Abadir, K.M.

Talmain, G.

English

Enlighten

     Abadir, K.M., and Talmain, G. (2009) Macro and financial markets: The memory of an elephant? In: 15th Spring Meeting of Young Economists 2009, 23-25 Apr 2009, Istanbul, Turkey.     Copyright © 2009 The Authors   A copy can be downloaded for personal non-commercial research or study, without prior permission or charge  Content must not be changed in any way or reproduced in any format or medium without the formal permission of the copyright holder(s)   When referring to this work, full bibliographic details must be given     http://eprints.gla.ac.uk/45444   Deposited on: 30 September 2013                   Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk MACRO AND FINANCIAL MARKETS:The memory of an elephant?Karim Abadir and Gabriel TalmainBackground, REStud 2002:•Model and solution:—Micro-founded macro model.— Standard RBC + heterogeneity: drop “representative firm” as-sumption.—General Equilibrium yields explicit dynamic equation for GDP etc.Background, REStud 2002:•Model and solution:—Micro-founded macro model.— Standard RBC + heterogeneity: drop “representative firm” as-sumption.—General Equilibrium yields explicit dynamic equation for GDP etc.•When ∃ heterogeneity, aggregation =⇒ long-memory; e.g. Robinson(1978), Granger (1980), and TS lit.•But in economics, ∃ an inherent nonlinearity. Decompose GDP asY ≡ Y1 + Y2 + · · · = elog Y1 + elog Y2 + · · · 6= elog Y1+log Y2+...•The result is a new auto-correlation function (ACF) ρτ :00.10.20.30.40.50.60.70.80.9104080120160200240280320360400tauACFAggregate Process AR(1) Root 0.9 AR(1) Root 0.99 AR(2) Roots 0.99 and 0.9 I(1/3)with diﬀerent macropolicy implications.•The result is a new auto-correlation function (ACF) ρτ :00.10.20.30.40.50.60.70.80.9104080120160200240280320360400tauACFAggregate Process AR(1) Root 0.9 AR(1) Root 0.99 AR(2) Roots 0.99 and 0.9 I(1/3)with diﬀerent macropolicy implications.•But if /∃ integration, what about the modification of co-integration?1 UIP and forward premium puzzle•Fisher (1930): “speculation” equates expected returns after conver-sion to the same currency (UIP); e.g.—£1 invested in domestic (UK) bond yields £(1 + It) at maturity;vs.—£1 invested in foreign (US) bond converted into $1/St, which yield$¡1 + I∗t¢/St = £¡1 + I∗t¢St+1/St at maturity.•Define i := log (1 + I), s := logS, andrt+1 := ∆st+1 + i∗t − it,as the excess return from investing in the foreign asset. Then, Et [rt+1] =0 and rt+1 should not be predictable.•Typical empirical implementation: regress rt+1 on the forward pre-mium (ft − st), where f and s are the forward and spot rates inlogs:rt+1 = α + β (ft − st) + ut+1,where α is the average risk premium and β is the informational con-tent of the forward premium.—UIP hypothesis implies H0 : α = 0 and β = 0.— If one believes ft = Et [st+1], then (ft − st) < 0 indicates that theUS$ should depreciate.—Routine finding: β < 0. As more US$ depreciation is expected,higher returns are actually made on the US$!Investors are ready to pay more for an asset which, according totheir expectations, should have become less attractive!— See Cumby and Obstfeld (1981, J Fin) or Engel (1996, J Emp Fin).•Running this regression with our data (3-month rates on US$ andUK£ deposits in London and 3-month forward rate)rˆt+1 = −0.0157 −3.26 (ft − st)t-ratio (−3.67) (−6.14)HAC t [−2.85] [−2.90]where HAC = heteroskedastic and autocorrelation-consistent,Durbin-Watson statistic : 0.65ARCH(7) test, F (7, 242) : 24.10 {0.0%}RESET (omitted nonlinearities), F (1, 255) : 9.66 {0.2%}• Scatter plot of datar t +1 = - 0.0157 -3.2615(f t-s t)R2 = 0.1283-0.3-0.2-0.100.10.20.3-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02forward premiumexcess return on foreign asset•For years, it has defied economic logic to find such a result: how couldinvestors’ expectations be so systematically wrong? Or are they?!2 Solution to the puzzle: coping with non-linear long-memory•Co-movements vs. own dynamics.