Estimating asymmetric output cost of lowering inflation for Australia.

Hyeon-seung Huh (*)

The purpose of this paper is to estimate the output cost associated with lowering inflation for Australia. The paper is particularly motivated by a strand of theoretical and empirical evidence in the literature suggesting nonlinearity in the output-inflation relationship, namely, a nonlinear Phillips curve. To accommodate this potentially important departure from linearity, a vector autoregression (VAR) model of output, inflation, and the terms of trade is augmented with logistic smooth transition autoregression specifications. My empirical results indicate that the model captures the nonlinear features present in the data well. Based on this nonlinear approximation, the output costs for reducing inflation are found to vary, depending critically on the state of the economy, the size of intended inflation change, and whether policymakers seek to disinflate or prevent inflation from rising. This implies that inferences based on the conventional linear Phillips curve may provide misleading signals about the cost of lowering inflation and thus the appropriate policy stance.

1. Introduction

One of the key costs of achieving low inflation is the short-term output loss that generally accompanies a permanent decline in inflation. Particularly stark examples of this output cost were seen in the early 1980s and 1990s where disinflations in both periods were accompanied by severe recessions. Obviously, policymakers' decisions on the timing and extent of inflation reduction depend on balancing the costs and benefits of moving to a new, lower level of inflation. The issue has become more relevant since the beginning of the 1990s, when several countries, including Australia, explicitly committed themselves to low inflation targets. With the advent of this inflation targeting regime, an increasingly important issue is the output cost of preventing inflation from rising. As incipient inflation pressures gain momentum, tighter monetary policy can slow the economy and thereby preemptively forestall the rise in actual inflation. This could avoid costly recessions down the track, but slower output growth would be the cost of resisting inflationary pressures. Together, these two output costs of fighting inflation play important roles in determining how to seek further disinflation toward price stability and how best to maintain low inflation. Undoubtedly, only with accurate measures can the net benefits of fighting inflation reliably be assessed.

A standard approach in the literature is to use the Phillips curve and estimate the so-called sacrifice ratio, which measures how much output would be lost by lowering inflation one percentage point. Traditionally, this short-run trade-off between output and inflation is assumed to be constant under the proposition that the shape of the Phillips curve is linear. However, a strand of the theoretical literature suggests the nonlinear nature of the Phillips curve. Empirical evidence supporting a variety of asymmetries in the output-inflation relationship has also mounted in recent times. Ball (1994) and Jordan (1997), for example, find that the output costs for reducing inflation vary with the states of the economy for the majority of OECD countries, including Australia. In these studies, they identified periods of disinflation and inflation episodes a priori and then estimated the output costs for each period separately. Recently, Filardo (1998) presents a more general approach by employing nonlinear modeling a long the line of Tong's (1983) threshold autoregression models. In this setup, the data itself define different regimes and determine the transition process between the regimes. The output costs of lowering inflation are allowed to vary with the signs and sizes of the shocks and the initial strength of the economy. With application to the U.S. data, he finds that the output costs of lowering inflation are indeed dependent on those factors.

The purpose of this paper is to extend Filardo's study to the case of Australia. There is a key difference, however. My vector autoregression (VAR) model is constructed to accommodate a potentially important departure from linearity through a logistic smooth transition autoregression (LSTAR) specification. The LSTAR specification allows the model to alternate between different regimes, with linear and discrete nonlinear cases as extreme ends. Importantly, the transition is carried out in a smooth manner so that there can be a continuum of states between the regimes. This contrasts with Filardo's model, where regime switching is discrete. The use of an LSTAR model in this paper is particularly justified by the fact that slow adjustments in inflation and consumers' expectations are the main reasons for the output cost of lowering inflation. Inflation tends to move slowly over time, generating a great deal of persistence and inertia. Consumers' expectations may also adjust slowly over time, perhaps being based o n some sort of adaptive mechanism. Because decisions about wages and prices depend on expectations of future changes, slow adaptation is self-fulfilling, creating inertia. Further, consumers may exert different degrees of inertia and so will adjust with different time lags. When considering aggregate behavior, the time path of regime changes is likely to be better captured by a model that permits gradual rather than instantaneous adjustment.

My baseline VAR model consists of real GDP, the inflation rate, and the terms of trade in line with Gordon and King (1982), Cecchetti (1994), King and Watson (1994), and Cecchetti and Rich (1999). The additional inclusion of the terms-of-trade variable in the model is motivated by Schelde-Anderson (1992) and Ball (1994), who identified the changes in that variable as a potentially important factor influencing the output costs through the inflationary process.' The model is assumed to be characterized by three structural shocks: a terms-of-trade shock, a domestic supply shock, and a domestic demand shock. Following previous studies, the output cost of lowering inflation is defined as a ratio of the output response relative to the inflation response with respect to an innovation to domestic demand. Hereafter, this is referred to as the cost of fighting inflation (COFI) ratio, following Filardo. Calculation of the COFI ratio then requires identification of the structural shocks in the model. In this paper, I ext end the identification procedure by Shapiro and Watson (1988) to the case involving LSTAR specifications. The underlying shocks in the model are identified by imposing long-run restrictions and exogeneity conditions of the type used for VAR models. In their paper, King and Watson (1994) find that structural estimates on the Phillips curve are quite sensitive to the directions of contemporaneous restrictions imposed for identification. Using long-run identifying restrictions will allow the data to determine the contemporaneous interactions between the two rather than imposing them a priori. This also differs from Filardo's paper, which adopted Cholesky-type contemporaneous identifying restrictions.

The remainder of this paper is organized as follows. Section 2 presents some preliminary analysis on the output-inflation relationship, together with discussions of the issues the paper seeks to answer. Section 3 discusses a structural model that constitutes my empirical analysis. The empirical results of this study are given in section 4. Section 5 draws policy implications from the empirical results and presents some conclusions.

2. Some Preliminary Considerations

In their paper, King and Watson (1994) find that for postwar U.S. data, the Phillips correlation between inflation and unemployment is alive and well, once one recognizes that it lives at the business cycle frequencies. After decomposing both data into three parts using the Baxter and King (1995) band-pass filter, there is a remarkably strong presence of this correlation at business cycle frequencies that is masked by low (i.e., trend) and high (i.e., irregular) frequency components. Figure 1 shows the results from passing real GDP and the (annualized) inflation rate of Australia through the same band-pass filter that isolates fluctuations at business cycle periodicities, six quarters to eight years. Also depicted in the figure is a dating scheme for Australian business cycle fluctuations constructed by the Melbourne Institute of Applied Economic and Social Research (MIAESR) at the University of Melbourne. (2) The shaded areas measure the time between the peaks and troughs of Australian growth cycles. Note th at by viewing the band-pass-filtered data as deviations from a local trend, I applied growth cycle peak and trough dates rather than classical cycle peak and trough dates. See Harding and Pagan (1999) for a detailed exposition of this distinction.

