1. Implications of asymmetry in the output-inflation process

In order to understand the implications of convexity in the Phillips curve, it is useful to assume for illustrative purposes that the inflation process can be described by the following quadratic function:

where π is the rate of inflation, gap^* = y - y^*, and β, λ > 0. ^2/ y^* is the level of output at which there is no tendency for inflation to either rise or fall in the absence of shocks to the economy. However, in the presence of shocks to the system, y^* will not be equal to the observed average level of output, as shown below.

Taking expectations of both sides of equation (1) gives:

In a sustainable equilibrium with a constant rate of inflation and no systematic error in inflation expectations, E[π] = E[π^e]. It follows that:

Making use of the definition of the variance (VAR) of gap^*, equation (3) can be expressed as:

It follows that the expected value of gap^* is equal to:

As long as there is some convexity in the Phillips curve, i.e., λ 0, and there is some variance in output, then the average level of gap ^* will be negative. Recalling that gap^* = y -y^*, and representing the right-hand side of (5) by α, we have:

i.e., y - y^* = α, or y - y^* + α where α < 0. Thus the important implication of a nonlinear Phillips curve is that the average level of output lies below the level that would be observed in the absence of shocks. The economic intuition underlying this result is that if output were maintained, on average, equal to y^*, then the nonlinearity in the response of prices to aggregate demand shocks would make it impossible to maintain a constant inflation rate. Because deviations of demand above y^* have a larger positive effect on inflation than deviations of the same magnitude below y^* have in reducing inflation, an attempt to keep y = y^* would lead to an acceleration in inflation without bound. Therefore the only way that this outcome can be avoided is if the average level of output lies below y^*. This implies that relative to the notional level of output attainable in the absence of shocks, periods of disinflation below this level will be more severe or of longer duration than periods of excess demand above this level. ^1/

2. An asymmetric model of the U.S. output-inflation process

In this section we present a simple model of the U.S. inflation process that captures certain key features of the interactions linking excess demand, inflation and monetary policy. The model consists of two estimated behavioral equations, one describing a Phillips curve and the other aggregate demand which is specified in terms of the output gap. In order to close the model, a monetary policy reaction function is specified in which the monetary authorities are assumed to vary the short-term interest rate to achieve their output and inflation objectives. This stylized model highlights the inflation generating process as primarily dependent on excess demand, i.e., the output gap, and the role of the monetary authorities in influencing the rate of inflation by controlling the short-term interest rate and thereby aggregate demand. The monetary control mechanism is not perfect, however, as the economy is subject to shocks which cannot be foreseen. Moreover, the influence of monetary policy on aggregate demand through interest rates operates with a lag. Thus, given the existence of short-run capacity constraints, the characteristics of the monetary policy reaction function have important implications for the dynamic behavior and overall performance of the economy.

Although the existence of short-run capacity constraints suggests that there may be asymmetries in the Phillips curve, economic theory does not provide much guidance in terms of a functional form. Furthermore, even if the true functional form were known with certainty, there would still be substantial difficulties in identifying its parameters in small samples. Indeed, one reason why statisticians prefer linear models is that they are far less sensitive to outliers than nonlinear models. In Laxton, Meredith, and Rose (1994), this problem was overcome by using a pooled-time series cross-section data set for the G-7 countries to estimate a nonlinear function. In this paper, by contrast, we are focusing only on one country, the United States, and therefore the number of observations of cyclical periods of excess demand and supply is quite restricted. Given the limited number of business cycles in the U.S. data, our estimation strategy is to take a linear approximation of a general convex function in which positive gaps have larger effects on inflation than negative gaps. This involves adding a separate term for the gap^* when it is positive, as shown in equation (6) below:

where:

• ${\pi }_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$ = weighted average of expected inflation over the next four quarters gap

• gap^* = y - y^*

• y^* = y - α where α < 0

• y = average value of y

• gappos^* = positive values of gap^*.

When y>y^*, there is inflationary pressure over and above the contribution of the linear term (gap^*), which also measures the disinflationary pressure when y < y^*. This estimation strategy is meant to strike a better balance between the statistical robustness that one achieves from assuming linearity compared to the policy errors that would be induced from assuming global linearity, i.e., γ = 0. ^1/ A major advantage of this approach is that it is easy to test the restriction of linearity, even though the “kinked” function may not be the best choice for a policy simulation model.

While the specification of the effect of the gap on inflation has been linearized, it is important to note that the estimates of β and γ involve the simultaneous estimation of a, which, as noted above, is the difference between y and y^*. Thus our estimate of the output gap which is relevant for explaining inflation, which we denote by gap^*, is equal to the difference between the actual level of output and the notional upper limit on output, y^*, which could only be achieved in the absence of stochastic shocks to the economy. As noted below, if α is omitted from the estimation, the estimate of γ will be biased downwards. This underscores the fact that the proper specification of the measure of excess demand is essential in identifying the effect that asymmetry has on inflation.

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. ^2/ In our work, expected future inflation is proxied by the Michigan survey measure of inflation expectations. The estimated weights on past (1 - δ) and expected future inflation (δ) determine the relative importance of the “forward” and “backward” looking components of the inflation process. Imposing the constraint that these weights sum to unity ensures that no long-run trade-off exists between the level of inflation and excess demand pressures; it is in this sense that the natural rate hypothesis holds in our model.

