NonCausal Vector AR Processes with .1in Application to...
Transcript of NonCausal Vector AR Processes with .1in Application to...
NonCausal Vector AR Processes withApplication to Economic Time Series
Richard A. DavisColumbia University
(Joint work with Li Song)
2nd Congreso De ActuariaUNAM, Mexico City
January 24, 2013
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 1
Example: Walmart
log(Volume) of Walmart stock 12/1/03-12/31/04
Louvain 2011 2
1. Motivating Example (cont)
day
log-
volu
me
0 50 100 150 200 250
15.0
15.5
16.0
16.5
17.0
Log(volume) of Walmart stock 12/1/03-12/31/04
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 2
Example: Walmart (ACF/PACF)
Analysis of the ACF and PACF of the time series (n=274) suggests that {Xt } follows anAR(1) or AR(2). A causal AR(2) fit (using Gaussian MLE) is
Xt = .4455Xt−1 + .1025Xt−2 + Zt
The estimated residuals are uncorrelated but dependent as seen in the plots of the ACF ofthe absolute values and squares of the residuals
⇒ residuals follow an allpass model
Louvain 2011 14
7. Walmart revisited---residuals from noncausal model
Analysis of the ACF and PACF of the time series (n=274) suggests that {Xt} follows an AR (1) or AR(2).
A causal AR(2) fit (using Gaussian MLE) is
Xt .4455 Xt-1 .1025 Xt-2 Zt
The estimated residuals were uncorrelated but dependent.
lag (h)
acf o
f abs
val
ues
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
lag (h)
acf o
f squ
ares
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 3
Example: Walmart (noncausal)
Maximum-likelihood model:
Xt = −2.0766Xt−1 + 2.0772Xt−2 + Zt (purely noncausal)
Zt ∼ IID Stable
α = 1.8335, β = .5650, σ = .4559, µ = 16.0030
Louvain 2011
lag (h)
acf o
f abs
val
ues
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
lag (h)
acf o
f squ
ares
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
CB’s simulatedCB’s simulated
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 4
Outline
Warm-up example with Walmart
Univariate AR models: causal and noncausal
Multivariate AR models: causal and noncausal
• Preliminaries
• Identifiability and likelihood function.
• Asymptotic properties of the estimators
• Examples
* term structure of interest rates
* fiscal foresight
Summary
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 5
AR Model:
The Model: Assume {Xt } is the stationary solution to the recursions
Xt = φ1Xt−1 + · · ·+ φpXt−p + Zt {Zt } ∼ IID(0, σ2)
(1 − φ1B − · · · − φpBp)Xt = Zt (B is backward shift operator)
φc(B)φnc(B)Xt = Zt ,
where φc(z) and φnc(z) are the respective causal and noncausal polynomials of the ARpolynomial, i.e.,
φc(z) causal means φc(z) , 0 for |z| ≤ 1.
φnc(z) noncausal means φc(z) , 0 for |z| > 1.
The process has the two-sided representation (one sided if the degrees of φc or φnc are 0)
Xt =∞∑
j=−∞
ψjZt−j
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 6
Univariate AR models: causal and noncausal models
Louvain 2011 10
4. AR models—causal and noncausal
Zt
Xt
Impulse response: causal & low frequency
jt-j
jt
ZX
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 7
Univariate AR models: causal and noncausal models
Louvain 2011 11
4. AR models—causal and noncausal
Impulse response: noncausal & high frequency
Zt
Xt
jt-j
jt
ZX
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 8
Univariate AR models: causal and noncausal models
Louvain 2011 12
4. AR models—causal and noncausal
Impulse response: mixed causal (low frequency) & noncausal (high frequency)
Zt
Xt
jt-j
jt
ZX
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 9
An Example: Muddy River–tributary to Sun River in Central Montana
Muddy Creek: surveyed every 15.24 meters, total of 5456m; 358 measurements
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 10
An Example: Muddy River–tributary to Sun River in Central Montana
Muddy Creek: surveyed every 15.24 meters, total of 5456m; 358 measurements
Best fitting (minimum AICC) ARMA model to residuals (after removal of quadratic trend):
Yt = .574Yt−1 + Zt − .311Zt−1, {Zt } ∼ WN(0, .0564)
.Residuals follow an allpass model AP(1), which suggests a noncausal model:
Noncausal ARMA(1,1) model:
Yt = 1.743Yt−1 + Zt − .311Zt−1, {Zt } ∼ WN(0, .0564)
.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 11
Noncausal VAR
A multivariate time series {Xt } is a VAR(p) process if it is stationary and satisfies therecursions
Φ(B)Xt = Xt −Φ1Xt−1 − · · · −ΦpXt−p = Zt ,
where Xt := (Xt ,1, . . . ,Xt ,m)T is an m-dimensional stochastic process, Zt := (Zt ,1, . . . ,Zt ,m)T
is an iid sequence of continuous random vectors with mean 0 and covariance matrix Σ∗0.
