a. Phillips curves

Perhaps the most common method of estimating the natural rate of unemployment is to infer it from the parameters of an estimated Phillips curve. If the natural rate and trend productivity growth (Δq^tr) are subsumed in the constant term (β₀, not included above), ^9/ equation [2] becomes a relatively standard Phillips curve from which the implicit natural rate can be calculated as:

U^NAT = {β₀ − (1−β₂)Δq^tr}/β₁

The economic interpretation of this estimate of the natural rate is that it is the rate of unemployment which is consistent with real wage growth being equal to the growth of trend labor productivity; i.e., the rate of unemployment consistent with stable income shares. Unless nominal wage growth fully reflects cyclical or short-run labor productivity growth (β₂=1), the calculated natural rate of unemployment will be inversely related to trend productivity growth. ^10/

This Phillips curve approach to estimating the natural rate of unemployment has the advantage of explicitly taking into account the relationship between wage inflation and the labor market gap. This is important because much of the policy interest in the natural rate of unemployment derives from its implications for wage inflation. An important disadvantage is that, in general, the structural determinants of the natural rate do not play an important role in its estimation. ^11/

b. Unemployment rate equations

An alternative approach to estimating the natural rate of unemployment, one which focuses explicitly on its structural determinants, is based on an unemployment rate equation. ^12/ The key requirement of this approach is to distinguish in estimation between cyclical and structural unemployment. Based on an estimated version of equation [1], and setting the output gap at its cyclically neutral level, the natural rate can be calculated as

U^NAT = α₀ + f(STRUCTURAL)

where all right-hand side parameters are at their long-run values. Structural variables would include, for example, demographic factors, unemployment insurance replacement ratios, relative minimum wages, and the degree of unionization of the work force. To the extent that statistically significant parameter, estimates are obtained for the structural variables, these estimates add empirical substance to the analysis of the possible effects of structural policies.

This approach, has the disadvantage of not incorporating any information on actual wage developments, and requiring proxies for trend or potential output. In practice, smoothing techniques are typically used to estimate trend output. The natural rate is then calculated based on a somewhat circular assumption about the level of the output gap consistent with “full employment”.

c. Okun’s Law

Another widely used method of calculating potential output is based on Okun’s law. ^13/ This “law” is an assumed relationship linking the gap between the actual and the natural rates of unemployment, and that between actual and potential output,

(y-y^pot)·100 = g(U-U^NAT)

Given estimates or assumptions about the natural rate of unemployment, potential output can be calculated as

y^pot = y − g(U−U^NAT)/100

where g is the Okun coefficient. Alternatively the natural rate of unemployment can be calculated given estimates or assumptions about potential output. ^14/

Aside from the possibility that the Okun coefficient may not be stable, this approach has a number of drawbacks. The most important is that these estimates of potential output depend entirely on the estimates of the natural rate of unemployment, and embody no further information on the structural determinants of potential output. In particular labor and capital inputs play no explicit role in the determination of potential output.

d. Production functions

A more structural approach to the estimation of potential output is to explicitly model output in terms of underlying factor inputs. ^15/ Given an estimated production function like [3], potential output can be calculated as that level of output which results when all factors are at their “normal” or “natural” levels. Determining the normal levels of factor inputs is an important step and is usually done by incorporating estimates of the natural rate of unemployment into the calculation of normal labor input, or by relating factor inputs, and hence output, to the deviation of output from its “normal” level. Alternatively, the “normal” levels of factor inputs can be determined from a fully specified model in which a general production function (or its dual, the cost function) is estimated jointly with a set of derived factor demand and supply functions. While such an approach has considerable theoretical appeal, and can deliver more efficient estimates of key relationships, its implementation can become very complex when allowance is made for adjustment costs. ^16/ Largely as a result of this complexity, and because the Cobb-Douglas form of the production function given by equation [4], appears adequate to describe the behavior of broad sectors of many economies, the more simple approach described earlier is often adopted.

This production function approach is now fairly common and has the advantage of explicitly relating potential output to factor inputs, including, in principle, factors such as research and development which might directly influence technical progress. Although incorporating estimates of the natural rate of unemployment into the calculation of potential output ensures some consistency, the implications of the estimates of potential output are not, in general, exploited in the determination of the natural rate of unemployment, wages, or prices. Moreover, any method in which potential output is estimated as a function of an assumed “trend”, which is itself a proxy for normal or potential output, involves some circularity. A further drawback of the production function approach when used in isolation is that, given its use as an indicator of inflationary pressures, the estimation procedure does not exploit the information contained in price developments.

a. Specification issues

A relatively general specification of the Phillips curve, one which subsumes many of the alternative specifications found in the literature, relates the growth of nominal wages (Δw) to excess demand in the labor market as measured by the unemployment rate (U-DEM); inflation expectations (Δp^exp); the growth of labor productivity (Δq and/or Δq^tr); and various supply shocks (equation [2] above). ^22/ This formulation is sufficiently general to apply either to atomistic segments of the labor market, where competitive pressures exert a direct impact on wage growth, or to more centralized bargaining environments where competitive pressures operate indirectly through income shares and relative bargaining powers.

Economic theory suggests a number of restrictions on the long-run relationship between wages, prices, and productivity. In equilibrium, when inflation expectations are realized, real wages should grow in line with labor productivity, implying stable income shares. Moreover, for such an equilibrium to be independent of the inflation rate, the coefficients on inflation expectations should sum to unity. The specification of equation [2] above is consistent with these a priori considerations.

By contrast, economic theory offers few, if any, restrictions on the short-term dynamics. In the short run, wage increases may deviate from their long-run rate as a result of expectational errors or deviations of productivity growth from its trend rate. The process whereby adjustment to long-run equilibrium occurs is generally left unspecified, but presumably reflects the response of wage increases to excess demand in the labor market and the adjustment of inflation expectations. In the United States, institutional features such as staggered wage contracts suggests that the lags in this adjustment process may be relatively long.

There are a number of issues regarding the specification of the estimated Phillips curve which were explored, and are only summarized here. Reflecting the importance of staggered and overlapping wage contracts in the unionized sector, as well as inertia in the response of wages to past inflation, inflation expectations were found to be well characterized by lags of about three years on past inflation. ^23/ A linear specification of the unemployment rate adjusted for demographic factors fitted the data as well or better than non-linear specifications such as the reciprocal or the logarithm of the unemployment rate. ^24/ An error correction formulation of the wage equation, where the deviation of real wages from their long-run level is included as an additional explanatory variable, was rejected by the data. Since the error-correction formulation effectively converts the equation to a relationship between the levels of wages, prices, the unemployment rate and productivity, the results suggest that the Phillips curve specification, which is in terms of rates of change, is appropriate.

b. Estimation results

A number of alternative wage equations are reported in Table 1. The measure of wages used is hourly compensation in the private nonfarm business sector. This relatively broad measure of labor costs includes wages and salaries as well as employers’ contributions to social security and private insurance and pension funds (nonwage labor costs). Labor productivity (q) is also measured on an hourly basis for the nonfarm business sector. Consumer (p^c) and output prices (p) are the implicit deflators for personal consumption expenditures and for output in the nonfarm business sector, respectively.