Figure 1 suggests that the Australian experience accords with King and Watson's observation. The cyclical components of the series vary, as the Phillips curve would suggest, with real GDP and the inflation rate falling in most cases during the MIAESR-dated growth recessions. To shed more light on this, Table 1 reports the cross-correlation coefficients of cyclical inflation at various lags and leads with cyclical real GDP. I also considered several subsamples to look at the stability of the relationship, including the break point of 1973:Q3, which was chosen to reflect the first major oil shock in October 1973. Looking at the full sample first, the presence of the Phillips curve relationship between GDP and inflation is empirically vindicated. The contemporaneous cross correlation is significant at the 5% level and has the correct sign (0.48 at k = 0). The evidence becomes stronger when cyclical GDP leads cyclical inflation (i.e., k < 0). They are also all positive in sign. Looking across the subsamples, simi lar results can be drawn in general. The cross correlations at k = 0 are all significant and positive. An exception is the period 1980:1-1989:4, but the correlation coefficients increase substantially when k < 0. Regardless of the subsample periods, the largest cross correlations (in bold) are all positive in sign and significantly high within the range of 0.61 and 0.86. (3) None of the subsample periods appears to substantiate evidence against the presence of the output and inflation relationship, as suggested by the Phillips curve.

Motivated by this finding, I model the output-inflation relationship formally in an attempt to estimate the output cost of lowering inflation. I do so particularly in a nonlinear framework, as there is a strand of the theoretical literature suggesting the asymmetric nature of the output-inflation relationship, namely, a nonlinear Phillips curve. In the monopolistic competitive model, for example, firms may be more reluctant to raise prices than to lower them in order to keep their market shares. Hence, they may respond to an increase in economic activity with more muted price changes and larger output changes than to a similar decrease in economic activity. The reduced sensitivity of inflation as the economy strengthens implies that the shape of the Phillips curve is concave. In the capacity constraints model, on the other hand, inflation will be increasingly sensitive to changes in output as the economy strengthens. Because of capacity constraints, an increase in demand will show up more as higher inflation than as higher output. Increased sensitivity of inflation to the economy's strength is consistent with the convex Phillips curve. There are also models suggesting that the relationship between output and inflation may vary with the level of inflation. In the menu cost model, for example, firms increase the frequency and size of price adjustment as inflation rises. Hence, demand shocks will have less effect on output and more effect on the price level, which is consistent with the convex Phillips curve (see Dupasquier and Ricketts 1998 for a comprehensive review of the microfoundations).

Different shapes of the nonlinear Phillips curve have different implications for the output cost of lowering inflation. If the Phillips curve is concave, as suggested by the monopolistic competitive model, the output cost of reducing inflation will rise with the strength of the economy. In contrast, the convex Phillips curve in the capacity constraints model implies that the output cost of reducing inflation will fall with the strength of the economy. The policy implications of the output cost of reducing inflation that is a function of the level of inflation can also be different from those of the output cost of reducing inflation that is a function of the economy's strength. Yet there is no consensus in the literature regarding the precise nonlinear form of the Phillips curve. Nonlinear shapes implied by explicit micro underpinnings may also be intractable in practice. Further, Dupasquier and Ricketts (1998) suggest that a model nesting more than one type of nonlinearity may be needed to fit the data better . For these reasons, I do not attempt to estimate directly one specific model or another. I take a rather broad approach through the VAR-LSTAR model. This specification is sufficiently flexible to allow various nonlinear Phillips curve shapes. It also allows shapes that are convex in one region and concave in another region (i.e., kinked curve).

In the model, the impulse response functions (hence COFI ratios) are dependent on the signs and sizes of the shocks and the initial states before these shocks hit the system. The results of the estimated LSTAR model are summarized along three dimensions. First, do positive shocks to domestic demand have different effects from negative shocks? Second, do domestic demand shocks have different effects at different points in the business cycle, for example, when output growth is initially low (high) or when inflation is initially falling (rising)? Third, do shocks of different magnitudes have disproportionate effects? A typical linear model produces a single coefficient, as it is independent of those factors. My specification can draw a richer picture of the output cost than a standard linear Phillips curve. Further, if the COFI ratio is indeed shown to have such asymmetries, inferences from the linear Phillips curve may provide misleading signals about the cost of reducing inflation and thus the appropriate poli cy stance to be taken.

The first and second questions are directly related to the conventional concepts of sacrifice and benefice ratios. The sacrifice ratio typically refers to the output cost for a permanent decline in inflation during a disinflation episode, whereas the benefice ratio typically refers to the output gain for a permanent rise in inflation during an episode of accelerating inflation. Both ratios are equal in absolute value for a linear case (i.e., a linear Phillips curve). In my impulse response framework, the response of output relative to the response of inflation following a negative shock to domestic demand can be regarded as a measure for the output cost of lowering inflation. The conventional sacrifice ratio corresponds to this measure when the economy is initially in a disinflation regime. On the other hand, the output response relative to the inflation response following a positive shock to domestic demand can be regarded as a measure for the output gain by allowing for higher inflation. The conventional be nefice ratio corresponds to this measure when the economy is initially in an accelerating inflation regime. Given that this paper focuses on the output cost of lowering inflation, I follow Filardo and relate the negative of this benefice ratio to the cost of lowering incipient inflation. The latter can be thought of as the forgone loss of output that would have accompanied the rise in inflation.

The third question is related to the macroeconomic policy central to the choice between "gradualism" and "cold turkey." One view is that gradualism is less costly because of wages and prices inertia and the need for time to adjust to monetary tightening. This view has been formalized by Taylor (1983), who presents a model of staggered wage adjustment in which quick disinflation reduces output but slow disinflation does not. A contrary view is that disinflation is less costly if it is quick, in favor of the cold turkey strategy. According to Sargent (1983) and Ball (1994), rapid disinflation produces credibility and hence a shift in expectations that makes disinflation less costly. While there is no consensus regarding the choice between gradualism and cold turkey, this paper sheds some light on the matter as the output cost of lowering inflation is allowed to vary with the speed of a given disinflation (or inflation).

