Title | Two-stage least-squares regression | |

Author | Vince Wiggins, StataCorp |

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: Dependencies between response
variables” section of [SEM] **intro 5 — Tour of models**.

Someone posed the following question:

I am estimating an equation:

Y = a + bX + cZ + dWI 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 (2011). See the whole discussion of 2SLS, particularly the paragraph after equation 11.40, on page 265. (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 | Coefficient 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 | |

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 | Coefficient 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 2.57e+14 1.36e+15 13791.56 1.12e+16 |

y1 | Coefficient Std. err. t P>|t| [95% conf. interval] | |

y2hat | -463.4688 745708.2 -0.00 1.000 -1487363 1486436 | |

x1 | -126.4979 691999.7 -0.00 1.000 -1379935 1379682 | |

_cons | 21051.36 4.11e+07 0.00 1.000 -8.18e+07 8.19e+07 | |

- Baltagi, B. H. 2011.
- Econometrics. New York: Springer.