Table 1.

Phillips Curve Equations ^25/

Table 1.

Phillips Curve Equations ^25/

	1	2	3	4	5	6
Constant	6.051	6.300	3.446	3.322	3.429	−0.202
	(0.68)	(0.72)	(1.02)	(1.02)	(0.94)	(0.26)
U-DEM	−0.827	−0.850	−0.468	−0.454	−0.531	−0.577
	(0.15)	(0.15)	(0.15)	(0.15)	(0.14)	(0.15)
Φ(L)Δpc	0.345	0.280	0.521 2/	0.358 2/	0.574 2/	0.640 2/
	(0.17)	(0.18)	(0.15)	(0.16)	(0.14)	(0.14)
θ(L)Δpc	0.656	0.698	0.479 2/	0.442 2/	0.426 2/	0.360 2/
	(0.22)	(0.23)
θ(L)Δq			0.385	0.388	0.350	0.559 2/
			(0.18)	(0.18)	(0.17)	(0.15)
Δqtr						0.441 2/
Δ(pc-p)				−0.124	−0.132	−0.158
				(0.09)	(0.08)	(0.08)
Δ(U-DEM)		0.527
		(0.45)
ΔNWLC					0.790	0.900
					(0.20)	(0.20)
D(68:I)			5.693	5.463	5.245	5.114
			(1.28)	(1.28)	(1.20)	(1.21)
D(70:III to 71:III)			0.827	0.800	0.634	0.970
			(0.59)	(0.59)	(0.55)	(0.56)
D(72:I, 73:I, 74:II)			2.748	2.822	1.674	1.421
			(0.75)	(0.75)	(0.75)	(0.77)
D(82:IV to 88:IV)			−0.640	−0.552	0.081
			(0.37)	(0.37)	(0.38)
TREND						0.081
						(0.03)
TREND2						−0.00048
						(0.0002)
R2	0.640	0.645	0.700	0.707	0.751	0.731
SEE	1.482	1.500	1.250	1.243	1.153	1.170
DW	2.11	2.17	2.07	2.09	2.17	2.13
F-statistics
Autocorrelation (1-4) (CV = 2.5)	0.45	0.61	1.36	1.38	0.59	0.26
ARCH (1-4) (CV = 2.5)	0.92	1.15	1.09	1.05	1.38	0.69
Break 85:IV (CV = 1.9)	0.54	0.49	0.64	0.63	0.90	0.94
Break 82:IV (CV = 1.7)	0.55	0.49	0.69	0.67	0.77	0.84

View Table

Table 1.

Phillips Curve Equations ^25/

	1	2	3	4	5	6
Constant	6.051	6.300	3.446	3.322	3.429	−0.202
	(0.68)	(0.72)	(1.02)	(1.02)	(0.94)	(0.26)
U-DEM	−0.827	−0.850	−0.468	−0.454	−0.531	−0.577
	(0.15)	(0.15)	(0.15)	(0.15)	(0.14)	(0.15)
Φ(L)Δpc	0.345	0.280	0.521 2/	0.358 2/	0.574 2/	0.640 2/
	(0.17)	(0.18)	(0.15)	(0.16)	(0.14)	(0.14)
θ(L)Δpc	0.656	0.698	0.479 2/	0.442 2/	0.426 2/	0.360 2/
	(0.22)	(0.23)
θ(L)Δq			0.385	0.388	0.350	0.559 2/
			(0.18)	(0.18)	(0.17)	(0.15)
Δqtr						0.441 2/
Δ(pc-p)				−0.124	−0.132	−0.158
				(0.09)	(0.08)	(0.08)
Δ(U-DEM)		0.527
		(0.45)
ΔNWLC					0.790	0.900
					(0.20)	(0.20)
D(68:I)			5.693	5.463	5.245	5.114
			(1.28)	(1.28)	(1.20)	(1.21)
D(70:III to 71:III)			0.827	0.800	0.634	0.970
			(0.59)	(0.59)	(0.55)	(0.56)
D(72:I, 73:I, 74:II)			2.748	2.822	1.674	1.421
			(0.75)	(0.75)	(0.75)	(0.77)
D(82:IV to 88:IV)			−0.640	−0.552	0.081
			(0.37)	(0.37)	(0.38)
TREND						0.081
						(0.03)
TREND2						−0.00048
						(0.0002)
R2	0.640	0.645	0.700	0.707	0.751	0.731
SEE	1.482	1.500	1.250	1.243	1.153	1.170
DW	2.11	2.17	2.07	2.09	2.17	2.13
F-statistics
Autocorrelation (1-4) (CV = 2.5)	0.45	0.61	1.36	1.38	0.59	0.26
ARCH (1-4) (CV = 2.5)	0.92	1.15	1.09	1.05	1.38	0.69
Break 85:IV (CV = 1.9)	0.54	0.49	0.64	0.63	0.90	0.94
Break 82:IV (CV = 1.7)	0.55	0.49	0.69	0.67	0.77	0.84

The first equation indicates that even a very simple specification including only the demographically adjusted unemployment rate (U-DEM) and consumer price inflation (a 5-quarter, Φ(L)dp^c, and 12-quarter, θ(L)dp^c, moving average) does a, good job of explaining wage developments. The estimated coefficients are well determined with the expected signs, and the sum of the coefficients on the change in consumer prices is indistinguishable from its expected value of unity. In this equation, neither productivity growth (Δq^tr or Δq) or the natural rate are entered explicitly. The constant term in the equation, which implicitly incorporates these two variables, is thus very important, as suggested by its relatively large estimated coefficient.

The second equation in Table 1 includes the change in the unemployment rate in the wage equation (Δ(U-DEM)). An estimated negative coefficient on the change, in conjunction with the level of the unemployment rate, would suggest that, for a given level of unemployment, wage pressures intensify when unemployment is falling. Alternatively, if the change but not the level of the unemployment rate were significant, this would suggest hysteresis-type effects, which have recently been discussed in the context of European unemployment rate developments. ^26/ The estimated coefficient on the change in the unemployment rate is incorrectly signed and not statistically different from zero when this term is entered, either alone or in conjunction with the level of the unemployment rate, which remains statistically significant. ^27/

c. The recent “moderate” behavior of wages

The moderate growth of labor compensation during the 1980s has been much commented upon. Not only has the growth of labor compensation been low relative to productivity growth (see Table 2), it also appears to have been weaker than predicted by previously estimated wage equations. ^28/ This suggests either that the natural rate of unemployment has shifted or that there has been some other change in the relationship between wages and its historical determinants.

Table 2.

Wage, Price, and Productivity Developments ^29/

Table 2.

Wage, Price, and Productivity Developments ^29/

	Compensation 30/	Consumer Prices 31/	Real Consumption Wages 32/	Real Product Wages 31/	Labor Productivity 29/
1965:I-1969:IV	5.86	3.45	2.41	2.27	1.59
1970:I-1973:IV	6.77	5.08	1.69	1.98	1.90
1974:I-1980:IV	8.85	8.05	0.80	0.72	0.53
1981:I-1988:IV	4.94	4.29	0.65	0.90	1.35

View Table

Table 2.

