Predicting Intra–daily Volume Shares for VWAP–based Trading Strategies: a GAS Approach
Francesco Calvori, Fabrizio Cipollini, Giampiero M. Gallo Dipartimento di Statistica, Informatica, Applicazioni (DiSIA) G. Parenti Università di Firenze
Outline
1 Motivation
2 Modeling Strategy
3 Empirical Application
4 Conclusions
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
1 / 17
Why a VWAP? I
Institutional investors do not pursue speculative behavior. Main concern: control transaction costs.
I
Benchmark in relationship with customers: reference to VWAP (Volume Weighted Average Price) as an end–of–day measure.
I
Large trades have a market impact (adverse price movements): try and minimize impact of a trade.
I
VWAP Trading Strategy An order is divided up in smaller chunks over the day aiming at achieving an average execution price as close as possible to the VWAP.
I
Theoretical justification: VWAP is an unbiased estimate (on both sides of the trade) of the prices facing a passive trader during the trading day (Berkowitz et al. (1988)).
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
2 / 17
Why a VWAP? I
Institutional investors do not pursue speculative behavior. Main concern: control transaction costs.
I
Benchmark in relationship with customers: reference to VWAP (Volume Weighted Average Price) as an end–of–day measure.
I
Large trades have a market impact (adverse price movements): try and minimize impact of a trade.
I
VWAP Trading Strategy An order is divided up in smaller chunks over the day aiming at achieving an average execution price as close as possible to the VWAP.
I
Theoretical justification: VWAP is an unbiased estimate (on both sides of the trade) of the prices facing a passive trader during the trading day (Berkowitz et al. (1988)).
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
2 / 17
Why a VWAP? I
Institutional investors do not pursue speculative behavior. Main concern: control transaction costs.
I
Benchmark in relationship with customers: reference to VWAP (Volume Weighted Average Price) as an end–of–day measure.
I
Large trades have a market impact (adverse price movements): try and minimize impact of a trade.
I
VWAP Trading Strategy An order is divided up in smaller chunks over the day aiming at achieving an average execution price as close as possible to the VWAP.
I
Theoretical justification: VWAP is an unbiased estimate (on both sides of the trade) of the prices facing a passive trader during the trading day (Berkowitz et al. (1988)).
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
2 / 17
Why a VWAP? I
Institutional investors do not pursue speculative behavior. Main concern: control transaction costs.
I
Benchmark in relationship with customers: reference to VWAP (Volume Weighted Average Price) as an end–of–day measure.
I
Large trades have a market impact (adverse price movements): try and minimize impact of a trade.
I
VWAP Trading Strategy An order is divided up in smaller chunks over the day aiming at achieving an average execution price as close as possible to the VWAP.
I
Theoretical justification: VWAP is an unbiased estimate (on both sides of the trade) of the prices facing a passive trader during the trading day (Berkowitz et al. (1988)).
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
2 / 17
Why a VWAP? I
Institutional investors do not pursue speculative behavior. Main concern: control transaction costs.
I
Benchmark in relationship with customers: reference to VWAP (Volume Weighted Average Price) as an end–of–day measure.
I
Large trades have a market impact (adverse price movements): try and minimize impact of a trade.
I
VWAP Trading Strategy An order is divided up in smaller chunks over the day aiming at achieving an average execution price as close as possible to the VWAP.
I
Theoretical justification: VWAP is an unbiased estimate (on both sides of the trade) of the prices facing a passive trader during the trading day (Berkowitz et al. (1988)).
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
2 / 17
What’s a VWAP? PJt
j=1 vt (j) pt (j) PJt j=1 vt (j)
VWAPt := PI =
i=1
PI
vt,i pt,i
i=1
I
vt,i bin volume
I
pt,i bin VWAP
I
wt,i = vt,i /
I X
vt,i
=
I X
wt,i pt,i
i=1
vt,i bin volume share
i=1 Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
3 / 17
What’s a VWAP? PJt
j=1 vt (j) pt (j) PJt j=1 vt (j)
VWAPt := PI =
i=1
PI
vt,i pt,i
i=1
I
vt,i bin volume
I
pt,i bin VWAP
I
wt,i = vt,i /
I X
vt,i
=
I X
wt,i pt,i
i=1
vt,i bin volume share
i=1 Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
3 / 17
What’s a VWAP? PJt
j=1 vt (j) pt (j) PJt j=1 vt (j)
VWAPt := PI =
i=1
PI
vt,i pt,i
i=1
I
vt,i bin volume
I
pt,i bin VWAP
I
wt,i = vt,i /
I X
vt,i
=
I X
wt,i pt,i
i=1
vt,i bin volume share
i=1 Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
3 / 17
Why a model for volume shares? PI VWAPt =
i=1
PI
vt,i pt,i
i=1
I I
vt,i
=
I X
wt,i pt,i
i=1
Price substantially unpredictable; take as given. Two approaches for VWAP 1. Predict volumes (Brownlees et al. (2011)) 2. Predict volume shares directly (this paper)
I
Why?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? PI VWAPt =
i=1
PI
vt,i pt,i
i=1
I I
vt,i
=
I X
wt,i pt,i
i=1
Price substantially unpredictable; take as given. Two approaches for VWAP 1. Predict volumes (Brownlees et al. (2011)) 2. Predict volume shares directly (this paper)
I
Why?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? PI VWAPt =
i=1
PI
vt,i pt,i
i=1
I I
vt,i
=
I X
wt,i pt,i
i=1
Price substantially unpredictable; take as given. Two approaches for VWAP 1. Predict volumes (Brownlees et al. (2011)) 2. Predict volume shares directly (this paper)
I
Why?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? PI VWAPt =
i=1
PI
vt,i pt,i
i=1
I I
vt,i
=
I X
wt,i pt,i
i=1
Price substantially unpredictable; take as given. Two approaches for VWAP 1. Predict volumes (Brownlees et al. (2011)) 2. Predict volume shares directly (this paper)
I
Why?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? PI VWAPt =
i=1
PI
vt,i pt,i
i=1
I I
vt,i
=
I X
wt,i pt,i
i=1
Price substantially unpredictable; take as given. Two approaches for VWAP 1. Predict volumes (Brownlees et al. (2011)) 2. Predict volume shares directly (this paper)
I
Why?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? PI VWAPt =
i=1
PI
vt,i pt,i
i=1
I I
vt,i
=
I X
wt,i pt,i
i=1
Price substantially unpredictable; take as given. Two approaches for VWAP 1. Predict volumes (Brownlees et al. (2011)) 2. Predict volume shares directly (this paper)
I
Why?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volumes I
Data must be normalized by the number of outstanding stocks (to stabilize the series)
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volumes I
Data display lot of features: daily, intra–daily periodic, intra-daily non–periodic
Calvori, Cipollini, Gallo
(a) Whole data
(b) Daily averages
(c) Intra-daily (ACF)
(d) Intra-daily non-periodic (ACF)
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volumes I
Multiplicative Error Model (Engle and Gallo, 2006) extended to a Component MEM by Brownlees et al. (2011): vt,i = ηt φi µt,i εt,i I I
I I
ηt = daily component φi = intra-daily periodic component reproducing the time–of–day pattern µt,i = intra-daily non-periodic component εt,i = i.i.d. non-negative multiplicative error term such that εt,i ∼ (1, σ 2 )
where ηt , φi and µt,i have suitable analytical formulations I
However. . .
