‘Déjà vol’ : رگرسیون پیشگویانه برای مجموع نوسانات بازار سهام با استفاده از متغیرهای اقتصاد کلان

Aggregate stock return volatility is both persistent and countercyclical. This paper tests whether it is possible to improve volatility forecasts at monthly and quarterly horizons by conditioning on additional macroeconomic variables. I find that several variables related to macroeconomic uncertainty, time-varying expected stock returns, and credit conditions Granger cause volatility. It is more difficult to find evidence that forecasts exploiting macroeconomic variables outperform a univariate benchmark out-of-sample. The most successful approaches involve simple combinations of individual forecasts. Predictive power associated with macroeconomic variables appears to concentrate around the onset of recessions.

What drives secular variation in stock return volatility? In a seminal paper, Schwert (1989) considers several potential explanations, including the possibility that volatility fluctuates with the level of economic activity. Although Schwert (1989) finds only limited support for links between volatility and macroeconomic activity, subsequent papers report more encouraging evidence. This large body of literature is difficult to digest, as different studies examine different forecasting variables and apply different econometric approaches.1 Understanding the robustness and magnitude of links between macroeconomic variables and volatility represents an important empirical question in finance. From a risk-management perspective, understanding how future aggregate stock market volatility responds to changing macroeconomic conditions is critical for stress-testing and computing value-at-risk over longer horizons. From an asset allocation perspective, quantities that forecast volatility become state variables in investors' portfolio decisions. Finally, characterizing the extent and pattern of time series variation in volatility is important for determining the appropriate stylized facts against which asset pricing models should be evaluated. Consistent with Schwert (1989) and other existing research, I find that stock return volatility behaves countercyclically. Empirical measures of business conditions, such as the growth rate of gross domestic product (GDP), co-move closely with sign-inverted measures of stock return volatility. From a forecasting perspective, the strong contemporaneous relation between volatility and business conditions implies that lagged volatility provides an efficient indicator of the economic state. Whether it is possible to improve forecast performance by conditioning on additional macroeconomic and financial variables is unclear. To be successful, such variables must capture information beyond that already contained in lagged volatility. This paper provides a broad assessment of the ability of macroeconomic and financial variables to improve volatility forecasts at monthly and quarterly horizons. Recent literature identifies several channels that could drive time variation in volatility. These include time-varying volatility in shocks to fundamentals (e.g., Bansal and Yaron, 2004), nonlinear relations between time-varying expected returns and the business cycle (Mele, 2007), learning effects related to investors' uncertainty about fundamentals (e.g., Veronesi, 1999), and amplification of shocks to asset markets via financial intermediation (Brunnermeier and Pedersen, 2009). This body of theoretical work motivates the set of forecasting variables considered in the paper. These include a measure of corporate payout, several interest rate and return spreads, a measure of changes in bank leverage, measures of current and expected economic growth, a direct proxy for time-varying expected returns, volatilities for two key macroeconomic series, and two ratios for the aggregate economy: consumption to wealth and investment to capital stock. I emphasize an out-of-sample econometric approach, although in-sample results appear for reference and comparison. This focus parallels the recent emphasis on out-of-sample inference in the literature on stock return predictability, where an active debate continues regarding the extent to which returns are predictable.2 The paper distinguishes between two alternative notions of out-of-sample forecast improvement. The first focuses on properties of the data generating process: Do macroeconomic variables Granger cause volatility, such that volatility depends upon these variables conditional on past volatility? The second interpretation adopts a normative stance: Do volatility models that incorporate macroeconomic variables improve the accuracy of out-of-sample forecasts? To see that these alternative interpretations are distinct, suppose that the conditional volatility of stock returns depends on some macroeconomic variable, so that this variable Granger causes volatility. Out-of-sample forecasts exploiting the variable could nevertheless under-perform forecasts based on a (misspecified) model that omits it. This is because there is a bias-variance trade-off at play. The conditional bias reduction afforded by including the macroeconomic predictor might not offset increased forecast variance related to parameter estimation. I conduct two econometric tests comparing the out-of-sample forecasting performance of a benchmark model with a model augmented with one or more of the predictor variables. Both tests involve the out-of-sample difference in mean square prediction error (MSPE) relative to the benchmark. The first test, proposed by Giacomini and White (2006), is equivalent to the Diebold and Mariano (1995) test for equal predictive ability. The second test, proposed by Clark and West (2007), adds an adjustment term to the out-of-sample difference in MSPE that accounts for parameter estimation noise. The key difference between the two testing frameworks lies in the specification of the null hypothesis. In the Clark and West (2007) framework, the null hypothesis involves the population difference in MSPE between the two nested models. By contrast, the null hypothesis in the Giacomini and White (2006) framework relates to the forecasting method and explicitly incorporates parameter estimation as a source of forecast error. The Clark and West (2007) test is appropriate when the underlying research question involves Granger causality, whereas the Giacomini and White (2006) test is appropriate for addressing the normatively oriented question of whether one forecast performs better than the