《金融经济学原理》课程教学资源（文献资料）When uncertainty and volatility are disconnected - Implications for asset pricing and portfolio performance.pdf_P6-P10

Journal of Econometrics xxx (xxxx) xxx 6 Y. Aït-Sahalia et al. 𝑑𝑣𝑡 = 𝜇𝑣,𝑡𝑑𝑡 + 𝜎𝑣,𝑡 (√ 1 − 𝜌 2 𝑆,𝑣𝑑𝑊 𝑣 𝑡 + 𝜌𝑆,𝑣𝑑𝑊 𝑆 𝑡 ) , 𝑣0 > 0, (4) where 𝑊 𝐷 𝑡 , 𝑊 𝑆 𝑡 and 𝑊 𝑣 𝑡 are Brownian motions, 𝜇𝐷 and 𝜎𝐷 are the drift and volatility of dividend growth, 𝑟𝑓,𝑡 is the risk-free rate, 𝜇𝑆,𝑡 is the expected total return of the risky asset and 𝜎𝑆 is a constant scaling stock return volatility.13 The stochastic volatility of stock return and dividends growth, 𝑣𝑡 , is a general stochastic process with drift 𝜇𝑣,𝑡 and diffusion 𝜎𝑣,𝑡 under the reference probability measure.14 The term 𝜌𝑆,𝑣 captures the correlation between the stock return and its volatility. In order to formally introduce model uncertainty, we define by P 𝜗 the robust probability measure, where 𝜗𝑡 is the change from the reference measure P to the robust measure P 𝜗 . This implies that a robust investor considers alternative models of the form:15 𝑑𝑆𝑡 𝑆𝑡 = ( 𝜇𝑆,𝑡 − 𝐷𝑡 𝑆𝑡 + 𝜎𝑆𝑣𝑡ℎ𝑡 ) 𝑑𝑡 + 𝜎𝑆𝑣𝑡𝑑𝑊 𝑆,𝜗 𝑡 , (5) where 𝑊 𝑆,𝜗 𝑡 is also a Brownian motion, but now under the robust probability measure P 𝜗 . Importantly, the drift of the stock return is now perturbed by the new term 𝜎𝑆𝑣𝑡ℎ𝑡 , driven by both stochastic volatility 𝑣𝑡 and the drift perturbation function ℎ𝑡 which will be optimally chosen by the representative agent.16 As can be seen from Eq. (5), the higher the perturbation function ℎ𝑡 becomes, the larger becomes the drift distortion of the risky asset and hence, the higher is the investor’s demand for robustness against potential model miss-specification. Conversely, if ℎ𝑡 = 0, then the representative investor has full confidence in his or her reference model. 3.2. The size of the alternative set of models and the role of disconnect While a robust investor considers a variety of alternative models, not all of them are plausible, i.e. they may be too distinct to be considered as reasonably close to the reference model. To discipline the size of the alternative model set, we make use of relative entropy which is a convenient measure of the distance between the reference model and the alternative models. The growth in entropy of P 𝜗 relative to P over the time interval [𝑡, 𝑡 + 𝛥𝑡] is defined as 𝐺(𝑡, 𝑡 + 𝛥𝑡) ≡ E 𝜗 𝑡 [ log ( 𝜗𝑡+𝛥𝑡 𝜗𝑡 )] , (8) and the instantaneous growth rate of relative entropy at time 𝑡 is then given by (𝜗𝑡 ) ≡ lim 𝛥𝑡→0 𝐺(𝑡, 𝑡 + 𝛥𝑡) 𝛥𝑡 = 1 2 ℎ 2 𝑡 , (9) where the last equality is proven in Appendix A.1. When ℎ𝑡 = 0, the relative entropy growth rate is zero, which implies that the two probability measures are equivalent. As ℎ𝑡 increases, so does the distance between the reference model and the alternative models. Next, we restrict the set of alternative models under consideration by the investor in the form of an upper bound on this distance. We assume that set of admissible alternative models is bounded from above as follows { 𝜗𝑡 ∶ (𝜗𝑡 ) ≤ 𝜖 2 2  2 𝑡 } , (10) where 𝑡 denotes the stochastic model uncertainty and the constant parameter 𝜖 measures the investor’s degree of uncertainty aversion. When 𝜖 = 0 the set of alternative models is empty and the investor has full confidence in the reference model. By contrast, an investor with higher 𝜖 expands the set of alternative models to include models that are statistically further away from the reference model. Finally, in order to formally introduce disconnect, we posit the following functional form for modeling uncertainty 𝑡 = 𝜂𝑡𝑣𝑡 . (11) 13 We will show, that in equilibrium, the stock price is proportional to dividends, so that 𝑊 𝐷 𝑡 = 𝑊 𝑆 𝑡 , ∀𝑡 ≥ 0. Therefore, instead of specifying the correlation between dividends 𝐷𝑡 and our state variables in the model, we specify the correlations with respect to the stock price process directly. 