Why Scale Is More Than a Notational Detail
An instrument might be recorded in dollars rather than cents, or a researcher might report one instrument in units that are one hundred times larger than another. Those changes alter the numerical entries in a data matrix, but they need not alter the information the instruments provide. In contrast, changing the units of an endogenous regressor changes the numerical size of its coefficient: an effect per cent is not numerically the same as an effect per dollar.
The distinction is important for interpreting and checking instrumental-variables calculations. A coefficient that changes merely because an instrument was multiplied by 100 is usually a warning about the estimator or its weighting. A coefficient that changes predictably when the regressor’s units change is expected.
Setup: 2SLS After Partialling Out Controls
Consider a structural equation with one endogenous regressor, \(x\), and outcome \(y\). Any exogenous controls, including an intercept if appropriate, can first be partialled out. In the equations below, \(y\) and \(x\) denote the resulting residualized vectors, and \(Z\) is the \(n \times q\) matrix of residualized instruments. This notation lets us focus on the scale question without changing how controls are handled.
The standard population conditions supporting an IV interpretation are that the structural equation is \(y=\beta x+u\), the instruments satisfy the exclusion and exogeneity restrictions represented by \(E[Z_i u_i]=0\), and they are relevant enough to identify \(\beta\). For one endogenous regressor, the population relevance condition is that the instrument moments have nonzero covariance with \(x\). These conditions matter for identification and consistency; they are not needed for the algebraic fact that two calculations on the same sample produce equal coefficients.
In an overidentified model, \(q\) can exceed one. 2SLS combines the instruments through their projection onto the data. Changing the instrument columns without changing their span leaves that projection unchanged.
Why Rescaling Instruments Leaves 2SLS Unchanged
Let \(A\) be any nonsingular \(q \times q\) matrix and replace \(Z\) by \(Z^*=ZA\). This includes multiplying each instrument by a nonzero constant, changing units, reordering instruments, or taking any invertible linear combination of them. Because \(A\) is nonsingular, \(Z\) and \(Z^*\) span the same column space.
The third line uses the inverse of a product of nonsingular matrices. Since the projection matrix is identical, substituting it into the 2SLS formula gives \(\hat{\beta}_{2SLS}^*=\hat{\beta}_{2SLS}\). The fitted values of the first-stage projection of \(x\) are identical too: \(P_{Z^*}x=P_Zx\). Thus a change of instrument scale does not change the IV coefficient or the projected variation used to calculate it.
Worked Example: Rescaling Two Instruments in an Overidentified Model
Suppose a hypothetical sample has four observations, one endogenous regressor, and two instruments. The variables below have already been residualized with respect to any controls. Assume, for illustration, that the population instrument conditions are appropriate; this small sample alone cannot establish instrument validity.
| Observation | \(z_1\) | \(z_2\) | \(x\) | \(y\) |
|---|---|---|---|---|
| 1 | 1 | 1 | 4 | 12 |
| 2 | -1 | 1 | -2 | -4 |
| 3 | 1 | -1 | 0 | -2 |
| 4 | -1 | -1 | -2 | -6 |
The two instrument columns are orthogonal and each has squared length 4. From the data, \(x=2z_1+z_2+w\) and \(y=5z_1+4z_2+3w\), where \(w=(1,-1,-1,1)'\) is orthogonal to both instruments. Therefore, their projections onto the instrument space are \(\hat{x}=P_Zx=2z_1+z_2\) and \(\hat{y}=P_Zy=5z_1+4z_2\). The coefficient is
Now record the first instrument in units 100 times as large and the second in units one tenth as large: \(z_1^*=100z_1\) and \(z_2^*=0.1z_2\). Both multipliers are nonzero, so the transformed instruments span exactly the same space. Their projection of \(x\) remains \(2z_1+z_2\), and the 2SLS coefficient remains \(2.8\). The arithmetic above and the instrument coordinates change, but the IV estimate does not.
Changing the Endogenous Regressor’s Units
Now multiply \(x\) by a nonzero constant \(c\), so that the recorded regressor is \(x^*=cx\). This changes the variable’s numerical scale, not the instrument space. Substitution into the 2SLS formula gives the new coefficient directly.
The coefficient’s units must change along with its value. If \(x^*=100x\), one unit of \(x^*\) represents one hundred units of the original regressor, so the coefficient per new unit is one hundredth as large. More generally, if the outcome is also rescaled to \(y^*=d y\), its coefficient becomes \((d/c)\hat{\beta}_{2SLS}\). These are unit conversions, not changes in the underlying relationship.
