# A Comprehensive Look at Financial Volatility Prediction by Economic Variables

**ABSTRACT** What drives volatility on financial markets? This paper takes a comprehensive look at the predictability of financial market volatility by macroeconomic and financial variables. We go beyond forecasting stock market volatility (by large the focus in previous studies) and additionally investigate the predictability of foreign exchange, bond, and commodity volatility by means of a data-rich modeling methodology which is able to handle a potentially large number of predictor variables. In line with previous research, we find relatively little economically meaningful predictability of stock market volatility. By contrast, volatility in foreign exchange, bond, and commodity markets appears predictable by macro and financial predictors both in-sample and out-of-sample.

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Electronic copy available at: http://ssrn.com/abstract=1737433

A Comprehensive Look at Financial Volatility

Prediction by Economic Variables∗

Charlotte Christiansen‡

Maik Schmeling§

Andreas Schrimpf∗∗

This draft: February 28, 2012

Abstract

We investigate if asset return volatility is predictable by macroeconomic and financial

variables and shed light on the economic drivers of financial volatility. Our approach is

distinct due to its comprehensiveness: First, we employ a data-rich forecast methodology

to handle a large set of potential predictors in a Bayesian Model Averaging approach,

and, second, we take a look at multiple asset classes (equities, foreign exchange, bonds,

and commodities) over long time spans. We find that proxies for credit risk and funding

(il)liquidity consistently show up as common predictors of volatility across asset classes.

Variables capturing time-varying risk premia also perform well as predictors of volatility.

While forecasts by macro-finance augmented models also achieve forecasting gains out-of-

sample relative to autoregressive benchmarks, the performance varies across asset classes

and over time.

JEL-Classification: G12, G15, G17, C53

Keywords: Realized volatility; Forecasting; Data-rich modeling; Bayesian model averaging; Model un-

certainty

∗We thank Tim Bollerslev (the co-editor), two anonymous referees, Eric Ghysels, Philippe Mueller, Peter

Schotman, Christian Speck (DGF discussant), Grigory Vilkov, Robert Vermeulen (EFA discussant), Christian

Upper and participants at the Arne Ryde Workshop in Financial Economics in Lund (2011), the European

Finance Association Meeting in Stockhom (2011), the ESEM in Oslo (2011), and German Finance Association

Meeting in Regensburg (2011) for helpful comments and discussions. Christiansen and Schrimpf acknowledge

support from CREATES funded by the Danish National Research Foundation and from the Danish Council

for Independent Research, Social Sciences. Schmeling gratefully acknowledges financial support by the German

Research Foundation (DFG). The views expressed in this paper are those of the authors and do not necessarily

reflect those of the Bank for International Settlements.

‡CREATES, Department of of Economics and Business, Business and Social Sciences, Aarhus University,

Bartholins Alle 10, DK-8000 Aarhus C, Denmark, e-mail: cchristiansen@creates.au.dk, phone: +45 8716

5576.

§Department of Economics, Leibniz Universit¨ at Hannover, K¨ onigsworther Platz 1, D-30167 Hannover, Ger-

many, e-mail: schmeling@gif.uni-hannover.de, phone: +49 511 762-8213.

∗∗Bank for International Settlements (BIS), Monetary and Economic Department, Centralbahnplatz 2, 4002

Basel, Switzerland. Tel: +41 61 280 8942. Email: andreas.schrimpf@bis.org.

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Electronic copy available at: http://ssrn.com/abstract=1737433

A Comprehensive Look at Financial Volatility

Prediction by Economic Variables

February 28, 2012

Abstract

We investigate if asset return volatility is predictable by macroeconomic and financial

variables and shed light on the economic drivers of financial volatility. Our approach is

distinct due to its comprehensiveness: First, we employ a data-rich forecast methodology

to handle a large set of potential predictors in a Bayesian Model Averaging approach,

and, second, we take a look at multiple asset classes (equities, foreign exchange, bonds,

and commodities) over long time spans. We find that proxies for credit risk and funding

(il)liquidity consistently show up as common predictors of volatility across asset classes.