3. Empirical Model

Baseline Linear VAR Model

Consider a three-variable VAR model that comprises the terms of trade (tt), real output (y), and the inflation rate ([pi]), all expressed in their first differences. There are assumed to be three structural disturbances in the system: a terms-of-trade shock ([[epsilon].sub.tt]), a domestic supply shock ([[epsilon].sub.yt]), and a domestic demand shock ([[epsilon].sub.nt]). These structural shocks are assumed to have a mean of zero and to be contemporaneously uncorrelated, that is, E([[epsilon].sub.it][[epsilon].sub.jt]) = 1 for i = j, and E([[epsilon].sub.it][[epsilon].sub.jt]) = 0 for i [not equal to] j. It is also assumed that the terms of trade are exogenous in the short and long run, which reflects the fact that Australia is a small open economy. Under this assumption, none of the domestic shocks, [[epsilon].sub.yt] and [[epsilon].sub.[pi]t] will have any impact on the terms of trade at all horizons. As a consequence, the dynamic interactions of real output and inflation in responses to [[epsilon].sub.yt] and [[epsilon].sub.[pi]t] which constitutes my major focus, can be equivalently evaluated from the following two variable model:

[DELTA][y.sub.t] = [summation over(p/i=0)] [a.sub.yt,i][DELTA][tt.sub.t-i] + [summation over (p/i=1)] [a.sub.yy,i][DELTA][y.sub.t-i] + [summation over(p/i=0)] [a.sub.y[pi],i][DELTA][[pi].sub.t-i] + [[epsilon].sub.yt] (1a)

[DELTA][[pi].sub.t] = [summation over(p/i=0)] [a.sub.[pi]t,i][DELTA][tt.sub.t-i] + [summation over(p/i=0)] [a.sub.[pi]y,i][DELTA][y.sub.t-i] + [summation over(p/i=1)] [a.sub.[pi][pi],i][delta][[pi].sub.t-i] + [[epsilon].sub.[pi]t], (1b)

Now, consider a vector moving average (VMA) expression corresponding to the system described by Equations 1a and 1b as

where [DELTA] = (1 - L) and L is the lag operator. By construction, [DELTA]t[t.sub.t] is uncorrelated with the domestic disturbances [[epsilon].sub.yt] and [[epsilon].sub.[pi]t]. Note that the previous model specification has a smaller number of free parameters relative to the unrestricted three-variable model. This feature will prove beneficial in dealing with the nonlinear version of the model later, in which the number of free parameters is doubled.

[MATHAMETICAL EXPRESSION NOT REPRODUCEBLE IN ASCII] (2)

where [GAMMA](L) is a polynomial function in the lag operator L. To exactly identify the two structural shocks, one additional restriction is needed. For this, it is assumed that domestic demand shocks have no long-run effects on the levels of real output by setting (1, 2) element of the long-run multiplier matrix [GAMMA](1) in Equation 2 to zero, that is [[GAMMA].sub.12](1) = 0. Cecchetti (1994) and Ceccheti and Rich (1999) used the same identifying restriction to estimate the U.S. sacrifice ratio in their two-variable structural VAR model. When the long-run multiplier matrix [GAMMA](1) is lower triangular as in this case, the Blanchard and Quah (1989) procedure can be used conveniently to accomplish the identification of the structural shocks. In this paper, however, I use an alternative procedure proposed by Shapiro and Watson (1988). In their method, the identifying restrictions are imposed directly on the VAR model itself without recourse to its VMA counterpart. This property allows us to perform the str uctural analysis (i.e., COFI ratio) in a straightforward manner even when the VAR is prompted with LSTAR specifications. This will be fully explained later. When the model is just identified, the results from Shapiro and Watson are identical to those from Blanchard and Quah.

Following Shapiro and Watson, the long-run restriction that [[GAMMA].sub.12](1) = 0 can be imposed by restricting the sum of the coefficients on [DELTA][[pi].sub.t] in Equation 1a to equal zero (i.e.,[summation over (p/i=0)] [a.sub.y[pi],i] = 0), which yields

[DELTA][y.sub.t] = [summation over(p/i=0)] [a.sub.yt,i][DELTA]t[t.sub.t-i] + [summation over (p/i=1)][a.sub.yy,i] [DELTA][y.sub.t-i] + [summation over (p-1/i=0)][b.sub.y[pi].i] [[DELTA].sup.2][[pi].sub.t-i] + [[epsilon].sub.yt], (1a')

where [b.sub.y[pi],i] are functions of [a.sub.y[pi],i] in Equation 1a. The variable [[pi].sub.t] only enters in second differences with the maximum number of lags setting to p - 1. Equation 1a' cannot be estimated by ordinary least squares (OLS) because of the contemporaneous value of [[DELTA].sup.2][[pi].sub.t] in the right-hand variables. However, the parameters in Equation 1a' can be estimated consistently using an instrumental variables (IV) procedure. An appropriate set of instruments is lags 1 through p of [DELTA][y.sub.t] and [DELTA][[pi].sub.t] and lags 0 through p of [DELTA]t[t.sub.t]. Equation 1b can also be estimated consistently using the same set of instruments as Equation 1a' plus [[epsilon].sub.yt], the estimated residual from Equation 1a'. (4) Once both Equation 1a' and Equation 1b are estimated, they are inverted to have [GAMMA](L) in Equation 2. It is then possible to use the estimated structural moving-average representation to examine the impact of changes in [[epsilon].sub.yt] and [[epsilon].sub.[pi]t] on output and inflation.

An LSTAR Representation

First, collect Equations 1a' and 1b in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or, in a compact notation.

[x.sub.t] = [PSI](L)[x.sup.*.sub.t] + [[epsilon].sub.i]. (3)

Then the set of structural equations (Eqn. 3) can be augmented with a LSTAR component such as

[x.sub.t] = [PSI](L)[x.sup.*.sub.t] + [XI](L)[x.sup.*.sub.t] F([z.sub.t]) + [[epsilon].sub.t], (4)

where [XI](L) has the exact same components as [PSI](L) of Equation 3. (5) The logistic function F([z.sub.t]) is assumed to have the following form:

F([z.sub.t]) = [(1 + exp{-[lambda]([z.sub.t] - c)/[[delta].sub.z]}).sup.-1] - 1/2, (5)

where F([z.sub.t]) lies in the range of - 1/2 and 1/2 and [lambda] > 0. The variable [z.sub.t] is a switching indicator that represents the state of the economy, and the parameter c represents the threshold around which the dynamics of the model change. The parameter [lambda] is the smoothness parameter. If [lambda] approaches zero, F([z.sub.t]) converges to a constant, and the model becomes linear. If [lambda] approaches infinity, the model becomes a threshold autoregression model along the lines of Tong (1983): The model's dynamics change abruptly, depending on whether [z.sub.i] is greater or less than c. The parameter [[delta].sub.z] is the standard deviation of the switching variable [z.sub.i]. The smoothness parameter [lambda] is not scale free, as its value depends on the magnitude of the switching variable [z.sub.t]. Dividing by [[delta].sub.z] normalizes the deviations of [z.sub.t] from the threshold value and facilitates interpretation of the smoothness parameter. This also makes it easier to find se nsible starting values in initiating the optimization process for estimation. See Granger and Terasvirta (1993) and Terasvirta (1998) for a detailed exposition on LSTAR models.