Wage, Price, and Productivity Developments ^29/

	Compensation 30/	Consumer Prices 31/	Real Consumption Wages 32/	Real Product Wages 31/	Labor Productivity 29/
1965:I-1969:IV	5.86	3.45	2.41	2.27	1.59
1970:I-1973:IV	6.77	5.08	1.69	1.98	1.90
1974:I-1980:IV	8.85	8.05	0.80	0.72	0.53
1981:I-1988:IV	4.94	4.29	0.65	0.90	1.35

The ex-post forecast errors from the first wage equation reported in Table 1, estimated from 1965:I to 1982:IV, are shown in Table 3. The equation overpredicts compensation growth in 20 out of the 24 quarters from 1983 to 1988 with an annual average forecast error of 1 percentage point. Although the equation passes the Chow test for structural breaks in 1982:IV and 1985:IV (see Table 1), there is some evidence of structural instability from recursive estimation and from dummy variables. ^33/

Table 3.

Forecast Errors From the Wage Equation ^34/

Table 3.

Forecast Errors From the Wage Equation ^34/

	Average Annual Forecast Error (Actual Less Predicted)	Number of Quarterly Over- predictions
1983	−1.72	4
1984	−1.02	3
1985	−0.47	4
1986	−0.20	3
1987	−1.45	3
1988	−1.30	3
Average error	−1.03
Total overpredictions		20

View Table

Table 3.

Forecast Errors From the Wage Equation ^34/

	Average Annual Forecast Error (Actual Less Predicted)	Number of Quarterly Over- predictions
1983	−1.72	4
1984	−1.02	3
1985	−0.47	4
1986	−0.20	3
1987	−1.45	3
1988	−1.30	3
Average error	−1.03
Total overpredictions		20

While there have been a number of attempts to explain the overprediction of wage inflation in the 1980s, its statistical significance and the underlying causes remain unclear. Possible explanations include the declining role of trade unions (Neumark (1989) and Mitchell (1987)), increased foreign competition (Vroman and Abowd (1988)), shifts in income distribution between labor and capital (Gordon (1988)), and a decline in the natural rate of unemployment. ^35/ Recently, Taylor (1988) has suggested that the moderate wage growth may be the result of wage earners persistently underpredicting the warranted growth of real wages during an exceptionally strong recovery of economic activity.

d. Further estimation results

The failure of the wage equation to accurately predict the growth of compensation may be the consequence of implicitly incorporating the natural rate unemployment and trend productivity growth in the constant, and thus constraining them to be related in an offsetting manner. Alternative specifications discussed below attempt to explain the over-prediction by incorporating additional explanatory variables, particularly labor productivity growth (Δq). The degree of unionization and other structural variables, including marginal tax rates, were also entered directly in the equations, but were not statistically important or were incorrectly signed.

The four right-most equations in Table 1 include a series of dummy variables to capture unusual wage behavior in the late 1960s and early 1970s, a period when a variety of policies attempted directly to influence wage developments. ^36/ A dummy variable is also included to explicitly test for the overprediction of wage growth after 1982:IV; the estimated coefficient is statistically significant in some of the equations suggesting that wage growth has indeed been unusually moderate.

In these equations the coefficients on inflation are constrained to sum to unity and a 12-quarter moving average of labor productivity growth, the lag specification which best contributed to explaining the growth of wages, is included. The estimated coefficient on productivity growth (Δq) is significant, correctly signed, and relatively robust across alternative specifications. Although productivity growth enters with a 12-quarter moving average, it is still somewhat procyclical and captures some of the impact of economic activity, as is indicated by the estimated coefficient on unemployment which is almost halved. Similarly, the size of the constant in the equation, which now incorporates trend productivity with a coefficient of about 0.5 rather than unity, is also reduced substantially.

In general, employees are concerned with wages relative to consumer prices (real consumption wages), while employers are concerned with wages relative to output prices (real product wages). This suggests that wage growth will be related to an average of consumer and output price inflation. The final three equations in Table 1 include the gap between the rate of growth of consumer and output prices (Δp^c - Δp). This variable is approximately equal to the change in the terms of trade and allows a test of whether changes in relative prices are shared between labor and capital. The more important is output price inflation relative to that for consumer prices, the more are profits and employment protected from adverse supply shocks. ^37/ The estimated coefficient is marginally significant and suggests weights of about 0.8 on consumer prices and 0.2 on output prices.

As noted, wages are defined here as hourly compensation in the private nonfarm business sector, and hence include nonwage labor costs (employers’ contributions for social security and private insurance and pension schemes) in addition to earnings. The specification of the equations discussed thus far implies that the wage bargain is for compensation, rather than earnings. This can be explicitly tested by including the change in nonwage labor costs as a percent of total wages and salaries (ANWLC) in the estimated equations, as is done in the final two equations in Table 1. The estimated coefficient is not significantly different from unity which suggests that the wage bargain is struck over earnings rather than total compensation. ^38/ This does not imply, of course, that nonwage benefits are unimportant to employees; it only implies that employers’ contributions are largely independent of the factors influencing wages—inflation, unemployment and labor productivity.

As discussed in the next section, nonwage labor costs as a share of wages and salaries (NWLC), increased steadily from the mid-1960s to the early 1980s and then declined somewhat. When this variable is entered in the equation, the declines tend to lower the predicted growth of wages during the 1980s, and the dummy variable for the post 1982:IV period is no longer important. Thus a part of the explanation for the moderate growth of compensation during the 1980s appears to be related to the decline in employers’ contribution rates. ^39/

As noted above, if the estimated coefficient on labor productivity growth in the wage equation is less than unity, then the growth of trend labor productivity is implicitly subsumed with the natural rate of unemployment in the constant term. The pattern of trend productivity growth (Δq^tr)—a decline in the 1970s followed by an upturn in the 1980s—together with the post-1982 dummy, implies in all of the estimated equations a hump-shaped pattern for the natural rate: after rising in the 1970s, the implicit natural rate declines in the early 1980s.

Over long periods of time, however, there does not appear to be a clear relationship between trend productivity growth and unemployment rates. ^40/ This suggests that estimating the natural rate in this way may incorrectly attribute movements in the natural rate related to structural factors to changes in trend productivity growth. The final equation in Table 1 constrains the coefficients on actual and trend productivity growth to sum to unity. A quadratic time trend is entered to allow the natural rate to have a turning point; without this quadratic trend, the equation overpredicts badly and the 1980s dummy is statistically significant. As expected, the size of the estimated constant is reduced substantially when the trend is included. Due to the presence of the quadratic trend, this equation gives a similar pattern for the natural rate of unemployment to that implied by the other equations.

a. Dynamic specification, cyclical, and supply shock variables

Unlike the wage and price equations, there are few, if any, institutional or theoretical priors as to the length of lags or the steady-state values of specific parameters in an unemployment rate equation. The hiring and firing practices of firms do, however, suggest that the unemployment rate is likely to react with a lag to changes in its determinants. For this reason the relationship of the unemployment rate to its cyclical and structural determinants was initially specified as a general autoregressive distributed lag model (of up to four quarters) and then tested down by dropping insignificant variables. ^41/ This relatively unrestricted estimation procedure captures the cyclical dynamics well, increasing the likelihood that the structural component of the unemployment rate will be correctly identified.

