Therefore if Z is not correlated with we can use Z as an instrument to

# Therefore if z is not correlated with we can use z as

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Zis not correlated with";we can useZas an instrument to consistently estimate±(and it°s also important to note thatZdoesn°t enter the demand function directly by assumption). Therefore, we would like to assumeZand"are uncorrelated. Also, we are assuming thatXis not correlated with"either.7. Consider an industry where two °rms are producing di/erentiated products. Demands are given byq1=75°5p1+p2q2=90°2p2+p1:Assuming that production costs are zero, solve for a Bertrand Nash equilibrium (compute each °rm±soutput and pro°t).Firm 1 maximizesp1q1=p1(75°5p1+p2)The FOC is(75 +p2)°10p1= 0This gives ±rm 1°s reaction function:p1=75 +p210:Firm 2 maximizesp2q2=p2(90°2p2+p1)The FOC is(90 +p1)°4p2= 0Therefore, the reaction function of ±rm 2 isp2=90 +p14:Solving this system, we getp1=10p2=25andq1=50q2=50and1=5002=1250:2
8. Consider a modelyi=a+b1x1i+b2x2i+"iwhere(yi; x1i; x2i)are observed fori= 1; :::; Nand"iis not observed. You want to estimate(a; b1; b2):Whilex2is exogenous, you expectx1to be endogenous. Supposezifor1; :::; Nis available. State theconditions under whichzis a valid instrument forx1:Explain the procedure of the 2-step Least Squaresto consistently estimate(a; b1; b2):There are two conditions under whichzis a valid instrument forx1:First,zshould be correlated withx1:Second,zshould be uncorrelated with":The 2-step Least Squares work as follows.First, we runOLS, regressingx1on a constant,zandx2:Then, compute the ±tted value^x1i= ^c+^dzzi+^d2x2i:where (^c;^dz;^d2) are the OLS estimates. Second, regressyon a constant,^x1;andx2:This will give usconsistent estimates of(a; b1; b2):9. An economist wants to show that competition decreases prices. He collects data on prices and the numberof active °rms in the market of a homogenous product over a long time period. Then, he considers thefollowing econometric model:pt=°+±nt+"t;whereptis the price of the product att; n

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