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

Persistence and Persistence and nonlinearities in nonlinearities in

Economics and FinanceEconomics and Finance““I built bustles for all Europe once, but I've been I built bustles for all Europe once, but I've been

badly hit,badly hit,

Things have decayed in the bustle making Things have decayed in the bustle making tradetrade

And that it the truth of it.”And that it the truth of it.”

from the poem: “I Was A Bustlemaker Once, Girls” by Patrick from the poem: “I Was A Bustlemaker Once, Girls” by Patrick BarringtonBarrington

2Presentation Chung Cheng, 13 April 2006

This presentationThis presentationis centred around: is centred around:

modelling and accounting for the rich non-linear modelling and accounting for the rich non-linear dynamics indynamics in

macroeconomicmacroeconomic and in financial time series.and in financial time series.

Purpose:Purpose: to uncover the underlying economic relationships between to uncover the underlying economic relationships between

variables,variables, to test empirically economic and finance theory,to test empirically economic and finance theory, to integrate these results in economic policy designto integrate these results in economic policy design

as diagnostic tools (e.g. early detection of economic as diagnostic tools (e.g. early detection of economic downturns or of price bubbles),downturns or of price bubbles),

as predictive and forecasting tools (e.g. predict more as predictive and forecasting tools (e.g. predict more accurately, over a forecast period, the impact of economic accurately, over a forecast period, the impact of economic policies),policies),

as modelling tools to help the design of policy at the as modelling tools to help the design of policy at the conceptual stage (i.e. defining and refining how the optimal conceptual stage (i.e. defining and refining how the optimal policy problem is set up).policy problem is set up).


This work builds on a joint This work builds on a joint Rev. Econ. Rev. Econ. Stud.Stud. 2002 paper with Karim Abadir 2002 paper with Karim Abadir we developed a general equilibrium we developed a general equilibrium

macromodel with heterogeneous firms,macromodel with heterogeneous firms, calculated its equilibrium and characterized calculated its equilibrium and characterized

the process of GDP/capita.the process of GDP/capita. We were able to make a prediction on the We were able to make a prediction on the

functional form of its ACF, and it was functional form of its ACF, and it was supported by US and UK data.supported by US and UK data.

Implication: ACF has an S-shape,Implication: ACF has an S-shape, initially, a plateau for which memory decreases very initially, a plateau for which memory decreases very

slowly,slowly, then a sharp drop, the memory decreases suddenly.then a sharp drop, the memory decreases suddenly.

Policy implication: an initial policy input will Policy implication: an initial policy input will build-up for the duration of the plateau, ...build-up for the duration of the plateau, ...


Autocorrelation function for the Autocorrelation function for the Log(GDP/Capita) UK, the fit for our AT Log(GDP/Capita) UK, the fit for our AT

process and for an AR(2)process and for an AR(2)

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 4 8 12 16 20 24 28 32 36 40

Lags

AC

F

Actual UK Data AT AR(2)


Autocorrelation function for the Autocorrelation function for the Log(GDP/Capita) US, the fit for our AT Log(GDP/Capita) US, the fit for our AT

process and for an AR(2)process and for an AR(2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

Lags

AC

F

Actual US Data AT AR(2)


Nature of the persistence in macro Nature of the persistence in macro and financial time seriesand financial time series

Aim of this researchAim of this research establish the nature of the dynamics establish the nature of the dynamics

encountered in macro and finance time encountered in macro and finance time series, (joint with Karim Abadir and series, (joint with Karim Abadir and Giovanni Caggiano)Giovanni Caggiano)

this dynamics turns out to be non-standard this dynamics turns out to be non-standard i.e. not well represented by AR processes,i.e. not well represented by AR processes,

hence the need for a methodology able to hence the need for a methodology able to handle such dynamics.handle such dynamics.


The functional form that we will fit was The functional form that we will fit was inspired from the previous inspired from the previous RESRES 2002 work, 2002 work, henceforth referred to as AT.henceforth referred to as AT. the Autocorrelation function (ACF) of a time the Autocorrelation function (ACF) of a time

series series zztt is its correlation with its lagged values. is its correlation with its lagged values. This ACF at lag This ACF at lag is defined as: is defined as:

The AT functional form of the ACF is a 4 The AT functional form of the ACF is a 4 parameters functionparameters function

The ACF associated with an AR(The ACF associated with an AR(pp) is given by ) is given by the so-called Yule-Walker equations, for the first the so-called Yule-Walker equations, for the first pp lags, and, for the rest, by the recursion lags, and, for the rest, by the recursion

= a= a11 + a + a22 + .... + a + .... + app pp..

.)var()var(

),cov(

tt

tt

zz

zz

.

1

cos11c

AT

b

a


0.85

0.87

0.89

0.91

0.93

0.95

0.97

0.990 4 8

12 16 20 24 28 32 36 40 44 48 52

Government Expenditure (real) AT_fit AR_fit

0.75

0.8

0.85

0.9

0.95

1

0 5 10 15 20 25 30 35 40 45 50 55

Tax Receipts (real) AT_fit AR_fit

0.75

0.8

0.85

0.9

0.95

1

0 3 6 9 12

15

18

21

24

27

30

33

Money stock (real) AT_fit AR_fit

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45 50 55

Bond yield (real) AT_fit AR_fit


0.88

0.9

0.92

0.94

0.96

0.98

1

0 5 10 15 20 25 30 35 40 45 50 55

GDP (real) AT_fit AR_fit

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 5 10 15 20 25 30 35 40 45 50 55