14 Given logarithmic preferences, the solution is independent of the specific choice of a drift and diffusion process for volatility. The investor is myopic with regards to the dynamics of volatility: he or she cares only about the current value of the state variables. However, the drift and volatility of volatility, 𝜇𝑣,𝑡 and 𝜎𝑡,𝑣, are not fully unrestricted since we require the volatility process to remain stationary and positive. Specific restrictions to be imposed are model-dependent. As an example, Feller’s square-root process in which the drift term is linearly mean reverting, i.e. 𝜇𝑣,𝑡 ∶= 𝜅 ( 𝜃 − 𝑣𝑡 ) with 𝜃 > 0 and 𝜅 > 0 and the volatility is given by 𝜎𝑣,𝑡 ∶= 𝜎 √ 𝑣𝑡 , 𝜎 > 0 requires that the parameters satisfy 𝜅𝜃 > 𝜎 2 2 for the process 𝑣𝑡 to be precluded from reaching zero. 15 A similar perturbed equation for the dividend process is omitted for brevity. Since it turns out that the stock price will be proportional to dividends in equilibrium, adding that equation here is not necessary. 16 This result follows immediately from an applications of Girsanov’s theorem which states that for any 𝑡 -measurable random variable 𝑍 and 𝑇 > 𝑡 we can write E 𝜗 [ 𝑍𝑇 | | 𝑡 ] = E [ 𝜗𝑇 𝜗𝑡 𝑍𝑇 | | | | 𝑡 ] . (6) so that the Brownian Motions are related by 𝑑𝑊 𝑆 𝑡 = 𝑑𝑊 𝑆,𝜗 𝑡 + ℎ𝑡𝑑𝑡 where 𝜗𝑡 is an exponential martingale, with dynamics 𝑑𝜗𝑡 𝜗𝑡 = ℎ𝑡𝑑𝑊 𝑆 𝑡 . (7)

Journal of Econometrics xxx (xxxx) xxx 7 Y. Aït-Sahalia et al. where we refer to 𝜂𝑡 as our stochastic disconnect process.17 This specification is motivated by the empirical relation between volatility and uncertainty observed in Fig. 1, showing that while volatility and uncertainty are in general positively correlated, there are situations in which one of them is significantly higher than the other, i.e., they are disconnected. The stochastic process 𝜂𝑡 measures the degree of disconnect, is positive and normalized to have mean one. This normalization is for ease of interpretation of the level of disconnect relative to its mean: when 𝜂𝑡 is far away from one (either above or below) there is high disconnect between uncertainty and volatility, and when 𝜂𝑡 is close to one disconnect is low. Rewriting Eq. (11) we have 𝜂𝑡 = 𝑡 𝑣𝑡 , (12) so high levels of 𝜂𝑡 capture situations in which the economy is in the ‘‘HL’’ regime, characterized by high uncertainty and low volatility, while low levels of 𝜂𝑡 are observed when the economy is in the ‘‘LH’’ regime, characterized by low uncertainty and high volatility. By contrast, in the two connected regimes ‘‘HH’’ and ‘‘LL’’, 𝜂𝑡 tends to be away from extreme values, and closer to its normalized mean value set to 1. Of course, the variables in Eq. (12) are continuous so the notion (and granularity) of any set of discrete regimes is simply a convenient low-dimensional representation of the investment environment that is useful for interpretation and aggregation, but plays no formal role in the analysis of the model. To fully specify the model, we assume that 𝜂𝑡 is a positive process that, like 𝑣𝑡 , is a general stochastic process with drift 𝜇𝑣,𝑡 and diffusion 𝜎𝑣,𝑡 under the reference probability measure.18 𝑑𝜂𝑡 = 𝜇𝜂,𝑡𝑑𝑡 + 𝜎𝜂,𝑡 (√ 1 − 𝜌 2 𝑆,𝜂𝑑𝑊 𝜂 𝑡 + 𝜌𝑆,𝜂𝑑𝑊 𝑆 𝑡 ) , 𝜂0 > 0. (13) The term 𝜌𝑆,𝜂 captures the correlation between the disconnect and stock price process. Furthermore, since volatility