The use of a survey measure of inflation expectations eliminates the problem of using fixed-parameter, reduced-form models and imposing inappropriate restrictions which arise when the typical procedure is used of regressing actual inflation on an information set known at the time expectations are formed. The survey measure, by contrast, can more flexibly accommodate the changing importance of different factors; in particular, it can incorporate more rapidly than other measures responses to changes in actual and prospective economic policies. In this way the survey measure would appear to be less subject to the Lucas critique, compared with other approaches that simply extrapolate in a mechanical fashion the impact of changes of variables in the past on expectations of future inflation.

There are numerous approaches to measuring the mean or trend level of output. We chose a simple and transparent approach, namely, the two-sided moving average of the logarithm of actual GDP which was used in Laxton, Meredith, and Rose (1994). In the estimation results shown in Table 1, we varied K, the number of quarters on each side of the center of the moving average, over a fairly wide range. ^1/ Table 1 shows that the goodness of fit is maximized at K = 12. It also shows the parameter estimates are not very sensitive to moderate variations in K. Also, in interpreting y as a measure of trend or capacity output, the fairly long moving-average process suggests that demand shocks--movements in the gap (y - y)--are more important for explaining the variation in observed output than movements in potential output.

Table 1.

A Simple Asymmetric Model of the U.S. Output-Inflation Process

(T-statistics in parentheses)

Table 1.

A Simple Asymmetric Model of the U.S. Output-Inflation Process

(T-statistics in parentheses)

Estimated
	equation:	${\pi }_t=\mathrm{\delta }{\overline{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}\text{\hspace{0.5em}+}(1-\delta \mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\text{\hspace{0.5em}+}\mathit{\beta }\mathrm{}{\text{gap}}_{\mathrm{t}}^{\mathrm{*}}\text{\hspace{0.5em}+}\mathrm{\gamma }{\mathrm{\text{gappos}}}_{\mathrm{t}}^{\mathrm{*}}\mathrm{}\text{\hspace{0.5em}+}{ϵ}_t^{\pi }$
where:		gap* = y - y* = y - y + α, gappos* = positive values of gap*
		${\overline{\mathrm{\pi }}}_{t+4}^e\text{\hspace{0.5em}=}.2\left({{}_t\mathrm{\pi }}_{t+4}^e\text{\hspace{0.5em}+}{{}_{t-1}\mathrm{\pi }}_{t+3}^e\text{\hspace{0.5em}+}{{}_{t-2}\mathrm{\pi }}_{t+2}^e\text{\hspace{0.5em}+}{{}_{t-3}\mathrm{\pi }}_{t+1}^e\text{\hspace{0.5em}+}{{}_{t-4}\mathrm{\pi }}_t^e\right)$
		π ≡ Percent change in the CPI at annual rates
		${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$ ≡ Michigan Survey Measure of Inflation Expectations
		$y\equiv \frac{1}{2k+1}[y_t+\stackrel{k}{\underset{i=1}{\mathrm{\Sigma }}}{y}_{t+1}+y_{t-t}]$
Data: U.S. Quarterly Data, 1964Q1-90Q4.
K		α	γ	β	δ	R2	σ	Wald Test: SL(α, γ = 0)
5		-0.354	0.878	0.458	0.566	.7776	1.6445	0.087
		(0.99)	(1.83)	(2.59)	(4.65)
6		-0.441	0.873	0.384	0.576	.7816	1.6296	0.032
		(1.23)	(2.19)	(2.64)	(4.91)
7		-0.547	0.813	0.337	0.582	.7821	1.6275	0.022
		(1.44)	(2.36)	(2.55)	(5.01)
8		-0.670	0.780	0.308	0.585	.7844	1.6192	0.015
		(1.73)	(2.64)	(2.61)	(5.20)
9		-0.770	0.758	0.282	0.586	.7850	1.6168	0.004
		(2.09)	(3.21)	(2.60)	(5.48)
10		-0.893	0.772	0.254	0.587	.7851	1.6164	0.010
		(2.72)	(3.00)	(2.52)	(5.87)
11		-1.149	0.918	0.218	0.591	.7873	1.6081	0.004
		(3.27)	(2.99)	(2.47)	(6.07)
12		-1.256	0.925	0.202	0.593	.7892	1.6010	0.001
		(3.66)	(3.16)	(2.43)	(6.13)
13		-1.369	0.916	0.189	0.600	.7890	1.6017	0.001
		(3.67)	(3.13)	(2.47)	(5.99)
14		-1.478	0.896	0.177	0.595	.7879	1.6058	0.007
		(3.13)	(2.72)	(2.43)	(5.88)
15		-1.661	0.919	0.166	0.593	.7862	1.6123	0.002
		(3.38)	(2.69)	(2.47)	(5.79)
16		-1.712	0.866	0.161	0.587	.7834	1.6229	0.010
		(3.02)	(2.56)	(2.49)	(5.58)

View Table

Table 1.

