The Basic Linear Model

The essential notion of the Phillips curve is that inflation is driven by inflation expectations and the output gap. The simplest form of the model can be written as

where π is inflation, π^e is expected inflation, gap is the output gap (or the unemployment gap), and ∈^π represents a stochastic disturbance term.⁷ The disturbance term also includes special factors (other than aggregate excess demand conditions and expectations) that can affect the inflation process from time to time. This could include, for example, cost-push factors such as an exogenous increase in wage demands or a change in the price of crude oil.

equation (1) is often motivated by appealing to costly price level adjustment, as in the models of Calvo (1983) or Rotemberg (1987), for example. Strictly speaking, however, these models consider price levels and not inflation rates. The simple theoretical framework has been extended by some authors (e.g., Cozier (1989)) by assuming that the changein prices is costly to adjust. In our view, however, Buiter and Miller (1985) and Roberts (1994c) describe this appropriately when they refer to it as “slipping a derivative” into the firm’s optimization problem.

The simple form of the model in equation (1) implies that the main source of inertia in inflation dynamics is rigidities in expectations. If combined with an assumption of model-consistent or rational expectations, the model implies that inflation can, in principle, be adjusted without cost. In other words, an announced reduction in the target inflation rate can be achieved without any output costs, as long as the expected inflation term in equation (1) moves one-for-one with the announced change in the target. For this to be so, however, one needs more than simple consistency with the predictions of a dynamic model; full credibility of the announced policy change is necessary.

Experience suggests that it is impossible to reduce inflation without creating a negative output gap. This can be rationalized through either one of two simple extensions to the above discussion. One extension adds the notion of intrinsic dynamics to equation (1), as would be suggested by a model with costly price adjustment. The sources of intrinsic dynamics could also take the form of overlapping wage contracts (e.g., Taylor (1980)). In either case, this would imply that an additional term in the lag(s) of inflation would have to be added to equation (1). The second extension focuses on how expectations are formed. If there is a backward-looking element to expectations formation, and we reinterpret expectations in equation (1) as the model-consistent component of overall expectations, then another rationalization of additional lags emerges. In other words, additional lags in equation (1) can arise either from intrinsic elements of inflation dynamics, which are independent of expectations, or from inertia in expectations formation itself (a backward-looking component), or both.⁸

Although there are differing interpretations of precisely why, there seems to be reasonably broad agreement among researchers working in this area that the simple model must be extended to account for inertia over and above what is possible in equation (1).⁹ The resulting model is often referred to as the “backward- and forward-looking components” model (Buiter and Miller (1985)). In this model, inflation in any time period is tied down, at least partially, by historical conditions (the backward-looking component), while the forward-looking component responds to new information about the future in a model-consistent fashion. The extended dynamic model can be written as

This version of the linear model provides the starting point for our extension to include asymmetry and the null hypothesis in tests of restriction to linearity of the asymmetric alternative.¹⁰

Extending the Model for Asymmetric Effects

The most straightforward test of state-dependent asymmetry is to add positive output gaps to the model and test if the estimated parameter(s) on the additional term(s) can help explain inflation. Robert Gordon (1994) provides some econometric evidence of this nature by augmenting his simple, backward-looking, inflation-unemployment model with positive unemployment gaps. Although Gordon focuses his attention on Eisner’s version of asymmetry—that inflation falls more when the economy is in a recession than it rises when the economy is booming—his statistical evidence can be easily reinterpreted as a test for asymmetry arising from capacity constraints. Gordon’s results reveal no statistical evidence in favor of either form of asymmetry.

These results stand in sharp contrast with some recent empirical results reported by Laxton, Meredith, and Rose (1995), who found strong empirical evidence for the major industrial countries that positive output gaps have more powerful effects on inflation than negative output gaps. One possible reconciliation of these results could be insufficient business cycle variation in the U.S. data for researchers to be able to say anything with confidence about the role of capacity constraints on the basis of U.S. experience alone. However, since Turner (1995) finds evidence of significant asymmetry in the U.S. data, and we report corroborating evidence in this paper, it appears that the explanation must be found in another element of the methodology.

One of the points argued by Laxton, Meredith, and Rose is that it is critical for econometricians to recognize the implications of asymmetry for the measurement of the output gaps that enter the Phillips curve in order to identify properly the asymmetric model. In particular, the mean value of the gap that enters the Phillips curve will be negative under the assumption of convexity and positive under the assumption of concavity in the functional form.¹¹ Laxton, Meredith, and Rose demonstrate, in the context of the pooled G-7 data, that failing to recognize this point will bias test results toward not rejecting a restriction to linearity. Turner takes this into account in his specification and tests, and we show below that the Laxton, Meredith, and Rose conclusions regarding tests for linearity carry over to applications with the U.S. data.