is correlated with the stock price, it follows that disconnect and the volatility are also correlated with correlation parameter equal to the product of 𝜌𝑆,𝜂 and 𝜌𝑆,𝑣. 4. Optimal portfolio allocation in partial equilibrium To solve the investor’s objective problem, we employ dynamic programming. We solve the resulting robust Hamilton–Jacobi– Bellman (HJB) equation under inequality constraints using the Lagrangian method. We then derive the optimal robust solution to the investor’s investment and consumption problem in closed form. Let 𝑋𝑡 denote the investor’s wealth and 𝜔𝑡 be the percentage of wealth (or portfolio weight) invested in the risky asset; with 1 − 𝜔𝑡 is invested in the risk-free asset. The investor consumes at an instantaneous rate 𝐶𝑡 and assumes the risky asset evolves according to the dynamics specified in Eq. (5). Accordingly, defining 𝜇 ℎ 𝑆,𝑡 ∶= 𝜇𝑆,𝑡 + 𝜎𝑆𝑣𝑡ℎ𝑡 , the dynamics of investor’s wealth 𝑋𝑡 follow 𝑑𝑋𝑡 = 𝜔𝑡𝑋𝑡 ( 𝑑𝑆𝑡 + 𝐷𝑡𝑑𝑡 𝑆𝑡 ) + ( 1 − 𝜔𝑡 ) 𝑋𝑡 𝑑𝐵𝑡 𝐵𝑡 − 𝐶𝑡𝑑𝑡 = ( 𝑋𝑡 [ 𝑟𝑓,𝑡 + 𝜔𝑡 ( 𝜇 ℎ 𝑆,𝑡 − 𝑟𝑓,𝑡)] − 𝐶𝑡 ) 𝑑𝑡 + 𝜔𝑡𝑋𝑡𝜎𝑆𝑣𝑡𝑑𝑊 𝑆,𝜗 𝑡 , (14) starting from an initial endowment 𝑋0 > 0. Under the robust probability measure P 𝜗 , the investor derives logarithmic utility from consumption with subjective discount rate 𝛽 > 0, and solves the infinite horizon problem19: sup {𝐶𝑠 ,𝜔𝑠 }𝑡≤𝑠<∞ inf {ℎ𝑠 }𝑡≤𝑠<∞ E 𝜗 𝑡 [ ∫ ∞ 𝑡 𝑒 −𝛽𝑠 log(𝐶𝑠 )𝑑𝑠] , (15) subject to the entropy growth constraint in Eq. (10) and the wealth dynamics in Eq. (14). In partial equilibrium, the investor solves this problem taking the dynamics of asset prices as given. The investor’s desire for robustness against model uncertainty is incorporated by evaluating the future evolution of the economy under the worst-case alternative model within the admissible set specified in Eq. (10). In order to solve the investor’s optimization problem, we make use of standard robust dynamic programming techniques. To this end, we define the value function 𝑉 (𝑡, 𝑋𝑡 , 𝑣𝑡 , 𝜂𝑡 ) = sup {𝐶𝑠 ,𝜔𝑠 }𝑡≤𝑠<∞ inf {ℎ𝑠 }𝑡≤𝑠<∞ E 𝜗 𝑡 [ ∫ ∞ 𝑡 𝑒 −𝛽𝑠 log(𝐶𝑠 )𝑑𝑠] , (16) 17 This approach to modeling a disconnect between uncertainty and volatility is not the only one possible. Other specifications, such as, for instance, an additive formulation can also be implemented and allow for explicit solutions within our framework provided that the investor has logarithmic utility. However, this simple multiplicative definition of uncertainty has two advantages: First, since disconnect is unobserved, it can only be identified once we have a proxy for volatility and uncertainty. By defining uncertainty 𝑡 as the product of both volatility and disconnect, we have a simple and a consistent (within our model) way of extracting an empirical measure for what we defined to be the disconnect process 𝜂𝑡 . Second, by imposing that 𝜂𝑡 is a strictly positive process, we do not have to be concerned with, for instance in the case that uncertainty decomposes additive into volatility and disconnect, how to interpret negative disconnect 𝜂𝑡 , and further technical issues as to whether the robust utility maximization problem is still well defined in the case when 𝜂𝑡 is allowed to change signs. 18 As was the case with volatility, the equilibrium is independent of the specific choice of the drift and diffusion for the disconnect process in the case of a logarithmic investor, who is myopic with respect to their specification. However, for the disconnect process to remain stationary and positive, just as in the case for the volatility process in Eq. (4), functional form and/or parameter restrictions on its drift and diffusive function are imposed. 19 Appendix A.4 derives the equilibrium when the investor has CRRA preferences using the martingale approach

