I think US is the world's biggest economy, so many other countries economies depend to a significant extent on the health of the US economy.
The US imports a lot of goods from China and outsources a lot of jobs and services to India. With our economy in bad shape, it affects other countries who do business with us, and you can see it in their lowered economic growth.
Also, China, India and other countries invested in US assets like mortgage-backed securities, other financial instruments and even real estate. The value of these have lowered as our real estate and stock markets have plunged in value.
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