With this issue in mind, we specify an asymmetric Phillips curve as follows:

where ${\mathrm{gap}}^{*}=y-\overline{y}+\alpha$ and gappos* represents the positive values of these adjusted gaps. The variable y is the log of output and we define $\overline{y}$ to be potential output, that is, the level of output that is attainable on average in an economy subject to continual shocks.¹² Thus, the gap measured as $y-\overline{y}$ will have a mean of zero in targe samples. In an economy with a linear symmetric Phillips curve, this would also be an appropriate property for the gap measure used in the Phillips curve. However, with asymmetry where positive gaps have larger effects than negative gaps—if γ>0 in equation (3)—and assuming that there is some variance in aggregate demand conditions, α must be less than zero for inflation to be bounded. In other words, the gap that enters a convex asymmetric Phillips curve must have a negative mean.

equation (3) embodies the simplest empirical form of asymmetry—a piecewise linear function with the possibility of a kink at the point where gap* begins to exert upward pressure on inflation. We think of this representation as an approximation of any convex functional form. Its advantage is that we can test for asymmetry in a very simple and direct manner. Laxton, Meredith, and Rose (1995) argue that there are advantages to more complicated functions with marginal convexity. We do not dispute this but, with the limited data available to identify the nature of the function, we prefer to keep the empirical exercise as simple as possible. equation (3) allows us to test for the presence of asymmetry, which is our prime objective here. Although we go on to use this particular model for some policy experiments, both here and in an earlier paper (Clark, Laxton, and Rose (1995a)), we consider this work illustrative. For policy analysis, it may be appropriate to assume alternative functional forms.

So far, we have not taken a stand on how one should measure potential output in practice. In any study of the Phillips curve, the choice of a measure of the output gap is very important. We shall show, however, that for the issues addressed here, it becomes less important once proper care has been taken to specify the functional form of the Phillips curve to be consistent with the hypothesis being tested. We report results using a variety of methodologies. To begin, however, we follow Laxton, Meredith, and Rose (1994) and measure the trend level of output using a simple two-sided moving average filter of actual output. In their study with annual data, they measure trend output as a five-year, centered-moving average of actual output (a two-year horizon, forward and backward). In this paper, we use the same approach but report results for a range of alternative horizons (the parameter k in equation (4) below) in the two-sided filter. We later test the sensitivity of our conclusions to this methodology by repeating the estimation with a number of alternative measures of the output gap.

With this filter, a small value of k implies that potential output is highly correlated with actual output and, at the limit, when k =0, potential output is equal to actual output at all points in time. This calibration would be consistent with an extreme real-business-cycle view of the world, where prices adjust instantaneously to changes in excess demand. In such a world, it obviously makes no sense to estimate a Phillips curve. At the other extreme, a large value of k would be consistent with a view that most of the variation in observed output at business cycle frequencies is associated with movements in the output gap and not with potential output. Because there is considerable uncertainty about the role of demand and supply shocks in the economy and because imposing a false restriction can result in invalid statistical inferences, one way to proceed is to report results, and hypothesis tests, for a wide range of values of k.¹³ This is the strategy we adopt.

As emphasized above, this (or any mean zero) measure of the output gap may be a seriously biased proxy for the gap that is appropriate for use in the Phillips curve. It is to correct for this potential bias that we introduce the level shift to the filter measures of the output gap for use in the identification of the Phillips curve. This level shift is assumed to be fixed over our sample period and is estimated as the parameter α¹⁴. If short-run capacity constraints are truly a feature of the U.S. economy, imposing a zero value on α will bias γ toward zero and bias standard tests for the presence of asymmetry toward false rejection.¹⁵ We will show that this point is of the utmost importance in assessing the empirical evidence.

Is There Asymmetry in the U.S. Phillips Curve?

In this subsection, we report estimates of the U.S. Phillips curve and tests for whether it has significant asymmetry. We begin with our specification and tests and then turn to supporting evidence in the form of documentation of what happens when the logical implications of asymmetry are not taken into account in the estimation and testing.

Unbiased Tests for Asymmetry in the U.S. Phillips Curve

Table 1 reports the results of estimating our model with a determined simultaneously along with the other parameters. We report estimates for values of kranging from 2 to 16 (quarters). Because we want to hold the estimation period fixed to see how alternative gap measures perform in explaining the same data on inflation, the sample for the dependent variable (for all regressions) must end in 1990:4. (The two-sided filter with k= 16, requires data on actual output for the years 1991 to 1994.)

The details of the model to be estimated are provided at the top of Table 1. We wanted to keep the model parsimonious, because in such a small sample it is difficult to identify more than a few key parameters with any precision. This led us, for example, to limit the output gap effects to contemporaneous measures. Second, we use the Michigan Survey measure of inflation expectations. We knew from the work of Roberts (1994b and 1994c) that this measure is useful in explaining movements in actual inflation, but the most important reason for using a survey measure is, we think, that having a forward-looking element to expectations is important in estimation as well as in policy analysis. The use of the survey measure allows us to avoid the difficult econometric issues implied by explicit model-consistent expectations (e.g., the use of instrumental variables, as in Laxton, Meredith, and Rose (1995)). We chose a specification that includes the contemporaneous value of one-year-ahead inflation expectations as well as four lags, where each lag is assumed to have the same weight. In addition, a lagged inflation term was added to allow for intrinsic inertia in inflation dynamics.¹⁶

Table 1.

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

(t -statistics in parentheses)

Table 1.

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

(t -statistics in parentheses)

Estimated equation: ${\pi }_t=\delta {\overline{\pi }}_{t+4}^e+(1-\delta ){\pi }_{t-1}+\beta {\mathrm{gap}}_t^{*}+\gamma {\mathrm{gappos}}_t^{*}+{ϵ}_t^{\pi },$

where ${\mathrm{gap}}^{*}=y-y^{*}=y-\overline{y}+\alpha ,$ gappos*= positive values of gap*

$\overline{\pi }\underset{t+4}{e}=.2\left({\pi }_{t+4}^e{+}_{t-1}{\pi }_{t+3}^e{+}_{t-2}{\pi }_{t+2}^e{+}_{t-3}{\pi }_{t+1}^e{+}_{t-4}{\pi }_t^e\right)$

π = percent change in the CPI at annual rates

πet+4= Michigan Survey measure of inflation expectations

$\overline{y}\equiv \frac{1}{2k+1}[y_t+\sum _{i=1}^k{y}_{t+i}+y_{t-i}]$

and

SL ≡ significance level.