• Incomplete modelling of dynamics can make the estimation of co-movements biased and inconsistent.•Example (just an illustration): consider autoregressive processyt = αyt−1 + βxt + ut,with ut = ρut−1 + εt and εt ∼ IID³0, σ2´.Running OLS on the first equation only =⇒ biased and inconsistentestimators; e.g. Maddala and Rao (1973, Ecta).•Usual approaches:—GLS on the augmented first equation; or— error correction mechanism (ECM), autoregressive distributed lag(ADL)yt = (α + ρ) yt−1 − αρyt−2 + βxt − βρxt−1 + εt,estimatingyt = a1yt−1 + a2yt−2 + a3xt + a4xt−1 + etthen testing for the restrictiona1 =a2a3a4− a4a3.— error correction mechanism (ECM), autoregressive distributed lag(ADL)yt = (α + ρ) yt−1 − αρyt−2 + βxt − βρxt−1 + εt,estimatingyt = a1yt−1 + a2yt−2 + a3xt + a4xt−1 + etthen testing for the restrictiona1 =a2a3a4− a4a3.• Long-memory times seriesmodels: very persistent time series (“Josepheﬀect”).•Problem of ECM/ADL for data with long memory: too many lags.•How about the GLS route?•We would like to estimate the relationzt = z˜t + ut, t = 1, 2, . . . T,where the T × 1 vector ez = Xβ is the “fundamental” value of z,but z and ez (hence possibly u) have long memory which needs to beaccounted for. A possible 2-step procedure (à la GLS):— decompose the autocorrelation matrix of z as R = LL0, where Lis lower-triangular and invertible;— the transformed dataL−1z andL−1ez do not contain long memoryand can be regressed by traditional methods.•Unfortunately, estimating R requires estimating T − 1 parameters:same as infeasible GLS!Solution: estimate the ACF of z, using a variant of the functionalform in Abadir and Talmain (2002, R E Stud)ρτ ≈1− a [1− cos (ωτ )]1 + bτc,with only 4 parameters to fit. (Note: denominator controls decay ofmemory.)•UIP example: fit is excellent for sActual vs fitted ACF of the spot rate $-£-100%-80%-60%-40%-20%0%20%40%60%80%100%024487296120144168lagsACFactual ACF spot $-£ fitted ACF spot $-£—The estimated T × T correlation matrix is thenbR ≡⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1 ρˆ1 ρˆ2 . . . ρˆT−2 ρˆT−1ρˆ1 1 ρˆ1 . . . . . . ρˆT−2ρˆ2 ρˆ1 . . . . . . . . . . . .. . . . . . . . . . . . ρˆ1 ρˆ2ρˆT−2 . . . . . . ρˆ1 1 ρˆ1ρˆT−1 ρˆT−2 . . . ρˆ2 ρˆ1 1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.—Find the Cholesky decomposition (matlab) bR = bLbL0; and— calculate sacf = bL−1s and facf = bL−1f .—The scatter plot of the transformed data is a nice spherical cloud-3-2-10123-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2forward premiumexcess return on foreign asset— and the regression with transformed data becomesrˆacft+1 = 0.00582 +0.0604³facft − sacft´t-ratio (0.14) (0.05)HAC t [0.19] [0.04]Durbin-Watson statistic : 2.20ARCH(7) test, F (7, 242) : 6.54 {0.0%}RESET (omitted nonlinearities), F (1, 255) : 0.82 {36.6%} .3 Stock market application44.555.566.5719461951195619611966197119761981198619911996YearS&P50077.588.599.5GDPSP Y•Unit roots?! (NB: unit root = permanent memory and no mean-reversion to any regular pattern or trend!)• Lack of unit roots shown for US andUKGDPs inAbadir andTalmain(2002, R E Stud).•How about Stock indices?— Jegadeesh and Titman (1993 and 2001, J Fin) find momentum;—De Bondt and Thaler (1985 and 1987, J Fin) find long cycles;—Cavaliere’s (2001, Ect J) non-parametric test detects long-memory.— Jegadeesh and Titman (1993 and 2001, J Fin) find momentum;—De Bondt and Thaler (1985 and 1987, J Fin) find long cycles;—Cavaliere’s (2001, Ect J) non-parametric test detects long-memory.•GE theory: long-run proportionality relationship between the aggre-gate real value of firms and GDP. On a balanced growth path:— real interest rate is a function of capital/output, which is constant⇒ rate at which future aggregate profits are discounted is fixed;— but share of aggregate profits in GDP is constant;— hence the discounted stream of future profits (i.e. capitalized valueof the stock market) is proportional to GDP.