Variables capturing time-varying risk premia also perform well as predictors of volatility.

While forecasts by macro-finance augmented models also achieve forecasting gains out-of-

sample relative to autoregressive benchmarks, the performance varies across asset classes

and over time.

JEL-Classification: G12, G15, G17, C53

Keywords: Realized volatility; Forecasting; Data-rich modeling; Bayesian model averaging;

Model uncertainty

Page 3

1. Introduction

Financial volatility is a crucial input for risk management, asset pricing, and portfolio manage-

ment and it may exert important repercussions on the economy as a whole as evinced forcefully

by the recent financial crisis. It is therefore of primary interest to learn more about the economic

drivers of volatility in financial markets. In this paper, we empirically investigate whether infor-

mation in a broad set of economic variables measuring financial and macro conditions is helpful

in predicting future volatility. We provide a comprehensive analysis of volatility predictability

for several major asset classes in a data-rich forecasting framework. While our main focus is

on studying the determinants of equity market volatility drawing on a long-term dataset which

covers more than 80 years of data, we also consider volatility in foreign exchange, bond, and

commodity markets over shorter time spans.

Economic theory suggests that variables capturing time-varying risk premia are primary

candidates for understanding and forecasting volatility (Mele, 2007). This implies that return

predictors from the extant literature (e.g. valuation ratios for equities, yield spreads for bonds,

interest rate differentials for foreign exchange etc.) qualify as promising volatility predictors

as well. However, countercyclical risk premia in financial markets do not mechanically imply

countercyclical return volatility (Mele, 2007).Thus, it is worthwhile to investigate sources

of volatility predictability separately from return predictability.In addition, several return

predictors also have a direct theoretical link to volatility forecasting. For example, structural

credit risk models such as Merton (1974) imply that equity volatility increases when leverage

increases, so that higher market-wide leverage should be positively related to future stock return

volatility.

Using information in financial and macroeconomic variables to forecast volatility in financial

markets is not entirely new to the literature but is far from having received the same attention as

the predictability of asset returns (see e.g. Goyal and Welch, 2003, 2008; Cochrane and Piazzesi,

2005; Ang and Bekaert, 2007; Ludvigson and Ng, 2009; Lustig, Roussanov, and Verdelhan, 2010,

for recent contributions to return predictability). Moreover, while there is a vast econometric lit-

erature on pure time-series modeling and forecasting of volatility (see e.g. Andersen, Bollerslev,

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Christoffersen, and Diebold, 2006, for a recent survey), the empirical literature on the economic

drivers of volatility is still fairly scarce.1

The seminal paper on the economic determinants of equity market volatility is Schwert

(1989). While his findings point towards countercyclical movements of stock market volatility,

the link between volatility and economic activity is not found to be very strong from a statistical

perspective. On a more positive note, Engle, Ghysels, and Sohn (2008) analyze the effect of

inflation and industrial production growth on daily stock return volatility, considering each

macroeconomic variable separately. They find that macro fundamentals do indeed matter for

stock return volatility. Diebold and Yilmaz (2010) consider a broad set of 40 international

equity markets and find that stock market volatility is cross-sectionally related to fundamental

macroeconomic volatility as measured by GDP volatility. In a recent paper, Paye (2012) studies

equity volatility predictability by macroeconomic and financial variables. His results suggest

meaningful and encouraging links between several economic variables and stock market volatility,

whereas improvements in terms of out-of-sample forecast accuracy are found to be fairly modest.

It is rather difficult, however, to draw general conclusions from the extant literature on

financial volatility predictability by macroeconomic and financial state variables. Most authors

employ different sample periods, different forecasting models and methods, different predictors,

different forecast evaluation criteria, and almost exclusively focus on stock market volatility.

Another important aspect is that“model uncertainty”is neglected in this branch of the literature.