Data: U.S. quarterly data, 1964:1–1990:4.

k	α	γ	β	δ	R 2	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 Nexus

(t -statistics in parentheses)

Estimated equation: ${\pi }_t=\delta {\overline{\pi }}_{t+4}^e+(1-\delta ){\pi }_{t-1}+\beta {\mathrm{gap}}_t^{*}+\gamma {\mathrm{gappos}}_t^{*}+{ϵ}_t^{\pi },$

where ${\mathrm{gap}}^{*}=y-y^{*}=y-\overline{y}+\alpha ,$ gappos*= positive values of gap*

$\overline{\pi }\underset{t+4}{e}=.2\left({\pi }_{t+4}^e{+}_{t-1}{\pi }_{t+3}^e{+}_{t-2}{\pi }_{t+2}^e{+}_{t-3}{\pi }_{t+1}^e{+}_{t-4}{\pi }_t^e\right)$

π = percent change in the CPI at annual rates

πet+4= Michigan Survey measure of inflation expectations

$\overline{y}\equiv \frac{1}{2k+1}[y_t+\sum _{i=1}^k{y}_{t+i}+y_{t-i}]$

and

SL ≡ significance level.

Data: U.S. quarterly data, 1964:1–1990:4.

k	α	γ	β	δ	R 2	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)

The results in Table 1 were obtained using nonlinear least squares in RATS. The value of kthat minimizes the standard error of the inflation equation is 12 quarters. This estimate suggests that variation in the output gap, as opposed to potential output, is the dominant source of variation in output at business cycle frequencies. Based on this preferred model, there is clear evidence that positive gaps have larger effects on inflation than do negative gaps. Indeed, the estimated total coefficient on the positive gaps (β + γ) is 1.1 or about five times greater than the coefficient on the negative gaps (ß alone). This is a very similar finding to that reported in Laxton, Meredith, and Rose (1995).

The estimated value of α is — 1.3; this implies that the gap* measures that are used in the Phillips curve are 1.3 percentage points smaller than the gaps based on our simple filter. Figure 1 illustrates the results for the asymmetric model. In this case, “potential” output in the top panel is raised by 0.013, relative to the trend measure from the filter, so that we can illustrate the “gaps” that enter the Phillips curve. It is important to remember that this level of output is not attainable in the stochastic economy.

The estimated value of δ is just under 0.6, leaving a weight of just over 0.4 on the lagged value of inflation. Regardless of how one chooses to interpret these coefficients (see the discussion in the previous subsection), the results suggest that there is important inertia (a backward-looking component) in inflation dynamics. There is also, however, an important forward-looking component that enters through the expectations.

Examination of the residuals from the preferred estimation indicated that there was some minor autocorrelation, primarily at the third lag. For this reason we have used the robust standard errors option in RATS to obtain the t -tests reported in parentheses. Note that the t -test for γ indicates that the simple restriction to linearity (γ = 0) is strongly rejected and that this result is robust over a wide range of values of k, Note, further, that α is strongly and significantly different from zero, a result that is also robust to substantial variation in k, Finally, we report a Wald test of the joint restriction, α, γ = 0, which imposes the full conditions of the linear null hypothesis. This test is performed on the condition that ß is strictly positive, a hypothesis that is not rejected by a direct test.¹⁷ It indicates that the joint restriction is rejected at the 99.9 percent confidence level for our preferred result.

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Interpreting U.S. Inflation with an Asymmetric Model

Citation: IMF Staff Papers 1996, 001; 10.5089/9781451957099.024.A008

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Biased Tests for Asymmetry in the U.S. Phillips Curve

In discussing the motivation for including the shift effect captured by the parameter α in our model, we argued that this was essential in order to obtain unbiased tests for the presence of asymmetry. We now return to this important issue. Table 2 reports some econometric estimates of the same model under the assumption that α = 0, again for a range of values of k, The standard errors for the t -statistics (reported in parentheses) are again computed using the robust errors option of RATS.

In this case, the value of k that minimizes the standard error of the inflation equation is 8. This still has substantial smoothing of the data in the implied profile of trend output, but less than in the model of Table 1. The relative weight estimated on the forward- and backward-looking components of the dynamics is not much affected by the restriction on a. However, the preferred model here has two noteworthy features that are very different. First, the econometrician would reject the hypothesis that γ >0 at the usual confidence levels and conclude that there was no compelling evidence of asymmetry in the U.S. Phillips curve. Second, the coefficient on the output gap is statistically significant, so there is evidence that inflation is related to the output gap—a Phillips curve exists. Again, these results are robust over a substantial range of values of k.

Figure 2 provides a graphical representation of the linear Phillips curve when we impose α, γ = 0 and choose the model that maximizes the fit of the inflation equation. The regression results are shown in the second line of Table 2 with k =9. which gives the best fit under these restrictions. The top panel of the chart plots our filter measure of potential output. The middle panel presents the percent change in the CPI measured at annual rates along with the Michigan Survey measure of one-year-ahead CPI inflation expectations. The bottom panel presents the difference between inflation and its backward- and forward-looking components, ${\pi }_t-\delta {\overline{\pi }}_t^e-(1-\delta ){\pi }_{t-1,}$ and the contribution of the output gap, as measured by the coefficient times the output gap, βgap,, As can be seen in the chart, there is a tendency for the linear model to underpredict inflation during the two inflationary episodes in the 1970s when there was large and persistent excess demand.