Output COFI Ratio

For the case of my linear baseline model, the COFI ratio can be computed on the basis of the structural impulse response functions from Equation 2 in the usual manner. For inflation, the sum of the coefficients in [[GAMMA].sub.22](L) measures the effects of a demand shock on its level. In the case of output, however, the COFI ratio requires us to consider the cumulative effect on its level resulting from the incidence of a demand shock. Taken together, the COFI ratio over the time horizon p can be calculated as

COFI([rho]) = [summation over ([rho]/j=0)] ([partial][y.sub.t+j]/[partial][[epsilon].sub.[pi]t]) / ([partial][[pi].sub.t+j]/[partial][[epsilon].sub.[pi]t]) = [summation over([rho]/j=0)] [summation over (i/j=0)] [[GAMMA].sub.12j] / [summation over ([rho]/j=0)] [[GAMMA].sub.22j] (6)

where [[GAMMA].sub.*2j] are the responses of the series to the demand shock at a horizon of j in Equation 2, that is, L = j.

An alternative measure I use here is to adopt the concept of generalized impulse response functions by Koop, Pesaran, and Potter (1996), which can be used in both linear and nonlinear cases. An impulse response function is defined as the effect of a one-time shock on the forecast of variables in a system. The response of a variable following a shock is then compared against a baseline "no shock" scenario. That is, the generalized impulse response function of a variable x, [GI.sub.x], is defined as the difference between two conditional expectations as

[GI.sub.x](n, [v.sub.t], [[omega].sub.t-1] = E[[x.sub.t+n]\[v.sub.t], [[omega].sub.t-1]] -- E[[x.sub.t+j]\[[[omega].sub.t-1]], n = 0, 1, 2, ..., (7)

where n is the forecast horizon, [v.sub.t] is the shock generating the response, [[omega].sub.t-1] is the history or initial values of the variables in the model, and E[.] is the expectations operator. Applying this, the COFI ratio can be expressed as

COFI([rho]) = [summation over (p/i=0)] [summation over (i/j=0)] [GI.sub.y] (j, [[epsilon].sub.[pi]t], [[omega].sub.t-1]) / [summation over (p/j=0)] [GI.sub.[pi]] (j, [[epsilon].sub.[pi]t], [[omega].sub.t-1]), (8)

where [GI.sub.y] (j, [[epsilon].sub.[pi]t], [[omega].sub.t-1]) is the generalized response of y to a demand shock [[epsilon].sub.[pi]t] at a horizon of j when the initial state before the shock was [[omega].sub.t-1]. The same notation applies to the case of [GI.sub.[pi]](j, [[epsilon].sub.[pi]t], [[omega].sub.t-1]). For a linear model, the impulses are invariant to history so that [[omega].sub.t-1] can be zeroed out. Hence, estimated GIs from Equation 7 will be identical to those estimated from the conventional impulse response function as in Equation 2. Both COFI ratios from Equations 6 and 8 will also be identical.

For the case of nonlinear models such as my LSTAR model, however, the responses are dependent on the signs and sizes of the shocks and the initial states before these shocks hit the system. In this context, Koop, Pesaran, and Potter treat the two conditional expectations in the right-hand side of Equation 7 as random variables, implying that [GI.sub.x] is also random. Accordingly, the GI functions and the COFI ratios must be computed by simulating the model. As alluded to earlier, the GI functions in adjunct with the Shapiro and Watson model may be usefully exploited in drawing structural interpretations such as the COFI ratio from my long-run identified LSTAR model. To see this, recall that in Equations 4 and 5, the restrictions required for identification are imposed directly on the VAR components. As a result, the forecast values made from this estimated LSTAR will also preserve those restrictions. These forecasted values are used to form the GI, which can be interpreted as the model's structural responses to the shocks. Following Equation 8, the COFI ratio can be obtained from forecasting the estimated LSTAR model directly without recourse to their VMA counterparts. For the simulation procedure, this paper follows the procedure used by Koop, Pesaran, and Potter (1996) and Weise (1999). The appendix in Weise, in particular, provides a detailed account of simulation of VARLSTAR models.

4. Empirical Results

Estimating the Baseline Linear Model

The baseline linear VAR model summarized in Equation 3 is estimated for quarterly data over the sample period 1961:1 to 1997:4. The lag length chosen was p = 3 on the basis of the Sims likelihood ratio test. Definitions of the data are as follows. The measure of real output (y) is the log of real GDP seasonally adjusted, in chained 1997/98 prices. The data on the terms of trade (tt) are defined as the log ratio of the implicit price deflator of exports of good and services to imports of goods and services and is seasonally adjusted at chained values (1997/98 = 100). The rate of inflation ([pi]) is measured as the quarterly percentage change in the consumer price index with the base year 1989/90 = 100 and is seasonally adjusted using the X-11 procedure. All data were drawn from DATASTREAM. The mnemonics for real output, export and import price deflators, and the consumer price index are AUGDP. . . D, AUIPDEXPE, AUIPDIMPE, and AUCP. . . F. (6)

Testing Linearity against LSTAR

The tests against a specific nonlinear alternative are usually formulated as Lagrange multiplier-type (LM) tests because they require only the estimation of the model under the null hypothesis, that is, the linear model. Applying this, the null hypothesis of linearity in Equations 4 and 5 is [H.sub.0] : [lambda] = 0 against [H.sub.1] : [lambda] > 0. However, there is a problem involved, as the model is identified under the alternative but not under the null hypothesis. When the null hypothesis is true, [XI](L), [lambda], and c in the nonlinear component can take any value. To circumvent this problem, Luukkonen, Saikkonen, and Terasvirta (1988) and Granger and Terasvirta (1993) suggest the use of the first-order Taylor series approximation for the LSTAR component.

