Realized Volatility and Business Cycle Fluctuations: A Mixed-Frequency VAR Approach Marcelle Chauvet 1 , Thomas G¨ otz 2 and Alain Hecq 2 1 Department of Economics, University of California Riverside 2 Maastricht University, SBE, Department of Quantitative Economics May 28, 2013 Department of Economics, University of California Riverside, Riverside CA 92507, Email: [email protected], Tel.: +1 951 827 1587 Corresponding author: Thomas G¨ otz, Department of Quantitative Economics, School of Business and Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands. Email: [email protected], Tel.: +31 43 388 3578, Fax: +31 43 388 2000 Department of Quantitative Economics, School of Business and Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands. Email: [email protected], Tel.: +31 43 388 3798, Fax: +31 43 388 4874 1
Transcript
Realized Volatility and Business Cycle
Fluctuations: A Mixed-Frequency VAR Approach
Marcelle Chauvet*1, Thomas Gotz2 and Alain Hecq2
1Department of Economics, University of California Riverside2Maastricht University, SBE, Department of Quantitative Economics
May 28, 2013
*Department of Economics, University of California Riverside, Riverside CA 92507, Email: [email protected],Tel.: +1 951 827 1587
Corresponding author: Thomas Gotz, Department of Quantitative Economics, School of Businessand Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands. Email:[email protected], Tel.: +31 43 388 3578, Fax: +31 43 388 2000
Department of Quantitative Economics, School of Business and Economics, Maastricht University, P.O. Box616, 6200 MD Maastricht, The Netherlands. Email: [email protected], Tel.: +31 43 388 3798, Fax:+31 43 388 4874
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Abstract
In this paper we analyze the link between uncertainty in financial markets and economicfluctuations. In particular, we test for Granger non-causality between the monthly growthrate of Industrial Production Index and the logarithm of daily bipower variation, an observedmeasure of volatility, in both directions. Due to the mismatch of sampling frequencies, weanalyze Granger non-causality testing in a mixed-frequency vector autoregressive model(VAR), originally proposed by Ghysels (2012) and extended to the non-stationary case byGotz et al. (2013). Due to the constituents of the regressand-vector, we define the termnowcasting causality as the mixed-frequency version of instantaneous causality (Lutkepohland Kratzig, 2004). We show that, starting from a mixed-frequency VAR, the presenceor absence of nowcasting causality has a crucial impact on testing for Granger causalityin standard mixed-frequency regression models. Furthermore, given a realistic sample size,the number of high-frequency observations per low-frequency period, 20 for our specific ex-ample, leads to parameter proliferation problems in case we attempt to estimate the VARunrestrictedly. Hence, we propose several parameter reduction techniques, among which arereduced rank conditions and the imposition of an ARX(1) structure on the high-frequencyvariables. The performance of these various approaches in terms Granger non-causalitytesting behavior is investigated in a Monte Carlo experiment. Subsequently, the techniquesare illustrated for the initial empirical research question.
JEL Codes: C32JEL Keywords: Granger Causality, Mixed Frequency VAR, Volatility, MIDAS, BusinessCycle
1 Introduction
The link between uncertainty in financial markets and economic fluctuations has been investi-gated for a long time. Generally, periods with increasing asset prices (bulls) are characterizedby positive return and low volatility, whereas periods with decreasing market prices (bears)are associated with negative return and high volatility. While the degree to which macroeco-nomic variables can help predict volatility has been investigated widely in the literature (seeSchwert, 1989b, Hamilton and Gang, 1996, or Engle and Rangel, 2008, among others), thereverse, i.e., whether the future path of the economy can be forecasted using return volatility,has been granted comparably few attention (examples are Schwert, 1989a, Mele, 2007, andAndreou et al., 2000). Independent from the role of volatility, regressand or predictor, it isalmost always computed as an aggregate, e.g. realized volatility, of an underlying observedmeasure of volatility. The reason is that the sampling frequency of volatility and economicactivity measures usually differ greatly. In a recent paper, Chauvet et al. (2012) use a longterm component of volatility to improve forecasts of future economic activity as well as businesscycle turning points. To this end, the authors employ quarterly or monthly realized volatilityin order to match the frequency of the macro variables involved.
In this paper, however, we use an observed measure of volatility to measure uncertainty.Instead of a monthly volatility measure, we use daily bipower variation (BV hereafter) computedon 5-minute returns by the Oxford-Man Institute of Quantitative Finance.1 Hence, we workwith a measure that is less sensitive to jumps compared to realized volatility, and that isavailable at a higher frequency than the most commonly reported indicators of business cyclefluctuations, which are sampled on a monthly basis. This implies that we work in a mixed-frequency framework instead of a common low-frequency one in which a monthly volatilitymeasure is computed beforehand.2 The latter is potentially based on daily or weekly regressors,e.g., squared returns, through so-called MIDAS volatility (see Ghysels et al., 2006, Engle, 1982,Bollerslev, 1986, or Andersen et al., 2002) or heterogeneous autoregressive realized volatilityregressions (Corsi, 2009).
As argued extensively in the mixed-frequency literature (Ghysels et al., 2007, Miller, 2012,or Gotz et al., 2012a), working in a mixed-frequency setup instead of a common low-frequencyone is advantageous due to the potential loss of information in the latter scenario and feasibilityof the former through Mi(xed) Da(ta) S(ampling) regressions (Ghysels et al., 2004) even in thepresence of many high-frequency variables compared to the number of observations, i.e., it doesnot suffer from parameter proliferation. Until recently, however, mixed-frequency problemswere limited to the simple regression framework, in which one of the low-frequency variablesis chosen as the dependent variable and the high-frequency ones are in the regressors. Sincethe work of Ghysels (2012) for stationary series and the extension of Gotz et al. (2013) forthe non-stationary and possibly cointegrated case, we can analyze the link between high- andlow-frequency series in a VAR system treating all variables as endogenous. More specifically,
1Heber, Gerd, Asger Lunde, Neil Shephard and Kevin Sheppard (2009), Oxford-Man Institute’s realizedlibrary (version 0.2), Oxford-Man Institute, University of Oxford.
2For a detailed survey on volatility models we refer the reader to Bauwens et al. (2012).
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in our case it allows us to investigate (non-)causalities of uncertainty in financial markets andbusiness cycle fluctuations in both directions.
In a recent paper, Ghysels et al. (2012) define causality in such a mixed-frequency VARand develop a corresponding test statistic. Using an underlying high-frequency process, theauthors derive the corresponding mixed- and low-frequency processes and analyze whethernon-causality from one variable (of high- or low-frequency) to another one (again of high-or low-frequency) is preserved. The authors, however, make an implicit assumption on thevariables involved, which does not hold in our specific case at hand and many other relatedstudies: The number of high-frequency observations within a low-frequency period is rathersmall, e.g., as in a year/quarter- or quarter/month-example. In this paper we analyze the finitesample performances with respect to Granger non-causality test behavior when the number ofhigh-frequency observations per low-frequency period is larger, e.g., in a month/working day-example. Two approaches to reduce the number of parameters to be estimated are considered,a rank reduction through common feature combinations and parameter restrictions throughspecifying the high-frequency process. Both approaches are compared with the low-frequencyapproach, obtained by temporally aggregating the high-frequency variable, and the unrestrictedmodel, which is assumed to suffer from parameter proliferation.
The rest of the paper is organized as follows. In Section 2 notation is introduced andthe mixed-frequency VAR for our specific case at hand is presented. Furthermore, the rele-vant Granger non-causality tests as well as nowcasting causality, the mixed-frequency analogof instantaneous causality (Lutkepohl and Kratzig, 2004), are defined. Section 3 discusses theapproaches to reduce the number of parameters to be estimated, the respective ways to estimatethe model and the test statistics corresponding to Granger non-causality. The finite sampleperformances of the Granger non-causality test behaviors is analyzed in Monte Carlo experi-ments in Section 4. An empirical example with U.S. data in Section 5 illustrates the results.Section 6 gives concluding remarks.