Table 2.

Some Biased Tests of Asymmetry

(t –statistics in parentheses)

Table 2.

Some Biased Tests of Asymmetry

(t –statistics in parentheses)

Estimated equation: ${\pi }_t=\delta {\overline{\pi }}_{t+4}^e+(1-\delta ){\pi }_{t-1}+\beta {\mathrm{gap}}_t^{*}+\gamma {\mathrm{gappos}}_t^{*}+{ϵ}_t^{\pi },$

where ${\mathrm{gap}}^{*}=y-y^{*}=y-\overline{y}+\alpha ,$ gappos* = positive values of gap*

$\overline{\pi }\underset{t+4}{e}=.2\left({\pi }_{t+4}^e{+}_{t-1}{\pi }_{t+3}^e{+}_{t-2}{\pi }_{t+2}^e{+}_{t-3}{\pi }_{t+1}^e{+}_{t-4}{\pi }_t^e\right)$

π= percent change in the CPI at annual rates

πet+4= Michigan Survey measure of inflation expectations

$\overline{y}\equiv \frac{1}{2k+1}[y_t+\sum _{i=1}^k{y}_{t+i}+y_{t-i}]$

Data: U.S. quarterly data, 1964:1–1990:4.

k	α	γ	β	δ	R 2	σ
5	0.00	0.285	0.593	0.547	.7739	1.6499
		(0.96)	(3.34)	(4.73)
6	0.00	0.302	0.497	0.556	.7761	1.6296
		(1.20)	(3.32)	(4.91)
7	0.00	0.281	0.435	0.562	.7769	1.6387
		(1.27)	(3.10)	(5.02)
8	0.00	0.231	0.414	0.564	.7787	1.6321
		(1.12)	(3.13)	(5.07)
9	0.00	0.192	0,391	0.564	.7785	1.6331
		(0.99)	(3.15)	(5.14)
9	0.00	0.00	0.484	0.552	.7765	1.6328
			(4.54)	(4.61)
10	0.00	0.174	0.362	0.563	.7774	1.6371
		(0.93)	(3.12)	(5.25)
11	0.00	0.154	0.341	0.560	.7765	1.6403
		(0.86)	(3.09)	(5.31)
12	0.00	0.134	0.326	0.557	.7761	1.6421
		(0.79)	(3.14)	(5.33)
13	0.00	0.110	0.316	0.555	.7747	1.6469
		(0.67)	(3.22)	(5.28)
14	0.00	0.907	0.308	0.552	.7736	1.6510
		(0.57)	(3.29)	(5.21)
15	0.00	0.745	0.301	0.549	.7725	1.6548
		(0.48)	(3.39)	(5.13)
16	0.00	0.059	0.296	0.545	.7713	1.6594
		(0,39)	(3.48)	(5.04)

View Table

Table 2.

Some Biased Tests of Asymmetry

(t –statistics in parentheses)

Estimated equation: ${\pi }_t=\delta {\overline{\pi }}_{t+4}^e+(1-\delta ){\pi }_{t-1}+\beta {\mathrm{gap}}_t^{*}+\gamma {\mathrm{gappos}}_t^{*}+{ϵ}_t^{\pi },$

where ${\mathrm{gap}}^{*}=y-y^{*}=y-\overline{y}+\alpha ,$ gappos* = positive values of gap*

$\overline{\pi }\underset{t+4}{e}=.2\left({\pi }_{t+4}^e{+}_{t-1}{\pi }_{t+3}^e{+}_{t-2}{\pi }_{t+2}^e{+}_{t-3}{\pi }_{t+1}^e{+}_{t-4}{\pi }_t^e\right)$

π= percent change in the CPI at annual rates

πet+4= Michigan Survey measure of inflation expectations

$\overline{y}\equiv \frac{1}{2k+1}[y_t+\sum _{i=1}^k{y}_{t+i}+y_{t-i}]$

Data: U.S. quarterly data, 1964:1–1990:4.

k	α	γ	β	δ	R 2	σ
5	0.00	0.285	0.593	0.547	.7739	1.6499
		(0.96)	(3.34)	(4.73)
6	0.00	0.302	0.497	0.556	.7761	1.6296
		(1.20)	(3.32)	(4.91)
7	0.00	0.281	0.435	0.562	.7769	1.6387
		(1.27)	(3.10)	(5.02)
8	0.00	0.231	0.414	0.564	.7787	1.6321
		(1.12)	(3.13)	(5.07)
9	0.00	0.192	0,391	0.564	.7785	1.6331
		(0.99)	(3.15)	(5.14)
9	0.00	0.00	0.484	0.552	.7765	1.6328
			(4.54)	(4.61)
10	0.00	0.174	0.362	0.563	.7774	1.6371
		(0.93)	(3.12)	(5.25)
11	0.00	0.154	0.341	0.560	.7765	1.6403
		(0.86)	(3.09)	(5.31)
12	0.00	0.134	0.326	0.557	.7761	1.6421
		(0.79)	(3.14)	(5.33)
13	0.00	0.110	0.316	0.555	.7747	1.6469
		(0.67)	(3.22)	(5.28)
14	0.00	0.907	0.308	0.552	.7736	1.6510
		(0.57)	(3.29)	(5.21)
15	0.00	0.745	0.301	0.549	.7725	1.6548
		(0.48)	(3.39)	(5.13)
16	0.00	0.059	0.296	0.545	.7713	1.6594
		(0,39)	(3.48)	(5.04)

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Interpreting U.S. Inflation with an Linear Model

