Notes on Measurement:

De�ating and Detrending Data

Guido MenzioUniversity of Pennsylvania

Spring 2006

De�ating Data

� On the one hand, our theoretical models make strong predictions aboutthe dynamics of quantity of output produced, quantity of goods consumed,quantities of goods invested, imported and exported

� On the other hand, NIPA gives us measures about the value (quanti-ties*prices) of output, consumption, investment, etc...

� In order to extract the dynamics of quantites from the dynamics of values,we have to de�ate the NIPA time-series

De�ating Data

� For example

According to NIPA, in 1930 the US GDP was 103.6 billion dollarsAccording to NIPA, in 2004 the US GDP was 11,734.3 billion dollarsGDP in 2004 was 113 times GDP in 1930

� In 1930 the price of all goods was lower than in 2004In 1930 the quantity of goods produced was lower than in 2004How much of growth in value is

� price growth (in�ation)?

� output growth (growth in quantities produced)?

Nominal GDP

0.0

2,000.0

4,000.0

6,000.0

8,000.0

10,000.0

12,000.0

14,000.0

1929

1934

1939

1944

1949

1954

1959

1964

1969

1974

1979

1984

1989

1994

1999

2004

Bill

ions

De�ating Data: GDP

� The basic principle for de�ating GDP is to re-evaluate all the economictransactions that enter the de�nition of GDP at constant prices

� Re-evaluate the year-� GDP on the basis of year-t prices:

GDPR� (pt) =X

x2Firms[Gov

24Xi

pi;t � yi;x;� �Xi

pi;t �mi;x;�

35yi;x;t is the quantity of good i sold by �rm x in year �mi;x;t is the quantity of intermediate good or import i bought by �rm x

in year �pi;t is the price of good i in year t

� De�ne the price of �GDP� in year-� relative to the price of �GDP� inyear�t (the base year) as

P�(pt) =GDP�

GDPR� (pt)

� Decompose the nominal GDP growth between year t and � as

GDP�

GDPt=

GDPR� (pt)

GDPRt (pt)

!| {z } P�(pt)| {z }Real Growth Price Growth

� This de�ation technique is conceptually neat, but practically problematic:

� the measures of output growth depends on the choice of the base year for prices

� for example �GDPRt+1(pt)

GDPRt (pt)

�6=�GDPRt+1(pt+1)

GDPRt (pt+1)

�� the inequality is caused by the fact that prices of di¤erent goods grow at di¤erent rates

� To mitigate this problem, we can take an average of di¤erent real growth estimates

De�ating Data: Fisher Index and Chain-Weighting

� The Fisher Index is a measure of the relative price of GDP in year�(t + 1) wrt GDP inyear�t and is de�ned as a the geometric average of Pt+1(pt) and 1=Pt(pt+1)

P Ft+1;t =

sPt+1(pt) �

1

Pt(pt+1)

Pt+1(pt) is price of GDP in (t+ 1) relative to GDP in t, when pt is the base year1

Pt(pt+1)is the price of GDP in (t+ 1) relative to GDP in t, when pt+1 is the base year

� The decomposition of GDP growth between t and (t+ 1) is

GDPRt+1(PFt+1;t)

GDPRt (PFt;t)

=

�GDPt+1

GDPt

�| {z } 1=P Ft+1;t| {z }

Real Growth Nominal Growth divided by Price Growth

� Substituting the de�nition of P Ft+1;t into the previous equation, we can conclude that

GDPRt+1(PFt+1;t)

GDPRt (PFt;t)

=

s�GDPRt+1(pt)

GDPRt (pt)

���GDPRt+1(pt+1)

GDPRt (pt+1)

�

� Having constructed the series of Fisher Indices fP F�+1;�g2004�=1930, we can compute

� the price of GDP in year � wrt the price of GDP in year�t as a chain of relative prices

P F�;t = PFt+1;t � P Ft+2;t+1 � :::P F�;��1

� the real growth of GDP between year-� and year-t as

GDPR� (PF�;t)

GDPRt (PFt;t)

=

�GDP�

GDPt

�1

P F�;t

� Using the Fisher Indeces and chain-weigthing, we can decompose nominal GDP growth be-tween 1930 and 2004 as

GDP2004

GDP1930= 113 =

=

GDPR2004(P

F2004;1930)

GDPR1930(PF1930;2000)

! P F2004;2000

P F1930;2000

!= (12:5) � (9:04)

Real and Nominal GDP

0.0

2,000.0

4,000.0

6,000.0

8,000.0

10,000.0

12,000.0

14,000.0

1929

1934

1939

1944

1949

1954

1959

1964

1969

1974

1979

1984

1989

1994

1999

2004

Real GDP Nominal GDP

De�ating Data: Consumption and Other Aggregates

� Following the same logic, we can construct de�ated series forPersonal consumption expendituresGross private domestic investmentNet exports of goods and servicesGovernment consumption expenditures

� In the process, we obtain measures forIn�ation in consumption goodsIn�ation in investment goodsIn�ation in exported and imported goods

Detrending Data

� Long-term growth in economic activity is likely to be determined by the legal framework,the market for lower and higher education, the tax and subsidy system, changes in thedemographic structure etc...

