For the time being, we consider one-sided filters. In Section 1.4.1, we first confirm the counter-cyclical behavior of long-term volatility for a broad set of macro variables. We then focus on the lead-lag-structure between the macro variables and stock market volatility and identify variables that require flexible unrestricted filters and, hence, lead long-term volatility. In Section 1.4.1, we analyze the question whether macro variables still contain predictive information on long-term volatility once one controls for lagged realized volatility.
The lead-lag-structure between macro variables and volatility
Estimation results for the parameters of the long-term volatility component of the various one-sided GARCH-MIDAS-X models are summarized in Table 1.2. An ex- tended version of the table containing all parameter estimates can be found in the Appendix. For each macro variable, the first/second line presents the estimates for the restricted/unrestricted weighting scheme. We choose K = 12 for all variables
which corresponds to three MIDAS lag years.5 To ensure comparability across the
one- and two-sided models (see Section 1.4.2), as well as models based on macroeco- nomic uncertainty measures (see Section 1.4.4), all models are estimated based on daily return data for the 1973Q1 to 2010Q4 period and quarterly macro data from
4There are a few missing observations of the four-quarters-ahead forecasts at the beginning of
the sample. Analogously to Eq. (1.8), we estimate Xt+4|tSP F = P4
i=1δiDit+P 4
i=1φiXt+4−i|tSP F + ξt
using the available data and replace the missing observations by the predictions ˆXt+4|tSP F.
5As long as the selected K is large enough, we find the estimation results to be robust with
1970Q1 onwards.6 The table also reports the estimates for the GARCH-MIDAS-RV as well as the one-component GARCH(1,1) model.
First, we note that the estimates of the GARCH parameters (µ, α, β, γ) are sig- nificant at the 1% level in all cases (see the Appendix). The estimates of α and β take the typical values and, consistent with the leverage effect, the estimate of the parameter γ is found to be positive.
Next, we have a closer look at the estimates of the long-run component τt. For all
variables except the GDP deflator the estimated θ is highly significant and has the expected sign. For example, for real GDP, the estimated θ is negative, meaning that an increase in the growth rate is associated with a decline in long-term volatility. Conversely, the positive θ for the unemployment rate indicates that a rise in unem- ployment is associated with higher long-term volatility. That is to say, in all cases the sign of the scale parameter confirms the counter-cyclical property of long-run volatility as observed in Engle and Rangel (2008) and Engle et al. (2013).
In Fig. 1.2 we plot the estimated restricted and unrestricted weighting schemes for the different macro variables. For six out of the eleven macro variables, e.g. the unemployment rate or the NAI, both schemes are declining from the beginning with almost identical shapes. As one would expect, for these variables a LRT (see
Table 1.2) does not reject the constraint (ω1 = 1) imposed by the restricted scheme.
In sharp contrast, for housing starts, the GDP deflator, consumer sentiments, real consumption, and the term spread the unrestricted schemes are hump-shaped and clearly different from the restricted ones. For these variables, the restricted scheme appears to be clearly misspecified. For example, for the term spread the unrestricted filter takes its maximum weight at a lag of five quarters, while the restricted scheme is characterized by an almost linear decay. Only in case of the GDP deflator, we find an extreme and somewhat unreasonable weighting scheme, putting almost all weight on the fifth lag. In line with these considerations, the LRT (see Table 1.2)
rejects the constraint that ω1 = 1 for these five variables.7 Accordingly, we classify
all variables that are characterized by hump-shaped weights as leading with respect to long-term volatility. Finally, note that the optimal weighting scheme for the GARCH-MIDAS-RV model is the restricted one.
6The first year of macro data (1969Q1-Q4) is used to construct ex-post macro volatility mea-
sures based on the AR(4) model in Eq. (1.8).
7Although the LRT rejects the restricted weighting scheme for the GDP deflator, the estimate of
θ is only marginally significant in the unrestricted filter. That is, the GDP deflator hardly explains any time variation in the conditional variance of the S&P 500 returns. Hence, all subsequent results with respect to the GDP deflator should be taken with a grain of salt.