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volumes I
Multiplicative Error Model (Engle and Gallo, 2006) extended to a Component MEM by Brownlees et al. (2011): vt,i = ηt φi µt,i εt,i I I
I I
ηt = daily component φi = intra-daily periodic component reproducing the time–of–day pattern µt,i = intra-daily non-periodic component εt,i = i.i.d. non-negative multiplicative error term such that εt,i ∼ (1, σ 2 )
where ηt , φi and µt,i have suitable analytical formulations I
However. . .
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volumes I
... now data tell a partially different story
(e) SPY, 2002-2006 (daily) I
(f) Citigroup, 2002-2012 (daily)
The daily level changes dramatically on long periods (trends, changes in levels, etc.)!
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volumes I
... now data tell a partially different story
(g) SPY, 2002-2006 (daily) I
(h) Citigroup, 2002-2012 (daily)
The daily level changes dramatically on long periods (trends, changes in levels, etc.)!
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volume Shares I
No normalization by the number of outstanding stocks
I
No trending patterns (by construction)
I
No need to formulate a daily dynamics
I
What really matters for VWAP is wt,i , not vt,i
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volume Shares I
No normalization by the number of outstanding stocks
I
No trending patterns (by construction)
I
No need to formulate a daily dynamics
I
What really matters for VWAP is wt,i , not vt,i
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volume Shares I
No normalization by the number of outstanding stocks
I
No trending patterns (by construction)
I
No need to formulate a daily dynamics
I
What really matters for VWAP is wt,i , not vt,i
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Why a model for volume shares? Modeling Volume Shares I
No normalization by the number of outstanding stocks
I
No trending patterns (by construction)
I
No need to formulate a daily dynamics
I
What really matters for VWAP is wt,i , not vt,i
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
4 / 17
Challenges I
wt,i ∈ [0, 1] and
I X
wt,i = 1
i=1 I I
wt,i cannot be evaluated before the daily market closure Strong intra-daily periodic pattern
(i) One week of data I
(j) Bin averages
Does a predictable dynamics exist in addition to the periodic pattern?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
5 / 17
Challenges I
wt,i ∈ [0, 1] and
I X
wt,i = 1
i=1 I I
wt,i cannot be evaluated before the daily market closure Strong intra-daily periodic pattern
(k) One week of data I
(l) Bin averages
Does a predictable dynamics exist in addition to the periodic pattern?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
5 / 17
Challenges I
wt,i ∈ [0, 1] and
I X
wt,i = 1
i=1 I I
wt,i cannot be evaluated before the daily market closure Strong intra-daily periodic pattern
(m) One week of data I
(n) Bin averages
Does a predictable dynamics exist in addition to the periodic pattern?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
5 / 17
Challenges I
wt,i ∈ [0, 1] and
I X
wt,i = 1
i=1 I I
wt,i cannot be evaluated before the daily market closure Strong intra-daily periodic pattern
(o) One week of data I
(p) Bin averages
Does a predictable dynamics exist in addition to the periodic pattern?
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
5 / 17
Maintained Assumption w t = (wt,1 , . . . , wt,I )0 |Ft−1 ∼ Dir(αt ) I
αt = (αt,1 , . . . , αt,I )0 (αt,i > 0) is made to depend on I I
I
Ft−1 : information up to day (t − 1) θ: p vector of parameters
Specifically: αt,i = exp( πi + βt,i ) where I
I
πi : value of the periodic component (a Fourier sine/cosine function as in Brownlees et al. (2011)) βt,i : time-varying parameter capturing the (possible) additional predictable dynamics
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
6 / 17
Maintained Assumption w t = (wt,1 , . . . , wt,I )0 |Ft−1 ∼ Dir(αt ) I
αt = (αt,1 , . . . , αt,I )0 (αt,i > 0) is made to depend on I I
I
Ft−1 : information up to day (t − 1) θ: p vector of parameters
Specifically: αt,i = exp( πi + βt,i ) where I
I
πi : value of the periodic component (a Fourier sine/cosine function as in Brownlees et al. (2011)) βt,i : time-varying parameter capturing the (possible) additional predictable dynamics
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
6 / 17
Maintained Assumption w t = (wt,1 , . . . , wt,I )0 |Ft−1 ∼ Dir(αt ) I
αt = (αt,1 , . . . , αt,I )0 (αt,i > 0) is made to depend on I I
I
Ft−1 : information up to day (t − 1) θ: p vector of parameters
Specifically: αt,i = exp( πi + βt,i ) where I
I
πi : value of the periodic component (a Fourier sine/cosine function as in Brownlees et al. (2011)) βt,i : time-varying parameter capturing the (possible) additional predictable dynamics
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
6 / 17
Maintained Assumption w t = (wt,1 , . . . , wt,I )0 |Ft−1 ∼ Dir(αt ) I
αt = (αt,1 , . . . , αt,I )0 (αt,i > 0) is made to depend on I I
I
Ft−1 : information up to day (t − 1) θ: p vector of parameters
Specifically: αt,i = exp( πi + βt,i ) where I
I
πi : value of the periodic component (a Fourier sine/cosine function as in Brownlees et al. (2011)) βt,i : time-varying parameter capturing the (possible) additional predictable dynamics
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
6 / 17
Maintained Assumption w t = (wt,1 , . . . , wt,I )0 |Ft−1 ∼ Dir(αt ) I
αt = (αt,1 , . . . , αt,I )0 (αt,i > 0) is made to depend on I I
I
Ft−1 : information up to day (t − 1) θ: p vector of parameters
Specifically: αt,i = exp( πi + βt,i ) where I
I
πi : value of the periodic component (a Fourier sine/cosine function as in Brownlees et al. (2011)) βt,i : time-varying parameter capturing the (possible) additional predictable dynamics
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
6 / 17
GAS Dynamics for βt I
I
I
GAS = Generalized Autoregressive Score: general approach for non-Gaussian observation driven time series models (Creal et al. (2012)). Crucial ingredient: the driver of the dynamics (i.e the innovation) is the conditional score GAS(1,1) formulation of βt βt = Ast−1 + Bβt−1 where I
A and B are (I, I) matrices (depending on θ)
I
st = S t ∇t : is the conditional scaled score ∂ ln f (w t |Ft−1 ) is the conditional score ∇t = ∂βt
I
I
S t = S(Ft−1 ) is a scaling matrix
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
7 / 17
GAS Dynamics for βt I
I
I
GAS = Generalized Autoregressive Score: general approach for non-Gaussian observation driven time series models (Creal et al. (2012)). Crucial ingredient: the driver of the dynamics (i.e the innovation) is the conditional score GAS(1,1) formulation of βt βt = Ast−1 + Bβt−1 where I
A and B are (I, I) matrices (depending on θ)
I
st = S t ∇t : is the conditional scaled score ∂ ln f (w t |Ft−1 ) is the conditional score ∇t = ∂βt
I
I
S t = S(Ft−1 ) is a scaling matrix
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
7 / 17
GAS Dynamics for βt I
I
I
GAS = Generalized Autoregressive Score: general approach for non-Gaussian observation driven time series models (Creal et al. (2012)). Crucial ingredient: the driver of the dynamics (i.e the innovation) is the conditional score GAS(1,1) formulation of βt βt = Ast−1 + Bβt−1 where I
A and B are (I, I) matrices (depending on θ)
I
st = S t ∇t : is the conditional scaled score ∂ ln f (w t |Ft−1 ) is the conditional score ∇t = ∂βt
I
I
S t = S(Ft−1 ) is a scaling matrix
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
7 / 17
GAS Dynamics for βt I
I
I
GAS = Generalized Autoregressive Score: general approach for non-Gaussian observation driven time series models (Creal et al. (2012)). Crucial ingredient: the driver of the dynamics (i.e the innovation) is the conditional score GAS(1,1) formulation of βt βt = Ast−1 + Bβt−1 where I
A and B are (I, I) matrices (depending on θ)
I
st = S t ∇t : is the conditional scaled score ∂ ln f (w t |Ft−1 ) ∇t = is the conditional score ∂βt
I
I
S t = S(Ft−1 ) is a scaling matrix
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
7 / 17
GAS Dynamics for βt I
I
I
GAS = Generalized Autoregressive Score: general approach for non-Gaussian observation driven time series models (Creal et al. (2012)). Crucial ingredient: the driver of the dynamics (i.e the innovation) is the conditional score GAS(1,1) formulation of βt βt = Ast−1 + Bβt−1 where I
A and B are (I, I) matrices (depending on θ)
I
st = S t ∇t : is the conditional scaled score ∂ ln f (w t |Ft−1 ) ∇t = is the conditional score ∂βt
I
I
S t = S(Ft−1 ) is a scaling matrix
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
7 / 17
Additional Details I
I
Parameterization of βt equation: I
M0 : βt = 0 → no GAS component.