2 Causality in a Mixed-frequency VAR
2.1 Notation
Let us start from a two variable mixed-frequency system,3 where yt, t = 1, . . . , T is the low-frequency variable (a monthly indicator of the business cycle fluctuations in this study, e.g.,
the growth rate of Industrial Production Index or IPI) and x(m)t−i/m are the high-frequency
variables (the logarithm of daily BV here) withm high-frequency observations per low-frequencyperiod t. In the particular month/day-example at hand we take m = 20 because it is theminimum number of working days per month in our 2000:01 to 2012:06 sample. This impliesthat T = 150 here. The value of i indicates the specific day under consideration, ranging from
approximately the beginning of each month (x(20)t−19/20 with i = 19) until the end of the month
3Extensions towards representations of higher dimensional multivariate systems are straightforward.
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(x(20)t with i = 0). Note that as far as the lag operator, L, and the difference operator, ∆,
are concerned, we make a distinction between high- and low-frequency operators. To be more
precise, L denotes the low-frequency lag operator, i.e., Lyt = yt−1 or Lx(m)t−i/m = x
(m)t−1−i/m,
whereas L1/m denotes the high-frequency counterpart, which is only applicable to the high-
frequency variables, i.e., L1/mx(m)t−i/m = x
(m)t−i/m−1/m = x
(m)t−(i+1)/m. Similar rules apply for the
difference operators, ∆ and ∆(1/m). Finally, note that L1/mx(m)t−(m−1)/m = x
(m)t−1 and, by the
same logic, ∆(1/m)x(m)t−(m−1)/m = x
(m)t−(m−1)/m−x
(m)t−1. These notational conventions have become
standard in the mixed-frequency literature and are similar to the ones in Gotz et al. (2012a,b),Clements and Galvao (2008, 2009) or Miller (2012). The following table illustrates the notationfor our month/working day-example:
Notation t = 2012 : 06, m = 20 Meaning
yt y2012:06 y in June, 2012No working day No working day x on June 30, 2012
x(m)t x
(20)2012:06 x on June 29, 2012
x(m)t−1/m x
(20)2012:06−1/20 x on June 28, 2012
......
...
x(m)t−(m−1)/m = x
(m)t−1+1/m x
(20)2012:06−19/20 x on June 4, 2012
yt−1 y2012:05 y in May, 2012
x(m)t−m/m = L1/mx
(m)t−(m−1)/m = x
(m)t−1 x
(20)2012:05 x on May 31, 2012
x(m)t−1−1/m x
(20)2012:05−1/20 x on May 30, 2012
In this particular case, it means that although there are 21 open days for June 2012, wedo not consider the first day, i.e., June 1. For May 2012 we do not consider the first threedays. An alternative (balanced) strategy would be to take the maximum number of days in amonth (i.e., 23) and to create additional values for non-existing days. As far as the treatment
of daily data is concerned we have also taken x(m)t = x
(m)t−1/m when there are no quotations on
the day x(m)t . The notational conventions presented above straightforwardly extend to different
mixed-frequency settings, whereby, contrary to the month/day-case, m is often a fixed numberas in a year/month-example.
Considering each high-frequency variable such that Xt = (x(m)t , x
(m)t−1/m, . . . , x
(m)t−(m−1)/m)′,
Ghysels (2012) proposes to estimate, after achieving stationarity of I(1) series (e.g., monthlygrowth rate of IPI), the VAR(p):
Zt = µ+ Γ1Zt−1 + . . .+ ΓpZt−p + εt, (1)
where Zt = (yt, X′t)′. Note that compared to Ghysels (2012) we put the low-frequency variable
first. Equation (1) is easy to estimate on small systems but relies on the assumption that thetime series are truly stationary. It is shown how to modify it in the presence of non-stationary
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and possibly cointegrated series in Gotz et al. (2012b). Although the following observations canbe easily generalized to a VAR(p), let us explicitly write a VAR(1) for the monthly growth rateof the business cycle index (∆lnIPI) and daily bipower variation (BV), yt and X ′t, respectively,as
∆lnIPIt
lnBV(20)t
lnBV(20)t−1/20...
lnBV(20)t−19/20
=
µ1µ2µ3...µ21
+
π1 φ1,1 φ1,2 · · · φ1,20π2 φ2,1 φ2,2 · · · φ2,20
π3 φ3,1 φ3,2... φ3,20
......
......
...π21 φ21,1 φ21,2 · · · φ21,20
∆lnIPIt−1
lnBV(20)t−1
lnBV(20)t−1−1/20...
lnBV(20)t−1−19/20
+
ε1,tε2,tε3,t
...ε21,t
,
(2)where εt ∼MVN(0(21×1),Σ) with
Σ =
σ1,1 σ1,2 . . . σ1,21σ2,1 σ2,2 . . . σ2,21
......
. . ....
σ21,1 . . . . . . σ21,21
. (3)
2.2 Information Sets and Granger Non-Causality
Let us denote by Ω all outcomes for the stochastic processes concerned and let the informationset generated by the collection of sigma-fields Ft = σ(Zs, s ≤ t), t ≥ 0 then be denoted by Ωt.Furthermore, let Zt be adapted to that filtration such that Ωt represents the set of informationavailable at moment t. Furthermore, let us define ΩW as the set of all outcomes for thestochastic processes under consideration except the process W . Similarly to before, ΩW
t is thecorresponding information set containing the information for all stochastic process but W up tomoment t. Now, let P [Xt+h|ΩW
t ] be the best linear forecast of Xt+h based on ΩWt and likewise
for P [yt+h|ΩWt ]. We are now able to define Granger non-causality:
Definition 1 y does not Granger cause X if
P [Xt+1|Ωyt ] = P [Xt+1|Ωt].
Similarly, X does not Granger cause y if
P [yt+1|ΩXt ] = P [yt+1|Ωt].
In other words, y does not Granger cause X if past information of the low-frequency variabledo not help in predicting current (or future) values of the high-frequency variables and viceversa. In terms of Equation (2) and our specific volatility/business cycle-example this impliesthe following null hypotheses:
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Volatility does not Granger cause business cycle fluctuations
H0 : φ1,1 = φ1,2 = . . . = φ1,20 = 0,
Business Cycle fluctuations do not Granger cause volatility
H0 : π2 = π3 = . . . = π21 = 0.
Ghysels et al. (2012) implicitly assume m to be not too large (m = 3 or 4 in their case) inorder to estimate the VAR and then test for Granger non-causality. They do so by computingthe usual Wald test, which follows a chi-squared distribution with q degrees of freedom underthe null hypothesis, where q is the number of restrictions tested. With m being rather largein the situations we have in mind and in the particular example we consider in this paper(m = 20), we would need a sample much larger than usually applicable for macroeconomicdatasets to estimate the parameters and test for causality properly. Additional lags wouldfurther complicate the issue. Section 3 presents three approaches to reduce the number ofparameters to estimate, a rank reduction through common feature combinations, parameterrestrictions through specifying the high-frequency process and the common low-frequency VAR.
2.3 Nowcasting Causality
It becomes clear from Definition 2.2 that Granger non-causality in both directions is definedin terms of the low frequency, i.e., in terms of index t. Given the mixed-frequency nature ofthe variables under consideration, it may be of interest to analyze whether knowing the values
of x(m)t−i/m, i = 0, . . . ,m − 1, helps to predict yt, which is referred to as instantaneous causality
(Lutkepohl and Kratzig, 2004, or Lutkepohl, 2005, p.42). Note, however, that instantaneouscausality is usually defined for common-frequency VARs, i.e., either both variables are sampledat the low or the high frequency, justifying the use of the word ”instantaneous”, because onetests for a causality pattern between yt and xt, where t refers to the common frequency underconsideration. Since we intend to predict yt using values of the high-frequency variables withinperiod t, we refer to instantaneous causality in the mixed-frequency case as nowcasting causality.Formally, we can define nowcasting causality using the notation from the previous subsection.
Definition 2 y does not nowcasting cause X if
P [Xt+1|Ωt ∪ ΩXt+1] = P [Xt+1|Ωt].
Similarly, X does not nowcasting cause y if
P [yt+1|Ωt ∪ Ωyt+1] = P [yt+1|Ωt].
