MIDAS Regressions:
Further Results and New Directions
∗
Eric Ghysels
†
Arthur Sinko
‡
Rossen Valkanov
§
First Draft: February 2002
This Draft: February 7, 2006
Abstract
We explore Mixed Data Sampling (henceforth MIDAS) regression models. The regressions
involve time series data sampled at different frequencies. Volatility and related processes are
our prime focus, though the regression method has wider applications in macroeconomics and
finance, among other areas. The regressions combine recent developments regarding estimation
of volatility and a not so recent literature on distributed lag models. We study various lag
structures to parameterize parsimoniously the regressions and relate them to existing models.
We also propose several new extensions of the MIDAS framework. The paper concludes with an
empirical section where we provide further evidence and new results on the risk-return tradeoff.
We also report empirical evidence on microstructure noise and volatility forecasting.
∗
We thank two Referees and an Associate Editor, Alberto Plazzi, Pedro Santa-Clara as well as seminar
participants at City University of Hong Kong, Emory University, the Federal Reserve Board, ITAM, Korea
University, New York University, Oxford University, Tsinghua University, University of Iowa, UNC, USC,
participants at the Symposium on New Frontiers in Financial Volatility Modelling, Florence, the Academia
Sinica Conference on Analysis of High-Frequency Financial Data and Market Microstructure, Taipei, the
CIREQ-CIRANO-MITACS conference on Financial Econometrics, Montreal and the Research Triangle
Conference, for helpful comments. All remaining errors are our own.
†
Department of Finance, Kenan-Flagler School of Business and Department of Economics University of
North Carolina, Gardner Hall CB 3305,Gardner Hall CB 3305, Chapel Hill, NC 27599-3305, phone: (919)
966-5325, e-mail: eghysels@unc.edu.
‡
Department of Economics, University of North Carolina, Gardner Hall CB 3305, Chapel Hill, NC 27599-
3305, e-mail: sinko@email.unc.edu
§
Rady School of Management, UCSD, Pepper Canyon Hall, 3rd Floor 9500 Gilman Drive, MC 0093 La
Jolla, CA 92093-0093, phone: (858) 534-0898, e-mail: rvalkanov@ucsd.edu.
1 Introduction
The availability of data sampled at different frequency always presents a dilemma for a
researcher working with time series data. On the one hand, the variables that are available
at high frequency contain potentially valuable information. On the other hand, the researcher
cannot use this high frequency information directly if some of the variables are available at
a lower frequency, because most time series regressions involve data sampled at the same
interval. The common solution in such cases is to “pre-filter” the data so that the left-hand
and right-hand side variables are available at the same frequency. In the process, a lot of
potentially useful information might be discarded, thus rendering the relation between the
variables difficult to detect.
1
As an alternative, Ghysels, Santa-Clara, and Valkanov (2002),
(2004) and (2005) have recently proposed regressions that directly accommodate variables
sampled at different frequencies. Their MIxed Data Sampling – or MIDAS – regressions
represent a simple, parsimonious, and flexible class of time series models that allow the
left-hand and right-hand side variables of time series regressions to be sampled at different
frequencies.
Since MIDAS regressions have only recently been introduced, there are a lot of unexplored
questions. The goal of this paper is to explore some of the most pressing issues, to lay
out some new ideas about mixed-frequency regressions, and to present some new empirical
results. Before we start, it is useful to introduce a simple MIDAS regression. Suppose that
a variable y
t
is available once between t − 1 and t (say, monthly), another variable x
(m)
t
is
observed m times in the same period (say, daily or m = 22), and that we are interested in
the dynamic relation between y
t
and x
(m)
t
. In other words, we want to project the left-hand
side variable y
t
onto a history of lagged observations of x
(m)
t−j/m
. The superscript on x
(m)
t−j/m
denotes the higher sampling frequency and its exact timing lag is expressed as a fraction of
the unit interval between t − 1 and t. A simple MIDAS model is
y
t
= β
0
+ β
1
B(L
1/m
; θ)x
(m)
t
+ ε
(m)
t
(1)
for t = 1, . . . , T and where B(L
1/m
; θ) =
P
K
k=0
B(k; θ)L
k/m
and L
1/m
is a lag operator such
that L
1/m
x
(m)
t
= x
(m)
t−1/m
, and the lag coefficients in B(k; θ) of the corresponding lag operator
L
k/m
are parameterized as a function of a small-dimensional vector of parameters θ.