A Simple Asymmetric Model of the U.S. Output-Inflation Process

(T-statistics in parentheses)

Estimated
	equation:	${\pi }_t=\mathrm{\delta }{\overline{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}\text{\hspace{0.5em}+}(1-\delta \mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\text{\hspace{0.5em}+}\mathit{\beta }\mathrm{}{\text{gap}}_{\mathrm{t}}^{\mathrm{*}}\text{\hspace{0.5em}+}\mathrm{\gamma }{\mathrm{\text{gappos}}}_{\mathrm{t}}^{\mathrm{*}}\mathrm{}\text{\hspace{0.5em}+}{ϵ}_t^{\pi }$
where:		gap* = y - y* = y - y + α, gappos* = positive values of gap*
		${\overline{\mathrm{\pi }}}_{t+4}^e\text{\hspace{0.5em}=}.2\left({{}_t\mathrm{\pi }}_{t+4}^e\text{\hspace{0.5em}+}{{}_{t-1}\mathrm{\pi }}_{t+3}^e\text{\hspace{0.5em}+}{{}_{t-2}\mathrm{\pi }}_{t+2}^e\text{\hspace{0.5em}+}{{}_{t-3}\mathrm{\pi }}_{t+1}^e\text{\hspace{0.5em}+}{{}_{t-4}\mathrm{\pi }}_t^e\right)$
		π ≡ Percent change in the CPI at annual rates
		${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$ ≡ Michigan Survey Measure of Inflation Expectations
		$y\equiv \frac{1}{2k+1}[y_t+\stackrel{k}{\underset{i=1}{\mathrm{\Sigma }}}{y}_{t+1}+y_{t-t}]$
Data: U.S. Quarterly Data, 1964Q1-90Q4.
K		α	γ	β	δ	R2	σ	Wald Test: SL(α, γ = 0)
5		-0.354	0.878	0.458	0.566	.7776	1.6445	0.087
		(0.99)	(1.83)	(2.59)	(4.65)
6		-0.441	0.873	0.384	0.576	.7816	1.6296	0.032
		(1.23)	(2.19)	(2.64)	(4.91)
7		-0.547	0.813	0.337	0.582	.7821	1.6275	0.022
		(1.44)	(2.36)	(2.55)	(5.01)
8		-0.670	0.780	0.308	0.585	.7844	1.6192	0.015
		(1.73)	(2.64)	(2.61)	(5.20)
9		-0.770	0.758	0.282	0.586	.7850	1.6168	0.004
		(2.09)	(3.21)	(2.60)	(5.48)
10		-0.893	0.772	0.254	0.587	.7851	1.6164	0.010
		(2.72)	(3.00)	(2.52)	(5.87)
11		-1.149	0.918	0.218	0.591	.7873	1.6081	0.004
		(3.27)	(2.99)	(2.47)	(6.07)
12		-1.256	0.925	0.202	0.593	.7892	1.6010	0.001
		(3.66)	(3.16)	(2.43)	(6.13)
13		-1.369	0.916	0.189	0.600	.7890	1.6017	0.001
		(3.67)	(3.13)	(2.47)	(5.99)
14		-1.478	0.896	0.177	0.595	.7879	1.6058	0.007
		(3.13)	(2.72)	(2.43)	(5.88)
15		-1.661	0.919	0.166	0.593	.7862	1.6123	0.002
		(3.38)	(2.69)	(2.47)	(5.79)
16		-1.712	0.866	0.161	0.587	.7834	1.6229	0.010
		(3.02)	(2.56)	(2.49)	(5.58)

In terms of the estimation results themselves, in the preferred equation with K = 12, all the coefficients have the expected signs and are statistically significant. ^2/ In particular, the sum of the estimated coefficients β and γ imply a significantly larger impact of positive excess demand on inflation than a comparable degree of excess supply would have in reducing inflation, which is equal to β. ^3/ In addition, the estimate of α indicates a plausible and non-negligible difference of 1.3 percent between the y^* and y, i.e., the average or trend level of output lies 1.3 percent below the notional level of output that is achievable in the absence of shocks. As discussed in more detail below, α provides a measure of the extent to which the average level of output was reduced over the sample period by fluctuations in aggregate demand.

Chart 2 shows in the top panel the adjusted level of potential output, which, as noted above, lies 1.3 percent above the average level of output used as the measure of potential output in Chart 1. The middle panel plots our measure of inflation expectations, the Michigan Survey, and actual inflation. It is noteworthy that following the high-inflation period of the 1970s and early 1980s, expected inflation typically exceeded actual inflation in subsequent years. The bottom panel plots the difference between actual and expected inflation, as well as the adjusted output gap, i.e., gap^*. It shows that after taking account of inflation expectations, actual inflation tends to rise (sometimes with a lag) when the output gap is positive and to fall when the output gap is negative. Because of the asymmetry in the Phillips curve, periods of excess supply (a negative output gap) are more persistent and severe than periods of excess demand. This is most noticeable in the 1980s and early 1990s.

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Inflation, Inflation Expectations, and Output: 1964q1-94q4

Citation: IMF Working Papers 1995, 075; 10.5089/9781451849677.001.A001

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As already noted above, the ability to obtain reasonable estimate of γ depends crucially on specifying the output gap appropriately to include a joint estimate of α. This intrinsic relationship can be demonstrated by estimating the equation (6) but setting α = 0. The results are shown as equation (1b) in Table 2. Ignoring a in the equation results in a misspecification of the output gap and the estimated coefficient of the positive value of the gap, γ, is only 0.13 and is not significantly different from zero. This shows that finding evidence of asymmetry in the output-inflation process requires careful attention to modeling the manner in which excess demand gets translated into upward price movements. Ignoring the a shift in a test for convexity is tantamount to ignoring a key implication of the hypothesis being tested. Our results show clearly that this has important implications for the power of econometric identification of the parameters and the test for convexity itself. ^1/

Table 2.

Tests with Alternative Measures of the Output Gap

(T-statistics in parentheses)

Table 2.