A number of alternative equations are reported in Table 4. Based on the standard criteria, the equations do a good job of explaining quarterly developments in the unemployment rate. There is no evidence of a systematic pattern in the errors or of instability, either of which would suggest the omission of relevant variables.

Table 4.

Unemployment Rate Equations ^1/

^1/See note to Table 1. The dependent variable is the unemployment rate minus the impact of changing demographic shares (U-DEM).

^2/Constrained to be equal.

Table 4.

Unemployment Rate Equations ^1/

	1	2	3	4	5	6
Constant	−1.616	−1.252	−1.929	12.127	−0.929	−0.679
	(0.75)	(0.68)	(0.86)	(3.84)	(1.0)	(0.79)
y-ytr	−0.188	−0.188	−0.189	−0.208	−0.182	−0.178
	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)
(y-ytr)−1	0.122	0.122	0.124	0.083	0.109	0.108
	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)
NWLC	0.069	0.058	0.080	0.176	0.058	0.054
	(0.02)	(0.02)	(0.02)	(0.08)	(0.02)	(0.02)
UIRR	0.016	0.014	0.017	0.019	0.008	0.011
	(0.01)	(0.01)	(0.01)	(0.01)	(0.01)	(0.01)
SL·RMW−1	0.016 2/	0.027		0.041		0.018
	(0.006)	(0.01)		(0.03)		(0.01)
UNN	0.016 2/		0.035	−0.176	0.039
	(0.006)		(0.01)	(0.07)	(0.01)
SL				0.034
				(0.04)
SF				−0.310
				(0.09)
pg-p						−0.003
						(0.002)
px-pm					−0.011
					(0.006)
(U-DEM)−1	1.160	1.164	1.164	1.007	1.110	1.142
	(0.09)	(0.09)	(0.09)	(0.10)	(0.10)	(0.10)
(U-DEM)−2	−0.315	−0.317	−0.321	−0.318	−0.279	−0.316
	(0.08)	(0.08)	(0.08)	(0.08)	(0.08)	(0.08)
R2	0.985	0.985	0.985	0.987	0.985	0.985
SEE	0.213	0.213	0.213	0.201	0.210	0.212
Mean lag (quarters)	5.5	5.5	5.4	2.2	4.9	4.7
F-statistics:
Autocorrelation (1-4) (CV = 2.5)	0.21	0.23	0.33	0.94	0.10	0.20
ARCH (4) (CV = 2.5)	0.70	0.75	0.64	0.53	1.20	0.72
Break 85:IV (CV = 1.9)	0.52	0.53	0.53	0.61	0.50	0.40
Break 82:IV (CV = 1.7)	1.00	1.00	1.01	1.00	0.93	0.93

^1/See note to Table 1. The dependent variable is the unemployment rate minus the impact of changing demographic shares (U-DEM).

^2/Constrained to be equal.

View Table

Table 4.

Unemployment Rate Equations ^1/

	1	2	3	4	5	6
Constant	−1.616	−1.252	−1.929	12.127	−0.929	−0.679
	(0.75)	(0.68)	(0.86)	(3.84)	(1.0)	(0.79)
y-ytr	−0.188	−0.188	−0.189	−0.208	−0.182	−0.178
	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)
(y-ytr)−1	0.122	0.122	0.124	0.083	0.109	0.108
	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)	(0.03)
NWLC	0.069	0.058	0.080	0.176	0.058	0.054
	(0.02)	(0.02)	(0.02)	(0.08)	(0.02)	(0.02)
UIRR	0.016	0.014	0.017	0.019	0.008	0.011
	(0.01)	(0.01)	(0.01)	(0.01)	(0.01)	(0.01)
SL·RMW−1	0.016 2/	0.027		0.041		0.018
	(0.006)	(0.01)		(0.03)		(0.01)
UNN	0.016 2/		0.035	−0.176	0.039
	(0.006)		(0.01)	(0.07)	(0.01)
SL				0.034
				(0.04)
SF				−0.310
				(0.09)
pg-p						−0.003
						(0.002)
px-pm					−0.011
					(0.006)
(U-DEM)−1	1.160	1.164	1.164	1.007	1.110	1.142
	(0.09)	(0.09)	(0.09)	(0.10)	(0.10)	(0.10)
(U-DEM)−2	−0.315	−0.317	−0.321	−0.318	−0.279	−0.316
	(0.08)	(0.08)	(0.08)	(0.08)	(0.08)	(0.08)
R2	0.985	0.985	0.985	0.987	0.985	0.985
SEE	0.213	0.213	0.213	0.201	0.210	0.212
Mean lag (quarters)	5.5	5.5	5.4	2.2	4.9	4.7
F-statistics:
Autocorrelation (1-4) (CV = 2.5)	0.21	0.23	0.33	0.94	0.10	0.20
ARCH (4) (CV = 2.5)	0.70	0.75	0.64	0.53	1.20	0.72
Break 85:IV (CV = 1.9)	0.52	0.53	0.53	0.61	0.50	0.40
Break 82:IV (CV = 1.7)	1.00	1.00	1.01	1.00	0.93	0.93

^1/See note to Table 1. The dependent variable is the unemployment rate minus the impact of changing demographic shares (U-DEM).

^2/Constrained to be equal.

The variables determining the cyclical movements of the unemployment rate are the current and lagged deviation of actual output from the estimated trend (y-y^tr). ^42/ Short-run movements in the unemployment rate are also captured by the dynamic specification. The offsetting pattern of the estimated coefficients on the two lagged dependent variables means that the lag distributions on the independent variables initially declines then rises somewhat before tailing off to zero. The dynamics implied by the offsetting parameters on the output gap are superimposed on this general dynamic pattern.

The terms of trade (specified as the price of exports relative to the price of imports, p^x-p^m) and the relative price of gasoline (p^g-p), two supply shock variables which might affect the unemployment rate, are included in the final two equations in Table 4. (Developments in these and other variables discussed below are presented in Chart 3.) The estimated coefficients on both variables are correctly signed, but only marginally significant while reducing the significance of some of the structural variables. For supply shocks to have long-lasting effects on unemployment, some degree of rigidity preventing the adjustment of relative or real wages is necessary. Although it may be possible to capture the impact of labor market rigidities indirectly through their interaction with supply shocks, it is preferable to include, if possible, structural variables more directly related to labor market rigidities.

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United States: Structural and Supply Factors Affecting Unemployment

Citation: IMF Working Papers 1989, 089; 10.5089/9781451952179.001.A001

¹Minimum wage multiplied by the share of youth in the labor force.¹Export prices relative to import prices.²Gasoline prices relative to the output deflator.

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b. Demographic impacts on unemployment

The impact of demographic factors on the natural rate of unemployment has often been emphasized. ^43/ If different labor market groups have different natural rates of unemployment, perhaps because of different reservation wages or other characteristics, the changing demographic composition of the labor force may affect the natural rate of unemployment. The importance of this compositional effect can be estimated by constructing an unemployment rate with constant labor market shares (Chart 1).