other. The empirical evidence from in-sample forecasting regressions is encouraging. Several variables appear to Granger cause volatility, including the commercial paper-to-Treasury spread, the default spread, a bond return spread, and the ratio of investment to capital in the aggregate economy. The null of no predictability is also rejected for a kitchen sink specification that includes the full set of predictors. Although the statistical evidence for Granger causality is compelling, the economic significance of the predictive power afforded by these variables is relatively small. These findings are robust to several alternative sample periods, with the strength of evidence for predictability being strongest between the 1950s and early 1980s. Out-of-sample evidence regarding Granger causality largely confirms the results from in-sample predictive regressions. The Clark and West (2007) test for Granger causality implicates essentially the same variables that are significant based on in-sample regressions. The null of no Granger causality is also typically rejected for the kitchen sink model. Results are somewhat sensitive; however, to inclusion of the economically volatile 1970s in the out-of-sample evaluation period. Specifically, the out-of-sample evidence for Granger causality is weaker over the period 1982–2010 relative to the equally long period 1972–2000 that includes the 1970s. The evidence for superior predictive ability (in the Giacomini and White sense) is mixed. Taken one at a time, the individual predictors fail to generate statistically significant improvements in forecast accuracy relative to the benchmark. The heavily parameterized kitchen sink model also fairs poorly. In some cases, this model significantly under-performs the benchmark. Not all of the evidence is negative, though. Combinations of the underlying univariate forecasts often statistically outperform the benchmark. A simple equal-weighted combination of the univariate forecasts often succeeds. These findings are consistent with Rapach, Strauss, and Zhou (2010), who consider similar combination schemes in the context of stock return forecasts. There are two caveats. First, the associated out-of-sample R2 improvements relative to the benchmark are relatively small. Second, evidence of superior performance is weaker when the out-of-sample evaluation period does not include the 1970s. The pattern of results highlights the distinction between the two types of out-of-sample tests applied in the paper. In many instances, the Clark and West test rejects, while the Giacomini and White test fails to reject, or vice versa. Results for the kitchen sink model provide a useful illustration. Under the null hypothesis of no Granger causality, this model is expected to substantially under-perform the benchmark, because it includes a large number of spurious variables. Despite underwhelming out-of-sample R2 values, the kitchen sink model often performs better than expected, and consequently the Clark and West test rejects the null. To expand upon this point, I develop a simple Monte Carlo experiment. Simulated data are calibrated to match the key empirical features of return volatility and a typical macroeconomic predictor. In one case, the simulated variable Granger causes volatility, while a second case imposes the null of no Granger causality. For empirically realistic sample sizes, the simulation results illustrate a pattern of divergence between the Giacomini and White (2006) and Clark and West (2007) tests consistent with that observed in the data. Weak forecasting results during the Great Moderation suggest connections between the business cycle and relative forecast performance. Time series plots of cumulative forecast performance relative to the benchmark display a striking pattern for the more successful predictors: Forecast improvements appear to concentrate around the onset of recessions. Relative forecast performance, therefore, is countercyclical.3 Unfortunately, the linkage between forecast performance and recessions is complex, as different variables exhibit different patterns of forecast performance around recessions. By exploiting information in multiple predictors, the combined forecasts generate a more consistent pattern of forecast improvements. The main findings survive a wide array of robustness checks. These include using alternative empirical proxies for stock return volatility, using alternative dynamic specifications for the benchmark forecasts, forecasting the level as opposed to the log of stock return volatility, and employing a recursive versus rolling estimation scheme. Stock return volatility is persistent, while stock returns are not. Notwithstanding this important difference, there are striking similarities between results established here and findings reported in the extensive literature on forecasting stock returns. Specifically, this paper shows that, from an ex post perspective, certain macroeconomic and financial variables help explain time variation in stock return volatility. At the same time, the additional predictive power afforded by these variables is small, and forecasting ability appears to concentrate during a subperiod from the 1950s through the early 1980s. Similar statements well-characterize recent findings in the equity premium literature.4 Viewed from this perspective, the results reported here conjure a distinct sense of déjà vu. This paper is among several recent studies that embody a resurgence of interest in connections between macroeconomic conditions and stock return volatility. Related work includes Adrian and Shin (2010), David and Veronesi (2009), Engle, Ghysels, and Sohn (2008), and Ludvigson and Ng (2007). Relative to these studies, the present paper focuses on applying recently developed econometric techniques to evaluate the extent to which macroeconomic and financial variables improve volatility forecasts out-of-sample. While the papers adopt different perspectives, important commonalities exist. The remainder of the paper proceeds as follows. Section 2 discusses the statistical properties of stock return volatility and analyzes connections between stock return volatility and the level of economic activity. Section 3 discusses theoretical explanations for time variation in stock return volatility and describes the set of forecasting variables used in the analysis. 4 and 5 present empirical results, the former from an in-sample perspective and the latter using an out-of-sample research design. Section 6 concludes.