Fitting the ECM to U.S Time Series Data

This time we begin by applying the ECM described above to U.S. time series data over the ‘full sample’ period (1960q1-2011q1) and consider the simulated data in the next section. While it is natural to use the federal funds rate to represent the nominal interest rate in the context of a Taylor-type interest rate rule (e.g. Clarida et al., 2000), empirical studies in the Fisher relation tradition often use a short- run Treasury bill rate to represent the nominal interest rate instead (e.g. Crowder and Hoffman, 1996; Arnwine and Yigit, 2008; see Neely and Rapach, 2008, for a review). We therefore consider the 3 month Treasury bill rate [FRED series code:

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TB3MS] in addition to the effective federal funds rate when implementing the ECM described above.38

As with Table 4.3 for simulated data, we apply one unit root test (ADF) and one stationarity test (KPSS) to each of the variables belonging to the long-run relationship (4.15); Mehra (1991) only considers an ADF test but we wish to implement a second test as a check on the results. The tests are applied to the nominal interest rate, the rate of inflation and the consumption growth rate and the first difference of each variable in order to explore the possibility that they might be I(2). The tests indicate that both the nominal interest rate series and the inflation series are I(1) – the KPSS tests in the top panel of Table 4.11 reject the null hypothesis that these two series are stationary in levels and the ADF test does not reject the null hypothesis that they are integrated of order one; the ADF tests in the bottom panel reject the null that the first difference of the nominal interest rate and inflation are I(1) and the KPSS test cannot reject the null that the first differences are stationary.39 _{Clearly, these findings immediately pose a further}

problem for the full sample estimates presented in Tables 4.5, 4.9 and 4.10. However, the tests are not so decisive with respect to the consumption growth rate – while both tests seem to agree that ̅ is not I(2), two of the ADF tests reject the null of a unit root at the 90% level of significance and the KPSS test with a constant and a trend fails to reject the null of stationarity. On balance, these test statistics suggest that the consumption growth rate is stationary over the full sample period. Given the mixed results for ̅ , it would seem prudent to adopt Mehra’s econometric framework which allows stationary and non-stationary variables to enter the long-run equilibrium relationship simultaneously.40

[Table 4.11 here]

38 _{The correlation coefficient between FEDFUNDS and TB3MS is 0.990 over the full sample period.} 39 _{The tax adjusted interest rates in Tables 4.11 and 4.12 should be ignored for the time being.} 40 _{As discussed in the previous section, the seminal study of Clarida et al. (2000) cites the}

unsatisfactory nature of such tests as grounds for omitting a formal analysis of the stationarity of the data.

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In light of the results reported in Table 4.11, we require cointegration between the non-stationary variables in (4.15) in order for equation (4.17) to represent a valid empirical specification. To evaluate this, we perform a series of ‘cointegrating regressions’ using the non-stationary variables; if the residual series from such regressions are found to be stationary then this suggests that a linear combination of the non-stationary series are stationary, i.e. the series are cointegrated. Following Mehra (1991, Table 2), we regress both empirical proxies for the nominal interest rate on inflation and also perform the regression with the dependent and independent variables reversed. Table 4.12 reports the slope coefficient obtained from this procedure along with ADF and KPSS test results for the residuals of the cointegrating regression, ςt.41 The ADF test suggests that the

null hypothesis that the estimated residual series has a unit root can be rejected at either the 90% or the 95% level depending on whether the nominal interest rate or inflation is used as the dependent variable for the cointegrating regression, while the KPSS test suggests that the null hypothesis that the estimated residual series is stationary cannot be rejected at conventional levels of significance. These results are not materially affected by whether the effective federal funds rate or the 3 month Treasury bill rate is used as the empirical proxy for the nominal interest rate. Overall, there is evidence to suggest that the nominal interest rate and inflation are cointegrated over the full sample of U.S. data considered here.

[Table 4.12 here]

Having confirmed that the pre-requisites for the ECM are satisfied for our sample of post-war U.S. data, we are now able to estimate the reduced form model (4.17) using OLS and GMM procedures. One issue we face is in selecting the lag lengths τ1, τ2 and τ3. Mehra (1991, 1993) is not clear as to how this feature of the

specification is determined. For present purposes, we apply a grid search over 27 empirical specifications which allow τ1, τ2 and τ3 to vary between 0 and 2. For the

41 _{Mehra (1991, Table 2) considers only the ADF test. He omits the constant and the trend when}

testing the residuals from the cointegrating regression despite the fact that he includes a constant and a trend when testing the stationarity of the series (Mehra, 1991, Table 1). As such, the trend is also omitted from the KPSS test here but the constant term cannot be dispensed with.

