The Phillips curve behavior over different horizons

Journal ag eEconomics and Finance �9 Volume 19, Number 3 . Fall 1995 �9 Pages M-69

The Phillips Curve Behavior Over Different Horizons J/m Lee

ABSTRACT

This article examines the historical behavior of the Phillips curve over frequency bands corresponding to the short run, the business cycle and long run horizons. Data transformed using band-pass filtering methods and the Hodrick-Prescott filter suggest that a negative correlation between inflation and unemployment and a positive correlation between inflation and output growth exist within the business cycle horizon of 3 to 8 years. During the post-war period, the relationships change signs at low frequencies, indicating a positive sloped Phillips curve over long horizons. Additionally, the Phillips curve is found structurally unstable not only across frequencies, but also over time.

Introduction

Today, there is renewed interest in the Phillips curve. The Federal Reserve's recent tight monetary policy was founded on the premise that too much output or job growth would be inflationary, and that high inflation would hinder long-term economic performance. Two questions have been raised against such a premise. First, has the Phillips curve trade-off relationship remained relevant to the U.S. economy since the 1970s when major oil shocks disrupted much of its historical regularity? Second, is there any correlation between price stability and output growth in the long run? This article is an attempt to explore these two issues in retrospect.

On analytical grounds, Friedman (1968) and Phelps (I967) convincingly argue that the Phillips curve trade-off is solely a short-run phenomenon. The negative correlation between unemployment and inflation occurs when inflation is imperfectly anticipated. When inflation becomes fully anticipated in the long run, their relationship vanishes, rendering a vertical Phillips curve. Friedman (1977) further argues that as inflation rises, inflation uncertainty rises as well, leading to permanent

Jim Lee, A.~istant Professor of Economics and F'm~r.e, Fort Hays State UniversiIy, Hays, KS 67601

51

JOURNAL OF ECONOMICS AND FINANCE Volume 19 Number 3 Fall 1995

resource misallocations. Accordingly, the Phillips curve slopes upward in the long run. On the contrary, Blanchard and Summers (1987) argue for a permanent trade- off relationship as a result of hysteresis in the unemployment rate.

Despite earlier empirieaI evidence (Gordon I985; Stockton and Glassman 1987) on the robustness of the link between inflation and unemployment, Judd and Beebe (1993) have documented anomalous features of the Phillips curve since the 1970s. Furthermore, Lueas (1976) argues that, since the Phillips curve is not a truly structural model, it is temporally unstable as individual expectations adjust according to changes in the underlying economic structure.

Recently, Fuhrer (1995), King and Watson (1994), and King, Stock and Watson (1995) reexamined the empirical short-run Phillips curve relationship and found it both robust and stable over post-war periods. In particular, the King, Stock, and Watson (1995) paper analyzes co-movements in unemployment and inflation by first extracting their cyclical components (i.e., those with a periodicity of less than 8 years) from raw data using the Hodriek-Prescott (1980), henceforth HP, filter. The HP filter has also been recently used by Sbordone and Kuttner (1994) to examine long-run effects of inflation on productivity.

The HP filter has been popularly applied in business cycle research to extract data signals within an 8-year horizon. In effect, it resembles a band-pass, henceforth BP, filter using spectral or frequency domain methods developed by Eagle (1974, 1980). Bullard (1994), Lueas (1980) and Summers (1983) have applied BP filtering techniques to examine long-run models at very low frequencies (above 20 years). Along this line, we explore the historical Phillips curve behavior over different horizons, particularly those beyond the business cycle.

The remainder of this paper proceeds as follows. The next section presents the estimation models, data and least-squares results. The third section examines the Phillips curve from spectral perspectives. The fourth section discusses estimates of a Phillips eurve model using filtered d~t~, and the last section concludes.

Models and Data

Following Gordon (1985), the traditional econometric model that describes the expectations-augmented Phillips curve can be written as:

K

=t = P0 + Pxi ,-1 + & x , + (1) i-1

where "rr t denotes the rate of inflation, x t reflects the amount of slack in economic activity which can be measured by the unemployment rate above its natural level, and s t is a random term. The lagged dependent terms reflect the ~dzptive nature of inflation expectations in the labor market. Variable x t can alternatively be represented by the demand pressure in the economy, or the output growth rate above its trend. The natural level of unemployment or output can be captured by the

52

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intercept B0, such that 82 (or the "Phillips coefficient") represents the Phillips curve trade-off (with a negative sign for unemployment and a positive sign for output growth).

