Factors that affect 10yr and 2yr treasury yield
How do you determine the factors that affect whether or not the treasury yield goes up or down? What would make one go up proportionally more than the other one?
supply and demand
You should think about what influences supply and demand for something that pays a coupon, has basically no default risk, and has extremely high liquidity. Then, think about what would influence differences between different terms for those instruments.
In short, supply/demand, but I guess thats not super helpful.
As a really crude start, you can say the N-year treasury yield is roughly the 'average' expected risk-free interest rate over the next N years. This is the expectations hypothesis (https://en.wikipedia.org/wiki/Expectations_hypothesis).
In reality, this works really well on the short end (so they say the Fed controls the short end by controlling the Fed Funds rate), but as you go further out the curve (10y+) it gets more influenced by inflation (or nominal GDP), 'term premium' (as a catch-all phrase for all the stuff not easily explained by the expectations hypothesis), and idiosyncratic supply demand. Ex: Insurance companies and pensions with long-dated liabilities (they owe more PV in liabilities if rates go down) so they buy long-dated bonds (to make money to offset this if rates go down). Reason being, as you go further out the curve, even if you know with certainty what short-term interest rates will be over N years, you are taking on far too much duration/market risk to make that sort of trade pragmatic, so other factors tend to dominate.
There is also other idiosyncratic supply demand- ex: if people panic, they tend sell US stocks to buy long-dated US treasuries (because these have more risk than short-dated bonds).