1.4.1 One-sided filters 25
The finding that some variables are leading with respect to stock market volatility while others are not is economically plausible. Variables such as industrial produc- tion (the unemployment rate) are typically considered as coincident (lagging) indica- tors for the business cycle. For these variables the most recent observations appear to matter most for predicting the counter-cyclical long-term volatility. In contrast,
the term spread or housing starts are usually considered as leading indicators.8 For
example, Estrella and Hardouvelis (1991), Estrella and Mishkin (1998) and Ang et al. (2006), among others, provide evidence that the term spread is a powerful pre- dictor of future economic activity and recessions. The predictive ability of the term spread is typically explained by the term spread’s relation to investors expectations about future economic activity, demand for credit and monetary policy (see, e.g., Estrella and Trubin, 2006). Similarly, Leamer (2007) and Kydland et al. (2012) show that housing starts lead real GDP. According to Kydland et al. (2012), the leading property of housing starts can be rationalized by the empirical observation of low interest rates for mortgages that precede economic upturns. Our results sug- gest that variables which lead the business cycle are also leading with respect to financial volatility and, therefore, require unrestricted weighting schemes.
Fig. 1.3 shows the quarterly aggregated long-term component, pN(t)τX
t , and
the quarterly conditional volatility, pτX
t gtX with gXt =
PN(t)
i=1 g
X
i,t, for all GARCH-
MIDAS-X models along with the realized volatility,√RVt. The figure clearly shows
the negative relation between √RVt and economic activity. The long-term compo-
nents of all macro variables, except the GDP deflator, mirror this counter-cyclical
pattern of stock market volatility.9 Nevertheless, there are also distinct differences.
While the long-term component of the term spread typically increases in advance of a recession, the long-term components of other variables, e.g. real GDP, seem
to increase during recessions. Finally, the long-run volatility component of the
GARCH-MIDAS-RV model is dominated by the 1987 stock market crash and the recent financial crisis. It hardly increases during the other recession periods.
Next, we compare the fit of the various models by means of the Bayesian informa- tion criterion (BIC). According to the BIC, the GARCH-MIDAS-X models based on housing starts, corporate profits, the NAI, new orders, and the term spread are
8Our classification of leading vs. coincident/lagging variables is in line with the fact that the
yield spread, housing permits, and also consumer expectations are included in the Conference Board’s leading economic index for the US, while industrial production is included in the coincident index.
9In line with the only weakly significant estimate of θ for the GDP deflator, the corresponding
preferred to the GARCH-MIDAS-RV (and to the nested GARCH specification). From an economic point of view, it is important to know how much of the variation in the expected quarterly variance of a specific GARCH-MIDAS-X model can be attributed to the variation in the corresponding macro variable (see Engle et al., 2013, p.794). In order to answer this question, we provide the value of a variance ratio (VR) statistic for each model. In general, we let the VR be defined as the fraction of the sample variance of the log of total quarterly conditional volatility,
d
Var(log(τX
t gXt )), that can be explained by the sample variance of the log long-
term component, dVar(log(τX
t )). For easier comparison across the various GARCH-
MIDAS-X models, we report
VR(X) = Var(log(τd X t )) d Var(log(τRV t gRVt )) , (1.9)
which relates the sample variance of the log of the long-term component of a specific GARCH-MIDAS-X model to the sample variance of the log of the total expected
variance of the baseline GARCH-MIDAS-RV model with restricted filter.10 It is
important to note that a small VR does not necessarily imply a poor model fit,
since a low dVar(log(τtX)) can also be an indication of smooth movements in the
underlying macro variable. However, in the extreme case, where dVar(log(τX
t )) ≈ 0,
the long-term component is constant and the GARCH-MIDAS-X reduces to the
simple GARCH model. Since τX
t dominates the multi-day/period-ahead volatility
forecast (see Eq. (1.7)), it is clear that only GARCH-MIDAS-X specifications with high VRs have the potential to outperform the simple GARCH model.
As Table 1.2 shows, the model based on housing starts (unrestricted weighting scheme) achieves the highest VR. Roughly 22% of the variation in expected quar- terly volatility is explained by housing starts. The specifications including new orders and the term spread (unrestricted weighting scheme) rank second and third. Most importantly, the models based on these variables achieve higher VRs than the benchmark GARCH-MIDAS-RV model. Interestingly, these are also the three models with the lowest BIC. As expected, the VR for the model based on the GDP deflator is by far the lowest.
10Although using dVar(log(τRV t g RV t )) instead of dVar(log(τ X t g X
t )) in the denominator does sim-