Citation: IMF Staff Papers 1996, 001; 10.5089/9781451957099.024.A008

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In conclusion, we find evidence of significant asymmetry in the U.S. Phillips curve, confirming the result in Turner (1995) (and the result in Laxton, Meredith, and Rose (1995) for the pooled G-7 data). A major distinguishing feature of this work, which sets it apart from other work that finds no such evidence for the U.S. data, is the explicit attention to the implications of asymmetry for the manner in which the gaps must enter the estimated Phillips curve. It is clear that procedures that do not allow for the logical implications of the alternative hypothesis can produce biased results in tests of restriction(s) to the null hypothesis. We maintain that the failure of some researchers to account for how the output gap should enter the Phillips curve in an asymmetric world has biased their test results toward false rejection of the presence of asymmetry. We have shown that the results of tests for asymmetry in the U.S. Phillips curve are changed dramatically toward not rejecting the linear null hypothesis if a is suppressed, echoing a similar result shown by Laxton, Meredith, and Rose (1995) for the major industrialized countries. We return to this issue in Section IV.

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

The equations of the model are listed in Table 3. In addition to the two versions of the Phillips curve and some definitions, the model has two equations: one that describes the dynamics of the output gap and the mechanism by which monetary policy influences aggregate demand, and another that describes a policy reaction function designed to keep inflation in the neighborhood of a chosen target level.

The equation for output dynamics has two features that are important for these experiments. First, there are lags in the effect of the monetary control variable. Second, there is an autoregressive propagation structure that, all else being equal, initially amplifies the effects of shocks to aggregate demand and then sustains them for some time. The monetary control variable in the model, i.e., the variable that provides the transmission mechanism for monetary policy, is a short-term real interest rate. We treat the Federal Reserve funds rate as the policy instrument, but it is an ex ante real interest rate that enters the demand equation. The model is specified with a two-quarter delay before changes in the real interest rate begin to have an influence on aggregate demand. Thereafter, the effect builds such that after five to six quarters, a persistent real interest rate hike of 100 basis points that is sustained for four quarters would reduce the output gap by 0.4 percentage points. The propagation of demand shocks is represented by a second-order autoregressive structure. All else being equal, the effects of a shock are amplified in the second period and then die out slowly. These properties are roughly consistent with the evidence from both reduced-form models of the transmission mechanism (e.g., Roberts (1994a)) and more structural models (e.g., Mauskopf (1990)).

Table 3.

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

Table 3.

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

Asymmetric Phillips curve:

${\pi }_t=.593\overline{\pi }\underset{t+4}{e}+(1-.593){\pi }_{t-1}+.202{gap}_t^{*}+.925{gap}_t^{*}+{∊}_t^{\pi }\overline{\pi }\underset{t+4=}{e}.2{(}_t{\pi }_{t+4}^e{+}_{t-1}{\pi }_{t+3}^e{+}_{t-2}{\pi }_{t+2}^e{+}_{t-3}{\pi }_{t+1}^e{+}_{t-4}{\pi }_t^e)$

Linear Phillips curve:

${\pi }_t=.548\overline{\pi }\underset{t+4}{e}+(1-.548){\pi }_{t-1}+.524{gap}_t+{∊}_t^{\pi }$

Real interest rate:

${rr}_t={rs}_t-{\pi }_{t+4}^e$

Inflation and inflation expectations:

${\pi }_t=\left[\right(P_t/P_{t-1}{)}^4-1]*100,{\pi }_{t+4}^e=(P_{t+4}/P_t-1)*100$

Aggregate demand equation:

${gap}_t=1.074{gap}_{t-1}-.290{gap}_{t-2}-.158{rr}_{t-2}+{∊}_t^{gap}$

Policy reaction function:

${rs}_t-{\pi }_{t+4}^e=2({\pi }_{t+3}-{\pi }^{*})+{gap}_t$

π ≡ CPI inflation at annual rates

gap ≡ output gap $(y-\overline{y})$

gap* ≡ $(y-\overline{y}+\alpha )=(y-y^{*})$

rr ≡ real interest rate

rs ≡ Federal Reserve funds rate

πet+4≡ one-year-a head inflation expectations

π* ≡ inflation target

Pt ≡ price level

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Table 3.

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

Asymmetric Phillips curve:

${\pi }_t=.593\overline{\pi }\underset{t+4}{e}+(1-.593){\pi }_{t-1}+.202{gap}_t^{*}+.925{gap}_t^{*}+{∊}_t^{\pi }\overline{\pi }\underset{t+4=}{e}.2{(}_t{\pi }_{t+4}^e{+}_{t-1}{\pi }_{t+3}^e{+}_{t-2}{\pi }_{t+2}^e{+}_{t-3}{\pi }_{t+1}^e{+}_{t-4}{\pi }_t^e)$

Linear Phillips curve:

${\pi }_t=.548\overline{\pi }\underset{t+4}{e}+(1-.548){\pi }_{t-1}+.524{gap}_t+{∊}_t^{\pi }$

Real interest rate:

${rr}_t={rs}_t-{\pi }_{t+4}^e$

Inflation and inflation expectations:

${\pi }_t=\left[\right(P_t/P_{t-1}{)}^4-1]*100,{\pi }_{t+4}^e=(P_{t+4}/P_t-1)*100$

Aggregate demand equation:

${gap}_t=1.074{gap}_{t-1}-.290{gap}_{t-2}-.158{rr}_{t-2}+{∊}_t^{gap}$

Policy reaction function:

