\u0631\u0628\u06c1 \u0686\u0644\u0627 \u0633\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062a\u0648 \u0627\u0633\u06d2 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<p>Perturbation \u0627\u06cc\u06a9 \u0628\u06c1\u062a\u0631 \u0631\u0648\u0679\u0646\u06af \u0633\u0633\u0679\u0645 \u06a9\u0627 \u0633\u0627\u062e\u062a\u06cc \u0688\u06cc\u0641\u0627\u0644\u0679 \u06c1\u06d2\u06d4 \u0645\u0639\u06cc\u0627\u0631\u06cc \u0631\u06cc\u06af\u0631\u06cc\u0634\u0646 \u0645\u0627\u0688\u0644 \u06a9\u06d2 \u0645\u0639\u06cc\u0627\u0631 \u0627\u0648\u0631 \u0627\u0646\u062f\u0631\u0648\u0646\u06cc \u0627\u0633\u062a\u0641\u0633\u0627\u0631 \u06a9\u06cc \u0645\u0634\u06a9\u0644 \u062f\u0648\u0646\u0648\u06ba \u06a9\u0648 \u0628\u06cc\u06a9 \u0648\u0642\u062a \u0645\u0627\u067e\u062a\u0627 \u06c1\u06d2 \u062c\u0628 \u0627\u0646\u06c1\u06cc\u06ba \u0627\u06cc\u06a9 \u0639\u062f\u062f \u0645\u06cc\u06ba \u0633\u0645\u0679 \u06a9\u0631 \u0627\u0644\u06af \u06a9\u06cc\u0627 \u062c\u0627\u0646\u0627 \u0686\u0627\u06c1\u06cc\u06d2\u06d4<\/p>\n<p>\u0634\u0631\u062d \u06a9\u0648 \u0645\u062d\u062f\u0648\u062f \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u0627 \u0641\u0627\u0644 \u0628\u06cc\u06a9 \u0631\u0648\u0679\u0646\u06af \u0633\u06af\u0646\u0644\u0632 \u0633\u06d2 \u0627\u0646\u0641\u0631\u0627\u0633\u0679\u0631\u06a9\u0686\u0631 \u06a9\u06d2 \u0634\u0648\u0631 \u06a9\u0648 \u0641\u0644\u0679\u0631 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u06cc\u06a9 \u0635\u0627\u0641 \u0627\u0648\u0631 \u0642\u062f\u0631\u062a\u06cc \u0679\u0648\u0644 \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0627\u0633 \u0646\u0642\u0637\u06c1 \u0646\u0638\u0631 \u0633\u06d2 \u0679\u06cc\u0645\u0648\u06ba \u06a9\u0648 \u0627\u0633 \u0628\u0627\u062a \u06a9\u0627 \u0642\u0627\u0628\u0644\u0650 \u062f\u0641\u0627\u0639 \u062a\u062e\u0645\u06cc\u0646\u06c1 \u062d\u0627\u0635\u0644 \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0627\u062c\u0627\u0632\u062a \u0645\u0644\u062a\u06cc \u06c1\u06d2 \u06a9\u06c1 \u0627\u0646 \u06a9\u0627 \u0645\u0627\u0688\u0644 \u0641\u0646 \u062a\u0639\u0645\u06cc\u0631 \u06a9\u0633 \u0637\u0631\u062d \u06a9\u0627\u0631\u0648\u0628\u0627\u0631\u06cc \u0642\u062f\u0631 \u067e\u06cc\u062f\u0627 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u0688\u06cc\u0679\u0627 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