It turns out that x(m)t−i/m, i = 0, . . . ,m− 1, is not nowcasting causal for yt if and only if the
corresponding errors of the DGP of Zt are uncorrelated (Lutkepohl and Kratzig, 2004). To see
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this, let us consider our specific volatility-business cycle example and start from the VAR in(2), where we partition Σ as follows:
Σ =
(σ1,1 σ′·1σ·1 Σ2:21
), where σ·1 = (σ2,1, σ3,1, . . . , σ21,1)
′ and Σ2:21 =
σ2,2 . . . σ2,21...
. . ....
σ21,2 . . . σ21,21
.
Note that σ′·1 appears in the top right cell of Σ because σi,1 = σ1,i, i = 2, . . . , 21. Also, theargument obviously extends to a higher lag order. Let us now factorize the unrestricted VAR
into the conditional model for ∆lnIPIt given all high-frequency variables, i.e., lnBV(20)t−i/20, i =
0, . . . , 19,, and the remaining marginal models. The conditional model actually becomes astandard mixed-frequency model:
∆lnIPIt = σ′·1Σ−12:21
lnBV
(20)t
lnBV(20)t−1/20...
lnBV(20)t−19/20
+ (µ1 − σ′·1Σ−12:21
µ2µ3...µ21
) + (π1 − σ′·1Σ−12:21
π2π3...π21
)∆lnIPIt−1
+(
φ1,1φ1,2
...φ1,20
′
− σ′·1Σ−12:21
φ2,1 . . . φ2,20...
. . ....
φ21,1 . . . φ21,20
)
lnBV
(20)t−1
lnBV(20)t−1−1/20...
lnBV(20)t−1−19/20
+(ε1t − σ′·1Σ−12:21
ε2tε3t...ε21t
).
(4)
So, with ψ0 = σ′·1Σ−12:21 being a (1×20) parameter vector, lnBVt = (lnBV
(20)t , lnBV
(20)t−1/20, . . . , lnBV
(20)t−19/20)
′
and other substitutions, which follow straightforwardly from (2) or (4), we have that
(5)This is indeed a standard mixed-frequency regression, the parameters of which could be es-timated unrestrictedly (what Foroni et al., 2012 call U-MIDAS) or using MIDAS restrictions
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(as proposed in Ghysels et al., 2004), which will be discussed in more detail in a later section.Note that the same result can be obtained by considering a structural VAR setup with suit-ably chosen parameter matrices (see Ghysels, 2012). Now, testing whether volatility does notnowcasting cause business cycle fluctuations corresponds to investigating
In other words, lnBV(20)t−i/20, i = 0, . . . , 19, is not nowcasting causal for ∆lnIPIt if and only if
the errors of ∆lnIPIt are uncorrelated with the ones from each BVt-variable. This can betested using a Wald test on the (1, j)-elements of Σ in (3), i.e., σ1,j = 1
T
∑Tt=1 ε1,tεj,t, j =
2, . . . , 21, where εj,t corresponds to the residual of equation j in the corresponding VAR (see,e.g., Hamilton, 1994, p. 301, or Gianetto and Raıssi, 2012). To be more specific, consider thevariance-covariance matrix in (3) and apply an operator analogous to the vec-operator to it(”vech” in Hamilton, 1994):
vech
σ1,1 σ1,2 . . . σ1,21σ2,1 σ2,2 . . . σ2,21
......
. . ....
σ21,1 . . . . . . σ21,21
=
σ1,1...
σ1,21σ2,2
...σ2,21σ3,3
...
...σ20,21σ21,21
One can show that √
T (vech(Σ)− vech(Σ))D→ (0,Ξ),
where the element of Ξ corresponding to the covariance between σi,j and σk,l is equal to (σi,kσj,l+σi,lσj,k)∀i, j, k, l = 1, . . . , 21 including i = j = k = l. Now, defining RNC as the matrix that
picks the corresponding elements in vech(Σ) that we want to test in (6), i.e., RNCvech(Σ) =(σ2,1, σ3,1, . . . , σ21,1)
′, allows us to test for nowcasting non-causality by computing
ξNC = T (RNCvech(Σ))′(RNCΞR′NC)−1(RNCvech(Σ)), (7)
which is asymptotically distributed as χ2rank(RNC)
.The above reasoning can, of course, be applied to testing whether business cycle fluctuation
nowcasting cause volatility. To this end, we factorize the unrestricted VAR into the conditional
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models for lnBVt given ∆lnIPIt and the marginal model for ∆lnIPIt. Calculations similar
to the ones above reveal that ∆lnIPIt does not nowcasting cause lnBV(20)t−i/20, i = 0, . . . , 19,
if and only if, again, the same corresponding errors are uncorrelated. Consequently, and aspointed out by Lutkepohl and Kratzig (2004), the concept is fully symmetric, i.e., if ∆lnIPItis nowcasting causal for lnBVt, then lnBVt is nowcasting causal for ∆lnIPIt. Since correlationdoes not imply causal direction, we need to know from another source that there can only bea unidirectional causality between the two variables. Only then is it interesting to check thepossibility of nowcasting causality.
2.4 Remark on standard mixed-frequency models and Causality
Equation (5) reveals an important fact about how different causality structures enter a stan-dard mixed-frequency regression. In fact, the standard mixed-frequency model obtained byconditioning on the high-frequency variables, i.e., by assuming that the regressor(s) is (are)strongly exogenous for the parameters of interest, ”mixes” Granger and nowcasting causality.The regression in (5) is repeated below in simplified manner.
∆lnIPIt = µ+ π∆lnIPIt−1 +19∑i=0
θi,0lnBV(20)t−i/20︸ ︷︷ ︸
ψ0lnBVt
+19∑i=0
θi,1lnBV(20)t−1−i/20︸ ︷︷ ︸
(φ1,1:20−ψ0φ2:21,1:20)lnBVt−1
+εt.
Indeed, as argued in section 2.2, Granger non-causality from lnBV to ∆lnIPI impliesφ1,k = 0, k = 1, . . . , 20, or simply φ1,1:20 = 0, where the φ-coefficients are the correspondingcoefficients in the autoregressive matrix of the underlying VAR in (2). Nowcasting non-causality,on the other hand, corresponds to θi,0 = 0, i = 0, . . . , 19, or simply ψ0 = 0. There are twoimportant issues to adress.
Firstly, because the standard mixed-frequency regression is obtained by conditioning on
the high-frequency variables, the coefficient on lnBV(20)t−1−i/20 depends not only on φ1,i, but
also on all the covariances between ε1,t and εj,t, j = 2, . . . , 21 (through ψ0). Consequently,
even if no Granger causality is found, i.e., φ1,1:20 = 0, the coefficients on lnBV(20)t−1−i/20 are
potentially non-zero if nowcasting causality exists. This implies potential ”spurious” Grangercausality in such a situation if one solely relies on the conditional (standard mixed-frequency)model, i.e., if one tests for Granger non-causality by checking θi,1 = 0, i = 0, . . . , 19 instead ofφ1,k = 0, k = 1, . . . , 20.
Secondly, as is going to be explained in Section 3.2, the standard mixed-frequency regressionmodel can be estimated unrestrictedly or after imposing MIDAS restrictions (see Ghysels et al.,2004). Without going into detail at this stage, it implies that one either computes one weightfunction for all high-frequency observations involved or several weight functions for each set ofhigh-frequency observations in a given t-period. The weights of each function sum to one inorder to identify a scale coefficient in front of the respective weight function (for details, see
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Section 3.2). For the example of p = 1 in the equation above, this boils down to either oneweight functions for all 40 lnBV -observations or two weight functions for the first and second20 observations, corresponding to period t and t − 1, respectively. Now, imagine there is nonowcasting causality, i.e., θi,0 = 0, i = 0, . . . , 19 ⇔ ψ0 = 0, but that there exists Grangercausality from lnBV to IPI-growth, i.e., φ1,k 6= 0, for some k = 1, . . . , 20 ⇔ φ1,1:20 6= 0. Interms of the weight function(s) (multiplied by their respective scale coefficient) this situationcorresponds to the expected picture given in Figure 1. Due to missing nowcasting causality,
Figure 1: Example of Expected Weight Function×Scale Coefficient in the Presence of GrangerCausality, but in the Absence of Nowcasting Causality
Note: This figure shows the coefficients, i.e., i.e., their respective weights×scale coefficient, of lnBV(20)
t−i/20 and
lnBV(20)
t−1−i/20, i = 0, . . . , 19. The weight function for the last 20 high-frequency observations is computed using
the two-dimensional Almon Lag Polynomial (to be introduced in Section 3.2, see, e.g., Ghysels et al., 2007) withparameters γ1 = 0 and γ2 = −0.00907. The first 20 high-frequency observations get assigned a value of zerobecause the scale coefficient is set to zero. It is an example of an expected weight function if there exists Grangercausality from lnBV to IPI-growth, but no nowcasting causality.
coefficients on lnBV(20)t−i/20, i = 0, . . . , 19 should equal zero, whereas the ones on the respective
observations in period t−1 should be non-zero due to existing Granger causality. The particularshape of second half of the weight function is merely an example implying that more recentobservations get assigned a larger weight. Different shapes are, of course, possible. In practice,one has to estimate the scale coefficient(s) as well as the parameters of the weight function(s).
