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Linear univariate modelling based on multivariate information Laurent Ferrara Master 2 EIMPC Universit´e Paris Ouest
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Introduction
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Additional information can be useful to autoprojective processes
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Especially for macroeconomic forecasting
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What type of information? Coincident macro indicators: hard data (IPI, retail sales, consumption ...) Leading indicators : opinion surveys, financial variables, composite indicators (OECD, US CLI ...)
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Correlations
How to measure the relationship between Yt and Xt ? t ,Yt ) Basic tool: contemporeanous correlation ρ(Xt , Yt ) = covσ(X X σY Alternative: cross-correlation ρ(Xt , Yt−k ) =
cov (Xt , Yt−k ) σX σY
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Example of cross-correlation: French housing market
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Objective: Etablish cyclical relationships between a set of macro and housing variables using correlation analysis
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Selected variables 1980q1 - 2009q2:
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Macro: GDP, Household investment, Employment in construction, IPI in construction
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Housing: Real prices, Sales, Permits, Starts, Survey by Property Developers
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Finance: Long (Gov. bonds 10 years) and Short (Euribor 3-months) interest rates
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gdp 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
1996
1998
2000
2002
2004
2006
2008
1996
1998
2000
2002
2004
2006
2008
1998
2000
2002
2004
2006
2008
housing prices 4 2 0 -2 -4 -6 -8 -10 1980
1982
1984
1986
1988
1990
1992
1994
sales 30 20 10 0 -10 -20 -30 -40 1980
1982
1984
1986
1988
1990
1992
1994
household investment 6 4 2 0 -2 -4 -6 -8 1980
1982
1984
1986
1988
1990
1992
1994
1996
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employment 2.4 1.6 0.8 -0.0 -0.8 -1.6 -2.4 -3.2 1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2003
2005
2006
2008
survey 40 20 0 -20 -40 -60 1985
1987
1989
1991
1993
1995
1997
1999
2001
2007
2009
short rate 40 20 0 -20 -40 -60 -80 -100 1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
1996
1998
2000
2002
2004
2006
2008
long rate 15 10 5 0 -5 -10 -15 -20 1980
1982
1984
1986
1988
1990
1992
1994
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20 15 10 5 0 -5 -10 -15
permits
1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008
housing starts 15 10 5 0 -5 -10 -15 -20 -25 -30 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008
6 4 2 0 -2 -4
ipi building
-6 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008
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Correlation Analysis
GDP Prices Sales Invest. Employ. Survey Short Long Permits Starts IPI
GDP 1 0.72 0.64 0.75 0.79 0.80 0.77 0.74 0.70 0.80 0.81
Prices 0.44 1 0.56 0.81 0.75 0.80 0.58 0.73 0.72 0.82 0.64
Sales 0.01 0.20 1 0.53 0.54 0.67 0.54 0.44 0.66 0.69 0.40
Invest. 0.80 0.60 0.23 1 0.79 0.81 0.65 0.69 0.70 0.87 0.72
Employ. 0.72 0.54 0.21 0.65 1 0.71 0.66 0.69 0.63 0.71 0.85
Survey 0.57 0.48 0.10 0.53 0.46 1 0.66 0.59 0.82 0.84 0.72
Short 0.68 0.18 -0.37 0.46 0.56 0.28 1 0.66 0.54 0.64 0.74
Long 0.53 0.19 -0.42 0.39 0.64 0.21 0.57 1 0.59 0.67 0.65
Permits 0.32 0.61 0.38 0.40 0.17 0.39 0.10 -0.13 1 0.86 0.56
Starts 0.45 0.62 0.40 0.57 0.22 0.40 0.09 0.05 0.67 1 0.63
Table: Concordance indexes for contemporaneous variables (lower diagonal) and contemporaneous correlation (upper diagonal) from 1980 Q1 to 2009 Q2.
IPI 0.60 0.26 -0.40 0.39 0.57 0.31 0.60 0.54 0.13 0.31 1
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Correlation Analysis
Results I
Strong correlations between GDP and Housing investment (0.80) and Employment (0.72)
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Overall, contemporaneous correlation coefficients appear appear quite small in comparison of what could be expected
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Focus on cross-correlations
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Cross-correlation with GDP prices
short
1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00
1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 -10
-5
0
5
10
-10
-5
sales
0
5
10
5
10
5
10
5
10
5
10
long
1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00
1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 -10
-5
0
5
10
-10
-5
invest
0
permits
1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00
1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 -10
-5
0
5
10
-10
-5
employment 1.00
0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00
0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 -10
-5
0
0
starts
1.00
5
10
-10
-5
survey
0
ipi building
1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00
1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 -10
-5
0
5
10
-10
-5
0
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Cross-correlation Analysis
GDP GDP Prices Sales Investment Employment Survey Short rate Long rate Permits Starts IPI
-2 -3 0 0 -1 0 -8 -4 -1 0
Prices 0.64 -2 +1 +2 +1 +3 +3 0 0 +2
Sales 0.48 0.36 +3 +7 +3 0 -1 +1 +2 -2
Invest. 0.80 0.70 0.60 +1 0 +1 +3 -3 -1 0
Employ. 0.72 0.71 0.27 0.69 -2 0 0 -4 -4 0
Survey 0.58 0.50 0.35 0.53 0.55 +2 -7 -2 -1 +2
Short 0.68 0.54 -0.37 0.50 0.56 0.37 -1 -5 -4 -1
Long -0.52 0.44 -0.49 0.54 0.64 -0.41 0.59 +6 +5 +8
Permits 0.58 0.61 0.44 0.63 0.52 0.42 0.44 -0.48
Starts 0.62 0.62 0.61 0.70 0.47 0.44 0.43 -0.47 0.69
+1 +4
Table: Highest cross-correlation coefficients among all leads and lags (upper diagonal, lags in parenthesis) and leads/leags (lower diagonal), from 1980 Q1 to 2009 Q2. A negative number indicates that the series in row leads the series in column with an advance equal to this number, and conversely.
