10 two stage least squares and demand estimation
om vi har Y = B0 + B1 college + u
vad är the first stage regression?
om instrument = z = dist
vad estimerar den?
côllêge = ø0 + ø1 dist
it estimates the best linear prediction som skrivs
(college)lin(dist) = côllêge
what does this tell us?
college = (college)lin(dist) + (college - (college)lin(dist) )
the right part of + tells us how far the best linear prediction is from the true value.
and can also be denited as W
how can we rewrite
college = (college)lin(dist) + (college - (college)lin(dist) )
college = ø0 + ø1dist + W
where côv(W,dist) = 0
what would be the second stage regression? and what is the exogeneity assumption?
regressing wage on côllêge
wâge = B0^ + B1^IV • côllêge
E[u|ø0 + ø1dist] = 0
what does the endogeneity problem bero på?
best linear prediction from previous question can be seen as expgenous, so it must be W that makes college an endogenous variable in the structural (original) equation.
what does 2SLS learn from?
it learns from exogenous variation and ignores endogenous variation
titta i häfte och second stage regression
.
what are the assumptions for second stage regression
exogeneity ols. E[u | (x1)lin(z)] = 0 which also means that
cov((x1)lin(z), ũ) = 0
OLS full rank:
var((x1)lin(z)) ≠ 0 och stoppa in bets linear prediction and solve for ø feta
end up with:
ø= cov(x,z) / var(z)
which means that cov(x,z) ≠ 0 which is the intrument relevance assumption
how does the two assumptions work to make sure that our second stage regression estimates the true causal effect?
relevance assumption is for the forst stage. it ensures that there is a behavioural change that generates variation that is both exogen and endogen.
the exogeneity assumption makes sure that the variation is only exogen.
which assumtion is testable anf not?
relevance is testable. we can set
H0: ø1 = 0 and test its significance in explaining the variable of interest. if we reject, the intrument relevance assumption holds.
exogen. assum. is not testable. we have to use expnomic arguemnt for why the instument cant predict u.
when do we have equilibrium?
when all agents are correctly accounting for all consequenses of their actions
how does endogeneity problem appear?
when the ols exogeneity assumptions fail because of economic equilibrium conditions
what is another name for population?
markets
explain:
Q(p) = B0 - B1•p + u
Q quantity demanded
p hypothetical price
B1 how demand react to price changes
u is the denand shocks
where is demand elasticity
if we put logs on output variable and regressor can we call B1 for demand elasticity
another name for hypothetical price and why
counterfactual price and the equation from previous question allows us to evaluate demand for prices that are nor factual (market) price.
what is RCT
random controlled trial
if we assaigb randomized price to each market, then p should be uninfomative about the demand shock, hence
E[u| p] = 0 which makes sure that we can use ols to estimate B1
however, RCT is impossible to implement since we cannot change prices like that.
we have to use market price P* that is determined in a given market and therefore is ENDOGENOUS to the market.
what is the new demand funtion, expgeneity adsumption and how do we get eqilibrium?
Q(P*) = B0 + B1•P* + u
E[u| P*] = 0
Q(P*) demand = Q(P*) supply
does our new E[u| P*] = 0 hold? why, why not? how can we see that?
it does not hold due to P* depending on the demand curve. we can see that by looking at an equilibrium graph. P* will change if there is a denand shock because of the demand curve shift.
= by looking at P* we can learn something about u. etc a higher price may indicate that there is a positive denand shock incressing the price
om vi sätter demand = supply och löser för P*. vi ser att P* beror på u.
sen sötter in funtionen för P* i Q* funtionen för att få Q*
is there any correlation between u snd P*?
yes since the exogeneity assumption does not hold. there is positive correlation since increased u also means increased P*
what can we say about a regression of Q* on P*?
it wont give the desired causal effect of price
varför uppstår the problem of endogeneity på markander?
demand shocks distinguish different markets
all markets are not the same, persistent heterogeneity between markets.
product differentiation, the orice cannot be the same everywhere due to different products
there may be monopoly power, imperfect competetion
consumers and producers make dynamic decision and do not act simultaneously as we have in our model