1. Alternative characterizations of the output-inflation tradeoff

In terms of flexible approximations to general nonlinear functions, the most common and easy to implement is a power series expansion. Representing the function relating the effect on inflation of the output gap as f(gap), such an expansion implies:

where gap is the deviation between actual output and its potential level. ^5/ In estimation, the testing down from the general expansion in equation (1) to a specific approximation can be based on statistical criteria. It is interesting though, to consider a priori the properties of some specific cases of this class of approximations. One is the function consisting only of the linear and quadratic terms:

As illustrated in the upper left-hand panel of Chart 1, this approximation implies a convex function, with the effect on inflation tapering off as the gap becomes negative. ^6/ Another, even more parsimonious, alternative would be to simply raise the gap to an odd integer—the power must be odd to preserve the appropriate sign of the inflation-output relationship. ^7/ An example is the cubic function: ^8/

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Alternative Nonlinear Functional Forms

Citation: IMF Working Papers 1994, 139; 10.5089/9781451929355.001.A001

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This specification would cause the effect on inflation to increase rapidly as the gap rose in size, while the slope of the function would approach zero when the gap was small. For negative gaps, the function would have the same shape as for positive gaps—that is, downward pressure on inflation would become large as the negative gap increased in size (Chart 1, upper right-hand panel). This limiting case, then, has the disadvantage that, while nonlinear, it is symmetric: positive and negative gaps have the same effect on inflation. Thus, it cannot explain apparent asymmetries in the inflation-activity relationship.

Alternatively, specific functional forms can be chosen ex ante that embody conceptual priors about the shape of the inflation-activity tradeoff. The latter approach was adopted by Chadha, Masson, and Meredith (1992)—henceforth CMM—in estimating alternative Phillips curves for the G-7 countries. The functional form employed by CMM to represent the nonlinearity is the (modified) hyperbola:

where β and ω are parameters to be estimated. This function is graphed in the lower left-hand panel of Chart 1. Its relevant properties can be derived by looking at the first derivative of f()—that is, the slope of the inflation-activity tradeoff:

The limiting values of this derivative (and of the function itself) for some specific values of ω and gap are:

Equation (5a) shows that, as the parameter ω becomes large, the CMM function approaches a linear relationship (as ω decreases, in contrast, it can be shown that the function approaches a reverse “L” shaped relationship). Equation (5b) indicates that the effect on inflation rises without bound as the output gap approaches ω: in other words, ω represents a “wall” beyond which output cannot increase in the short run. As the gap becomes negative, the slope of the function decreases; equation (5c) shows that, at the limit, there is a lower bound to the effect on inflation of -βω as the gap becomes highly negative. When the gap is zero, the effect on inflation is also zero, and the slope of the tradeoff is β (equation (5d)).

Another example of a specific functional form is Laxton, Rose, and Tetlow (1993)—henceforth LRT—using Canadian data. LRT used a “kinked” function to represent the nonlinearity, with a discontinuity in its slope as the output gap changes sign: ^9/

where α₁ and α₂ are parameters to be estimated, and the value of α₂ exceeds that of α₁ (Chart 1, lower right-hand panel).

The limited theoretical literature on nonlinearities provides little guide as to the preferred specification. On conceptual and empirical grounds, however, some discrimination is possible. The function consisting only of the cubic term, for instance, has the conceptual drawback that, while nonlinear, it is symmetric: negative gaps reduce inflation by as much as positive gaps raise it. Empirically, the implication that downward pressure on inflation increases exponentially as economic slack rises appears to conflict with the experience of large industrial countries that have experienced deep recessions in recent years. ^10/ The “kinked” function used by LRT is both transparent and consistent with the proposition that excess supply has a smaller effect on inflation than does excess demand. However, it has the weakness that no upper bound is imposed on output. Operationally, the discontinuity implied by the kink makes some aspects of estimation problematic. It is also difficult to rationalize the abrupt change in the value of α when the gap changes sign, given that the aggregate gap reflects the average of conditions in many different markets. The quadratic function also fails to impose an upper bound on output, while the upward-sloping region in the area of significant excess supply is implausible on conceptual grounds.

From these points of view, the CMM function seems the most suitable of the nonlinear alternatives: it is asymmetric, has a continuously differentiable slope, and implies an upper bound on output. Nevertheless, in the absence of clear predictions from theory, considerable weight must be given to empirical evidence in selecting the appropriate functional form. As shown below, the CMM function performs well in fitting the data for the major industrial countries. We therefore focus on it in the following discussion and in the simulation exercises in the next section.

2. Specification of the price equation

Our price equation is a reduced-form representation of wage and price dynamics, in which the response of wages to labor market conditions is subsumed in the response of inflation to deviations in output from potential. To focus on the broadest possible measure of output prices, but to exclude the supply shocks associated with movements in oil prices in the 1970s and 1980s, we use the rate of change in the non-oil GDP deflator (π) as the dependent variable. Inflation is explained by: a weighted average of past and expected future growth in output prices; growth in the contemporaneous absorption deflator (π^a) relative to output prices; and terms in contemporaneous and lagged output gaps. We allow the slope parameter β in equation (4) to differ between the two output gap terms, whereas the “wall” parameter ω is constrained to be the same. The general form of the equation is then:

To make equation (7) suitable for estimation, the contemporaneous value of π on the right-hand side can be eliminated by bringing it to the left-hand side and dividing through by (1+λ), yielding:

Here, we summarize the rationale for this specification; a more detailed discussion is provided in Chadha, Masson, and Meredith (1992).

Traditional, “backward-looking,” Phillips curves relied on past inflation to reflect inertia in the wage and price contracting process as well as to proxy for expectations of future inflation. In contrast, more recent theoretical models of overlapping contracts with forward-looking agents (such as Calvo (1983)) represent inflation as a function of its expected future realization based on all available information about the state of the economy. The inclusion in our specification of a weighted average of past and expected future inflation reflects elements of both approaches, with the importance of each determined empirically. ^11/ Expected future inflation is constructed by regressing actual inflation on a set of past information known at the time expectations are formed; the fitted values from this regression are then used as proxies for inflation expectations. The estimated weights on past and expected future inflation determine the relative importance of the “forward” and “backward” looking components of the inflation process. The constraint that the parameters sum to unity ensures that no long-run tradeoff exists between the level of inflation and excess demand pressures.