Note that an abrupt jump in the coefficients, as the one we expect between lnBV(20)t−19/20 and
lnBV(20)t−1 , can be easily and accurately achieved when estimating separate weight functions
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for each set of high-frequency observations in a given t-period. An estimated scale coefficientof zero for the first 20 lnBV -observations is sufficient. This is, however, much harder whenestimating merely one weight function for all high-frequency observations. Hence, in ordernot to introduce ”spurious” nowcasting causality when estimating the conditional model usingMIDAS restrictions, users should employ multiple weight functions (corresponding to eachlow-frequency period) instead of estimating merely one weight function for the whole set ofhigh-frequency observations.
The two issues just discussed lead to the following general step-by-step procedure to detecta causality pattern between low-frequency variable y and high-frequency variables X:
1. Determine the integration order of all variables concerned. If both variables are I(1), checkfor cointegration between the variables. If cointegration is found, apply the techniques inGotz et al. (2013) to get a VAR involving stationary terms only. If no cointegration existsbetween y and X, or if only one variable is I(1), take first differences of the correspondingvariables. If the variables are I(0), no transformations are required.
2. Determine the lag order using multivariate information criteria (Akaike, Schwarz orHannan-Quinn) and estimate the VAR in equations (1) or (2). Note that different ways toreduce the number of parameters to be estimated are presented in the following section.
3. Test for Granger non-causality in both directions.
4. Test for nowcasting non-causality using the Wald test in (7).
5. If a conditional model is considered instead of a VAR, do not test for Granger causalityunless nowcasting causality is rejected.
6. If the conditional model for the low-frequency variable is estimated after MIDAS re-strictions are imposed, estimate separate weight functions for each set of high-frequencyobservations in a given t-period in order not to introduce ”spurious” nowcasting causality.
3 Parameter Reduction
3.1 Reduced Rank Conditions
In order to reduce the number of parameters to estimate in the 21-dimensional mixed-frequencyVAR, we propose the following:
∆lnIPIt
lnBV(20)t
lnBV(20)t−1/20...
lnBV(20)t−19/20
=
µ1µ2µ3...µ21
+
π1π2π3...π21
,
α1
α2
α3...α21
×[δ1 δ2 · · · δ20
]
∆lnIPIt−1
lnBV(20)t−1
lnBV(20)t−1−1/20...
lnBV(20)t−1−19/20
+
ε1,tε2,tε3,t
...ε21,t
,
12
where πi, i = 1, . . . , 21 are scalar coefficients, but αi, i = 1, . . . , 21 are potentially 1× (20− s)vectors depending on the reduced rank conditions. Consequently, δi, i = 1, . . . , 20 are (20−s)×1vectors such that we could call
[δ1 δ2 · · · δ20
]
lnBV(20)t−1
lnBV(20)t−1−1/20...
lnBV(20)t−1−19/20
=
Factor1Factor2
...Factor(20−s)
the high-frequency factors. This enables us to simplify the VAR(1) under reduced rank condi-tions to
∆lnIPIt
lnBV(20)t
lnBV(20)t−1/20...
lnBV(20)t−19/20
=
µ1µ2µ3...µ21
+
π1π2π3...π21
∆lnIPIt−1+
α1
α2
α3...α21
Factor1Factor2
...Factor(20−s)
+
ε1,tε2,tε3,t
...ε21,t
.
In terms of the number of parameters, the unrestricted VAR(1) in (2) requires 21×21 = 441coefficients to be estimated in the autoregressive matrix, whereas the VAR(1) under reducedrank conditions needs 21 + 21(20− s) + 20(20− s) = 21 + 41(20− s) parameter estimates. So,if s = 19, 18 or 17 and if there is a unique transmission mechanism there are only 62, 103 or144 coefficients to be estimated in the partially reduced rank coefficient matrix. Note that wedo not require yt−1 to be included in the same transmission mechanism as the x variables. Theapproach could be made less flexible if one relaxes the hypothesis that the common transmissionmechanism also hits yt, but we do not follow that approach here.
We can implicitly test for the number of high-frequency factors in a maximum likelihoodframework by using a Full Information Maximum Likelihood approach or a canonical correlationanalysis (CCA hereafter) by analyzing the eigenvalues and the eigenvectors of
Σ−1ZZ
ΣZXΣ−1XX
ΣXZ , (8)
or, similarly, in the symmetric matrix Σ−1/2ZZ
ΣZXΣ−1XX
ΣXZΣ−1/2ZZ
(Anderson, 1951, or for the
application to common dynamics, Vahid and Engle, 1993). Note that Σij represent empiri-cal covariance matrices and that Zt and Xt denote that Zt and Xt, respectively, have beenconcentrated out by the variables that do not enter in the reduced rank regression, i.e., theintercept and yt−1, or ∆lnIPIt−1 in our case. For i.i.d. normally distributed disturbances, thelikelihood ratio test statistic for the null hypothesis that there exist at least s common feature
13
combinations within Xt is given by
ζLR(s) = −Ts∑j=1
ln(1− λj) s = 1, . . . , 20. (9)
We also consider partial least squares (PLS hereafter) to extract factors (Groen and Kapetanios,2008). Indeed, the estimation of the eigenvectors δi obtained from the canonical correlationanalysis in (8) may perform poorly with high-dimensional systems because inversions of largevariance matrices Σ−1
ZZand Σ−1
XXare required. Hence, we use a PLS algorithm similar to the
one used in Cubadda and Hecq (2011) as an alternative to CCA. In order to make the solutionof this eigenvalue problem invariant to scale changes of individual elements, we compute theeigenvectors associated with the largest eigenvalues of the matrix
D−1ZZ
ΣZXD−1XX
ΣXZ ,
with DZZ and DXX being diagonal matrices having the diagonal elements of, respectively, Σ−1ZZ
and Σ−1XX
as their entries.Note that the aforementioned approach is, of course, not the only possible avenue to pursue
in order to reduce the number of parameters in a model. General solution methods to estimateor forecast economic time series in a data rich environment are, as mentioned above, principalcomponents (after imposing a factor structure), Lasso (Tibshirani, 1996) as well as Bayesianor ridge regressions (Banbura et al., 2010). Cubadda and Guardabascio (2012) analyze a so-called ”medium-N” approach arguing that many of the results in the literature favor a numberof predictors (N) that is considerably larger than in usual small-scale forecasting problems,but not too large for being forced to rely on double (T and N) asymptotic methods. Theauthors find, using Monte Carlo simulations, that, under a certain condition,4 both principalcomponents and PLS provide consistent estimates in such a medium-N framework as only thesample size diverges.
Principal components, however, do not necessarily preserve the same dynamics under thenull. To be more precise, nothing prevents, e.g., the first and only principal component to beloading only on ∆lnIPIt implying that the remaining dynamics enter the error term. In otherwords, unlike the CCA presented above, principal components may and will most likely notpreserve the autoregressive matrix in (2). This affects, in a possibly drastic manner, the blockof parameters we test on for Granger non-causality.