Following their method, the LM tests can be performed equation by equation in three steps. In step 1, collect residuals [[micro].sub.it] for i = [DELTA]y and [DELTA][pi] from estimating Equation 3 and compute the residual sum of squares [SSR.sup.0.sub.i] = [SIGMA] [[micro].sup.2 7.sub.it]. In step 2, run the following auxiliary regression of [[micro].sub.it] on [PSI][(L).sub.i][x.sup.*.sub.t] and [z.sub.t][XI][(L).sub.i][x.sup.*.sub.t], where [PSI][(L).sub.i] and [XI][(L).sub.i] are the rows of [PSI](L) and [XI](L) corresponding to equation i in Equation 4 and [z.sub.t] is a switching variable. As before, collect residuals [v.sub.it] and calculate [SSR.sup.1.sub.i] = [SIGMA] [v.sup.2.sub.it]. In the final step (3), compute the test statistics [LM.sub.i] T([SSR.sup.0.sub.i] - [SSR.sup.1.sub.i])/[SSR.sup.0.sub.i] for each equation i, where T is the number of observations. Under the null hypothesis, [LM.sub.i] is distributed [x.sup.2]([m.sub.i]), where [m.sub.i] is the number of the regressors in the term [z.sub .t][XI][(L).sub.i][x.sup.*.sub.t]. In small samples, Granger and Terasvirta recommend the use of an F-test in order to improve size and power properties. The equivalent F-statistic is [F.sub.i] = [([SSR.sup.0.sub.i] - [SSR.sup.1.sub.i])/[m.sub.i]]/[[SSR.sup.1.sub.i]/(T - 2m - 1)]. While these tests are performed within a single-equation context, Weise (1999) recently used a log-likelihood ratio test for the test of linearity in the system as a whole, that is, [H.sub.0] [lambda] = 0 in all of the equations. Let [[OMEGA].sup.0] = [SIGMA] [[micro].sub.t][[micro]'.sub.t]/T and [[OMEGA].sup.1] = [v.sub.t][v'.sub.t]/T be the estimated variance-covariance matrices of residuals from the restricted and unrestricted regressions, respectively, where [[micro]'.sub.t] = ([[micro].sub.1t], [[micro].sub.2t]) and [v'.sub.t] = ([v.sub.1t], [v.sub.2t]). Then the log-likelihood statistic is LR = T{log\[[OMEGA].sup.0]\ - log\[[OMEGA].sup.1]\}, which is asymptotically distributed [chi square] with the degree of freedom ([m.sub.1] + [m.sub.2]). (8)

To perform these tests, the switching variable z, is assumed a priori. (9) Several candidates are considered for this switching variable. They are the growth rate of real output ([DELTA]y) and the change in the rate of inflation ([DELTA][pi]), one to five lagged values. I also experiment with the contemporaneous and one to five lagged values of changes in the terms of trade ([DELTA]tt) as possible switching variables. The results of these linearity tests are reported in Table 2 in terms of their marginal significance levels (p-values). When the change in terms of trade is used as the switching variable, the evidence against linearity is very weak. Except for one case in the inflation rate equation, the null hypothesis of linearity is never rejected in both equations and the system as a whole. By contrast, linearity is rejected in favor of a LSTAR specification when output growth and the change rate in inflation are used as switching variables. The evidence is particularly strong with [DELTA][y.sub.t-1] and [D ELTA][[pi].sub.t-1] as the switching variables. In each case, linearity is rejected unanimously by the LR test for the system as a whole and by the LM test for individual equations in the system. Overall, these tests provide solid evidence against linearity in favor of LSTAR specifications.

A different LSTAR model can be estimated for a host of switching variables, including other variables not considered in this section. Obviously, one cannot estimate every possible model, and there is no strong theoretical consensus on which variable to choose. In this sense, I restrict my consideration to the results of Table 2 in search of a plausible candidate for a switching variable. In particular, I follow Granger and Terasvirta (1993) and Terasvirta (1998), who suggest using the variable associated with the smallest p-value if that value is sufficiently small to reject the null hypothesis of linearity. The rationale behind this decision rule is that the linearity test has the highest power against the correctly specified alternative. If an inappropriate switching variable is selected, the resulting test may still have power against the alternative, but the power is less than if the correct switching variable is used. Thus, the strongest rejection of the null hypothesis suggests that the corresponding sw itching variable should be selected. Based on this criterion, I use [DELTA][y.sub.t-1] and [DELTA][[pi].sub.t-1] as the switching variables for the subsequent analysis.

Estimating the Structural LSTAR Model

In principle, the parameters of the LSTAR can be estimated by applying nonlinear least squares, which is equivalent to the maximum likelihood estimation when the errors are normal and independent. In practice, however, there could be convergence problems because some of the parameters are redundant and their estimates highly associated with those of other parameters. This problem may be more severe in a VAR type of LSTAR models like ours, which could result in the possible danger of underidentification. One way of circumventing this problem is to use the method by Terasvirta, Tjostheim, and Granger (1994) and Terasvirta (1998), which imposes coefficient restrictions that set certain elements of [PSI](L) and [XI](L) equal to zero or each other. However, this suggest ion may not be very helpful for VAR-type LSTAR models, as it could distort dynamic interactions among the variables in the model. Weise (1999) raises a similar concern, arguing that the results with such a method may be sensitive to which restricti ons are imposed, and the choice of restrictions cannot be guided by economic theory.

Weise used an alternative procedure to estimating a VAR type of LSTAR model. First, the threshold parameter c is fixed at a constant that is sensible from an economic standpoint. The LSTAR model is then estimated by equation-by-equation OLS using this value for c, allowing X to vary. The value of [lambda] that minimizes the log of the determinant of the variance-covariance matrix of residuals from these regressions is selected for the final regressions. This method avoids unjustifiable coefficient restrictions but sacrifices efficiency. In addition, standard errors are not computed for c and [lambda]. (10) I follow this method but also take up several plausible values of c for consideration. For the switching variable [DELTA][y.sub.t-1], the parameter c is selected in the range of -3.0% to 3.0% with an incrementing order of 0.2 percentage points. While others can be considered, this prespecified range is reasonable given that the switching variable is expressed in quarterly changes. It amounts to covering abo ut 98% of the quarterly growth rates observed in the Australian GDP data. With the switching variable [DELTA][[pi].sub.t-1], the parameter c is selected from -2.0 to 2.0 percentage points in an incrementing order of 0.1 percentage points. This range covers about 95% of the quarterly changes in the Australian inflation rate.

Estimation of LSTAR (Eqns. 4 and 5) is then carried Out by the instrumental variables (IV) procedure in sequential order, starting from the output equation to the inflation rate equation. The pair (c, [lambda]) that minimizes the log determinant of the variance-covariance matrix of residuals is selected. The second panel of Table 3 reports the estimates of [lambda] along with the selection of c produced by this procedure. When the switching variable [DELTA][y.sub.t-1] is used, the selected choice is c = 1.0 and [lambda] = 12.1. When [DELTA][[pi].sub.t-1] is used as the switching variable with the selection of c to zero, the smoothness parameter is much smaller at 1.3, indicating a fairly smooth transition from one regime to another. (11) As a way of checking the sensitivity of the estimates, the third panel of Table 3 reports the results obtained from estimating a threshold autoregression ([lambda] [right arrow] [infinity]). The parameter c was chosen as the value for which the log determinant of the variance -covarmance matrix of residuals was minimized. The threshold values for [DELTA][y.sub.t-1] are indeed close to the maintained values of 1.0. Accordingly, with the switching variable [DELTA][y.sub.t-1], little would be lost by estimating this system using a threshold autoregression specification. For the case of however, the threshold value differs significantly from the selected value of c in the second panel. Hence, threshold autoregression specification is likely to produce different results from those reported using the LSTAR model.