In addition to this compositional effect of demographic changes, the relative minimum wage variable (RMW) has been multiplied by the labor-force share of youth aged 16 to 24 (SL), the group most likely to be affected by the minimum wage. This implies that the impact on unemployment of a given change in the minimum wage will be greater the larger is the proportion of youth in the labor force.

The total impact of both of these demographic effects turns out to be relatively small. The fourth equation in Table 4 tests for more direct effects on unemployment by also including the share of the labor force which is young (SL) and female (SF) (Chart 3). The proportion of youth, which has returned to about the same levels as in the mid-1960s, does not appear to have a significant effect on unemployment over and above the two effects noted above. The estimated coefficient on the share of the labor force which is female, which has increased steadily from the mid-1960s, is significant and negative. ^44/ Given that unemployment rates for women are typically higher than for men, implying, if anything, a higher natural rate, it is difficult to rationalize a negative impact on unemployment rates from the increased proportion of women in the labor force.

c. Structural determinants of the natural rate

Many, perhaps most, of the structural features of labor markets are constant or evolve very slowly; and those which do change, may be difficult to quantify. In the estimated equations in Table 4, the main structural variables affecting, the natural rate of unemployment are union members as a percent of total private sector employment (UNN), relative minimum wages (RMW), the average weekly unemployment insurance replacement ratio adjusted for the proportion of employment covered by the unemployment insurance system (UIRR), and employers’ contributions as a percent of total wages and salaries (NWLC) (Chart 3). Average and marginal income tax rates were also entered in the equations but the estimated coefficients were insignificant and/or incorrectly signed.

The amount of time the unemployed search, an important determinant of the duration of unemployment, can be expected to depend, at least in part, on a comparison between the offered and reservation wage. The latter will be related to, among other factors, minimum wages, unemployment insurance benefits, and, perhaps, the degree of unionization. In the empirical analysis reported here, minimum wages and average weekly unemployment insurance benefits have been expressed relative to average wages of employees of private households, a relatively low paying service “industry” which is likely to be more relevant for a typical unemployed person, and works better empirically, than a measure of aggregate wages.

Developments in relative minimum wages (adjusted for the share of youth) and unionization, particularly the decline in the 1980s (Chart 3), are sufficiently similar that it is not possible to get precise estimates of the contribution of each variable. ^45/ In view of this multicollinearity, which can be seen by comparing the first three equations presented in Table 4, ^46/ the coefficients on these two variables are constrained to be identical in the first equation.

The estimated coefficients on employers’ contributions as a percent of total wages and salaries (NWLC) are statistically significant in each of the reported equations, and large relative to the other estimated structural coefficients. Except for the fourth equation in Table 4, the size of the estimated parameters is also robust across the different specifications. In line with developments in most industrialized countries, ^47/ there were steady increases in employers’ contributions as a percent of wages and salaries from the early 1960s until the early 1980s (Chart 3). In the United States, these developments reflected increases in employers’ contributions for social security and for private insurance and pension funds, both of which increased from about 5 percent of wages and salaries in 1965 to about 10 percent in the early 1980s (Table 5). ^48/

Table 5.

Employers’ Contributions

(As a percent of total wages and salaries)

Sources: U.S. Department of Commerce, 1988, Survey of Current Business, July 1988, National Income and Product Accounts Tables 1.14 and 6.13; and United States Historical Abstract, various issues.

Table 5.

Employers’ Contributions

(As a percent of total wages and salaries)

			1965	1970	1975	1980	1983	1987
Total			9.9	12.1	16.5	19.4	20.5	19.3
	Social security		5.0	6.2	8.4	9.3	10.2	10.1
	Other contributions		4.9	5.9	8.1	10.1	10.4	9.2
		Health insurance	1.6	2.2	3.1	4.3	5.3	5.1
		Pensions and profit sharing	2.1	2.4	3.4	4.0	3.4	2.3
		Other 49/	1.2	1.3	1.6	1.8	1.7	1.8
Memorandum:
Employers’ social security contribution rate (percent)			3.62	4.80	5.85	6.13	6.70	7.15
Maximum taxable earnings (dollars per year)			4,800	7,800	14,100	25,900	35,700	43,000

Sources: U.S. Department of Commerce, 1988, Survey of Current Business, July 1988, National Income and Product Accounts Tables 1.14 and 6.13; and United States Historical Abstract, various issues.

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Table 5.

Employers’ Contributions

(As a percent of total wages and salaries)

			1965	1970	1975	1980	1983	1987
Total			9.9	12.1	16.5	19.4	20.5	19.3
	Social security		5.0	6.2	8.4	9.3	10.2	10.1
	Other contributions		4.9	5.9	8.1	10.1	10.4	9.2
		Health insurance	1.6	2.2	3.1	4.3	5.3	5.1
		Pensions and profit sharing	2.1	2.4	3.4	4.0	3.4	2.3
		Other 49/	1.2	1.3	1.6	1.8	1.7	1.8
Memorandum:
Employers’ social security contribution rate (percent)			3.62	4.80	5.85	6.13	6.70	7.15
Maximum taxable earnings (dollars per year)			4,800	7,800	14,100	25,900	35,700	43,000

Sources: U.S. Department of Commerce, 1988, Survey of Current Business, July 1988, National Income and Product Accounts Tables 1.14 and 6.13; and United States Historical Abstract, various issues.

The presence of employers’ contributions as a structural variable in the unemployment equation implies that nonwage labor costs are different, in some important respect, from “wage” labor costs. As discussed above, the evidence from the compensation equations suggests that one difference is that the wage bargain appears to be struck over earnings, not compensation. ^50/ To the extent that increases in employers’ contributions are not offset by wages, employee compensation will increase and profits will decline. ^51/ There is also a fixed cost aspect to nonwage labor costs, ^52/ and income ceilings mean that the average contribution rate paid by the employer declines as the level of wages increase. More generally, many benefits are interlocked with tenure, particularly those which involve deferred pay, and hence with firm-specific human capital. ^53/ Consequently increased benefits may go hand-in-hand with increased screening and higher recruitment costs as firms search longer to fill vacancies and become more reluctant to hire low paid workers or workers likely to stay with the firm for a relatively short period. This implies that the duration of vacancies and the equilibrium unemployment rate will increase. ^54/

Thus, increases in employers’ contributions may tend to increase the equilibrium rate of unemployment for a number of reasons: they increase the rigidity of compensation with regard to labor market conditions, inflation, and productivity developments; they lower profits and profit expectations; ^55/ and, they may make employers more reluctant to hire.

d. An equation for the natural rate of unemployment

By solving for the long-run relationship between the unemployment rate and its structural determinants, an equation for the natural rate of unemployment can be derived from the dynamic equations reported in Table 4. This long-run focus seems particularly appropriate for an equation describing one of the key structural features of the economy. Based on the first equation in Table 4, for example, the natural rate can be calculated as (standard errors in parentheses):

In order to calculate the constant in this equation, it is necessary to make an explicit assumption about the cyclically neutral level of the output gap. This will not, in general, be zero. ^56/ Moreover, as noted above, the calculation of the level of the natural rate becomes somewhat circular if the full-employment level of output is already known or has to be assumed. In the systems estimates of the natural rate presented in Section 4, the constant as well as the structural parameters in the natural rate equation are estimated jointly with the wage, price, and output equations.