Some studies (Benderly and Zwick 1985; and I-laslag and Ozment 1991) also have found that money growth, particularly M2, helps predict inflation in the context of the above Phillips curve model. Incorporating the lagged effects of money growth (mr) into Equation (1) would yield: t

K K

i=l i=l

There are several ways to obtain long-run Phillips curve estimates. The first is to restrict the sum of the coefficients on lagged inflation to sum up to unity (Gordon 1985; and Stockton and Struekmeyer 1989). In effect, the restriction reflects full anticipation of inflation in the long run. This model-based method, however, is sensitive to possible structural instability and, thus, misspecifications. Another approach is the use of cross-country data. However, the empirical results are sensitive to modifications to the sample of countries and the time period. The third approaeh is based on cointegration and error-correction models to capture long- and short-run dynamics among series which contain stochastie trends or unit roots. However, as shown below, the inflation data do not support the presence of a unit root.

As an alternative, we explore results in light of band-pass data filtering methods, which have gained popularity in empirical researeh. To assess the robustness of our inferences, we further investigate two dat8 sets. The first set contains annual data over a century-long period of 1891-1992, and the second set contains quarterly data over the post-World War II period of 1955:1-1992:4. 2 The inflation series is measured by the percentage change of the consumer price index (CPI), while the output growth series is measured by the percentage change of GDP. To account for supply shocks from food and energy prices that were evident mostly in the post-war period, we further examine the core inflation rate (CPI inflation less food and energy) as opposed to the CPI inflation rate over 1957:1-1992:4.

Since either time or frequency domain analysis requires that data be stationary to avoid spurious inferences, we first conduct tests for unit roots. Table 1 reports unit- root test results for the inflation, unemployment, output growth and M2 growth data based alternatively on augmented Dickey-Fuller 0981) and Phillips-Perron (PP, 1988) t-statistics: Except for the levels of unemployment in PP tests, the null hypothesis of a unit root is rejected at the .05 significance level in most cases. Therefore, we conclude that all but the unemployment series are either mean or trend stationary. Given the mixed results for the unemployment series, we consider both their (annual and quarterly) levels and first differences.

53


I . 1 i ADF4 PP4 Trend Lags Without With Lags W'~hout W ' ~ Test

Trend Tread Trend Trend A. Annum Data

CPI Inflation 1 -5.82* -6.00* 6 -4.16" -4.37* 1.75 Unemployment 7 -3.77* -3.57* 6 -3.23* -3.32 0.11 Output Growth 8 -4.85* -4.67* 6 -7.07* -7.08* -0.18 M2 Growth 0 -9.28* -8.64* 6 -9.33* -9.38* 0.47

R. auarter~ Oata

CPI Inflation 7 -2.98* -3.68* 6 -4.86* -5.23* 2.62* Core Inflation 0 -3.39* -3.70* 6 -3.25* -3.58* 2.23* Unemployment 1 -3.32* -3.69* 6 -2.26 -2.63 2.17" Output Growth 0 -8.93* -8.07* 6 -9.01" -9.09* -0.42 M2 Growth 0 -4.89* -5.39* 6 -4.82* -4.81" 2.81"

Note: The augmented Dickey-Fuller (ADF) t-test for a unit root (with constant and trend) is based on the t-statistic for &2 in the regression:

M AZt = ~'0 + ~r + a~2Zt-I i~la~3Azt-i + Et

where z t is a relevant 4~t, series. The second term on the right hand side of the equation is omitted for tests without a trend. The lag order M is determined by the Akaike Information Criterion. The Phillips-Perron (PP) t-test (with constant and trend) is based on the regression: A z t = r~ 0 + a lt + AzAzt_ 1 + e t, and the number o flags used to estimate the long-run variance is fixed at 6, which is in line with Said and Diekey's (1984)T tn rule where T is the number of observations. The annual d~_t~_ span from 1891 through 1992. Except for the core inflation data which cover the period 1957:1-1992:4, the quarterly data cover the period 1955:1-1992:4. The 5% critical values for testing the null hypothesis of a unit root is -2.88 without trend and -3.45 with trend (Davidson and MacKirmon 1993). Figures under "Trend Test ~ are t-statistics for &l in the above ADF regression. (*) Indicates rejection of the null hypothesis at the 5% level.