Worked Example: Converting a Coefficient From Dollars to Cents
Suppose the hypothetical regressor in the previous example is expenditure measured in dollars and the outcome is measured in outcome units. Its IV estimate is \(2.8\) outcome units per dollar. Record the same expenditures in cents instead, so that \(x_{\text{cents}}=100x_{\text{dollars}}\).
Using the scale rule with \(c=100\), the new coefficient is
For example, a 100-cent change corresponds to a one-dollar change, and \(100(0.028)=2.8\) outcome units. The numerical coefficient is smaller because the new regressor unit is smaller; the implied effect for the same one-dollar change is unchanged.
Instrument Scaling and GMM Weighting
A further qualification matters in overidentified instrumental-variables estimation. 2SLS uses the projection onto the instrument space, so any nonsingular change of instrument coordinates leaves its estimate unchanged. More generally, a GMM estimator minimizes a weighted quadratic form in the sample moment vector. Rescaling the instruments transforms that vector, and the weighting matrix must transform compatibly if the estimator is to remain the same.
For example, write the sample moments as \(g_n(\beta)=Z'(y-x\beta)/n\) and the GMM objective as \(g_n(\beta)'Wg_n(\beta)\), where \(W\) is a positive-definite weighting matrix. Under \(Z^*=ZA\), the moments become \(g_n^*(\beta)=A'g_n(\beta)\). The matching weighting matrix is \(W^*=A^{-1}W(A')^{-1}\). Then
The objective is identical at every value of \(\beta\), so its minimizer is unchanged. The 2SLS weighting induced by projection has this invariance. An arbitrary GMM weighting matrix held numerically fixed after changing instrument units need not.
Worked Example: Why a Fixed GMM Weight Can Break Invariance
Consider two hypothetical overidentifying moment functions, \(g(\beta)=(1-\beta,\ 4-2\beta)'\), and initially use the identity matrix \(W=I\). The objective is \((1-\beta)^2+(4-2\beta)^2\). Differentiating and setting the derivative to zero gives
Now rescale the first instrument by 10. The transformed moment vector is \(g^*(\beta)=(10(1-\beta),\ 4-2\beta)'\). To preserve the original objective, use \(W^*=\operatorname{diag}(0.01,1)\). The transformed objective is
so its minimizer remains \(1.8\). But if the identity weight is instead kept unchanged, the objective becomes \(100(1-\beta)^2+(4-2\beta)^2\). Its minimizer is
This difference is not a contradiction to 2SLS invariance: it comes from changing the relative weight of the moments while leaving the weighting matrix fixed. In a scale-invariant GMM comparison, the moment coordinates and their weighting must be considered together.
Assumptions, Interpretation, and a Common Pitfall
The invariance proofs are finite-sample algebra. They require the relevant inverses and identifying variation to exist, but they do not show that the instruments are valid. If an instrument is correlated with the structural error, rescaling it will not repair that problem; the resulting coefficient is still calculated from the same instrument space. Likewise, a coefficient’s numerical scale alone does not establish the substantive meaning of the relationship.
A common mistake is to say, “The instrument is now measured in hundreds, so the estimated IV effect must be one hundredth as large.” That reasoning confuses the instrument’s scale with the regressor’s scale. With 2SLS, multiplying an instrument by a nonzero constant leaves the projection and coefficient unchanged. Multiplying the endogenous regressor by 100 makes its coefficient one hundredth as large, with the units changing to effect per 100 original units.
Check Your Understanding
Use the formulas and distinctions developed above to answer these questions.
- If every column of a full-rank instrument matrix is multiplied by a nonzero constant, why does its 2SLS projection matrix stay the same?
- An endogenous regressor is converted from hours to minutes. If \(x_{\text{minutes}}=60x_{\text{hours}}\), how does the coefficient per minute relate to the coefficient per hour?
- Does an unchanged 2SLS estimate after rescaling an instrument establish that the instrument satisfies the exclusion restriction? Explain.
- In overidentified GMM, what must happen to the weighting matrix when the moment vector is transformed by \(A'\) for the objective to remain unchanged?
- Why might an overidentified GMM estimate change if an instrument is rescaled but the identity weighting matrix is kept fixed?