6.2 Instrument Properties
This sub-section considers the econometric merits of each instrument. For the case of a single instrumented variable, Δlog (1 - τ), and instrument, over-identification tests are not feasible. However it is possible to examine the relative strength of each separate instrument in the first stage of the regressions, using parameter t-values, regression R2s and partial-R2s associated with each case.[18] Further, adding two, or all three, instruments to the first stage regression allows their relative contributions to be assessed and compared using an F-test for the validity of the instruments.
For the case of a single instrument the partial correlation among Δlog (1 - τ) and the three instrumenting variables is also of interest. The partial correlation between Δlog (1 - τ) and the expected income, or expected tax rate, instruments are much higher, at around 0.15, compared to that with the standard instrument, at 0.06. Also, while the standard instrument is correlated at +0.50 with the expected income instrument, the expected tax rate instrument is not highly positively correlated with the other two (-0.51, -0.05). The -0.05 correlation suggests that, using both the expected income and tax rate instruments, each instrument should contribute independent information.
| Dependent variable: Δlog (1 - τ) | ||||
|---|---|---|---|---|
| Instrument included | Parameter t-values |
Regression Adj-R2 |
Partial R2 |
F-test (p-value) of over-identification.a |
| (1): 'Standard' | -0.1 | 0.023 | 0.0000 | n.a. |
| (2): Expected income | 25.7 | 0.039 | 0.0168 | n.a. |
| (3): Expected tax rate | 62.05 | 0.111 | 0.0905 | n.a. |
| (1), (2), (3) | -1.43, 4.69, 55.4 | 0.112 | 0.0910 | 4.00 (0.018) |
| (1), (2) | -9.26, 27.34 | 0.041 | 0.0189 | 6.53 (0.011) |
| (1), (3) | 0.28, 62.05 | 0.111 | 0.0905 | 6.14 (0.013) |
| (2), (3) | 4.47, 56.20 | 0.112 | 0.0909 | 0.14 (0.710) |
Note a: Critical values are: 5.99 (95%) and 9.21 (99%) with 2 df, and 3.84 (95%) and 6.63 (99%) with 1 df.
Table 5 shows a set of additional diagnostics for the various instruments (all first stage regressions also include the exogenous variables Age, Age2, log y99, log y99 - log y98, and the 'other income' dummy). The top half of the table shows that the standard instrument performs poorly (for example, t = -0.1) while the other two instruments appear statistically strong (t = 25.7 and 62.05). In addition when 2 or 3 instruments are included in the vector of instruments and the standard instrument is included (instrument '(1)' above), the instrument vector fails the over-identification F-test. That is, at least one instrument is invalid. However, for the combination of instruments (2) and (3), the F-test fails to reject the null hypothesis of instrument validity, further supporting the inclusion of either or both of the new instruments. The t-ratios in Table 5 also confirm that, when all three instruments are included, the expected income and tax rate instruments are highly statistically significant (t = 4.69 and 55.4, respectively), while inclusion of the standard tax rate instrument is rejected at usual confidence levels (t = -1.43). Hence, both new instruments are valid and potentially useful, but the expected tax rate instrument is expected to perform much better.
Notes
- [18] Hausman-Wu tests also confirm that the marginal tax rate variable, Δ log (1- τ), is endogenous in OLS regressions.
