Partial R-Squared as a Strength Measure

Why Measure Instrument Strength?

In an instrumental-variables model, excluded instruments must explain variation in the endogenous regressor beyond the variation explained by included controls. The partial R-squared summarizes how much of that remaining sample variation the excluded instruments explain. It complements the first-stage F statistic: the two are algebraically linked, but they emphasize different features of the sample design.

A related concept, the concentration parameter, describes the first-stage signal relative to the disturbance variance. It is a model-based quantity, not simply another name for the sample partial R-squared. Keeping these distinctions clear helps prevent a descriptive sample statistic from being mistaken for a complete diagnosis of weak identification.

Definition After Controlling for Included Regressors

Consider one endogenous regressor $x$, an $n\times q$ matrix $Z$ of excluded instruments, and an $n\times k$ matrix $C$ of included regressors. Assume $C$ has full column rank and that the columns of $M_C Z$ have rank $q$, where $M_C$ removes the linear projection on the columns of $C$. Write $P_{M_C Z}$ for the orthogonal projection onto the column space of $M_C Z$. The sample first stage regresses $x$ on $C$ and $Z$.

Definition: The partial R-squared for the excluded instruments is the fraction of the variation in $x$ left after controlling for $C$ that is explained by adding $Z$: $$ R^2_{\mathrm{partial}} = \frac{x'M_C P_{M_C Z}M_Cx}{x'M_Cx} = \frac{x'M_Cx-x'M_{[C,Z]}x}{x'M_Cx}. $$ Here $M_C=I-P_C$, and $M_{[C,Z]}$ removes the projection on the combined columns of $C$ and $Z$. The definition requires $x'M_Cx>0$.

The numerator is the reduction in residual sum of squares when the excluded instruments are added to the first stage. The denominator is the restricted residual sum of squares, from regressing $x$ on the included regressors alone. Thus $R^2_{\mathrm{partial}}$ lies between 0 and 1. A value of zero means the excluded instruments add no fitted variation in this sample; a value near one means they account for nearly all the variation remaining after the controls.

Worked Example: Computing Partial R-Squared From Sums of Squares

Suppose the first-stage regression has restricted residual sum of squares 60 and unrestricted residual sum of squares 42. The excluded instruments reduce the residual sum of squares by $60-42=18$. Therefore,

R^2_{\mathrm{partial}}=\frac{60-42}{60} =\frac{18}{60}=0.30.

The excluded instruments explain 30% of the sample variation in $x$ that remains after the included regressors have been accounted for. They do not necessarily explain 30% of the total, unadjusted variation in $x$; the denominator is specifically the variation left after controlling for $C$.

The Exact Relationship to the First-Stage F Statistic

Let $\nu=n-k-q$ be the residual degrees of freedom in the unrestricted first stage. With full column rank for $[C,Z]$, the conventional homoskedastic first-stage F statistic for the joint null that all excluded-instrument coefficients are zero is

F = \frac{(RSS_R-RSS_U)/q}{RSS_U/\nu},

where $RSS_R=x'M_Cx$ and $RSS_U=x'M_{[C,Z]}x$. This F statistic is commonly used to summarize the joint contribution of the excluded instruments. The following relationship is an algebraic identity for the ordinary least-squares first stage; it does not require normality.

Theorem: If $RSS_R>0$, $RSS_U>0$, and the unrestricted first-stage design has rank $k+q$, then $$ F = \frac{\nu}{q} \frac{R^2_{\mathrm{partial}}}{1-R^2_{\mathrm{partial}}}, \qquad R^2_{\mathrm{partial}} = \frac{qF}{qF+\nu}. $$

Proof. From the definition, $R^2_{\mathrm{partial}}=(RSS_R-RSS_U)/RSS_R$. Subtracting this quantity from one gives $1-R^2_{\mathrm{partial}}=RSS_U/RSS_R$. Consequently,

\frac{R^2_{\mathrm{partial}}}{1-R^2_{\mathrm{partial}}} = \frac{(RSS_R-RSS_U)/RSS_R}{RSS_U/RSS_R} = \frac{RSS_R-RSS_U}{RSS_U}.

Multiplying the last ratio by $\nu/q$ gives exactly the stated formula for $F$. Solving that formula for $R^2_{\mathrm{partial}}$ gives the inverse relationship. $\square$

The identity shows why a first-stage F statistic cannot be interpreted without its degrees of freedom: the same partial R-squared can produce different F values when $q$ or $\nu$ changes. Conversely, a reported F statistic and the relevant degrees of freedom determine the partial R-squared exactly.

Worked Example: Converting Partial R-Squared to F

Suppose $n=120$, there are $k=4$ included regressors and $q=3$ excluded instruments, and the partial R-squared is $0.12$. The unrestricted residual degrees of freedom are

\nu=120-4-3=113.

The first-stage F statistic is therefore

F=\frac{113}{3}\frac{0.12}{1-0.12} =\frac{113}{3}\frac{0.12}{0.88} =\frac{113}{3}\frac{3}{22} =\frac{113}{22} \approx 5.14.

The calculation uses the partial, not the unadjusted, R-squared. It also uses three restrictions in the numerator and 113 residual degrees of freedom in the denominator.

Worked Example: Recovering Partial R-Squared From F

Suppose a first stage has $n=200$, $k=5$, and $q=2$, and reports $F=10$. Then $\nu=200-5-2=193$. The inverse formula gives

R^2_{\mathrm{partial}} = \frac{qF}{qF+\nu} = \frac{2(10)}{2(10)+193} = \frac{20}{213} \approx 0.0939.