Model uncertainty is the ex ante uncertainty of an economic agent with regard to the right choice

of macro-finance variables that are best suited for volatility prediction. While theory provides

some motivation for why some economic variables might qualify as predictors, it offers little

guidance on which specific variable (or particular combination of variables) should enter the

forecasting model for volatility. In this paper, we consider a Bayesian model averaging framework

1Modeling and forecasting time-varying volatility has its foundations in the class of (G)ARCH models (Engle,

1982; Bollerslev, 1986). More recently, the literature has expanded substantially drawing on the concept of

realized volatility and high-frequency modeling (See, e.g. Andersen, Bollerslev, Diebold, and Labys, 2003). This

literature is typically interested in high-frequency movements of volatility and time series aspects, while this

paper is mainly interested in low frequency variation and its link to macroeconomic and financial conditions.

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which takes account of this model uncertainty.2In essence, the approach we take is to let the

data speak about the usefulness of specific predictors.

The recent paper by Paye (2012) comes closest to ours in that it also models realized volatil-

ity in a predictive regressions setting. In this paper we go beyond Paye’s work in several regards.

First, we study a larger set of macro-finance predictors (38 as opposed to 13). Second, we go

beyond equity market volatility to consider other major asset classes such as foreign exchange,

bond markets and commodities for which (to the best of our knowledge) the economic determi-

nants of volatility have not been systematically analyzed before. Third and most important, we

explicitly consider the effects of model uncertainty when dealing with a large number of potential

predictors. We do this by a Bayesian model averaging (BMA) approach which has been shown

to be an adequate tool for stock return and exchange rate predictability (see e.g. Avramov, 2002;

Wright, 2008) and we show its virtues for studying the economic determinants of volatility as

well.3This allows us also to go beyond na¨ ıve forecast combination approaches and to consider

optimal Bayesian foreacsts for out-of-sample forecasting. In a nutshell, it is the comprehensive

approach, both in terms of scope as well as in terms of the applied econometric methodology,

which is the unique feature of this paper.

Understanding volatility movements is important since it is a consequential input for in-

vestment and asset allocation decisions. Moreover, understanding the macroeconomic causes of

financial market volatility is interesting in itself since it helps to uncover linkages between price

movements in financial markets and underlying risk factors or business cycle state variables.

This is even more important since there is a growing body of evidence showing that risks asso-

ciated with volatility are priced in option, stock, bond, and foreign exchange markets (e.g. Ang,

Hodrick, Xing, and Zhang, 2006, Da and Schaumburg, 2009, Menkhoff, Sarno, Schmeling, and

Schrimpf, 2012, Christiansen, Ranaldo, and S¨ oderlind, 2011 among others). Volatility-based

measures have also been shown to predict future stock market returns (see, e.g., Bollerslev,

2BMA is originally due to Leamer (1978). For treatments of model uncertainty in financial forecasting setups

similar to this paper, see e.g. Avramov (2002), Cremers (2002), or Wright (2008); Faust, Gilchrist, Wright, and

Zakrajsek (2011) recently use a BMA approach to predict U.S. business cycle fluctuations by credit spreads.

3Independent contemporaneous work by Cakmakli and van Dijk (2010) also considers the issue of stock return

and volatility predictability based on many (macro-)economic variables. Unlike the Bayesian Model Averaging

framework considered here, the authors extract information from macroeconomic series by dynamic factor models.

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Tauchen, and Zhou, 2009). Furthermore, recent evidence in Mele (2008) and Fornari and Mele

(2010) shows that stock market volatility is informative about future business cycle fluctuations

so that a better understanding of the driving forces of financial volatility is important for pol-

icy makers and monetary authorities. In the same vein, Chauvet, Senyuz, and Yoldas (2010)

find that financial volatility helps predict future economic activity when they consider volatility

measures that cover both stock and bond markets.

Our results indicate that economic predictor variables add significant explanatory power in

our forecasting exercises. Importantly, these results hold when controlling for the informational

content in lagged asset return volatility. It is also important to point out that our main fore-

casting approach shows the strongest predictive ability for predictors that are associated with

time-varying risk premia, leverage, or financial distress. Default spreads, for instance, stand

out as useful predictors not only for equity market volatility but also for other asset classes.