— Jegadeesh and Titman (1993 and 2001, J Fin) find momentum;—De Bondt and Thaler (1985 and 1987, J Fin) find long cycles;—Cavaliere’s (2001, Ect J) non-parametric test detects long-memory.•GE theory: long-run proportionality relationship between the aggre-gate real value of firms and GDP. On a balanced growth path:— real interest rate is a function of capital/output, which is constant⇒ rate at which future aggregate profits are discounted is fixed;— but share of aggregate profits in GDP is constant;— hence the discounted stream of future profits (i.e. capitalized valueof the stock market) is proportional to GDP.•Empirically-testable version: log of stock market index (S&P500)corrected for inflation, st, should be in a long-term proportionalitywith the log of real GDP, yt.•Error-correction model of st on yt,∆st = α + (β1∆st−1 + · · · + βm∆st−m) + (γ0∆yt + · · · + γn∆yt−n)−δ (st−1 − yt−1) + δ1yt−1 + εt.—The term −δ (st−1 − yt−1) + δ1yt−1 is the ECM.— It represents the long-run ‘equilibrium’ relationship between s andy:se =µ1 +δ1δ¶ye, δ 6= 0.—H0: δ1 = 0 for long-run proportionality between Se and Ye.—Define dt−1 := st−1−se as the deviation of st−1 from its long-termvalue se.—ECM: this deviation will pull st back towards its long-term equi-librium value by δdt−1, where δ > 0.•For S&P500 over 1958-2000, we obtained the regressiond∆st = −0.728 +0.539 ∆st−4 +0.380 ∆st−6(−2.33) (4.35) (2.83)+3.11 ∆yt −1.72 ∆yt−1 +1.84 ∆yt−2 −1.15 ∆yt−6(4.30) (−2.41) (2.58) (−1.77)−0.112 (st−1 − yt−1) +0.0396 yt−1(−2.11) (0.92)where the t-ratios are in parentheses, and we have R2 = 57.5%,AR(2) test, F (2, 32) : 0.43 {65.4%}ARCH(1) test, F (1, 32) : 0.17 {68.1%}RESET, F (1, 33) : 1.36 {25.3%} .•H0: δ1 = 0 is supported, but:— by the end of the period, 1995-2000, the fit is poor;— the coeﬃcient of yt−1 is unstable and H0 would be rejected on asample ending in 1994;— including more lags of ∆s in the regression worsens rather thanimproves stability, while not improving the fit.The recursive parameter estimates are...1960 1970 1980 1990 2000012 Original datadsp_4 × +/-2SE 1960 1970 1980 1990 20000.00.51.0 dsp_6 × +/-2SE 1960 1970 1980 1990 20000.02.55.07.5 dy × +/-2SE 1960 1970 1980 1990 2000-2.50.02.55.0 dy_1 × +/-2SE 1960 1970 1980 1990 2000051015 dy_2 × +/-2SE 1960 1970 1980 1990 2000-2.50.02.5 dy_6 × +/-2SE 1960 1970 1980 1990 2000-10y_1 × +/-2SE 1960 1970 1980 1990 2000-202 spy_1 × +/-2SE •Remember memory?!—Fit the ACF of s;Actual ACF of SP and fitted ACF-100%-80%-60%-40%-20%0%20%40%60%80%100%0 2 4 6 8 10 12 14 16 18 20 22 24 26 28lagsACFactual ACF SP fitted ACF SP— run the regression with transformed variables over 1960-2000[∆sacft = −0.363 +2.52 ∆yacft −1.41³sacft−1 − yacft−1´−0.301 yacft−1(−0.27) (3.67) (−10.0) (−0.49)where R2 = 76.3%,AR(2) test, F (2, 35) : 2.47 {9.9%}ARCH(1) test, F (1, 35) : 1.72 {19.9%}RESET, F (1, 33) : 0.72 {40.2%} .—H0: δ1 = 0 is supported throughout the sample1970 1975 1980 1985 1990 1995 20000.02.55.0Corrected dataDYc × +/-2SE 1970 1975 1980 1985 1990 1995 2000-2.50.02.5 Yc_1 × +/-2SE 1970 1975 1980 1985 1990 1995 2000-2.0-1.5-1.0 SPYc_1 × +/-2SE — deviations from cycles around the long-run fundamental values arerestored well within a year;— good fit.4 Extensions•The two-step procedure is not the most eﬃcient estimation method:—we provide formulae for full GLS, QMLE, etc.;— but qualitative results remain unchanged.Example: QMLE case. For any given R, definebβR ≡ ³X 0R−1X´−1X 0R−1zas a function of R. The QMLE of R is obtained by maximizing− log¯¯¯¯³z −XbβR´0R−1 ³z −XbβR´R¯¯¯¯with respect to the parameters of the ACF: the optimization of thejoint likelihood (forR andβ) now depends on only 4 parameters thatdetermine the whole autocorrelation matrix R. Once the optimalvalue bR of R is obtained, the MLE of β is bβ ≡ bβ bR.