The r ight-most co lumn o f Table i displays the t-statistics on the t rend coefficients

in A D F tests. Except for output growth , a linear trend is f ound statistically

signif icant fo r all pos t -war quarterly data series. Fo r this reason, these pos t -war data

(along wi th the post -war subsample o f the annual data over 1946-1992) are detrended

pr ior to fur ther analysis.* To mot iva te our study, we first present ordinary least-squares (OLS) results fo r

the phi l l ips curve models depicted alternatively by Equations (1) and (2). Lag lengths

are chosen according to the Akaike Informat ion Criterion, and long- run estimates are

54


obtained by restricting the sum of the coefficient estimates on the lagged dependent variable to unity. The lag length for the dependent variable in the CPI or core inflation equation is K = 3 (for both annual and quarterly data). For the annual M2 growth data, the lag length is L=6; and for the quarterly data, it is L = 1.5

Table 2 reports estimation results using annual data over 1891-1992. Alternative estimates for the Phillips coefficient/3 2 all bear the expected signs. By contrast, the sums of the coefficient estimates on money growth are largely insignificant. Figures on rows (1') and (2') are long-run estimates obtained by coefficient restrictions, i.e.,

K

fill = 1 i - I

Imposing long-run restrictions renders the coefficient estimates on unemployment and output growth statistically insignificant, while the coefficient estimate on the change (first-difference) in the unemployment rate remains significant with a negative sign.

To test for structural stability, Chow tests are performed by dividing the sample into pre- and post-World Wax 1I periods, i.e., 1891-1945 and 1946-1992, respectively. The F-statistics on the right-most column of Table 2 indicate that the null hypothesis of structural stability is rejected when output growth or unemployment change is used as an explanatory variable.

Table 3 presents analogous results using quarterly data over 1955:1-1992:4 for CPI inflation, and 1957:1-1992:4 for core inflation. To reduce the simultaneity bias found in preliminary estimations, one-period lagged values of unemployment and output growth are used in place of their current values. Evidently, the sums of coefficient estimates on lagged inflation are much closer to unity than their counterparts in Table 2, reflecting a higher persistence in inflation during the post- war period. This is reinforced by stronger evidence of serial correlation indicated by Ljung-Box Q-statistics. Furthermore, the Phillips coefficient estimates become less sensitive to long-run restrictions.

The output growth coefficient estimates in the core inflation equation differ remarkably from their counterparts in the CPI inflation equation. This reflects that, compared to unemployment, output is more sensitive to supply shocks. Chow tests for structural stability are performed using 1974:1 as a break date (which bisects the 1955:1-1992:4 sample). The null hypothesis of structural stability is rejected primarily for the inflation relationship with unemployment, but not its relationship with output growth or the first-difference of unemployment.

Taken together, the OLS estimates of the Phillips curve trade-off are robust to different measures of real economic activity and change tittle as the money growthvariable is included. Historical evidence on its long-run behavior, on the other hand, is inconclusive. Inconsistent long-run estimates between the two data sets might be attributable to different degrees of persistence in inflation or structural instability over time.

55


TABLE 2 OLS Results for Annual Data

K L / R e g r e ~ O l l ; "tr t = J~O 4- i-lY~'~li'lrt-i + ~ 2 x t + i~l ~ 3 i n l t - i = + t~t

Vat. xt: Coefficient Estimates Summary Statistics

Adj . /~ Q Chow F K

i=l

~ 2 L

I - I

:: ::::: ::: ::::::: ::::::::::::::: :: :::: ::::: ::::::::: ::: :: ::::::::::::: :::::::::::::::::::::::::::: :::::::::: :i:: ::,::,:i:: :i:i:i::: :::::::::i: ::::::::::::::::::::::::::::::::::: :i: :i:i: ::::::::::::::::::::::::::::::::::::::::::::::: ::!::::::::::::

(1) 0.53 (6.85)*

(1') 1.00

(2) 0.54 (6.31)*

(2') 1.oo

-0.15 (2.38)* -0.01 (0.12) -0.16 (1.99)* 0.01 (0.18)

0.03 (O.74) 0.10 (1.98)*

0.52

0.53

19.79

21.68

1.48

1.36

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0.52 19.76 4.45* (1)

(1')

(2)

(2')

0.62 (8.61)* 1.00

0.60 (7.71)* 1.00

0.16 (2.70)* 0.18 (1.7O) 0.13 (2.13)* 0.12 (1.62)

0.01 (0.15) 0.01 (1.32)

0.53 20.58 2.29*

(1)

(1')

(2)

(2')