The term in relative absorption inflation captures the effect on wage demands of changes in consumption prices. Theoretically, it could enter the equation in either levels or growth rates, depending on the assumptions of the model. In the Calvo (1983) model, for instance, wages are fixed in levels for the life of the contract, implying that the relevant variable is the level of the absorption price. In other models, where contract wages grow over time, it is the growth rate that is relevant. Following the CMM finding that the growth rate specification works better empirically for the G-7 countries, we retain that specification here.

The output gap term is a (nonlinear) function of the deviation of actual output from its “potential” level. ^12/ A complication arises in implementing this approach given our assumption of a nonlinear output-inflation tradeoff. Specifically, as shown formally in Appendix I, the average observed level of output will generally be below the potential level of output. Constructing proxies for potential by detrending movements in actual output will result in downward-biased estimates of potential. To correct for this bias, we introduce a parameter, a, in the output gap term that indicates the average deviation of trend output from the conceptual level of potential in the absence of stochastic demand shocks:

where y is real GDP, y^* is potential output, ỹ is trend GDP, and α is the percent deviation between y^* and ỹ.

Both the contemporaneous and lagged values of the output gap are included in equation (7’). Such dynamics in the response of inflation to demand conditions can be motivated by traditional backward-looking models of inflation. ^13/ They are also consistent with forward-looking behavior with adjustment costs. A particular case arises when costs are associated with changing the level of output, as reflected in an upward-sloping marginal cost curve. This would imply that the change in the markup of output prices over variable costs depends on the change in the level of capacity utilization.

3. Estimation results for the G-7 countries

Data for the estimation of the inflation equation were obtained from the MULTIMOD database for the G-7 industrial countries. ^14/ Specifically, annual data were available from 1965 to 1993 for: the non-oil GDP deflator; the deflator for total domestic demand (i.e., absorption); and real GDP. In addition, it was necessary to construct a measure of trend output for each country. Two methods were used for this purpose. The first involved smoothing (the logarithm of) actual output using the filter developed by Hodrick and Prescott (1980) in the context of the analysis of real business cycles. The second—more direct—method involved taking a two-sided moving average of the logarithm of actual output; a five-year “window” was chosen for this purpose. ^15/ Both methods yielded plausible measures of output gaps based on the stylized facts of G-7 business cycles. While the estimation results were also similar, those using the moving-average filter yielded somewhat more precise parameter estimates; as it is also the more direct of the two measures to construct (and easily replicated by other researchers), we use it in the estimation results described below.

In addition, it was necessary to estimate auxiliary equations using instrumental variables to deal with the endogeneity of absorption inflation, and to construct proxies for expected future inflation. Following the strategy pursued by CMM, the instruments used were: the lagged output gap; lagged growth in the non-oil GDP deflator; the second lag of growth in the absorption deflator; lagged money growth; and the lagged ratio of real government expenditures to GDP. ^16/ Auxiliary equations were estimated using these variables as regressors—the fitted values from these regressions were then used in the estimation of equation (7’).

After allowing for leads and lags in the constructed data, 25 observations remained for each G-7 country, extending from 1967-1991. The data on actual and trend output for the G-7 countries are shown in Chart 2(a), while Chart 2(b) compares the associated output gaps with the inflation rates observed in these countries. For most countries there appears to be a systematic relationship between output gaps and changes in the inflation rate. However, given the limited number of observations for each country, it is not surprising that empirical research using individual country data has failed to find consistent evidence in favor of nonlinearities. This problem has been compounded by imprecise measures of the gap. ^17/ However, even if the output gaps were measured precisely, there are too few disinflationary/inflationary episodes in the individual country experiences to develop reliable estimates of the degree of convexity. However, taken together, the experiences of all countries in the G-7 may be sufficient to detect significant convexity in the inflation-output process. Consequently, to enhance the efficiency of the estimation process, data for the G-7 countries were pooled and equation (7’) was estimated as a nonlinear system with common parameters across countries. ^18/

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Real GDP and Trend GDP in the G-7 Countries

(In natural logarithms)

Citation: IMF Working Papers 1994, 139; 10.5089/9781451929355.001.A001

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Inflation and Output Gaps in the G-7 Countries

(In percent)

Citation: IMF Working Papers 1994, 139; 10.5089/9781451929355.001.A001

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Table 1 compares results for the CMM model with those for the linear alternative. ^19/ In the case of the CMM function, the nonlinearity in the functional form implied by ω—the “wall” parameter—prohibited its joint estimation with the other parameters using conventional nonlinear methods. Instead, a grid search was used where ω was varied from 0.04 to 0.10 in increments of 0.001. Line (1) shows the results for the unconstrained model with nonlinear CMM terms in the contemporaneous and lagged output gaps. The parameters on both terms are economically large and statistically significant. The value of the “wall” parameter, ω, that maximizes the log likelihood function is 0.049, implying that the effect on inflation becomes unbounded as the output gap, measured relative to potential output, approaches 5 percent of GDP. However, the parameter α has a value of 0.006, indicating an average difference of 0.6 percent between the hypothetical level of output at which the effect on inflation is zero and the constructed “trend” level of output. ^20/ Thus, measured relative to trend output, inflation becomes infinite as the output gap approaches 5 1/2 percent of GDP.

Table 1.

Estimation Results for CMM Versus Linear Functional Forms

(t-statistics in parentheses)

Estimated equation:

${\mathrm{\pi }}_{\mathrm{t}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathrm{\lambda }}^{\mathrm{\prime }}\mathrm{)}\mathrm{(}\mathrm{\delta }{\hat{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\delta }\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}{\mathrm{\lambda }}^{\mathrm{\prime }}{{\hat{\mathrm{\pi }}}^{\mathrm{a}}}_{\mathrm{t}}\mathrm{+}{{\mathit{\beta }}_{\mathrm{1}}}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}}\mathrm{\right)}\mathrm{+}{{\mathit{\beta }}_2}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

where:

$\mathit{\text{gap}}\mathrm{=}\mathrm{\text{log}}\mathrm{(}\mathrm{y}\mathrm{/}{\mathrm{y}}^{\mathrm{*}}\mathrm{)}$

$y^{*}=(1+\alpha )\overline{y}$

$\mathrm{\text{f}}\mathit{\left(}\mathit{\text{gap}}\mathit{\right)}\mathit{=}({\mathit{\omega }}^{\mathit{2}}\mathit{/}\mathit{(}\mathit{\omega }\mathit{-}\mathit{\text{gap}}\mathit{)})\mathit{-}\mathit{\omega }$

π = change in the log of the non-oil GDP deflator

π^a = change in the log of the absorption deflator

y = real GDP

y = trend real GDP

y^* = level of GDP at which the effect on inflation is zero

α = gap between trend GDP (y) and level of GDP at which the effect on inflation is zero (y^*)

ω = parameter identifying maximum short-run level of GDP

δ = weight on forward-looking component of inflation expectations

λ^’ = weight on absorption deflator

^ = denotes fitted value from auxiliary regression (see text)

Data: Annual, 1967-91, pooled G-7 countries.