Causal: All the roots of detΦ(z) are outside the unit circle.
Purely noncausal: All the roots of detΦ(z) are inside the unit circle.
Mixed: detΦ(z) has roots both inside and outside the unit circle.
Univariate vs multivariate: A multivariate AR(1) can be mixed!
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 12
Identifiability
A general VAR(p) model given by
Φ(B)Xt = Xt −Φ1Xt−1 − · · · −ΦpXt−p = Zt ,
may have zeros of detΦ(z) = det(I −Φ1z − · · · −Φpzp) that are both inside andoutside the unit circle.
Hannan (1970) showed that the spectral density matrix of Xt always has arepresentation which is the spectral density matrix of a causal VAR(p) process.
Noncausal models cannot be distinguished from causal models in the Gaussian case.
Adapting Theorem 1 from Chan and Tong (2006) on reversibility of vector-valued linearprocesses, one can show identifiability of possibly noncausal VAR models undersuitable conditions on the distribution of Zt .
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 13
Identifiability
From Chan and Tong (2006), we have
Lemma
Let {Xt } and {Xt } be two nonGaussian m-dimensional linear processes defined by
Xt =∞∑
j=−∞
CjZt−j and Xt =∞∑
j=−∞
Cj Zt−j ,
where
Cj and Cj square-summable
{Zt } ∼ IID(0,Σ) and {Zt } ∼ IID(0, Σ)
Then under some technical conditions on the distribution of Zt ,
{Xt }d= {Xt }
if and only if there exist an integer q and a matrix H such that for all t,
Zt−qd= HZt and Ct−q = Ct H . (1)
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 14
State-space representation
As is well known, a VAR(p) model can be re-expressed as a VAR(1) model.
Define two new processes {Yt } and Z∗t by
Yt =
Xt
Xt−1
...Xt−p+1
pm×1
and Z∗t =
Zt
0...0
pm×1
.
ThenYt = ΦY Yt−1 + Z∗t , (2)
where
ΦY =
Φ1 Φ2 · · · · · · Φp
Im Om · · · · · · Om
Om Im. . .
......
. . .. . .
...Om Om · · · Im Om
pm×pm
,
and Om is the m ×m matrix of zeros.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 15
State-space representation
Zeros of det Φ(z) correspond to the reciprocals of the eigenvalues of ΦY .
causal(noncausal) roots of det Φ(z) correspond to eigenvalues of ΦY that areinside(outside) the unit circle.
Jordan canonical form of ΦY implies there exists a pm × pm invertible matrix A suchthat ΦY A = AJ, where
J =
λ1
s1. . .
. . .. . .
sl−1 λl
0 λl+1
sl+1. . .