I
M1 : βt = ast−1 + bβt−1 → GAS component with a simple parameterization.
I
M2 : βt = diag(a1 ; a; . . . ; a; aI )st−1 + bβt−1 → GAS component with an intermediate parameterization (different parameters for the first and last bins).
I
M3 : βt = diag(a)st−1 + bβt−1 → GAS component with a richer parameterization (a different parameter for each bin).
Scaling matrix (S t ): I
Different alternatives tested
I
Best one (used here): the inverse of the conditional information matrix I t = E (∇t ∇0t |Ft−1 ).
I
The structure of the Dirichlet p.d.f. allows for important simplifications in computing the scaled score st = I −1 t ∇t
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
8 / 17
Additional Details I
I
Parameterization of βt equation: I
M0 : βt = 0 → no GAS component.
I
M1 : βt = ast−1 + bβt−1 → GAS component with a simple parameterization.
I
M2 : βt = diag(a1 ; a; . . . ; a; aI )st−1 + bβt−1 → GAS component with an intermediate parameterization (different parameters for the first and last bins).
I
M3 : βt = diag(a)st−1 + bβt−1 → GAS component with a richer parameterization (a different parameter for each bin).
Scaling matrix (S t ): I
Different alternatives tested
I
Best one (used here): the inverse of the conditional information matrix I t = E (∇t ∇0t |Ft−1 ).
I
The structure of the Dirichlet p.d.f. allows for important simplifications in computing the scaled score st = I −1 t ∇t
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
8 / 17
Data
I
Volume shares at 15 minutes (I = 26)
I
January 2006 – July 2012
I
ANF (Abercrombie & Fitch), BAC (Bank of America), C (Citigroup), F (Ford Motor), GE (General Electric), JNJ (Johnson & Johnson)
I
Tick-by-tick data (used for computing the shares) cleaned according to Brownlees & Gallo (2006)
I
Days lacking of one or more bins (because of trading halts or anticipated closures) removed from the analysis
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
9 / 17
Data
I
Volume shares at 15 minutes (I = 26)
I
January 2006 – July 2012
I
ANF (Abercrombie & Fitch), BAC (Bank of America), C (Citigroup), F (Ford Motor), GE (General Electric), JNJ (Johnson & Johnson)
I
Tick-by-tick data (used for computing the shares) cleaned according to Brownlees & Gallo (2006)
I
Days lacking of one or more bins (because of trading halts or anticipated closures) removed from the analysis
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
9 / 17
Data
I
Volume shares at 15 minutes (I = 26)
I
January 2006 – July 2012
I
ANF (Abercrombie & Fitch), BAC (Bank of America), C (Citigroup), F (Ford Motor), GE (General Electric), JNJ (Johnson & Johnson)
I
Tick-by-tick data (used for computing the shares) cleaned according to Brownlees & Gallo (2006)
I
Days lacking of one or more bins (because of trading halts or anticipated closures) removed from the analysis
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
9 / 17
Data
I
Volume shares at 15 minutes (I = 26)
I
January 2006 – July 2012
I
ANF (Abercrombie & Fitch), BAC (Bank of America), C (Citigroup), F (Ford Motor), GE (General Electric), JNJ (Johnson & Johnson)
I
Tick-by-tick data (used for computing the shares) cleaned according to Brownlees & Gallo (2006)
I
Days lacking of one or more bins (because of trading halts or anticipated closures) removed from the analysis
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
9 / 17
Data
I
Volume shares at 15 minutes (I = 26)
I
January 2006 – July 2012
I
ANF (Abercrombie & Fitch), BAC (Bank of America), C (Citigroup), F (Ford Motor), GE (General Electric), JNJ (Johnson & Johnson)
I
Tick-by-tick data (used for computing the shares) cleaned according to Brownlees & Gallo (2006)
I
Days lacking of one or more bins (because of trading halts or anticipated closures) removed from the analysis
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
9 / 17
Preliminary Analyses I
I
Based on one novel GAS-LM and two novel score tests for autocorrelation In all cases: I
I
Accepting H0 provides evidence that αt does not depend on t and each of its components is a deterministic function of the time of day (i-th bin) Rejecting H0 points to the presence of an additional predictable dynamics
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
10 / 17
Preliminary Analyses I
I
Based on one novel GAS-LM and two novel score tests for autocorrelation In all cases: I
I
Accepting H0 provides evidence that αt does not depend on t and each of its components is a deterministic function of the time of day (i-th bin) Rejecting H0 points to the presence of an additional predictable dynamics
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
10 / 17
Preliminary Analyses I
GAS-LM Test I
I
Consider the GAS equation βt = Ast−1 + Bβt−1 with H0 : A = B = 0 and H1 : A 6= 0, B = 0. Then d b −1 ∇ b (θ0 ) → b (θ0 )0 D χ2 (df ) LMGAS = ∇ T X
b (θ0 ) ∇ b (θ0 )0 ∇ t t
I
b= D
I
b (θ0 ) = score under H1 evaluated at the ML estimate under ∇ H0 df = # of parameters set to zero under H0 , but not under H1
t=1
I
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
10 / 17
Preliminary Analyses I
GAS-LM Test I
I
Consider the GAS equation βt = Ast−1 + Bβt−1 with H0 : A = B = 0 and H1 : A 6= 0, B = 0. Then d b −1 ∇ b (θ0 ) → b (θ0 )0 D χ2 (df ) LMGAS = ∇ T X
b (θ0 ) ∇ b (θ0 )0 ∇ t t
I
b= D
I