1
This situation is becoming more frequent now as dramatic improvements in information gathering have
produced new, high-frequency datasets, particularly in the area of financial econometrics.
1
In the mixed-frequency framework (1), the number of lags of x
(m)
t
is likely to be significant.
For instance, if monthly observations of y
t
is affected by six months’ worth of lagged daily
x
(m)
t
’s, we would need 132 lags (K = 132) of high-frequency lagged variables. If the
parameters of the lagged polynomial are left unrestricted (or B(k) does not depend on
θ), then there would be a lot of parameters to estimate. As a way of addressing parameter
proliferation, in a MIDAS regression the coefficients of the polynomial in L
1/m
are captured
by a known function B(L
1/m
; θ) of a few parameters summarized in a vector θ. We will
discuss several alternative specifications of B(L
1/m
; θ) in the paper. Finally, the parameter
β
1
captures the overall impact of lagged x
(m)
t
’s on y
t
. We identify β
1
by normalizing the
function B(L
1/m
; θ) to sum up to unity. While the normalization and the identification of β
1
are not strictly necessary in a MIDAS regression, they will be very useful for our applications
later in the paper.
In some specific cases, the results from the MIDAS regressions can be obtained using high-
frequency regressions alone. We work out one such example in the context of volatility
forecasting. While we are able to derive an explicit relation between the MIDAS parameters
and the purely high-frequency model, the relation is already quite complicated in this simple
case. For more interesting applications, such as these we conduct later in the paper, such
a relation is difficult to derive. This finding illustrates another advantage of our approach:
the MIDAS specification captures a very rich dynamic of the high-frequency process in a
very simple and parsimonious fashion. The MIDAS models benefit from several strands
of econometric models. The parameterization of the polynomial is similar in spirit to the
distributed lag models (see e.g. Griliches (1967), Dhrymes (1971) and Sims (1974) for surveys
on distributed lag models). Mixed data sampling regression models share some features
with distributed lag models but also have unique features. For instance, while we use a
parameterization of B(k; θ) that is common in distributed lag models, we also introduce a
new one called beta polynomial and that appears well suited in the applications that we
consider. We also discuss MIDAS regressions with stepfunctions introduced in Forsberg and
Ghysels (2004). Their appeal is the use of OLS estimation methods, but this comes at a
cost, namely that parsimony may not be preserved.
A convenient parametric function of B(L
1/m
; θ) also allows us to directly deal with lag
selection. In an unrestricted case, we have to design a lag selection procedure which can be
particularly difficult in this setup, where we will have to make the choice whether to include,
say, 66 or 67 daily lags in forecasting of a monthly observation y
t
. The parameterizations
2
of B(L
1/m
; θ) that we propose are quite flexible. For different value of θ, they can take
various shapes. In particular, the parameterized weights can decrease at different rates
as the number of lags increases. Therefore, by estimating θ, we effectively allow the data
to select the number of lags that are needed in the mixed-data relation between y
t
and x
t
.
Hence, once we choose the appropriate functional form of B(L
1/m
; θ), the lag length selection
in MIDAS is purely data-driven.
