MIT Microeconomics 14.32 Final Review

8/12/2019 MIT Microeconomics 14.32 Final Review

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e.g. to test H 0 : 2 = 3 = 0 in (1), RSS r is the residual sum of squares from the regression of y on just x1 . RSS ur is the residual sum of squaes from (1), q = 2, and k = 3.

An F-statistic testing only one restriction is just the square of the corresponding t-statistic!

Differences-in-differences: have some observations where a policy change took place, others where it didnot. Have data on an outcome of interest before and after change in both types of places. Estimate effectof policy as the difference in the difference in mean outcome before and after the change across placeswith and without the policy change. Key assumption is that only thing that caused systematic differencein changes across places is the policy. Regression version:

yit = 0 + 1Post t + 2Treatment i + 3(Post t Treatment i ) + it

Diff-in-diff estimate is 3 .

Extended version (state-level example):

yst = 0 + s + t + X st + DD (Policy st ) + st

s are state xed effects, t are year xed effects, X st allows for state-specic trends (i.e. demographic

changes), and the coefficient on Policyst provides our differences-in-differences-style estimate.

2 Asymptopia

S n converges in probability to c (plim S n = c) if Lim n P (|S n c| ) = 0 for all > 0

Under random sampling alone, we can show plim = (i.e. is a consistent estimator of )

is asymptotically normally distributed by the CLT: AV ( ) = E [(X i X )

2 2

i ]n ( 2x )

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3 Problems with Residuals and GLS

Heteroskedasticity: V ( |x) = E [ 2 |x] varies as a function of x. Usual OLS standard errors incorrect and OLS not efficient, but estimates still consistent. Use heteroskedasticity robust standard errors If know functional form of heteroskedasticity, can do FGLS.

e.g. suppose know that V ( |x) = 0 + 1x2 . Then: (1) estimate original model by OLS, (2)regress 2 = 0 + 1x2i to estimate , (3) form V ( |x) = 0 + 1x2i , (4) reweight observations byV ( |x) 1/ 2 , (5) Run OLS on re-weighted data (this is WLS, a special case of GLS).

FGLS is asymptotically efficient (has lowest asymptotic variance)

Serial correlation:

E [ t s ] = 0 for some s = t Usual OLS standard errors incorrect, but estimates still consistent (again, OLS is not BLUE)

Intuition: Correlated observations provide less information than independent observations be-cause they do not contain entirely fresh evidence on the relationship in question.

Use HAC (aka Newey-West) standard errors

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Do FGLS if willing to take stance of form of serial correlation:

e.g. for AR(1) where we assume the form of serial correlation is t = t 1 + ut and assumeu t is homoskedastic and serially uncorrelated. Then: (1) estimate model using OLS, (2) regress t = t 1 + ut , (3) quasi-difference data yt = yt yt 1 , x t = xt x t 1 , (4) Run OLS onquasi-differenced data.

FGLS is asymptotically efficient (has lowest asymptotic variance)

4 Instrumental Variables

Motivating example: omitted variable bias in OLS

Model: yi = xi + i , (and for simpler formulas, let x = y = 0) then

=x i yix2i

=x i (x i + i )

x2i

plim = + plim 1n x i i plim 1n x

2i

= + C ov(x, )

V (x)

Usual omitted variable bias formula; If i = a i + u i with E [x i u i ] = 0, then plim = + Cov (x, )V (x ) = + where is the coefficient from regressing on x.

Need to use IV when we want to estimate a causal effect. A good way to think about causal effects is toimagine the ideal randomized experiment that you would perform to answer your question. Then thinkabout the data you have. If you can tell stories about why OLS might not estimate the experimentaleffect, then you need an instrument.

Consistency of IV: need z such that E [z ] = 0 (exogeneity) and E [zx ] = 0 (relevance)

IV =s ZY/s 2Zs ZX /s 2Z

(reduced form estimates divided by rst stage estimates)

Given binary instrument z, we can construct an IV estimate as the ratio of differences in means (this isthe Wald estimator!)