GDP (nominal) AT_fit AR_fit

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 5 10 15 20 25 30 35 40 45 50 55

CPI AT_fit AR_fit

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40 45 50 55

S&P 500 (nominal) AT_fit AR_fit


0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 5 10 15 20 25 30 35 40 45 50 55

Investment (real) AT_fit AR_fit

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 5 10 15 20 25 30 35 40 45 50 55

Exports (real) AT_fit AR_fit

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45 50 55

Imports (real) AT_fit AR_fit

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 4 8 12

16

20

24

28

32

36

40

Unemployment AT_fit AR_fit


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45 50

Inflation AT_fit AR_fit

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 3 6 9 12 15 18 21 24 27 30

Money growth (nominal) AT_fit AR_fit

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 3 6 9 12 15 18 21 24 27

Wages growth (nominal) AT_fit AR_fit

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 3 6 9 12 15 18 21 24 27

Wages growth (real) AT_fit AR_fit


Implications of the dynamics:Implications of the dynamics:movements and co-movementsmovements and co-movements

design an econometric methodology design an econometric methodology which can distil the co-movements which can distil the co-movements between variables from their own between variables from their own dynamics. (joint with Karim Abadir)dynamics. (joint with Karim Abadir) What happens if variables have the What happens if variables have the

previous type of dynamics?previous type of dynamics? Inertia will mask the true relationship Inertia will mask the true relationship

between variables.between variables. Consider a relationship of the typeConsider a relationship of the type

yy = = XX + + uu, where , where uu ~ D(0, ~ D(0, ), ), generated by the previous dynamics.generated by the previous dynamics.


Application I: the UIP Application I: the UIP anomalyanomaly

Efficient market theory posits that the Efficient market theory posits that the market exploit all available information market exploit all available information to price the assets.to price the assets. In particular, for the exchange rate, In particular, for the exchange rate,

consider 2 investment strategies: invest at consider 2 investment strategies: invest at time time tt $1 in US denominated asset, let its $1 in US denominated asset, let its log return be log return be iitt. Or invest the same amount . Or invest the same amount in in £. The log return, after exchanging your £. The log return, after exchanging your 1 dollar in pounds at 1 dollar in pounds at sstt, and exchanging the , and exchanging the pounds back into $ at time pounds back into $ at time tt+1 is+1 isii**t t + + sstt+1+1 - - sstt..


The difference between the two is called the The difference between the two is called the excess return on investing in foreign assetsexcess return on investing in foreign assets

rrtt+1+1 = i = i**tt – – iitt + + sstt+1+1.. Consider the contemporaneous variable the Consider the contemporaneous variable the

forward premium: forward premium: fftt – – sstt.. The (future) excess return should be The (future) excess return should be

independent of this forward premium.independent of this forward premium. In practice, one finds a negative relationship.In practice, one finds a negative relationship. If one takes into account the AT type of If one takes into account the AT type of

dynamics, one can remove the effect of this dynamics, one can remove the effect of this dynamics by pre-multiplying the whole dynamics by pre-multiplying the whole equation by a the Choleski decomposition of equation by a the Choleski decomposition of the covariance matrix the covariance matrix . .


Excess return vs.. forw ard premium for the £-$ exchange rate, Jan 1979 - Feb 2004

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

forward premium

exce

ss r

etu

rns

on

fo

reig

n a

sset

Excess return vs. forward premium for the £-$ exchange rate after ACF transformation

-0.15

-0.1

-0.05

0

0.05

0.1

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

forward premium

exce

ss r

etu

rns

on

fo

reig

n a

sset


ACF of log(£-$) exchange rate

-1

-0.5

0

0.5

1

0 24 48 72 96 120 144

ACF £-$ exchange rate fitted ACF

ACF of log(S&P 500)

-0.5

0

0.5

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

ACF of log(real S&P500) fitted ACF


Application II: Stock Application II: Stock market.market.

Efficient market theory is also Efficient market theory is also usually thought to predict that usually thought to predict that equity prices should follow be equity prices should follow be martingale.martingale.

Looking at the S&P 500 data and Looking at the S&P 500 data and GDP, in real terms and in logs, GDP, in real terms and in logs, suggests that actually the stock suggests that actually the stock index grows in long cycles around a index grows in long cycles around a trend given by GDP.trend given by GDP.


Real S&P 500 and real GDP Real S&P 500 and real GDP in logsin logs

7.1

7.6

8.1

8.6

9.1

9.6

1946 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002

Rea

l G

DP

in

lo

gs

4.5

5

5.5

6

6.5

7

Rea

l S

&P

500

in

lo

gs

Real GDP real S&P 500


The idea is to use an error The idea is to use an error correction mechanism (ECM) to correction mechanism (ECM) to model the relation between S&P and model the relation between S&P and GDP.GDP.

Running such an ECM without Running such an ECM without taking into account the AT long taking into account the AT long dynamics produces a regression that dynamics produces a regression that is fragile and the explanatory power is fragile and the explanatory power is not great.is not great.


fit is fit is RR22 = 37.4%. = 37.4%. parameters are unstable:parameters are unstable:


Performing the analysis using the AT Performing the analysis using the AT procedure, the stability of the regression procedure, the stability of the regression is much improved:is much improved:


The explanatory power is also improved to The explanatory power is also improved to 50.8%. The p-value of this improvement is 3.9%.50.8%. The p-value 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

0.1

0.2

0.3

0.4

1953 1959 1965 1971 1977 1983 1989 1995 2001

DSP DSP prediction

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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