Moreover, the most robust predictors include valuation ratios (e.g. dividend yields in case of

equity volatility), a measure of interest rate differentials vis ` a vis the U.S. (in case of foreign

exchange volatility), and a measure of funding market (il)liquidity and heightened counterparty

credit risk (the TED spread as in Brunnermeier, Nagel, and Pedersen, 2009) which matter for

several asset classes.

In a nutshell, our results therefore suggest that there are economically meaningful relations

between variables measuring financial conditions and future volatility of different asset classes.

Purely macroeconomic variables (as opposed to financial variables) hardly show up as impor-

tant predictors of financial volatility. These results are fairly robust to a number of additional

checks and methodological variations. Our findings on out-of-sample predictability show that

including macro-finance predictors are able to enhance the forecast performance relative to sim-

ple autoregressive benchmarks in particular in the case of forecast combination methods. These

improvements do not hold for all asset classes and sample periods uniformly, however, and,

as in the case of return predictability (Goyal and Welch, 2008; Timmermann, 2008), forecast

performance can vary strongly over time.

The remaining part of the paper is structured as follows. Section 2 describes the data,

Section 3 details the econometric framework, Section 4 presents the empirical results, and Section

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5 concludes. Various additional results and details are delegated to an Online Appendix.

2. Data and Volatility Measurement

We base our main analysis on monthly observations of macroeconomic and financial variables

and realized volatilities computed from daily return observations. We focus on a long sample

for the U.S. equity market (represented by the S&P500), which runs from December 1926 to

December 2010, a total of 1,009 monthly observations.4In addition, we also investigate a shorter

sample for all four asset classes which covers the period from January 1983 to December 2010.

The starting point of the shorter sample is guided by having a comprehensive and common

dataset for all asset classes, both in terms of predictors and dependent variables. Thus, in the

case of the short-term multi-asset class sample we have 366 monthly time series observations for

each variable.

2.1. Measuring Financial Volatility

The main variables of interest are volatilities of the different asset classes which serve as depen-

dent variables in our predictive regressions. We compute the realized variance for asset class

i in month t as the sum of squared intra-period (daily) returns:

?Mt

τ=1r2

i;t;τwhere ri;t;τ is the

τth daily continuously compounded return in month t for asset i and Mtdenotes the number

of trading days during month t. In our empirical analysis, we define the realized volatility to

be the log of the square root of the realized variance since it is better behaved (i.e. closer to

normality) than the raw series:

RVi;t= ln

?

?

?

?

Mt

?

τ=1

r2

i;t;τ, t = 1,...,T.(1)

Realized volatility is an accurate proxy for the true, but latent, integrated volatility as the

number of intra-period observations becomes large (See, e.g. Andersen, Bollerslev, Diebold, and

Labys, 2003). We then proceed by using realized volatilities computed this way as observable

4We are most grateful to G. William Schwert for providing these data.

5

Page 8

dependent variables (See, e.g. Andersen, Bollerslev, Christoffersen, and Diebold, 2006, for a

survey).

For the construction of realized stock market volatility (RVS,t) we compute the realized

volatility measure according to Eq. (1) based on daily returns on the S&P500. Realized bond

market volatility (RVB,t) is computed from returns on the 10-year Treasury note futures contract

traded on the Chicago Board of Trade (CBOT).5In the same manner, we employ Standard &

Poor’s GSCI commodity index to construct our proxy for commodity market volatility (RVC,t).6

Our construction of aggregate foreign exchange (FX) market volatility is somewhat less

standard and draws on a portfolio approach for the global foreign exchange market as suggested

in recent work by Lustig, Roussanov, and Verdelhan (2011), among others. Hence, we use

a basket of currencies against the U.S. dollar for our main analyses. We do so to obtain an

aggregate measure of foreign exchange volatility (from the perspective of a U.S. investor) similar

to the aggregate stock market index, bond index, and commodity index we use for the other

asset classes. For robustness, however, we also report results for four major individual exchange

rates (German mark/euro, Japanese yen, British pound, and Swiss franc against the U.S. dollar)

in the Online Appendix. To construct the aggregate FX volatility measure we form an equally

weighted portfolio consisting of all currencies with available data at a given point in time.7For

the aggregate FX portfolio we calculate the time series of the daily spot rate changes which are

then used to construct realized FX volatility (denoted by RVFX,t) according to Eq. (1).

