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transcript
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