b (θ0 ) = score under H1 evaluated at the ML estimate under ∇ H0 df = # of parameters set to zero under H0 , but not under H1
t=1
I
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
10 / 17
Preliminary Analyses I
GAS-LM Test I
I
Consider the GAS equation βt = Ast−1 + Bβt−1 with H0 : A = B = 0 and H1 : A 6= 0, B = 0. Then d b −1 ∇ b (θ0 ) → b (θ0 )0 D χ2 (df ) LMGAS = ∇ T X
b (θ0 ) ∇ b (θ0 )0 ∇ t t
I
b= D
I
b (θ0 ) = score under H1 evaluated at the ML estimate under ∇ H0 df = # of parameters set to zero under H0 , but not under H1
t=1
I
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
10 / 17
Preliminary Analyses I
Score Based Tests I
I
Test if some moment conditions implied on ∇t by the model under correct specification, are satisfied by the data. Moment conditions (E(mt |Ft−1 ) = 0) proposed E (∇0t ∇t−1 |Ft−1 ) = 0
I
Then
I
I I I Calvori, Cipollini, Gallo
E (∇t ∇t−1 |Ft−1 ) = 0
b0 V b −1 b d 2 ST (df ) = T m T T mT → χ (df )
b T = T −1 m
T X
b t estimate of E(mt |Ft−1 ) = 0 m t=1 b T = T −1 m b −1 ∇ b 0m b ∇ b 0 ∇) b 0m b −m b 0 ∇( b V b = (∇ b 1; . . . ; ∇ b T) b = (m b 1; . . . ; m b T ), ∇ m df = 1 (df = p) for the first (second) moment condition Volume Shares Predictions
Venezia, September 12, 2013
10 / 17
Preliminary Analyses I
Score Based Tests I
I
Test if some moment conditions implied on ∇t by the model under correct specification, are satisfied by the data. Moment conditions (E(mt |Ft−1 ) = 0) proposed E (∇0t ∇t−1 |Ft−1 ) = 0
I
Then
I
I I I Calvori, Cipollini, Gallo
E (∇t ∇t−1 |Ft−1 ) = 0
b0 V b −1 b d 2 ST (df ) = T m T T mT → χ (df )
b T = T −1 m
T X
b t estimate of E(mt |Ft−1 ) = 0 m t=1 b T = T −1 m b −1 ∇ b 0m b ∇ b 0 ∇) b 0m b −m b 0 ∇( b V b = (∇ b 1; . . . ; ∇ b T) b = (m b 1; . . . ; m b T ), ∇ m df = 1 (df = p) for the first (second) moment condition Volume Shares Predictions
Venezia, September 12, 2013
10 / 17
Preliminary Analyses I
Score Based Tests I
I
Test if some moment conditions implied on ∇t by the model under correct specification, are satisfied by the data. Moment conditions (E(mt |Ft−1 ) = 0) proposed E (∇0t ∇t−1 |Ft−1 ) = 0
I
Then
I
I I I Calvori, Cipollini, Gallo
E (∇t ∇t−1 |Ft−1 ) = 0
b0 V b −1 b d 2 ST (df ) = T m T T mT → χ (df )
b T = T −1 m
T X
b t estimate of E(mt |Ft−1 ) = 0 m t=1 b T = T −1 m b −1 ∇ b 0m b ∇ b 0 ∇) b 0m b −m b 0 ∇( b V b = (∇ b 1; . . . ; ∇ b T) b = (m b 1; . . . ; m b T ), ∇ m df = 1 (df = p) for the first (second) moment condition Volume Shares Predictions
Venezia, September 12, 2013
10 / 17
Preliminary Analyses I
Score Based Tests I
I
Test if some moment conditions implied on ∇t by the model under correct specification, are satisfied by the data. Moment conditions (E(mt |Ft−1 ) = 0) proposed E (∇0t ∇t−1 |Ft−1 ) = 0
I
Then
I
I I I Calvori, Cipollini, Gallo
E (∇t ∇t−1 |Ft−1 ) = 0
b0 V b −1 b d 2 ST (df ) = T m T T mT → χ (df )
b T = T −1 m
T X
b t estimate of E(mt |Ft−1 ) = 0 m t=1 b T = T −1 m b −1 ∇ b 0m b ∇ b 0 ∇) b 0m b −m b 0 ∇( b V b = (∇ b 1; . . . ; ∇ b T) b = (m b 1; . . . ; m b T ), ∇ m df = 1 (df = p) for the first (second) moment condition Volume Shares Predictions
Venezia, September 12, 2013
10 / 17
Preliminary Analyses I
Empirical evidence Presence of a strongly significant predictable dynamics in addition to the periodic component (p-values in table) Ticker ANF BAC C F GE JNJ
Calvori, Cipollini, Gallo
LMGAS 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
ST (1) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Volume Shares Predictions
ST (p) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Venezia, September 12, 2013
10 / 17
Parameter Estimates GAS parameters (M1 parameterization) Coeff. a b I I I I I
ANF 0.0230 (0.0009) 0.9926 (0.0007)
BAC 0.0247 (0.0008) 0.9945 (0.0005)
C 0.0314 (0.0008) 0.9945 (0.0004)
F 0.0253 (0.0008) 0.9959 (0.0003)
GE 0.0300 (0.0009) 0.9901 (0.0006)
JNJ 0.0215 (0.0006) 0.9954 (0.0004)
Similar estimates across series b (persistence parameter) above 0.99 in all cases a (parameter of the lagged score) from 0.02 to 0.03 Largely significant (TI > 34000) M2 and M3 parameterizations (results not shown): ai parameters within (0.02, 0.04), with an end–of–the–day effect b again above 0.99
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
11 / 17
Parameter Estimates GAS parameters (M1 parameterization) Coeff. a b I I I I I
ANF 0.0230 (0.0009) 0.9926 (0.0007)
BAC 0.0247 (0.0008) 0.9945 (0.0005)
C 0.0314 (0.0008) 0.9945 (0.0004)
F 0.0253 (0.0008) 0.9959 (0.0003)
GE 0.0300 (0.0009) 0.9901 (0.0006)
JNJ 0.0215 (0.0006) 0.9954 (0.0004)
Similar estimates across series b (persistence parameter) above 0.99 in all cases a (parameter of the lagged score) from 0.02 to 0.03 Largely significant (TI > 34000) M2 and M3 parameterizations (results not shown): ai parameters within (0.02, 0.04), with an end–of–the–day effect b again above 0.99
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
11 / 17
Parameter Estimates GAS parameters (M1 parameterization) Coeff. a b I I I I I
ANF 0.0230 (0.0009) 0.9926 (0.0007)
BAC 0.0247 (0.0008) 0.9945 (0.0005)
C 0.0314 (0.0008) 0.9945 (0.0004)
F 0.0253 (0.0008) 0.9959 (0.0003)
GE 0.0300 (0.0009) 0.9901 (0.0006)
JNJ 0.0215 (0.0006) 0.9954 (0.0004)