5We use the daily closing price of the bond futures contract which is available from Datastream (mnemonic

symbol is CTYCS00(PS)). This is the contract used by Fleming, Kirby, and Ostdiek (1998) who consider volatility

linkages between equity and bond markets. The advantages of using futures data are that these contracts are

highly liquid and that we can compute bond returns straight away without having to rely on return approximations

based on yields.

6These data are available from Datastream. In place of the GSCI index, it may have been preferable to use

data on the GSCI futures contract as this is actively traded at the CME (See e.g. Fong and See, 2001). Yet, the

GSCI futures only start trading in 1992. Still, the correlation between the realized volatility for the GSCI index

and the GSCI futures amounts to 0.97 during the period 1992-2009, so we deem it reasonable to use the GSCI

index to obtain a longer time-series.

7The foreign exchange rates are available from Thomson Financial Datastream. We use the currencies of

the following countries (all quoted against U.S. Dollar): Australia, Austria, Belgium, Brazil, Bulgaria, Canada,

Croatia, Cyprus, Czech Republic, Denmark, Egypt, Euro area, Finland, France, Germany, Greece, Hong Kong,

Hungary, Iceland, India, Indonesia, Ireland, Israel, Italy, Japan, Kuwait, Malaysia, Mexico, Netherlands, New

Zealand, Norway, Philippines, Poland, Portugal, Russia, Saudi Arabia, Singapore, Slovakia, Slovenia, South

Africa, South Korea, Spain, Sweden, Switzerland, Taiwan, Thailand, Ukraine, and United Kingdom. Not all

currencies are available during the entire sample period, as some currencies enter or exit the sample.

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[Insert Table 1 about here]

Table 1 shows summary statistics for the realized volatility series. The average volatilities

for commodities and stocks are much larger than the average volatilities for foreign exchange

and bonds. The same holds for the standard deviations of the realized volatility series. As is

well known, realized volatility is highly persistent and we find this behavior for all four asset

markets under investigation as indicated by the autocorrelation coefficients.

[Insert Figures 1 and 2 about here]

Figures 1 and 2 plot our realized volatility measures for the different asset classes. The

time series are highly variable and they do not appear to follow an identical pattern across asset

classes. This is also reflected in the pair-wise correlation coefficients that are reported in Panel

B of Table 1 which are generally not very high in absolute terms, i.e. below 48%. Given this

heterogeneous behavior, one may suspect that the volatility of different asset classes is at least

partly driven by different economic variables.

2.2. Macroeconomic and Financial Predictors

Overall, we rely on a comprehensive set of 38 macroeconomic and financial predictive variables.

The results for the long-term sample period draw upon a reduced set of 13 predictors (indicated in

Table A.1) which are available as of December 1926. Since many of those variables are motivated

via the risk-premium channel by Mele (2007), there is some overlap with the predictive variables

used in the comprehensive study on stock return predictability by Goyal and Welch (2008).

Table A.1 provides an overview and summary statistics of the predictors whereas the Online

Appendix provides further details on data sources and construction.