2SLS: example model

y = + x + w +

where E [x ] = 0 but E [w ] = 0 and we have z as an instrument.

First stage: Regress x on z and w, form tted values, x Regress y on x and w. Estimated coefficient for x is 2SLS .

In bivariate model, y = + x + , with one instrument, 2SLS = (z i z )yi

(z i z )x i Examples: QOB instrument to estimate economic return to schooling, draft lottery, charterschool lotteries

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Overidentication:

More instruments than endogenous variables. e.g. xi scalar, and have z1i , z2i . Can test overidentifying restrictions

Ex./Run 2SLS and then regress resids on instruments to test whether they estimate the samecausal parameter (i.e. coefficients are not signicantly different from 0).

Can combine estimates that would get from just using one or the other instrument to get moreefficient estimate (this is what 2SLS (and 3SLS) does automatically)

5 Simultaneous Equations

When prices and quantities are determined by solving two (or more) equations simultaneously, OLSestimates are inconsistent for supply and demand elasticities; this is called simultaneous equations bias

Suppose we have jointly determined variables, q t and pt

q dt ( p) = 0 + 1 pt + 2zt +dt

q st ( p) = 0 + 1 pt + 2x t + st

Identication: when can we estimate and ?

Lets start by writing down reduced form equations for q t and pt (i.e. equations in which the regressorsare uncorrelated with the errors):

p = 0 0 1 1

2

1 1z +

2 1 1

x + s d

1 1= 10 + 11 z + 12 x + 1

q = 1 0 1 0 1 1 + 2 1 1 1 z

1 2 1 1 x + 1

s

1d

1 1= 20 + 21 z + 22 x + 2

As a sidenote, suppose we had run OLS regression of q t on pt :

1 = pt q t pt 2

= 1 + pt dt pt 2

Since ptdt

pt 2 = 0 based on our reduced form equation for pt , our estimate of 1 is biased.

However, we can run OLS on these reduced form equations and solve for the structural parameters (the

s and s). This is Indirect Least Squares! For example, 21 11 = 1 , 12 ( 1 1) = 2 , etc.

In the simultaneous equations framework, an equation is identied by the exclusion of exogenous variablesthat are included in other equations.

An equation is overidentied if there are more excluded exogenous variables than endogenous vari-ables, just-identied if there are as many excluded exogenous variables as endogenous variables, andunder-identied if there are fewer excluded exogenous variables than endogenous variables.

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Note the link between simultaneous equations and IV. Using ILS, note that 1 = 22 12 , but we can re-writethis ratio as the sample analog of C (q, x )C ( p, x ) , which is just the ratio of reduced form to rst stage!

If the system is overidentied, 3SLS, which is a version of GLS that takes into account the correlationbetween u1 and u2 is asymptotically more efficient than 2SLS

For just identied equations, ILS=2SLS=3SLS For over identied equations, 2SLS is a weighted combination of the ILS estimates constructed by

regressing pt on both potential instruments and then using the tted values as the single instrument

6 Regression Discontinuity

Basic model of sharp RD: Treatment status D i is a deterministic and discontinuous function of covariatex i (so D i = 1 if x i x0 and D i = 0 if x i < x 0)

E [Y 0i |xi ] = + x i

Y 1i = Y 0i +

and so we can estimate the causal effect of crossing the cutoff with the following regression:

Y i = + x i + D i + i

Fuzzy RD: Jump in the probability D i = 1 at the cutoff is less than one. To estimate the causal impact of D i on Y i , we can make use of the following ratio in a neighborhood around the cutoff:

E [Y i |x0 < x i < x o + ] E [Y i |x0 < x i < x o]E [D i |x0 < x i < x o + ] E [D i |x0 < x i < x o]

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MIT Microeconomics 14.32 Final Review

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