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OLS estimator, the ‘preferred specification’ is deemed to be the one which minimises the Akaike Information Criterion. However, information criterion such as these cannot be applied under the GMM procedure so the lag structure is chosen simply in order to relieve the estimated residuals of serial correlation. The GMM selection therefore inevitably involves a greater degree of judgement but we shall show that estimates of the key inflation coefficient (ρ1) are not sensitive to

the estimation technique adopted. Following Mehra (1993) and Arnwine and Yigit (2008), the instrument set for the GMM estimator comprises of a constant, two lagged first differences of the nominal interest rate, four lagged first differences of inflation, four lagged first differences of consumption growth and one lagged level of the nominal interest rate, inflation and consumption growth.

Table 4.13 presents the results obtained from the OLS and GMM estimators, using both the effective federal funds rate and the 3 month Treasury bill rate as the empirical proxy for the nominal interest rate. The estimates of primary interest are those for the coefficients of the long-run equilibrium condition, ρ1 and ρ2 in

equation (4.15); these estimates can be found in the uppermost portion of Table 4.13. The results show that the estimated coefficient on inflation is similar regardless of whether the federal funds rate or the Treasury bill rate is used to represent the nominal interest rate and regardless of whether the OLS or GMM estimator is employed. Furthermore, the Q-statistic suggests that the estimated residual series is not afflicted by serial correlation and the J-statistic suggests that the instrument set used for the GMM procedure is valid. The R-squared statistics are in the region of 0.20-0.25, which is rather low, but consistent with the R- squared statistics reported by Arnwine and Yigit (2008, Table 1) for a similar regression.

[Table 4.13 here]

The estimates for ρ1 and ρ2 presented in Table 4.13 can be compared to the

analytical form for the ‘long-run Taylor Condition’, equation (4.7). Take the GMM estimates produced using the 3 month Treasury bill as the empirical proxy for the nominal interest rate for example, the estimated coefficient on the constant term

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is not statistically significant, the estimated coefficient on inflation is 1.210 and the estimated coefficient on consumption growth is 1.040. The estimates for ρ0

and ρ2 therefore conform to equation (4.7) with θ≃1 but, at first sight, the point

estimate for the coefficient on inflation appears to be ‘too high’. But as discussed in the introduction, an inflation coefficient in excess of one would be consistent with a long-run Fisher relation if tax effects are important. Disregarding the form of investor irrationality suggested by Tanzi (1980), if capital income is subject to taxation then investors have an incentive to adjust their asset portfolios up until the point at which a one-for-one relationship exists between the after tax nominal interest rate and inflation. As Crowder and Wohar (1999, p.310) explain, in this case the coefficient on inflation in the Fisher relation would not take a value of unity but (1–ξk)-1, where ξk is the tax rate on capital income (the return on

nominal government bonds here). As such, Neely and Rapach (2008, p.614) note that estimates in the region of 1.3-1.4 for the coefficient on inflation would be reasonable because estimates of this order imply a plausible tax rate in the region of 20%-30%. Similarly, Summers (1982) and Crowder and Hoffman (1996) state that one would expect a coefficient estimate in the range of 1.3-1.5 if no provision is made for taxation.

Upon obtaining an estimated inflation coefficient of this magnitude, some studies conclude with the conjecture that the results are consistent with a ‘tax adjusted Fisher relation’ (e.g. Crowder, 2003) but others attempt to calculate the appropriate adjustment to apply to the nominal interest rate in order to test for the expected one-for-one relation in tax adjusted data (e.g. Crowder and Hoffman, 1996). We estimate the appropriate tax adjustment by following the procedure set out by Padovano and Galli (2001). Although they did not examine the Fisher relation per se, others have used their estimates in this context (see Neely and Rapach, 2008, for a review). We prefer to calculate our own tax rates rather than adopt Padovano and Galli’s estimates for two reasons: firstly, because they do not produce estimates beyond 1990 and secondly, because we wish to include only federal tax receipts in the calculation given that U.S. Treasury bills are exempt from state and local taxes. To estimate the appropriate tax adjustment to apply to the nominal interest rate, we regress annual federal tax receipts on a constant and