. . .. . .
spm−1 λpm
,
and |λ1| ≤ |λ2| ≤ · · · ≤ |λl | < 1 < |λl+1| ≤ · · · ≤ |λpm |, {si} are 0 or 1
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 16
State-space representation
From Yt = ΦY Yt−1 + Z∗t and the fact that ΦY = AJA−1 we have
(I − JB)A−1Yt := (I − JB)Yt = A−1Z∗t := Zt
Since J is block diagonal (corresponding to the causal and noncausal components),we decompose Yt and Zt into the first l and last pm − l components, i.e.,
Yt = (YTt ,1, Y
Tt ,2)T and Zt = (ZT
t ,1, ZTt ,2)T
and obtain Yt ,1 − J1Yt−1,1 = Zt ,1
Yt ,2 − J2Yt−1,2 = Zt ,2
This implies Yt ,1 is a purely causal AR process and Yt ,2 is a purely noncausal ARprocess.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 17
State-space representation
Representation for Yt :
Yt =∞∑
i=−∞
AFiA−1Z∗t−i ,
where
Fi =
Ji1
Opm−l
, i ≥ 0, Ol
−Ji2
, i ≤ −1.
Representation for Xt : Let Mi be the upper-left sub-matrix of AFiA−1, then
Xt =∞∑
i=−∞
MiZt−i .
Impulse response coefficients: {Mi} or {MiPL } where PL is the lower triangular matrix inthe Cholesky decomposition of Σ∗0.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 18
Impulse response coefficients
Model (Xt ,1
Xt ,2
)=
(0.8 0.60.6 1.7
) (Xt−1,1
Xt−1,2
)+
(Zt ,1
Zt ,2
),
for which det(I −Φ1z) has zeros at z = .5 and z = 2. The corresponding component series{Yt ,1} and {Yt ,2} (found from Yt = A−1Xt ) are univariate noncausal and causal AR(1)processes given by
Yt ,1 = .5Yt−1,1 + Zt ,1
Yt ,2 = 2Yt−1,2 + Zt ,2
where {(Zt ,1, Zt ,2)′} is an iid sequence.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 19
Impulse response coefficients(Xt ,1
Xt ,2
)=
(0.8 0.60.6 1.7
) (Xt−1,1
Xt−1,2
)+
(Zt ,1
Zt ,2
).
−10 −5 0 5 10−0.4
−0.2
0
0.2
0.4
0.6
0.8M11
−10 −5 0 5 10−0.4
−0.2
0
0.2
0.4
0.6
0.8M12
−10 −5 0 5 10−0.4
−0.2
0
0.2
0.4
0.6
0.8M21
−10 −5 0 5 10−0.4
−0.2
0
0.2
0.4
0.6
0.8M22
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 20
Non-Gaussian distributions and parameters
We consider a general elliptical class of distributions for Zt which has a density of theform fZ (z; ν) = det(Σ)−1/2f(zT Σ−1z; ν). Multivariate t-distribution is also included in thisclass.
Parameters
φ =
φ1
φ2
...φpm2
=
vec(Φ1)vec(Φ2)
...vec(Φp)
and σ = vech(Σ).
Let θ be the vector that contains all the unknown parameters, namely
θ =
φσν
.The state-space representation of the VAR(p) process described earlier plays a keyrole in deriving the likelihood function.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 21
Non-Gaussian likelihood function
A complete likelihood function of Xt is given by
ln(θ) = log(L(X1, . . . ,Xn))
= log(p1(Yp,1)p2(Yn,2) · | det(A)|−1) +n∑
i=p+1
(log fZ (Zi(φ); ν) + κ(φ)) ,
where Yp,1 only depends on {Z−∞, . . . ,Zp}, Yn,2 only depends on {Zn+1, . . . ,Z∞} and Aonly depends on {Φ1, . . . ,Φp}.
The function κ(φ) is the reciprocal of the product of all roots of detΦ(z) = 0 that areinside the unit circle.
This complete likelihood can be approximated by
ln(θ) =n∑
i=p+1
(log fZ (Zi(φ); ν) + κ(φ)) :=n∑
i=p+1
gi(θ).