� Short-term �uctuations in economic activity are likely to be determined by shocks to thesupply of inputs factors, news about the pro�tability of a new technology, etc...

� As a �rst approximation, it is reasonable to study growth and business cycle as independentphenomena

� In order to study independently growth and business cycles, we want to �lter the time-seriesof GDP and obtain

� a time-series capturing the trends in GDP

� a time-series of the short-term �uctuations in GDP

� From the time-series of trends, we derive the statistical regularities against which test theoriesof growth

� From the time-series of short-term �uctuations, we derive the statistical regularities againstwhich test theories of business cycles

Hodrick-Prescott Filter

Statistical tool created by John Hodrick and Edward Prescott in 1980

Step 1. Construct the time-series of the natural logarithms of real GDP over the period of interest

fytgTt=0 = flog YtgTt=0

Remark. Natural logs are a useful transformation of data because the di¤erence between log Yt+1and log Yt is approximately equal to the growth rate of Y between t and t+ 1

log Yt+1 � log Yt 'Yt+1 � Yt

Yt

Step 2. Construct the time-series of trends f�tgTt=0 by solving the following minimization problem

minf�tgTt=0

(TXt=0

(yt � �t)2 + �T�1Xt=1

[(�t+1 � �t)� (�t � �t�1)]2)

Remarks(i) the �rst term is the sum of squared deviations between the terms fytgTt=0 and the terms inf�tgTt=0 (it measures how far the trends time-series is from the original time series)(ii) the second term is the sum of squared deviations in the slope of the time-series f�tgTt=0 (itmeasures how �choppy� is the trends time-series)(iii) for �!1, then f�tgTt=0 is the linear trend(iv) for �! 0, then f�tgTt=0 is equal to the original time-series fytg

Tt=0

(v) for quarterly data, we typically set � to 1600

Step 3. Construct the time-series of cyclical �uctuations fytgTt=0 as the di¤erence between fytgTt=0

and f�tgTt=0yt = yt � �t

Remarks

(i) yt is the deviation rate of real GDP from the trend

yt = yt � �t = log Yt � log(e�t) 'Yt � e�te�t

(ii) the same �ltering procedure can be used to derive the trends time-series and the short-term�uctuations time-series for consumption, investment, net exports, government spending, hoursworked, etc...

Volatility and Comovement

After the time-series are constructed, we derive the statistical moments of the data to summarizethe key regularities of the phenomenon of interest

1. Volatility. The frequency and magnitude of movements in a time-series fxtgTt=0 can bemeasured by its standard deviation

�x =

TXt=1

(xt � x)2

T

!1=2

Remarks(i) we say that the time-series fxtgTt=0 is more (less) volatile of the time-series fytg

Tt=0 if

�x=�y > 1(< 1 )

(ii) if x is the natural logarithm of X, then �x is independent from the unit of measure of X(iii) if x is the deviation rate ofX from trend, then �x is independent from the unit of measure ofX

2. Comovement In order to measure the extent to which two time-series fxtgTt=0 and fytgTt=0

move together, we can use their correlation

�x;y =

PTt=1 [(xt � x) � (yt � y) =T ]

�x � �y

Remarks(i) we say that fxtgTt=0 moves together with (against) fytg

Tt=0 if

�x;y > 0 (< 0 )

(ii) if x and y are natural logarithms, �x;y is independent from the unit of measure of X and Y(iii) if x and y are deviation rates, �x;y is independent from the unit of measure of X and Y

3. Leading/Lagging Variable

We say that fxtgTt=0 leads fytgTt=0 if

�x;L(y;1) =

PT�1t=0 [(xt � x) � (L(yt; 1)� y) =(T � 1)]

�x � �y> 0

L(yt; 1) = yt+1

We say that fxtgTt=0 lags fytgTt=0 if

�x;L(y;�1) =

PTt=1 [(xt � x) � (L(yt;�1)� y) =(T � 1)]

�x � �y> 0

L(yt;�1) = yt�1

Application of Detrending and Filtering: The Statistical Properties of the Business Cycle

• From the NIPA dataset– we collect the time-series for GDP, C, I, NX and G– we derive the deflated quarterly series (Fisher Index and chain-

weighted)– we derive the time-series of fluctuations using the HP filter (λ=1600)

• From the CPS dataset – we collect the time-series for employment– we derive the time-series of fluctuations using the HP filter (λ=100,000)