0.64 (10.01)* 1.00

0.61 (8.71)* 1.00

-0.68 (5.84)* -0.74 (5.62)* -0.72 (5.o9)* -0.73 (4.52)*

0.02 (0.54) 0.03 (0.68)

0.62

0.61

14.89

15.60

2.65*

1.97

Note: Estimates for (1') and (2') are long-run estimates obtained by restricting~ ~i i = I in (1) and (2), respectively. Absolute t-statistics are reported in parentheses. Q-valuesi%[re Ljung-Box test statistics for autoeorrelation using 25 lags. F-value~ are Chow test statistics for structural stability between the periods 1891-1945 and 1946-1992. (*) repnments stalistieal significance at the .05 level.

56


Spectral Analys

The relationship between two series across different time horizons can also be shown in the frequency domain. Figure la illustrates the degree of correlation, or squared coherence (analogous t o R~ i n time-domain analysis), between the annual CPI inflation and unemployment data for the pre- and post-war samples. The grids highlight frequencies corresponding to the business cycle periodicity of 3 to 8 years (King, Stock and Watson 1995).6 The peaks of both coherence functions are located within the business cycle horizon. In addition, their patterns beyond a 5-year horizon a re in tandem to each other. Figure lb displays the corresponding coherence functions between inflation and output growth. Similarly, their peaks lie around a &year r od.

The critical value for testing the significance of correlation is approximately .5 at the 5 % level, as given by Amos and Koopmans (1963). In addition to the peaks within business cycle frequencies, some coherence values at very low frequencies (over 20 years in duration) remain statistically significant. This suggests the existence of a Phillips curve relationship at long horizons.

1.00 +. Inflzmon utd U~ ,mC~-'mms ]

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0~0 ', ; ., , '

�9 'I ~~ �9 ~ "' II ' , t ; I , . " , : :~ I : "" :

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F I G U R E 1A - Inf lat ion and U n e m p l o y m e n t

3.o9 - I~. ~ +zd Ore:pint ~rm+.m "- i

F I G U R E 1B - Inf lat ion and O u t p u t Growth

Given the varying degrees of correlation between inflation and economic activity across frequencies and that much of its power falls within the business cycle periodicity, conventional time-series methods hi~hlight primarily a cyclical phenomenon. More specifically, the OLS results reported in Tables 2 and 3 might

57


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Table 4: Estimation Results for Annual Data with BP and l i p F'dters

V a t . X t:

K L

Regression: "tr t = flo + t~lflH'n't-a § fl2xt + ~fl3imt-i + e~ i-I

Summary Statistics

z f12 L Adj. Chow F ~ f l l t ~ f l 3 i R 2 iffil i=l

Unemployment BP Filter Frequency:

High (Below 3 yrs.) Cyclical (3-8 yrs.) LOw (Above 8 yrs.) Very Low (Above 20 yrs.)

0.66 -0.34 0.02 0.74 1.74 ~877). (393). ~lg)

-0.11 0.84 1.53 ~1.3~i87)* (3.99)* ~2~)78)*

-0.02 . 0.91 3.87* ~1.~91)* (2.36)* (3.94)*

-.003 .001 0.99 5.45* (51.69)* (1.58) (1.82)*

HP Filter Frequency: Cyclical (Below 8 yrs.) 0.53 -0.30 0.01 0.55 1.25 Low (Above 8 yrs.) ~)5.~19)* (2.17)* (0.57)

-0.001 .001 0.99 22.13" (43.23)* (0.23) (I,1~

Output Growth BP Filter Frequency:

High (Below 3 yrs.) 0.70 0.12 0.02 0.76 4.29* Cyeli~(3-Sy~.) ~1.~73)* !?~)* ~i.0~3)* 0.82 4.70* Low (Above 8 yrs.) (15.50)* (0.89) (2.06)* Very LOw (Above 20 yrs.) 0.87 0.08 0.01 0.92 4.46*

t% 68)* ~ g 7 (2.77)* .001 0.99 7.43* (68.12)* (4.86)* (5.70)*

HP Filter Frequency: Cyclical (Below 8 yrs.) 0.61 0.19 0.01 0.58 2.45* LOw (Above 8 yrs.) ~6~9)* (2.97)* (1.23)

-0.01 .001 0.99 20.93* (68.29)* (I .44) (0.46)

Unemployment Change BP Filter Frequency:

High (Below 3 yrs.) 0.72 -0.55 0.02 0.81 2.37* Cyclieal(3-Syrs.) ~)1.1~97)* !.66~)* (2.91)*