Macro and financial markets: The memory of an elephant?

Enlighten: Publications

  
 
 
 
Abadir, K.M., and Talmain, G. (2009) Macro and financial markets: The 
memory of an elephant? In: 15th Spring Meeting of Young Economists 
2009, 23-25 Apr 2009, Istanbul, Turkey. 
 
  
 
Copyright © 2009 The Authors 
 
 
A copy can be downloaded for personal non-commercial research or 
study, without prior permission or charge 
 
Content must not be changed in any way or reproduced in any format 
or medium without the formal permission of the copyright holder(s) 
 
 
When referring to this work, full bibliographic details must be given 
 
 
  
http://eprints.gla.ac.uk/45444 
 
 
Deposited on: 30 September 2013 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Enlighten – Research publications by members of the University of Glasgow 
http://eprints.gla.ac.uk 
MACRO AND FINANCIAL MARKETS:
The memory of an elephant?
Karim Abadir and Gabriel Talmain
Background, REStud 2002:
•Model and solution:
—Micro-founded macro model.
— Standard RBC + heterogeneity: drop “representative firm” as-
sumption.
—General Equilibrium yields explicit dynamic equation for GDP etc.
Background, REStud 2002:
•Model and solution:
—Micro-founded macro model.
— Standard RBC + heterogeneity: drop “representative firm” as-
sumption.
—General Equilibrium yields explicit dynamic equation for GDP etc.
•When ∃ heterogeneity, aggregation =⇒ long-memory; e.g. Robinson
(1978), Granger (1980), and TS lit.
•But in economics, ∃ an inherent nonlinearity. Decompose GDP as
Y ≡ Y1 + Y2 + · · · = elog Y1 + elog Y2 + · · · 6= elog Y1+log Y2+...
•The result is a new auto-correlation function (ACF) ρτ :
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
4
0
8
0
1
2
0
1
6
0
2
0
0
2
4
0
2
8
0
3
2
0
3
6
0
4
0
0
tau
A
C
F
Aggregate Process AR(1) Root 0.9 AR(1) Root 0.99 AR(2) Roots 0.99 and 0.9 I(1/3)
with different macropolicy implications.
•The result is a new auto-correlation function (ACF) ρτ :
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
4
0
8
0
1
2
0
1
6
0
2
0
0
2
4
0
2
8
0
3
2
0
3
6
0
4
0
0
tau
A
C
F
Aggregate Process AR(1) Root 0.9 AR(1) Root 0.99 AR(2) Roots 0.99 and 0.9 I(1/3)
with different macropolicy implications.
•But if /∃ integration, what about the modification of co-integration?
1 UIP and forward premium puzzle
•Fisher (1930): “speculation” equates expected returns after conver-
sion to the same currency (UIP); e.g.
—£1 invested in domestic (UK) bond yields £(1 + It) at maturity;
vs.
—£1 invested in foreign (US) bond converted into $1/St, which yield
$
¡
1 + I∗t
¢
/St = £
¡
1 + I∗t
¢
St+1/St at maturity.
•Define i := log (1 + I), s := logS, and
rt+1 := ∆st+1 + i∗t − it,
as the excess return from investing in the foreign asset. Then, Et [rt+1] =
0 and rt+1 should not be predictable.
•Typical empirical implementation: regress rt+1 on the forward pre-
mium (ft − st), where f and s are the forward and spot rates in
logs:
rt+1 = α + β (ft − st) + ut+1,
where α is the average risk premium and β is the informational con-
tent of the forward premium.
—UIP hypothesis implies H0 : α = 0 and β = 0.
— If one believes ft = Et [st+1], then (ft − st) < 0 indicates that the
US$ should depreciate.
—Routine finding: β < 0. As more US$ depreciation is expected,
higher returns are actually made on the US$!
Investors are ready to pay more for an asset which, according to
their expectations, should have become less attractive!
— See Cumby and Obstfeld (1981, J Fin) or Engel (1996, J Emp Fin).
•Running this regression with our data (3-month rates on US$ and
UK£ deposits in London and 3-month forward rate)
rˆt+1 = −0.0157 −3.26 (ft − st)
t-ratio (−3.67) (−6.14)
HAC t [−2.85] [−2.90]
where HAC = heteroskedastic and autocorrelation-consistent,
Durbin-Watson statistic : 0.65
ARCH(7) test, F (7, 242) : 24.10 {0.0%}
RESET (omitted nonlinearities), F (1, 255) : 9.66 {0.2%}
• Scatter plot of data
r t +1 = - 0.0157 -3.2615(f t-s t)
R2 = 0.1283
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02
forward premium
e
x
c
e
s
s
 