${rs}_t-{\pi }_{t+4}^e=2({\pi }_{t+3}-{\pi }^{*})+{gap}_t$

π ≡ CPI inflation at annual rates

gap ≡ output gap $(y-\overline{y})$

gap* ≡ $(y-\overline{y}+\alpha )=(y-y^{*})$

rr ≡ real interest rate

rs ≡ Federal Reserve funds rate

πet+4≡ one-year-a head inflation expectations

π* ≡ inflation target

Pt ≡ price level

These properties of output dynamics and the monetary policy transmission mechanism have an important implication for the design of monetary policy rules—monetary policy cannot offset completely the effects of shocks. There will be cycles in economic activity and there will be temporary deviations of inflation from its target level. This makes it especially important for the monetary authority to be forward-looking in its actions. The particular forward-looking policy reaction function that we use is a variant of the rule considered by Bryant, Hooper, and Mann (1993). In setting the short-term interest rate, the monetary authority acts to raise the real rate that enters the output equation when inflation is expected to be above the target level three quarters ahead or there is excess demand in the economy. The particular calibration adopted is designed to ensure that inflation remains in the region of the target level and returns to the target level within a reasonably short period (two years) following a shock to aggregate demand.

The Effects of Delaying Monetary Response to a Demand Shock

The simulations show what happens when there is an impulse shock of 1 percent to aggregate demand, i.e., when a 1 percentage point output gap opens on impact and there are no further shocks.¹⁸ We compare the consequences of delaying the monetary policy response by just one quarter for the two versions of the model. The delay is implemented by simply holding the U.S. Federal Reserve funds rate at its control value for one quarter (i.e., the normal reaction function is turned off for one quarter and then operates normally).¹⁹ The results from the model with a linear Phillips curve are shown in Figures 3 (no delay) and 4 (delay). Figures 5 and 6 show the comparable results from the model with asymmetry.

If the economy is linear, the effects of delay are relatively innocuous. With an immediate response amounting to a 270 basis point increase in the short-term interest rate in the first quarter, inflation peaks at about 1 percentage point above control, and there is a cumulative gain of output of about 2 percent. With a delayed monetary response, there is a larger hike in short-term interest rates, and inflation edges slightly higher, peaking at about 1.3 percentage points above control, but the cumulative gain in output is even larger, amounting to about 2.5 percent by the end of the simulation. These results show that if the world is linear, there is no strong case for aggressive monetary resistance to inflationary demand shocks. While the monetary authority may be concerned to see inflation rise above the target by more and for a longer period of time, it would be difficult to argue that there are any real costs to delaying interest rate hikes when output exceeds potential output.

The picture looks quite different if there are capacity constraints that result in asymmetry in inflation dynamics. Despite a much stronger policy response in the no-delay scenario, with short-term interest rates up by over 400 basis points in the first quarter, inflation peaks at about 2 percentage points above control. While the course of output is roughly similar over the first few quarters, the task of reining in the higher inflationary consequences of the same shock in this model requires a much deeper secondary contraction and the elimination of any cumulative gain in output. In fact, there is a small cumulative loss of output.

The consequences of delaying the monetary response to the shock are much more dramatic in this case. Short-term interest rates must be raised substantially higher to combat the cumulating inflationary pressures, as expectations respond to the more sustained excess demand and rising inflation. Inflation now peaks at 3 percentage points above control. To bring inflation back under control in this environment requires a much more severe contraction. Indeed, the cumulative change in output is now substantially negative because a large contraction is needed to counteract the inflationary effects that are caused by the initial temporary boom.

The model with asymmetry has many of the predictions that most central bankers highlight when discussing the fundamental role of monetary policy. It provides a clear and compelling reason for forward-looking action by the monetary authority to forestall any tendency for the economy to move into excess demand. To fail to do so necessitates even tougher action later on to keep inflation in check, with unfortunate but unavoidable macroeconomic consequences. Monetary authorities’ apparent preoccupation with inflation reflects a concern about the consequences for the real economy: to temporize on inflation is to force the economy to suffer both a cumulative loss of output and more extreme variation in economic conditions.

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Linear Model Responses to a Temporary I Percent Positive Demand Shock

Citation: IMF Staff Papers 1996, 001; 10.5089/9781451957099.024.A008

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Linear Model Responses to a Temporary 1 Percent Positive Demand Shock: Delayed Monetary Policy Response

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Asymmetric Model Responses to a 1 Percent Positive Demand Shock

Citation: IMF Staff Papers 1996, 001; 10.5089/9781451957099.024.A008

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Asymmetrie Model Responses to a 1 Percent Positive Demand Shock: Delayed Monetary Policy Response

Citation: IMF Staff Papers 1996, 001; 10.5089/9781451957099.024.A008

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On Methods of Measuring Potential Output and Their Implications

This study might be criticized on the grounds that it has relied on an ad hoc, two-sided, moving average filter to measure potential output. In principle, it would be preferable to specify a dynamic structural model that nested both the linear and asymmetric views of the world, and to derive model-consistent measures of the output gap under both hypotheses. This would place the models on an equal statistical footing and, in principle, permit a clear that it would be possible to identify many more parameters and to discriminate between more complex models, given the limited number of observations of business cycles available in the data. We leave this as an interesting topic for future research.²⁰ We would be remiss, however, if we did not check whether our results stand up if we use other univariate techniques to measure potential output. This could be important; Harvey and Jaeger (1993), for example, have shown that reliance on simple filters may induce spurious relationships in regressions.²¹ A related example is provided in Laxton, Shoom, and Tetlow (1992). They show that the use of simple univariate filters to measure gaps can cause spurious change-in-the-gap terms to have significant coefficients in estimated Phillips curves in cases where the true data generating process has only pure level gap effects.²² Thus, there is a risk that our use of the two-sided moving average filter to measure potential output has somehow biased our results in favor of the asymmetric model.