. 0.03 0.83 1.86"* Low (Above8 yrs.) ~1.~15)* (2.54)* ~l.b ?) Very Low (Above 20 yrs.) -0.30 0.93 4.36*

~.~0~*. -0.01(428)* !~?). 0.99 6.77- (72.86)* (3.88)* (4.36)*

HP Filter Frequency: Cyclical (Below 8 yrs.) 0.61 -0.60 0.02 0.71 1.90"* Low (Above 8 yrs.) ~8.#)* (5.43)* ! ~ *

-0.02 0.99 4.22* (87.83)* (2.81)* (0.83)

Note: Absolute t-stAtistics are reported in parentheses. F-values are Chow test statistics for structural stability between the periods 1891-1945 and 1946-1992. (*) and (**) represent statistical significance at the .05 and .I0 levels, respectively.

60


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63


be unreliable for tracking the Phillips curve behavior over horizons beyond the business cycle.

Data Filtering Results

In this section, we seek to explore the Phillips curve behavior in various time horizons using data filtering methods. The first method is in line with Engle's (1974) band-spectral regression procedure. Briefly, data are first decomposed through a BP filter in the frequency domain into signals specific to high (below 3 years), business cycle (3 to 8 years), low (above 8 years) and very low (above 20 years) frequency bands. Frequency-specific coefficient estimates are sequentially obtained by estimating the filtered data in the time domain. The second data decomposition method stems from the HP filter. 7 King and Rebelo (1993) show that the I-IP filter is an approximation of a BP filter that extracts low frequency signals of over 8 years, with the residuals treated as the cyclical component. Because of serial correlation, which has been found remarkably high in low frequency estimations, standard errors are estimated using Hannah's (1963) efficient procedure.

Estimation results for Equation (2) with filtered annual data over 1891-1992 are reported in Table 4. s Several aspects of the results are of special interest. First, the goodness-of-fit represented by adjusted R 2 increases as the frequency decreases (or the period lengthens). This is because the lagged values of the dependent variable approach its current value as the frequency declines. Second, the coefficient estimates on unemployment are negative at the high, cyclical and low frequency bands. On the other hand, the corresponding very low BP frequency estimate as well as the low I-IP frequency estimate are insignificant. When the change in unemployment instead of its level is used, the Phillips coefficient (f12) estimates are negative across all frequency bands. The patterns of band-spectral coefficient estimates on output growth also differ from those on unemployment. The estimate is insignificant at the business cycle frequency band. Based on the HP filter, the coefficient estimates on output growth are consistent with those on unemployment (with an opposite sign).

Analogous to the OLS results in Table 2, a sprit-sample Chow test for temporal stability is performed for each estimation with the filtered data. F-statistics, which are listed on the right-most column of Table 4, indicate particularly strong evidence of structural instability across World War 11 at low and very low frequency bands.

Table 5 reports estimation results for quarterly data. The left-hand side of the table displays estimates using CPI inflation data over 1955:1-1992:4. First, the coefficient estimates on the lagged dependent variable and money growth are qualitatively similar to their counterparts in Table 4. However, the adjusted R2's at

the high frequency band are much lower. This is largely due to greater high frequency variations in the quarterly as opposed to the annual inflation data and, thus, more estimation errors.

64


Second, estimates of the Phillips curve relationship vary across frequency bands. At the high frequency band, in particular, the coefficient estimates on output growth and unemployment change are insignificant. At the low or very low frequency band, on the contrary, the coefficient estimates on unemployment and its change become positive while that on output growth becomes negative. Similar low frequency estimates on unemployment and its change are found with data decomposed by the HP filter.

The right-hand side of Table 5 reports results for core inflation as opposed to CPI inflation over 1957:1-1992:4. When food and energy prices are factored out from CPI inflation, there are qualitative changes in some estimates of the Phillips coefficient. First, for the unemployment coefficient, the low BP frequency estimate becomes positive. Second, for the output growth coefficient, its high BP frequency estimate becomes positive, while its estimate over the cyclical frequency band becomes negative.

Analogous to the OI.,S results in Table 3, Chow tests are further performed using period 1974:1 as a break date. As shown in Table 5, the F-statistic rises as the frequency declines, indicating stronger evidence of structural instability at low frequencies than high frequencies. The results with the HP filter are similar, with evidence of structural instability only for the low frequency data.

