9 intrument variable regression (IV)
what is IV regression?
it is an alternative way of estimating causal effects that can be used if the regressors are endogenous (when exog. assun. is not satisfied)
when to use IV?
omiyted importabt variabelse
measurement error when we have to use a proxy
is equilibrium conditions that determine jointly regressor and our outcome variable
our variable of interest is a dummy variable (binary), how do we calculate B1^ where x1 is our dummy
B1^ = côv(Y,d1) / vâr(d1)
=>
E[Y| x1 = 1] - E[Y| x1 = 0]
/
E[x1| x1 = 1] - E[x1| x1 = 1]
this is the average outcome difference between two groups dicided by an input difference between two groups
to get the treatment effect, B1^
suppose that we now have two groups, where in each group we have some observations x1=1 and some =0. so there are mixed dummys within each group, what is the fomeula for the effect estimator? B1^IV
B1^IV = Ê[Y| red] - Ê[Y| blue]
/
Ê[x1| red] - Ê[x1| blue]
so this is still the output difference divided by the input difference, for each of the two groups
(this is the example where the input difference is smaller in between the values 0-1 on the x-axis) there are blue and red observations for when x1 is either 1 or 0
what is the structural equation?
the regular
Y = B0 + B1X1 + u
we assume that we can sample fron population (Y,x,z), what is z? how do we calculate B1^IV?
the instrument variable which in this case is a dummy
we caclulate in the same way as before with outcome dicideed by input different får grouo red and blue
what assumptions are needed for IV?
1) E[u|z] the instrument cannot be used to predict u. called intrument exogeneity assumption
2) intrument relevance assumtion
E[x1| z=1] ≠ E[x1| z=0]
om dom skulle vara lika can vi inte drive an input difference och dörmed inte heller få ett värde för B1^IV för att vi divide med 0
ge ett exempel på x1 och z
x1 är more kids
som i sin tur drivs av z som anger om barnen har samma kön eller inte. om de inte har samma kön, ör det mer sannolikt stt dom förösker få fler barn.
samesex is random (z) så vi can kot predict u with z. intrunent exo. assum. is fulfilled
if z = 1, we are more likely to have x1=1
if z=0 så är vi more likely att ha x1=0
this satisfies the intrument relevance assumption since we would have different averages of x1 depending on which value z has.
so by other words:
if 1) and 2) assunptions are satisfied, we can sort observations into 2 groups. if we compare the twi groups we can reveal the ceteris paribus causal effect
vad kan man säga istöllet för
E[x1| z=1] ≠ E[x1| z=0]
cov(x1,z) ≠ 0 vilket betyder att de ha correlation
what is IV regression with a continous instrument?
we develop a general IV estimator that does nor require a binary instrument (dummy)
wage = B0 + B1 college + u
u innehåller unobservable ability which both affect the wage at work and college decision.
=> endogenous problem since exog. assuntiob does not work and the structural equation does not satisfie for ols.
solution?
create a new variable dist that is a continous intrument, that gives the distance in km to the nearest college.
it affect behavioural changes and impacts the choice of college. intriment relevancs ok sonce we now have created dist, which will give digferent averages of x1 for different values if z, so E[x1| z=1] ≠ E[x1| z=0] holds.
intrument exogen. assum. also holds since we cant use z to predicg u. the value of z changes the expected value of college (endogenous variable). we cannot see how to use dist to learn about ability and therefore not u.
there is cov(z,u) = 0
so E[u|z] = 0
jur kan vi förtydliga felevance och expgeneity assumptions or instruments with covariances?
cov(u,z) = 0 exogeneity
cov(x1,z) ≠ 0 relevance
all we need is that the insturment correlates with the variable of interest for the instument IV calculation to work
hur kommer man fram till fomeln för B1^ när det är continous instrument?
1. lös ut u ur the structural equation
2. plug u into cov(z,u) = 0 (exog.ass)
3. bryt ut så mycket det går och förenkla
4. solve for B1. we need to stgue for the relevance assumption here.
5. apply method ofmoments vilket är ^(hattar)
B1^ = côv(z,Y) / côv(z,x1)
hur kan man annrs stölla upp B1^IV
=
slope coeficient when regressing Y on z1 (reduced form)
/
slope coefficient when regressing x1 on z1 (first stage)
(so it is the outcome diff divided by the input diff)
delta^ / feta^ (0^)
kolla i blocket vid dubbelvikt sidning.
gäller en liknande representation fast för the ols estimstor.
om z är ett väldigt bra intrument så att x=z, är nämnaren (feta^, 0^) väldigt nära 1. så B1^IV ≈ B1^OLS
.