Causality in Economics
Topics on Instrumental Variable Regression Techniques
Alex Klein
April 2016
Motivation
► Causality - a crucial issue in economics (maybe more than in other social sciences)
► Non-experimental nature of data as opposed to experiments such as laboratory experiments or randomized controlled trials
► Estimation techniques developed over the past 70 or so years to estimate a causal effect of variables on the outcome of interest
► Development of 'instrumental variable estimation techniques' is an attempt to account for causality in non-experimental data
Basic set-up I
► Consider a basic regression y — x,-jß- + u, i = 1, K
► Key condition of consistency of OLS estimator is that the error term is uncorrelated with each of the regressors: cov (x/, u) — 0, / = 1,K
► Sufficient condition for cov (x,-,    = 0 is £ (ü|x,-) = 0
► An explanatory variable is endogenous if it is correlated with the error term which is caused by
1. omitted variables
2. measurement error
3. simultaneity
-«[f^^ <       >•      -E      0 0,0
Basic set-up I
► p0LS = (x'xy1 x'y = (x'xy1 x'x$ + (x'xy1 x'u = p +(x'xy1 x'u
► = £ + (N^X'Xy1 N^X'u - renormalization to allow the use of large numbers to be applied to X'X
► plim/3^ = jS+ (p\\m N^X'Xy1 (plim A/^X'u) (Slutsky's theorem)
OLS is consistent if plim A/_1X/u = 0
► a necessary condition for the above equality to hold is that
E [X'u] = 0
Instrumental Variable Regression I
► To obtain consistent estimates of /3 when cov (x,-, u) 7^ 0, we need to find a variable - call it z\ - which satisfies two conditions:
1. Instrument relevance: cov(z;,Xj) 7^ 0
2. Instrument exogeneity: cov (z,-, u) — 0
► failure of the first condition leads to weak instrumental variable problem, but we can deal with it (somehow)
► failure of the second condition is fatal and we can't interpret the estimated relationship as causal (only as a sophisticated correlation)
Instrumental Variable Regression I
► we will deal with a single equation model
► number of instruments can be the same as the number of endogenous variables (Just-identified model) or larger (overidentified model)
► just-identified model:
= (z'xy1 zfy = p + (z'xy1 z'u = (N^z'xy1 N~lZ
► consistency of IV estimator requires plim N~lZru — 0 and plim N~lZ'X ^ 0
► variance of J8^: V(f^) = (Zf X)~l Zf CiZ (Zf X)~l where O = Diag(uj2)
► though consistent, IV estimators exhibit efficiency loss
Instrumental Variable Regression I
► over-identified model requires Two-Stage Least Square estimator (TSLS/2SLS)
^ = [X>Z(Z'Z)-iZ>X]-i[X>Z(Z'Z)-iZ>y]
► in just-identified model 2SLS=IV
► Stage 1: obtain predicted values of X from a regression of X on Z: X = Z(Z'Z)-lZ'X
► Stage 2: run OLS with predicted values X
► again, 2SLS causes efficiency loss relative to OLS, but, it is effecienct estimator in the class of all instrumental variable estimators using instrument linear in z
Instrumental Variable Regression I
► Even though 2SLS is a consistent estimator when instruments satisfy the conditions of relevance and exogeneity, it is biased in finite samples
► In fact, we must rely on large sample analysis to derive the properties of 2SLS (mean of just-identified 2SLS does not even exist)
► When instruments are weak, 2SLS is biased even in very large sample
► Consider the 'degree of inconsistency' - there is some, though very mild, correlation between instruments are error terms
► When instruments are weak, the degree of inconsistency increases
Instrumental Variable Regression I
► consider a simple model with one endogenous variable:
Yu = Oil + j6x Y2j + €j and Y2\ = oc2 + j62Z/ + ]i]
► assume that Var(e,-)=1 and Var(^/)=1 =>cov(e/, Jij)=p where p is the correlation coefficient
► if we assume that Z/ is exogenous, then p measures the degree to which y2\ is correlated with €;
► Hahn and Hausman (2005) showed that in this simple case, the finite sample bias of 2SLS in overidentified case is, to a second-degree approximation
► / is the number of instruments, n is sample size, R2 is R2 from the regression of Z/ on Y2/ and measures the strength of instruments
Instrumental Variable Regression I
► the bias of 2SLS in finite samples is toward inconsistent OLS
► a fundamental question arises: if a consistent 2SLS estimator is biased in finite samples toward inconsistent OLS, is 2SLS bias smaller or larger then that of OLS?