Journal of Econometrics xxx (xxxx) xxx 8 Y. Aït-Sahalia et al. associated with the optimal stochastic robust control problem in Eq. (15). Then, as above, we define 𝜇 ℎ 𝑣,𝑡 ∶= 𝜇𝑣,𝑡 + 𝜌𝑆,𝑣𝜎𝑣,𝑡ℎ𝑡 and similarly 𝜇 ℎ 𝜂,𝑡 ∶= 𝜇𝜂,𝑡 + 𝜌𝑆,𝜂𝜎𝜂,𝑡ℎ𝑡 , the perturbed Hamilton–Jacobi–Bellman (HJB) equation characterizing the optimal solution is 0 = sup {𝐶𝑡 ,𝜔𝑡 } inf {ℎ𝑡 } { 𝑒 −𝛽𝑡 log(𝐶𝑡 ) + 𝜕𝑉 𝜕𝑡 + 𝜕𝑉 𝜕𝑋 ( 𝑋𝑡 [ 𝑟𝑓,𝑡 + 𝜔𝑡 ( 𝜇 ℎ 𝑆,𝑡 − 𝑟𝑓,𝑡)] − 𝐶𝑡 ) + 1 2 𝜕 2𝑉 𝜕𝑋2 𝜔 2 𝑡 𝑋 2 𝑡 𝜎 2 𝑆 𝑣 2 𝑡 + 𝜕𝑉 𝜕𝑣 𝜇 ℎ 𝑣,𝑡 + 1 2 𝜕 2𝑉 𝜕𝑣2 𝜎 2 𝑣,𝑡 + 𝜕𝑉 𝜕𝜂 𝜇 ℎ 𝜂,𝑡 + 1 2 𝜕 2𝑉 𝜕𝜂2 𝜎 2 𝜂,𝑡 (17) + 𝜕 2𝑉 𝜕𝑋𝜕𝑣 𝜔𝑡𝑋𝑡𝜎𝑆𝑣𝑡𝜌𝑆,𝑣𝜎𝑣,𝑡 + 𝜕 2𝑉 𝜕𝑋𝜕𝜂 𝜔𝑡𝑋𝑡𝜎𝑆𝑣𝑡𝜌𝑆,𝜂𝜎𝜂,𝑡 + 𝜕 2𝑉 𝜕𝜂𝜕𝑣 𝜌𝑆,𝜂𝜌𝑆,𝑣𝜎𝑣,𝑡𝜎𝜂,𝑡} , subject to the relative entropy growth constraint20 (𝜗𝑡 ) = ℎ 2 𝑡 2 ≤ 𝜖 2 2  2 𝑡 . (18) The robust optimal control problem is solved in two steps. First, the investor solves the inner optimization problem, deriving the optimal worst-case drift perturbation ℎ ∗ 𝑡 . Second, the investor solves the outer problem, selecting the optimal consumption and portfolio holdings that maximize his or her expected utility of consumption under the worst-case model. The solution for the optimal robust policies is characterized in the following proposition. Proposition 1. The optimal consumption, drift perturbation, and portfolio policy are given, respectively, by Consumption ∶ 𝐶 ∗ 𝑡 = 𝛽𝑋𝑡 . (19) Perturbation ∶ ℎ ∗ 𝑡 = −𝜖𝑡 = −𝜖𝜂𝑡𝑣𝑡 . (20) Portfolio weight ∶ 𝜔 ∗ 𝑡 = 𝜇𝑆,𝑡 − 𝑟𝑓,𝑡 𝜎 2 𝑆 𝑣 2 𝑡 ⏟⏞⏞⏞⏟⏞⏞⏞⏟ reference demand − 𝜖 𝜎𝑆 𝜂𝑡 ⏟⏟⏟ . uncertainty correction (21) The investor’s optimal consumption is a constant fraction of wealth equal to the subjective discount rate 𝛽, which is a standard result with logarithmic utility. The optimal drift perturbation ℎ ∗ 𝑡 is negative, and driven by the product of uncertainty aversion and model uncertainty.21 Because model uncertainty is the product of stochastic disconnect and volatility, for a given level of uncertainty aversion, the drift adjustment can be high even when volatility is low, if there is high disconnect (corresponding to the HL regime). It is negative and increasing in magnitude in the investor’s degree of uncertainty aversion and in the level of disconnect. Finally, as Eq. (21) shows, the optimal portfolio holdings decompose into a standard myopic term equal to the Sharpe ratio, plus an uncertainty correction term. The correction term is largest when 𝜂𝑡 is largest, that is, in the HL regime where uncertainty is significantly higher than volatility, and lowest when 𝜂𝑡 is lowest, i.e., in the LH regime characterized by uncertainty being significantly lower than volatility.22 Conversely, in the case where 𝜂𝑡 is low, which implies high disconnect as volatility is high relative to uncertainty, the uncertainty correction term 𝜖𝜂𝑡∕𝜎𝑆 is small, but so is the portfolio holdings of the robust investor because the myopic term (𝜇𝑆,𝑡 − 𝑟𝑓,𝑡)∕𝜎 2 𝑆 𝑣 2 𝑡 shrinks as well. 5. Equilibrium asset prices Given the optimal demand for the assets expressed by the investor, we now solve for the equilibrium asset prices. An equilibrium is a specification of the dynamics of the risky and risk-less asset prices, including 𝜇𝑆,𝑡 = 𝜇𝑆 ( 𝑣𝑡 , 𝜂𝑡 ) and 𝑟𝑓,𝑡 = 𝑟𝑓 ( 𝑣𝑡 , 𝜂𝑡 ) , combined with a set of optimal robust consumption and investment policies that support continuous clearing in the markets for the consumption good and the risky asset. The two market clearing conditions are 𝐶𝑡 = 𝐷𝑡 , (22) 𝜔𝑡 = 1. (23) The next proposition characterizes the equilibrium risk-free rate 𝑟𝑓,𝑡 and the conditional equity premium under the reference measure, which can be expressed as 1 𝑑𝑡 E𝑡 [ 𝑑𝑆𝑡 + 𝐷𝑡𝑑𝑡 𝑆𝑡 ] − 𝑟𝑓,𝑡 = 𝜇𝑆,𝑡 − 𝑟𝑓,𝑡, (24) given the dynamics for the stock price in Eq. (3). 