APPENDIX I. Linearity Testing
Equation (7) in a difference form can be rewritten for testing linearity:
Testing the null hypothesis of α′2 = γ′2 proceeds in the following steps, as in Granger and Terasvirta (1993). First, run the following regression by least squares (LS) ΔrNt = βΔπt + 1 + et; then compute the residual
Following Hansen (1996), /realizations of the LM statistics are generated for each grid in the grid set Γ. We generate (ωjt, t = 1,..T) i.i.d. N(0, l) random variables; under the assumption of homoskedastic error, generate
where #Γ is the number of grid points in Γ.
APPENDIX II. Estimating Switching Regressions with Perfect Discrimination
The switching regression with perfect discrimination in a difference form is given by
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Woon Gyu Choi is a Senior Economist in the Asian Division of the IMF Institute. The author is grateful to Robert Flood, Reza Vaez–Zadeh, and two anonymous referees for helpful comments and suggestions. He thanks Costas Azariadis, Michael R. Darby, Dmitriy Gershenson, Lung–fei Lee, Axel Leijonhufvud, Seonghwan Oh, Simon Potter, Keunkwan Ryu, and seminar participants at the University of California, Los Angeles and the Hong Kong University of Science and Technology for helpful comments and suggestions on earlier drafts. He also thanks Jae Young Kim for the use of a program for the Phillips fractional integration estimation and Benjamin C. Fung for research assistance.
Carmichael and Stebbing argue that one should take into account the substitutability between money and bonds in determining the effect of inflation on interest rates, especially because the data mainly pertain to the return on financial assets rather than to the return on capital.
Klein argues that, under specie standards that provided an anchor for the price level, there was considerable short–term unpredictability, but much less long–term unpredictability, of inflation. Under the fiduciary monetary system of the post–World War II period, there is no anchor for the price level. Agents have come to regard prices as largely affected by policy, and short–term unpredictability is less than before. For example, Hutchison and Keeley (1989) show that inflation evolved from a white noise process in the pre–World War I period to a highly persistent, nonstationary process in the post–1960 period, which strengthened the Fisher effect.
Estimating the nominal rate equation to test the IFH should account for the nominal rate response to E(πt+1), which involves the use of arbitrary data about expected inflation. Nonetheless, one can check whether the nominal rate is independent of actual inflation (see, for the result of this test, the robustness check section).
Gallagher (1986) points out that this testing pertains to the maintained hypothesis that (πt+1) and ξt are only contemporaneously uncorrelated. Following earlier studies including Carmichael and Stebbing, this study interprets the IFH as claiming that inflation and nominal interest rates are contemporaneously uncorrelated in the sense of Gallagher.
Switching model (7) assumes heteroskedasticity across regimes but homoskedasticity within a regime, whereas a model of GARCH (generalized autoregressive conditional heteroskedasticity, Bollerslev, 1986) (with regime shifts) allows for autocorrelation in the second moment of errors, implying history–dependent variability in errors.
As inflation moves more persistently, it can be better forecast using the past history of inflation. Autocorrelations up to several orders of the residual, which contain information about persistence, will be reflected in this index. In general, agents would utilize indicators for future inflation, too.
Observations with moderate forecastability may be grouped between the two extreme regimes. The smooth transition treats these observations as a mix of the two regimes, weighted by the distance from each. Thereby the degree to which inflation is reflected in interest rates is a monotonic function of inflation persistence.
Davies (1987) and Granger and Terasvirta (1993) suggest using the supremum of statistics over a grid set, whereas Andrews and Ploberger (1994) suggest using the average and the exponential average of statistics.
Using the mnemonics on the FRED of the Federal Reserve Bank of St. Louis, the variable definitions are TB3MS (the three–month treasury bill rate), CPIAUSL (the CPI–U: whole items), and PCE/PCEC92 (the personal consumption expenditure deflator). The average for each quarter is used to measure the quarterly series. All except interest rates are seasonally adjusted data.
The use of a monthly model may cause serial correlation in errors because the maturity of yield on a bill is longer than the data frequency. Suppose that the error term of equation (3′) (in a difference form) for the one–month rate is serially uncorrelated. Based on this equation, the monthly model of the threemonth rate can be expressed—for example, via the expectations hypothesis. Time aggregation results in serial correlation in errors.
Different lag lengths are suggested by different information criteria (over rolling samples). However, choosing an alternative lag length (13 lags, for example) does not affect the results of this study qualitatively.
Barth and Bradley (1988) and Gupta (1991) find that estimation results for equation (3′) are not sensitive to whether or not the real rate is adjusted for taxes. Indeed, as Barth and Bradley point out, “since taxes are levied on the nominal rate, irrespective of tax adjustments, the real interest rate should vary inversely and one–for–one with the inflation rate (under the hypothesis),” unless the tax rate is correlated with inflation.
13The plots of the likelihood ratio as a function of τf indicate a unique major dip in the likelihood ratio around the estimate, suggesting that two regimes are sufficient to describe the nature of the threshold effect.
To deal with the seasonality issue, we use the period–average rate of interest rather than the rate at a specific date for quarter t, taking the rate of inflation as the period–average figure.
We consider the fractional difference process of the form (1 – L)d(π t – μ) = vt, where L is the lag operator, d is the fractional integration parameter, and vt is a mean zero stationary error. This process is stationary for | d | < 0.5.
As noted by King and Watson (1997), Mishkin (1992) and Evans and Lewis (1995) find that nominal rates do not respond fully to permanent changes in inflation and attribute this to a small–sample bias associated with shifts in the inflation process.
Fischer (1981) argues that high inflation raises the variability, but not necessarily the uncertainty, of inflation. According to Ball and Cecchetti (1990), inflation uncertainty pertains to the variance of unanticipated inflation, whereas inflation variability pertains to the variance of inflation. Since high inflation tends to be largely anticipated, unanticipated inflation will not be large relative to inflation, supporting Fischer′s argument. Also, Ball and Cecchetti show that inflation has much larger positive effects on inflation uncertainty at long horizons.
This is similar to the Mundell–Tobin effect that represents a portfolio–substitution effect of inflation on the steady–state capital–labor ratio. The Mundell–Tobin effect, however, assumes that real balances and capital are substitutes (direct financing for capital investments) so that an increase in inflation increases the portfolio demand for capital and thus the capital–labor ratio, which, in turn, lowers the real rate of return on assets.
The change points of inflation persistence can also be related to shifts in the Fed′s operating mechanism (see, for the Fed′s operating mechanism, Choi, 1999a). In particular, the nonborrowed reserve targeting for 1979:10–1982:10, which involves high and volatile money growth, matches the persistent inflation regime, whereas federal funds rate targeting since 1987 matches the nonpersistent inflation regime. (Under the Bretton Woods system and before the Vietnam War, interest rates were the focus of monetary policy and inflation persistence was low.)