Tests with Alternative Measures of the Output Gap

(T-statistics in parentheses)

Estimated
	equation:	${\pi }_t=\mathrm{\delta }{\overline{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}\text{\hspace{0.5em}+}(1-\delta \mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\text{\hspace{0.5em}+}\mathit{\beta }\mathrm{}{\text{gap}}_{\mathrm{t}}^{\mathrm{*}}\text{\hspace{0.5em}+}\mathrm{\gamma }{\mathrm{\text{gappos}}}_{\mathrm{t}}^{\mathrm{*}}\mathrm{}\text{\hspace{0.5em}+}{ϵ}_t^{\pi }$
Where:		gap* = y - y* = y - y + α, gappos* = positive values of gap*
		${\overline{\mathrm{\pi }}}_{t+4}^e\text{\hspace{0.5em}=}.2\left({{}_t\mathrm{\pi }}_{t+4}^e\text{\hspace{0.5em}+}{{}_{t-1}\mathrm{\pi }}_{t+3}^e\text{\hspace{0.5em}+}{{}_{t-2}\mathrm{\pi }}_{t+2}^e\text{\hspace{0.5em}+}{{}_{t-3}\mathrm{\pi }}_{t+1}^e\text{\hspace{0.5em}+}{{}_{t-4}\mathrm{\pi }}_t^e\right)$
		π ≡ Percent change in the CPI at annual rates
		${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$ Michigan Survey Measure of Inflation Expectations
Model #1: Case of k = 12 from Table 1
Model #2: Harvey and Jaeger (1993)’s Trend Plus Cycle Model
Model #3: Hodrick-Prescot Filter with λ=1600
Model #4: Hodrick-Prescot Filter with λ=5000
Model #5: Hodrick-Prescot Filter with λ=10000
Model #6: Hodrick-Prescot Filter with λ=100000
Model #7: Quadratic Time Trend (SMPL 1961Q1-94Q4)
Inflation equation estimation period: SMPL 1964Q1-90Q4
Mod.		α	γ	β	δ	R2	σ	Wald Test SL(α, γ = 0)
1a		-1.256 (3.66)	0.925 (3.16)	0.202 (2.43)	0.593 (6.13)	.7892	1.6010	0.001
1b		0.000	0.134 (0.79)	0.326 (3.14)	0.557 (5.33)	.7761	1.6421
2a		-0.791 (2.48)	1.092 (2.80)	0.281 (2.40)	0.611 (6.18)	7927	1.5877	0.018
2b		0.000	0.331 (1.52)	0.403 (2.75)	0.583 (5.58)	.7849	1.6094
3a		-0.848 (2.34)	0.943 (3.05)	0.248 (2.46)	0.611 (5.88)	.7927	1.5874	0.008
3b		0.000	0.283 (1.52)	0.344 (2.94)	0.581 (5.40)	.7827	1.6175
4a		-1.326 (4.02)	1.073 (3.17)	0.192 (2.52)	0.600 (5.44)	.7874	1.6077	0.000
4b		0.000	0.194 (1.12)	0.311 (3.36)	0.568 (5.12)	.7778	1.6358
5a		-1.498 (4.28)	1.081 (3.03)	0.175 (2.67)	0.589 (5.25)	.7833	1.6231	0.000
5b		0.000	0.148 (0.88)	0.298 (3.63)	0.558 (5.00)	.7745	1.6477
6a		-0.506 (0.53)	0.176 (0.70)	0.226 (2.71)	0.530 (4.69)	.7652	1.6895	0.782
6b		0.000	0.046 (0.29)	0.266 (4.00)	0.528 (4.62)	.7649	1.6826
7a		1.458 (1.74)	-0.205 (1.57)	0.335 (3.28)	0.515 (4.15)	.7581	1.7148	0.160
7b		0.000	-0.013 (0.10)	0.222 (3.86)	0.512 (4.29)	.7567	1.7117

View Table

Table 2.

Tests with Alternative Measures of the Output Gap

(T-statistics in parentheses)