The variables considered in this paper are motivated by theory, mostly focussing on the

time-varying risk-premium channel emphasized by Mele (2007). This implies that specifically

those variables that have been shown to be useful predictors of returns and hence drivers of

risk premia should qualify as potential predictors. Given the scope of our study with its multi-

asset class focus, we do not only consider popular forecasting variables for equity returns (such

7

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as valuation ratios, industrial production growth etc.) but we also take into account potential

drivers of countercyclical risk premia in other markets. In addition, we also consider variables

associated with market and funding (il)liquidity, heightened credit and counterparty risk, as one

may suspect that a specific variable capturing risk premia or market conditions in one market is

also influential for volatility in another market.8For instance, during the recent financial crisis

stress in money markets quickly spilled over to affect market conditions and returns in other

asset classes (Baba, 2009). Our tests allow for such a possibility.9

In the following, we provide a brief overview and some further details regarding the moti-

vation behind some specific variables. For ease of exposition, we classify the predictive variables

according to five broad economic categories:

(1) Equity Market Variables and Risk Factors: Our list of explanatory variables in-

cludes well-known equity market valuation ratios such as the dividend price ratio (D-P) and the

earnings-price ratio (E-P), commonly considered in predictive regressions for stock returns (e.g.

Campbell and Shiller, 1988; Goyal and Welch, 2008). We also include lagged equity market re-

turns (MKT) to capture the well-established leverage effect (Black, 1976; Glosten, Jagannathan,

and Runkle, 1993; Nelson, 1991), i.e. the finding that negative returns are associated with higher

subsequent volatility. Other equity variables include the risk factors by Fama and French (1993),

a short-term reversal factor (STR) which is related to market volatility and distress as analyzed

in Nagel (2012). We also consider S&P500 turnover (TURN), which is commonly viewed as a

proxy for difference in opinion (Scheinkman and Xiong, 2003; Baker and Wurgler, 2007) and

hence potential uncertainty about future market valuations.

(2) Interest Rates, Spreads, and Bond Market Factors: Our set of bond market variables

include variables for instance the T-bill rate (T-B) which has shown to be a useful predictor of

equity excess returns (e.g. Ang and Bekaert, 2007). In addition, we include prominent bond re-

turn predictors such as the term spread (Campbell and Shiller, 1991) and the factor by Cochrane

and Piazzesi (2005) based on the term structure of forward rates. These variables are intended

8We thank an anonymous referee for pointing this out.

9In unreported tests, we also tested if volatilities in one specific market Granger-cause volatilities in other

markets, so-called volatility spillover effects (See e.g. Diebold and Yilmaz, 2009). We do not find evidence for

these kinds of lead-lag relationships, which is most likely due to the low frequency of the data considered in this

paper.

8

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to capture the evolution of risk premia in bond markets.

(3) Foreign Exchange Variables and Risk Factors: We consider three foreign exchange

specific forecasting variables. The average forward discount (AFD) – which measures interest

rate differentials vis ` a vis the U.S. for a broad range of currencies – might be particularly useful

given the findings on its ability to capture countercyclical FX risk premia in Lustig, Roussanov,

and Verdelhan (2010). In addition, we also include the dollar risk factor (DOL) and a carry

trade factor (C-T) from Lustig, Roussanov, and Verdelhan (2011) which have been shown to

capture a large fraction of common FX return variation.

(4) Liquidity and Credit Risk Variables: To proxy for heightened credit risk we rely on

the yield spread between BAA and AAA rated bonds (often labeled the default spread, DEF).

Credit risk tends to be higher in situations where leverage rises, which – according to models

such as Merton (1974) – should influence volatility. Furthermore, we include the TED spread

(difference between the 3-month LIBOR rate and T-Bill rate), a common measure of funding

(il)liquidity in interbank markets (e.g. Brunnermeier, Nagel, and Pedersen, 2009). We also

consider an aggregate measure of bid-ask spreads in foreign exchange markets to proxy for FX

market (il)liquidity (Menkhoff, Sarno, Schmeling, and Schrimpf, 2012) as well as the measure of

stock market liquidity in equity markets by Pastor and Stambaugh (2003).

(5) Macroeconomic Variables: We also consider general macroeconomic variables, such as

inflation and industrial production growth (either computed in monthly or annual growth rates).

The latter variable is central in the recent return predictability of excess returns in bonds and

foreign exchange (see e.g. Ludvigson and Ng, 2009; Lustig, Roussanov, and Verdelhan, 2010).

Output-based measures have also been found to be successful predictors of equity returns (e.g.