Table 4 reports results of several diagnostic tests to check the statistical adequacy of the estimated LSTAR models in terms of their marginal significance levels (p-values). The F-test confirms evidence against linearity and in favor of the type of nonlinearity represented by the LSTAR specification. All individual equations are statistically significant at standard levels regardless of which switching variables are used. However, the choice of the switching variables appears to matter when each equation in the system passes through a battery of misspecification tests. When [DELTA][[pi].sub.t-1] is used as the switching variable, the test results point to no evidence of any serious model inadequacy. The residual variances of the equations are only 72% and 89%, respectively, of those from the corresponding linear models. Results of the LM test in the fourth column indicate no serial correlation, nor is there any evidence of ARCH effects in the fifth column. I also checked for remaining nonlinearity in residua ls using the LSTAR test described previously. No further evidence to support nonlinearity was found in any equation. The situation changes somewhat when the switching variable is [DELTA][y.sub.t-1]. There is strong evidence against the normality of the errors, while remaining ARCH effects in the inflation rate equation pose another problem. The residual variances of the estimated models did decrease, though, by around 20% compared with those from the corresponding linear models.

Evaluating dynamic interactions among the variables is conditional on the statistical adequacy of the estimated models. I interpret the failure of passing those diagnostic tests as a result of a poor approximation of the estimated models to data. Accordingly, the subsequent analysis commences with the selection of [delta][[pi].sub.t-1] as the switching variable (referred to as model 1). Note, though, that my LSTAR specification is simply one way of approximating the true feature of nonlinearity and that there may be other alternatives that could lead to different implications. As discussed briefly in section 2, there are also some studies supporting the proposition that the asymmetric nature of the output-inflation relationship varies with the strength of the economy. In this context, I also report the case of [delta][y.sub.t-1] as the switching variable (referred to as model 2).

Asymmetric COFI Ratio Estimates

Table 5 reports the estimated COFI ratios using Equation 8 over the horizon of 20 quarters, that is, [rho] = 20. The COFI ratios are calculated for three initial states when the switching variable is [delta][y.sub.t-1]: low growth ([delta][y.sub.t-1] < 0), moderate growth (0 < [delta][y.sub.t-1] < 2.25), and high growth ([delta][y.sub.t-1] > 2.25). (12) In the case of [delta][[pi].sub.t-1] as the switching variable, the ratios are calculated for two regimes: falling inflation ([delta][[pi].sub.t-1] < 0) and rising inflation ([delta][[pi].sub.t-1] > 0). The figures under "1%" are computed from the impulse response functions to the demand shock of 0.87. This size of the shock is set to the standard deviation of demand shocks computed from their corresponding linear models. The figures under "2%" are computed from the impulse response functions to the demand shock of 1.74. These impulse responses are then divided by two, respectively, so that they can be compared to the responses to one-standard-error shocks. Ta ble 5 also reports the results from estimating the linear models as in Equation 3 for the sake of comparison.

Simulations show that COFI ratios vary systematically depending on the signs of the shocks. For both models 1 and 2, the COFI ratios for deliberate disinflations (negative shocks) are different from those for preemptive interventions against incipient inflation pressures (positive shocks). They are additionally dependent on the initial state of the economy. Looking at model 1 in the case of disinflation first, the output cost in a rising inflation state is much lower than that in a falling inflation state. A 1% disinflation costs 5.12% of output if initiated when the economy is in a rising inflation state and 7.21% when the economy is in a falling inflation state. In a rising inflation state, a tighter monetary policy could stifle economic activity to a lesser extent than in a falling inflation state, making resultant output costs less in this case. Looking at the case of disinflation with model 2, the output cost decreases as the initial strength of the economy increases. A 1% disinflation costs 2.13% of out put if initiated when the economy is in a high-growth state, 5.01% when the economy is in a medium-growth state, and 7.63% when the economy is in a low-growth state. Intuitively, the stronger the economy, the less the effect on output of a tighter monetary policy, which results in fewer output costs to achieve a given disinflation.

The output cost of preemptively resisting incipient inflation varies with the state of the economy in a similar manner. In the case of model 1, the output cost in a rising inflation state is much lower than that in a falling inflation state. Preventing a one-percentage-point increase in inflation costs 4.98% of output when the economy is in a rising inflation state but 7.03% when the economy is in a falling inflation state. The authority must relax monetary policy more extensively in a falling inflation state than in a rising inflation state if both would produce a given rise in inflation. Accordingly, the forgone output as a result of preemptive monetary policy will be larger in a falling inflation state than in a rising inflation state. In the case of model 2, the output cost of preemptively resisting incipient inflation decreases as the initial strength of the economy increases. Preventing a one-percentage-point increase in inflation costs 2.06% of output if initiated when the economy is in a high-growth s tate, 4.85% when the economy is in a medium-growth state, and 6.90% when the economy is in a low-growth state. The strength of economic forces associated with a weak economy must exceed those associated with a strong economy if both are to produce a given rise in inflation. Then this requires a tighter preemptive monetary policy in a low-growth state in order to prevent rising inflation, thus leading to a larger forgone output.

Simulation results in the table also indicate that the COFI ratio increases as the size of the desired inflation change increases. This finding is shown to be unanimous regardless of the model specifications. In the case of disinflation, for example, model 1 suggests that the output cost per percentage point reduction in inflation is 5.12% (7.21%) for a one-percentage-point and 5.79% (8.46%) for a two-percentage-point reduction when the economy is in a rising (falling) inflation. Model 2 draws a similar line, indicating that the cost for a one-percentage-point disinflation is lower than for each percentage point reduction of a two-percentage-point disinflation, regardless of the initial strength of the economy. The same result can also be inferred in the case of a preemptive intervention against incipient inflation. Regardless of the initial states of the economy, models 1 and 2 both indicate that preventing a potential one-percentage-point rise in inflation costs less per percentage point of incipient inflat ion than a two-percentage-point rise.

5. Policy Implications and Concluding Remarks

Accurately assessing the output cost of reducing inflation plays an important role in determining the appropriate policy to achieve and ultimately maintain price stability. As its contribution, this paper takes explicit account of possible nonlinearity in estimating the output cost of lowering inflation. While the precise nonlinear form of the Phillips curve implied by explicit micro underpinnings may be intractable, this paper shows that its distinct features may be parsimoniously represented in terms of an LSTAR model. The adoption of the LSTAR specification as a nonlinear component was justified by its flexibility in allowing slow adjustments in association with price/wage rigidities and consumers' expectations. The findings of this paper suggest that the nonlinear model provides a richer picture of output cost than a standard linear Phillips curve. Specifically, the COFI ratio is related in a nonlinear way, depending critically on the state of the economy, the size of the inflation change, and whether pol icymakers seek to disinflate or prevent inflation from rising. This evidence has valuable policy implications for traditional views on the output cost of fighting inflation.