Estimated
	equation:	${\pi }_t=\mathrm{\delta }{\overline{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}\text{\hspace{0.5em}+}(1-\delta \mathrm{)}{\mathrm{\pi }}_{\mathrm{t}\mathrm{-}\mathrm{1}}\text{\hspace{0.5em}+}\mathit{\beta }\mathrm{}{\text{gap}}_{\mathrm{t}}^{\mathrm{*}}\text{\hspace{0.5em}+}\mathrm{\gamma }{\mathrm{\text{gappos}}}_{\mathrm{t}}^{\mathrm{*}}\mathrm{}\text{\hspace{0.5em}+}{ϵ}_t^{\pi }$
Where:		gap* = y - y* = y - y + α, gappos* = positive values of gap*
		${\overline{\mathrm{\pi }}}_{t+4}^e\text{\hspace{0.5em}=}.2\left({{}_t\mathrm{\pi }}_{t+4}^e\text{\hspace{0.5em}+}{{}_{t-1}\mathrm{\pi }}_{t+3}^e\text{\hspace{0.5em}+}{{}_{t-2}\mathrm{\pi }}_{t+2}^e\text{\hspace{0.5em}+}{{}_{t-3}\mathrm{\pi }}_{t+1}^e\text{\hspace{0.5em}+}{{}_{t-4}\mathrm{\pi }}_t^e\right)$
		π ≡ Percent change in the CPI at annual rates
		${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$ Michigan Survey Measure of Inflation Expectations
Model #1: Case of k = 12 from Table 1
Model #2: Harvey and Jaeger (1993)’s Trend Plus Cycle Model
Model #3: Hodrick-Prescot Filter with λ=1600
Model #4: Hodrick-Prescot Filter with λ=5000
Model #5: Hodrick-Prescot Filter with λ=10000
Model #6: Hodrick-Prescot Filter with λ=100000
Model #7: Quadratic Time Trend (SMPL 1961Q1-94Q4)
Inflation equation estimation period: SMPL 1964Q1-90Q4
Mod.		α	γ	β	δ	R2	σ	Wald Test SL(α, γ = 0)
1a		-1.256 (3.66)	0.925 (3.16)	0.202 (2.43)	0.593 (6.13)	.7892	1.6010	0.001
1b		0.000	0.134 (0.79)	0.326 (3.14)	0.557 (5.33)	.7761	1.6421
2a		-0.791 (2.48)	1.092 (2.80)	0.281 (2.40)	0.611 (6.18)	7927	1.5877	0.018
2b		0.000	0.331 (1.52)	0.403 (2.75)	0.583 (5.58)	.7849	1.6094
3a		-0.848 (2.34)	0.943 (3.05)	0.248 (2.46)	0.611 (5.88)	.7927	1.5874	0.008
3b		0.000	0.283 (1.52)	0.344 (2.94)	0.581 (5.40)	.7827	1.6175
4a		-1.326 (4.02)	1.073 (3.17)	0.192 (2.52)	0.600 (5.44)	.7874	1.6077	0.000
4b		0.000	0.194 (1.12)	0.311 (3.36)	0.568 (5.12)	.7778	1.6358
5a		-1.498 (4.28)	1.081 (3.03)	0.175 (2.67)	0.589 (5.25)	.7833	1.6231	0.000
5b		0.000	0.148 (0.88)	0.298 (3.63)	0.558 (5.00)	.7745	1.6477
6a		-0.506 (0.53)	0.176 (0.70)	0.226 (2.71)	0.530 (4.69)	.7652	1.6895	0.782
6b		0.000	0.046 (0.29)	0.266 (4.00)	0.528 (4.62)	.7649	1.6826
7a		1.458 (1.74)	-0.205 (1.57)	0.335 (3.28)	0.515 (4.15)	.7581	1.7148	0.160
7b		0.000	-0.013 (0.10)	0.222 (3.86)	0.512 (4.29)	.7567	1.7117

Table 2 also presents some estimation results when we use alternative measures of the output gap. These include the Hodrick-Prescott filter under a number of alternative assumptions regarding the curvature restriction. Since ad hoc filters such as the Hodrick-Prescott filter or the simple two-sided moving average filter employed in Table 1 are in general suboptimal for developing trend estimates of output, one can use more sophisticated models which can be optimized to explain the time series properties of U.S. output. Indeed, Harvey and Jaeger (1993) show that estimates from the Hodrick-Prescott filter will only be optimal--in the class of univariate filters--under very specific parameterizations. Their unobserved components (UC) model of trend-plus-cycle is quite flexible and can be thought of as a simple reduced-form of a more general dynamic stochastic structural model of the U.S. economy. They argue that their simple UC model has several advantages over simple time trends or segmented time trends. Using several series as examples, they also use the UC model to show how the H-P filter can result in biased estimates of the cyclical component and can even induce spurious correlations if two series are detrended with the H-P filter. Despite this general argument--which is true in principle--they show that at least for U.S. GNP the UC model produces estimates that are very similar to the HP1600 estimates.

Table 2 reports estimates of the inflation model which are based on these alternative estimates of the output gap. Again, to give some indication of the potential bias associated with using unadjusted measures of the output gap, we also report results where α is imposed to be equal to zero. Equation (2a) reports the coefficients using the UC model of Harvey-Jaeger to construct the output gap. It is interesting that the fit of the model is slightly better than the best fit which was obtained when we varied the horizon in the simple two-sided filter to maximize the fit of the inflation equation, with both γ and β are slightly larger than in equation (1a). All the parameters are statistically significantly different from zero and a Wald test indicates that we can reject the joint restriction that α, γ = 0 with considerable confidence. Equation 2b reports the results when α is imposed to be equal to zero; in this case the estimate of the coefficient on the positive output gaps (γ) falls from 1.092 in the unrestricted model to 0.331. In addition, the coefficient on the output gap (β) increases from 0.281 in the unrestricted model to 0.403 in the restricted model.

Equation 3a reports the results when we use the HP1600 filter to measure the output gaps. The results confirm the earlier indication that these gaps are not significantly different from the gaps produced by the UC model. Indeed, the model’s parameter estimates are almost identical to Equation 2a. Again, the case for asymmetries weakens when we impose α = 0 in equation (3b).

In general, all univariate techniques can be expected to produce very uncertain estimates of the true relative variance of the gap term. Consequently, it may be reasonable to presume that potential output is smoother than what is implied by the HP1600 or the UC model. To test the sensitivity of our results to this assumption we reestimated our model with larger weights on the curvature part of the HP penalty function. In the limit, an extremely high weight implies that potential output could be proxied with a linear time trend. Equation (4a) - (6a) provide additional estimates when λ is set equal to 5,000, 10,000 and 100,000 respectively. Note that the fit of the inflation equation deteriorates monotonically as larger values of λ are imposed. Thus, a view that supply shocks are irrelevant for changing the growth rate of potential output is inconsistent with explaining both the time series properties of both U.S. output and inflation.