Cooper and Priestley, 2009). Including these variables also allows us to assess if macroeconomic

conditions are causal (in a post hoc ergo propter hoc sense) for volatility or by contrast whether

causality runs the other way as emphasized in papers such as Fornari and Mele (2010) or Chauvet,

Senyuz, and Yoldas (2010).

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3. Econometric Framework

We now outline our econometric approach.10Note that we use a univariate framework through-

out the paper which aims at predicting financial volatility for each asset class separately. We

use standard predictive regressions for the future realized volatility of asset i

RVi;t= α +

L

?

?=1

ρlRVi;t−?+ β?

jzj;t−1+ ut, (2)

where βjdenotes the kj-dimensional vector of regression coefficients on the predictive variables

and i indexes the asset type. The subscript j indicates that the composition of the vector of

predictive variables zj;tdepends on the particular model Mj. As we have a large number of

potentially relevant predictor variables, we investigate j = 1,...,2κmodels, where κ denotes the

overall number of predictive variables under consideration.

Since volatility is fairly persistent, it is important to include autoregressive terms in the

predictive regression to investigate if there is additional predictive content of the macroeconomic

and financial variables that goes beyond the information contained in the time-series history of

volatility. We therefore also report results from fitting an autoregressive model for the RV series

as the relevant benchmark case. While we largely focus on one autoregressive lag (L = 1), we

also discuss models with higher order terms in Section 4.2 and report additional results in the

Online Appendix.

Since the number of potential models is very large, it is computationally infeasible to evaluate

all possible models analytically. With κ = 38+1 potentially useful predictive variables we have

239= 549,755,813,888 possible model specifications. Given these considerations, we rely on two

approaches in this paper. First, we make use of a Bayesian model averaging approach with a

stochastic model search algorithm (MC3). Second, we employ a model selection approach based

on information criteria. We detail these two approaches next.

10See e.g. Avramov (2002) and Ludvigson and Ng (2009) for related approaches in the literature on stock

return predictability and bond return predictability. Wright (2008) studies the predictability of exchange rates

in a similar data-rich forecasting environment.

10

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3.1. Bayesian Model Averaging and MC3

Our baseline results draw on a Bayesian model averaging (BMA) approach. We briefly outline the

approach in the following, while some further technical details are discussed in the Appendix. A

particularly attractive feature of the BMA approach is that model uncertainty can be addressed

in a coherent way.Results in the stock return predictability literature suggest that model

uncertainty can be substantial when forecasting returns, see for instance Avramov (2002) or

Schrimpf (2010). In our context, model uncertainty refers to a situation where it is not clear ex

ante what the important predictive variables might be or which combination of variables may be

useful for prediction purposes. Unlike the classical approach, BMA does not posit the existence

of a true model and is therefore particularly suited to deal with a setup where model uncertainty

plays a role. Moreover, the BMA approach can be used to obtain optimal weights for forecast

combination (See e.g. Timmermann, 2006). To handle the large number of potential models we

rely on Markov Chain Monte Carlo Model Composition (MC3), a sampling approach drawing

from the model space which is particularly suited for high-dimensional problems such as the

one encountered here (See, e.g. Fernandez, Ley, and Steel, 2001; Koop, 2003). We outline this

approach in section B.2 of the appendix. The results are based on 500,000 draws and a burn-in

period of 50,000 draws.

In the Bayesian framework it is common to derive posterior probabilities p(Mj|D) for a par-

ticular model, where different models are defined by inclusion or exclusion of specific explanatory

variables. These posterior model probabilities, which reflect the usefulness of a particular model

after having seen the data D, are used in the BMA framework as weights in a composite model:

E[β|D] =

2κ

?

j=1

p(Mj|D)βj|D, (3)

where βj|D denotes the posterior mean of the predictive coefficients in the jth model. Likewise,

combined forecasts of BMA can be obtained by weighting the forecasts of the individual models

by the corresponding posterior model probabilities. Thus, in line with the Bayesian tradition,

the data allow us to learn by updating our belief about the quality of a particular model. The

posterior model probability is given by

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p(Mj|D) =

p(D|Mj)p(Mj)

