Transcript Distilling co-movements from persistent macro and

Persistence and nonlinearities
in Economics and Finance
“I built bustles for all Europe once, but I've been badly hit,
Things have decayed in the bustle making trade
And that it the truth of it.”
from the poem: “I Was A Bustlemaker Once, Girls” by Patrick Barrington
This presentation
is centred around:

modelling and accounting for the rich non-linear dynamics in



macroeconomic
and in financial time series.
Purpose:

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
to uncover the underlying economic relationships between variables,
to test empirically economic and finance theory,
to integrate these results in economic policy design


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as diagnostic tools (e.g. early detection of economic downturns or of price
bubbles),
as predictive and forecasting tools (e.g. predict more accurately, over a forecast
period, the impact of economic policies),
as modelling tools to help the design of policy at the conceptual stage (i.e.
defining and refining how the optimal policy problem is set up).
Presentation Chung Cheng, 13 April 2006
1

This work builds on a joint Rev. Econ. Stud. 2002 paper
with Karim Abadir
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
we developed a general equilibrium macromodel with
heterogeneous firms,
calculated its equilibrium and characterized the process of
GDP/capita.
We were able to make a prediction on the functional form of
its ACF, and it was supported by US and UK data.
Implication: ACF has an S-shape,
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initially, a plateau for which memory decreases very slowly,
then a sharp drop, the memory decreases suddenly.
Policy implication: an initial policy input will build-up for the
duration of the plateau, ...
Presentation Chung Cheng, 13 April 2006
2
Autocorrelation function for the Log(GDP/Capita)
UK, the fit for our AT process and for an AR(2)
1
0.98
0.96
ACF
0.94
0.92
0.9
0.88
0.86
0.84
0
4
8
12
16
20
24
28
32
36
40
Lags
Presentation Chung Cheng, 13 April 2006
Actual UK Data
AT
AR(2)
3
Autocorrelation function for the Log(GDP/Capita)
US, the fit for our AT process and for an AR(2)
1
0.9
0.8
0.7
ACF
0.6
0.5
0.4
0.3
0.2
0.1
0
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
Lags
Actual US Data
Presentation Chung Cheng, 13 April 2006
AT
AR(2)
4
Nature of the persistence in macro and financial
time series

Aim of this research

establish the nature of the dynamics encountered in
macro and finance time series, (joint with Karim
Abadir and Giovanni Caggiano)
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
this dynamics turns out to be non-standard i.e. not well
represented by AR processes,
hence the need for a methodology able to handle such
dynamics.
Presentation Chung Cheng, 13 April 2006
5

The functional form that we will fit was inspired from
the previous RES 2002 work, henceforth referred to as
AT.

the Autocorrelation function (ACF) of a time series zt is its
correlation with its lagged values. This ACF at lag t is defined
as:
cov( zt , zt -t )
rt 
.
var( zt ) var( zt -t )


The AT functional form of the ACF is a 4 parameters function
1 - a1 - cost 
rtAT 
.
c
1  bt
The ACF associated with an AR(p) is given by the so-called
Yule-Walker equations, for the first p lags, and, for the rest, by
the recursion
 rt = a1 rt-1 + a2 rt-2 + .... + ap rt-p.
Presentation Chung Cheng, 13 April 2006
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1
0.99
0.97
0.95
0.93
0.95
0.9
0.91
0.89
0.87
0.85
0.85
Government Expenditure (real)
AT_fit
Tax Receipts (real)
1
AT_fit
55
50
45
40
35
30
25
20
15
10
0
AR_fit
5
0.75
52
48
44
40
36
32
28
24
20
16
12
8
4
0
0.8
AR_fit
1
0.8
0.95
0.6
0.9
0.4
0.85
0.2
0.75
55
50
45
40
35
30
25
20
15
10
5
-0.2
0
0
0.8
Money stock (real)
AT_fit
Presentation Chung Cheng, 13 April 2006
AR_fit
33
30
27
24
21
18
15
12
9
6
3
0
-0.4
-0.6
Bond yield (real)
AT_fit
AR_fit
7
1
1
0.98
0.98
0.96
0.96
0.94
0.94
0.92
0.92
GDP (real)
AT_fit
AR_fit
GDP (nominal)
1
0.95
0.9
AT_fit
55
50
45
40
35
30
25
20
15
10
0
55
50
45
40
35
30
25
20
15
0.86
10
0.88
5
0.88
0
0.9
5
0.9
AR_fit
1
0.9
0.8
0.85
0.8
0.75
0.7
0.65
0.6
0.7
0.6
0.5
0.4
CPI
AT_fit
AR_fit
Presentation Chung Cheng, 13 April 2006
S&P 500 (nominal)
AT_fit
AR_fit
8
55
50
45
40
35
30
25
20
15
10
5
0
55
50
45
40
35
30
25
20
15
10
5
0
0.3
Investment (real)
AT_fit
55
50
45
40
35
30
25
20
15
10
5
0
Unemployment
AR_fit
AT_fit
40
36
32
28
24
20
16
12
8
4
0.85
0.8
0.75
0.7
0.65
0.6
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
1
0.95
0.9
AR_fit
1
1
0.95
0.9
0.8
0.85
0.8
0.75
0.7
0.65
0.6
0.6
0.4
0.2
Exports (real)
AT_fit
AR_fit
Presentation Chung Cheng, 13 April 2006
Imports (real)
AT_fit
55
50
45
40
35
30
25
20
15
10
5
0
55
50
45
40
35
30
25
20
15
10
5
0
0
AR_fit
9
1
0.8
0.6
0.4
30
27
24
21
18
15
12
9
-0.4
-0.6
Inflation
AT_fit
Money growth (nominal)
AR_fit
1
1
0.8
0.8
0.6
0.6
AT_fit
AR_fit
0.4
0.4
0.2
0.2
27
24
21
18
15
12
9
6
3
0
-0.4
27
24
21
18
15
12
9
6
3
-0.2
0
0
0
-0.2
6
-0.2
3
0
0
50
45
40
35
30
25
20
15
10
5
0.2
0
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-0.4
-0.6
Wages growth (nominal)
AT_fit
Presentation Chung Cheng, 13 April 2006
AR_fit
Wages growth (real)
AT_fit
AR_fit
10
Implications of the dynamics:
movements and co-movements