20 It can be shown that, under mild regularity assumptions, the solution we provide satisfies the transversality condition: lim𝑡→∞ E 𝜗 [ 𝑉 (𝑡, 𝑋𝑡 , 𝜂𝑡 , 𝑣𝑡 ) ] = 0. A formal proof of this sufficient condition is available from the authors upon request. 21 While in principle there are two roots for the candidate solution of ℎ ∗ 𝑡 = ±𝜖𝑡 , only the negative solution is consistent with the minimization in Eq. (15). Intuitively, the minimization means that the investor evaluates his or her policies under the worst-case alternative in the set of models under consideration. 22 Disconnect has two effects in the demand function for the risky asset in Eq. (21). First, in the partial equilibrium setting considered here, disconnect directly affects the optimal portfolio holdings of the robust investor. As Eq. (21) shows, the robust investor optimally reduces his or her exposure in the risky asset. Second, in general equilibrium, as we will later show, disconnect increases the risk premium, i.e. the equity premium appearing in the first term in the demand function will be itself a function of volatility and disconnect

Journal of Econometrics xxx (xxxx) xxx 9 Y. Aït-Sahalia et al. Proposition 2. In equilibrium, the price–dividend ratio is 𝑆𝑡∕𝐷𝑡 = 1∕𝛽. Using the optimal perturbation function ℎ ∗ 𝑡 in Eq. (20), the equilibrium equity premium under the reference measure P and the risk-free rate are given by Equity premium ∶ 𝜇𝑆,𝑡 − 𝑟𝑓,𝑡 = 𝜎 2 𝐷 𝑣 2 𝑡 + 𝜖𝜎𝐷𝜂𝑡𝑣 2 𝑡 . (25) Risk-free rate ∶ 𝑟𝑓,𝑡 = 𝛽 + 𝜇𝐷𝑣𝑡 − 𝜎 2 𝐷 𝑣 2 𝑡 − 𝜖𝜎𝐷𝜂𝑡𝑣 2 𝑡 . (26) As the results in Proposition 2 show, the equity premium and risk-free rate are time-varying and non-linear functions of volatility and disconnect. In particular, the uncertainty term embedded in disconnect generates a flight-to-quality-like correlation among asset returns. In periods of increasing uncertainty (increasing disconnect, for a given level of volatility) the demand for the risky asset decreases and the demand for the safe asset increases. Accordingly, in periods of increasing uncertainty the investor requires a higher equity premium to hold the risky asset, and is willing to accept a lower risk-free rate to hold the risk-less asset in equilibrium. The presence of stochastic volatility may amplify or diminish these effects depending on whether it is connected or disconnected from uncertainty, respectively. In the following sections we show that, because both volatility and disconnect jointly drive asset prices, our model is more consistent with the observed dynamics of asset prices, including during high-disconnect episodes. The structure of the equilibrium equity premium and risk-free rate can be also understood in the context of the optimal portfolio policy obtained in Proposition 1. The negative uncertainty correction implies that the representative investor is more conservative and prefers to hold less of the risky asset and more of the risk-free asset compared to a reference investor. In general equilibrium, the investor must allocate all his or her wealth to the risky asset (see Eq. (23)). Therefore, the last term in the equity premium boosts up the reward from holding the risky asset just enough so that the investor optimally chooses to allocate all his or her wealth to the risky asset. By the same token, the risk-free rate has to decrease in general