First, the traditional linear model tends to either overestimate or underestimate the COFI ratio, depending on the initial states of the economy. For example, model 1 indicates that the COFI ratio implied by the linear model overestimates when the economy is in a rising inflation state but underestimates when the economy is in a falling inflation state. If the economy were in the former regime, the nonlinear model's lower cost of reducing inflation would hence suggest a more aggressive stance of monetary policy than would have been implied by a linear Phillips curve model. A less aggressive fight is recommended if the economy is in the latter regime. Looking across the other model specifications reported in Table 5, a similar inference can be drawn in that my nonlinear COFI ratios vary case by case. This suggests that inferences based on the linear Phillips curve may provide misleading signals about the cost of lowering inflation and hence the appropriate policy stance.

The second implication is related to the benefit of preemptive policy. My empirical evidence reveals that the COFI ratios for the case of deliberate disinflation are higher than those calculated from the case of preemptively preventing incipient inflation. That is, a policy to preemptively prevent rising inflation of a given size turns out to be less costly than a policy to deliberately disinflate. Preemptive policies are often justified by the familiar finding that monetary policy affects the economy with long and variable lags. This study provides a further incentive for preemptive monetary response to inflationary pressures. A preemptive tightening helps prevent the economy from moving too far up to the levels where inflation begins to rise more rapidly, thereby avoiding the need for a more aggressive tightening in the future to reverse this large rise in inflation.

The final implication is due to the fact that the output cost of disinflation may vary with the speed of a given disinflation, central to the choice between gradualism and cold turkey. My study shows that the average cost of a one-percentage-point disinflation is in all cases less costly per percentage point of disinflation than a two-percentage-point disinflation. The higher output cost of rapid disinflation indicates that gradualism is a lower output cost strategy than cold turkey. This finding corroborates Taylor (1983) on the cost advantage of the gradualist approach.

(*.) Department of Economics, Hallym University, Okehon-dong 1 Ga, Chunchon Kangwondo 200-702, South Korea; E-mail [e-mail].

The author wishes to thank two anonymous referees, Don Harding, Peter Summers, Peter Dawkins, Tim Kam, John Janssen, and participants at the 2000 Australian Macroeconomics Workshop for their very helpful comments on earlier versions of the paper. This research was supported by the Hallym Academy of Sciences at Hallym University. The usual disclaimer applies.

Received August 2000; accepted June 2001.

(1.) A rise in import prices, for example, will feed into consumer price inflation, possibly with a lag, which would increase output costs to achieve a given disinflation. Further, if it sparks consumers' inflation expectations through the associated depreciation in the exchange rate, lowering inflation could be more costly. Given that Australia is a small open economy, those effects of terms of trade on the output costs may be particularly relevant.

(2.) Available at MIAESR's Web Site (http://www.ecom.unimelb.edu.au/iaesrwww/bdates.html). Also see Boehm (1998) for this construction of Australian business cycle chronology.

(3.) Table 1 indicates that the largest cross correlations occurred at different horizons, depending on the sample periods. For example, cyclical inflation is contemporaneous with the cycle of GDP in the subsample of before the first major oil shock while it lags thc cycle of real GDP in the subsample of after the first major oil shock. There is no case in which cyclical inflation leads the cycle of GDP.

(4.) Equation lb can be equivalently estimated by OLS after substituting Equation la' into Equation lb.

(5.) Note that the intercept terms are suppressed in both [psi](L) and [XI](L).

(6.) Before estimating a linear VAR model, augmented Dickey-Fuller tests were used to determine the order of integration of the series. The test indicated that real GDP, the terms of trade, and the inflation rate are characterized as an I(1) process. I then applied the Johansen procedure to test for evidence of cointegration between output and inflation with the terms of trade entering as an exogenous variable. Both the trace and the maximum eigenvalue tests indicated no cointegrating relationships in the model at standard significance levels.

(7.) As explained in section 3, estimation should be carried in a sequential manner, that is, from Equation 1a' to Equation 1b.

(8.) Granger and Terasvirta point out that an LM-type test mentioned previously would have low power against the alternative hypothesis when [XI](L) is close to zero such that the only nonlinear element in Equation 4 is the constant. They describe an adjustment proposed by Luukkonen, Saikkonen, and Terasvirta that involves a third-order Taylor approximation to the regression in step 2. In a multivariate setup, however, the addition of higher-order terms is very costly, and given the typical size of macroeconomic samples, it is not practical. Further, estimates of [XI](L) reported in the following section are far from zero.

(9.) If [z.sub.t] is assumed unknown, the test statistics must be generalized to account for this. See Tsay (1986) and Luukkonen, Saikkonen, and Terasvirta (1988) for this case.

(10.) Terasvirta, Tjostheim, and Granger (1994) also considered this procedure in a univariate LSTAR model.

(11.) To see this, designate F([Z.sub.t] = 0 to be a pure "falling inflation" regime and F([Z.sub.t]) = 1 to be a pure "rising inflation" regime. Then a smoothness parameter [lambda] = 1.3 implies that when [DELTA][[pi].sub.t-1] moves one standard deviation above (below) zero, F([Z.sub.t]) equals approximately 0.79 (0.21). This means that the regime is a linear combination of the pure rising and falling inflation regimes, with a weight of 0.79 (0.21) on the former and 0.21 (0.79) on the latter. Compare this with the value of [lambda] = 12.1, for which minute deviations of the switching variable from the threshold level essentially place the economy entirely on one regime or the other.

(12.) The moderate-growth state is classified as the period that its GDP growth rate lies in the range of 15% and 85% when the whole-sample GDP growth rates are ordered from lowest to highest. The low-growth state corresponds to the period having its growth rate in the lower 15% band. while the high-growth state corresponds to the period having its growth rate in the upper 15% band. This way of devising the initial states is arbitrary, but it may not be avoidable in nature.

References

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[Figure 1 Omitted]

Table 1

Cross Correlations at Business Cycle Frequencies

                                              Cross Correlation of Real
                                            Output (t) with Inflation (t
                                                        - k)
Sample Period               k                            0

1961:1-1997:4  k [less than or equal to] 0             0.48 (*)
               k > 0
1961:1-1973:3  k [less than or equal to] 0             0.66 (*)
               k > 0
1973:4-1997:4  k [less than or equal to] 0             0.37 (*)
               k > 0
1961:1-1969:4  k [less than or equal to] 0             0.74 (*)
               k > 0
1970:1-1979:4  k [less than or equal to] 0             0.60 (*)
               k > 0
1980:1-1989:4  k [less than or equal to] 0             0.03
               k > 0
1990:1-1997:4  k [less than or equal to] 0             0.68 (*)
               k > 0

                Cross Correlation of Real Output (t) with Inflation (t -
                                           k)
Sample Period    1                                   2          3