Similar estimates across series b (persistence parameter) above 0.99 in all cases a (parameter of the lagged score) from 0.02 to 0.03 Largely significant (TI > 34000) M2 and M3 parameterizations (results not shown): ai parameters within (0.02, 0.04), with an end–of–the–day effect b again above 0.99
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
11 / 17
Parameter Estimates GAS parameters (M1 parameterization) Coeff. a b I I I I I
ANF 0.0230 (0.0009) 0.9926 (0.0007)
BAC 0.0247 (0.0008) 0.9945 (0.0005)
C 0.0314 (0.0008) 0.9945 (0.0004)
F 0.0253 (0.0008) 0.9959 (0.0003)
GE 0.0300 (0.0009) 0.9901 (0.0006)
JNJ 0.0215 (0.0006) 0.9954 (0.0004)
Similar estimates across series b (persistence parameter) above 0.99 in all cases a (parameter of the lagged score) from 0.02 to 0.03 Largely significant (TI > 34000) M2 and M3 parameterizations (results not shown): ai parameters within (0.02, 0.04), with an end–of–the–day effect b again above 0.99
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
11 / 17
Parameter Estimates GAS parameters (M1 parameterization) Coeff. a b I I I I I
ANF 0.0230 (0.0009) 0.9926 (0.0007)
BAC 0.0247 (0.0008) 0.9945 (0.0005)
C 0.0314 (0.0008) 0.9945 (0.0004)
F 0.0253 (0.0008) 0.9959 (0.0003)
GE 0.0300 (0.0009) 0.9901 (0.0006)
JNJ 0.0215 (0.0006) 0.9954 (0.0004)
Similar estimates across series b (persistence parameter) above 0.99 in all cases a (parameter of the lagged score) from 0.02 to 0.03 Largely significant (TI > 34000) M2 and M3 parameterizations (results not shown): ai parameters within (0.02, 0.04), with an end–of–the–day effect b again above 0.99
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
11 / 17
Parameter Estimates GAS parameters (M1 parameterization) Coeff. a b I I I I I
ANF 0.0230 (0.0009) 0.9926 (0.0007)
BAC 0.0247 (0.0008) 0.9945 (0.0005)
C 0.0314 (0.0008) 0.9945 (0.0004)
F 0.0253 (0.0008) 0.9959 (0.0003)
GE 0.0300 (0.0009) 0.9901 (0.0006)
JNJ 0.0215 (0.0006) 0.9954 (0.0004)
Similar estimates across series b (persistence parameter) above 0.99 in all cases a (parameter of the lagged score) from 0.02 to 0.03 Largely significant (TI > 34000) M2 and M3 parameterizations (results not shown): ai parameters within (0.02, 0.04), with an end–of–the–day effect b again above 0.99
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
11 / 17
In–Sample Diagnostics Likelihood Ratio Tests (p-values) Ticker ANF BAC C F GE JNJ
M1 vs. M0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
M2 vs. M1 0.0147 0.0000 0.0002 0.0000 0.0000 0.0000
M3 vs. M2 0.0000 0.0000 0.0742 0.0224 0.0000 0.0000
I
Larger models seem better (M1 beats M0 , etc.)
I
However. . .
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
12 / 17
In–Sample Diagnostics Likelihood Ratio Tests (p-values) Ticker ANF BAC C F GE JNJ
M1 vs. M0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
M2 vs. M1 0.0147 0.0000 0.0002 0.0000 0.0000 0.0000
M3 vs. M2 0.0000 0.0000 0.0742 0.0224 0.0000 0.0000
I
Larger models seem better (M1 beats M0 , etc.)
I
However. . .
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
12 / 17
In–Sample Diagnostics Likelihood Ratio Tests (p-values) Ticker ANF BAC C F GE JNJ
M1 vs. M0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
M2 vs. M1 0.0147 0.0000 0.0002 0.0000 0.0000 0.0000
M3 vs. M2 0.0000 0.0000 0.0742 0.0224 0.0000 0.0000
I
Larger models seem better (M1 beats M0 , etc.)
I
However. . .
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
12 / 17
In–Sample Diagnostics Score based autocorrelation tests (p-values)
Ticker ANF BAC C F GE JNJ
M1 ST (1) ST (p) 0.7033 0.3555 0.0413 0.0044 0.0018 0.0705 0.1606 0.1333 0.0607 0.0158 0.0391 0.0002
M2 ST (1) ST (p) 0.5585 0.5828 0.0147 0.0061 0.0027 0.1221 0.0775 0.0171 0.2713 0.2755 0.2175 0.0074
M3 ST (1) ST (p) 0.5369 0.0558 0.0654 0.0040 0.0064 0.1994 0.2528 0.0291 0.4871 0.3627 0.3266 0.0044
I
All GAS formulations capture well the dynamics of the data
I
No strong support for M2 and M3 against M1
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
12 / 17
In–Sample Diagnostics Score based autocorrelation tests (p-values)
Ticker ANF BAC C F GE JNJ
M1 ST (1) ST (p) 0.7033 0.3555 0.0413 0.0044 0.0018 0.0705 0.1606 0.1333 0.0607 0.0158 0.0391 0.0002
M2 ST (1) ST (p) 0.5585 0.5828 0.0147 0.0061 0.0027 0.1221 0.0775 0.0171 0.2713 0.2755 0.2175 0.0074
M3 ST (1) ST (p) 0.5369 0.0558 0.0654 0.0040 0.0064 0.1994 0.2528 0.0291 0.4871 0.3627 0.3266 0.0044
I
All GAS formulations capture well the dynamics of the data
I
No strong support for M2 and M3 against M1
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
12 / 17
In–Sample Diagnostics Score based autocorrelation tests (p-values)
Ticker ANF BAC C F GE JNJ
M1 ST (1) ST (p) 0.7033 0.3555 0.0413 0.0044 0.0018 0.0705 0.1606 0.1333 0.0607 0.0158 0.0391 0.0002
M2 ST (1) ST (p) 0.5585 0.5828 0.0147 0.0061 0.0027 0.1221 0.0775 0.0171 0.2713 0.2755 0.2175 0.0074
M3 ST (1) ST (p) 0.5369 0.0558 0.0654 0.0040 0.0064 0.1994 0.2528 0.0291 0.4871 0.3627 0.3266 0.0044
I
All GAS formulations capture well the dynamics of the data
I
No strong support for M2 and M3 against M1
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
12 / 17
Out-of-Sample Forecasts Diebold & Mariano tests (p-values) I
I
Out-of-sample, 1-day ahead forecasts (86 days between b t,i = α bt Aug. 1 to Dec. 5, 2012) w bt,i /10 α (·)
Three loss functions L
=
T +τ X
(·)
Lt :
t=T +1
" 0 bt ) − LLL t = − ln Γ(1 α
I X
ln Γ(b αt,i ) +
i=1
LSLICING =− t LMSE = t