r
e
t
u
r
n
 
o
n
 
f
o
r
e
i
g
n
 
a
s
s
e
t
•For years, it has defied economic logic to find such a result: how could
investors’ expectations be so systematically wrong? Or are they?!
2 Solution to the puzzle: coping with non-linear long-
memory
•Co-movements vs. own dynamics.
• Incomplete modelling of dynamics can make the estimation of co-
movements biased and inconsistent.
•Example (just an illustration): consider autoregressive process
yt = αyt−1 + βxt + ut,
with ut = ρut−1 + εt and εt ∼ IID
³
0, σ2
´
.
Running OLS on the first equation only =⇒ biased and inconsistent
estimators; e.g. Maddala and Rao (1973, Ecta).
•Usual approaches:
—GLS on the augmented first equation; or
— error correction mechanism (ECM), autoregressive distributed lag
(ADL)
yt = (α + ρ) yt−1 − αρyt−2 + βxt − βρxt−1 + εt,
estimating
yt = a1yt−1 + a2yt−2 + a3xt + a4xt−1 + et
then testing for the restriction
a1 =
a2a3
a4
− a4
a3
.
— error correction mechanism (ECM), autoregressive distributed lag
(ADL)
yt = (α + ρ) yt−1 − αρyt−2 + βxt − βρxt−1 + εt,
estimating
yt = a1yt−1 + a2yt−2 + a3xt + a4xt−1 + et
then testing for the restriction
a1 =
a2a3
a4
− a4
a3
.
• Long-memory times seriesmodels: very persistent time series (“Joseph
effect”).
•Problem of ECM/ADL for data with long memory: too many lags.
•How about the GLS route?
•We would like to estimate the relation
zt = z˜t + ut, t = 1, 2, . . . T,
where the T × 1 vector ez = Xβ is the “fundamental” value of z,
but z and ez (hence possibly u) have long memory which needs to be
accounted for. A possible 2-step procedure (à la GLS):
— decompose the autocorrelation matrix of z as R = LL0, where L
is lower-triangular and invertible;
— the transformed dataL−1z andL−1ez do not contain long memory
and can be regressed by traditional methods.
•Unfortunately, estimating R requires estimating T − 1 parameters:
same as infeasible GLS!
Solution: estimate the ACF of z, using a variant of the functional
form in Abadir and Talmain (2002, R E Stud)
ρτ ≈
1− a [1− cos (ωτ )]
1 + bτc
,
with only 4 parameters to fit. (Note: denominator controls decay of
memory.)
•UIP example: fit is excellent for s
Actual vs fitted ACF of the spot rate $-£
-100%
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
100%
0
2
4
4
8
7
2
9
6
1
2
0
1
4
4
1
6
8
lags
A
C
F
actual ACF spot $-£ fitted ACF spot $-£
—The estimated T × T correlation matrix is then
bR ≡
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
1 ρˆ1 ρˆ2 . . . ρˆT−2 ρˆT−1
ρˆ1 1 ρˆ1 . . . . . . ρˆT−2
ρˆ2 ρˆ1 . . . . . . . . . . . .
. . . . . . . . . . . . ρˆ1 ρˆ2
ρˆT−2 . . . . . . ρˆ1 1 ρˆ1
ρˆT−1 ρˆT−2 . . . ρˆ2 ρˆ1 1
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
.
—Find the Cholesky decomposition (matlab) bR = bLbL0; and
— calculate sacf = bL−1s and facf = bL−1f .
—The scatter plot of the transformed data is a nice spherical cloud
-3
-2
-1
0
1
2
3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
forward premium
e
x
c
e
s
s
 
