Empirical Evidence on Inflation and
Unemployment in the Long Run
Alfred A. Haug
(University of Otago)
and
Ian P. King
(University of Melbourne)
July 2011
1
Goal:


Analyze the time-series properties for:

inflation

unemployment
Establish stylized facts about the relation
between the two variables in the long run
2
Theoretical Issues:

Friedman’s 1977 Nobel lecture: long-run “Phillips
curve” may be positively sloped

Generally accepted in mainstream macro nowadays: a
vertical long-run Phillips curve

Empirical evidence is not so clear: King & Watson
(1997)

Recent theoretical developments
◦Berentsen,
Menzio & Wright (2011, AER) argue for a positive
long-run relation between inflation and unemployment
3
Econometric Tools:

As much as possible free from statistical and
economic models

Theory of spectral analysis provides rigorous
framework


extract components with specific frequencies
we are interested in co-variability of series over
frequencies lower than the business cycle (the long-run
behavior)
4
Data:

Postwar U.S. data from 1952Q1 to 2010Q1 from
Fed of St. Louis FRED data base

Civilian unemployment rate, 16 years of age and
older

CPI-based inflation and GDP-deflator based
inflation

(ln(Pt-Pt-4))100 and (ln(Pt-Pt-1))400
5
Figure 1. CPI-Based Inflation Rate, Percentages, Year-on-Year, 1953Q1 to 2010Q1
14
12
10
8
6
4
2
0
-2
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
6
Figure 2. Unemployment Rate, Quarterly, 1952:Q1 to 2010:Q1
11
10
9
8
7
6
5
4
3
2
1960
1970
1980
1990
2000
2010
7
Basic Idea:


Search for regularities in the relationship between inflation and
unemployment
Main frequency bands

8 – 50 years per cycle

also: 8 - 25 years per cycle
8
What we do in this paper:

Apply the band-pass filter of Christiano and Fitzgerald (2003; CF)

Compare results to those from the Baxter-King (BK) filter

Study dynamic cross-correlation patterns of CF filtered
components of inflation and unemployment contemporaneously
and at leads and lags

Test for structural breaks in the raw data and filtered data

Carry out a sensitivity analysis
9
Methodology:

Spectrum s y ( )

Coherences coh()

Ideal (but not feasible) filter in the time domain is:

yt   a j yt  j  a( L) yt
*
j  
10
11
Methodology:

Spectrum of the filtered series y t* :
s y* ( )  A( ) s y ( )
2
where A( ) is the gain of the ideal filter given by a(L)
12
Methodology:

For symmetric filters it can be shown that the phase
shift is zero for all

The ideal filter is not feasible as one would need
infinite observations

Various approximations of the ideal filter have been
proposed

13
The Hodrick-Prescott filter:

Finite sample problems:

Is asymmetric and can produce phase-shift

“spurious” cycles : no - see Pedersen (2001)

but: the long-run or trend component inherits the
nonstationarity of the raw series

HP filter cannot be used for long-run analysis

it is not a band-pass filter
14
The Baxter-King filter:

Ideal filter is approximated by a symmetric finite MA

Filter weights are chosen in the frequency domain by
minimizing the following loss function:

Q  (2 )  A( )  B( ) d
1
2

15
The Baxter-King filter:

This is a band-pass filter

Filter weights are adjusted to ensure that the filtered
series are covariance stationary

BK-filter renders stationary series that are I(1) and I(2),
and also series with a linear or quadratic deterministic
trend
16
Problems:

Trade-off between a better approximation to the ideal
filter by making k larger and a loss of 2k observations
for further analysis

It is not consistent because the approximation errors
does not go to zero as T goes to infinity (k is fixed);
would need k as an increasing function of T
17
Christiano-Fitzgerald filter:

This filter minimizes the mean squared error criterion
instead:

E[( y  y )
*
t

* 2
t
y]
In the frequency domain the problem can be stated as:

min  A( )  B

p, f
( ) s y ( ) d
2
18
Christiano-Fitzgerald filter:

The solution depends on the spectrum of y

Filter weights are adjusted according to the
importance of the spectrum at a given frequency

Christiano and Fitzgerald assume a random walk
process (avoids estimating the spectrum every time)
19
Figure 4. Filtered Inflation and Unemployment, 8 to 50 Year Cycles
5
Inflation component
Unemployment component
4
3
2
1
0
-1
-2
-3
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
20
Empirical Analysis
Table 1. Confidence Bands for the Largest Autoregressive Root
Model with
constant only:
inflation
constant and deterministic trend:
unemployment
inflation
unemployment
95% confidence
banda
0.951 – 1.022
0.917 – 1.019
0.952 – 1.022
0.913 – 1.018
21
Empirical Analysis
22
Empirical Analysis
23
Empirical Analysis

The filtered components are generated series so that
standard critical values and standard confidence
bands can not be used for the cross-correlations

We calculate critical values with the bootstrap method

20,000 replications with Gaussian errors generated
under the null hypothesis of zero cross-correlations in
the data generating process
24
25
26
27
Structural breaks:

Apply break tests of Bai and Perron (1998, 2003) to the dynamic
cross-correlations; filtered components are covariance stationary

First apply the double maximum test UDmax, then Sup-F if there
is a break at all; estimate break dates; allow for multiple breaks

10,000 replications to calculate critical values for the UDmax test

UDmax test for the stability of relationship between the filtered
components of inflation and the unemployment rate for cycles of
8 to 50 years at lead 13: cannot reject null of no break

Same results for other significant cross-correlations
28
Summary of the results:

Highest level of cross-correlation occurs when cycles are 8-50
years in length, and where unemployment responds to inflation
after 13 quarters (3.25 years): +0.8338

Only correlations that are significant at or better than the 10%
level are those where inflation leads unemployment by 1 to 6
years; all are positive

There are no breaks in significant 8-50 year correlations

Our results are robust to modifications of the cycle length and
different filtering methods (CF and BK); no breaks despite
different fiscal and monetary policies over time
29