Σ2κ

i=1p(D|Mi)p(Mi), (4)

where p(D|Mj) is the marginal likelihood and p(Mj) denotes the prior probability of model j

(as determined by inclusion and exclusion of specific predictive variables). The expression for

the marginal likelihood is obtained as

p(D|Mj) =

?

p(D|Mj,βj)p(βj|Mj)dβj, (5)

where p(βj|Mj) refers to the prior on the parameters of model j and p(D|Mj,βj) is the likelihood

of the model.11

3.2.Model Selection Based on Information Criteria

For completeness, we provide comparisons to a classical model selection approach that neglects

model uncertainty. Given the large amount of predictors, some standard pretesting is necessary

before estimating and evaluating the different models. To this end, we reduce the initial set of

potential predictors by only considering variables with a t-statistic greater than two in absolute

value in a predictive regression containing the respective macro-finance predictor and the lagged

dependent variable. In this way, we end up with a smaller set of predictors such that an analytical

evaluation of all models is computationally feasible. This is a common approach and is also used

by e.g. Ludvigson and Ng (2009) in the context of bond return predictability. For each of the

different model specifications, the Schwarz Information Criterion (BIC) is computed. Then the

models are ranked according to the BIC. The BIC favors models that provide a good fit while at

the same time penalizing highly parameterized models. Our tables report estimation results for

the three best model specifications according to the BIC and we report coefficients, Newey and

West (1987) HAC standard errors with optimal lag length selection by Andrews (1991), and the

adjusted R2.

11We focus on a 1-month forecasting horizon in the paper since the Bayesian approach does not allow for longer

horizons with overlapping observations. However, we also consider longer horizons (based on quarterly data) with

detailed results reported in the Online Appendix (Table IA.8). These results do not yield much additional insight.

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3.3.Out-of-Sample Forecast Evaluation

We also evaluate how forecasting models augmented by macro-finance predictors perform in an

out-of-sample context. As a general rule, we always evaluate the out-of-sample performance of

the forecast against the benchmark forecast of an autoregressive model. We basically employ

the same procedure as in our in-sample tests but we now estimate our models recursively and

evaluate the resulting out-of-sample forecasts. More specifically, we start with an initialization

period, estimate predictive regressions in the same way as above to produce the first out-of-

sample forecast. We then expand the estimation window and repeat the above steps to obtain

out-of-sample forecasts for the next period and continue in this way until we reach the end of

the sample period. In the following, we denote the forecast by the macro-finance augmented

model by fM

i,t+1and the forecast of the autoregressive benchmark model by fB

i,t+1.

We report Theil’s U (TU) which is given by the root mean square error (RMSE) of our

macro-finance augmented model relative to the RMSE of the benchmark model such that a value

smaller than one indicates that the model beats the benchmark in terms of forecast accuracy. In

addition, we report out-of-sample R2s as in Campbell and Thompson (2008). The out-of-sample

R2is computed as

R2

OOS= 1 −

?T−1

t=R(RVi,t+1− fM

?T−1

i,t+1)2

t=R(RVi,t+1− fB

i,t+1)2

(6)

where T denotes the overall sample size, and R is the initialization period.

Besides these purely descriptive forecast evaluation criteria, we provide bootstrap-based

statistical inference in order to assess if models augmented by macro-finance predictors signifi-

cantly outperform the benchmark forecast. Since the benchmark model is nested by the model

of interest, the asymptotic test put forth by Clark and West (2007) may be used. However,

the theoretical setup considered in Clark and West (2007) does not cover our case where the

forecasts are generated by forecast combination and where a model search over a large amount

of models is conducted. Hence, we rely on a bootstrap approach instead of asymptotic tests.12

12We are grateful to Todd E. Clark for this suggestion. In a similar vein, Wright (2008) relies on a bootstrap

approach to evaluate the out-of-sample accuracy of BMA generated forecasts.

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- Available from Andreas Schrimpf · May 29, 2014
- Available from SSRN