r
e
t
u
r
n
 
o
n
 
f
o
r
e
i
g
n
 
a
s
s
e
t
— and the regression with transformed data becomes
rˆacft+1 = 0.00582 +0.0604
³
facft − sacft
´
t-ratio (0.14) (0.05)
HAC t [0.19] [0.04]
Durbin-Watson statistic : 2.20
ARCH(7) test, F (7, 242) : 6.54 {0.0%}
RESET (omitted nonlinearities), F (1, 255) : 0.82 {36.6%} .
3 Stock market application
4
4.5
5
5.5
6
6.5
7
1
9
4
6
1
9
5
1
1
9
5
6
1
9
6
1
1
9
6
6
1
9
7
1
1
9
7
6
1
9
8
1
1
9
8
6
1
9
9
1
1
9
9
6
Year
S
&
P
5
0
0
7
7.5
8
8.5
9
9.5
G
D
P
SP Y
•Unit roots?! (NB: unit root = permanent memory and no mean-
reversion to any regular pattern or trend!)
• Lack of unit roots shown for US andUKGDPs inAbadir andTalmain
(2002, R E Stud).
•How about Stock indices?
— Jegadeesh and Titman (1993 and 2001, J Fin) find momentum;
—De Bondt and Thaler (1985 and 1987, J Fin) find long cycles;
—Cavaliere’s (2001, Ect J) non-parametric test detects long-memory.
— Jegadeesh and Titman (1993 and 2001, J Fin) find momentum;
—De Bondt and Thaler (1985 and 1987, J Fin) find long cycles;
—Cavaliere’s (2001, Ect J) non-parametric test detects long-memory.
•GE theory: long-run proportionality relationship between the aggre-
gate real value of firms and GDP. On a balanced growth path:
— real interest rate is a function of capital/output, which is constant
⇒ rate at which future aggregate profits are discounted is fixed;
— but share of aggregate profits in GDP is constant;
— hence the discounted stream of future profits (i.e. capitalized value
of the stock market) is proportional to GDP.
— Jegadeesh and Titman (1993 and 2001, J Fin) find momentum;
—De Bondt and Thaler (1985 and 1987, J Fin) find long cycles;
—Cavaliere’s (2001, Ect J) non-parametric test detects long-memory.
•GE theory: long-run proportionality relationship between the aggre-
gate real value of firms and GDP. On a balanced growth path:
— real interest rate is a function of capital/output, which is constant
⇒ rate at which future aggregate profits are discounted is fixed;
— but share of aggregate profits in GDP is constant;
— hence the discounted stream of future profits (i.e. capitalized value
of the stock market) is proportional to GDP.
•Empirically-testable version: log of stock market index (S&P500)
corrected for inflation, st, should be in a long-term proportionality
with the log of real GDP, yt.
•Error-correction model of st on yt,
∆st = α + (β1∆st−1 + · · · + βm∆st−m) + (γ0∆yt + · · · + γn∆yt−n)
−δ (st−1 − yt−1) + δ1yt−1 + εt.
—The term −δ (st−1 − yt−1) + δ1yt−1 is the ECM.
— It represents the long-run ‘equilibrium’ relationship between s and
y:
se =
µ
1 +
δ1
δ
¶
ye, δ 6= 0.
—H0: δ1 = 0 for long-run proportionality between Se and Ye.
—Define dt−1 := st−1−se as the deviation of st−1 from its long-term
value se.
—ECM: this deviation will pull st back towards its long-term equi-
librium value by δdt−1, where δ > 0.