For a given s, we test for Granger non-causality by defining R as the matrix that picks theelements of Γ1 : ... : Γp in (1) depending on the direction of causality we intend to test (seeSection 2.2), i.e., Rvec(Γ1 : ... : Γp). For a general construction of the matrix R in the presenceof several low- and high-frequency variables, we refer the reader to Ghysels (2012). The Waldtest is then constructed as
ξWald =[Rvec(Γ1 : ... : Γp)
]′(RΩR′)−1
[Rvec(Γ1 : ... : Γp)
], (10)
4The so-called Helland and Almoy condition (Helland, 1990, or Helland and Almoy, 1994).
14
withΩ = (W ′W )−1 ⊗ Σ,
where Σ = 1T−KW
ε′ε is the empirical covariance matrix of the disturbance terms and whereW denotes the regressors, i.e., an intercept, ∆lnIPIt−1 and Factori, i = 1, . . . , (20− s) in thisspecific case and where KW is the number of regressors in W . As illustrated in Ghysels et al.(2012), ξWald is asymptotically χ2
rank(R). Additionally, to account for the potential presence of
a time varying multivariate process, we compute a robust estimator of Ω (see Ravikumar et al.,2000) as
Note that these test statistics are computable with any usual econometric software packagesas, e.g., EViews.5
3.2 Specifying the High-Frequency Process
It is obvious that no restrictions are placed on the coefficients in the autoregressive matrix in(2). While this seems reasonable for the first equation, i.e., the one for ∆lnIPIt, it is less
clear for the remaining ones. To be more precise, why should each lnBV(20)t−i/20, i = 0, . . . , 19,
depend on all high-frequency observations of the previous low-frequency period, i.e., on alldaily observations of lnBV of the previous month? Likewise, the preceding section reduces theamount of parameters to estimate by computing factors for the high-frequency observations.However, why should all factors appear in each equation, especially for the lnBV -equations?
Ghysels (2012) discusses the issue of parsimony after introducing mixed-frequency VARmodels. It is argued that a sensible specification of the high-frequency process allows us toobtain a number of parameters that is independent from m, i.e., the number of high-frequencyobservations per t-period. In particular, let us disregard the constant term for now and let usassume that lnBV follows an ARX(1) process with a possibly non-constant effect of ∆lnIPIt−1
on lnBV(20)t−i/20, i = 0, . . . , 19:
lnBV(20)t−i/20 = ρBV lnBV
(20)t−(i+1)/20 + π2+i∆lnIPIt−1 + εi+2,t for i = 0, . . . , 19. (12)
Furthermore, writing down the first equation of the VAR in (2) with π1 = ρIPI and without aconstant term yields
∆lnIPIt = ρIPI∆lnIPIt−1 +
19∑i=0
φ1,i+1lnBV(20)t−1−i/20 + ε1,t, (13)
5Note that depending on the software package used, the degrees of freedom adjustments in Σ, ΩR or S0 maychange. This has, however, only a marginal impact on the results.
15
which looks very similar to equation (5). In fact, it is a standard mixed-frequency regressionwithout contemporaneous (in terms of t) values of lnBV . As indicated before, we may estimatethe parameters φ1,i+1 unrestrictedly or using MIDAS restrictions (Ghysels et al., 2004). Wechoose the latter option in this paper because it allows us to reduce the number of parameterseven further. This implies that we impose the following:
φ1,i+1 = βwi(γ) for i = 0, . . . , 19, (14)
where wi is the weight corresponding to BV(20)t−i/20. As already sketched in Section 2.4, weight
functions are usually calculated with a low-dimensional lag polynomial such as the Beta orthe Exponential Almon lag polynomial (see Ghysels et al., 2007 for a detailed discussion)in order to hyperparameterize the polynomial lag structure and thereby solve the parameterproliferation problem, especially for large m. If more than one lag in terms of t is considered,two approaches for designing weight functions are dominating. One either summarizes all high-frequency observations in one weight function or one models all high-frequency observationsfalling into one t-period by a separate weighting scheme (see Gotz et al., 2012a). In this paper,we employ the latter option whenever applicable for the reasons given in Section 2.4 and becauseit generally allows us to differentiate between the influences of observations from different t-periods, i.e., months. Note that while we allow for further lags of both, ∆lnIPI and lnBV , in(13), we do not do so in (12), which implies that no ∆lnIPIt−p for p > 1 appears on the righthand sides. We present the estimation methodology for the case with only one lag (in terms oft), the extensions to high lag order are immediate. The weights of each separate weight functionsum up to one in order to identify the corresponding scale coefficient, β in the example above.We use the two-dimensional Exponential Almon Lag Polynomial, i.e.,
throughout this paper for its superiority over alternative choices reported in Ghysels and Valka-nov (2006).
Note that with respect to the error terms in (12) and (13), we assume, as is done in Ghy-sels (2012), that E(εi,tεi,t) = σBV,BV and E(ε1,tεi,t) = σIPI,BV for i = 2, . . . , 21 and thatE(ε1,tε1,t) = σIPI,IPI . Combining all of these assumptions we end up with the following re-stricted VAR:
These restrictions cause the number of parameters to be estimated in the autoregressive matrixto reduce to 25, two autoregressive parameters, ρIPI and ρBV , two parameters determining theweight function, γ, the scale coefficient, β, and the effects of ∆lnIPIt−1 on the high-frequencyvariables, πi, i = 2, . . . , 21.
As far as estimation of the restricted system in (15) is concerned, we propose the followingtwo-step procedure. First, run the regression for the low-frequency variable in (13) with theMIDAS restrictions (14) imposed in order to obtain the weight function, i.e., by getting esti-mates for γ. Then, include the weight function as a regressor, call it Weightt, and estimate thefollowing SUR (see, e.g., Greene, 2011) regression (including a constant term to be in line withthe original VAR in (2)):
∆lnIPIt
lnBV(20)t
lnBV(20)t−1/20...
lnBV(20)t−19/20
︸ ︷︷ ︸
ySURt
=
XSUR
1,t 0 . . . 0
0 XSUR2,t . . . 0
......
. . ....
0 . . . 0 XSUR2,t
︸ ︷︷ ︸
XSURt
µ1ρIPIβµ2∑19
i=0 ρiBV πi+2
ρ20BV...µ21π21ρBV
︸ ︷︷ ︸
βSUR
+
u1,tu2,tu3,t
...u21,t
︸ ︷︷ ︸
uSURt
,
(16)
where XSUR1,t = (1,∆lnIPIt−1,Weightt) and XSUR
2,t = (1,∆lnIPIt−1, lnBV(20)t−1 ) and which can
be written in usual vector notation as
ySUR = XSURβSUR + uSUR. (17)
17
Consequently,
E(uSURu′SUR|XSUR) = ΩSUR = Σu ⊗ IT and Ω−1SUR = Σ−1u ⊗ IT .
The FGLS estimator is then obtained in the standard way (see, e.g., Greene, 2011):
βSUR =[X ′SURΩ−1SURXSUR
]−1X ′SURΩ−1SURySUR =
[X ′SUR(Σ−1u ⊗ IT )XSUR
]−1X ′SUR(Σ−1u ⊗IT )ySUR,
where Σu = 1T u′tut with ut being the standard least squares residuals. Alternatively, one can
estimate the restricted mixed-frequency VAR in (15) using a maximum likelihood based esti-mator (Hamilton, 1994) or Bayesian estimation approaches (see, e.g., Rodriguez and Puggioni,2010). Both approaches are also discussed in Ghysels (2012).
To subsequently test for Granger non-causality in our setup, we have to adapt the Waldtest to the specific SUR setting we face. In particular,
ξW =[RβSUR)
]′(R(X ′SURΩ−1SURXSUR)−1R′)−1
[RβSUR)
]. (18)
Note that we do not compute a heteroscedasticity robust estimator as in (11) in this casebecause the according theory developed in Ravikumar et al. (2000) or Greene (2011) appliesonly for SUR systems with identical regressors, i.e., where the model can be estimated by OLS.