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 22
Asymptotic behavior of the score
Theorem 2:
Suppose that Zt has a nonGaussian density function and satisfying some regularity condi-tions. Then,
1√
n − p∂ln(θ0)
∂θ=
1√
n − p
n∑i=p+1
∂gi(θ0)
∂θ
d→ N(0,Iθθ(θ0)),
where the matrix Iθθ(θ0) is given by −E(∂2gp+1(θ0)
∂θ∂θT
)and is positive definite.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 23
Technical complements
Identifiability of θ0 and consistency:The independence of Yp,1 and Yn,2 from Zp+1(φ0), . . . ,Zn(φ0) implies
ln(θ) = log(L(X1, . . . ,Xn))
= log(p1(Yp,1)p2(Yn,2) · | det(A)|−1) +n∑
i=p+1
(log fZ (Zi(φ); ν) + κ(φ)) ,
is also a complete likelihood for any choice of initial distributions p1(·) and p2(·) for Yp,1 andYn,2.
In fact, one can choose p1(·) and p2(·) independent of θ, and for such a complete likelihoodwe will have the following properties:
E ∂ln(θ0)
∂θ
= 0,
where E(·) denotes the expectation under the measure with the new choices of p1(·) andp2(·).
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 24
Technical complements (cont)
It follows that:
0 =1
n − pE
∂ln(θ0)
∂θ
=
E(∂ log(p1(Yp,1)p2(Yn,2) · | det(A)|−1)/∂θ
)n − p
+1
n − pE
(∂ln(θ0)
∂θ
)=
1n − p
n∑i=p+1
E(∂gi(θ0)
∂θ
).
But using an asymptotic stationarity argument (i.e., using stationarity of {∂gi(θ0)/∂θ} underthe stationary measure), we obtain the following relationship,
E(∂gi(θ0)
∂θ
)= lim
n→∞E
(∂gbn/2c(θ0)
∂θ
)= lim
n→∞E
(∂gbn/2c(θ0)
∂θ
)= lim
n→∞
1n − p
n∑i=p+1
E(∂gi(θ0)
∂θ
)= 0, (3)
which is the key result for identifiability and consistency.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 25
Simulation example
Recall the mixed model given by(Xt ,1
Xt ,2
)=
(0.8 0.60.6 1.7
) (Xt−1,1
Xt−1,2
)+
(Zt ,1
Zt ,2
).
We simulated the {Xt } process based on bivariate t noise {Zt } with Σ0 = I2 and ν = 6.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 26
Simulation results
True value Sample size 100 Sample size 200 Sample size 500 Sample size 1000φ1 0.8 0.8435 (0.3851) 0.8220 (0.2394) 0.8068 (0.1345) 0.8026 (0.0932)
(0.2985) (0.2106) (0.1330) (0.0940)φ2 0.6 0.5274 (0.2612) 0.5883 (0.1338) 0.5992 (0.0665) 0.5989 (0.0456)
(0.1487) (0.1049) (0.0662) (0.0468)φ3 0.6 0.5359 (0.5491) 0.5938 (0.3695) 0.5988 (0.2102) 0.5989 (0.1444)
(0.4614) (0.3254) (0.2055) (0.1452)φ4 1.7 1.6297 (0.5576) 1.6987 (0.3228) 1.7067 (0.1693) 1.7021 (0.1186)
(0.3839) (0.2708) (0.1710) (0.1029)σ1 1 1.1382 (0.5579) 1.0561 (0.2693) 1.0159 (0.1421) 1.0075 (0.0987)
(0.3115) (0.2197) (0.1387) (0.0980)σ2 0 -0.0663 (0.3529) -0.0115 (0.2060) -0.0024 (0.1130) -0.0024 (0.0770)
(0.2444) (0.1724) (0.1089) (0.0769)σ3 1 1.0226 (0.5093) 1.0264 (0.3137) 1.0108 (0.1760) 1.0040 (0.1233)
(0.4063) (0.2866) (0.1810) (0.1279)ν 6 6.7925 (3.5194) 6.4947 (2.2386) 6.1386 (1.1061) 6.0816 (0.7712)
(2.4436) (1.7235) (1.0884) (0.7692)
Table: The true and empirical mean of the MLE’s of φ1, φ2, φ3, φ4, σ1, σ2, σ3 and ν. The empiricalstandard errors are given in (· · · ) theoretical standard errors are given in (· · · ).