1961:1-1997:4   0.55 (*)                            0.64 (*)   0.59 (*)
                0.22 (*)                            0.03      -0.16
1961:1-1973:3   0.62 (*)                            0.45 (*)   0.25
                0.44 (*)                            0.15      -0.08
1973:4-1997:4   0.42 (*)                            0.56 (*)   0.63 (*)
                0.11                               -0.05      -0.20
1961:1-1969:4   0.81 (*)                            0.67 (*)   0.45 (*)
                0.46 (*)                            0.15      -0.06
1970:1-1979:4   0.61 (*)                            0.51 (*)   0.34
                0.45 (*)                            0.15      -0.24
1980:1-1989:4   0.20                                0.48 (*)   0.74 (*)
               -0.22                               -0.36 (*)  -0.47 (*)
1990:1-1997:4   0.64 (*)                            0.55 (*)   0.40 (*)
                0.58 (*)                           -0.47 (*)   0.34

                Cross Correlation of
                Real Output (t) with
                 Inflation (t - k)
Sample Period    4          5

1961:1-1997:4   0.48 (*)   0.34 (*)
               -0.32 (*)  -0.38 (*)
1961:1-1973:3   0.11       0.04
               -0.19      -0.18
1973:4-1997:4   0.59 (*)   0.41 (*)
               -0.34 (*)  -0.41 (*)
1961:1-1969:4   0.25       0.16
               -0.12      -0.12
1970:1-1979:4   0.12      -0.14
               -0.57 (*)  -0.33
1980:1-1989:4   0.86 (*)   0.77 (*)
               -0.55 (*)  -0.55 (*)
1990:1-1997:4   0.20      -0.02
                0.19       0.03

Entry in row k [less than or equal to] 0 (k > 0) is the cross-
correlation coefficient of cyclical inflation at lead (lag) k with
cyclical real GDP. The exception is for the entry in k = 0, which
reports the contemporaneous cross-correlation coefficient between the
two. Figures in bold refer to largest cross correlations of the cyclical
movements in the GDP and inflation at each sample period. Figures with
asterisk are significant at the 5% level where the confidence bands are
approximated [+ or -]2/ [square root of (T)] (T = number of effective
observations).
Table 2

The p-Values of the LM-Type Linearity Test against LSTAR

                               LM Tests           LR Tests
Switching Variables     [DELTA]y   [DELTA][pi]   System

[DELTA][tt.sub.t]         0.20        0.11        0.12
[DELTA][tt.sub.t-1]       0.35        0.09        0.22
[DELTA][tt.sub.t-2]       0.32        0.17        0.29
{DELTA][tt.sub.t-3]       0.49        0.29        0.33
[DELTA][tt.sub.t-4]       0.49        0.34        0.47
[DELTA][tt.sub.t-5]       0.55        0.49        0.50
[DELTA][y.sub.t-1]        0.05        0.02        0.05
[DELTA][y.sub.t-2]        0.12        0.09        0.14
[DELTA][y.sub.t-3]        0.23        0.20        0.20
[DELTA][y.sub.t-4]        0.29        0.30        0.25
[DELTA][y.sub.t-5]        0.40        0.37        0.38
[DELTA][[pi].sub.t-1]     0.01        0.00        0.01
[DELTA][[pi].sub.t-2]     0.06        0.03        0.09
[DELTA][[pi].sub.t-3]     0.15        0.09        0.14
[DELTA][[pi].sub.t-4]     0.27        0.19        0.21
[DELTA][[pi].sub.t-5]     0.35        0.30        0.34

Figures reported are the marginal significance levels (p-values).
Table 3

Estimates of Smoothness ([lambda]) and Threshold (c) Parameters

Switching
Variables                  Estimates of [lambda] with c Being Fixed

[DELTA][Y.sub.t-1]                c = 1.0              [lambda] = 12.1
[DELTA][[pi].sub.t-1]             c = 0.0              [lambda] = 1.3

Switching
Variables                 Estimates of c with [lambda] = [infinity]

[DELTA][Y.sub.t-1]          [lambda] = [infinity]        c = 1.12
[DELTA][[pi].sub.t-1]       [lambda] = [infinity]        c = 0.92
Table 4

Diagnostic Tests for the Structural LSTAR Model

Dependent
Variables    F-test
                   Switching Variable =
                    [DELTA][y.sub.t-1]

[DELTA]y      0.01
[DELTA][pi]   0.05

Dependent
Variables    [[sigma].sup.2]/[[sigma].sup.2.sub.L]  Auto (4)  ARCH (4)
                       Switching Variable = [DELTA][y.sub.t-1]

[DELTA]y     0.79                                     0.32      0.17
[DELTA][pi]  0.80                                     0.31      0.03

Dependent
Variables    Normality  LSTAR
             Switching Variable =
              [DELTA][y.sub.t-1]

[DELTA]y       0.01     0.19
[DELTA][pi]    0.03     0.12
                     Switching Variable = [DELTA][[pi].sub.-1]

[DELTA]y     0.03                            0.89           0.40
[DELTA][pi]  0.01                            0.72           0.61

                  Switching Variable =
                  [DELTA][[pi].sub.-1]

[DELTA]y     0.29      0.14      0.32
[DELTA][pi]  0.18      0.21      0.56

Figure reported are the marginal significance levels (p-values). Figures
in the second column are from the F-tests for the null hypothesis that
the coefficients on the f([Z.sub.t]) terms are jointly equal to zero.
The third column, denoted by [[sigma].sup.2]/[[sigma].sup.2.sub.L],
reports the ratio of the estimated variance of the model relative to
that of the corresponding linear model. For the following two columns,
the terms Auto (4) and ARCH (4) refer the F versions of LM test for
fourth-order serial correlation and the LM test for fourth-order ARCH
effects, respectively. The sixth column reports the results of the
Jarque-Bera test for normality. The last column,termed as LSTAR, reports
the results the testing linearity against the LSTAR Specification using
the estimated residuals. See section 4 for details.
Table 5

Average Output Cost of Lowering Inflation

                       [delta][[pi].sub.t-1] (Model 1)
Shock Size  [delta][[pi].sub.t-1] < 0  [delta][[pi].sub.t-1] > 0
                           Deliberate Disinflation

1%                    7.21                       5.12
2%                    8.46                       5.79

                            [delta][y.sub.t-1] (Model 2)
Shock Size    [delta][y.sub.t-1] < 0     0 < [delta][y.sub.t-1] < 2.25
                              Deliberate Disinflation

1%                     7.63                          5.01
2%                     9.55                          6.32

             [delta][y.sub.t-1] (Model
                        2)
Shock Size  [delta][y.sub.t-1] > 2.25  Linear Model
                     Deliberate Disinflation

1%                    2.13                 5.69
2%                    2.96                 5.69
                  Preventing Incipient Inflation Pressures

1%                 7.03           4.98                  6.90
2%                 8.22           5.36                  8.87

          Preventing Incipient Inflation
                    Pressures

1%        4.85      2.06      5.69
2%        5.96      2.32      5.69