3.3 A Common Low-Frequency VAR
Before the introduction of MIDAS regression models by Ghysels et al. (2004), high-frequencyvariables were usually aggregated to the low frequency in order to obtain a common frequencyfor all variables appearing in a regression. Likewise for systems, such as VAR models, the obser-vations of a high-frequency, say monthly, variable are usually aggregated to one low-frequency,say quarterly, observation such that the VAR is estimated in the resulting common low fre-quency, quarterly in our case. Recent work by Ghysels (2012) or Gotz et al. (2013), however,allows the user to estimate systems of equations without aggregating the variables, and withoutrelying on parameter-driven models (Cox et al., 1981) such as state space models, and therebypotentially preserve all information present in the high-frequency observations. Naturally, thequestion arises in how far such a strategy is superior to the classical approach. Indeed, e.g.,Gotz et al. (2012a) for the standard linear regression case, and, e.g., Ghysels (2012) for theVAR case, analyze the impact of temporal aggregation on the short-run forecasting perfor-mances or on Granger (non-)causality compared to an underlying common high-frequency ormixed-frequency DGP. Interestingly, the latter work reports higher power for mixed-frequencycausality tests than for the low-frequency ones, especially when a certain high-frequency vari-able is not Granger caused by any variable, high- or low-frequency, in the underlying commonhigh-frequency DGP.
Coming back to our specific case at hand, the classical approach of temporally aggregatinghigh-frequency variables also leads to a great reduction in parameters which need to be esti-mated. After all, each set of m, or in our case 20, high-frequency variables per low-frequency
18
period t is aggregated into one low-frequency, here monthly, observation. In terms of theVAR(1) in (2), instead of 441 parameters to estimate in the autoregressive matrix, a commonlow-frequency VAR as in(
∆lnIPIt
lnBV(aggr)t
)=
(µ1µ2
)+
(π1 φ1π2 φ2
)(∆lnIPIt−1
lnBV(aggr)t−1
)+
(ε1,tε2,t
)(19)
requires only 4 parameter estimates. Of course, this comes at the cost of potentially disregardinginformation present in the high-frequency observations. Additionally, causality among othersis a property which is not invariant to temporal aggregation as noted in Marcellino (1999)and Sims (1971), implying that aggregating a high-frequency variable may lead to ”spurious”causality in the common low-frequency setup (as in Ghysels, 2012). It will be of interest tosee how a common low-frequency VAR approach performs in terms of Granger causality testbehavior compared to the previous two strategies for reducing the number of parameters.
In the temporal aggregation literature, two aggregation schemes are dominating: Skip orPoint-in-time sampling and Averaging or Average sampling (Marcellino, 1999, or Gotz et al.,2012a). While the former is usually applied to stock variables, the latter is more suitable for flowvariables. For a detailed survey on temporal aggregation see Silvestrini and Veredas (2008).With daily bipower variation as high-frequency regressors, we employ an average samplingscheme in our specific case at hand.
As far as testing for Granger non-causality is concerned, we can rely on the Wald statisticin (10) and its robust version derived from (11). Note, however, that the set of regressors, Win (10) and (11), as well as Γ1 : ... : Γp and thereby R need to be adjusted to the commonlow-frequency VAR setup in (19).
3.4 The Unrestricted Model
Finally, we can attempt to estimate the full VAR in (1), or (2) for our particular applicationand a lag order of one, ignoring the fact that, given the sample sizes usually available, theamount of parameters is probably too large for estimating them properly or for testing anycausality patterns adequately. Still, it serves as benchmark case allowing us to compare theaforementioned alternatives in terms of their relative performances to this benchmark. To thisend we simply estimate the mixed-frequency VAR using ordinary least squares disregarding theparameter proliferation problem we inevitably run into.
Similar to the common low-frequency VAR case described in the previous section, we cantest for Granger non-causality using the Wald statistic in (10) or the one derived from (11)after adjusting the set of regressors and parameters accordingly, i.e., for the unrestricted modelit should be adjusted consistent with equations (1) or (2).
4 A Monte Carlo Study
In order to assess the finite sample performance of our different parameter reduction techniquesin terms of Granger non-causality testing behavior, we conduct a Monte Carlo experiment.
19
Unlike Ghysels et al. (2012), we do not start with a common high-frequency DGP, but rather amixed-frequency one. The former implies the assumption that the low-frequency variable hashigh-frequency observations that are missing. Given such a situation it seems natural to castthe model in state space form and estimate the parameters using the Kalman filter. However,this approach leads us to parameter-driven models (Cox et al., 1981), which we intended toavoid when setting up our mixed-frequency VAR. Note that even if we worked with a commonhigh-frequency VAR, there is no reason to believe that the data we deal with is not trulysampled at a much higher frequency, or even continuously, and the data we observe is merelydaily, for example. If we continued this thought experiment, it would imply that we ultimatelyneed to cast the model in continuous state space form.
Alternatively, we can rely on observable data. After all, when researchers go online anddownload a data set, they obtain variables sampled at their various frequencies ignoring anyconsiderations about missing high-frequency observations. Hence, in contrast to parameter-driven models, our mixed-frequency VAR is observation-driven (Ghysels, 2012). This impliesthat the whole model is expressed in terms of observable data, freeing us from potential taskssuch as interpreting the impact of latent shocks in an impulse response analysis. With respectto the latter argument, imagine a change in a policy instrument; This is usually a policy shock,which is, of course, observable.
As far as the Granger non-causality test behavior is concerned, we start by considering thesize of the test. In other words, we first assume that the data are generated as a mixed-frequencywhite noise process. We check for T = 50, 250 and 1000, corresponding to roughly 4, 20 and83 years of monthly data. The reason to consider the latter, rather unrealistic, sample size isto explore whether any size distortions are due to small sample issues. Note that an additional100 monthly observations are used to initialize the process. The mixed-frequency white noiseprocess that is generated looks as follows:
∆lnIPIt
lnBV(20)t
lnBV(20)t−1/20...
lnBV(20)t−19/20
=
µ1µ2µ3...µ21
+
ε1,tε2,tε3,t
...ε21,t
, (20)
where µ1 = −15 and µj = −9.75, j = 2, . . . , 21, which are just the sample means of the datadescribed in section 2.1. As far as the error term is concerned, we assume εt ∼MVN(0(21×1),Σ)and take the sample variance of ∆lnIPI as an estimator of σ1,1, the mean of the sample
variances of lnBV(20)t−i/20 as the one for σi,i, i = 0, . . . , 19 and the mean of all covariances between
lnBV(20)t−j/20 and lnBV
(20)t−i/20, i, j = 0, . . . , 19; i 6= j for σi+2,j+2. Finally, for σ1,i+2, i = 0, . . . , 19,
we consider two cases: (i) No nowcasting causality, i.e., σ1,i+2 = 0, and (ii) nowcasting causality,
where we estimate σ1,i+2 by the mean of the covariances between ∆lnIPI and lnBV(20)t−i/20, i =
20
0, . . . , 19. So, the two versions of the matrix Σ are:
(21)where NC and noNC refer to ”nowcasting causality” and ”no nowcasting causality”, respec-tively. Subsequently, we apply the three parameter reduction techniques described in Sections3.1, 3.2 and 3.3 and test for Granger non-causality. For the method in Section 3.2 T = 500is considered instead T = 1000 due to insufficient computer memory when the data have thelatter amount of observations. Furthermore, we only display the NC-case for Granger causalitytesting, because the results are similar to the noNC-case. The lag length of the estimated VARsis equal to 1 for all approaches under consideration. The figures in the tables below representthe percentage amount of rejections at the 10%, 5% and 1% levels. Subsequently, we also testfor nowcasting causality using the test in equation 7. We do so, however, for the NC- andthe noNC-case in order to investigate size and power of this test of the data are generatedas a mixed-frequency white noise process. All figures are based on 2,500 replications and arecomputed using GAUSS12.
Table 1 contains the results for the unrestricted and the common low-frequency VAR. (Sec-tion in progress...)
Table 2 displays the outcomes for the VARs after reduced rank conditions have been im-posed. Note that the factors are computed using canonical correlations on the one hand, andpartial least squares on the other hand. (Section in progress...)
When testing for Granger causality from IPI to BV in the unrestricted VAR and theVARs obtained after reduced rank conditions have been imposed, we compute a joint test on20p parameters, where p is the lag length. Especially for small T it becomes obvious thatthis fact distorts our size results. In order to address this issue, we recompute the rejectionfrequencies using the Bonferroni correction (Dunn, 1961). The corresponding results for T = 50and T = 250 are summarized in Table 3:
Finally, Table 4 presents the results for the VAR after the high-frequency process hasbeen specified as an ARX(1) process with a possibly non-constant effect of ∆lnIPIt−1 on
lnBV(20)t−i/20, i = 0, . . . , 19. (Section in progress...)
