Does the signal predict the market? A chi-squared test of independence

Question

Over 200 days you record whether a sentiment signal was positive or negative and whether the next day's market was up or down: Up Down ---------------------- Positive 60 40 Negative 50 50 Test whether the signal and the next-day direction are independent. Give the statistic, the degrees of freedom, and the conclusion.

Accepted Answer

Expected counts under independence come from the margins: E = \frac{	ext{row total} 	imes 	ext{column total}}{	ext{grand total}}. Row totals are 100 and 100; column totals are 110 (Up) and 90 (Down); grand total 200. So every "Up" cell expects \frac{100 	imes 110}{200} = 55 and every "Down" cell expects \frac{100 	imes 90}{200} = 45. Statistic. \chi^2 = \sum \frac{(O - E)^2}{E} = \frac{(60-55)^2}{55} + \frac{(40-45)^2}{45} + \frac{(50-55)^2}{55} + \frac{(50-45)^2}{45}. Each deviation is 5, so this is \frac{25}{55} + \frac{25}{45} + \frac{25}{55} + \frac{25}{45} = 0.455 + 0.556 + 0.455 +…

Does the signal predict the market? A chi-squared test of independence

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