Thus the excluded instruments explain about 9.39% of the variation in the endogenous regressor remaining after the five included regressors are controlled for. This is not 9.39% of the total variation in the regressor.

Connection to the Concentration Parameter

For a precise connection, consider the scalar first-stage model with fixed regressors, $x=C\gamma+Z\pi+v$, where $v\sim N(0,\sigma_v^2 I_n)$, $\sigma_v^2>0$, and the design satisfies the rank conditions above. Here $\pi$ is the vector of excluded-instrument coefficients. The concentration parameter for this scalar first stage is

Definition: The concentration parameter is $$ \lambda = \frac{\pi'Z'M_CZ\pi}{\sigma_v^2}. $$ It is the squared length of the conditional first-stage signal, measured in units of the disturbance variance.

Under these fixed-design Gaussian assumptions, the conventional first-stage F statistic has a noncentral $F$ distribution under the alternative, with numerator degrees of freedom $q$, denominator degrees of freedom $\nu$, and noncentrality parameter $\lambda$. Under the null $\pi=0$, $\lambda=0$ and the distribution is central $F$.

To see why, let $W=M_CZ$. The unrestricted residualized outcome is $M_Cx=W\pi+M_Cv$. Its sum of squares explained by the excluded instruments is $(M_Cx)'P_W(M_Cx)$. Since $P_WW\pi=W\pi$, this is the squared length of a $q$-dimensional Gaussian projection whose mean is $W\pi$. Dividing that sum of squares by $\sigma_v^2$ gives a noncentral chi-squared variable with $q$ degrees of freedom and noncentrality parameter

\frac{(W\pi)'(W\pi)}{\sigma_v^2} = \frac{\pi'Z'M_CZ\pi}{\sigma_v^2} = \lambda.

The unrestricted residual sum of squares is the squared length of the projection of $v$ onto the orthogonal complement of $[C,Z]$. Under the Gaussian model, this sum divided by $\sigma_v^2$ is a central chi-squared variable with $\nu$ degrees of freedom. The numerator and denominator use orthogonal projections, so they are independent. Their scaled ratio is the noncentral $F_{q,\nu}(\lambda)$ distribution, as claimed.

There is a useful way to express the connection with a population-style partial R-squared. Define the design signal per observation as $s_n=\pi'Z'M_CZ\pi/n$, and define its signal fraction relative to this signal plus the disturbance variance as $\rho_n^2=s_n/(s_n+\sigma_v^2)$. These definitions imply

\lambda=\frac{n s_n}{\sigma_v^2} = \frac{n\rho_n^2}{1-\rho_n^2}.

This identity relates a normalized design signal fraction to the concentration parameter. It does not say that the observed sample $R^2_{\mathrm{partial}}$ equals $\rho_n^2$, or that the observed F statistic equals $\lambda$. The sample partial R-squared depends on realized first-stage disturbances, whereas $\lambda$ is determined by the design, first-stage coefficients, and disturbance variance.

Worked Example: Interpreting a Concentration Parameter

Suppose a fixed-design first stage has $n=500$ and concentration parameter $\lambda=25$. The normalized design signal fraction just defined is

\rho_n^2=\frac{\lambda}{n+\lambda} =\frac{25}{500+25} =\frac{25}{525} \approx 0.0476.

Under the stated normalization, the design signal is about 4.76% of the signal-plus-disturbance quantity. The parameter $\lambda=25$ is also the noncentrality parameter of the conventional first-stage F distribution under the fixed-design Gaussian model. It is not an observed F value: the observed F also varies with the realized disturbance and the denominator degrees of freedom.

What These Measures Can and Cannot Tell You

Partial R-squared is a scale-free description of sample relevance conditional on the included regressors. The F statistic expresses the same fit improvement relative to its degrees of freedom. The concentration parameter describes signal strength relative to noise in the fixed-design model. All three concern the first-stage relationship; none by itself establishes that an instrument is exogenous or satisfies the exclusion restriction.

A common mistake is to treat a low partial R-squared as automatically equivalent to a particular F statistic, or to compare partial R-squared values without considering the number of excluded instruments and sample degrees of freedom. The correction is to report the partial R-squared together with the specification and to use the degrees-of-freedom identity when translating to the conventional F statistic. In addition, the exact noncentral-F interpretation of the concentration parameter requires the fixed-design, homoskedastic Gaussian assumptions stated above. Robust first-stage diagnostics require separate treatment.

Key takeaway: Partial R-squared measures the share of residualized first-stage variation explained by excluded instruments. Its exact algebraic link to the conventional F statistic is $F=(\nu/q)R^2_{\mathrm{partial}}/(1-R^2_{\mathrm{partial}})$. The concentration parameter is a model-based signal-to-noise quantity, not the realized sample partial R-squared or F statistic.

Check Your Understanding

Use the definitions and relationships above to answer these questions.

In a first stage with $n=150$, $k=3$, and $q=2$, what residual degrees of freedom enter the F-to-partial-R-squared conversion?
If $R^2_{\mathrm{partial}}=0.20$, $q=4$, and $\nu=96$, calculate the conventional first-stage F statistic.
Explain why partial R-squared is not generally the share of the endogenous regressor’s total variation explained by the excluded instruments.
Under the fixed-design Gaussian first-stage model, what role does the concentration parameter play in the distribution of the first-stage F statistic?
Does a large partial R-squared establish that the excluded instruments satisfy the exclusion restriction? Explain.