Real GDP 2000 US$

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

1947

1949

1952

1954

1957

1959

1962

1964

1967

1969

1972

1974

1977

1979

1982

1984

1987

1989

1992

1994

1997

1999

2002

2004

Per

cent

Dev

iatio

n fo

rm T

rend

Real Consumption 2000 US$

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

1947

1949

1952

1954

1957

1959

1962

1964

1967

1969

1972

1974

1977

1979

1982

1984

1987

1989

1992

1994

1997

1999

2002

2004

Per

cent

Dev

iatio

n fr

om T

rend

GDP C

Real Investment 2000 US$

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

1947

1949

1952

1954

1957

1959

1962

1964

1967

1969

1972

1974

1977

1979

1982

1984

1987

1989

1992

1994

1997

1999

2002

2004

Per

cent

Dev

iatio

n fr

om T

rend

GDP I

Real Public Consumption 2000 US$

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

1947

1949

1952

1954

1957

1959

1962

1964

1967

1969

1972

1974

1977

1979

1982

1984

1987

1989

1992

1994

1997

1999

2002

2004

Per

cent

Dev

iatio

n fr

om T

rend

GDP G

Average Productivity of Labor

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

1947

1949

1952

1954

1957

1959

1962

1964

1967

1969

1972

1974

1977

1979

1982

1984

1987

1989

1992

1994

1997

1999

2002

2004

Per

cent

Dev

iatio

n fr

om T

rend

GDP APL

Employment

-4

-3

-2

-1

0

1

2

3

1948

1950

1952

1954

1957

1959

1961

1963

1966

1968

1970

1972

1975

1977

1979

1981

1984

1986

1988

1990

1993

1995

1997

1999

2002

2004

Perc

enta

ge D

evia

tion

from

Tre

nd

Employment

Table 2: Cyclical Behavior of the US Economy: Deviations from Trend of Expenditure Components, 1954:I-1991:II

Cross-Correlation of Output with:

Variable

SD X(-3) X(-2) X(-1) X X(+1) X(+2) X(+3)

GDP 1.72 .38 .63 .85 1 .85 .63 .38

C 1.27 .57 .72 .82 .83 .67 .46 .22

I 8.24 .38 .59 .79 .91 .76 .50 .22

G 2.04 -.03 -.01 -.01 .04 .08 .11 .16

Exp 5.53 -.29 -.10 .15 .37 .50 .54 .54

Imp 4.88 .31 .45 .62 .72 .71 .52 .28

Data Source: NIPA. All data are deflated and HP filtered

Table 1: Cyclical Behavior of U.S. Labor Market Aggregates, 1954:I-1991:II

Cross-Correlation of Real GDP with:

VariableVolatilit

y (%SD)

X(-5) X(-4) X(-3) X(-2) X(-1) X X(+1)

X(+2)

X(+3)

X(+4)

X(+5)

Real Gross Domestic Product 1.72 -.02 .16 .38 .63 .85 .85 .63 .38 .16 -.02

Hours (Household Survey) 1.49 -.10 .05 .25 .46 .70 .86 .85 .74 .58 .38 .17

Employment 1.09 -.17 -.03 .16 .38 .63 .83 .88 .80 .65 .46 .25

Hours per Worker 0.54 .07 .20 .36 .49 .64 .70 .58 .42 .28 .12 -.02

GDP/Hours 0.87 .12 .23 .33 .47 .50 .51 .22 -.01 -.24 -.32 -.34

Average Hourly Real Compensation(Business Sector)

0.93 .35 .39 .41 .43 .41 .35 .25 .16 .05 -0.7 -.18

Real Employee Compensation (NIPA)/Hours (Household Survey)

0.65 -.11 -.11 -.13 .06 .02 .10 .13 .14 .10 .08 .04

Cross-Correlation of *:

Employment and Average Labor Productivity** (X)

1.09 .73 .68 .57 .35 .09 -.15 -.32

Vacanciesand Unemployment (X) 12.54 -.36 -.61 -.82 -.95 -.93 -.77 -.54

GNP and Labor Share (X) 1.07 -.61 -.73 -.78 -.74 -.48 -.22 -.00

Source: Finn E. Kydland (1995), (*) Source: M. Merz (1995) using CITIBASE data for the period 1959:I-1988:II, (**) Average Labor Productivity is defined as Real GNP over Employment

The Stylized Facts about the Business Cycle

• Output– fluctuations in GDP are persistent

• Expenditure– consumption is procyclical and less volatile than GDP– investment is procyclical and 5 times as volatile as GDP– government expenditures are acyclical

• Productivity– the average output per hour of work is somewhat procyclical and leads

the cycle

The Stylized Facts about the Business Cycle

• Labor Markets– employment volatility accounts for 2/3 of the volatility of total hours– hours-per-worker volatility accounts for 1/3 of the volatility of total hours– employment is procyclical lags the cycle– hors-per-worker are procyclical and lead the cycle– total hours are procyclical and almost as volatile as GDP

References

• Chapters 2 and 3 in Williamson

• Check-out the website of the National Bureau of Economic Research http://www.nber.org

• Check-out the book "Frontier of Business Cycle Research," ed. T. Cooley, 1995, Princeton University Press