+3
0 0 -0 0 0 0 0 -0 0 0
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Cross-correlation with GDP
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A leading pattern in housing variables (Real prices, Sales, Permits and Starts)
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Residential investement is strongly related to the economic cycle, in coincident manner
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Employent and IPI in construction are coincident with GDP
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Short rate (3m): a positive correlation with a short delay (0-1 quarters)
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Long rate (10y): a negative correlation with lead of 2 years.
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ARDL models Definition yt = α +
m X j=0
βj0 xt−j
+
p X
φj yt−j+1 + εt ,
(1)
j=1
where: xt is the n-vector of explanatory variables (x1t , . . . , xnt )0 , m is the lag of the explanatory variables, εt strong WN, for a given lag j, βj = (βj1 , . . . , βjn )0 is the coefficient vector for explanatory variables of length n. The model specification is generally carried out using information criteria such as AIC or BIC. The mn + p + 1 parameters of the model can be estimated by ordinary least-squares
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Forecasting based on ARDL
1. Iterative forecasting: Conditional forecasting of explanatory variables 2. Scenario forecasting: Judgemental forecasting of explanatory variables 3. Direct forecasting: a specific regression for each horizon h
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Iterative forecasting based on ARDL
Assume n = 1 explanatory variable, m = 1, p = 1, h = 1 : yt = α + βxt + φyt−1 + εt
(2)
ˆ (xt+1 |It ) + φy ˆ t yˆt (1) = E (yt+1 |It ) = α ˆ + βE
(3)
How to compute E (xt+1 |It )? Auxiliary models
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Scenario forecasting based on ARDL
Assume n = 1 explanatory variable, m = 1, p = 1, h = 1. Example of 3 scenarii for E (xt+1 |It ) : − 1. negative growth of -2% (xt+1 ) 0 ) 2. stability : 0% (xt+1 + 3. positive growth of +2% (xt+1 )
ˆ − + φy ˆ t ˆ + βx yˆt− (1) = α t+1
(4)
0 ˆ t+1 ˆ t yˆt0 (1) = α ˆ + βx + φy
(5)
ˆ + + φy ˆ t yˆt+ (1) = α ˆ + βx t+1
(6)
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Direct forecasting based on ARDL
For any h > 0 yt+h = αh +
m X
0 βhj xt−j
j=0
+
p X
φhj yt−j+1 + εt+h ,
(7)
j=1
1 , . . . , β n )0 is the coefficient where, for a given lag j, βhj = (βhj hj vector for financial variables of length n. The h-step-ahead forecast is thus given by
yˆt (h) = α ˆh +
m X j=0
0 βˆhj xt−j +
p X j=1
φˆhj yt−j+1 .
(8)
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References related GDP forecasting based on financial variables
US data: Estrella and Hardouvelis (1991), Hamilton and Kim (2002), Estrella et al. (2003) or Giacomini and Rossi (2006). Gilchrist and Zakrajsek (2012): a new credit spread index to predict US GDP growth. Estrella and Mishkin (1997): usefulness of various term spreads and monetary variables for the US GDP Euro area: Andersson and d’Agostino (2008) use sectoral stock prices to predict the euro area GDP. Duarte et al. (2005) spread between 10-year sovereign yeld and the 3-month interbank rate
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Examples of applications: French GDP forecasting Components modelled 1. Supply side decomposition: I Agri-food I Manufacturing I Energy I Construction I Market services 2. Demand side decomposition: I Household consumption (agrifood, energy, manufacturing, services) I Government consumption I Investment (corporate, household, government) I Exports and Imports Inventories bridge the gap between both approaches.
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Predicting with bridge equations in real-time Denote the quarterly GDP component growth as ytQ and the K Q selected auxiliary monthly variables as xk,t : ytQ
=µ+
K X
Q βk (B)xk,t + εQ t
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k=1 m1 m2 m3 Q where xk,t = 1/3(xk,t + xk,t + xk,t ). Predictions of GDP growth are obtained in two steps. m are forecast using 1. The K auxiliary monthly variables xi,t ARIMA models over the remainder of the quarter to obtain Q forecasts of their quartely aggregates, xi,t .
2. The resulting values are used as regressors in the bridge equation to obtain the GDP forecast.
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Component
First
Second
Third
AR
Naive
GDP
0.27
0.25
0.23
0.38
0.51
Production Agri-food
0.49
0.47
0.45
0.57
0.68
Production Manufactured
0.82
0.79
0.71
1.28
1.73
Production Energy
1.44
1.34
1.21
1.44
2.52
Production Construction
0.62
0.57
0.55
0.67
0.76
Production Services
0.41
0.39
0.34
0.45
0.59
Household Consumption
0.26
0.19
0.19
0.33
0.45
Government Consumption
0.23
0.28
Investment
0.8
0.23 0.77
0.71
0.87
1.24
Imports
1.23
1.13
1.13
1.31
1.54
Exports
1.46
1.32
1.27
1.62
2.07
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Impulse Response Function
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See example of EA GDP (RATS example)
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Need of VAR model
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