As indicated above, we also test for nowcasting causality using the Wald test in (7) for thevarious approaches. As far as this test is concerned, we consider size and power of the test byallowing the variance-covariance matrix of the error term in the data generating process to beonce ΣNC (power) and once ΣnoNC (size). Table 5 contains the results for the unrestrictedand the low-frequency VARs. (Section in progress...)
Table 6 displays the outcomes for the VARs after reduced rank conditions have been im-posed. Again, the factors are computed using canonical correlations on the one hand, and
21
Table 1: Size of Granger Causality Tests for the low-frequency and the unrestricted VAR
Sample Size Test Statistic Low-frequency VAR Unrestricted VAR Test Direction10% 5% 1% 10% 5% 1%
Note: The figures represent the percentage of rejections of the test statistics in (10) for the usual Wald test andin (11) for the robust test statistic of White at the 10%, 5% and 1% levels. Prior to estimation, the initial VARhas been transformed into a common low-frequency VAR (left part of the table) as discussed in Section 3.3 andhas been left unrestricted (right part of the table) as discussed in Section 3.4. The lag length of the estimatedVARs is equal to 1. The variance-covariance matrix of the error term in the data generation process is equal toΣNC .
partial least squares on the other hand. (Section in progress...)Finally, Table 7 presents the results for the VAR after the high-frequency process has been
specified as an ARX(1) as described before. (Section in progress...)
22
Table 2: Size of Granger Causality Tests after reduced rank conditions (factor by either canon-ical correlations or partial least squares) have been imposed
Sample Size Test Statistic Factor by CC Factor by PLS Test Direction10% 5% 1% 10% 5% 1%
Note: The figures represent the percentage of rejections of the test statistics in (10) for the usual Wald testand in (11) for the robust test statistic of White at the 10%, 5% and 1% levels. Prior to estimation, reducedrank conditions have been imposed on the initial VAR as discussed in Section 3.1. The factors are computedonce using canonical correlations (left part of the table) and once using partial least squares (right part of thetable). The number of factors and the lag length of the estimated VARs are equal to 1 in both cases. Thevariance-covariance matrix of the error term in the data generation process is equal to ΣNC .
Table 3: Size of Granger Causality Tests using the Bonferroni correction for the unrestrictedVAR and after reduced rank conditions (factor by either canonical correlations or partial leastsquares) have been imposed
Sample Size Test Statistic Unrestricted VAR Factor by CC Factor by PLS10% 5% 1% 10% 5% 1% 10% 5% 1%
Note: The figures represent the percentage of rejections of the test statistics in (10) for the usual Wald testand in (11) for the robust test statistic of White at the 10%, 5% and 1% levels using the Bonferroni correction.Prior to estimation, the VAR is either left unrestricted (left part of the table) or reduced rank conditions havebeen imposed on the initial VAR as discussed in Section 3.1. The factors are computed once using canonicalcorrelations (middle part of the table) and once using partial least squares (right part of the table). The numberof factors and the lag length of the estimated VARs are equal to 1 in both cases. The variance-covariance matrixof the error term in the data generation process is equal to ΣNC .
23
Table 4: Size of Granger Causality Tests after specifying the high-frequency process as anARX(1)
Sample Size Test Statistic ARX(1) HF process Test Direction10% 5% 1%
Note: The figures represent the percentage of rejections of the test statistic in (18) for the usual Wald test ina SUR setting at the 10%, 5% and 1% levels. Prior to estimation, the high-frequency process is specified as anARX(1) process with a possibly non-constant effect of ∆lnIPIt−1 on lnBV
(20)
t−i/20, i = 0, . . . , 19 as discussed inSection 3.2. The lag length of the estimated VARs is equal to 1. The variance-covariance matrix of the errorterm in the data generation process is equal to ΣNC .
Table 5: Size of Nowcasting Causality Tests for the low-frequency and the unrestricted VAR
Sample Size Low-frequency VAR Unrestricted VAR Σ10% 5% 1% 10% 5% 1%
Note: The figures represent the percentage of rejections of the test statistic in (7) at the 10%, 5% and 1% levels.Prior to estimation, the initial VAR has been transformed into a common low-frequency VAR (left part of thetable) as discussed in Section 3.3 and has been left unrestricted (right part of the table) as discussed in Section3.4. The lag length of the estimated VARs is equal to 1.
24
Table 6: Size of Nowcasting Causality Tests after reduced rank conditions (factor by eithercanonical correlations or partial least squares) have been imposed
Sample Size Factor by CC Factor by PLS Σ10% 5% 1% 10% 5% 1%
Note: The figures represent the percentage of rejections of the test statistic in (7) at the 10%, 5% and 1% levels.Prior to estimation, reduced rank conditions have been imposed on the initial VAR as discussed in Section 3.1.The factors are computed once using canonical correlations (left part of the table) and once using partial leastsquares (right part of the table). The number of factors and the lag length of the estimated VARs are equal to1 in both cases.
Table 7: Size of Nowcasting Causality Tests after specifying the high-frequency process as anARX(1)
Sample Size ARX(1) HF process Σ10% 5% 1%
T=50 0 0 0no NCT=250 0.032 0.0088 0
T=500 0.07 0.024 0.001
T=50 0 0 0NCT=250 0.9036 0.812 0.5032
T=500 0.999 0.999 0.995
Note: The figures represent the percentage of rejections of the test statistic in (7) at the 10%, 5% and 1% levels.Prior to estimation, the high-frequency process is specified as an ARX(1) process with a possibly non-constant
effect of ∆lnIPIt−1 on lnBV(20)
t−i/20, i = 0, . . . , 19 as discussed in Section 3.2. The lag length of the estimatedVARs is equal to 1.
25
5 Empirical Results
In the section we apply the parameter reduction approaches described in section 3 to themixed-frequency VAR consisting of monthly industrial production index, a measure of businesscycle fluctuations, and daily bipower variation, a measure of volatility. Subsequently, we testfor Granger non-causality in the various cases and if uni-directional causality is found in aninstance, we also test for nowcasting non-causality. We, however, restrict ourselves to theunrestricted VAR, the VARs obtained by imposing reduced rank conditions and by aggregatingthe high-frequency variables, i.e., a common low-frequency VAR. The reason not to considerthe specification of the high-frequency process as an ARX(1) is its poor finite sample Grangercausality testing performance in terms of test size. The figures presented in the tables to comerepresent p-values.
As far as parameter reduction by reduced rank conditions is concerned, canonical correlationtests favor n−s = 2 factors whereas the approach of Cubadda and Hecq (2011) selects n−s = 1.Consequently, we present the results for 1, 2 as well as 3 factors, i.e., f = 1, 2, 3. Informationcriteria lead to p = 1 in the unrestricted VAR (1), we, however, also look at p = 2, 3 and 4. Asbefore, lnBV stands for the logarithm of bipower variation, ∆lnIPI for the growth rate of thebusiness cycle index. Table 8 represents the p-values associated with the Wald test statistic inequation (10), labeled ’Wald’, and the one derived from equation (11), labeled ’White’.