In sum, the pest-war data suggest a robust Phillips curve trade-off relationship within the business cycle periodicity of 3 to 8 years. Estimation results over the cyclical frequency baud closely resemble the OLS counterparts in Table 2. Beyond the business cycle horizon, higher inflation appears to coexist with higher unemployment and lower output growth, implying a positive sloped Phillips curve. The latter finding contradicts results using data that cover the past century, suggesting that the Phillips curve over long horizons is uniquely a post-war episode.

While the overall results from alternative filtering methods are similar, significant temporal instability in parameter estimates at business cycle frequencies is found in the data transformed with the BP filter, but not with the HP filter employed here as well as in King and Watson (1994) and King, Stock, and Watson (1995). Therefore, in light of band-spectral regression results, we reverse their finding of a stable Phillips curve over the business cycle.

Conclusion

In this paper, we have examined the historical behavior of the Phillips curve relationship over a wide range of frequeney bands or time horizons. In essence, we have found varying degrees of correlation between inflation and real activity across frequencies. Much of their correlation lies within the business cycle horizon of 3 to 8 years. This has provided motivation for exploring the Phillips curve behavior over other horizons, particularly the long run, using methods alternative to conventional regression analysis.

65


Besides being unstable across frequencies, the Phillips curve also shifts across World War II and the mid-1970s. The evidence of short-run instability contradicts earlier findings by Fuhrer (1995). On the other hand, our results reinforce King, Stock and Watson's (1995) finding that the Phillips curve is more temporally unstable in the long run than the short run. For the post-war period, in particular, we have found a robust Phillips curve trade-off relationship within the business cycle horizon, beyond which the relationship changes sign. The positively sloped long-run Phillips curve, which lends support to Friedman's (1977) view, appears to be uniquely a post-war episode.

At first glance, our results are of immediate importance for the conduct of stab'flization policy. For instance, when a Phillips curve trade-off relationship exists in the short run or the business cycle, the Fed can choose between expansionary policies, which would increase output and employment at the expense of higher inflation, and contractionary policies, which would reduce inflation along with lower output growth and employment. A positively sloped Phillips curve in the long run, however, makes long-term policymaking more problematic. For instance, to achieve a long-run objective of perpetual economic growth would require the Fed to combat inflation. Yet, low or zero inflation in the short run must be accompanied by output and job losses:

Unfortunately, it should also be emphasized that the above policy implications are also susceptible to the Lucas (1976) critique. In essence, temporal instability present in the Phillips curve, particularly over horizons beyond the short run, would make it less relevant as a benchmark for policy evaluation.

NOTES

1. See Haslag and Oment (1991) for a detailed theoretical exposition of the expanded model.

2. The data for the CPI (quarterly), M2, real GDP, and unemployment are obtained from Gordon (1993, Appendix A). The annual CPI data over the period 1891- 1989 are given by Maddison (1991, Tables E.2-E.4) with corresponding data over the period 1990-1992 and the quarterly CPI and CPI less food and energy data obtained from the Federal Reserve Bank of St. Louis's National Economic Trend. All estimations have been replicated for the GDP deflator as opposed to the CPI, the cyclical unemployment (the unemployment rate above the natural rate) as opposed to the unemployment rate, and the output gap (the output level above the potential level) as opposed to output growth. The results are qualitatively similar to those reported here.

3. See Davidson and MacKinnon (1993) for a detailed discussion of the ADF and PP tests.

4. In line with the quarterly data, a linear time trend is found statistically significant for the post-war subsample of the annual inflation, unemployment and M.2 growth data.

5. The inconsistency in the lag lengths between the annual and quarterly data can be a result of an aggregation bias due to data averaging, or structural differences before and after World War IT.

66


6. Spectral analysis is discussed in detail by Granger and Newbold (1986). A frequency ordinate is equal to 1/T where T is the time length of a cycle, such that a higher frequency is equivalent to a shorter time period and vice versa. The (squared) coherence functions for the first-difference of the unemployment rate are similar to those for its level.

7. See King and Rebelo (1993) for a detailed comparison between the BP and ~ filters. The smoothing parameter in the I-IP filter is set to 1600 for quarterly data, and 400 for annual data.

8. The quarterly data and the post-war (1946-1992) portion of the annual data for all variables except output growth are prefiltered with a linear time trend. Estimates for Equation (1), i.e., without the money growth variable, are qualitatively similar to those for Equation (2) and, therefore, are not reported here.

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The Phillips curve behavior over different horizons