Movements in the natural rate will be related to developments in the structural variables. The natural rate which is implied by the above equation increases by about 3 to 4 percentage points from the mid-1960s to the early 1980s, and then declines by about 1 to 2 percentage points in the period to 1988. This hump-shaped time profile is characteristic of all of the alternative specifications of the unemployment rate equations. Thus, a substantial part of the increase in actual unemployment rates from 1965 to their peak in 1983 may be attributed to increases in the natural rate of unemployment. A fuller discussion of the estimated natural rate of unemployment and its determinants is presented in Section 4 based on the system estimation results.

a. Specification issues and explanatory variables

The production function has been estimated in the form of a multifactor productivity equation. This approach has been adopted to facilitate discussion of the outlook for productivity growth as well as discussion of the productivity slowdown. There is, of course, an enormous literature on the productivity slowdown and, although a large number of possible reasons have been identified, there is not a consensus on their empirical significance. ^57/ Among the more prominent reasons advanced for the productivity slowdown in the United States are the large increases in energy prices, a slowdown in the advance of relevant knowledge, and a decline in the quality of labor input. Explanatory variables related to each of these possible explanations of the slowdown are included in the estimated multifactor productivity equations reported below.

The growth of multifactor productivity is generally taken as an indicator of technical progress, and a time trend is often included as a proxy for this in estimated equations. ^58/ Although this may work well empirically, it is preferable to include, if possible, explanatory variables more directly related to technical progress and the generation of relevant knowledge. This has been done by including estimates of the stock of research and development expenditures, distinguishing between the part financed by the Federal government and the part privately financed, in the estimated equations. ^59/

Multifactor productivity is defined here as output (y) minus a weighted average of the inputs of labor (manhours, h) and capital (k) (see Chart 4). Thus, in terms of the specification of equation [3] above, the coefficient on labor is constrained to be the share of labor compensation in national income (λ), with its compliment (1-λ) the constrained coefficient on capital. ^60/ Actual income shares in the private nonfarm sector, which for labor are in the range 0.63 to 0.71, are used; thus, strictly speaking, the equation should not be interpreted as a Cobb-Douglas production This measure of multifactor productivity, including the use of moving weights, is very similar to that regularly reported by the Bureau of Labor Statistics in the Monthly Labor Review. ^61/

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United States: Output and Factor Inputs

Citation: IMF Working Papers 1989, 089; 10.5089/9781451952179.001.A001

¹In constant 1982 dollars.

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The intensity with which factors, including capital, are utilized varies procyclically. For this reason, a measure of capacity utilization or the output gap is usually included as an additional explanatory variable in estimated production functions or multifactor productivity equations. ^62/ This may cause econometric problems given that actual output is the dependent variable, or in multifactor productivity form is included in the dependent variable, and is also in the numerator of the output gap. Another, potentially more serious problem with including the output gap might arise in the system estimation: the expression for potential output, which is itself derived from the estimated production function, would then have to be substituted back into the production function.

The fact that the denominator of the output gap is usually based on a smoothing of actual output, suggests that it may be possible to capture the impact of varying factor utilization rates in the dynamic specification. ^63/ For this reason, and in light of the potential problem with the system estimates, the output gap was not included as an explanatory variable in the multifactor productivity equations. Rather, the same estimation strategy used for the unemployment rate equations was adopted here: variables were initially entered contemporaneously and with three discrete lags and then tested down.

Estimating the production function as a multifactor productivity equations not only constrains the dynamic relationship between output and labor and capital inputs, it also imposes constant returns to scale on the underlying production function. While the latter may or may not be true, the former is difficult to justify. To relax both of these restrictions, current and lagged values of labor and capital inputs were also included as explanatory variables in the estimated multifactor productivity equations. ^64/

In addition to energy prices, the quality of labor input is another factor which is often highlighted in growth accounting analyses of the productivity slowdown. This is sometimes proxied by years of schooling or measures of performance on standardized tests. ^65/ In the equations reported below, labor input is interacted with the employment shares of youth and females as a proxy for the impact on the productivity of labor input of demographic changes and/or changes in labor quality.

b. Estimation results

A number of alternative specifications of multifactor productivity equations are reported in Table 6. ^66/ All equations include the stock of privately financed R&D (rd) lagged two periods, and current and lagged man hours (h) multiplied by a demographic share variable (youth, SE, or female, SF). The estimated equations track developments in multifactor productivity well, with no evidence of autocorrelated errors or instability.

Table 6.

Multifactor Productivity Equations ^1/

^1/See note to Table 1. The dependent variable is the logarithim of multifactor productivity (y-λh-(1-λ)k) defined with moving factor shares as geometric weights. All equations are estimated by ordinary least squares on quarterly data from 1965:I to 1987:IV (92 observations).

Table 6.

Multifactor Productivity Equations ^1/

	1	2	3	4	5
Constant	−0.002	−0.003	0.017	−0.097	−0.010
	(0.002)	(0.002)	(0.02)	(0.03)	(0.003)
rd-2	0.057	0.021	0.019	−0.056	0.017
	(0.02)	(0.007)	(0.01)	(0.03)	(0.007)
${\mathrm{rd}}_{\mathrm{-}\mathrm{2}}^{\mathrm{f}}$	−0.057
	(0.02)
SE·h	0.019	0.024		0.018	0.0265
	(0.005)	(0.005)		(0.005)	(0.005)
(SE·h)-1	−0.024	−0.026		−0.024	−0.0261
	(0.004)	(0.004)		(0.004)	(0.004)
SF·h			0.011
			(0.002)
(SF·h)-1			−0.012
			(0.002)
pg-p					−0.017
					(0.006)
TREND				0.001
				(0.0004)
(y-λh-(1-λ)k)-1	0.907	0.843	0.830	0.913	0.737
	(0.05)	(0.05)	(0.05)	(0.05)	(0.06)
R2	0.944	0.944	0.939	0.949	0.948
SEE (multiplied by 100)	0.713	0.713	0.742	0.679	0.689
Mean lag (quarters)	9.8	5.4	4.9	10.5	2.8
F-statistics:
Autocorrelation (1-4) (CV = 2.5)	0.21	1.04	1.42	0.09	1.17
ARCH (4) (CV = 2.5)	0.33	0.20	0.17	0.39	0.26
Break 84:IV (CV = 1.9)	1.54	1.83	1.80	1.24	1.31
Break 81:IV (CV = 1.7)	0.85	0.96	0.99	0.74	0.70

^1/See note to Table 1. The dependent variable is the logarithim of multifactor productivity (y-λh-(1-λ)k) defined with moving factor shares as geometric weights. All equations are estimated by ordinary least squares on quarterly data from 1965:I to 1987:IV (92 observations).

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Table 6.