Table 1.

Estimation Results for CMM Versus Linear Functional Forms

(t-statistics in parentheses)

Estimated equation:

${\mathrm{\pi }}_{\mathrm{t}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathrm{\lambda }}^{\mathrm{\prime }}\mathrm{)}\mathrm{(}\mathrm{\delta }{\hat{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\delta }\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}{\mathrm{\lambda }}^{\mathrm{\prime }}{{\hat{\mathrm{\pi }}}^{\mathrm{a}}}_{\mathrm{t}}\mathrm{+}{{\mathit{\beta }}_{\mathrm{1}}}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}}\mathrm{\right)}\mathrm{+}{{\mathit{\beta }}_2}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

where:

$\mathit{\text{gap}}\mathrm{=}\mathrm{\text{log}}\mathrm{(}\mathrm{y}\mathrm{/}{\mathrm{y}}^{\mathrm{*}}\mathrm{)}$

$y^{*}=(1+\alpha )\overline{y}$

$\mathrm{\text{f}}\mathit{\left(}\mathit{\text{gap}}\mathit{\right)}\mathit{=}({\mathit{\omega }}^{\mathit{2}}\mathit{/}\mathit{(}\mathit{\omega }\mathit{-}\mathit{\text{gap}}\mathit{)})\mathit{-}\mathit{\omega }$

π = change in the log of the non-oil GDP deflator

π^a = change in the log of the absorption deflator

y = real GDP

y = trend real GDP

y^* = level of GDP at which the effect on inflation is zero

α = gap between trend GDP (y) and level of GDP at which the effect on inflation is zero (y^*)

ω = parameter identifying maximum short-run level of GDP

δ = weight on forward-looking component of inflation expectations

λ^’ = weight on absorption deflator

^ = denotes fitted value from auxiliary regression (see text)

Data: Annual, 1967-91, pooled G-7 countries.

		δ	λ’	α	β1	β2’	ω	LLF
(1)	Nonlinear	0.414	0.193	0.006	0.196	0.276	0.049
	CMM function	(7.33)	(1.96)	(3.04)	(3.55)	(3.96)	(7.96)	495.28
(2)	Linear	0.417	0.258	0.002	0.224	0.180	large
	function	(6.18)	(2.56)	(0.67)	(3.00)	(2.28)		489.43
	(ω=∞)
(3)	Nonlinear
	CMM function	0.444	0.194	—	0.091	0.160	0.053
		(7.76)	(1.94)		(2.24)	(2.60)	(11.38)	492.59
(4)	Linear
	function	0.423	0.248	—	0.219	0.173	large
	(ω=∞, α=0)	(6.39)	(2.48)		(2.91)	(2.19)		489.21

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Table 1.

Estimation Results for CMM Versus Linear Functional Forms

(t-statistics in parentheses)

Estimated equation:

${\mathrm{\pi }}_{\mathrm{t}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathrm{\lambda }}^{\mathrm{\prime }}\mathrm{)}\mathrm{(}\mathrm{\delta }{\hat{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\delta }\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}{\mathrm{\lambda }}^{\mathrm{\prime }}{{\hat{\mathrm{\pi }}}^{\mathrm{a}}}_{\mathrm{t}}\mathrm{+}{{\mathit{\beta }}_{\mathrm{1}}}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}}\mathrm{\right)}\mathrm{+}{{\mathit{\beta }}_2}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

where:

$\mathit{\text{gap}}\mathrm{=}\mathrm{\text{log}}\mathrm{(}\mathrm{y}\mathrm{/}{\mathrm{y}}^{\mathrm{*}}\mathrm{)}$

$y^{*}=(1+\alpha )\overline{y}$

$\mathrm{\text{f}}\mathit{\left(}\mathit{\text{gap}}\mathit{\right)}\mathit{=}({\mathit{\omega }}^{\mathit{2}}\mathit{/}\mathit{(}\mathit{\omega }\mathit{-}\mathit{\text{gap}}\mathit{)})\mathit{-}\mathit{\omega }$

π = change in the log of the non-oil GDP deflator

π^a = change in the log of the absorption deflator

y = real GDP

y = trend real GDP

y^* = level of GDP at which the effect on inflation is zero

α = gap between trend GDP (y) and level of GDP at which the effect on inflation is zero (y^*)

ω = parameter identifying maximum short-run level of GDP

δ = weight on forward-looking component of inflation expectations

λ^’ = weight on absorption deflator

^ = denotes fitted value from auxiliary regression (see text)

Data: Annual, 1967-91, pooled G-7 countries.

		δ	λ’	α	β1	β2’	ω	LLF
(1)	Nonlinear	0.414	0.193	0.006	0.196	0.276	0.049
	CMM function	(7.33)	(1.96)	(3.04)	(3.55)	(3.96)	(7.96)	495.28
(2)	Linear	0.417	0.258	0.002	0.224	0.180	large
	function	(6.18)	(2.56)	(0.67)	(3.00)	(2.28)		489.43
	(ω=∞)
(3)	Nonlinear
	CMM function	0.444	0.194	—	0.091	0.160	0.053
		(7.76)	(1.94)		(2.24)	(2.60)	(11.38)	492.59
(4)	Linear
	function	0.423	0.248	—	0.219	0.173	large
	(ω=∞, α=0)	(6.39)	(2.48)		(2.91)	(2.19)		489.21

The resulting nonlinearity in the relationship between output gaps and the effect on inflation, based on the estimated parameters, is illustrated in the upper two panels of Chart 3. Looking, for example, at the contemporaneous effect, a positive output gap of 3 percent would tend to raise the inflation rate by about 1 1/2 percentage points, while a negative gap of the same size would reduce inflation by less than 1/2 percentage point. The effects of the lagged gap are larger given that β₂’ exceeds β₁’, but the overall shape of the function is the same.