I X
I X
# α bt,i − 1 ln wt,i
i=1 I X
b t,i wt,i ln w
i=1
b t,i wt,i − w
2
i=1 Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
13 / 17
Out-of-Sample Forecasts Diebold & Mariano tests (p-values) I
I
Out-of-sample, 1-day ahead forecasts (86 days between b t,i = α bt Aug. 1 to Dec. 5, 2012) w bt,i /10 α (·)
Three loss functions L
=
T +τ X
(·)
Lt :
t=T +1
" 0 bt ) − LLL t = − ln Γ(1 α
I X
ln Γ(b αt,i ) +
i=1
LSLICING =− t LMSE = t
I X
I X
# α bt,i − 1 ln wt,i
i=1 I X
b t,i wt,i ln w
i=1
b t,i wt,i − w
2
i=1 Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
13 / 17
Out-of-Sample Forecasts Diebold & Mariano tests (p-values) I
I
Out-of-sample, 1-day ahead forecasts (86 days between b t,i = α bt Aug. 1 to Dec. 5, 2012) w bt,i /10 α (·)
Three loss functions L
=
T +τ X
(·)
Lt :
t=T +1
" 0 bt ) − LLL t = − ln Γ(1 α
I X
ln Γ(b αt,i ) +
i=1
LSLICING =− t LMSE = t
I X
I X
# α bt,i − 1 ln wt,i
i=1 I X
b t,i wt,i ln w
i=1
b t,i wt,i − w
2
i=1 Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
13 / 17
Out-of-Sample Forecasts Diebold & Mariano tests (p-values) I
I
Out-of-sample, 1-day ahead forecasts (86 days between b t,i = α bt Aug. 1 to Dec. 5, 2012) w bt,i /10 α (·)
Three loss functions L
=
T +τ X
(·)
Lt :
t=T +1
" 0 bt ) − LLL t = − ln Γ(1 α
I X
ln Γ(b αt,i ) +
i=1
LSLICING =− t LMSE = t
I X
I X
# α bt,i − 1 ln wt,i
i=1 I X
b t,i wt,i ln w
i=1
b t,i wt,i − w
2
i=1 Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
13 / 17
Out-of-Sample Forecasts Diebold & Mariano tests (p-values) I
I
Out-of-sample, 1-day ahead forecasts (86 days between b t,i = α bt Aug. 1 to Dec. 5, 2012) w bt,i /10 α (·)
Three loss functions L
=
T +τ X
(·)
Lt :
t=T +1
" 0 bt ) − LLL t = − ln Γ(1 α
I X
ln Γ(b αt,i ) +
i=1
LSLICING =− t LMSE = t
I X
I X
# α bt,i − 1 ln wt,i
i=1 I X
b t,i wt,i ln w
i=1
b t,i wt,i − w
2
i=1 Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
13 / 17
Out-of-Sample Forecasts Diebold & Mariano tests (p-values)
Ticker ANF BAC C F GE JNJ
I
H1 : L(M0 ) > L(M1 ) LLL LSLICING LMSE 0.0163 0.0025 0.0016 0.0000 0.0000 0.0000 0.0116 0.0023 0.0086 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
H1 : L(M1 ) > L(M2 ) LLL LSLICING LMSE 0.9115 0.9274 0.9291 0.3444 0.4009 0.4885 0.1996 0.3182 0.3863 0.5578 0.3369 0.5108 0.0851 0.1423 0.1998 0.1177 0.0054 0.0178
H1 : L(M2 ) > L(M3 ) LLL LSLICING LMSE 0.9914 0.9730 0.9953 0.3896 0.3748 0.3050 0.0805 0.0741 0.0392 0.0787 0.0196 0.0424 0.0092 0.0089 0.0008 0.9678 0.8546 0.5012
Similar results across loss functions
I
M1 improves significantly over M0
I
M1 not worse than the richer M2 and similarly for M2 against M3
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
13 / 17
Out-of-Sample Forecasts Diebold & Mariano tests (p-values)
Ticker ANF BAC C F GE JNJ
I
H1 : L(M0 ) > L(M1 ) LLL LSLICING LMSE 0.0163 0.0025 0.0016 0.0000 0.0000 0.0000 0.0116 0.0023 0.0086 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
H1 : L(M1 ) > L(M2 ) LLL LSLICING LMSE 0.9115 0.9274 0.9291 0.3444 0.4009 0.4885 0.1996 0.3182 0.3863 0.5578 0.3369 0.5108 0.0851 0.1423 0.1998 0.1177 0.0054 0.0178
H1 : L(M2 ) > L(M3 ) LLL LSLICING LMSE 0.9914 0.9730 0.9953 0.3896 0.3748 0.3050 0.0805 0.0741 0.0392 0.0787 0.0196 0.0424 0.0092 0.0089 0.0008 0.9678 0.8546 0.5012
Similar results across loss functions
I
M1 improves significantly over M0
I
M1 not worse than the richer M2 and similarly for M2 against M3
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
13 / 17
Out-of-Sample Forecasts Diebold & Mariano tests (p-values)
Ticker ANF BAC C F GE JNJ
I
H1 : L(M0 ) > L(M1 ) LLL LSLICING LMSE 0.0163 0.0025 0.0016 0.0000 0.0000 0.0000 0.0116 0.0023 0.0086 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
H1 : L(M1 ) > L(M2 ) LLL LSLICING LMSE 0.9115 0.9274 0.9291 0.3444 0.4009 0.4885 0.1996 0.3182 0.3863 0.5578 0.3369 0.5108 0.0851 0.1423 0.1998 0.1177 0.0054 0.0178
H1 : L(M2 ) > L(M3 ) LLL LSLICING LMSE 0.9914 0.9730 0.9953 0.3896 0.3748 0.3050 0.0805 0.0741 0.0392 0.0787 0.0196 0.0424 0.0092 0.0089 0.0008 0.9678 0.8546 0.5012
Similar results across loss functions
I
M1 improves significantly over M0
I
M1 not worse than the richer M2 and similarly for M2 against M3
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
13 / 17
Out-of-Sample Forecasts Diebold & Mariano tests (p-values)
Ticker ANF BAC C F GE JNJ
I
H1 : L(M0 ) > L(M1 ) LLL LSLICING LMSE 0.0163 0.0025 0.0016 0.0000 0.0000 0.0000 0.0116 0.0023 0.0086 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
H1 : L(M1 ) > L(M2 ) LLL LSLICING LMSE 0.9115 0.9274 0.9291 0.3444 0.4009 0.4885 0.1996 0.3182 0.3863 0.5578 0.3369 0.5108 0.0851 0.1423 0.1998 0.1177 0.0054 0.0178
H1 : L(M2 ) > L(M3 ) LLL LSLICING LMSE 0.9914 0.9730 0.9953 0.3896 0.3748 0.3050 0.0805 0.0741 0.0392 0.0787 0.0196 0.0424 0.0092 0.0089 0.0008 0.9678 0.8546 0.5012
Similar results across loss functions
I
M1 improves significantly over M0
I
M1 not worse than the richer M2 and similarly for M2 against M3
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
13 / 17
Better VWAP Forecasts? I
VWAP: VWAPt =
I X
wt,i pt,i
i=1 I
VWAP forecast: \t = VWAP
I X
(L) b t,i pt,i w
i=1 (L)
pt,i = last price recorded in the bin (fair benchmark for prices?) I
Trading strategy U (the same fraction across bins) added for comparisons to strategies based on M0 , M1 , M2 .