equilibrium, so the investor optimally chooses not to hold the risk-free asset at all. Finally, it is likely that alternative approaches to model robustness (as discussed in footnote ) would also lead to an agent’s optimal portfolio holdings that are more conservative and command a higher risk premium (an additional positive ambiguity or uncertainty adjustment) and a lower equilibrium risk free rate. 6. Empirical analysis 6.1. Data The S&P 500 (logarithmic) returns including dividends obtained from the CRSP database serve as a proxy for the risky-asset return.23 The risk-free rate is the three-month (constant maturity) Treasury bill rate, which we obtain from the Fred St. Louis Database (Ticker ‘‘DGS3MO’’). Both series are available at the daily frequency from January 1, 1986 until December 31, 2020; we aggregate them to the weekly frequency. We construct our weekly measure of uncertainty 𝑡 using the daily news-based economic policy uncertainty (EPU) index developed by Baker et al. (2016).24 The daily EPU index is constructed by counting in the archives of well over one thousand US newspapers the number of articles that contain at least one term related to economic policy uncertainty from the following list:‘‘uncertain’’, ‘‘uncertainty’’, ‘‘economic’’, ‘‘economy’’, ‘‘Congress’’, ‘‘deficit’’, ‘‘Federal Reserve’’, ‘‘legislation’’, ‘‘regulation’’, and ‘‘White House’’. The daily EPU time series is volatile, and we smooth it by computing its moving average over the past week (5 trading days) to construct the uncertainty measure we use at the weekly frequency. The diffusive term of the stock price process is 𝜎𝑆,𝑡 ∶= 𝜎𝑆𝑣𝑡 where 𝜎𝑆 = 1, since we match 𝑣𝑡 to the square root of realized variance, which is computed daily from aggregated intraday returns sampled at five minutes frequency (78 observations per trading day). To obtain a weekly measure of realized variance, we sum the daily realized variance estimates. Based on the definition of disconnect in Eq. (12), we construct an empirical proxy for 𝜂𝑡 at the weekly frequency by computing the ratio of uncertainty as measured by EPU to realized volatility, and we then normalize this ratio to have an average value over the full sample equal to one. 6.2. Time-series properties of volatility and uncertainty In Fig. 2, we plot the time series of volatility and uncertainty, as well as the resulting disconnect process. From Panel A, we observe that uncertainty is substantially more volatile than volatility: the standard deviation of the EPU index is 50.1% compared to only 7.3% for realized volatility. While less volatile, realized volatility is more persistent than uncertainty: the first order autocorrelation of volatility is 0.87, compared to 0.74 for the EPU index. In Panel B, we plot the associated normalized disconnect time series. Since it is constructed as a normalized ratio of uncertainty and volatility, it is also volatile and persistent. The shaded areas in the charts mark the three high-disconnect periods described above: the financial crisis, the US 2016 election, and the Covid-19 pandemic. High disconnect occurs whenever 𝜂𝑡 is far away from 1. Consistent with Fig. 1, the correlation between volatility and uncertainty is only 0.44: while the two series co-move on average, they are often disconnected. Regressing uncertainty on volatility and a constant yields an adjusted 𝑅2 of 25%. Therefore, 75% of the time-series variation in uncertainty cannot be explained by volatility—the gap which we attribute to disconnect in the model. 23 Center for Research in Security Prices, US Stock Database, obtained from Wharton Research Data Services. 24 Section 6.5 below shows that our results continue to hold when considering alternative empirical measurements of uncertainty