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Estimated Nonlinear Functional Forms Versus Linear Model Contemporaneous and Lagged Effects on Inflation

Citation: IMF Working Papers 1994, 139; 10.5089/9781451929355.001.A001

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The weight on the forward-looking component of the inflationary process is 0.414, while that on the backward-looking component is 0.586 (i.e., 1-0.414); both are (highly) significantly different from zero, with values similar to those found in other studies embodying forward- and backward-looking elements in the inflationary process. The parameter on contemporaneous absorption inflation is 0.193 with a t-statistic of slightly below 2, suggesting a limited impact of shocks to the relative price of absorption on output prices.

Line (2) shows the results obtained when the linear version of the output-inflation tradeoff is substituted for the CMM function (equivalent to imposing a large value for ω in the estimation procedure). For testing the hypothesis of nonlinearity, the most important aspect of these results is the drop in the value of the log of the likelihood function (LLF) when linearity is imposed. In the event, twice the difference in the LLF gives a test statistic of 11.8, which is distributed χ²(1) under the null hypothesis of a linear tradeoff. As the critical value of the χ²(1) distribution at the 99 percent confidence level is 6.6, the linear hypothesis is overwhelmingly rejected by the data. Otherwise, the parameters are similar to those estimated using the CMM function, although the parameters on the lagged output gap (β₂’) and the difference between the potential and trend levels of output (α) are both somewhat smaller and less significant.

The above results, indicating a strong rejection of the linear model in favor of the nonlinear alternative, raise the question of why earlier estimation exercises have not found conclusive evidence of nonlinearities. For example, CMM (1992) were able to reject the linear model at the 95 percent confidence level, but not at the 97.5 percent level. An indication of how these earlier results can be reconciled with the current evidence is provided in Lines (3) and (4) of Table 1, which show estimation results for the nonlinear and linear models when α is constrained to zero. This parameter, which measures the gap between the potential and trend levels of output, allows for a horizontal “shift” in the output-inflation tradeoff such that inflation tends to fall when output is at its trend level. In the earlier work of CMM, α was implicitly constrained to zero. As indicated by the drop in the LLF in Line (3) from that in Line (1), this constraint significantly worsens the fit of the nonlinear specification. The linear specification, in contrast, fits roughly as well with or without the constraint on a, as shown by a comparison of Line (4) with Line (2). ^21/ Comparing Lines (3) and (4), then, the evidence in favor of nonlinearities is much less strong when α is constrained to zero than when it is a free parameter. From this, we conclude that the empirical case for nonlinearity becomes more robust when the specification of the functional form is made fully consistent with theoretical priors.

Further evidence on this point is provided in Table 2, which shows the value of the log likelihood function for a grid of values for α and ω. Consistent with the above results, the value of the likelihood function is maximized at 495.26 when α is 0.006 and ω is 0.05. However, the slope of the likelihood function does not remain constant as we move away from this point: when α is at its optimal value, the LLF declines faster as the relationship becomes more linear (i.e., as ω becomes larger) than when α is zero. For example, the LLF falls to 491.44 when α equals 0.006 and ω moves from 0.05 to 0.10, producing a χ² statistic of 7.64. When α is zero, in contrast, the decline in the LLF as ω moves from 0.05 to 0.10 is only 1.62 (492.49-490.87), and the test statistic falls to 3.24. This confirms the earlier result, that the power of tests of linear versus nonlinear models are weakened by misspecification of the level of potential. ^22/ Since traditional detrending techniques tend to produce estimates of excess demand that are, on average, too large when the world is truly nonlinear, their naive use can result in false rejections of nonlinear models.

Table 2.

Values of the Log Likelihood Function for Various Values of ALPHA and OMEGA

(t-statistics in parentheses)

Estimated equation:

${\mathrm{\pi }}_{\mathrm{t}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathrm{\lambda }}^{\mathrm{\prime }}\mathrm{)}\mathrm{(}\mathrm{\delta }{\hat{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\delta }\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}{\mathrm{\lambda }}^{\mathrm{\prime }}{{\hat{\mathrm{\pi }}}^{\mathrm{a}}}_{\mathrm{t}}\mathrm{+}{{\mathit{\beta }}_{\mathrm{1}}}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}}\mathrm{\right)}\mathrm{+}{{\mathit{\beta }}_2}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

where:

$\mathit{\text{gap}}\mathrm{=}\mathrm{\text{log}}\mathrm{(}\mathrm{y}\mathrm{/}{\mathrm{y}}^{\mathrm{*}}\mathrm{)}$

$y^{*}=(1+\alpha )\overline{y}$

$\mathrm{\text{f}}\mathit{\left(}\mathit{\text{gap}}\mathit{\right)}\mathit{=}({\mathit{\omega }}^{\mathit{2}}\mathit{/}\mathit{(}\mathit{\omega }\mathit{-}\mathit{\text{gap}}\mathit{)})\mathit{-}\mathit{\omega }$

π = change in the log of the non-oil GDP deflator

π^a = change in the log of the absorption deflator

y = real GDP

y = trend real GDP

y^* = level of GDP at which the effect on inflation is zero

α = gap between trend GDP (y) and level of GDP at which the effect on inflation is zero (y*)

ω = parameter identifying maximum short-run level of GDP

δ = weight on forward-looking component of inflation expectations

λ^’ = weight on absorption deflator

^ = denotes fitted value from auxiliary regression (see text)

Data: Annual, 1967-91, pooled G-7 countries.

Table 2.

Values of the Log Likelihood Function for Various Values of ALPHA and OMEGA

(t-statistics in parentheses)

Estimated equation:

${\mathrm{\pi }}_{\mathrm{t}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathrm{\lambda }}^{\mathrm{\prime }}\mathrm{)}\mathrm{(}\mathrm{\delta }{\hat{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\delta }\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}{\mathrm{\lambda }}^{\mathrm{\prime }}{{\hat{\mathrm{\pi }}}^{\mathrm{a}}}_{\mathrm{t}}\mathrm{+}{{\mathit{\beta }}_{\mathrm{1}}}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}}\mathrm{\right)}\mathrm{+}{{\mathit{\beta }}_2}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

where:

$\mathit{\text{gap}}\mathrm{=}\mathrm{\text{log}}\mathrm{(}\mathrm{y}\mathrm{/}{\mathrm{y}}^{\mathrm{*}}\mathrm{)}$

$y^{*}=(1+\alpha )\overline{y}$

$\mathrm{\text{f}}\mathit{\left(}\mathit{\text{gap}}\mathit{\right)}\mathit{=}({\mathit{\omega }}^{\mathit{2}}\mathit{/}\mathit{(}\mathit{\omega }\mathit{-}\mathit{\text{gap}}\mathit{)})\mathit{-}\mathit{\omega }$

π = change in the log of the non-oil GDP deflator

π^a = change in the log of the absorption deflator

y = real GDP

y = trend real GDP

y^* = level of GDP at which the effect on inflation is zero

α = gap between trend GDP (y) and level of GDP at which the effect on inflation is zero (y*)

ω = parameter identifying maximum short-run level of GDP

δ = weight on forward-looking component of inflation expectations

λ^’ = weight on absorption deflator

^ = denotes fitted value from auxiliary regression (see text)

Data: Annual, 1967-91, pooled G-7 countries.