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Better VWAP Forecasts? I
VWAP: VWAPt =
I X
wt,i pt,i
i=1 I
VWAP forecast: \t = VWAP
I X
(L) b t,i pt,i w
i=1 (L)
pt,i = last price recorded in the bin (fair benchmark for prices?) I
Trading strategy U (the same fraction across bins) added for comparisons to strategies based on M0 , M1 , M2 .
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Better VWAP Forecasts? I
VWAP: VWAPt =
I X
wt,i pt,i
i=1 I
VWAP forecast: \t = VWAP
I X
(L) b t,i pt,i w
i=1 (L)
pt,i = last price recorded in the bin (fair benchmark for prices?) I
Trading strategy U (the same fraction across bins) added for comparisons to strategies based on M0 , M1 , M2 .
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Better VWAP Forecasts? Ticker ANF BAC C F GE JNJ
U 0.1908 0.1227 0.1188 0.0895 0.0714 0.0439
M0 0.1646 0.0880 0.0847 0.0687 0.0409 0.0217
Calvori, Cipollini, Gallo
MAPE M1 0.1609 0.0877 0.0858 0.0716 0.0399 0.0210
M2 0.1629 0.0877 0.0854 0.0716 0.0395 0.0208
M3 0.1640 0.0886 0.0852 0.0716 0.0397 0.0203
U 0.0687 0.0110 0.0393 0.0092 0.0153 0.0304
Volume Shares Predictions
M0 0.0594 0.0079 0.0280 0.0071 0.0088 0.0150
MAE M1 0.0583 0.0079 0.0284 0.0074 0.0086 0.0145
M2 0.0590 0.0079 0.0282 0.0074 0.0085 0.0144
M3 0.0594 0.0080 0.0282 0.0074 0.0085 0.0140
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Better VWAP Forecasts? Ticker ANF BAC C F GE JNJ
U 0.1908 0.1227 0.1188 0.0895 0.0714 0.0439
M0 0.1646 0.0880 0.0847 0.0687 0.0409 0.0217
MAPE M1 0.1609 0.0877 0.0858 0.0716 0.0399 0.0210
M2 0.1629 0.0877 0.0854 0.0716 0.0395 0.0208
M3 0.1640 0.0886 0.0852 0.0716 0.0397 0.0203
U 0.0687 0.0110 0.0393 0.0092 0.0153 0.0304
M0 0.0594 0.0079 0.0280 0.0071 0.0088 0.0150
MAE M1 0.0583 0.0079 0.0284 0.0074 0.0086 0.0145
M2 0.0590 0.0079 0.0282 0.0074 0.0085 0.0144
M3 0.0594 0.0080 0.0282 0.0074 0.0085 0.0140
I
All Mj ’s overperform U by an economically relevant amount (the commission fee for guaranteed VWAP by Interactive Brokers Group Inc is 0.017$ per stock).
I
M1 , M2 and M3 do not improve over the model including only the periodic component (M0 ).
I
Interpretation: the additional noise induced, in forecasts, (L) by the true prices pt,i (in place of the bin VWAP pt,i ), hides gains from more accurate shares’ predictions (cf. Brownlees et al. (2011)).
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Better VWAP Forecasts? Ticker ANF BAC C F GE JNJ
U 0.1908 0.1227 0.1188 0.0895 0.0714 0.0439
M0 0.1646 0.0880 0.0847 0.0687 0.0409 0.0217
MAPE M1 0.1609 0.0877 0.0858 0.0716 0.0399 0.0210
M2 0.1629 0.0877 0.0854 0.0716 0.0395 0.0208
M3 0.1640 0.0886 0.0852 0.0716 0.0397 0.0203
U 0.0687 0.0110 0.0393 0.0092 0.0153 0.0304
M0 0.0594 0.0079 0.0280 0.0071 0.0088 0.0150
MAE M1 0.0583 0.0079 0.0284 0.0074 0.0086 0.0145
M2 0.0590 0.0079 0.0282 0.0074 0.0085 0.0144
M3 0.0594 0.0080 0.0282 0.0074 0.0085 0.0140
I
All Mj ’s overperform U by an economically relevant amount (the commission fee for guaranteed VWAP by Interactive Brokers Group Inc is 0.017$ per stock).
I
M1 , M2 and M3 do not improve over the model including only the periodic component (M0 ).
I
Interpretation: the additional noise induced, in forecasts, (L) by the true prices pt,i (in place of the bin VWAP pt,i ), hides gains from more accurate shares’ predictions (cf. Brownlees et al. (2011)).
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Better VWAP Forecasts? Ticker ANF BAC C F GE JNJ
U 0.1908 0.1227 0.1188 0.0895 0.0714 0.0439
M0 0.1646 0.0880 0.0847 0.0687 0.0409 0.0217
MAPE M1 0.1609 0.0877 0.0858 0.0716 0.0399 0.0210
M2 0.1629 0.0877 0.0854 0.0716 0.0395 0.0208
M3 0.1640 0.0886 0.0852 0.0716 0.0397 0.0203
U 0.0687 0.0110 0.0393 0.0092 0.0153 0.0304
M0 0.0594 0.0079 0.0280 0.0071 0.0088 0.0150
MAE M1 0.0583 0.0079 0.0284 0.0074 0.0086 0.0145
M2 0.0590 0.0079 0.0282 0.0074 0.0085 0.0144
M3 0.0594 0.0080 0.0282 0.0074 0.0085 0.0140
I
All Mj ’s overperform U by an economically relevant amount (the commission fee for guaranteed VWAP by Interactive Brokers Group Inc is 0.017$ per stock).
I
M1 , M2 and M3 do not improve over the model including only the periodic component (M0 ).
I
Interpretation: the additional noise induced, in forecasts, (L) by the true prices pt,i (in place of the bin VWAP pt,i ), hides gains from more accurate shares’ predictions (cf. Brownlees et al. (2011)).