► Hahn and Hausman (2005) offer the following equation
Bi*s(fiSLS) _ Bias(p°LS)  ~ nR2
► as long as the denominator is larger than the nominator, 2SLS bias is smaller than OLS bias
► ceteris paribus, the bias of 2SLS grows with the number of instruments
► weak instruments (low R2) increase the bias of 2SLS toward inconsistent OLS!!!
Instrumental Variable Regression I
weak instruments and 'mild inconsistency'
cov(Z,u) _ au
corr(Z ,u) corr(Z,X)
► relative inconsistency of 2SLS
p\\mß2SLS— ß_corr(X,u) 1
plimißOlS-iß — corr(X,u) R2p
► if instruments are weak and moderately correlated with error term (mildly endogenous), instrumental variable estimator is even more inconsistent than OLS
Instrumental Variable Regression I
► unless we have a perfect natural experiment of a perfectly exogenous instrument, weak instrument is more fatal than running a simple OLS even when a correlation between instrument and error term is very small
► this result is due to Bound, Jaeger and Baker (1995) and h not received much attention in the literature
► literature on weak instruments assumes that instruments satisfy exogeneity assumption and the only problem is their weak correlation with endogenous variables
Weak Instruments
► how to detect it:
1. Shea's partial R2 from the first stage regression
2. F-statistics from the first stage regression
► logic of R2 from the first stage regressions: consider y = j61xi + j62x2 + u where xi is endogenous and x2 exogenous, and let z be a vector of instruments (includes x2)
► we need a measure of the correlation between z and xi which purges out x2
► R2measure adjusted for the presence of x2 proposed by Bound, Jaeger, and Baker (1995)
► R2measure adjusted for the presence of x2 and another endogenous variables proposed Shea (1997)
Weak Instruments
► F-statistics from the first-stage regression; the test statistics are not drawn from the standard F-distribution
► Stock and Yogo (2005) offer critical values which depend on the number of instruments and endogenous variables
► Null hypothesis: the bias in 2SLS is less than some percentage of the bias of OLS
► for example, for one endogenous variable and three instruments, and HO stating the bias being less than 10%, the critical value of F-statistic is 9.08
Weak Instruments - Solution(s) I
► alternative estimators to 2SLS which exhibit better properties in the presence of weak instruments
► test statistics which are robust to weak-instrument problem
An Example - Housing Expenditures I
► the model allows for household fixed effects
dit = 1 (n'xit + rj; - uit > 0) yon = jSjx/t + ocoi + eoit if dit = 0 ynt = jSix/t + OCi; + £i/t if dit = 1
► the selection variable dlt is a choice between owning a property (djt = 1) and renting a property (djt = 0)
► Xjt is a vector of explanatory variables (total expenditures, square of total expenditures, prices, household characteristics)
* ynt and yon are budget shares spent on housing for renters and owners respectively
► ocqi, ocu, t]i are unobservable household specific time-invariant effects
An Example - Housing Expenditures I
► xj is decomposed into xa/- (log of total expend, square of total expend),      (log of hh income, square of hh income), x^; (prices, hh characteristics), xc/ are exclusion restrictions
► selection equation includes x^\ and x^/, the budget equation
xai and xci
► taking the difference between period t and r yields:
Ypit ~ Ypir = jSpa (xa/t - Xair) + $pc (xdt - Xcjr) + (epjt ~ Spir) if
dn = dir = p, p=0,l
d/s = 1 (tt^x^ + zr^xj/t + rji - ult > 0) , s= t, r
An Example - Housing Expenditures I
► we can rewrite the above equation as
Ypit ~ Ypir — ftpa (xait ~ xair) + fi'pc (xcit ~ xcir) + SptT (xbit> xbir> xditi xdir) + ZpitT
► the function gptr, p — 0,1 is given by
SptT (xbit> xbiTi xdit, xdir) — E (£pit ~ £pir xbit> xbiT> xdit> xdir> djt — djs = p)
► and Spitz satisfies
E (£pitr\xbit, xbir, xdit, Xdir, dit = di5 = p) = 0, p = 0, 1
=
An Example - Housing Expenditures I
► we can assume no sample selection after differencing => gptr — 0, p = 0,1 which is equivalent to saying that t] ■ — ult is independent of eo/t and      f°r aH t
► in other words, possible selection effect on budget shares operate only through correlation between oc\ and {rj- Ujt)