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Must I use all of my exogenous variables as instruments when estimating instrumental variables regression?

Title   Two-stage least-squares regression
Author Vince Wiggins, StataCorp
Date July 2000; updated June 2013

Note: This model could also be fit with sem, using maximum likelihood instead of a two-step method.
You can find examples for recursive models fit with sem in the “Structural models 2: Dependencies between endogenous variables” section of [SEM] intro 5 — Tour of models.

Someone posed the following question:

I am estimating an equation:
        Y = a + bX + cZ + dW 
I then want to instrument W with Q. I know the first-stage regression is supposed to be
        W = e + fX + gZ + hQ 
(i.e., use all the exogenous variables in the first stage). Actually this is automatically done if I use the ivregress command. However, I only want to use Q to instrument W without using X and Z in the first stage. Is there a way I can do it in Stata? I can regress W on Q and get the predicted W, and then use it in the second-stage regression. The standard errors will, however, be incorrect.

ivregress will not let you do this and, moreover, if you believe W to be endogenous because it is part of a system, then you must include X and Z as instruments, or you will get biased estimates for b, c, and d.

Consider the system

        Y1 = a0 + a1*Y2 + a2*X1 + a3*X2 + e1               (1)

        Y2 = b0 + b1*Y1 + b2*X3 + b3*X4 + e2               (2)

Warning: Assume we are estimating structural equation (1); if X1 and X2 are exogenous, then they must be kept as instruments or your estimates will be biased. In a general system, such exogenous variables must be used as instruments for any endogenous variables when the instrumented value for the endogenous variables appears in an equation in which the exogenous variable also appears.

Consider the reduced forms of your two equations:

        Y1 = e0 + e1*X1 + e2*X2 + e3*X3 + e4*x4 + u1        (1r)

        Y2 = f0 + f1*X1 + f2*X2 + f3*X3 + f4*x4 + u2        (2r)

where e# and f# are combinations of the a# and b# coefficients from (1) and (2) and u1 and u2 are linear combinations of e1 and e2.

All exogenous variables appear in each equation for an endogenous variable. This is the nature of simultaneous systems, so efficiency argues that all exogenous variables be included as instruments for each endogenous variable.

Here is the real problem. Take (1): the reduced-form equation for Y2, (2r), clearly shows that Y2 is correlated with X2 (by the coefficient f2). If we do not include X2 among the instruments for Y2, then we will have failed to account for the correlation of Y2 with X2 in its instrumented values. Since we did not account for this correlation, when we estimate (1) with the instrumented values for Y2, the coefficient a3 will be forced to account for this correlation. This approach will lead to biased estimates of both a1 and a3.

For a brief reference, see Baltagi (2002). See the whole discussion of 2SLS, particularly the paragraph after equation 11.40, on page 277. (I have no idea why this issue is not emphasized in more books.)

Failing to include X4 affects only efficiency and not bias.

However, there is one case where it is not necessary to include X1 and X2 as instruments for Y2. That is when the system is triangular such that Y2 does not depend on Y1, but you believe it is weakly endogenous because the disturbances are correlated between the equations. You are still consistent here to do what ivregress does and retain X1 and X2 as instruments. They are, however, no longer required. Then you could do what you suggested and just regress on the predicted instruments from the first stage.

If you do use this method of indirect least squares, you will have to perform the adjustment to the covariance matrix yourself. Consider the structural equation

        y1 = y2 + x1 + e 

where you have an instrument z1 and you do not think that y2 is a function of y1.

The following example uses only z1 as an instrument for y2. Let’s begin by creating a dataset (containing made-up data) on y1, y2, x1, and z1:

. sysuse auto (1978 Automobile Data) . rename price y1 . rename mpg y2 . rename displacement z1 . rename turn x1

Now we perform the first-stage regression and get predictions for the instrumented variable, which we must do for each endogenous right-hand-side variable.

. regress y2 z1
Source SS df MS Number of obs = 74
F( 1, 72) = 71.41
Model 1216.67534 1 1216.67534 Prob > F = 0.0000
Residual 1226.78412 72 17.0386683 R-squared = 0.4979
Adj R-squared = 0.4910
Total 2443.45946 73 33.4720474 Root MSE = 4.1278
y2 Coef. Std. Err. t P>|t| [95% Conf. Interval]
z1 -.0444536 .0052606 -8.45 0.000 -.0549405 -.0339668
_cons 30.06788 1.143462 26.30 0.000 27.78843 32.34733
. predict double y2hat (option xb assumed; fitted values) * perform IV regression . regress y1 y2hat x1
Source SS df MS Number of obs = 74
F( 2, 71) = 12.41
Model 164538571 2 82269285.5 Prob > F = 0.0000
Residual 470526825 71 6627138.38 R-squared = 0.2591
Adj R-squared = 0.2382
Total 635065396 73 8699525.97 Root MSE = 2574.3
y1 Coef. Std. Err. t P>|t| [95% Conf. Interval]
y2hat -463.4688 117.187 -3.95 0.000 -697.1329 -229.8046
x1 -126.4979 108.7468 -1.16 0.249 -343.3328 90.33697
_cons 21051.36 6451.837 3.26 0.002 8186.762 33915.96

Now we correct the variance–covariance by applying the correct mean squared error:

. rename y2hat y2hold . rename y2 y2hat . predict double res, residual . rename y2hat y2 /* put back real y2 */ . rename y2hold y2hat . replace res = res^2 (74 real changes made) . summarize res
Variable Obs Mean Std. Dev. Min Max
res 74 7553657 1.43e+07 117.4375 1.06e+08
. scalar realmse = r(mean)*r(N)/e(df_r) /* much ado about small sample */ . matrix bmatrix = e(b) . matrix Vmatrix = e(V) . matrix Vmatrix = e(V) * realmse / e(rmse)^2 . ereturn post bmatrix Vmatrix, noclear . ereturn display
Coef. Std. Err. t P>|t| [95% Conf. Interval]
y2hat -463.4688 127.7267 -3.63 0.001 -718.1485 -208.789
x1 -126.4979 118.5274 -1.07 0.289 -362.8348 109.8389
_cons 21051.36 7032.111 2.99 0.004 7029.73 35072.99

Reference

Baltagi, B. H. 2002.
Econometrics. New York: Springer.
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