Multifactor Productivity Equations ^1/

	1	2	3	4	5
Constant	−0.002	−0.003	0.017	−0.097	−0.010
	(0.002)	(0.002)	(0.02)	(0.03)	(0.003)
rd-2	0.057	0.021	0.019	−0.056	0.017
	(0.02)	(0.007)	(0.01)	(0.03)	(0.007)
${\mathrm{rd}}_{\mathrm{-}\mathrm{2}}^{\mathrm{f}}$	−0.057
	(0.02)
SE·h	0.019	0.024		0.018	0.0265
	(0.005)	(0.005)		(0.005)	(0.005)
(SE·h)-1	−0.024	−0.026		−0.024	−0.0261
	(0.004)	(0.004)		(0.004)	(0.004)
SF·h			0.011
			(0.002)
(SF·h)-1			−0.012
			(0.002)
pg-p					−0.017
					(0.006)
TREND				0.001
				(0.0004)
(y-λh-(1-λ)k)-1	0.907	0.843	0.830	0.913	0.737
	(0.05)	(0.05)	(0.05)	(0.05)	(0.06)
R2	0.944	0.944	0.939	0.949	0.948
SEE (multiplied by 100)	0.713	0.713	0.742	0.679	0.689
Mean lag (quarters)	9.8	5.4	4.9	10.5	2.8
F-statistics:
Autocorrelation (1-4) (CV = 2.5)	0.21	1.04	1.42	0.09	1.17
ARCH (4) (CV = 2.5)	0.33	0.20	0.17	0.39	0.26
Break 84:IV (CV = 1.9)	1.54	1.83	1.80	1.24	1.31
Break 81:IV (CV = 1.7)	0.85	0.96	0.99	0.74	0.70

^1/See note to Table 1. The dependent variable is the logarithim of multifactor productivity (y-λh-(1-λ)k) defined with moving factor shares as geometric weights. All equations are estimated by ordinary least squares on quarterly data from 1965:I to 1987:IV (92 observations).

The first equation in Table 6 also includes the stock of R&D financed by the Federal Government (rd^f). The estimated coefficient is incorrectly signed. Other studies often find a smaller impact from federally, as opposed to privately, financed R&D. This may reflect the fact that much government sponsored R&D is for defense, which may have a lower payoff than other research in terms of measured productivity growth. ^67/

Whether labor input (h) is multiplied by the employment share of youth (age 16 to 24, SE) or females (SF), as in the third equation in Table 6, the effect (after two quarters) of increases in either share is to reduce multifactor productivity. The specification with youth is marginally better, and the argument that new entrants to the labor force are less productive than more experienced workers seems more defensible than the argument that women are less productive than men.

When lag distributions on labor and capital inputs are included in the estimated equations in addition to labor input adjusted by the share of youth, the estimated coefficients are not significantly different than zero. Labor input and labor input adjusted by the share of youth are, of course, strongly correlated. If labor input is entered without any demographic interaction, the estimated coefficient is significantly negative and reduces by about one-half the constrained coefficient based on factor shares. The interpretation of the equations reported in Table 6, with a demographic/labor-quality adjustment to labor input, is more straightforward than the alternative with a very small estimated impact from labor input.

A time trend is included in the fourth equation as a proxy for that part of technical progress not related to the R&D stock. The fit of the equation is improved slightly with the inclusion of the trend but the estimated coefficient on the stock of research and development changes sign. This result may reflect multicollinearity between the two variables since the stock of privately financed research and development itself has a pronounced trend (Chart 4). The information content in the research and development variable is clearly greater, although its estimated coefficient may also be picking up the impact of other omitted trend variables, such as technological advance unrelated to R&D.

The final equation in Table 6 includes the relative price of gasoline (p^g-p). The estimated coefficient is significant but suggests an extremely large impact from the increase in oil prices, one far greater than would be consistent with the importance of energy as an intermediate input. ^68/ Moreover, given that the estimated equations are based on value-added output, it is not clear how increases in the price of energy, an intermediate input, could have such large effects.

c. An expression for potential output

The long-run version of the estimated equation, together with auxiliary assumptions about “potential” levels of factor inputs, can be used to determine multifactor productivity at potential output. Based on the second equation reported in Table 6, the long-run equation for multifactor productivity is (standard errors in parentheses):

To calculate potential output or, with appropriate transformations productivity at potential output, each variable must be expressed at its “natural” or potential level. This is discussed below based on the system estimation results. The long-run elasticity of output with respect to the stock of R&D capital is somewhat higher than typically found in studies based on firm and industry level data. ^69/ As noted above, this is probably because the R&D stock variable is also serving as a proxy for other factors, such as the steady advance of knowledge which is independent of R&D expenditures.

a. Specification issues

Much of the empirical work on price determination at the macro level has been based on some variant of the markup hypothesis. ^70/ According to this hypothesis, prices are set as a markup over unit costs, with the markup varying with the state of excess demand in goods markets. A dynamic form of the markup hypothesis is specified in equation [4] above where the increase in the (value added) deflator in the nonfarm business sector is related to the rise in normalized unit labor costs (Δw-Δq^tr) and to excess demand as measured by the gap between actual and trend output (y-y^tr). Given that these are value-added prices, the coefficient on normalized unit labor costs would be expected to be unity.

In equilibrium, with actual output equal to trend, inflation will equal the growth of normalized unit labor costs, implying a constant markup and stable income shares. It is worth noting that this equilibrium condition (Δp = Δw - Δq^tr) is consistent—indeed, is a restatement—of the equilibrium condition discussed above in connection with the Phillips curve, i.e., the increase in real wages is equal to the growth of trend productivity (Δw - Δp = Δq^tr).

In both the price equation and the Phillips curve there are strong theoretical priors about some long-run parameters. Many of the specification issues are also similar. One issue is whether the price equation should specify a level as opposed to a rate-of-change relationship between prices and normalized unit labor costs. This was tested by including an error-correction term in the equation; ^71/ the estimated coefficient was not significantly different from zero suggesting a growth-rate formulation is appropriate, the same result obtained with the wage equation. Another issue common to both equations is the role of changes in, as opposed to the level of, the gap terms.

b. Estimation results

A number of estimates of the price equation are presented in Table 7. Based on a preliminary search over alternative lag distributions, normalized unit labor costs are defined as a 12-quarter moving average of actual compensation growth minus either a 12-quarter moving average of actual labor productivity growth (θ(L)(Δw-Δq)) or trend productivity growth (θ(L)(Δw-Δq^tr)). All variables refer to the private nonfarm business sector (Table 2, Charts 1 and 2).

Table 7.

Price Equations ^1/

^1/See note to Table 1. The dependent variable is the annual percentage change in the deflator for private nonfarm output.

^2/Coefficient constrained to unity.

Table 7.