	Values of ω
α	.04	.05	.06	.07	.08	.09	.10
0.000	487.11	492.49	492.24	491.71	491.34	491.07	490.87
0.001	488.04	492.95	492.82	492.33	491.95	491.65	491.42
0.002	489.17	493.46	493.42	492.94	492.51	492.16	491.87
0.003	490.71	492.01	493.99	493.45	492.93	492.50	492.15
0.004	488.95	494.55	494.43	493.74	493.10	492.58	492.17
0.005	490.86	495.01	494.61	493.72	492.96	492.37	491.91
0.006	491.94	495.26	494.44	493.35	492.51	491.90	491.44
0.007	492.87	495.18	493.91	492.71	491.86	491.26	490.83
0.008	493.66	494.72	493.10	491.89	491.10	490.56	490.17
0.009	494.20	493.92	492.15	491.02	490.32	489.84	489.51

View Table

Table 2.

Values of the Log Likelihood Function for Various Values of ALPHA and OMEGA

(t-statistics in parentheses)

Estimated equation:

${\mathrm{\pi }}_{\mathrm{t}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}{\mathrm{\lambda }}^{\mathrm{\prime }}\mathrm{)}\mathrm{(}\mathrm{\delta }{\hat{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\delta }\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}{\mathrm{\lambda }}^{\mathrm{\prime }}{{\hat{\mathrm{\pi }}}^{\mathrm{a}}}_{\mathrm{t}}\mathrm{+}{{\mathit{\beta }}_{\mathrm{1}}}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}}\mathrm{\right)}\mathrm{+}{{\mathit{\beta }}_2}^{\mathrm{\prime }}\mathrm{\text{f}}\mathrm{\left(}{\mathit{\text{gap}}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

where:

$\mathit{\text{gap}}\mathrm{=}\mathrm{\text{log}}\mathrm{(}\mathrm{y}\mathrm{/}{\mathrm{y}}^{\mathrm{*}}\mathrm{)}$

$y^{*}=(1+\alpha )\overline{y}$

$\mathrm{\text{f}}\mathit{\left(}\mathit{\text{gap}}\mathit{\right)}\mathit{=}({\mathit{\omega }}^{\mathit{2}}\mathit{/}\mathit{(}\mathit{\omega }\mathit{-}\mathit{\text{gap}}\mathit{)})\mathit{-}\mathit{\omega }$

π = change in the log of the non-oil GDP deflator

π^a = change in the log of the absorption deflator

y = real GDP

y = trend real GDP

y^* = level of GDP at which the effect on inflation is zero

α = gap between trend GDP (y) and level of GDP at which the effect on inflation is zero (y*)

ω = parameter identifying maximum short-run level of GDP

δ = weight on forward-looking component of inflation expectations

λ^’ = weight on absorption deflator

^ = denotes fitted value from auxiliary regression (see text)

Data: Annual, 1967-91, pooled G-7 countries.

	Values of ω
α	.04	.05	.06	.07	.08	.09	.10
0.000	487.11	492.49	492.24	491.71	491.34	491.07	490.87
0.001	488.04	492.95	492.82	492.33	491.95	491.65	491.42
0.002	489.17	493.46	493.42	492.94	492.51	492.16	491.87
0.003	490.71	492.01	493.99	493.45	492.93	492.50	492.15
0.004	488.95	494.55	494.43	493.74	493.10	492.58	492.17
0.005	490.86	495.01	494.61	493.72	492.96	492.37	491.91
0.006	491.94	495.26	494.44	493.35	492.51	491.90	491.44
0.007	492.87	495.18	493.91	492.71	491.86	491.26	490.83
0.008	493.66	494.72	493.10	491.89	491.10	490.56	490.17
0.009	494.20	493.92	492.15	491.02	490.32	489.84	489.51

Turning to the estimation of approximations to general functional forms, Table 3 provides results for the power series expansion discussed above. The “general” cubic model shown in Column (1) incorporates contemporaneous and lagged values of three terms in the output gap: linear, quadratic, and cubic (higher-order terms were statistically insignificant). Compared with the pure linear model, this specification contains four additional free parameters. Taking twice the difference in the value of the log likelihood function for this model versus the linear alternative (Column (4)) yields a test statistic of 11.5 distributed ξ2(4): as the critical value at the 99 percent confidence level is 13.3, the linear model cannot be conclusively rejected in favor of this general alternative. However, the critical value at the 97.5 percent confidence level is 11.1, so the evidence against the linear model remains strong by classical standards. The estimated relationship between output gaps and inflation using this general nonlinear approximation is shown in the middle two panels in Chart 3—the shape of the function is similar to the estimated CMM relationship, supporting the view that the latter provides a useful characterization of the convexity.

Table 3.