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 27
Term structure of interest rates
The raw data (1970:1 to 1998:4 [116 observations]) are taken from the website ofGregory Duffee.
The bivariate time series is constructed by
• ∆r = Change in the three-month interest rate (quarter-end yields on U.S.zero-coupon bonds);
• S = Spread between the ten-year and three-month interest rates.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 28
Time series plot
Change in three−month interest rate
Time
V1
0 20 40 60 80 100 120
−6
−4
−2
02
Spread between the ten−year and three−month interest rates
Time
V2
0 20 40 60 80 100 120
−4
−2
02
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 29
ACF of residuals from VAR(3) causal fit
0 5 10 15
−0.
50.
00.
51.
0
Lag
AC
F
V1
0 5 10 15
−0.
50.
00.
51.
0
Lag
V1 & V2
−15 −10 −5 0
−0.
50.
00.
51.
0
Lag
AC
F
V2 & V1
0 5 10 15
−0.
50.
00.
51.
0
Lag
V2
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 30
ACF of squares of residuals from VAR(3) causal fit
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
AC
F
V1
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V1 & V2
−15 −10 −5 0
−0.
20.
00.
20.
40.
60.
81.
0
Lag
AC
F
V2 & V1
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V2
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 31
Noncausal fitted models
Model assumption CG CN PNN MXLog-likelihood -256.624 -237.345 -235.481 -229.054
Table: Comparison of log-likelihood.
The following table summarizes the results of the best fitted VAR(3) model of the data. Itturns out the detΦ(z) = 0 corresponding to the best fit has only one root inside the unitcircle and five roots outside the unit circle.
Φ1 Φ2 Φ3 Σ ν
0.789 0.009 0.434 0.110 0.728 -0.268 0.774 -0.448 2.806(0.289) (0.255) (0.202) (0.327) (0.128) (0.232) (0.245) (0.145) (0.715)-0.548 0.818 -0.209 0.014 -0.552 0.136 -0.448 0.393(0.170) (0.174) (0.128) (0.240) (0.079) (0.175) (0.145) (0.106)
Table: The MLE’s of the parameters and their associated standard errors for the interest rate data.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 32
ACF of residuals from noncausal fit
0 5 10 15
−0.
50.
00.
51.
0
Lag
AC
F
V1
0 5 10 15
−0.
50.
00.
51.
0
Lag
V1 & V2
−15 −10 −5 0
−0.
50.
00.
51.
0
Lag
AC
F
V2 & V1
0 5 10 15
−0.
50.
00.
51.
0
Lag
V2
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 33
ACF of squares of residuals from noncausal fit
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
AC
F
V1
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V1 & V2
−15 −10 −5 0
−0.
20.
00.
20.
40.
60.
81.
0
Lag
AC
F
V2 & V1
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V2
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 34
Impulse response coefficientsTerm structure of interest rates
−20 −10 0 10 20
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5M11
−20 −10 0 10 20
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5M12
−20 −10 0 10 20
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5M21
−20 −10 0 10 20
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5M22
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 35
Fiscal foresight
Fiscal foresight (see Lanne and Saikkonen (2009)) is the notion that agents receivesignals in advance about prospective changes fiscal policy issues that may anticipatemovements in some macro economic time series.
Data are quarterly from 1955:1 to 2000:4 [184 observations] and are taken from theNational Income and Product Accounts (NIPA) tables.