It becomes clear from the figures in Table 8 that Granger non-causality from business cycle
Table 8: P-values of Granger non-causality tests after reduced rank conditions are imposed;factors computed using canonical correlations
p = 1 p = 2 p = 3 p = 4Wald White Wald White Wald White Wald White
Note: The figures represent the p-values associated with the Wald test statistic in equation (10), labeled ’Wald’,and the one derived from equation (11), labeled ’White’. Reduced rank conditions have been imposed on theunrestricted VAR(p), p = 1, 2, 3 or 4 in (1), where the factors have been computed using canonical correlations,consistent with section 3.1. Due to the low number of factors found using canonical correlation tests, the resultsfor 1, 2 and 3 factors are presented.
fluctuations to volatility can be rejected at a significance level of roughly 1% whatever thenumber of lags and the number of transmission mechanisms, except for the combinations p = 1,f = 1 or 2 for the usual Wald test. Slightly less clear cut is the opposite direction (lnBV 9∆lnIPI). We reject the null hypothesis of no causality in almost all cases at 10% signficancelevel (except when p = 2 and f = 1), but the p-values are generally a bit larger than in the case oftesting ∆lnIPI 9 lnBV . However, overall, the results suggest bi-directional Granger causality
26
between business cycle fluctuations and volatility, which renders the search for a causal linkvia nowcasting causality testing pointless, as argued in section 2.3. Still, computing the Waldstatistic in (7) yields a non-rejection of nowcasting non-causality for almost all combinations ofp and f , only for p = 4 and f > 1 do we conclude nowcasting causality at the 5% level. Notethat the results using the Bonferroni correction, although not displayed here for representationalease, clearly point towards bi-directional causality between the two variables (the null hypothesisis rejected for all combinations of p and f at the 1% level).
Table 9 shows the results for Granger non-causality testing when the factors are computedusing partial least squares instead of canonical correlations. As for the latter case, we analyzethe cases p = 2, 3 and 4 and f = 1, 2 and 3. It turns out the results are not much differentfrom the ones for Table 8, i.e., we conclude bi-directional Granger causality between IPI andBV . Again, the results using the Bonferroni correction strongly confirm the bi-directional linkbetween lnIPI and lnBV . As far as nowcasting causality is concerned, we do not reject thenull of no nowcasting causality for any combination of p and f .
Table 10 contains the outcomes for Granger non-causality testing after the initial mixed-
Table 9: P-values of Granger non-causality tests after reduced rank conditions are imposed;factors computed using partial least squares
p = 1 p = 2 p = 3 p = 4Wald White Wald White Wald White Wald White
Note: The figures represent the p-values associated with the Wald test statistic in equation (10), labeled ’Wald’,and the one derived from equation (11), labeled ’White’. Reduced rank conditions have been imposed on theunrestricted VAR(p), p = 1, 2, 3 or 4 in (1), where the factors have been computed using partial least squares,consistent with section 3.1. Due to the low number of factors found using canonical correlation tests, the resultsfor 1, 2 and 3 factors are presented.
frequency VAR in (1) or (2) has been transformed into a common low-frequency VAR as in(19) by temporally aggregating the high-frequency variable lnBV . To be specific, we computethe monthly observation of lnBV in each period t as the average of the corresponding high-
frequency observations in the same month, i.e., lnBV(aggr)t = ln( 1
20
∑19i=0BV
(20)t−i/20). Granger
non-causality tests are than computed using the Wald statistic (10) and its heteroscedasticityconsistent variant obtained by (11) for the specific low-frequency VAR in (19). Clearly, thefigures in Table 10 suggest bi-directional Granger causality, except for the heteroscedasticityconsistent test when p = 3 or 4. Once again, we do not reject nowcasting non-causality for eachp under consideration.
Finally, Table 11 presents the outcomes for testing for Granger non-causality in the unre-
27
Table 10: P-values of Granger non-causality tests in a common low-frequency VAR
p = 1 p = 2 p = 3 p = 4Wald White Wald White Wald White Wald White
Note: The figures represent the p-values associated with the Wald test statistic in equation (10), labeled ’Wald’,and the one derived from equation (11), labeled ’White’, for the common low-frequency VAR in (19). Note thataggregated bipower variation, i.e., BV (aggr) in (19), is computed as the average of the high-frequency observations
for each respective low-frequency period t. To be precise, lnBV(aggr)t = ln( 1
20
∑19i=0BV
(20)
t−i/20). The lag order pis again equal to 1, 2, 3 or 4.
stricted original VAR in (1) or (2), ignoring the fact that the amount of parameters is probablytoo large compared to the sample size given. Again, the Wald statistic in (10) and the onederived from (11) are computed and are labeled similarly to before. The outcomes for the un-
Table 11: P-values of Granger non-causality tests in the unrestricted VAR
p = 1 p = 2 p = 3 p = 4Wald White Wald White Wald White Wald White
Note: The figures represent the p-values associated with the Wald test statistic in equation (10), labeled ’Wald’,and the one derived from equation (11), labeled ’White’, for the unrestricted VAR(p), p = 1, 2, 3 or 4, in (1) or(2).
restricted model differ immensely if we consider the standard Wald test in (10), labeled ’Wald’,or the one derived from (11), labeled ’White’. Based on the latter we clearly reject Grangernon-causality in both directions. Based on the former, however, we only reject Granger non-causality from IPI-growth to lnBV for p > 1 and from volatility to business cycle fluctuationsfor p = 2. Overall, bi-directional Granger causality can be concluded for ’White’, but onlyuni-directional one from IPI-growth to lnBV if looking at ’Wald’. Employing the Bonferronicorrection yields once again similar results for ∆lnIPI 9 lnBV . Also, when computing theWald statistic in (7) we find again a non-rejection of nowcasting non-causality for each laglength considered here.6
6 Conclusion
In this paper we analyzed the link between uncertainty in financial markets and economic fluc-tuations. In particular, we tested for Granger non-causality between the monthly growth rate of
6Precise results for nowcasting causality and the Bonferroni correction are available upon request.
28
Industrial Production Index and the logarithm of daily bipower variation, an observed measureof volatility, in both directions. We investigated Granger non-causality testing in a mixed-frequency VAR, originally proposed by Ghysels (2012) and extended to the non-stationary caseby Gotz et al. (2013), because of the mismatch between the sampling frequencies of the variablesunder consideration.
We adapted the term instantaneous causality (Lutkepohl and Kratzig, 2004) to the mixed-frequency setup naming it nowcasting causality. We showed that, starting from a mixed-frequency VAR, the presence or absence of nowcasting causality has a crucial impact on testingfor Granger causality in standard mixed-frequency regression models. Indeed, by conditioningon the high-frequency variables, the conditional model for the low-frequency model becamethe standard mixed-frequency regression model. We outlined that in the presence of nowcast-ing causality it is possible to obtain ”spurious” Granger causality when solely relying on theconditional model for the low-frequency variable. Also, we emphasized that separate weightfunctions, one for each set of high-frequency variables per low-frequency period, should be es-timated when imposing MIDAS restrictions in order not to ”mix” Granger and nowcastingcausality or introduce ”spurious” nowcasting causality in case of its absence.
We proposed several parameter reduction techniques to avoid parameter proliferation prob-lems that would surely emerge given the size of the mixed-frequency VAR. These approacheswere reduced rank conditions, the imposition of an ARX(1) structure on the high-frequencyvariables and the transformation of the mixed-frequency into a common low-frequency VAR.The performance of these various approaches in terms Granger non-causality testing behaviorwas investigated via Monte Carlo simulations. As far as the size of the test is concerned, partic-ularly good results were found the VAR obtained after reduced rank conditions were imposedand the factors were computed using partial least squares.
Subsequently, these techniques were illustrated for the initial empirical research question,i.e., whether there is a link between business cycle fluctuations and volatility. It turned out thatreduced rank conditions and the common low-frequency VAR lead to bi-directional causalitybetween both variables. In the benchmark case, i.e., when the initial mixed-frequency VARwas estimated unrestrictedly, the same uni-directional Granger causality was found when basedon the original Wald test, but bi-directional Granger causality was detected when the het-eroscedasticity consistent variant of the Wald test was considered. Nowcasting causality wasrejected for each parameter reduction method and each combination of p, the lag length, andf , the number of factors. The use of the Bonferroni correction strongly confirmed the existenceof bi-directional Granger causality between the two variables.
A number of avenues should be pursued in future work. For example, a formal comparison ofthe canonical correlation approach with alternative options, e.g., principal components, wouldbe useful. Furthermore, it should be investigated whether the results obtained in the MonteCarlo experiment can be generalized to different data generating processes, e.g., a commonhigh-frequency DGP. Nevertheless, the approaches presented in this paper indicate that manymore empirical research questions, where the sampling frequencies of the variables involveddiffer considerably and where a link between these variables is of interest, can be addressed in
29
the future. More importantly, these problems can be tackled without the need to temporallyaggregate the high-frequency variable and thereby potentially create non-existing or concealexisting causality patterns.
30
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