Estimation Results with Quadratic and Cubic Functional Forms

(t-statistics in parentheses)

Estimated equation:

$\begin{array}{cc}{\mathrm{\pi }}_{\mathrm{t}}\hfill & \mathrm{=}\mathrm{\alpha }\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\lambda }\mathrm{)}\mathrm{(}\mathit{\delta }{\hat{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\delta }\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}\mathrm{\lambda }{\hat{\mathrm{\pi }}}^{\mathrm{a}}\hfill \\ & \mathrm{+}{\mathit{\tau }}_{\mathrm{1}}{\mathrm{\text{gap}}}_{\mathrm{t}}\mathrm{+}{\mathit{\tau }}_{\mathrm{2}}{\mathrm{\text{gap}}}_{\mathrm{\text{t}}}^{\mathrm{2}}\mathrm{+}{\mathit{\tau }}_3{\mathrm{\text{gap}}}_{\mathrm{\text{t}}}^{\mathrm{3}}\hfill \\ & \mathrm{+}{\mathit{\beta }}_{\mathrm{1}}{\mathrm{\text{gap}}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathit{\beta }}_{\mathrm{2}}{\mathrm{\text{gap}}}_{\mathrm{\text{t-1}}}^{\mathrm{2}}\mathrm{+}{\mathit{\beta }}_{\mathrm{3}}{\mathrm{\text{gap}}}_{\mathrm{\text{t-1}}}^{\mathrm{3}}\hfill \end{array}$

where: π = change in the log of the non-oil GDP deflator

π^a = change in the log of the absorption deflator

y = real GDP

y = trend real GDP

gap = log(y/y)

δ = weight on forward-looking component of inflation expectations

λ = weight on absorption deflator

α = constant term

Λ = fitted value from auxiliary regression (see text)

Data: Annual, 1967-91, pooled G-7 countries.

Table 3.

Estimation Results with Quadratic and Cubic Functional Forms

(t-statistics in parentheses)

Estimated equation:

$\begin{array}{cc}{\mathrm{\pi }}_{\mathrm{t}}\hfill & \mathrm{=}\mathrm{\alpha }\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\lambda }\mathrm{)}\mathrm{(}\mathit{\delta }{\hat{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\delta }\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}\mathrm{\lambda }{\hat{\mathrm{\pi }}}^{\mathrm{a}}\hfill \\ & \mathrm{+}{\mathit{\tau }}_{\mathrm{1}}{\mathrm{\text{gap}}}_{\mathrm{t}}\mathrm{+}{\mathit{\tau }}_{\mathrm{2}}{\mathrm{\text{gap}}}_{\mathrm{\text{t}}}^{\mathrm{2}}\mathrm{+}{\mathit{\tau }}_3{\mathrm{\text{gap}}}_{\mathrm{\text{t}}}^{\mathrm{3}}\hfill \\ & \mathrm{+}{\mathit{\beta }}_{\mathrm{1}}{\mathrm{\text{gap}}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathit{\beta }}_{\mathrm{2}}{\mathrm{\text{gap}}}_{\mathrm{\text{t-1}}}^{\mathrm{2}}\mathrm{+}{\mathit{\beta }}_{\mathrm{3}}{\mathrm{\text{gap}}}_{\mathrm{\text{t-1}}}^{\mathrm{3}}\hfill \end{array}$

where: π = change in the log of the non-oil GDP deflator

π^a = change in the log of the absorption deflator

y = real GDP

y = trend real GDP

gap = log(y/y)

δ = weight on forward-looking component of inflation expectations

λ = weight on absorption deflator

α = constant term

Λ = fitted value from auxiliary regression (see text)

Data: Annual, 1967-91, pooled G-7 countries.

	(1) General Cubic Model	(2) General Quadratic Model	(3) Constrained Cubic Model	(4) Linear Model
α	-0.004 (2.97)	-0.004 (3.33)	-0.001 (0.56)	-0.001 (0.67)
γ	0.187 (1.87)	0.188 (1.93)	0.300 (2.88)	0.257 (2.56)
δ	0.500 (7.24)	0.436 (7.63)	0.384 (5.46)	0.417 (6.19)
τ1	0.277 (2.09)	0.245 (3.51)	…	0.223 (2.97)
τ2	5.449 (1.73)	6.480 (2.19)	…	…
τ3	69.234 (0.53)	…	268.011 (3.09)	…
β1	0.148 (1.32)	0.281 (3.82)	…	0.181 (2.29)
β2	9.003 (3.05)	9.070 (3.17)	…	…
β3	171.451 (1.34)		276.458 (3.07)	…
LLF	495.02	494.29	489.46	489.27

View Table

Table 3.

Estimation Results with Quadratic and Cubic Functional Forms

(t-statistics in parentheses)

Estimated equation:

$\begin{array}{cc}{\mathrm{\pi }}_{\mathrm{t}}\hfill & \mathrm{=}\mathrm{\alpha }\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\lambda }\mathrm{)}\mathrm{(}\mathit{\delta }{\hat{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\delta }\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}\mathrm{\lambda }{\hat{\mathrm{\pi }}}^{\mathrm{a}}\hfill \\ & \mathrm{+}{\mathit{\tau }}_{\mathrm{1}}{\mathrm{\text{gap}}}_{\mathrm{t}}\mathrm{+}{\mathit{\tau }}_{\mathrm{2}}{\mathrm{\text{gap}}}_{\mathrm{\text{t}}}^{\mathrm{2}}\mathrm{+}{\mathit{\tau }}_3{\mathrm{\text{gap}}}_{\mathrm{\text{t}}}^{\mathrm{3}}\hfill \\ & \mathrm{+}{\mathit{\beta }}_{\mathrm{1}}{\mathrm{\text{gap}}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathit{\beta }}_{\mathrm{2}}{\mathrm{\text{gap}}}_{\mathrm{\text{t-1}}}^{\mathrm{2}}\mathrm{+}{\mathit{\beta }}_{\mathrm{3}}{\mathrm{\text{gap}}}_{\mathrm{\text{t-1}}}^{\mathrm{3}}\hfill \end{array}$

where: π = change in the log of the non-oil GDP deflator

π^a = change in the log of the absorption deflator

y = real GDP

y = trend real GDP

gap = log(y/y)

δ = weight on forward-looking component of inflation expectations

λ = weight on absorption deflator

α = constant term

Λ = fitted value from auxiliary regression (see text)

Data: Annual, 1967-91, pooled G-7 countries.

	(1) General Cubic Model	(2) General Quadratic Model	(3) Constrained Cubic Model	(4) Linear Model
α	-0.004 (2.97)	-0.004 (3.33)	-0.001 (0.56)	-0.001 (0.67)
γ	0.187 (1.87)	0.188 (1.93)	0.300 (2.88)	0.257 (2.56)
δ	0.500 (7.24)	0.436 (7.63)	0.384 (5.46)	0.417 (6.19)
τ1	0.277 (2.09)	0.245 (3.51)	…	0.223 (2.97)
τ2	5.449 (1.73)	6.480 (2.19)	…	…
τ3	69.234 (0.53)	…	268.011 (3.09)	…
β1	0.148 (1.32)	0.281 (3.82)	…	0.181 (2.29)
β2	9.003 (3.05)	9.070 (3.17)	…	…
β3	171.451 (1.34)		276.458 (3.07)	…
LLF	495.02	494.29	489.46	489.27

Setting the parameters on the cubic terms to zero yields the general quadratic model shown in Column (2) (these estimates are graphed in the lower panels of Chart 3). Comparing the LLF for this model with the general model indicates that the exclusion of the cubic terms implies only a small loss of fit. Not surprisingly, then, the evidence in favor of the quadratic model versus the linear model is stronger than that for the general cubic model: twice the difference in the LLF yields a test statistic of 10.0 distributed χ²(2), exceeding the critical level of 9.2 at the 99 percent confidence level.