All the components of national income are in real per capita terms and are transformedfrom their nominal values by dividing them by the gdp deflator and the populationmeasure.The trivariate time series is constructed by the differences of the GDP, the TotalGovernment Expenditure and the Total Government Revenue. More specifically,
1 GDP = Quarterly U.S. GDP data;2 Total Government Expenditure = Federal Defense Consumption Expenditures + Federal
Non Defense Consumption Expenditures + State and Local Consumption Expenditures +Federal Defense Gross Investment + Federal Non Defense Gross Investment + State andLocal Gross Investment;
3 Total Government Revenue = Total Government Receipts - Net Transfers Payments - NetInterest Paid.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 36
Time series plot
Differenced US GDP
Time
V1
0 50 100 150
−5
05
Total government expenditure
Time
V2
0 50 100 150
−1.
00.
01.
0
Total government revenue
Time
V3
0 50 100 150
−6
−2
02
46
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 37
ACF of squares of residuals from VAR(2) causal fit
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
AC
F
V1
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V1 & V2
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V1 & V3
−15 −10 −5 0
−0.
20.
00.
20.
40.
60.
81.
0
Lag
AC
F
V2 & V1
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V2
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V2 & V3
−15 −10 −5 0
−0.
20.
00.
20.
40.
60.
81.
0
Lag
AC
F
V3 & V1
−15 −10 −5 0
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V3 & V2
0 5 10 15
−0.
20.
00.
20.
40.
60.
81.
0
Lag
V3
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 38
Noncausal fitted models
Model assumption CG CN PNN MXLog-likelihood -819.230 -802.865 -800.355 -791.270
Table: Comparison of log-likelihood.
The following table summarizes the results of the best fitted VAR(2) model of the data. Itturns out the detΦ(z) = 0 corresponding to the best fit has three roots inside the unit circleand three roots outside the unit circle.
Φ1 Φ2 Σ ν
3.516 3.710 -3.799 1.789 -6.268 -1.162 49.108 -12.746 -14.666(1.226) (3.631) (1.527) (0.924) (2.946) (0.673) (9.654) (5.728) (7.215)-0.512 -3.104 0.679 -0.586 2.658 0.295 -12.746 5.135 4.019 5.751(0.456) (1.267) (0.625) (0.311) (1.081) (0.312) (5.728) (3.373) (3.494) (1.399)-1.099 -1.811 2.918 -0.505 1.149 1.257 -14.666 4.019 9.376(0.475) (1.889) (0.762) (0.435) (1.534) (0.314) (7.215) (3.494) (5.203)
Table: The MLE’s of the parameters and their associated standard errors for the GDP data.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 39
ACF of squares of residuals from noncausal fit
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
V1
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
Lag
V1 & V2
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
Lag
V1 & V3
−15 −10 −5 0
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
V2 & V1
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
Lag
V2
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
Lag
V2 & V3
−15 −10 −5 0
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
V3 & V1
−15 −10 −5 0
0.0
0.2
0.4
0.6
0.8
1.0
Lag
V3 & V2
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
Lag
V3
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 40
Impulse response coefficientsFiscal foresight
−10 −5 0 5 10
−1
−0.5
0
0.5
M11
−10 −5 0 5 10
−1
−0.5
0
0.5
M12
−10 −5 0 5 10
−1
−0.5
0
0.5
M13
−10 −5 0 5 10
−1
−0.5
0
0.5
M21
−10 −5 0 5 10
−1
−0.5
0
0.5
M22
−10 −5 0 5 10
−1
−0.5
0
0.5
M23
−10 −5 0 5 10
−1
−0.5
0
0.5
M31
−10 −5 0 5 10
−1
−0.5
0
0.5
M32
−10 −5 0 5 10
−1
−0.5
0
0.5
M33
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 41
Summary
We have developed a state-space representation for VAR models that can handle bothcausal and noncausal cases.
Developed a likelihood procedure for estimation of parameters in a possibly noncausalVAR framework.
Noncausal models may be useful in producing more independent looking residuals.
Noncausal models may provide additional insight (and creative explanations!) into theunderlying dynamics of a multivariate time series that are not otherwise detected viacausal models.
Richard A. Davis (Columbia) 2nd Congreso De Actuaria January 24, 2013 42