•For S&P500 over 1958-2000, we obtained the regressiond∆st = −0.728 +0.539 ∆st−4 +0.380 ∆st−6
(−2.33) (4.35) (2.83)
+3.11 ∆yt −1.72 ∆yt−1 +1.84 ∆yt−2 −1.15 ∆yt−6
(4.30) (−2.41) (2.58) (−1.77)
−0.112 (st−1 − yt−1) +0.0396 yt−1
(−2.11) (0.92)
where the t-ratios are in parentheses, and we have R2 = 57.5%,
AR(2) test, F (2, 32) : 0.43 {65.4%}
ARCH(1) test, F (1, 32) : 0.17 {68.1%}
RESET, F (1, 33) : 1.36 {25.3%} .
•H0: δ1 = 0 is supported, but:
— by the end of the period, 1995-2000, the fit is poor;
— the coefficient of yt−1 is unstable and H0 would be rejected on a
sample ending in 1994;
— including more lags of ∆s in the regression worsens rather than
improves stability, while not improving the fit.
The recursive parameter estimates are...
1960 1970 1980 1990 2000
0
1
2 Original datadsp_4 × +/-2SE 
1960 1970 1980 1990 2000
0.0
0.5
1.0 dsp_6 × +/-2SE 
1960 1970 1980 1990 2000
0.0
2.5
5.0
7.5 dy × +/-2SE 
1960 1970 1980 1990 2000
-2.5
0.0
2.5
5.0 dy_1 × +/-2SE 
1960 1970 1980 1990 2000
0
5
10
15 dy_2 × +/-2SE 
1960 1970 1980 1990 2000
-2.5
0.0
2.5 dy_6 × +/-2SE 
1960 1970 1980 1990 2000
-1
0
y_1 × +/-2SE 
1960 1970 1980 1990 2000
-2
0
2 spy_1 × +/-2SE 
•Remember memory?!
—Fit the ACF of s;
Actual ACF of SP and fitted ACF
-100%
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
100%
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
lags
A
C
F
actual ACF SP fitted ACF SP
— run the regression with transformed variables over 1960-2000
[∆sacft = −0.363 +2.52 ∆yacft −1.41
³
sacft−1 − yacft−1
´
−0.301 yacft−1
(−0.27) (3.67) (−10.0) (−0.49)
where R2 = 76.3%,
AR(2) test, F (2, 35) : 2.47 {9.9%}
ARCH(1) test, F (1, 35) : 1.72 {19.9%}
RESET, F (1, 33) : 0.72 {40.2%} .
—H0: δ1 = 0 is supported throughout the sample
1970 1975 1980 1985 1990 1995 2000
0.0
2.5
5.0
Corrected data
DYc × +/-2SE 
1970 1975 1980 1985 1990 1995 2000
-2.5
0.0
2.5 Yc_1 × +/-2SE 
1970 1975 1980 1985 1990 1995 2000
-2.0
-1.5
-1.0 SPYc_1 × +/-2SE 
— deviations from cycles around the long-run fundamental values are
restored well within a year;
— good fit.
4 Extensions
•The two-step procedure is not the most efficient estimation method:
—we provide formulae for full GLS, QMLE, etc.;
— but qualitative results remain unchanged.
Example: QMLE case. For any given R, definebβR ≡ ³X 0R−1X´−1X 0R−1z
as a function of R. The QMLE of R is obtained by maximizing
− log
¯¯¯¯³
z −XbβR´0R−1 ³z −XbβR´R¯¯¯¯
with respect to the parameters of the ACF: the optimization of the
joint likelihood (forR andβ) now depends on only 4 parameters that
determine the whole autocorrelation matrix R. Once the optimal
value bR of R is obtained, the MLE of β is bβ ≡ bβ bR.


http://eprints.gla.ac.uk/45444/1/45444.pdf

Macro and financial markets: The memory of an elephant?

Abstract

Similar works

Full text

Available Versions

Enlighten

Enlighten: Publications