One implication of these results using flexible approximations is that the rejection of the linear model using the CMM function is not due to idiosyncracies of that specification: the linear model is also rejected using more general nonlinear alternatives. At the same time, comparing the fit of the quadratic model with that of the CMM function indicates that the latter fits the data better with a more parsimonious specification. In addition, the CMM function does not present the conceptual problem discussed earlier: in some regions, the quadratic function implies that the downward effect on inflation declines, rather than increases, as excess supply rises (see bottom panels of Chart 3). For these reasons, we retain the CMM function as the preferred nonlinear specification.

Finally, Column (3) of Table 3 shows the results for the constrained cubic model with no linear or quadratic terms. The fit of this model is similar to that of the linear model, with the same number of free parameters. The marginal improvement in the fit of this cubic model—which is symmetric in terms of the effects of positive and negative output gaps—compared with the large improvement shown by other nonlinear models suggests that an important role is played by asymmetries, and not just nonlinearities, in the output-inflation relationship. This result, then, supports the view that the effect of positive output gaps on inflation exceeds that of negative gaps. ^23/

1. A small simulation model

As discussed above, the core of our model consists of an aggregate demand equation, a Phillips curve, and a policy reaction function. In order to examine the implications of nonlinear price adjustment for forward-looking asset prices, we also include an auxiliary equation that links the five-year bond yield to expected future one-year yields.

The aggregate demand equation determines the deviation of output from its supply-determined value (the latter is assumed to be exogenous). Our specification reflects two assumptions that are critical to the ability of policymakers to control the economy. First, there are assumed to be significant lags between changes in interest rates and their full effects on aggregate demand. Second, there is persistence in movements in the output gap, implying that shocks to aggregate demand propagate to future periods. These assumptions are important because they make the model economy more difficult to control than if the dynamics linking demand shocks, interest rates, and output were purely contemporaneous. ^24/

Specifically, the output equation is:

• where: gap = output gap,

  • rr = short-term real interest rate, (measured as a deviation from equilibrium)

  • ∊ = a stochastic disturbance to aggregate demand.

Rather than attempting to estimate equation (9) directly from observed annual data, the parameter values were obtained by simulating a quarterly equation of the relationship between the output gap and the short-term real interest rate using U.S. data. (Details are provided in Appendix II). ^25/ The quarterly responses were then converted to an annual frequency by time-averaging the simulation results obtained from a number of shocks to the real interest rate. These results were consistent with those obtained from simulations of the Federal Reserve Board’s quarterly MPS model of the U.S. economy. They show that, while there is some contemporaneous effect of real interest rates on output, most of the response occurs with a one-year lag. In addition, there is inertia in the output gap, as reflected in the coefficient of 0.304 on the lagged gap. Taken together, these coefficients imply that output would eventually fall about 0.6 percent below potential in the face of a persistent 100 basis points rise in short-term real interest rates above their equilibrium value.

The model is completed with equations that define the ex ante real rate, inflation expectations, and a policy rule. The definition of the ex ante real interest rate is:

• where: rs = short-term nominal interest rate,

  • π^e_t+1 = expected inflation in period t+1.

For inflation expectations, we rely on the estimates of the backward- and forward-looking components model implied by the estimated Phillips curve. This representation is used widely in policy simulation models. ^26/ For the CMM specification, this implied the following weights on the model-consistent solution for future inflation and lagged inflation: ^27/

In keeping with the traditional approach to implementing policy feedback rules in simulation models, our policy reaction function is represented by an interest rate rule: interest rates rise when inflation is above target and fall when it is below. To incorporate developments in real activity (which signal changes in the future inflation rate) in the reaction function, the gap between actual and potential output is also included. The resulting policy reaction function is slight generalization of the one employed extensively in model simulation work reported by Bryant, Hooper and Mann (1993). This basic rule has been shown to have desirable properties compared to other simple alternatives such as money control and fixed exchange rates in a wide class of macroeconomic models. The parameters were chosen to reestablish the initial level of inflation within three years following a shock to aggregate demand, without inducing ongoing cycles in the economy. In our model, experiments pointed to weights of 3.0 on the deviation of inflation from the target and 1.0 on the output gap:

• where: π* = inflation target,

  • rsbar = baseline interest rate.

In experiments where we want to delay the interest rate response of the authorities, the reaction function is temporarily “turned off” in the first year of the shock by exogenizing the level of the short-term interest rate. Of course, pegging the short-term interest rate is only possible in the short run; attempting to fix interest rates indefinitely in accelerationist models results in indeterminacy because no mechanism exists to bound future inflation expectations. In this sense, the monetary authority must eventually take actions to anchor longer-run inflation expectations.

Long-term rates are linked to short-term rates according to the expectations theory of the term structure. We could add a fixed term premium, but nothing is lost here by ignoring this complication. Thus, the long-term (five-year) rate is a geometric average of expected short-term rates:

where r1 is the long-term interest rate. The simulation model then has equations (9)-(13) and the Phillips curve estimated in the previous section. For convenience, these equations are repeated in Table 4.

Table 4.

A Small Simulation Model

Table 4.