Variations of the MIDAS regression (1) have been used by Ghysels, Santa-Clara, and
Valkanov (2002), Ghysels, Santa-Clara, and Valkanov (2003). More complex specifications
are certainly possible and, in this paper, we propose several natural extensions of the basic
MIDAS regressions. First, on the right-hand side we can include variables sampled at various
frequencies. Second, non-linearities are easy to introduce as demonstrated by Ghysels, Santa-
Clara, and Valkanov (2005) who use one such model. In this paper, we discuss more general
non-linear MIDAS regressions. Third, MIDAS can accommodate tick-by-tick data that are
observed at unequally spaced intervals. Finally, multivariate MIDAS regressions are also
possible. All of these models are new and still unexplored. Some of them present unique
challenges, others are straightforward to estimate.
We revisit two empirical applications that related to prior studies, (1) the risk-return trade-
off and (2) volatility prediction. Regarding the risk-return trade-off, we present a variation
of the results in Ghysels, Santa-Clara, and Valkanov (2005) and Ghysels, Santa-Clara, and
Valkanov (2003). The first paper uses a MIDAS regression to show that there is a positive
relation between market volatility and return. Expected returns are proxied using monthly
averages while the variance is estimated using daily squared returns over the last year. The
second paper shows that while squared daily returns are good forecasts of future monthly
variances, there are predictors that clearly dominate. Here, we combine the insights from
both papers. First, we look at the risk-return relation at different frequencies, one, two,
three, and four weeks. Second, we use a different polynomial specification from the one
used in Ghysels, Santa-Clara, and Valkanov (2005).
2
Third, we use several predictors that
Ghysels, Santa-Clara, and Valkanov (2003) show are good at forecasting future volatility
in a MIDAS context. Finally, we use a different dataset from Ghysels, Santa-Clara, and
Valkanov (2005).
2
For further evidence on the risk-return trade-off using MIDAS, see e.g.
´
Angel, Nave, and Rubio (2004),
Wang (2004) and Charoenrook and Conrad (2005). Models of idiosyncratic volatility using MIDAS appear
in e.g. Jiang and Lee (2004) and Brown and Ferreira (2004).
3
We find that there is a robustly positive and statistically significant risk-return tradeoff
across horizons and across predictors. Remarkably, the tradeoff is significant even for weekly
returns, even though they are noisy proxies of expected returns. However, the relation is
clearer at the two to four week horizon. Surprisingly, we find that variables that are better
at predicting the variance do not necessarily produce better forecasts of expected returns or
better estimates of the risk-return tradeoff. Hence, they must be capturing a component of
the variance that is not priced by the market and consequently that is unrelated to expected
returns.
We also include empirical evidence on the impact of microstructure noise on volatility
prediction. While using high frequency data has some clear advantages, there are some
costs. High frequency sampling may be plagued by microstructure noise. Several papers
have tried to shed light on this: A¨ıt-Sahalia, Mykland, and Zhang (2005), Bandi and Russell
(2005b), Bandi and Russell (2005a), Hansen and Lunde (2004), Zhang, Mykland, and A¨ıt-
Sahalia (2005a), among others have suggested corrections for microstructure noise. We assess
how much these corrections improve forecasting.
The paper is structured as follows. Section two discusses various polynomial specifications.
Section three shows that the MIDAS framework is very flexible and captures a rich set
of dynamics that would be difficult to obtain using standard same-frequency regressions.
Section four presents various extensions of MIDAS models, such as a generalized MIDAS
regression, non-linear MIDAS regressions, tick-by-tick MIDAS regressions, and multivariate
MIDAS. In section five, we apply some of the generalizations to estimate the relation between
conditional expected return and risk using ten years of daily Dow Jones index return data.
Some of our results confirm previous findings, others are quite surprising and offer new
directions for research. In section six, we offer concluding remarks.
2 Polynomial Specifications
The parameterization of the lagged coefficients of B(k; θ) in a parsimonious fashion is one
of the key MIDAS features. In this section, we discuss various specifications of MIDAS
regression polynomials. A first subsection is devoted to finite polynomials and we discuss
in particular two parameterizations that were used in previous papers and that we will use
in the empirical section of this paper. A second subsection deals with infinite polynomials
4