# When is an instrument too strong in the first stage?

Instrumental variable regression is estimated with ivreg in R. ivreg provides diagnostics with the diagnostics option in summary. The first stage is similar to an OLS regression where the endogenous variable is regressed on all instruments and exogenous variables to check instrument relevance. The weak instrument test is often said to be higher than 10 or 20 to indicate good instruments. It is 17.254 in the example below.

My issue is that I often get weak instrument test statistics of about 100.000. I am looking for advice on causes and recommendations for "very high" weak instrument statistics.

library(ivreg)
data(mtcars)

summary(ivreg(mpg ~ vs + am | disp | cyl, data = mtcars), diagnostics = TRUE)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 29.19516    5.30464   5.504 7.01e-06 ***
disp        -0.04204    0.01506  -2.791  0.00936 **
vs           0.43059    2.61832   0.164  0.87056
am           0.99878    2.18173   0.458  0.65063

Diagnostic tests:
df1 df2 statistic  p-value
Weak instruments   1  28    17.254 0.000278 ***
Wu-Hausman         1  27     2.346 0.137213
Sargan             0  NA        NA       NA
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.312 on 28 degrees of freedom
Multiple R-Squared: 0.7272, Adjusted R-squared: 0.698
Wald test: 26.07 on 3 and 28 DF,  p-value: 2.948e-08


I found that the this F-test is calculated as the square of the coefficients t-statistic. Thus my question can be translated into what are potential causes of very high t test statistics in the first stage?

summary(lm(disp ~ cyl + vs + am, data=mtcars))

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   -55.34      97.95  -0.565 0.576616
cyl            50.85      12.24   4.154 0.000278 ***
vs            -20.55      37.52  -0.548 0.588146
am            -48.24      26.02  -1.854 0.074294 .

4.154^2 = 17.25572