A Small Simulation Model

Nonlinear Inflation Model:
	Phillips Curve: πct	$\begin{array}{l}{\mathrm{\pi }}_{\mathrm{t}}\mathit{=}{{\mathrm{\pi }}^{\mathrm{e}}}_{\mathrm{t}\mathit{+}\mathrm{1}}\mathit{+}\mathit{.196}\mathrm{f}\mathit{\left(}{\mathit{\text{gap}}}_{\mathrm{\text{t}}}\mathit{\right)}\mathit{+}\mathit{.276}\mathrm{\text{f}}\mathit{\left(}{\mathit{\text{gap}}}_{\mathrm{\text{t}}\mathit{-}\mathrm{1}}\mathit{\right)}\end{array}$
	Inflation Expectations:	${{\mathrm{\pi }}^{\mathrm{e}}}_{\mathrm{t}\mathit{+}\mathrm{1}}=\mathit{.414}{\mathrm{\pi }}_{\mathrm{t}+1}\mathit{+}(1-.414){\mathrm{\pi }}_{\mathrm{t}-1}$
	Convex Function:	$\mathrm{f}\left(\mathit{\text{gap}}\right)={\omega }^2/(\omega -\mathit{\text{ga}}p)-\omega ,\omega =.049$
	Real Interest Rate:	${rr}_t={rs}_t-{{\mathrm{\pi }}^{\mathrm{e}}}_{t+1}$
	Aggregate Demand Equation:	$\begin{array}{l}{\mathit{\text{gap}}}_{\mathrm{t}}=.304{\text{gap}}_{\mathrm{t}-1}-.098r_{\mathrm{t}}-.315r{r}_{\mathrm{t}-1}+{ϵ}^{\mathit{\text{gap}}}\end{array}$
	Long-Term Interest Rates:	$\mathit{r}{\mathit{1}}_{\mathit{t}}\mathit{=}\underset{\mathit{i}\mathit{=}\mathrm{0}}{\overset{\mathrm{5}}{\mathrm{\Pi }}}\mathit{(}\mathrm{1}\mathit{+}\mathit{r}{\mathit{s}}_{\mathit{t}\mathit{+}\mathrm{1}}\mathit{)}$
	Policy Reaction Function:	${\mathit{r}\mathit{s}}_{\mathit{t}}\mathrm{=}\mathrm{\lambda }{\overline{\mathit{r}\mathit{s}}}_{\mathit{t}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\lambda }\mathrm{)}\mathrm{3}\mathrm{(}{\mathrm{\pi }}_{\mathrm{t}}\mathrm{-}{\mathrm{\pi }}_{\mathrm{t}}^{\mathrm{*}}\mathrm{)}\mathrm{+}\mathrm{\lambda }{\mathit{\text{gap}}}_{\mathit{t}}$
	π = inflation	λ = variable to make rs exogenous
	gap = output gap (yd-ys)	rr = real interest rate
	rs = short-term interest rate	${\mathrm{\Pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{e}}$ = inflation expectations
	r1 = long-term interest rate	ω = wall capacity constraint
	Π* = inflation target
Estimated Model with a Linear Phillips Curve:
	Phillips Curve:	${\mathrm{\pi }}_{\mathrm{t}}\mathrm{=}{\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{e}}\mathrm{+}\mathrm{.223}{\mathit{\text{gap}}}_{\mathit{t}}\mathrm{+}\mathrm{.181}{\mathit{\text{gap}}}_{\mathit{t}\mathit{-}\mathrm{1}}$
Inflation Expectations:		${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{e}}\mathrm{=}\mathrm{.417}{\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{.417}\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}$

View Table

Table 4.

A Small Simulation Model

Nonlinear Inflation Model:
	Phillips Curve: πct	$\begin{array}{l}{\mathrm{\pi }}_{\mathrm{t}}\mathit{=}{{\mathrm{\pi }}^{\mathrm{e}}}_{\mathrm{t}\mathit{+}\mathrm{1}}\mathit{+}\mathit{.196}\mathrm{f}\mathit{\left(}{\mathit{\text{gap}}}_{\mathrm{\text{t}}}\mathit{\right)}\mathit{+}\mathit{.276}\mathrm{\text{f}}\mathit{\left(}{\mathit{\text{gap}}}_{\mathrm{\text{t}}\mathit{-}\mathrm{1}}\mathit{\right)}\end{array}$
	Inflation Expectations:	${{\mathrm{\pi }}^{\mathrm{e}}}_{\mathrm{t}\mathit{+}\mathrm{1}}=\mathit{.414}{\mathrm{\pi }}_{\mathrm{t}+1}\mathit{+}(1-.414){\mathrm{\pi }}_{\mathrm{t}-1}$
	Convex Function:	$\mathrm{f}\left(\mathit{\text{gap}}\right)={\omega }^2/(\omega -\mathit{\text{ga}}p)-\omega ,\omega =.049$
	Real Interest Rate:	${rr}_t={rs}_t-{{\mathrm{\pi }}^{\mathrm{e}}}_{t+1}$
	Aggregate Demand Equation:	$\begin{array}{l}{\mathit{\text{gap}}}_{\mathrm{t}}=.304{\text{gap}}_{\mathrm{t}-1}-.098r_{\mathrm{t}}-.315r{r}_{\mathrm{t}-1}+{ϵ}^{\mathit{\text{gap}}}\end{array}$
	Long-Term Interest Rates:	$\mathit{r}{\mathit{1}}_{\mathit{t}}\mathit{=}\underset{\mathit{i}\mathit{=}\mathrm{0}}{\overset{\mathrm{5}}{\mathrm{\Pi }}}\mathit{(}\mathrm{1}\mathit{+}\mathit{r}{\mathit{s}}_{\mathit{t}\mathit{+}\mathrm{1}}\mathit{)}$
	Policy Reaction Function:	${\mathit{r}\mathit{s}}_{\mathit{t}}\mathrm{=}\mathrm{\lambda }{\overline{\mathit{r}\mathit{s}}}_{\mathit{t}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\lambda }\mathrm{)}\mathrm{3}\mathrm{(}{\mathrm{\pi }}_{\mathrm{t}}\mathrm{-}{\mathrm{\pi }}_{\mathrm{t}}^{\mathrm{*}}\mathrm{)}\mathrm{+}\mathrm{\lambda }{\mathit{\text{gap}}}_{\mathit{t}}$
	π = inflation	λ = variable to make rs exogenous
	gap = output gap (yd-ys)	rr = real interest rate
	rs = short-term interest rate	${\mathrm{\Pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{e}}$ = inflation expectations
	r1 = long-term interest rate	ω = wall capacity constraint
	Π* = inflation target
Estimated Model with a Linear Phillips Curve:
	Phillips Curve:	${\mathrm{\pi }}_{\mathrm{t}}\mathrm{=}{\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{e}}\mathrm{+}\mathrm{.223}{\mathit{\text{gap}}}_{\mathit{t}}\mathrm{+}\mathrm{.181}{\mathit{\text{gap}}}_{\mathit{t}\mathit{-}\mathrm{1}}$
Inflation Expectations:		${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{e}}\mathrm{=}\mathrm{.417}{\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{.417}\mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}$