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the level of annual real GDP using OLS.42 _{The estimated coefficient on real GDP}

subsequently provides an estimate of the change in tax revenue due to a change in aggregate income, i.e. the ‘effective marginal tax rate’ (Padovano and Galli, 2001). Separate regressions are run for each decade and the marginal tax rate obtained is applied to all quarters within that decade. Although this approach severely limits the sample size, the results (Table 4.14) show that the coefficient on real GDP is statistically significant at the 1% level and that the regression produces a high R-squared statistic for all decades with the exception of the final period (2000-2011). We therefore adjust the nominal interest rate series using our estimates for the first four decades of the sample period and use the average tax rate (revenue-to-GDP ratio) to make this adjustment for the 2000-2011 period.43, 44 _{Overall, our estimated marginal tax rates are smaller than Padovano and Galli’s}

(2001) estimates, which seems reasonable if they have included tax revenues at all levels of government whereas we have considered federal tax receipts only.45

[Table 4.14 here]

We now re-estimate (4.17) using the tax adjusted nominal interest rate series. The same pre-requisites for the ECM must also apply to these new series. Table

42 _{Tax revenue data are obtained from OMB Historical Table 1.3 and are expressed in 2005 dollars.}

This data is only available on an annual basis, hence the frequency used for the regressions. Real GDP is FRED series GDPC1. For simplicity we omit the dummy variables used by Padovano and Galli (2001) to account for major tax reforms. Theirs is a multi-country study but for the U.S. they apply a dummy variable only to the 1980s. This appears to have a quantitatively small impact upon their estimated tax rate.

43 _{Padovano and Galli (2001) note that empirical studies routinely use the revenue-to-GDP ratio to}

proxy for the effective tax rate. They argue that the effective marginal tax rate is a more appropriate measure theoretically. We use their preferred measure where it has been possible to produce a statistically significant estimate (1960s, 1970s, 1980s, 1990s) but revert to the revenue- to-GDP ratio where this has not been possible (2000-2011). Using the revenue-to-GDP ratio for the whole period would provide an estimated tax rate which varies very little because this ratio has been remarkably stable for post-war U.S. data, despite notable declines in top marginal and average tax rates. Gillman and Kejak (2013) offer a theoretical explanation for this apparent contradiction.

44 _{There is a discrepancy between the revenue-to-GDP ratio calculated from OMB and FRED data}

and the receipts-to-GDP ratio reported in Historical Table 1.3. Table 4.14 reports both of these figures for each decade for completeness. However, for 2000-2011, the period for which we use the revenue-to-GDP ratio instead of the estimated marginal tax rate to adjust the nominal interest rate, there is only a trivial difference between the two estimates (0.171 and 0.173).

45 _{It is unclear whether their measure of tax revenue includes state and local taxes. The only}

information they provide is that their series are obtained from the IMF's Government Financial Statistics and International Financial Statistics database.

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4.11 considers the time series properties of the tax adjusted nominal interest rate and Table 4.12 presents a new set of cointegrating regressions. The tax adjustment makes no material difference to the conclusions drawn from the unadjusted series. The final step is to re-run the ECM (4.17) using both measures of the nominal interest rate and the same two econometric procedures. As Table 4.15 shows, the estimated inflation coefficient is now consistent with the Fisher relation but only if the 3 month Treasury bill rate is used as the empirical proxy for the nominal interest rate. Taking the OLS estimate under the 3 month rate, for example, the constant term is not statistically significant, the estimate of ρ1 is

1.070 and the estimate of ρ2 is 1.052. In the context of the ‘long-run Taylor

Condition’, equation (4.7), these estimates would again correspond to a log utility function (CRRA with θ=1). The estimate for ρ2 falls slightly to 0.820 under the

GMM estimator but the coefficient on inflation is still consistent with the Fisher relation (ρ1=1.041). On the other hand, the estimates for ρ1 obtained using the

federal funds rate fall only slightly compared to the estimates presented in Table 4.13 for the unadjusted nominal interest rate and are thus still ‘too high’ relative to the predicted Fisher relation. Other aspects of this tax adjusted regression – including R-squared statistics, Q statistics and J statistics for instrument validity – remain satisfactory and similar to the results presented in Table 4.13.

[Table 4.15 here]