Roberts (1994) has estimated inflation models which use quadratic time trend approximations to measure potential output. These estimates are reported in equation (7a) of Table 2. When we estimate our model with these gaps, we obtain a positive estimate of α and a negative estimate of γ. This result is very similar to the finding of Eisner (1994) and has the exact opposite policy implications of short-run capacity constraints. However, this particular model has the worst fit of any of the other models.

The next stage in constructing the model involves estimating the link between the instrument controlled by the monetary authorities, the short-term interest rate, and aggregate demand. We follow the approach recently described by Fuhrer and Moore (1995), namely, we estimate an equation for the output gap as a function of lagged values of the gap and the real short-term interest rate. We do not attempt to identify any nonlinearities on the demand side; consequently, we assume that the aggregate demand curve is linear (in logs) and the gap is the conventional measure of y - y, where y is the 25 quarter centered moving average. The aggregate demand equation determines the deviation of output from its supply-determined value which is assumed to be exogenous. Our specification reflects two stylized facts that are critical to the ability of policymakers to control the economy. First, there are presumed 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 features are important because they make the economy more difficult to control than if the dynamics linking demand shocks, interest rates and output were purely contemporaneous.

In estimating the response of aggregate demand to interest rates, a choice must be made between a short-term versus a long-term measure. A longer maturity would clearly be most relevant for such interest-sensitive expenditures as residential investment and business-fixed investment. A medium-term maturity, e.g., three to five years, would be appropriate for consumer durables, especially automobiles. However, we have chosen to use the nominal interest rate controlled directly by the Federal Reserve, namely, the Federal funds rate. This simplifies the model by avoiding the need to specify a term-structure equation, but there is a cost in terms of introducing a specification error. ^1/ This nominal rate is converted to a real interest rate by subtracting the one-year ahead inflation expectations as measured by the Michigan Survey.

The estimation results are reported in Table 3. The results of the estimated regression with two lags on the gap and eight on real interest rates are shown in column (1). The only significant interest rate effects show up with a lag of two quarters. Indeed, if we constrain all other interest rate coefficients to zero we end up with a parsimonious representation of the monetary transmission mechanism shown in column (2). We chose to estimate this equation over a shorter sample than that used for the Phillips curve because the results over the entire sample, shown in column (3), gave what appear to us to be an implausibly low estimate of the interest elasticity of aggregate demand. The shorter sample was chosen to avoid including the effect of the rise in real interest rates between the 1960s and the 1980s due to the substantial increase in government debt in the latter period. As our aim is to isolate the effect on aggregate demand of changes in real interest rates caused by monetary policy actions, we wished to exclude possible effects arising from fiscal policy. It should be noted that using the estimated coefficients from the equation in column (3) do not change the qualitative results below, but the variation in the interest rate is, of course, much larger.

Table 3:

Estimation Results for Quarterly U.S. Output Gap Equation

(T-statistics in parentheses)

Table 3:

Estimation Results for Quarterly U.S. Output Gap Equation

(T-statistics in parentheses)

Estimated
	equation:		${\text{gap}}_{\mathrm{t}}\text{\hspace{0.5em}=}{\alpha }_0\text{\hspace{0.5em}+}{\mathrm{\gamma }}_{\mathrm{1}\mathrm{}}{\text{gap}}_{\mathrm{t}\mathrm{-}\mathrm{1}}+{\mathrm{\gamma }}_{\mathrm{2}}{\text{gap}}_{\mathrm{t}\mathrm{-}\mathrm{2}}\text{\hspace{0.5em}+}{\mathrm{\beta }}_1{\mathit{\text{\hspace{0.5em}rr}}}_{t-1}\text{\hspace{0.5em}+}\mathrm{.}\mathrm{.}\mathrm{.}\text{\hspace{0.5em}+}{\mathrm{\beta }}_8{\mathit{\text{\hspace{0.5em}rr}}}_{t-8}\text{\hspace{0.5em}+}{ϵ}_t^{gap}$
where:			α0 ≡ constant term
			gap ≡ 100*(y - y)
			y ≡ log of real GDP
			y ≡ 25-quarter centered moving average estimate of trend GDP
			rrt ≡ real interest rate (rst - ${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$)
			rst ≡ federal funds rate
			${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$ ≡ one-year-ahead measure of inflation expectations taken from the Michigan Survey data
			ct = disturbance term
Quarterly U.S. Data				#1: 1978Q1-91Q4		#2:1978Q1-91Q4		#3: 1964Q1-91Q4
α				0.513	(1.78)	0.579	(3.06)	0.228	(2.16)
τ 1				1.023	(7.00)	1.074	(8.72)	1.110	(12.21)
τ 2				-0.240	(-1.60)	-0.290	(-2.45)	-0.268	(2.97)
β 1				0.049	(0.57)	-0.157	(-3.61)	-0.095	(-2.95)
β 2				-0.181	(-1.71)
β 3				-0.004	(-0.03)
β 4				0.042	(0.42)
β 5				-0.058	(-0.59)
β 6				-0.101	(-1.02)
β 7				-0.035	(0.34)
β 8				0.082	(1.03)
Standard Error					0.776		0.756		0.825
R2					0.846		0.854		0.832

View Table

Table 3:

Estimation Results for Quarterly U.S. Output Gap Equation

(T-statistics in parentheses)