Harvey and Jaeger (1993) present a trend-plus-cycle or unobserved components (UC) model that is quite flexible and encompasses many other popular detrending procedures. For our purposes, their model can be thought of as a simple reduced form of a more general dynamic stochastic structural model of the U.S. economy. They show that the Hodrick and Prescott (1981) filter and the two-sided moving average filter used in this paper are generally suboptimal for estimating trends of economic variables.²³ Indeed, they show that estimates from the Hodrick-Prescott (HP) filter are optimal—in the class of univariate filters—only under very specific parameterizations, which vary from case to case. Using several economic series as examples, they show how the arbitrary use of the HP filter can result in biased estimates of the cyclical component and can even induce spurious correlations and regression results when the variables are detrended this way. These potential problems also apply to our filter.

Harvey and Jaeger also argue that their model has several advantages over simple time trends or segmented time trends. Despite the fact that it is impossible to discriminate between deterministic segmented time trends and stochastic representations of output in small samples, they argue that the latter is preferable because segmented time trends are based on arbitrary assumptions on when the trends break.²⁴

The validity of their general argument notwithstanding, Harvey and Jaeger show that in the particular case of the U.S. GNP their model produces estimates that are very similar to the HP (λ= 1,600) estimates. They conclude (p, 236) that this “suggests that the HP filter is tailor-made for extracting the business cycle component from U.S. GNP.” The bottom panel of Figure 7 shows the estimates obtained from the Harvey and Jaeger unobserved components (UC) model along with the estimates obtained from the HP filter with X set equal to 1,600.²⁵ It is evident that the Harvey-Jaeger conclusion is not altered by the change in the output measure; the two methods produce very similar estimates of the U.S. GDP output gap.

Another common procedure for providing trend estimates and gap measures is to fit polynomial, often quadratic, trend lines to the data. For example, Roberts (1994c) reports estimates of inflation models that use quadratic time trend approximations to measure potential output.

Table 4 reports estimates of our inflation model based on these alternative estimates of the output gap. Again, to give some indication of the bias associated with using pure measures of central tendency in deriving gap measures for estimating the Phillips curve, we report results with and without the α shift effect. Equation (2a) in Table 4 reports the estimates for our model with the Harvey-Jaeger output gaps. Note that the fit of the model is slightly better with the Harvey-Jaeger gaps than in our best result. In our estimation, both γ and β are slightly larger than in Table 1. Moreover, all the parameters are statistically significantly different from zero, and a Wald test indicates that we can reject the joint restriction that α,β = 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 the restricted model. In addition, the coefficient on the output gap (β) rises from 0.281 in the unrestricted model to 0.403 in the restricted model. This confirms the direction of bias from the Monte Carlo evidence reviewed above. Moreover, as in our exercise using the simple filter, an econometrician would not be able to reject the hypothesis γ = 0 at the traditional high levels of confidence, although with these results an objective researcher might think long and hard about the risks of assuming that this gave license to throw the gappos* variable out of the model.

Equation (3a) reports the results when we use the HP (λ = 1,600) filter to measure the output gaps. The results confirm the earlier indication that these gaps are not significantly different from the Harvey-Jaeger gaps from the unobserved components model. Indeed, the parameter estimates are almost identical to those in equation (2a). And, again, the case for asymmetry weakens when we impose α = 0.

In general, 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 is implied by the HP (λ = 1,600) filter or the unobserved components model. To test the sensitivity of our results to this assumption we re-estimated our model with larger weights on the smoothness parameter of the HP filter. In the limit, an extremely high weight on smoothness implies that potential output is treated as a linear time trend. Equations (4a) and (5a) provide additional estimates when λ the HP smoothing parameter, is set equal to 10,000 and 100,000, respectively. Note that the fit of the inflation equation deteriorates 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 the time series properties of both U.S. output and inflation.

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Harvey and Jaeger’s Measure of Potential Output

Citation: IMF Staff Papers 1996, 001; 10.5089/9781451957099.024.A008

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Table 4.

Tests with Alternative Measures of the Output Gap

(t-statistics in parentheses)

Table 4.

Tests with Alternative Measures of the Output Gap

(t-statistics in parentheses)

Estimated equation: ${\pi }_t=\delta {\overline{\pi }}_{t+4}^e+(1-\delta ){\pi }_{t-1}+\beta {\mathrm{gap}}_t^{*}+\gamma {\mathrm{gappos}}_t^{*}+{ϵ}_t^{\pi },$

where: ${\mathrm{gap}}^{*}=y-y^{*}=y-\overline{y}+\alpha ,$ gappos* = positive values of gap*

$\overline{\pi }\underset{t+4}{e}=.2\left({\pi }_{t+4}^e{+}_{t-1}{\pi }_{t+3}^e{+}_{t-2}{\pi }_{t+2}^e{+}_{t-3}{\pi }_{t+1}^e{+}_{t-4}{\pi }_t^e\right)$

π ≡ percent change in the CPI at annual rates

πet+4≡ Michigan Survey measure of inflation expectations

SL ≡ significance level.