Price Equations ^1/

	1	2	3	4	5	6
Constant	0.331	0.570	0.627	0.826	−0.169	−0.203
	(0.64)	(0.37)	(0.35)	(0.31)	(0.16)	(0.15)
θ(L)Δw-Δqtr	0.938	0.857	0.846		1.000 2/
	(0.09)	(0.06)	(0.06)
θ(L)Δw-Δq)				0.802		1.000 2/
				(0.05)
y-ytr	0.300	0.345	0.370	0.447	0.371	0.511
	(0.08)	(0.06)	(0.06)	(0.06)	(0.06)	(0.06)
Δ(y-ytr)			−0.457
			(0.16)
Δ2pg		0.020	0.021	0.023	0.019	0.023
		(0.01)	(0.01)	(0.01)	(0.01)	(0.01)
D(68:I)		3.513	3.491	3.295	3.747	3.589
		(1.43)	(1.37)	(1.32)	(1.45)	(1.41)
D(74:II to 75:I)		6.195	5.554	6.617	5.892	6.274
		(0.75)	(0.76)	(0.69)	(0.76)	(0.73)
D(72:I to 73:I)		−2.167	−1.571	−1.335	−2.192	−1.168
		(0.65)	(0.66)	(0.61)	(0.66)	(0.64)
D(66:IV) - D(67:II)		3.740	3.679	3.782	3.738	3.789
		(0.99)	(0.96)	(0.93)	(1.02)	(0.99)
R2	0.539	0.787	0.806	0.816	0.580	0.647
SEE	2.024	1.413	1.357	1.312	1.444	1.400
DW	1.49	1.94	2.03	2.15	1.86	1.89
F-statistics
Autocorrelation (1-4) (CV = 2.5)	3.17	1.63	0.39	0.48	2.50	1.29
ARCH (4) (CV = 2.5)	7.89	0.99	1.02	0.83	0.42	0.67
Break 85:IV (CV = 1.9)	0.49	0.91	0.81	0.98	0.86	0.85
Break 82:IV (CV = 1.7)	0.34	0.66	0.51	0.75	0.61	0.79

^1/See note to Table 1. The dependent variable is the annual percentage change in the deflator for private nonfarm output.

^2/Coefficient constrained to unity.

View Table

Table 7.

Price Equations ^1/

	1	2	3	4	5	6
Constant	0.331	0.570	0.627	0.826	−0.169	−0.203
	(0.64)	(0.37)	(0.35)	(0.31)	(0.16)	(0.15)
θ(L)Δw-Δqtr	0.938	0.857	0.846		1.000 2/
	(0.09)	(0.06)	(0.06)
θ(L)Δw-Δq)				0.802		1.000 2/
				(0.05)
y-ytr	0.300	0.345	0.370	0.447	0.371	0.511
	(0.08)	(0.06)	(0.06)	(0.06)	(0.06)	(0.06)
Δ(y-ytr)			−0.457
			(0.16)
Δ2pg		0.020	0.021	0.023	0.019	0.023
		(0.01)	(0.01)	(0.01)	(0.01)	(0.01)
D(68:I)		3.513	3.491	3.295	3.747	3.589
		(1.43)	(1.37)	(1.32)	(1.45)	(1.41)
D(74:II to 75:I)		6.195	5.554	6.617	5.892	6.274
		(0.75)	(0.76)	(0.69)	(0.76)	(0.73)
D(72:I to 73:I)		−2.167	−1.571	−1.335	−2.192	−1.168
		(0.65)	(0.66)	(0.61)	(0.66)	(0.64)
D(66:IV) - D(67:II)		3.740	3.679	3.782	3.738	3.789
		(0.99)	(0.96)	(0.93)	(1.02)	(0.99)
R2	0.539	0.787	0.806	0.816	0.580	0.647
SEE	2.024	1.413	1.357	1.312	1.444	1.400
DW	1.49	1.94	2.03	2.15	1.86	1.89
F-statistics
Autocorrelation (1-4) (CV = 2.5)	3.17	1.63	0.39	0.48	2.50	1.29
ARCH (4) (CV = 2.5)	7.89	0.99	1.02	0.83	0.42	0.67
Break 85:IV (CV = 1.9)	0.49	0.91	0.81	0.98	0.86	0.85
Break 82:IV (CV = 1.7)	0.34	0.66	0.51	0.75	0.61	0.79

^1/See note to Table 1. The dependent variable is the annual percentage change in the deflator for private nonfarm output.

^2/Coefficient constrained to unity.

The first equation in Table 7 includes only the increase in normalized unit labor costs and the level of the output gap, both of which are highly significant and have the expected signs. The estimated coefficient on the increase in unit labor costs is not statistically different from its expected value of unity. There is, however, evidence of autocorrelation and heteroskedasticity in the residuals, and there are some very large residuals in the late 1960s and early 1970s.

The second equation in Table 7 includes a number of dummy variables and the acceleration of gasoline prices. As in the wage equations, the dummy variables are for periods of unusual inflation developments in the late 1960s and the early 1970s, a period that included exogenous price shocks and the Nixon price controls. Including the dummy variables does not significantly change the estimated coefficients on the other variables but does contribute to a substantial improvement in the summary and diagnostic statistics.

In addition to gasoline prices, other supply variables such as the terms of trade were included in the equations. Possible impacts on profit margins of various measures of foreign competition were also tried. The estimated coefficients on these variables were either insignificant arid/or incorrectly signed. Although the estimated coefficient on the acceleration of gasoline prices is correctly signed and statistically significant, it is relatively small and implies that gasoline prices have had only a small impact on the rise in (value added) output prices.

The third equation reported in Table 7 includes the change as well as the level of the output gap. If, at a given level of the output gap, inflationary pressures are stronger when the gap is rising its change would enter the equation with a positive sign. The results, which are not affected by whether the dummy variables are included in the equation, suggest a statistically significant but perversely signed impact from the change in the output gap. ^72/ This result is similar to that reported above for the wage equation.

The second and fourth equations in Table 7 are the same except for the definition of trend productivity. Using a 12-quarter moving average of actual productivity growth improves the fit of the equation modestly relative to the use of trend productivity growth. The size of the estimated coefficient on the output gap is also increased. The last two equations reported in Table 7 impose homogeneity by constraining the estimated coefficients on the increase in normalized unit labor costs to unity. While the restriction that the estimated coefficient on unit labor costs equals unity is not rejected for either equation, the estimated impact of the output gap is larger in the final equation where the moving average rather than trend productivity is used. ^73/ There is some evidence of serial correlation in the equation with trend productivity. ^74/

a. Specification of the system

Table 8 reports the equation specifications for the system estimates. ^75/ Compared to the single-equation estimates based on the specification of equations [1]-[4] above, the primary differences in the specifications reported in Table 8 are:

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—	trend output (ytr) in the unemployment rate and price equations has been replaced by the expression for potential output derived from the multifactor productivity equation;
—	trend labor productivity growth (Δqtr) in the Phillips curve and price equations has been replaced by the expression for labor productivity growth at potential output;
—	the implicit natural rate of unemployment in the constant term in the Phillips curve has been replaced by the expression for the natural rate derived from the unemployment equation;
—	the constant terms in the Phillips curve and price equations have been incorporated into the expressions for the natural rate and potential output, reflecting the homogeneity of these equations when the unemployment, and output gaps are zero.

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—	trend output (ytr) in the unemployment rate and price equations has been replaced by the expression for potential output derived from the multifactor productivity equation;
—	trend labor productivity growth (Δqtr) in the Phillips curve and price equations has been replaced by the expression for labor productivity growth at potential output;
—	the implicit natural rate of unemployment in the constant term in the Phillips curve has been replaced by the expression for the natural rate derived from the unemployment equation;
—	the constant terms in the Phillips curve and price equations have been incorporated into the expressions for the natural rate and potential output, reflecting the homogeneity of these equations when the unemployment, and output gaps are zero.