Estimated
	equation:		${\text{gap}}_{\mathrm{t}}\text{\hspace{0.5em}=}{\alpha }_0\text{\hspace{0.5em}+}{\mathrm{\gamma }}_{\mathrm{1}\mathrm{}}{\text{gap}}_{\mathrm{t}\mathrm{-}\mathrm{1}}+{\mathrm{\gamma }}_{\mathrm{2}}{\text{gap}}_{\mathrm{t}\mathrm{-}\mathrm{2}}\text{\hspace{0.5em}+}{\mathrm{\beta }}_1{\mathit{\text{\hspace{0.5em}rr}}}_{t-1}\text{\hspace{0.5em}+}\mathrm{.}\mathrm{.}\mathrm{.}\text{\hspace{0.5em}+}{\mathrm{\beta }}_8{\mathit{\text{\hspace{0.5em}rr}}}_{t-8}\text{\hspace{0.5em}+}{ϵ}_t^{gap}$
where:			α0 ≡ constant term
			gap ≡ 100*(y - y)
			y ≡ log of real GDP
			y ≡ 25-quarter centered moving average estimate of trend GDP
			rrt ≡ real interest rate (rst - ${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$)
			rst ≡ federal funds rate
			${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$ ≡ one-year-ahead measure of inflation expectations taken from the Michigan Survey data
			ct = disturbance term
Quarterly U.S. Data				#1: 1978Q1-91Q4		#2:1978Q1-91Q4		#3: 1964Q1-91Q4
α				0.513	(1.78)	0.579	(3.06)	0.228	(2.16)
τ 1				1.023	(7.00)	1.074	(8.72)	1.110	(12.21)
τ 2				-0.240	(-1.60)	-0.290	(-2.45)	-0.268	(2.97)
β 1				0.049	(0.57)	-0.157	(-3.61)	-0.095	(-2.95)
β 2				-0.181	(-1.71)
β 3				-0.004	(-0.03)
β 4				0.042	(0.42)
β 5				-0.058	(-0.59)
β 6				-0.101	(-1.02)
β 7				-0.035	(0.34)
β 8				0.082	(1.03)
Standard Error					0.776		0.756		0.825
R2					0.846		0.854		0.832

To compare the estimated quarterly models more formally, and also to compare them with structural models, we conducted some simple simulations. Table 4 reports the effects on real GDP for a temporary monetary-induced 100 basis points increase in the Federal funds rate derived from two versions of the MPS model, as well as the results for the same experiment conducted using Roberts’ reduced-form equation. ^1/ The results for the MPS model and Robert’s (1994) model are for a temporary four-quarter increase of 100 basis points in the Federal funds rate. The results for our equation are for the same shock applied to the real short-term interest rate. For this reason our results tend to be smaller because we are holding inflation expectations fixed. In other words, a monetary-induced shock to the nominal interest rate implies lower inflation expectations and a growing shock to the real interest rate.

Table 4.

Lags in the Transmission Mechanism in the MPS and Simple Reduced-Form Models

(Percent response of real GDP)

^1/Results based on the reduced-form equation reported in column (3) of Table 1 in Roberts (1994).

^2/Results based on the reduced-form equation reported in column (2) of Table 3.

Table 4.

Lags in the Transmission Mechanism in the MPS and Simple Reduced-Form Models

(Percent response of real GDP)

Quarter	Federal Reserve Board’s MPS Model		Reduced-form Models
	Version #1	Version #2	Roberts (1994) 1/	Our Equation 2/
0	0.0	0.0	0.0	0.0
1	0.0	0.0	0.0	0.0
2	-0.1	-0.1	-0.3	-0.2
3	-0.2	-0.1	-0.4	-0.3
4	-0.3	-0.2	-0.5	-0.4
5	-0.4	-0.2	-0.7	-0.4
6	-0.5	-0.3	-0.5	-0.3
7	-0.6	-0.3	-0.5	-0.3
8	-0.7	-0.3	-0.5	-0.2

^1/Results based on the reduced-form equation reported in column (3) of Table 1 in Roberts (1994).

^2/Results based on the reduced-form equation reported in column (2) of Table 3.

View Table

Table 4.

Lags in the Transmission Mechanism in the MPS and Simple Reduced-Form Models

(Percent response of real GDP)

Quarter	Federal Reserve Board’s MPS Model		Reduced-form Models
	Version #1	Version #2	Roberts (1994) 1/	Our Equation 2/
0	0.0	0.0	0.0	0.0
1	0.0	0.0	0.0	0.0
2	-0.1	-0.1	-0.3	-0.2
3	-0.2	-0.1	-0.4	-0.3
4	-0.3	-0.2	-0.5	-0.4
5	-0.4	-0.2	-0.7	-0.4
6	-0.5	-0.3	-0.5	-0.3
7	-0.6	-0.3	-0.5	-0.3
8	-0.7	-0.3	-0.5	-0.2

^1/Results based on the reduced-form equation reported in column (3) of Table 1 in Roberts (1994).

^2/Results based on the reduced-form equation reported in column (2) of Table 3.

We have included results for two versions of the MPS model because there is some uncertainty about the strength of real-financial linkages in the current version of the model. The results for the first version are obtained from a full-model simulation of the standard model. The results for the second version are obtained by excluding the effects of the price-earnings ratio on investment, as there is some disagreement as to whether these effects are too large in the standard model. Roberts’ reduced-form equation produces estimates that fall between the two MPS estimates. In terms of timing, all four models suggest that there are significant lags in the monetary transmission mechanism.

The complete model is described in Table 5. The only new element is the monetary policy reaction function. 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. As the policymaker also is assumed to give some weight to developments in real activity, the gap between actual and potential output is included in the reaction function. The resulting policy reaction function is a 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 two years following a shock to aggregate demand. Experiments using this model indicated that weights of 2.0 on the deviation of inflation from the target and 1.0 on the output gap would bring inflation back to control roughly within this time span.

