Signal Processing - May 2017 - 26
with the information-processing process as shown in Figure 6(a). They know that it often takes time (however short)
for market participants to process complex information,
whether it comes from human behaviors or trading/communication systems. Such transition processes are not well studied
quantitatively in finance literature, posing opportunities for SP
researchers. It is possible to use SP models and methodologies
to find impulse response to certain events, exploring the
microstructure of trading systems, especially in HFT systems.
Without ground-truth models and controlled experiments/
studies in modeling economic systems, the validity of SP models relies on data as well as economic explanations. We note
several traps in dealing with the model and data as follows.
■ Traps in data. First, survivorship bias: only data from successful companies or funds are used in analysis. For example,
stock data downloaded from popular websites like Yahoo
Finance do not include historical delisted companies. Second,
forward-looking bias: future (test) data are used to estimate
the model or inform the model construction. Third, selection
bias: data are selectively reported in a database, e.g., only
mutual funds/hedge funds performing well report results.
■ Solution: use unbiased data sets, e.g., Center for Research
in Security Prices (CRSP) data set, a gold standard in academic research on U.S. market daily stock returns. See [43]
for more details.
When evaluating and interpreting empirical results,
researchers need to be careful to select meaningful criteria and
statistical test methodology.
■ Test against alternative models, e.g., can you beat the FF
three-factor or the random-walk model?
■ Minimum mean-squared errors are not enough. Note that
asset-pricing model residuals are often considered idiosyncratic risks or the unknown risks that are not captured by
known factors. The important thing may be whether a model
properly attributes different types of risks rather than minimizes the residual. The explanatory power of a model should
be given more consideration depending on the applications.
■ Conduct extensive out-of-sample tests, and prove with statistical significance that your results are not due to chance.
Again, we caution that out-of-sample backtesting is helpful
as much as out-of-sample testing in SP and machine-learning
applications to check model fitness and robustness for known
data over multiple time periods. However, multiple models may
have similar fit into a set of data, and the past data may not perfectly represent the future in an open economic system. There
is generally no gold criterion to know the ground-truth model
without controlled experiments. Indeed, we do not even know
whether there exists a ground-truth model. The joint hypothesis problem always exists in discovering a model or a theory.
Therefore, economic justifications are always an integral part
of economic and business studies.
communities. Understanding basic concepts in econometrics
will help propel SP researchers in their work and allow them to
appreciate application issues in economics and business.
ARMA
In time-series econometrics [14], [44], the definitions of models have similar forms to those in SP. An autoregressive (AR)
process with the first-order, AR(1) process, is defined as
Yt = c + zYt -1 + f t,
AR( p) processes are defined as
p
Yt = c + / z i Yt -i + f t,
i =1
and ARMA( p, q) processes are defined as
q
i =1
i =1
where f t is white noise with zero mean and variance v 2 and
E [f t f x] = 0 when t ! x. Note that f t are indeed unpredictable innovations (shocks).
In econometrics, in place of the Z transform used in SP, a lag
operator L is used to represent time shift. Thus, the ARMA(p, q)
process can be represented by lag operator polynomials as
p
q
i =1
i =1
e 1 - / z i Li o Yt = c + e 1 + / i i Li o f t .
When observing an economic system, researchers do not
have control of the innovation or the unexpected shock, f t, which
can be various events, such as a sudden decrease of crude oil
prices. Meanwhile, there may be other exogenous variables that
are not uncorrelated white noise, such as the advertising investment of a company or interest rates set by central banks. Thus, to
analyze an economic system, the ARMA model needs to be generalized to the ARMA processes with the exogenous variables
(ARMAX) model. ARMAX(p, q, r) can be defined as
p
q
r
i =1
i =1
i =1
Yt = c + / z i Yt -i + f t + / i i f t -i + / h i X t -i ,
where X t represents exogenous variables. A special case of
the ARMAX model is AR distributed lag models (ARDL). An
ARDL( p, r) model is defined as
p
r
i =1
i =1
Yt = c + / z i Yt -i + / h i X t -i + f t .
A vector AR (VAR) model can be used to examine the interaction of a set of n economic variables. VAR(p) is specified as
p
y t = c + / U i y t - i + f t,
Basic econometric models,
time-series analysis, HFT, and SP
i =1
SP shares many similar terms and mathematics with econometrics, even though there are few interactions between the two
26
p
Yt = c + / z i Yt -i + f t + / i i f t -i,
where y t, c, and f t are n # 1 vectors and U i is an n # n
matrix.
IEEE Signal Processing Magazine
|
May 2017
|
Table of Contents for the Digital Edition of Signal Processing - May 2017
Signal Processing - May 2017 - Cover1
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Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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