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Considering that markets open in different time zones, in general the opening and closing times are different. I suppose that a stock market opening later (in the same day) receive information on the other markets opening earlier.

I think I should take into account this if I want to implement i.e. Granger causality analysis and I have daily stock indices prices. What could be a proper method to approach the problem?

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I'm skeptical that this will work. I'm concerned that for the sort of shock that is large enough to be interesting and large enough to study that stock prices will move before they open. You could potentially measure this by looking at the changes in stock price from close to open, but I'm not sure what you'd be measuring. Are changes in Tokyo stock prices in any meaningful sense causing the changes in NYSE price? Or is it simply that all stock prices are moving at once, but you can't easily see the NYSE stock prices until the exchange opens?

In support of this concern, consider that there are (thin but functioning) after hours trading markets in many securities. In addition, through the use of American Depositary Receipts, it is also possible to trade US stocks on foreign stock exchanges. As such, for many or even most securities, markets are almost always open for those willing to pay higher transaction costs.

That said, there is some evidence that when stock exchanges are closed that markets are meaningfully affected. See, for example, Stock returns and the weekend effect (French 1980). Stock Market Structure and Volatility (Stoll and Whaley (1990)) contribute to the litterature showing the expected returns and return volatility are different when stock exchanges are closed, including at night. While both important papers, they provide dated evidence. More recent evidence from Overnight stock returns and realized volatility (Ahoniemia and Lanneb (2013)) shows that shares, while stock markets are closed, continue to show different return volatility from when stock markets are open through 2007.

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