Table 5.

A Small Simulation Model of the U.S. Output-Inflation Process

Table 5.

A Small Simulation Model of the U.S. Output-Inflation Process

Phillips Curve:
	${\mathrm{\pi }}_t=.593{\overline{\mathrm{\pi }}}_{t+4}^e\text{\hspace{0.5em}+}(1-.593){\mathrm{\pi }}_{t-1}\text{\hspace{0.5em}+}.202ga{p}^{{*}_t}\text{\hspace{0.5em}+}.925gappo{s}_t^{*}\text{\hspace{0.5em}+}{\mathrm{\varepsilon }}_t^{\mathrm{\pi }}$
	${\overline{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}\mathrm{=}\mathrm{.2}\mathrm{(}{{}_t\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}\mathrm{+}{{}_{t-1}\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{3}}^{\mathrm{e}}\text{\hspace{0.5em}+}{{}_{t-2}\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{2}}^{\mathrm{e}}\text{\hspace{0.5em}+}{{}_{t-3}\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{e}}\text{\hspace{0.5em}+}{{}_{t-4}\mathrm{\pi }}_{\mathrm{t}}^{\mathrm{e}}\mathrm{)}$
Real Interest Rate:
	$r{r}_t=r{s}_t-{\mathrm{\pi }}_{t+4}^e$
Inflation and Inflation Expectations:
	${\mathrm{\pi }}_t=\left[{\left(P_t/P_{t-1}\right)}^4-1\right]*100,{\mathrm{\pi }}_{t+4}^e=\left(P_{t+4}/P_t-1\right)*100$
Aggregate Demand Equation:
	$ga{p}_t\text{\hspace{0.5em}=}1.074ga{p}_{t-1}\text{\hspace{0.5em}-}.290ga{p}_{t-2}\text{\hspace{0.5em}-}.157r{r}_{t-2}+{\varepsilon }_t^{gap}$
Forward-Looking Policy Reaction Function:
	$r{s}_t-{\mathrm{\pi }}_{t+4}^e=2\mathrm{\left(}{\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{3}}-{\mathrm{\pi }}^{*}\right)+ga{p}_t$
Myopic Policy Reaction Function:
	$r{s}_t-{\mathrm{\pi }}_{t+4}^e=2\left({\mathrm{\pi }}_t-{\mathrm{\pi }}^{*}\right)+ga{p}_t$
π ≡ CPI inflation at annual rates
gap ≡ output gap (y-y)
gap* ≡ y - y* = y - y + α
rr ≡ real interest rate
rs ≡ Federal funds rate
${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$ ≡ one-year-ahead inflation expectations
π* ≡ inflation target
Pt ≡ price level

View Table

Table 5.

A Small Simulation Model of the U.S. Output-Inflation Process

Phillips Curve:
	${\mathrm{\pi }}_t=.593{\overline{\mathrm{\pi }}}_{t+4}^e\text{\hspace{0.5em}+}(1-.593){\mathrm{\pi }}_{t-1}\text{\hspace{0.5em}+}.202ga{p}^{{*}_t}\text{\hspace{0.5em}+}.925gappo{s}_t^{*}\text{\hspace{0.5em}+}{\mathrm{\varepsilon }}_t^{\mathrm{\pi }}$
	${\overline{\mathrm{\pi }}}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}\mathrm{=}\mathrm{.2}\mathrm{(}{{}_t\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}\mathrm{+}{{}_{t-1}\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{3}}^{\mathrm{e}}\text{\hspace{0.5em}+}{{}_{t-2}\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{2}}^{\mathrm{e}}\text{\hspace{0.5em}+}{{}_{t-3}\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{e}}\text{\hspace{0.5em}+}{{}_{t-4}\mathrm{\pi }}_{\mathrm{t}}^{\mathrm{e}}\mathrm{)}$
Real Interest Rate:
	$r{r}_t=r{s}_t-{\mathrm{\pi }}_{t+4}^e$
Inflation and Inflation Expectations:
	${\mathrm{\pi }}_t=\left[{\left(P_t/P_{t-1}\right)}^4-1\right]*100,{\mathrm{\pi }}_{t+4}^e=\left(P_{t+4}/P_t-1\right)*100$
Aggregate Demand Equation:
	$ga{p}_t\text{\hspace{0.5em}=}1.074ga{p}_{t-1}\text{\hspace{0.5em}-}.290ga{p}_{t-2}\text{\hspace{0.5em}-}.157r{r}_{t-2}+{\varepsilon }_t^{gap}$
Forward-Looking Policy Reaction Function:
	$r{s}_t-{\mathrm{\pi }}_{t+4}^e=2\mathrm{\left(}{\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{3}}-{\mathrm{\pi }}^{*}\right)+ga{p}_t$
Myopic Policy Reaction Function:
	$r{s}_t-{\mathrm{\pi }}_{t+4}^e=2\left({\mathrm{\pi }}_t-{\mathrm{\pi }}^{*}\right)+ga{p}_t$
π ≡ CPI inflation at annual rates
gap ≡ output gap (y-y)
gap* ≡ y - y* = y - y + α
rr ≡ real interest rate
rs ≡ Federal funds rate
${\mathrm{\pi }}_{\mathrm{t}\mathrm{+}\mathrm{4}}^{\mathrm{e}}$ ≡ one-year-ahead inflation expectations
π* ≡ inflation target
Pt ≡ price level