design an econometric methodology which can
distil the co-movements between variables from
their own dynamics. (joint with Karim Abadir)
What happens if variables have the previous type of
dynamics?
 Inertia will mask the true relationship between
variables.
 Consider a relationship of the type
y = Xb + u, where u ~ D(0, S), S generated by the
previous dynamics.
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Presentation Chung Cheng, 13 April 2006
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Application I: the UIP anomaly
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Efficient market theory posits that the market
exploit all available information to price the
assets.
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In particular, for the exchange rate, consider 2
investment strategies: invest at time t $1 in US
denominated asset, let its log return be it. Or invest
the same amount in £. The log return, after
exchanging your 1 dollar in pounds at st, and
exchanging the pounds back into $ at time t+1 is
i*t + st+1 - st.
Presentation Chung Cheng, 13 April 2006
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The difference between the two is called the excess
return on investing in foreign assets
rt+1 = i*t – it + Dst+1.
 Consider the contemporaneous variable the forward
premium: ft – st.
 The (future) excess return should be independent of
this forward premium.
 In practice, one finds a negative relationship.
 If one takes into account the AT type of dynamics,
one can remove the effect of this dynamics by premultiplying the whole equation by a the Choleski
decomposition of the covariance matrix S.
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Presentation Chung Cheng, 13 April 2006
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Excess return vs.. forw ard premium for the £-$ exchange rate, Jan 1979 - Feb 2004
0.15
0.05
0
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
-0.05
-0.1
-0.15
forward premium
Excess return vs. forward premium for the £-$ exchange rate after ACF transformation
0.1
excess returns on foreign asset
excess returns on foreign asset
0.1
0.05
0
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
-0.05
-0.1
Presentation Chung Cheng, 13 April 2006
-0.15
forward premium
14
ACF of log(£-$) exchange rate
1
0.5
0
0
24
48
72
96
120
144
-0.5
-1
ACF £-$ exchange rate
fitted ACF
ACF of log(S&P 500)
1
0.5
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
-0.5
ACF of log(real S&P500)
Presentation Chung Cheng, 13 April 2006
fitted ACF
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Application II: Stock market.
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Efficient market theory is also usually thought to
predict that equity prices should follow be
martingale.
Looking at the S&P 500 data and GDP, in real
terms and in logs, suggests that actually the
stock index grows in long cycles around a trend
given by GDP.
Presentation Chung Cheng, 13 April 2006
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Real S&P 500 and real GDP in logs
9.6
7
9.1
8.6
6
8.1
5.5
7.6
5
7.1
4.5
1946
1950
1954
1958
1962
1966
1970
1974
Real GDP
Presentation Chung Cheng, 13 April 2006
1978
1982
1986
1990
1994
1998
2002
real S&P 500
17
Real S&P 500 in logs
Real GDP in logs
6.5
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The idea is to use an error correction
mechanism (ECM) to model the relation
between S&P and GDP.
Running such an ECM without taking into
account the AT long dynamics produces a
regression that is fragile and the explanatory
power is not great.
Presentation Chung Cheng, 13 April 2006
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fit is R2 = 37.4%.
parameters are unstable:
Panel 1
0.0
Panel 2
0.0
-0.5
-0.5
-1.0
1970
1980
1990
2000
1970
1980
1990
2000
1980
1990
2000
1980
1990
2000
Panel 4
Panel 3
0.0
0.0
-0.5
-0.2
-1.0
1970
1980
1990
2000
Panel 5
1970
5.0
Panel 6
0.10
0.05
2.5
0.00
0.0
-0.05
1970
1980
1990
Presentation Chung Cheng, 13 April 2006
2000
1970
19

Performing the analysis using the AT procedure, the
stability of the regression is much improved:
Panel 1
Panel 2
-0.5
-0.5
-1.0
-1.0
1970
1980
1990
2000
1970
1980
1990
2000
1980
1990
2000
1980
1990
2000
Panel 4
Panel 3
0.00
-0.5
-0.05
-1.0
-0.10
1970
0.000
1980
1990
2000
1970
Panel 6
Panel 5
0
-1
-0.005
-2
-3
-0.010
1970
1980
1990
Presentation Chung Cheng, 13 April 2006
2000
1970
20

The explanatory power is also improved to 50.8%. The pvalue of this improvement is 3.9%.
Actual and predicted changes in log S&P 500 using the ACF-based methodolgy
0.4
0.3
0.2
0.1
0
1953
1959
1965
1971
1977
1983
1989
1995
2001
-0.1
-0.2
-0.3
-0.4
DSP
Presentation Chung Cheng, 13 April 2006
DSP prediction
21