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Better VWAP Forecasts? Ticker ANF BAC C F GE JNJ
Calvori, Cipollini, Gallo
Probability of a loss (Long) U M0 M1 M2 M3 37.2 36.1 34.9 33.7 33.7 10.5 5.8 4.7 4.7 4.7 44.2 30.2 30.2 30.2 30.2 7.0 3.5 5.8 5.8 5.8 15.1 4.7 3.5 3.5 3.5 39.5 12.8 10.5 9.3 9.3
Probability of a loss (Short) U M0 M1 M2 M3 37.2 33.7 34.9 33.7 33.7 10.5 5.8 4.7 3.5 3.5 29.1 27.9 31.4 31.4 30.2 5.8 7.0 7.0 7.0 7.0 12.8 4.7 4.7 4.7 4.7 29.1 17.4 15.1 14.0 14.0
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Better VWAP Forecasts? Ticker ANF BAC C F GE JNJ
Probability of a loss (Long) U M0 M1 M2 M3 37.2 36.1 34.9 33.7 33.7 10.5 5.8 4.7 4.7 4.7 44.2 30.2 30.2 30.2 30.2 7.0 3.5 5.8 5.8 5.8 15.1 4.7 3.5 3.5 3.5 39.5 12.8 10.5 9.3 9.3
Probability of a loss (Short) U M0 M1 M2 M3 37.2 33.7 34.9 33.7 33.7 10.5 5.8 4.7 3.5 3.5 29.1 27.9 31.4 31.4 30.2 5.8 7.0 7.0 7.0 7.0 12.8 4.7 4.7 4.7 4.7 29.1 17.4 15.1 14.0 14.0
I
Percentage probabilities of a loss (commission fees included)
I
All probabilities < 0.5
I
Model based approaches (M0 , . . . , M3 ) better than the naive U allocation strategy
I
Sophisticated model based approaches (M1 , . . . , M3 ) improve only marginally over the simple M0 -based strategy
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Better VWAP Forecasts? Ticker ANF BAC C F GE JNJ
Probability of a loss (Long) U M0 M1 M2 M3 37.2 36.1 34.9 33.7 33.7 10.5 5.8 4.7 4.7 4.7 44.2 30.2 30.2 30.2 30.2 7.0 3.5 5.8 5.8 5.8 15.1 4.7 3.5 3.5 3.5 39.5 12.8 10.5 9.3 9.3
Probability of a loss (Short) U M0 M1 M2 M3 37.2 33.7 34.9 33.7 33.7 10.5 5.8 4.7 3.5 3.5 29.1 27.9 31.4 31.4 30.2 5.8 7.0 7.0 7.0 7.0 12.8 4.7 4.7 4.7 4.7 29.1 17.4 15.1 14.0 14.0
I
Percentage probabilities of a loss (commission fees included)
I
All probabilities < 0.5
I
Model based approaches (M0 , . . . , M3 ) better than the naive U allocation strategy
I
Sophisticated model based approaches (M1 , . . . , M3 ) improve only marginally over the simple M0 -based strategy
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Better VWAP Forecasts? Ticker ANF BAC C F GE JNJ
Probability of a loss (Long) U M0 M1 M2 M3 37.2 36.1 34.9 33.7 33.7 10.5 5.8 4.7 4.7 4.7 44.2 30.2 30.2 30.2 30.2 7.0 3.5 5.8 5.8 5.8 15.1 4.7 3.5 3.5 3.5 39.5 12.8 10.5 9.3 9.3
Probability of a loss (Short) U M0 M1 M2 M3 37.2 33.7 34.9 33.7 33.7 10.5 5.8 4.7 3.5 3.5 29.1 27.9 31.4 31.4 30.2 5.8 7.0 7.0 7.0 7.0 12.8 4.7 4.7 4.7 4.7 29.1 17.4 15.1 14.0 14.0
I
Percentage probabilities of a loss (commission fees included)
I
All probabilities < 0.5
I
Model based approaches (M0 , . . . , M3 ) better than the naive U allocation strategy
I
Sophisticated model based approaches (M1 , . . . , M3 ) improve only marginally over the simple M0 -based strategy
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
14 / 17
Conclusions I
I
I
I
I
I
We provide motivations for forecasting intra–daily volume shares. Volume shares present a strong intra–daily periodic pattern but also a significant additional dynamics. We propose a novel model for the additional dynamics, based on the GAS approach. We propose novel score based tests to check model adequacy in reproducing the pattern of the data. The model captures well the in-sample dynamics of the data. The model including a GAS component with a simple parameterization beats in out-of-sample forecasting accuracy the model without GAS effects, without losing against a more richly parameterized GAS formulation.
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
15 / 17
Conclusions I
I
I
I
I
I
We provide motivations for forecasting intra–daily volume shares. Volume shares present a strong intra–daily periodic pattern but also a significant additional dynamics. We propose a novel model for the additional dynamics, based on the GAS approach. We propose novel score based tests to check model adequacy in reproducing the pattern of the data. The model captures well the in-sample dynamics of the data. The model including a GAS component with a simple parameterization beats in out-of-sample forecasting accuracy the model without GAS effects, without losing against a more richly parameterized GAS formulation.
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
15 / 17
Conclusions I
I
I
I
I
I
We provide motivations for forecasting intra–daily volume shares. Volume shares present a strong intra–daily periodic pattern but also a significant additional dynamics. We propose a novel model for the additional dynamics, based on the GAS approach. We propose novel score based tests to check model adequacy in reproducing the pattern of the data. The model captures well the in-sample dynamics of the data. The model including a GAS component with a simple parameterization beats in out-of-sample forecasting accuracy the model without GAS effects, without losing against a more richly parameterized GAS formulation.
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
15 / 17
Conclusions I
I
I
I
I
I
We provide motivations for forecasting intra–daily volume shares. Volume shares present a strong intra–daily periodic pattern but also a significant additional dynamics. We propose a novel model for the additional dynamics, based on the GAS approach. We propose novel score based tests to check model adequacy in reproducing the pattern of the data. The model captures well the in-sample dynamics of the data. The model including a GAS component with a simple parameterization beats in out-of-sample forecasting accuracy the model without GAS effects, without losing against a more richly parameterized GAS formulation.
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
15 / 17
Conclusions I
I
I
I
I
I
We provide motivations for forecasting intra–daily volume shares. Volume shares present a strong intra–daily periodic pattern but also a significant additional dynamics. We propose a novel model for the additional dynamics, based on the GAS approach. We propose novel score based tests to check model adequacy in reproducing the pattern of the data. The model captures well the in-sample dynamics of the data. The model including a GAS component with a simple parameterization beats in out-of-sample forecasting accuracy the model without GAS effects, without losing against a more richly parameterized GAS formulation.
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
15 / 17
Conclusions I
I
I
I
I
I
We provide motivations for forecasting intra–daily volume shares. Volume shares present a strong intra–daily periodic pattern but also a significant additional dynamics. We propose a novel model for the additional dynamics, based on the GAS approach. We propose novel score based tests to check model adequacy in reproducing the pattern of the data. The model captures well the in-sample dynamics of the data. The model including a GAS component with a simple parameterization beats in out-of-sample forecasting accuracy the model without GAS effects, without losing against a more richly parameterized GAS formulation.
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
15 / 17
References
I Berkowitz, S. A., Logue, D. E., and Noser, E. A. J. (1988). The total cost of transactions on the NYSE. The Journal of Finance, 43, 97–112. I Brownlees, C. T., Cipollini, F., and Gallo, G. M. (2011). Intra-daily volume modeling and prediction for algorithmic trading. Journal of Financial Econometrics, 9, 489–518. I Brownlees, C. T. and Gallo, G. M. (2006). Financial econometric analysis at ultra–high frequency: Data handling concerns. Computational Statistics and Data Analysis, 51, 2232–2245. I Creal, D., Koopman, S. J., and Lucas, A. (2012). Generalized autoregressive score models with applications. Journal of Applied Econometrics, (available online). I Diebold, F. X. and Mariano, R. S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics, 13, 253–263.
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
16 / 17
Thank you
Calvori, Cipollini, Gallo
Volume Shares Predictions
Venezia, September 12, 2013
17 / 17