Journal of Econometrics xxx (xxxx) xxx 10 Y. Aït-Sahalia et al. Fig. 2. Time series of volatility, uncertainty and disconnect. Notes: Panel A shows uncertainty (proxied by the economic policy uncertainty index EPU𝑡 ) and volatility (proxied by realized volatility). Panel B shows the disconnect process (𝜂𝑡 ) defined as the ratio of the EPU index divided by realized volatility, normalized to have a mean of 1. The sample period is from January 1986 until December 2020, and the data is sampled at the weekly frequency. The shaded areas correspond to five sub-periods associated with high disconnect: the mid 80’s (from January 1986 until November 1986), the early 90’s (from June 1991 until February 1996), the financial crisis (from July 2007 until March 2009), the US 2016 election (from July 2016 until January 2018), and the Covid-19 pandemic (from January 2020 until December 2020). Table 1 Average stock excess return and risk-free rate in selected high-disconnect regimes. Full Sample Mid 80’s Early 90’s Financial Crisis US 2016 election Covid-19 Time Period Begin Jan. 1986 Jan. 1986 Jun. 1991 Jul. 2007 Jul. 2016 Jan. 2020 Time Period End Dec. 2020 Nov. 1986 Feb. 1996 Mar. 2009 Jan. 2018 Dec. 2020 Stock excess return (%) 7.70 18.8 7.77 −33.40 17.29 16.97 Risk-free rate (%) 3.20 6.2 4.22 2.02 0.67 0.38 Uncertainty 𝑡 101.21 126.79 111.68 130.02 85.00 284.30 Stock return volatility 𝑣𝑡 (%) 17.70 9.78 7.32 28.16 10.51 28.94 Disconnect 𝜂𝑡 = 𝑡∕𝑣𝑡 (scaled) 1.00 1.52 1.77 0.59 1.29 1.49 Notes: This table presents the average annualized stock excess return, risk-free rate, uncertainty (𝑡 ), annualized volatility (𝑣𝑡 ), and disconnect (𝜂𝑡 ). These averages are computed over the full sample (from January 1986 until December 2020) and during five sub-periods associated with high disconnect: the mid 80’s (from January 1986 until November 1986), the early 90’s (from June 1991 until February 1996), the financial crisis (from July 2007 until March 2009), the US 2016 election (from July 2016 until January 2018), and the Covid-19 pandemic (from January 2020 until December 2020). All the data is nominal and sampled at the weekly frequency. 6.3. Average excess returns in high-disconnect periods Table 1 reports the average stock excess return and risk-free rate, along with the average uncertainty, volatility, and disconnect for the full sample and for the following high-disconnect sub-periods: the mid 80’s (from January 1986 until November 1986), the early 90’s (from June 1991 until February 1996), the financial crisis (from July 2007 until March 2009), the US 2016 election (from July 2016 until January 2018), and the Covid-19 pandemic (from January 2020 until December 2020). Over the full sample, the average stock excess return is 7.7%, with a relatively high level of the average risk-free rate compared to recent values. Given the paths of uncertainty 𝑡 and volatility 𝑣𝑡 , the normalized measure of disconnect 𝜂𝑡 has an average value of 1 over the full sample by construction. Before analyzing the most recent disconnect sub-periods, over which we will examine in more detail the performance of the model, we briefly describe two older high-disconnect periods: the mid 80’s and the early 90’s. Compared to the full sample results, both of these periods are characterized by a very high risk-free rate (about double its level in the full sample for the mid 80’s and