Seemingly Unrelated Regressions

Let us pretend you have a system of M equations, with N observations for each equation. For example, if we are estimating supply and demand independently over 20 years, M=2 and N=20.

If each of the regressors is predetermined in each equation and we have an exclusion restriction, we can use the Seemingly Unrelated Regressions (SUR) methodology to improve the efficiency of the estimates. SUR is simply computing the generalized least squares (GLS) estimate in the multivariate case. A more detailed explanation is given here.

An example is the following:

• PTS = α₀ + α₁EXP + α₂MIN + u
• REB = β₀ + β₁EXP + β₂MIN + β₃HT + v

Let us assume each basketball player’s points are a function of only their years of experience (EXP), the number of minutes they play per game (MIN) and a constant. The number of rebounds they get per game is also a function of a constant, EXP, and MIN but the person’s height (HT) also affects their rebounding totals. This system of equations would be the same as:

• PTS = α₀ + α₁EXP + α₂MIN + α₃HT + u
• REB = β₀ + β₁EXP + β₂MIN + β₃HT + v

where α₃ was constrained to be 0. The fact that α₃ is constrained to be 0 is our exclusion restriction. SUR uses a typically instrumental variables approach but our vector of instruments, z, is equal to the union of the regressors from all equations.

• z=union of (x₁,…x_M)

In this example, M=2 so: x₁=(1, EXP, MIN)’; x₂=(1, EXP, MIN, HT)’; z=(1, EXP, MIN, HT)’. Our orthogonality conditions are that E(zu)=0 and E(zv)=0. Our parameter estimates become:

• δ= …[σ¹¹A¹¹ , σ¹²A¹² ]^-1[σ¹¹c¹¹ , σ¹²c¹² ]
• ……..[σ²¹A²¹ , σ²²A²²]….[σ²¹c²¹ , σ²²c²²]

A^mh=n^-1Σ_i x_imx_ih.

c^mh=n^-1Σ_i x_imy_ih.

OLS can also be used because the regressors are predetermined. In fact, if each equation is just identified, SUR is mathematically equivalent to OLS. If at least one equation is overidentified—which would be the case in the first (PTS) equation in our example—then SUR is more efficient than equation-by-equation OLS.

For more information of Seemingly Unrelated Regressions, see Hayashi (2000) Econometrics, pp. 279-283.