Model #1: Case of k = 12 from Table 1

Model #2: Harvey and Jaeger’s (1993) Trend Plus Cycle Model

Model #3: Hodrick-Prescott filter with λ = 1,600

Model #4: Hodrick-Prescott filter with λ = 10,000

Model #5: Hodrick-Prescott filter with λ = 100,000

Model #6: Hodrick-Prescott filter with λ = 500

Model #7: Quadratic time trend (sample period = 1961:1–1994:4)

Inflation equation estimation period: sample 1964:1–1990:4

Model	α	γ	β	δ	R2	σ	Wald test: SL(α, γ = 0)
1a	–1.256	0.925	0.202	0.593	.7892	1.6010	0.001
	(3.66)	(3.16)	(2.43)	(6.13)
1b	0.000	0.134	0.326	0.557	.7761	1.6421
		(0.79)	(3.14)	(5.33)
2a	–0.791	1.092	0.281	0.611	.7927	1.5877	0.018
	(2.48)	(2.80)	(2.40)	(6.18)
2b	0.000	0.331	0.403	0.583	.7849	1.6094
		(1.52)	(2.75)	(5.58)
3a	–0.848	0.943	0.248	0.611	.7927	1.5874	0.008
	(2.34)	(3.05)	(2.46)	(5.88)
3b	0.000	0.283	0.344	0.581	.7827	1.6175
		(1.52)	(2.94)	(5.40)
4a	–1.498	1.081	0.175	0.589	.7833	1.6231	0.000
	(4.28)	(3.03)	(2.67)	(5.25)
4b	0.000	0.148	0.298	0.558	.7745	1.6477
		(0.88)	(3.63)	(5.00)
5a	–0.506	0.176	0.226	0.530	.7652	1.6895	0.782
	(0.53)	(0.70)	(2.71)	(4.69)
5b	0.000	0.046	0.266	0.528	.7649	1.6826
6a	–0.713	1.066	0.303	0.611	.7920	1.5902	0.013
	(2.48)	(2.92)	(2.53)	(5.80)
6b	0.000	0.328	0.424	0.583	.7842	1.6119
		(1.49)	(2.80)	(5.47)
7a	1.458	–0.205	0.335	0.515	.7581	1.7148	0.160
	(1.74)	(1.57)	(3.28)	(4.15)
7b	0.000	–0.013	0.222	0.512	.7567	1.7117
		(0.10)	(3.86)	(4.29)

View Table

Table 4.

Tests with Alternative Measures of the Output Gap

(t-statistics in parentheses)

Estimated equation: ${\pi }_t=\delta {\overline{\pi }}_{t+4}^e+(1-\delta ){\pi }_{t-1}+\beta {\mathrm{gap}}_t^{*}+\gamma {\mathrm{gappos}}_t^{*}+{ϵ}_t^{\pi },$

where: ${\mathrm{gap}}^{*}=y-y^{*}=y-\overline{y}+\alpha ,$ gappos* = positive values of gap*

$\overline{\pi }\underset{t+4}{e}=.2\left({\pi }_{t+4}^e{+}_{t-1}{\pi }_{t+3}^e{+}_{t-2}{\pi }_{t+2}^e{+}_{t-3}{\pi }_{t+1}^e{+}_{t-4}{\pi }_t^e\right)$

π ≡ percent change in the CPI at annual rates

πet+4≡ Michigan Survey measure of inflation expectations

SL ≡ significance level.

Model #1: Case of k = 12 from Table 1

Model #2: Harvey and Jaeger’s (1993) Trend Plus Cycle Model

Model #3: Hodrick-Prescott filter with λ = 1,600

Model #4: Hodrick-Prescott filter with λ = 10,000

Model #5: Hodrick-Prescott filter with λ = 100,000

Model #6: Hodrick-Prescott filter with λ = 500

Model #7: Quadratic time trend (sample period = 1961:1–1994:4)

Inflation equation estimation period: sample 1964:1–1990:4

Model	α	γ	β	δ	R2	σ	Wald test: SL(α, γ = 0)
1a	–1.256	0.925	0.202	0.593	.7892	1.6010	0.001
	(3.66)	(3.16)	(2.43)	(6.13)
1b	0.000	0.134	0.326	0.557	.7761	1.6421
		(0.79)	(3.14)	(5.33)
2a	–0.791	1.092	0.281	0.611	.7927	1.5877	0.018
	(2.48)	(2.80)	(2.40)	(6.18)
2b	0.000	0.331	0.403	0.583	.7849	1.6094
		(1.52)	(2.75)	(5.58)
3a	–0.848	0.943	0.248	0.611	.7927	1.5874	0.008
	(2.34)	(3.05)	(2.46)	(5.88)
3b	0.000	0.283	0.344	0.581	.7827	1.6175
		(1.52)	(2.94)	(5.40)
4a	–1.498	1.081	0.175	0.589	.7833	1.6231	0.000
	(4.28)	(3.03)	(2.67)	(5.25)
4b	0.000	0.148	0.298	0.558	.7745	1.6477
		(0.88)	(3.63)	(5.00)
5a	–0.506	0.176	0.226	0.530	.7652	1.6895	0.782
	(0.53)	(0.70)	(2.71)	(4.69)
5b	0.000	0.046	0.266	0.528	.7649	1.6826
6a	–0.713	1.066	0.303	0.611	.7920	1.5902	0.013
	(2.48)	(2.92)	(2.53)	(5.80)
6b	0.000	0.328	0.424	0.583	.7842	1.6119
		(1.49)	(2.80)	(5.47)
7a	1.458	–0.205	0.335	0.515	.7581	1.7148	0.160
	(1.74)	(1.57)	(3.28)	(4.15)
7b	0.000	–0.013	0.222	0.512	.7567	1.7117
		(0.10)	(3.86)	(4.29)